Applied Science, Faculty of
Civil Engineering, Department of
DSpace
UBCV
Seidalinov, Gaziz
Doctor of Philosophy - PhD
University of British Columbia
Natural clays are anisotropic in their in-situ state and have an undisturbed shear strength in excess of the remoulded strength. In addition, most of the structures founded on clay deposits must be designed to withstand cyclic loads such as seismic ground motions or ocean waves. When subjected to earthquake or wind induced cyclic loadings, clay exhibits a complex response. A realistic modeling of clay response under irregular cyclic loading requires an appropriate stress--strain relationship described by a constitutive model. This thesis extends the formulation of an existing constitutive model, namely Simple ANIsotropic CLAY plasticity (SANICLAY) model, by incorporation of a, well-established in geomechanics, bounding surface formulation for successful simulations of clay response under cyclic loading. The most important aspects of the proposed formulation are the position of a projection center and the ability to capture continuous stiffness degradation. The proposed projection center is established in the instant of any stress reversal, and it realistically reflects the experimentally observed plastic strains. A damage parameter is also adopted to better simulate the continuous stiffness reduction during the course of applied cyclic loading. The proposed model is developed with the aim of maintaining the simplicity, and yet including an adequate level of sophistication for successful modeling of the key features of clay response. The model formulation is presented in detail, followed by details of its implementation for applications in boundary value problems. Verification of the model implementation and validation of its performance are also presented. Verification of the model implementation is required in order to build confidence prior to its validation. Followed model validation demonstrates the capabilities of the model in capturing a number of important characteristic features of clay response in cyclic loading. Further exploration of model response in multi-directional cyclic shear is performed demonstrating its extension into more complex multi-directional cyclic shear. Development of the model, its implementation, verification of its implementation, validation of its performance, and exploration of model response in multi-directional cyclic shear provide a tool that can be used in modeling clay response under cyclic loadings. Limitations and recommendations for future work are discussed as well.
Xon“titutivz vny cumzrixvl boyzling of Xlvyhuwjzxtzy to Xyxlix aovyingbyGaziz SeidalinovM.A.Sc., The University of British Columbia, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Civil Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2018c© Gaziz Seidalinov 2018Vw“trvxtNatural clays are anisotropic in their in-situ state and have an undisturbed shear strengthin excess of the remoulded strength. In addition, most of the structures founded on claydeposits must be designed to withstand cyclic loads such as seismic ground motions or oceanwaves. When subjected to earthquake or wind induced cyclic loadings, clay exhibits a com-plex response. A realistic modeling of clay response under irregular cyclic loading requires anappropriate stress–strain relationship described by a constitutive model. This thesis extendsthe formulation of an existing constitutive model, namely Simple ANIsotropic CLAY plastic-ity (SANICLAY) model, by incorporation of a, well-established in geomechanics, boundingsurface formulation for successful simulations of clay response under cyclic loading. Themost important aspects of the proposed formulation are the position of a projection centerand the ability to capture continuous stiffness degradation. The proposed projection centeris established in the instant of any stress reversal, and it realistically reflects the experi-mentally observed plastic strains. A damage parameter is also adopted to better simulatethe continuous stiffness reduction during the course of applied cyclic loading. The proposedmodel is developed with the aim of maintaining the simplicity, and yet including an adequatelevel of sophistication for successful modeling of the key features of clay response. The modelformulation is presented in detail, followed by details of its implementation for applicationsin boundary value problems. Verification of the model implementation and validation ofits performance are also presented. Verification of the model implementation is required inorder to build confidence prior to its validation. Followed model validation demonstratesthe capabilities of the model in capturing a number of important characteristic features ofclay response in cyclic loading. Further exploration of model response in multi-directionalcyclic shear is performed demonstrating its extension into more complex multi-directionalcyclic shear. Development of the model, its implementation, verification of its implementa-iiAvstruwttion, validation of its performance, and exploration of model response in multi-directionalcyclic shear provide a tool that can be used in modeling clay response under cyclic loadings.Limitations and recommendations for future work are discussed as well.iiiavy hummvryCyclic loading imposed on clay deposits during such events as earthquakes, tsunamis orstorms can lead to serious damage of infrastructure and loss of lives in the vicinity of suchdeposits. In order to be able to avoid such consequences, it is necessary to first understand thenature of clay response when subjected to cyclic loads and then develop numerical tools thatcan be used for reproduction of the observed response. A part of such a tool is developedin this thesis, which proposes a constitutive model that can replicate clay response whensubjected to cyclic loads. When the constitutive model is combined with widely availablenumerical platforms, they represent a tool that can be used for modeling the impacts thatcyclic loads would induce to infrastructures founded on clay deposits.iverzfvxzI, Gaziz Seidalinov, am the principle contributor to all seven chapters of this thesis. I wasresponsible for all major areas of formulation, verification and validation, and evaluation ofthe model and its implementation, and writing the chapters. Some parts of the findings ofthis thesis have been published in a journal and three conferences so far, some parts havebeen submitted to a journal, and some parts are being prepared for submission to a journal.Published or submitted material to journals:• Seidalinov, G., and Taiebat, M. (2014), ‘Bounding surface SANICLAY plasticity modelfor cyclic clay behavior.’, Intyrnutionul Journul zor bumyriwul unx Unulytiwul aythoxsin Gyomywhuniws HM(7), 702–24.This paper includes chapter 3, and section 5.3 of chapter 5. The idea of the researchstarted with the initial work by Taiebat et al. (2010a) on the incorporation of thebounding surface formulation into SANICLAY family of models. My further detailedresearch incorporates two main features (update of projection center during stress re-versals and effect of accumulated damage parameter on plastic modulus) which signifi-cantly improve model performance in simulating undrained cyclic loading. Dr. YannisDafalias contributed with his idea of moving projection center with the evolution of thebounding surface in order to preserve the requirement of the unique image stress in thebounding surface formulation. The paper demonstrates development of the model andits validation against available conventional laboratory data. I and Dr. Mahdi Taiebatplanned the paper together. I conducted all of the analyses and data collection, andprepared the first draft. Dr. Mahdi Taiebat provided technical assistance in revisingthe manuscript.• Yang, M., Seidalinov, G., and Taiebat, M. (accepted in April 2018), ‘Multi-directionalvdryzuwycyclic shearing of clays and sands: evaluation of two bounding surface plasticity mod-els.’, goil Dynumiws unx Eurthquuky Enginyyring.This paper includes chapter 6. Mr. Ming Yang conducted data collection, and I wasresponsible for simulation and evaluation of the results for the SANICLAY model. Ialso contributed with an idea on evaluation of proximity of simulated loading path tothe neutral loading. The paper explores SANICLAY-B model performance in multi-directional cyclic shear against complex laboratory data that became available in recentyears. Mr. Ming Yang, I and Dr. Mahdi Taiebat planned the paper together. The firstdraft of the paper was prepared by Mr. Ming Yang and myself. Dr. Mahdi Taiebatprovided technical assistance in revising the manuscript.• Seidalinov, G., Barciaga, T., Taiebat, M., and Schanz, T. (in preparation), ‘SANICLAYmodel with bounding surface plasticity: implementation, verification and application.’.This paper includes chapter 4, and section 5.2 in chapter 5 and continues the initialwork on propagation of seismic waves through saturated soft clay included in Appen-dices B and C. The paper demonstrates the model implementation in finite elementcode (OpenSEES) and its application in simulations of seismic waves propagation in asloping ground. I and Mr. Barciaga collaborated in details of the model implementa-tion into finite element codes with me focusing on OpenSEES finite element code. I,Mr. Barciaga, Dr. Schanz and Dr. Taiebat planned the paper together. I conducted themodel implementation into OpenSEES and currently I am performing analyses of prop-agation of seismic waves, and preparing the first draft. Mr. Barciaga, and Dr. Taiebatprovide technical assistance in revising the manuscript.• Barciaga, T., Seidalinov, G., Datcheva, M., Taiebat, M., and Schanz., T. (in prepara-tion), ‘Mechanized tunneling in natural soft soil.’.The paper demonstrates the model implementation and its application in simulatingmechanized tunneling in natural soft soil. I and Dr. Taiebat provided assistance inthe model formulation and implementation. I and Mr. Barciaga collaborated in detailsof the model implementation into finite element codes with Mr. Barciaga focusing onPLAXIS finite element code. Mr. Barciaga, myself, Dr. Datcheva and Dr. Taiebatvidryzuwyplanned the paper together. Mr. Barciaga conducted the model implementation intoPLAXIS and currently is preparing the first draft. I, Dr. Datcheva and Dr. Taiebatprovide technical assistance in revising the manuscript.Published material in confererences:• Seidalinov, G., and Taiebat, M. (2013), ‘SANICLAY-B: a plasticity model for cyclicresponse of clays.’, in Proceedings of Sixty Sixth Canadian Geotechnical Conference,Montreal, QC, Canada.This paper includes parts of section 3.2 and 3.4 of chapter 3, and parts of section 5.3 ofchapter 5. I and Dr. Mahdi Taiebat planned the paper together. I conducted all of theanalyses and data collection, and wrote the first draft of the manuscript. Dr. MahdiTaiebat provided technical assistance in revising the manuscript.• Taiebat, M., and Seidalinov, G. (2014), ‘A bounding surface extension of SANI-CLAY plasticity model for nonlinear site response analysis.’, in Numerical Methods inGeotechnical Engineering, Delft, Netherlands, pp. 1295–1300.• Seidalinov, G., and Taiebat, M. (2014), ‘Propagation of seismic waves through satu-rated soft clay deposits: constitutive and numerical modeling’, in Proceedings of the14th International Association for Computer Methods and Advances in Geomechanics,Kyoto, Japan, pp. 285–290.The two papers above include the results of Appendix B and C demonstrating the useof the SANICLAY-B model, proposed in this thesis, for simulations of propagationof harmonic and earthquake induced seismic waves through saturated clay. I andDr. Mahdi Taiebat planned the paper together. I conducted all of the analyses, andwrote the first draft of the manuscript. Dr. Mahdi Taiebat provided technical assistancein revising the manuscript.• Seidalinov, G., Yang, M., and Taiebat, M. (2017), ‘Multi-directional cyclic shearing ofclays and sands: evaluation of two bounding surface plasticity models.’, in Proceed-ings of the 3rd International Conference on Performance-based Design in EarthquakeEngineering, Vancouver, Canada.viidryzuwyThis paper includes parts of chapter 6. Mr. Ming Yang, I and Dr. Mahdi Taiebatplanned the paper together. Mr. Ming Yang conducted data collection, and I wasresponsible for simulation and evaluation of the results for the SANICLAY model.The first draft of the paper was prepared by Mr. Ming Yang and myself. Dr. MahdiTaiebat provided technical assistance in revising the manuscript.• Palmieri, F., Seidalinov, G., and Taiebat, M. (2017), ‘Exploring a bounding surfaceplasticity model in capturing the cyclic behavior of clays.’, in Proceedings of SeventiethCanadian Geotechnical Conference, Ottawa, ON, Canada.This paper explores capabilities of the SANICLAY-B model, proposed in this thesis, inreproducing cyclic behavior of clays. Mrs. Francesca Palmieri and Dr. Mahdi Taiebatplanned the paper together. Mrs. Francesca Palmieri conducted all of the analyses,and wrote the first draft of the manuscript. I provided technical assistance in usingthe model for the simulations. Dr. Mahdi Taiebat and I provided technical assistancein revising the manuscript.viiiivwlz of Xontznt“Vw“trvxt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiavy hummvry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iverzfvxz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vivwlz of Xontznt“ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixai“t of ivwlz“ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiai“t of Figurz“ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvai“t of hymwol“ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiVxknofilzygzmznt“ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxixF Introyuxtion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation of research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Scope of work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Objectives and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8G erzviou“ lork“ on boyzling Xlvy gz“pon“z unyzr Xyxlix aovying . . . 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Laboratory element tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.1 “Fast” and “slow” undrained cyclic loading . . . . . . . . . . . . . . 122.2.2 Undrained cyclic triaxial loading . . . . . . . . . . . . . . . . . . . . 13ixhuvly oz Wontynts2.2.3 Undrained cyclic multi-directional shear . . . . . . . . . . . . . . . . 202.3 Theoretical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.1 Overview of constitutive modeling . . . . . . . . . . . . . . . . . . . 282.3.2 Bounding surface formulation . . . . . . . . . . . . . . . . . . . . . . 342.3.3 Bounding surface models for cyclic response of clay . . . . . . . . . . 422.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.4.1 Experimentally observed response . . . . . . . . . . . . . . . . . . . 492.4.2 Constitutive modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 49H Yzvzlopmznt of hVcIXaVnBW boyzl for Xyxlix gz“pon“z of Xlvy . . . 523.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2 Model formulation in triaxial space . . . . . . . . . . . . . . . . . . . . . . . 553.2.1 Bounding surface and plastic potential surface . . . . . . . . . . . . 563.2.2 Image stress and projection center . . . . . . . . . . . . . . . . . . . 573.2.3 Plastic modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2.4 Destructuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3 Generalization to multiaxial space . . . . . . . . . . . . . . . . . . . . . . . 643.3.1 Bounding surface and plastic potential surface . . . . . . . . . . . . 643.3.2 Image stress and projection center . . . . . . . . . . . . . . . . . . . 663.3.3 Plastic modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.4 Improved constitutive features of SANICLAY-B . . . . . . . . . . . . . . . . 733.4.1 Stress path and stress-strain variation . . . . . . . . . . . . . . . . . 743.4.2 Variations of GsecRGmax and damping ratio with shear strain . . . . . 753.4.3 Accumulation of peak "a with number of cycles . . . . . . . . . . . . 783.4.4 Variations of CSR with number of cycles . . . . . . . . . . . . . . . . 823.4.5 Response envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.5 Summary of proposed SANICLAY-B model . . . . . . . . . . . . . . . . . . 87I hVcIXaVnBW cumzrixvl Implzmzntvtion . . . . . . . . . . . . . . . . . . 904.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.2 Constitutive driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95xhuvly oz Wontynts4.2.1 Refined explicit integration . . . . . . . . . . . . . . . . . . . . . . . 964.2.2 Elastic response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.2.3 Elastoplastic response . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.3 Linearized constraints of laboratory tests . . . . . . . . . . . . . . . . . . . 1204.3.1 Linearized constraints and driving variables . . . . . . . . . . . . . . 1214.3.2 Application to some laboratory element tests . . . . . . . . . . . . . 1224.4 Finite element implementation . . . . . . . . . . . . . . . . . . . . . . . . . 1284.4.1 General description of OpenSEES finite element code . . . . . . . . . 1284.4.2 OpenSEES interaction with constitutive driver . . . . . . . . . . . . 1294.5 Summary of numerical implementation . . . . . . . . . . . . . . . . . . . . . 1315 kzri xvtion of cumzrixvl Implzmzntvtion vny kvliyvtion of boyzl ezrBformvnxz in Xyxlix hhzvr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.2 Verification of numerical implementation . . . . . . . . . . . . . . . . . . . . 1365.2.1 Verification of reduced model against exact solution . . . . . . . . . 1375.2.2 ISO-error maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.3 Validation in undrained cyclic triaxial test . . . . . . . . . . . . . . . . . . . 1455.3.1 Model calibration procedure . . . . . . . . . . . . . . . . . . . . . . . 1485.3.2 Reconstituted Georgia kaolin clay . . . . . . . . . . . . . . . . . . . 1505.3.3 Structured Cloverdale clay . . . . . . . . . . . . . . . . . . . . . . . 1575.3.4 Reconstituted Ariake clay . . . . . . . . . . . . . . . . . . . . . . . . 1745.4 Summary of Verification and Validation . . . . . . . . . . . . . . . . . . . . 177K hVcIXaVnBW gz“pon“z in Xyxlix bultiByirzxtionvl hhzvr . . . . . . . . 1806.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1806.2 Description of Gulf of Mexico clay . . . . . . . . . . . . . . . . . . . . . . . 1826.3 Calibration of model parameters . . . . . . . . . . . . . . . . . . . . . . . . 1836.4 Simulation of cyclic multi-directional shear . . . . . . . . . . . . . . . . . . 1906.4.1 1-D and 2-D linear cyclic shear . . . . . . . . . . . . . . . . . . . . . 1916.4.2 Circular cyclic shear . . . . . . . . . . . . . . . . . . . . . . . . . . . 195xihuvly oz Wontynts6.4.3 Figure-8 cyclic shear . . . . . . . . . . . . . . . . . . . . . . . . . . . 1996.4.4 Accumulated pore pressure ratio and shear strains . . . . . . . . . . 2086.5 Exploring model simulation results . . . . . . . . . . . . . . . . . . . . . . . 2096.5.1 Hypoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2106.5.2 Neutral loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2116.6 Summary of simulated cyclic multi-directional shear . . . . . . . . . . . . . 215L hummvryA Xonxlu“ionA vny gzxommznyvtion“ for Futurz lork . . . . . 2177.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2177.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2207.3 Recommendations for future research . . . . . . . . . . . . . . . . . . . . . . 221Wiwliogrvphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222Vppznyix V cumzrixvl Implzmzntvtion in dpznhEEh . . . . . . . . . . . . 238A.1 Requirements for subclasses of NDMaterial class . . . . . . . . . . . . . . . 238A.2 Generation of dynamically linked library . . . . . . . . . . . . . . . . . . . . 239A.3 Model related additional features . . . . . . . . . . . . . . . . . . . . . . . . 240A.3.1 Update of projection center during stress reversal . . . . . . . . . . . 240A.3.2 Linear elastic stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248A.3.3 Update of material stage and other model parameters . . . . . . . . 249A.3.4 Initialization of SANICLAY-B state variables . . . . . . . . . . . . . 249Vppznyix W eropvgvtion of Hvrmonix hzi“mix lvvz“ through hvturvtzy hoftXlvy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251Vppznyix X eropvgvtion of Evrthquvkz inyuxzy hzi“mix lvvz“ through hvtBurvtzy hoft Xlvy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255xiiai“t of ivwlz“2.1 Experimental database for cyclic multi-directional shear tests on clays . . . . 263.1 Model parameters used for illustration of improved constitutive features . . . 743.2 Initial internal variables used for illustration of improved constitutive features 744.1 Summary of pseudo-codes for implementation scheme . . . . . . . . . . . . . 964.2 Automatic substepping and error control . . . . . . . . . . . . . . . . . . . . 1004.3 Modified Euler increments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.4 Elastic nucleus intersection state . . . . . . . . . . . . . . . . . . . . . . . . . 1104.5 Intersection stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.6 Trial intersection stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.7 Ratio N difference for iterations of elastic nucleus intersection . . . . . . . . 1114.8 Strain intersection factor for elastic nucleus . . . . . . . . . . . . . . . . . . . 1134.9 Elastoplastic increment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.10 Bounding surface intersection factor . . . . . . . . . . . . . . . . . . . . . . . 1164.11 Trial bounding surface intersection factor . . . . . . . . . . . . . . . . . . . . 1164.12 Ratio N difference for iterations of bounding surface cross . . . . . . . . . . 1164.13 Ratio b between the bounding surface and the loading surface . . . . . . . . 1174.14 Trial ratio b between the bounding surface and the loading surface . . . . . . 1184.15 Ratio N difference for iterations of image stress . . . . . . . . . . . . . . . . 1184.16 Evolution of projection center . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.17 Drift correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.18 Newton-Raphson iteration for stress-controlled element test simulations . . . 1285.1 Model parameters used in verification of model implementation . . . . . . . 136xiiiList oz huvlys5.2 Initial internal variables used in verification of model implementation . . . . 1375.3 Numerical integration parameters used in verification of model implementation1375.4 Model constants for the proposed SANICLAY-B model . . . . . . . . . . . . 1505.5 Initial internal variables for the proposed SANICLAY-B model . . . . . . . . 1505.6 Model parameters for Georgia kaolin clay . . . . . . . . . . . . . . . . . . . . 1565.7 Initial internal variables for Georgia kaolin clay . . . . . . . . . . . . . . . . 1565.8 Model parameters for Cloverdale clay . . . . . . . . . . . . . . . . . . . . . . 1665.9 Initial internal variables for Cloverdale clay . . . . . . . . . . . . . . . . . . . 1665.10 Model parameters for Ariake clay . . . . . . . . . . . . . . . . . . . . . . . . 1775.11 Initial internal variables for Ariake clay . . . . . . . . . . . . . . . . . . . . . 1776.1 Model parameters for Gulf of Mexico clay . . . . . . . . . . . . . . . . . . . 1886.2 Initial internal variables for Gulf of Mexico clay . . . . . . . . . . . . . . . . 1886.3 Summary of cyclic multi-directional shear tests for Gulf of Mexico clay . . . 191A.1 State at stress reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245A.2 Pseudo-time increment at neutral loading . . . . . . . . . . . . . . . . . . . . 246A.3 Pseudo-time increment at stress reversal . . . . . . . . . . . . . . . . . . . . 247A.4 Ratio N difference for iterations of the variable "cum for stress reversal detection247B.1 Model parameters used for the analysis . . . . . . . . . . . . . . . . . . . . . 252xivai“t of Figurz“1.1 Example of shallow valley and 1-D amplification in Mexico City 1985 . . . . 21.2 Actual acceleration recorded at Lotung, Taiwan . . . . . . . . . . . . . . . . 31.3 Schematic illustration of linked research steps (numbered in parenthesis) andcontributions (numbered as Roman numerals) . . . . . . . . . . . . . . . . . 52.1 Undrained cyclic triaxial data on Newfield clay . . . . . . . . . . . . . . . . . 152.2 Undrained cyclic triaxial data on Lower Cromer Till . . . . . . . . . . . . . . 162.3 Undrained cyclic triaxial data on Georgia kaolin clay . . . . . . . . . . . . . 182.4 Undrained cyclic triaxial data on Cloverdale clay . . . . . . . . . . . . . . . 192.5 Comparison of accumulation of pore water pressure in “fast” and “slow”undrained cyclic triaxial loading tests . . . . . . . . . . . . . . . . . . . . . . 192.6 Cyclic shearing paths for cyclic multi-directional shearing . . . . . . . . . . . 242.7 Illustration of migration of shear strains in the direction of initial shear stressfor circular and figure-8 type A cyclic shear tests . . . . . . . . . . . . . . . . 252.8 Radial mapping rule from bounding surface fromulation: I . . . . . . . . . . 382.9 Radial mapping rule from bounding surface fromulation: II . . . . . . . . . . 402.10 Radial mapping rule from bounding surface fromulation: III . . . . . . . . . 413.1 Evolution of the SANICLAY family of models . . . . . . . . . . . . . . . . . 543.2 Schematic illustration of the loading, bounding, and plastic potential surfacesof the SANICLAY-B model in the triaxial stress space . . . . . . . . . . . . 563.3 Schematic illustration of the moving projection center inside the boundingsurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.4 Schematic illustrations of the model surfaces in the principal stress space . . 673.5 Illustration of correct solution for ratio b using maximum root . . . . . . . . 69xvList oz Figurys3.6 Illustration of an unaltered Lode angle for the projection center evolutionduring rotational hardening of the bounding surface . . . . . . . . . . . . . . 713.7 Illustration of the bounding surface and the damage parameter effects in sim-ulation of undrained cyclic triaxial loading . . . . . . . . . . . . . . . . . . . 763.8 Modulus reduction curves GsecRGmax versus cyclic shear strain amplitude . 793.9 Damping ratio curves versus cyclic shear strain amplitude . . . . . . . . . 803.10 Development of the axial strain "a at peaks of cyclic stress with the numberof cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.11 Cyclic stress ratio amplitudes CSR = xyRSu versus number of cycles requiredto reach "a = 3% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.12 Stress response envelopes produced using "˙tot = 1% for the initial stress statedefined by OCRm = 1 and M = 0:5bc, 1bc, and 2bc . . . . . . . . . . . . . 853.13 Stress response envelopes produced using "˙tot = 1% demonstrate the require-ment for limit of p given in Eq. 3.14 . . . . . . . . . . . . . . . . . . . . . . 863.14 Stress response envelopes produced using "˙tot = 1% demonstrate the effect ofinitial location of the projection center (initial bounding surface shown in gray) 874.1 Illustration of the strain increment "˙ passed from a finite element code into aconstitutive driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.2 Geometric interpretation of the integration algorithms used in numerical im-plementation of a general constitutive relation . . . . . . . . . . . . . . . . . 944.3 Schematic interpretation of the two-step modified Euler stress integrationscheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.4 Illustration of the Lode angle for the elastic nucleus coinciding with the Lodeangle for the bounding surface in the deviatoric stress ratio plane . . . . . . 1044.5 Schematic diagram for finding int . . . . . . . . . . . . . . . . . . . . . . . 1054.6 Illustration of correct solution for v using maximum root . . . . . . . . . . 1074.7 Illustration of the Lode angle dependency of the bounding surface and theelastic nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108xviList oz Figurys4.8 Illustration of the trial N values established along the trial nc for the iterativesolution of the intersection stress int . . . . . . . . . . . . . . . . . . . . . . 1094.9 Examples of multi-directional shear stress paths . . . . . . . . . . . . . . . . 1275.1 The analytical solution and reduced SANICLAY-B model simulation results 1395.2 Zones and initial stress states for generation of ISO-error maps . . . . . . . . 1415.3 ISO-error maps showing the reduction of the error with reduced tolerance . . 1425.4 The effect of the rotational hardening on the contours of ISO-error maps . . 1435.5 CPU time for different tolerance values . . . . . . . . . . . . . . . . . . . . . 1445.6 ISO-error maps for the initial stress states inside the bounding surface . . . . 1465.7 ISO-error maps showing effect of initial stress ratio . . . . . . . . . . . . . . 1475.8 Calibration of Xc and Xr in z– log p plane for Georgia kaolin clay . . . . . . . 1515.9 Calibration of bc and be in the triaxial plane for Georgia kaolin clay . . . . 1525.10 Calibration of X for Georgia kaolin clay . . . . . . . . . . . . . . . . . . . . . 1545.11 Initial hardening parameter calibrated to hM = 50 for Georgia kaolin clay . . 1555.12 Simulation of isotropic consolidation test for Georgia kaolin clay . . . . . . . 1575.13 Simulation of CIUC and CIUE tests on normally consolidated Georgia kaolinclay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1585.14 Undrained cyclic triaxial loading stress paths with experimental results andmodel simulations for Georgia kaolin clay . . . . . . . . . . . . . . . . . . . . 1595.15 Undrained cyclic triaxial loading stress–strain paths with experimental resultsand model simulations for Georgia kaolin clay . . . . . . . . . . . . . . . . . 1605.16 Calibration of Xc and Xr in z– log p plane for Cloverdale clay . . . . . . . . . 1625.17 Calibration of ϕc for Cloverdale clay . . . . . . . . . . . . . . . . . . . . . . . 1635.18 Calibration of hi and ki for Cloverdale clay . . . . . . . . . . . . . . . . . . . 1645.19 Calibration of X for Cloverdale clay . . . . . . . . . . . . . . . . . . . . . . . 1655.20 Simulation of oedometer test on Cloverdale clay . . . . . . . . . . . . . . . . 1675.21 Simulations of CIUC and CIUE tests on Cloverdale clay . . . . . . . . . . . 1685.22 Undrained cyclic triaxial loading stress path with experimental results andmodel simulations for Cloverdale clay . . . . . . . . . . . . . . . . . . . . . . 169xviiList oz Figurys5.23 Undrained cyclic triaxial loading stress–strain path with experimental resultsand model simulations for Cloverdale clay . . . . . . . . . . . . . . . . . . . 1705.24 Simulations of peak "a versus number of cycles in undrained cyclic triaxialloading for the range of available cyclic stress amplitudes on Cloverdale clay 1735.25 Simulations of peak "a versus number of cycles in undrained cyclic triaxialloading for the range of available cyclic stress amplitudes with the originalSANICLAY model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1745.26 Calibration of hM and vd for Ariake clay . . . . . . . . . . . . . . . . . . . . . 1765.27 Simulation of cyclic stress ratio CSR versus number of cycles for Ariake clay 1786.1 Calibration of Xc and Xr for Gulf of Mexico clay . . . . . . . . . . . . . . . . 1846.2 Calibration of bc and be for Gulf of Mexico clay . . . . . . . . . . . . . . . 1856.3 Calibration of Nc for Gulf of Mexico clay . . . . . . . . . . . . . . . . . . . . 1856.4 Calibration of X for Gulf of Mexico clay . . . . . . . . . . . . . . . . . . . . 1876.5 Experiment vs. simulation results of 1-D linear cyclic shear test, GOM-5, forGulf of Mexico clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1896.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1906.7 Schematic illustration of multi-directional loading for Gulf of Mexico clay . . 1926.8 Experiment vs. simulation results of 1-D linear cyclic shear test, GOM-6, forGulf of Mexico clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1936.9 Experiment vs. simulation results of 1-D linear cyclic shear test, GOM-7, forGulf of Mexico clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1946.10 Experiment vs. simulation results of 2-D linear cyclic shear test, GOM-8, forGulf of Mexico clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1966.11 Summarized experiments and simulations of 1-D and 2-D linear cyclic sheartests for Gulf of Mexico clay . . . . . . . . . . . . . . . . . . . . . . . . . . . 1976.12 Experiment vs. simulation results of circular cyclic shear test, GOM-10 , forGulf of Mexico clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1986.13 Experiment vs. simulation results of circular cyclic shear test, GOM-11, forGulf of Mexico clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200xviiiList oz Figurys6.14 Experiment vs. simulation results of circular cyclic shear test, GOM-12, forGulf of Mexico clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2016.15 Initially negative accumulation of simulated pore pressure for circular cyclicshear tests GOM-11 and GOM-12 on Gulf of Mexico Clay . . . . . . . . . . 2026.16 Summarized experiments and simulations of circular cyclic shear tests for Gulfof Mexico clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2036.17 Experiment vs. simulation results of figure-8 cyclic shear test, GOM-13, forGulf of Mexico clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2046.18 Experiment vs. simulation results of figure-8 cyclic shear test, GOM-14, forGulf of Mexico clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2056.19 Experiment vs. simulation results of figure-8 cyclic shear test, GOM-15, forGulf of Mexico clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2066.20 Summarized experiments and simulations of figure-8 cyclic shear tests for Gulfof Mexico clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2076.21 Simulations vs. experiments for pore pressure ratio, and shear strains at theend of the tests for Gulf of Mexico clay . . . . . . . . . . . . . . . . . . . . . 2096.22 Comparison between SANICLAY-B simulation results for Gulf of Mexico claywith the model hypoelasticity and constant elasticity . . . . . . . . . . . . . 2116.23 Schematic illustration of neutral loading tangential to the loading surface f =0 for SANICLAY-B model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2126.24 The angle Θ for cyclic tests simulations on Gulf of Mexico clay . . . . . . . . 214A.1 Illustration of stress increments and neutral loading path . . . . . . . . . . . 241B.1 Time history of the input acceleration at the base. . . . . . . . . . . . . . . . 251B.2 Simulation results for time histories of shear stress and shear strain atdepth of 5 m in the soil column using the SANICLAY model. . . . . . . . . . 253B.3 Simulation results for time histories of shear stress and shear strain atdepth of 5 m in the soil column using the SANICLAY-B model. . . . . . . . 253B.4 Simulation results for shear stress-strain response at depth of 5 m in the soilcolumn using the SANICLAY (left) and SANICLAY-B (right) models. . . . . 254xixList oz FigurysC.1 Imperial Valley record scaled to PGA of 0.35g . . . . . . . . . . . . . . . . . 256C.2 Simulation results for time histories of shear stress and shear strain atdepth of 5.5 m in the soil column using the SANICLAY model. . . . . . . . . 256C.3 Simulation results for time histories of shear stress and shear strain atdepth of 5.5 m in the soil column using the SANICLAY-B model. . . . . . . 257C.4 Residual displacement with depth at the end of seismic loading. . . . . . . . 257C.5 Acceleration response spectra of the base input motion and the predictedground surface response (SANICLAY & SANICLAY-B). . . . . . . . . . . . 258xxai“t of hymwol“vd rate of y evolutionvcross factor for stress increment crossing bounding surfacev" strain intersection factor for elastic nucleusv stress intersection factor for elastic nucleusV destructuration distributing model parameterb similarity ratio between the bounding surface F = 0 and the loading surface f = 0beln ratio b between the bounding surface F = 0 and the elastic nucleus Feln = 0B matrix strain operatorX rate of evolutionCKM KM consolidationCK K consolidation, with normalized initial consolidation shear stress = cRvcCSR cyclic stress ratioCSRx cyclic stress ratio along x-directionCSRy cyclic stress ratio along y-directioncount number of consecutive times minimum allowable pseudo-time i˙min has been assignedXc compression line slope in z– log p planeXp model parameter controlling position of projection centerXr unloading–reloading line slope in z– log p planeCe fourth-order elastic modulus tensorCep fourth-order elastoplastic modulus tensory damage parameterDe fourth-order elastic modulus tensorDesec fourth-order secant elastic modulus tensorxxiList oz gymvolsDep fourth-order elastoplastic modulus tensorz void ratiozin initial void ratiozint void ratio z at intzincl void ratio z at p = 1 kPa on isotropic normal consolidation linezp0 void ratio z at pM on isotropic normal consolidation linez deviatoric strain tensorzp plastic deviatoric strain tensorE strain constraints matrixEPS machine epsilonE error between the first and the second order Euler estimates of stress state andinternal variablesf = 0 loading surface in bounding surface formulation, or yield surface in classical plas-ticityf initial anisotropy factorF = 0 bounding surfaceFeln = 0 elastic nucleusFext external forces vectorFint internal forces vectorg = 0 plastic potentialG shear modulusGelKstK shear modulus G for linear elastic stageGs specific gravity of soil particlesGsec secant shear modulusGmax maximum tangential shear modulush bounding surface hardening functionhM initial value of hH Heaviside step functionI N + 2 + 3, first invariant of stress tensorIx I related to projection centerxxiiList oz gymvolsIo size of bounding surface along I-axisISL inelastic stiffness locusICC intrinsic compression curveI identity tensorJ second invariant of deviatoric stress tensorki rate of destructurationK bulk modulusKelKstK buld modulus K for elastic stageKp plastic modulusKsec secant bulk modulus KK¯p plastic modulus for the stress states on the bounding surfaceKM earth pressure coefficient at rest (with lateral strain constrained)K earth pressure coefficient at rest (with lateral strain constrained and presence ofnormalized initial consolidation shear stress = cRvc)K stiffness matrixa loading indexb critical state ratio in p–q spacebc critical state ratio b in triaxial compressionbe critical state ratio b in triaxial extensionm ratio beRbcn unit tensornc unit tensor along rc − N number of cyclesN stress ratio at peak q on the bounding surfaceNc strsss ratio N in triaxial compressionNe strsss ratio N in triaxial extensionNSSRA nonlinear seismic site response analysisn ratio NeRNcOCR overconsolidation ratioOCRm modified overconsolidation ratio estimated ‖Bp‖R‖‖xxiiiList oz gymvolsPI plasticity indexPC projection centerp hydrostatic counterpart of pa atmospheric pressurepBp hydrostatic counterpart of Bppc hydrostatic counterpart of cpM size of bounding surface along p-axispM;d destructured value of pMp size of plastic potential along p-axisp¯ hydrostatic counterpart of ¯ = “¯+ p¯Iq deviatoric stressq¯ image deviatoric stress q on bounding surfaceqb deviatoric stress qc projection onto bounding surfaceqc deviatoric stress q at projection centerqcy amplitude of uniform cyclic loadingq set of pM and internal variablesqn set of n internal variablesq qc projection on r distance from projection center to image stressr back stress-ratio tensor “Rprc back stress-ratio r of cg relative error between the first and the second order Euler estimatesg g for  estimatesgp0 g for pM estimatesg g for estimatesgy g for y estimatess indirect measure of the radius of the elastic nucleus“ deviatoric counterpart of , “ =  − pI“Bp deviatoric counterpart of Bph stress constraints matrixxxivList oz gymvols“c deviatoric counterpart of c“b projection of “c on the bounding surface along nchi initial structuration factor“ projection of “c on at pc“¯ deviatoric counterpart of ¯SRE stress response envelopeSSR static stress ratioSSRx static stress ratio along x-directionSSRy static stress ratio along y-directionSSRA seismic site response analysisSTOL allowable tolerance for relative error gSu undrained shear strengthSuc undrained shear strength in compressioni pseudo-timei˙ pseudo-time incrementi˙ el pseudo-time increment for elastic responsei˙min minimum allowable pseudo-timeTOL allowable tolerance for stress state around elastic nuclues and bounding surfaceu pore pressureu displacement vectorUC undrained compression testUE undrained extension testx model parameter for saturation during KM consolidationn˙ loading increment vector deviatoric back-stress ratio deviatoric back-stress ratio tensor controlling rotation of the bounding surface andthe plastic potential"a axial strain"cum cumulative pseudo-strain for detection of stress reversal"aA double amplitude strainxxvList oz gymvols"pd plastic destructuration strain"q deviatoric strain"pq plastic deviatoric strain"r radial strain"tot total strain"pv plastic volumetric strain"v volumetric strain" strain tensor"e elastic strain tensor"p plastic strain tensor distance from current stress to image stress∆u excess pore pressurein, o length inaij evolving anisotropy in Fig. 2.10  generalized to multiaxial stress spacein tensorial representation of distance from current stress point to image point at themoment of stress reversal stress ratio qRpM initial stress ratioK0  for KM-loading Lode angleΘ angle between gradient of the loading surface and stress increment shear strainx shear strain in x-direcitony shear strain in y-direciton slope of unloading–reloading line in z– ln p plane slope of compression line in z– ln p planeΛ damping ratio, Poisson’s ratio,elKstK Poisson’s ratio , for elastic stagexxviList oz gymvolsa axial stressc consolidation stressh horizontal stressr radial stressv vertical stressvc vertical consolidation stressv;max maximum vertical stresse equivalent c stress tensorBp  radially mapped from origin of stress space onto bounding surfacepred stress tensor predicted using elastic responseint stress tesnor at intersection of elastic nucleusc projection center tensorN major principal stress2 intermediate principal stress3 minor principal stress¯ image stress tensor shear stress c consolidation shear stress cy cyclic shear stress cy;x cyclic shear stress in x-direction cy;y cyclic shear stress in y-direction x shear stress in x-direction y shear stress in y-directionϕ critical state friction angleϕc friction angle in triaxial compression@FR@¯ bounding surface gradient〈〉 Macauley brackets˙ time derivative or rate¯ association with stress state on bounding surfacexxviiList oz gymvols: inner product of two tensors⊗ outer product of two tensorstr trace operator#0 tensor with all zero componentsxxviiiVxknofilzygzmznt“I offer my sincerest gratitude to my supervisor, Dr. Mahdi Taiebat, who has supported mewith his guidance throughout my research. His encouragement and high level of competenceinspired me to explore this subject. I am truly indebted and thankful to him for the time hedevoted to discussions on the subject matter. His sometimes critical comments helped meto become a better researcher. I would like to thank Professor Yannis Dafalias for his helpin formulating the motion of the projection center. Many thanks to my supervisory commit-tee members Prof. Jonathan Fannin, Prof. Arcesio Lizcano, and Prof. Dirk van Zyl whosedevotion provided guidance, support and necessary comments to shape this dissertation.Thanks to the Department of Civil Engineering faculty and staff members, the Universityof British Columbia (UBC), including the library staff for the resources provided. I wouldalso like to thank the people who have become my friends during my studies at UBC andmade this journey memorable.I would also like to thank Prof. Boris Jeremic for his support and guidance during mystay as a Visiting Scholar at the University of California, Davis, supported by the Go Globalprogram at UBC. I am very grateful to have met a number of great people who became myclose friends during this memorable time in Davis, and have stayed friends throughout theseyears.Many thanks to the Ruhr-Universita¨t Bochum (RUB) group in Bochum, Germany, ledby Prof. Tom Schanz who hosted me as a Visiting Scholar supported by RUB ResearchSchool through their Ph.D. Exchange Scholarship. Special thanks to Mr. Thomas Barciagafor initiating our collaboration and being a supportive good friend all throughout. Timespent in Bochum, Germany gave me another opportunity to explore the world and makemany new friends.Thanks will not be enough for my parents, Ilyas and Karlygash, from whom I have beenxxixAwknowlyxgymyntsconstantly away. Special thanks to my lovely brother, Arystan, who has been taking careof them and tried to fill my missing spot. I am also very grateful to my aunt Aliya for hercontinuous support and encouragement to take this journey.Financial support provided by the Natural Sciences and Engineering Research Councilof Canada (NSERC) is gratefully acknowledged.xxxFC IntroyuxtionFCF botivvtion of rz“zvrxhGeomechanics treats soil media as a continuum based on the observation that real physi-cal particles are very small when compared to the dimensions of the body to be analyzed.In continuum mechanics, macroscopic physical observations are described with considera-tion of equilibrium, kinematics, and the link between stress change and strain change. Thestress-strain relationship in geomechanics is represented by a constitutive model. A highlynonlinear stress-strain relationship requires advanced constitutive modeling of geomaterials.When such advanced constitutive models are linked with advanced numerical platformsNusing efficient computational procedures, complex static and transient boundary value prob-lems can be solved numerically. Complexities in constitutive models depend on the typeof problem that needs to be analyzed. Observations on the response of clay subjected tocyclic loading require important features that a constitutive model should possess in orderto replicate the observed response. Proper representation of the stress-strain response wouldallow numerical analyzes of such problems as modeling the response of clay subjected toloading induced by seismic, wind and tidal waves.Damage to structures during earthquake events is directly related to the local site effects.The complex and nonlinear response of clay can change the character of syismiw wuvys, and insome cases result in amplification and more destructive excitation at the ground surface. Thiswas one of the main causes of damage to buildings and bridges during a number of destructiveearthquakes such as the 1985 Mexico city, and the 1999 Kocaeli earthquakes. Recordedalteration of seismic waves is shown in Fig. 1.1 during the 1985 Mexico city earthquake:1 nitz zlzmznt vny nitz yiffzrznxz xoyzs thvt vllofi simulvtion of thrzzByimznsionvl prowlzms vny fullyxouplzy soliyBuiy rzsponsz1EBEB aotivution oz rysyurwhsignificant amplification and change of frequency content of ground shaking occurred asa result of seismic waves traveling through the soft clay deposit. Multi-directionality ofseismic excitation is often omitted when analyzing the impact of earthquakes; however, insome cases the multi-directionality could be significant as shown by an example of horizontalcomponents of acceleration in Fig. 1.2 recorded during an earthquake of b = 7:0 in Lotung,Taiwan in 1989. Recent studies (Matsuda et al., 2013; Nie et al., 2015; Rutherford, 2012)suggest that clay subjected to multi-directional loading exhibits a much softer response thanthose subjected to uni-directional loading.value of RRSmax in Fig. 8 ranges between about 3 and 20, with a clear trend for RRSmax to decrease as the shear wave velocity increases. Fortunately, as discussed later herein, both the extreme softness of the soil as well as other characteristics of the Mexico City clay inducing these very high amplifications are rather unusual. Figure 7 : Soil conditions and recorded acceleration spectra for several sites in Mexico City during the 1985 earthquake (modified after Seed et al., 1988). Figure 8 : Maximum Ratio of Response Spectra, (RRS)max, versus V s, recorded at Mexico City clay sites during five earthquakes between 1985 and 1990 (Ordaz and Arciniegas, 1992, Dobry et al., 1999). Useful insight into the factors controlling the value of RRSmax and the amplification phenomenon at soft clay sites revealed by Figs. 1-8 is provided by the 1D amplification model of Fig. 9 and the results in Fig. 10 (Roesset, 1977). Both the horizontal acceleration on rock at point B, aB, and the corresponding soil Figure 1.1: Example of shallow valley and 1-D amplification in Mexico City 1985 (fromDobry and Iai, 2000)Growing consumption of the world’s energy resources (oil & gas, wind, and tidal), pushesthe boundaries of engineering into deep waters. Wind and tidal waves directly imposeloads passed through marine structures (e.g., wind turbine towers, tension legs) into thesoil deposits. In addition, potential failure of slopes formed in the marine environment,where clay is deposited, imposes a major hazard on the marine structures during seismicevents or storms. Equally important, failure of submarine slopes involves landslides of largeamounts of the deposited material traveling at a high speed, thereby causing tsunamis thatimpose a major hazard for marine structures and coastal communities. Major submarinelandslides triggered by earthquakes and causing tsunamis were recorded at the Grand Banks,2EBFB gwopy oz work(\ Proceedings: Third International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, W Apri12-7, 1995, Volume II, St. Louis, Missouri The Role of Rotational Shear in Site Response Analyses Paper No. 7.24 X.S. Li and C.K. Shen Assistant Professor and Professor, Hong Kong University of Science and Technology, Hong Kong SYNOPSIS Rotational shear is a class of loading under which the second invariant of the deviatoric stress tensor, J, is kept constant during shear. A limited number of undrained tests on saturated loose sand have shown that rotational shear yields more pore pressure than other shear paths of the same J magnitude. By intuition, this experimental finding seems to suggest that soil subjected to multi-directional earthquake loading, which bears the characteristics of the rotational shear, has lower liquefaction resistance than that under unidirectional shaking. However, in contrast to intuition, a preliminary but careful examination of the field stress and boundary conditions indicates that rotation shear seems to have very little impact on the seismic response of natural soil deposits. This paper presents the theoretical evidence that supports the preliminary conclusion. INTRODUCTION When a soil deposit is subjected to an upward propagating seismic shake, the soil in the deposit experiences cyclic simple shear in two horizontal directions, thus resulting in the change of the resultant shear stress in both magnitude and direction. As a vector, the resultant shear yields irregular stress paths in its component plane. The characteristics of the stress paths is similar to that of the horizontal acceleration trajectories. Fig. I shows s·1ch an acceleration trajectory which was recorded in Lotung, Taiwan, in I936, during an earthquake of M=7 .0. It is evident that the resultant shear exhibited a widespread change in both magnitude and direction. Acceleration in g, E-W direction Fig. I Trajectory of Acceleration Recorded in Lotung, Taiwan, Nov., 1986 Earlier studies on soil behavior under seismic loading were essentially based on the results of unidirectional cyclic shear tests, and the influences of change in shear direction were simply ignored. In view of this, Ishihara and Yamazaki (I 980) conducted a series of undrained tests on sand using a two directional simple shear apparatus. The apparatus had two mutually perpendicular horizontal actuators for cyclic shear applications. The soil samples were initially consolidated at an isotropic effective confining pressure of 200 kpa and during shear the total confining pressure was kept unchanged. By combining two 603 shear components, a variety of shearing schemes characterized by the loading paths in the horizontal plane could be achieved. The investiga-tors studied two types of shearing schemes: the Rotational Simple Shear and the Alternate Simple Shear. For the rotational simple shear tests, two sinusoidal shears with 90° out of phase to each other were applied simultaneously in two horizontal directions. Depending on the relative amplitudes of the two shear components, the resultant loading path could either cirr.ula, or elliptic. For the tests of the alternate simple shear, cyclic simple shear was applied alternately in two horizontal directions cycle by cycle, and the amplitudes of the two components could also be different. The typical stress paths of those tests are shown in Fig. 2. The test results of the two types of shearing Circular loading Elliptic loading {a) Rotational Shear (a) Alternate Shear Fig. 2 Patterns of Bi-directional Simple Shear (After Ishihara and Yamazaki, I980) ' schemes are reproduced and shown in Figs. 3a and 3b. It is obvious that when 't/t1=I the stress pattern is a circle, when 't,f't1=0, the shear loading becomes unidirectional. From Fig. 3a it can be seen that the Figure 1.2: Actual acceleration recorded at Lotung, Taiwan (from Li and Shen, 1995)Newfoundland in 1929 (Hampton et al., 1996), Papua New Guinea in 1998 (Tappin et al.,2001), and at the offshore platforms in the Gulf of Mexico in 1969 (Bea, 1971) triggered byHurricane Camille.Driven by the above motivational examples, this dissertation is aimed for preparation ofa numerical tool that can be used in future research and applications related to geotechnicalearthquake engineering in clay deposits.FCG hxopz of fiorkThis dissertation describes development of a constitutive model aimed to replicate stress-strain response of clay subjected to cyclic loading. The model is implemented in a robustmanner allowing its application in boundary value problems. Validation of model perfor-mance is made against conventional laboratory element testing, demonstrating the effectof the incorporated features in the model. Exploring the model response in cyclic multi-directional shear element tests gives further insights into its predictive capabilities.3EBGB cvjywtivys unx aythoxologyFCH dwjzxtivz“ vny bzthoyologyThe main two objectives of the dissertation are: (1) to develop a constitutive model thatcan be used for description of clay response subjected to cyclic loading; (2) to preparea constitutive model for its application in numerical solution of boundary value problemsthrough the process of its implementation, validation and exploration of its performanceagainst laboratory element testing. When necessary, the model can be further refined withinthe prepared package.The research consists of four major parts that are also linked into sequential researchsteps as shown in Fig. 1.3 along with highlighted main contributions:1. Constitutive model development: model formulation is developed in multi-axial stressand strain space. The model incorporates, well-established in geomechanics, boundingsurface formulation into an existing Simple ANIsotropic CLAY (SANICLAY) plasticitymodel. The resultant model, named SANICLAY-B, incorporates additional featuresin form of updating projection center and damage evolution for successful simulationsof clay response subjected to cyclic loading.2. Implementation: the constitutive equations of the model are transferred into codinglines using C++ programming language and a constitutive driver that can be usedin any numerical platform. To ensure robust integration of the equations, refinedexplicit integration with substepping and automatic error control is used, enablingintegration of material response over any input strain increment. Bardet’s constraintsare also implemented for ease of simulating laboratory element tests. In addition, theconstitutive driver interaction with OpenSEES finite element platform is described.3. Verification & Validation: Verification of the implemented constitutive driver is per-formed in comparison with exact solution, which is followed by demonstration of itsrobustness to handle any input strain increments using ISO-error maps and demon-stration of its response using response envelopes. Validation of the proposed model isthen performed against three clays: Georgia kaolin, Cloverdale, and Ariake clays.4. Exploration: Performance of the SANICLAY-B model is explored in simulating cyclic4EBGB cvjywtivys unx aythoxology"˙⇐⇒ ˙I. Development (1)Numerical Integration (2)∆"⇐⇒ ∆Implementation into FE code (2); Dep∆"Fint =∫kWqykK =∫kWqDepWyk∆" = Wq∆uVerification (3)Validation (3)&Exploration (4)II. PackageforFutureResearchandApplicationsFigure 1.3: Schematic illustration of linked research steps (numbered in parenthesis) andcontributions (numbered as Roman numerals)multi-directional shear tests in comparison with experiments on Gulf of Mexico clay.The tests are performed using various cyclic shear loading paths applied along the tophorizontal plane of a tested specimen.Theoretical developments of a constitutive model that can be used for replicating re-sponse of clay subjected to cyclic loading is the main contribution (I) in this dissertation.Its further numerical integration and implementation into a finite element code provides anumerical tool. Prior to its use, the numerical tool is verified, validated and its capabilitiesare also explored under complex loading conditions. This process builds up credibility ofthe numerical tool which can be used as a package in future research and applications forgeotechnical earthquake engineering in clay deposits (e.g., nonlinear SSRA, design of windturbines, tension leg platforms). When necessary, further refinements can be implementedand transferred to the constitutive driver, thereby allowing future researchers to focus onthe developments of needed features. Preparation of the numerical tool that can be usedby researchers and engineers is considered a second major contribution (II) in this disserta-tion. The contributions are highlighted with red color in Fig. 1.3 describing the steps of theresearch.5EBHB cutlinyFCI dutlinzThe research is outlined in seven chapters of this thesis. The chapters are organized insequential order starting from the development of constitutive model, and then followedby implementation of the constitutive equations, verification of the implementation andvalidation of its performance in undrained cyclic triaxial tests, and finally exploration ofmodel response in cyclic multi-directional shear.Chapter 2 describes previous laboratory element tests and theoretical works in model-ing clay response subjected to cyclic loading. The laboratory element tests are focused onavailable triaxial and cyclic multi-directional shear tests. The importance of reliable exper-imental results for development of a constitutive model is discussed. In particular, reliablemeasurements of pore pressure during tests are required for correct interpretation of effec-tive stresses in element tests. Triaxial test data with reliable measurements of pore pressureis selected to investigate observed clay response that can be used for model development.Multi-direction shear tests however are quite limited and all of them are performed in a“fast” manner thereby their results cannot be used for development of a constitutive model.Instead, they can be used to evaluate model performance in complex cyclic shear loading.The theoretical modeling is focusing on investigation of available constitutive models thatwere used for the description of clay response subjected to cyclic loading. The models arereviewed with the focus on establishing their advantages and disadvantages that can be usedfor model development.Chapter 3 presents a formulation of a developed constitutive model for description ofcyclic clay response subjected to cyclic loading. Illustration of the effect of proposed modelfeatures are demonstrated in the sensitivity analysis. The main features of the model areupdate of projection center during stress reversals and damage parameter. The former isrequired to reduce large elastic region of stress-strain response in unloading and reloadingsequences. The latter is introduced to replicate accumulation of plastic shear strains observedin undrained cyclic loading tests. Although these features are not strictly novel2, they were2fihilz thz iyzv of upyvting projzxtion xzntzr is not novzlA its zvolution rzquirzy to kzzp it fiithin vwounying surfvxz is novzlP fihilz thz iyzv of vxxumulvting yvmvgz fiith plvstix yzvivtorix strvin inxrzmzntsis vlso not novzlA vpplixvtion of thz yvmvgz pvrvmztzr to thz wounying surfvxz formulvtion is vyoptzyspzxi xvlly for thz formulvtionP thusA thz vyoptzy fzvturzs vrz not strixtly novzl6EBHB cutlinypreviously used in constitutive models that lack other important features, thereby makingthem impossible to use as to be discussed in Section 2.4.In chapter 4 the constitutive equations proposed in chapter 3 are coded into the linesof C++ programming language providing a complete constitutive driver that can be usedin numerical platforms such as finite element or finite difference for solution of boundaryvalue problems. The method chosen for the integration of the equations is refined explicitintegration with substepping and automatic error control, which reassures that intendedstress-strain response is integrated correctly for any input strain increment. Details of thesteps undertaken for the development of the constitutive driver are presented in the chapteralong with pseudo-codes. A number of model specific details of integration are presented. Inaddition, for model validation process Bardet’s constraints presented in the chapter are usedto simulate laboratory element tests. This approach is chosen to separate potential issuesassociated with the finite element from the newly implemented model. The constitutivedriver interaction with OpenSEES finite element platform is also provided as an example ofmodel implementation in a finite element code.Verification of the implementation and validation of model performance in undrainedcyclic triaxial loading is illustrated in chapter 5. The verification of the implementation isfirst performed for a reduced model against an exact solution of the constitutive equationsfor undrained triaxial compression and extension tests. The next step for the verificationof the implementation is conducted using ISO-error maps illustrating the robustness of thechosen integration algorithm. Response envelopes are used as the last step of verificationalso providing insights into the model response. Validation of the model is then performedin comparison with undrained cyclic triaxial loading tests on reconstitutive Georgia kaolin,structured Cloverdale, and reconstitutive Ariake clays.In chapter 6 the model is explored in comparison with cyclic multi-directional shear testson Gulf of Mexico clay. Simulated shear paths included 1-D linear, 2-D linear, circular, andfigure-8 paths. The experiments are performed in “fast” undrained cyclic loading; therefore,pore pressure measurements are perhaps not reliable, thereby providing effective stresses thatcannot be fully relied on. However, out of available cyclic multi-directional shear tests, thisexperimental data is the only one of its kind. As a result, the model shear response, instead7EBIB botutionsof being validated, is explored in comparison with the available data. This chapter exploresmodel performance when subjected to complex loading conditions, thereby providing furtherinsights into the model’s ability to predict observed shear response.Finally, chapter 7 outlines summary, conclusion, and recommendations for future re-search.FC5 cotvtion“Triaxial stress quantities are p = (a + 2r) R3; q = (a − r), and strain quantities are"v = ("a + 2"r) ; "q = 2 ("a − "r) R3. The subscripts a and r denote the axial and radialdirections, and the subscripts v and q denote the volumetric and deviatoric components,respectively. Note that in this work all stress components are considered effective and asusual in geomechanics, both stress and strain quantities are assumed positive in compression.A superposed dot denotes, henceforth, the material time derivative or the rate.In multiaxial stress and strain spaces, the second-order tensors are denoted by boldfacem symbols, while the fourth-order tensors are denoted by open face X symbols. Triaxialstress quantities are generalized to multiaxial space, given a stress tensor  that can bedecomposed into its hydrostatic pI and deviatoric “ components, as follows:p =tr3; “ =  − pI (1.1)where tr is a trace operator and I is an identity tensor;Triaxial strain quantities are similarly generalized to multiaxial space, given that a straintensor " can be decomposed into its volumetric "vIR3 and deviatoric z parts:"v = tr"; z = "− "vI3(1.2)A systematic generalization of the triaxial constitutive relations is based on the observa-tion that under triaxial conditions the following holds true:x = [(3R2) x : x]N=2 (1.3)8EBIB botutionswhere x is any deviatoric stress tensor and x is its triaxial counterpart; the symbol : denotesthe trace of the product of adjacent tensors.On the basis of Eq. 1.3:q = [(3R2) “ : “]N=2 (1.4)"q = [(2R3) z : z]N=2 (1.5) = [(3R2) : ]N=2 (1.6)For convenience, implementation of the SANICLAY-B model is performed using vecto-rial and matricial notations representing symmetric second order and fourth order tensors,respectively. Hereafter, symmetric tensors of second order are represented by vectors withsix independent components:˙ =˙NN˙22˙33˙N2˙N3˙23; "˙ ="˙NN"˙22"˙33"˙N2"˙N3"˙23(1.7)Consequently, fourth-order tensors that relate two symmetric tensors of second order aredenoted by 6 x 6 matrices. For instance, ˙ = D : "˙ becomes˙NN˙22˙33˙N2˙N3˙23=YNNNN YNN22 YNN33 2YNNN2 2YNNN3 2YNN23Y22NN Y2222 Y2233 2Y22N2 2Y22N3 2Y2223Y33NN Y3322 Y3333 2Y33N2 2Y33N3 2Y3323YN2NN YN222 YN233 2YN2N2 2YN2N3 2YN223YN3NN YN322 YN333 2YN3N2 2YN3N3 2YN323Y23NN Y2322 Y2333 2Y23N2 2Y23N3 2Y2323"˙NN"˙22"˙33"˙N2"˙N3"˙23(1.8)The factor 2 in the above equation accounts for the minor symmetry of D and the9EBIB botutionsrepetitive shear terms of the implied sum: Yijkl"˙kl = YijNN"˙NN+Yij22"˙22+Yij33"˙33+2YijN2"˙N2+2YijN3"˙N3 + 2Yij23"˙23.10GC erzviou“ lork“ on boyzling Xlvygz“pon“z unyzr Xyxlix aovyingGCF IntroyuxtionConstitutive models are predominantly developed based on a single type of testing datasuch as direct simple shear or triaxial tests. Investigation of reliable test data is the initialstep in the process of the development in order to build up confidence in modeling the re-sponse of a clay subjected to cyclic loading. It is known that the direct simple shear testis more representative of in-situ plane strain conditions with rotation of principal stresses;however, it is recognized that stresses and strains are inherently non-uniform (e.g., Aireyet al., 1985) in this test. Although the triaxial test is limited to axisymmetric stress condi-tions, it is likely to remain the most widely used soil testing device (Wood, 2004) that, to acertain degree, can represent element test with uniform distribution of stresses and strains.In this chapter, the literature review consists of two parts necessary for development of aconstitutive model. The first part consists of investigation of available experimental datafor clays subjected to cyclic loading. The focus is attempted to be made on more reliableundrained cyclic triaxial tests that can be used for development of a constitutive model. Inaddition, cyclic multi-directional shear3 response that involves rotation of principle stressesand multi-directionality of cyclic shearing is also explored. The second part focuses on avail-able constitutive models emphasizing their features and observation made for enabling cyclicloading simulations. These important features required for successful cyclic loading simula-tions are to be incorporated into an extended version of an existing model in the following3in thz xontzflt of this yisszrtvtionA xyxlix multiByirzxtionvl shzvr rzfzrs to shzvring fiith tfio xomponzntson v singlz plvnzA iCzCA xyxlix multiByirzxtionvl shzvr is similvr to yirzxt simplz shzvr wut xonsiyzrs tfioxomponznts of shzvr strzss vxting on v horizontvl plvnz11FBFB Luvorutory ylymynt tystschapter.GCG avworvtory zlzmznt tz“t“The majority of laboratory element tests4 are performed under either drained or undrainedconditions. The rapid nature of seismic excitation does not allow enough time for dissipa-tion of excess pore pressure in the in-situ state of clays, thereby requiring that laboratoryexperiments are performed under undrained conditions to mimic the drainage condition inthe field. Literature review on available experimental data is therefore made on undrainedcyclic loading. Below a discussion is first made on the differences between “fast” and “slow”undrained cyclic loading. Then available experimental data on conventional triaxial andcyclic multi-directional loading is discussed.GCGCF qFv“t7 vny q“lofi7 unyrvinzy xyxlix lovyingImposed undrained cyclic loading on clays can be divided into two categories: “slow” and“fast” tests. “Slow” tests are those in which the rate of applied loading is small enough toachieve equalization of pore pressure in the whole specimen. The rate of applied loading thatensures the equalization depends on the material. It is important to understand advantagesand disadvantages of “slow” and “fast” undrained cyclic loading tests. During cyclic loading,pore pressure is usually measured on the boundaries of tested specimens. Measured porepressure is then deducted from applied total stresses to obtain effective stresses, governingthe response of a tested material. In this way, positive pore pressure is associated with thereduction of effective stresses, while negative pore pressure with its increase. An elementtest requires that measured data on specimen boundaries to be representative of the entirespecimen. The main advantage of “slow” undrained cyclic loading is that equalization of porepressure throughout the specimen is allowed, which ensures measurements of approximatelyaverage pressure in the entire specimen. Thus, interpreted effective stresses from “slow”undrained cyclic loading are more reliable. This is especially important for clays, where4zlzmznt tzsts vrz thosz thvt vrz vimzy to mvintvin uniform yistriwution of strzsszs vny strvins throughoutvn zlzmznt12FBFB Luvorutory ylymynt tystsdissipation of pore pressure requires a considerable amount of time (e.g., Andersen, 2009).“Fast” cyclic loading tests miss this important feature and become less reliable in terms ofrepresentative effective stress measurements for the soil element.The rate effect of applied loading also plays an important role and becomes more pro-nounced for clays. Some investigators (e.g., Li et al., 2011; Oka et al., 2004), in the effort tomatch both volumetric and shear response of clays, rely on the supposition that measuredresponse is solely a result of the rate of applied loading. While this is true for the shearresponse, the volumetric response is also a result of measuring un-equalized pore pressureon the boundaries of the specimen (e.g., Andersen et al., 2008; Zergoun and Vaid, 1994).Of course, the rate effect is present in both volumetric and shear response, but it is verydifficult or even impossible to differentiate the volumetric response of a clay due to the rateeffect from the volumetric response due to the pore pressure inequalities within the specimen.In constitutive modeling, coupling between volumetric and shear responses is an essentialfeature. Erroneous interpretation of volumetric response would impact credibility of a con-stitutive model and is avoided by not considering “fast” undrained cyclic loading for thestage of a constitutive model development in this dissertation.Based on the discussion above and availability of experimental data the following isconsidered. “Slow” undrained cyclic triaxial loading is used for understanding the volumet-ric and shear response of clays subjected to cyclic loading. “Fast” cyclic multi-directionalshear tests are used for understanding the shear response of clays subjected to cyclic multi-directional loading.GCGCG jnyrvinzy xyxlix trivxivl lovying“Slow” undrained cyclic triaxial loading tests are studied below in order to highlight observedclay response required for development of a constitutive model. For stress-controlled teststhe rate of loading is given in terms of frequency (Hz) of applied stress history, while forstrain-controlled tests the rate of loading is given in terms of strain rate (%/hr). Four claysconsidered below are: Newfield, Lower Cromer Till, Georgia kaolin, and Cloverdale clays.These data sets are obtained from “slow” undrained cyclic triaxial tests.13FBFB Luvorutory ylymynt tystsczfi zly xlvySangrey et al. (1969) paid close attention to reliable pore water pressure measurementsin undrained cyclic triaxial tests. They performed a series of one-way (compression) cyclicloading tests with constant stress amplitude on isotropcially and anisotropically consolidatedundisturbed Newfield clay to different OCR values. Tested clay was classified as silty clayand had PI=10. To ensure pore water pressure equalization within tested specimens, alltests were conducted with an axial strain rate of about 0:012%Rhr. Their main findingswere:• for each OCR there existed a “critical cyclic stress level” (CCSL); tests with cyclic stressamplitudes below CCSL attained equilibrium, and resultant strains reached 0:7% ofaxial strain in 100 cycles as shown in Fig. 2.1(a–b) for isotropicaly normaly consolidatedspecimen, whereas applied loading with stress amplitude above CCSL reached 14%of axial strains in just 10 cycles as shown in Fig. 2.1(c–d) for isotropicaly normalyconsolidated specimen• while normally consolidated specimens subjected to undrained cyclic triaxial loadingexhibited stress path migration towards the origin with decreasing rate of progressionwith the number of cycles, overconsolidated samples showed an increase of p withinthe initial cycle; the increase of p was then followed by its reduction that was lesspronounced for larger OCRsaofizr Xromzr iillTakahashi et al. (1980) discussed observations from undrained cyclic triaxial tests with dif-ferent frequencies on a low-plasticity sandy clay, Lower Cromer Till. The loading was appliedin constant stress amplitude tests with sinusoidal waveform. The authors considered one-way (compression or extension) and two-way (compression followed by extension) undrainedcyclic triaxial loading on isotropically and anisotropically consolidated specimens to variousOCR values. In addition to conventional pore water pressure transducers, piezometer probesat mid-height of tested specimens were used. The period of applied cyclic loading was re-14FBFB Luvorutory ylymynt tystsp (kPa)0 100 200 300 400q (kPa)0100200300400experiment (1-3, 100 cycles)=v)a (%)0 0.25 0.5 0.75 1q (kPa)0100200300400experiment (1-3, 100 cycles)=w)p (kPa)0 100 200 300 400q (kPa)0100200300400experiment (1-2, 10 cycles)=x)a (%)0 2 4 6 8 10 12 14 16q (kPa)0100200300400experiment (1-10 cycles)=y)Figure 2.1: Undrained cyclic triaxial data on Newfield clay (data from Sangrey et al., 1969)for cyclic stress amplitudes: (a–b) qcy = 180 kPa, and (c–d) qcy = 323 kPastricted to ensure reliable measurements of pore water pressure at the mid-height of testedspecimens.The following observations were made in examination of undrained cyclic triaxial testson normally consolidated specimen shown in Fig. 2.2 with 0:005 Hz frequency and appliedstress magnitude of (a − r) R (2Suc) = 0:76, where Suc is an undrained shear strength incompression:• a migration of the stress path towards the origin induced by the rise of pore waterpressure within a constant volume specimen15FBFB Luvorutory ylymynt tysts• the stress path changed its shape due to more dilative response with the number ofcycles as it approached the critical stress ratio in both compression and extension; moredilative response in extension than in compression was observed with the number ofcycles• as the stress path reached the critical stress ratio, axial strains development acceleratedand a larger magnitude of axial strains was observed in extension• the stress path traveled beyond the critical stress ratio in extension as the rate ofapplied loading was increasedp (kPa)0 50 100 150 200 250 300 350q (kPa)-150-100-50050100150 experiment (1-6, 8 cycles)=v)a (%)-14 -12 -10 -8 -6 -4 -2 0 2 4q (kPa)-150-100-50050100150 experiment (1, 5, 8 cycles)=w)Figure 2.2: Undrained cyclic triaxial data on Lower Cromer Till (data from Takahashi et al.,1980)The authors further aimed to investigate what influenced the rate of stress path migrationtowards the origin. From their findings they concluded that the main factors were: cyclicstress ratio (CSR), consolidation history, and rate of applied loading. A larger CSR initiallyinduced more rapid reduction of p. However, once the critical stress ratio was reached, preduction occurred at a decreasing rate. A lower CSR, apparently, approached the criticalstress ratio at lower p. As a result, overall reduction of p for lower CSRs was larger oncethe stress path was stabilized. Specimens sheared from high OCRs exhibited an increase inp, i.e., a dialative response during initial cycles, which was then followed by reduction of pwith the magnitude of reduction inversely proportional to OCR.16FBFB Luvorutory ylymynt tystsGzorgiv kvolin xlvySheu (1984) performed undrained cyclic triaxial tests with applied low frequency constantstress amplitude. The frequency of loading considered by the author as sufficient for reliablepore pressure measurements was 0:001 Hz applied with sinusoidal waveform. The authorconsidered one-way (compression or extension) and two-way (compression followed by ex-tension) undrained cyclic triaxial loading on isotropically normally consolidated specimens.The following observations are made based on the obtained experimental results:• two way cyclic loading is more destructive than one way cyclic loading• there exists a “critical level of repeated loading” (CLRL); for constant stress amplitudesbelow CLRL, tested specimen reaches an equilibrium level in which stress and stress-strain paths are nearly stabilized as shown for qcy = 121:4 kPa in Fig. 2.3(a–b); forconstant stress amplitudes above CLRL, tested specimens reach failure state with thenumber of cycles as shown for qcy = 140:7 kPa in Fig. 2.3(c–d)• for two way cyclic loading, the response is influenced by the portion of loading withinthe extension side for the cyclic loading with constant stress amplitudes below CLRL;thus, the stress-strain loops migrate towards the extension side; for the cyclic loadingwith constant stress amplitudes above CLRL, the response is influenced equally bycompression and extension portions of applied cyclic loadingXlovzryvlz xlvyZergoun (1991) and Zergoun and Vaid (1994) justified the need for “slow” undrained cyclictriaxial loading of clays. The authors considered two-way (compression followed by exten-sion or extension followed by compression) undrained cyclic triaxial loading on isotropicallynormally consolidated specimens. They mentioned that non-uniformities within a specimenwere dependent on the degree of end restraints and the rate of loading. As the center of thespecimen was believed to be the most representative of the whole specimen, the only way toobtain reliable pore water pressure measurements was to perform “slow” undrained cyclictriaxial loading tests with 0:05%/hr for strain-controlled test.17FBFB Luvorutory ylymynt tystsp (kPa)0 100 200 300 400 500q (kPa)-200-1000100200experiment (6 cycles)=v)ǫa (%)-15 -10 -5 0 5 10q (kPa)-200-1000100200experiment (1-6,19-22 cycles)=w)p (kPa)0 100 200 300 400 500q (kPa)-200-1000100200experiment (1-6,15 cycles)=x)ǫa (%)-15 -10 -5 0 5 10q (kPa)-200-1000100200experiment (1-7,13-15 cycles)=y)Figure 2.3: Undrained cyclic triaxial data on Georgia kaolin clay (data from Sheu, 1984) forcyclic stress amplitudes: (a–b) qcy = 121:4 kPa, and (c–d) qcy = 140:7 kPaThe soil tested was structured Cloverdale clay with PI = 24%. The results from “slow”test are shown in terms of stress and stress-strain paths in Fig. 2.4 with applied constantstress amplitudes of (a − r) R (2Suc) = 0:75. The observations for this test are similar tothose made for Lower Cromer Till by Takahashi et al. (1980). The results from “slow” testswith different applied stress amplitudes (a − r) R (2Suc) are shown in terms of accumulatedpeak |"a| with the number of cycles in Fig. 2.5. Their additional findings are:• there is a “threshold stress amplitude”; axial strains decelerated if the applied constantstress amplitude was below the threshold and accelerate if it was above; larger axialstrains developed in extension for all constant stress amplitudes with a decreasingdifference between strains in compression and extension as the amplitude of appliedloading was decreased18FBFB Luvorutory ylymynt tysts• initial direction of applied cyclic loading played a significant role; two undrained testswith the same constant stress amplitude, one starting from compression and the otherfrom extension, were compared; the cyclic test starting from extension exhibited moredeformations than the test starting from compression; this implied the importance ofan initial anisotropic state for tested clayp (kPa)0 50 100 150 200 250q (kPa)-100-50050100 experiment (13 cycles)=v)a (%)-10 -8 -6 -4 -2 0 2 4 6 8 10q (kPa)-100-50050100 experiment (13 cycles)=w)Figure 2.4: Undrained cyclic triaxial data on Cloverdale clay (data from Zergoun and Vaid,1994)Number of cycles0 20 40 60 80 100Peak |ǫ a| (%)10-210-1100101compressionextension0.47 0.36Experiment0.750.690.790.62 0.570.54 τcy/Suc=0.27Figure 2.5: Accumulation of "a with the number of cycles for a range of applied stressamplitudes, xyR2Suc (data from Zergoun and Vaid, 1994)19FBFB Luvorutory ylymynt tystsGCGCH jnyrvinzy xyxlix multiByirzxtionvl “hzvrUndrained cyclic multi-directional shear tests, available to-date, are studied below in orderto highlight observed clay response under complex loading conditions. For stress and strain-controlled tests the rate of loading is given in terms of frequency (Hz) of applied stress andstrain history, respectively. Four clays considered below are: Gulf of Mexico, Kaolinite,Tokyo Bay, Kitakyushu, and Soft clays. Up to date, all of the available data is obtainedfrom “fast” undrained multi-directional shear tests. Therefore, the emphasize is made onmeasured shear response due to most likely measurement of un-equalized pore pressureduring the tests.Xyxlix multiByirzxtionvl “hzvr yzvixzMulti-directional shear device departs from the Direct Simple Shear (DSS) apparatus withthe main difference in that the shear stress acting on the horizontal plane is multi-directional.DSS apparatus has been developed with consideration of its specific applications and itsshortcomings are inherited by the multi-directional shear device.Historically, the shape of a DSS specimen is either square or circular in horizontal plane.A square specimen shape with straight lateral boundaries was used in a series of Cambridgeapparatus initially developed by Roscoe (1953). In this device, the plane strain conditionrequired for a DSS specimen is maintained by fixed side walls parallel to the direction ofshearing. Side walls perpendicular to the direction of shearing are connected to the bottomplate by hinges, and the top plate is allowed to move parallel to the bottom plate while beingfixed vertically. A circular specimen enclosed in a rubber membrane and surrounded by alu-minum rings was proposed by Kjellman (1951). Confinement of the specimen by aluminumrings maintains zero lateral displacement and thereby plane strain condition perpendicularto the direction of shearing. Shearing of the specimen is applied through the top plate witheither vertical load or height of the specimen constant. However, undrained conditions couldnot be considered in these devices. An alternative for undrained conditions was proposed byBjerrum and Landva (1966). They used a reinforced rubber membrane, enclosing tested spec-imen. As excess pore water pressure developed during undrained test would induce bulging20FBFB Luvorutory ylymynt tystsof the specimen and thereby a change in volume, they performed drained test with constantvolume maintained by adjustment of vertical stress so that the height of the specimen iskept constant. In this case, change in the vertical load during constant volume drained testis equivalent to change in excess pore water pressure during undrained test. Franke et al.(1979) developed DSS device in which truly undrained conditions could be applied. A spec-imen in this device is laterally enclosed by a rubber membrane, sealed with top and baseplates using O rings, and placed into a cylindrical pressure cell filled with water. Adjustmentof radial stress h by applying hydrostatic cell pressure during the undrained test is used tomake sure that lateral strain is kept close to zero. Back pressure cell ensures full saturationof a specimen and accurate measurements of pore water pressure. Dyvik et al. (1987) alsodeveloped DSS device with the specimen placed in a pressurized cell. In their comparisonof truly undrained tests with constant volume tests, they confirmed the hypothesis used forequivalent pore water pressure interpretation in NGI device for normally consolidated claysheared up to = 10%. Prevost and Ho¨eg (1976) suggested that, due to non-developmentof complimentary shear stresses, for constant volume conditions the incompressibility is onlysatisfied in an average sense for the entire specimen, while local compressibility will takeplace.Representation of in situ stress state acting on a single point requires uniform distribu-tion of stresses and strains in a laboratory element test. While uniform distribution of thesevariables is always hypothesized in element tests, practically it is difficult, if not impossible,to impose idealized boundary conditions that would assure such uniformity. The main de-ficiency of the DSS device is that complimentary shear stress acting on a side of specimencannot be applied and thereby a homogeneous stress state cannot be achieved. The only de-vice where complimentary shear stresses can be transmitted was developed by Arthur et al.(1977), but has never been used in simple shear apparatus under undrained or constant vol-ume conditions. Many researchers studied this problem experimentally (e.g., Franke et al.,1979; Vucetic and Lacasse, 1982), analytically (e.g., Roscoe et al., 1958), and numerically(e.g., Budhu and Britto, 1987). One of the conclusions from these studies is that at the coreof DSS specimen the stress state is homogeneous. However, measurements during DSS testcan only be taken at specimen’s boundaries. A general agreement between the researchers21FBFB Luvorutory ylymynt tystsis that increased diameter to height ratio reduces non-uniformity of distribution of stressesand strains. Another important consideration for DSS test is that the shear stress actingon the sides of DSS specimens and along its perimeter during the test cannot be measuredand therefore is not known. Wright et al. (1978) investigated the assumption that the shearstress is zero. In their study they provided Saint Venant solution (linear elastic) for a fixedend beam and experimental evidence from 3D photoelastic method, both suggesting that themagnitude of this shear stress is negligible for a square shaped specimen while it is significantfor a circular specimen.Yang et al. (2016) presented a comprehensive list of available multi-directional simpleshear tests. In these tests, a circular shape of specimen in the horizontal plane is chosen,and used devices are capable to impose true undrained conditions and measurements of porewater pressure using cell pressure and back pressure. As mentioned above, for circular shapespecimens, the shear stress acting on a side of specimen and along its perimeter is not zeroand therefore has to be considered in modeling element tests. Also it is mentioned abovethat constant volume tests might be impossible to perform and therefore truly undrainedconditions used in these tests are more reliable. To minimize non-uniformity of stresses andstrains, the devices are adopting large enough diameter to height ratio. Experimental resultsfrom available multidirectional simple shear tests are discussed next.Vvvilvwlz zxpzrimzntvl yvtvwv“zAvailable cyclic multi-directional shear tests in the literature are considered for discussionon the observations made by researchers. In terms of two horizontal cyclic shear compo-nents during the cyclic shearing stage, different shearing paths can be generated. With theinformation on compression stage, eight types of cyclic multi-directional shear tests in theliterature are presented in Fig. 2.6 in the space of the shear components of stress or strain.Here, point O represents zero stress or strain states, prior to the initial compression stage.While the specimen is consolidated under CKM or CK R conditions, the state moves frompoint O to point C along x-direction and the end of the path revealing normalized magnitude5CK is vn vnisotropix xonsoliyvtion fihzrz R c=vc fiith thz initivl xonsoliyvtion shzvr strzss c vnyvzrtixvl strzss vc22FBFB Luvorutory ylymynt tystsof the initial consolidation shear stress given in terms of static stress ratio, SSRx, if this stressexists; otherwise, SSRx = 0. Details on the exact initial state for each experiment on clay aregiven in Table 2.1. Undrained cyclic shearing stage starts from point C. Yang et al. (2016)have elaborated how the eight shearing paths can be produced based on mathematical ex-pressions. The 1-D linear path, as shown in Fig. 2.6(a), is the traditional unidirectional cyclicshear test (commonly known as DSS) with an initial offset shear stress in the same directionto the subsequent cyclic shear stress path. It has been conducted by many researchers ondifferent types of soil. When the subsequent cyclic shearing is applied perpendicular to theinitial compression shear stress, the shearing path is denoted by 2-D linear path, as shownin Fig. 2.6(b). This type of shearing dates back to Boulanger et al. (1991) on SacramentoRiver sand and later was adopted by Kammerer, J.M. (2002) on Monterey No. 0/30 sandand Rutherford (2012) on Gulf of Mexico clay. Circular/oval path in Fig. 2.6(c) was firstreported by Ishihara and Yamazaki (1980) on Fuji River sand samples under CI conditionprior to the cyclic multi-directional shear test. Along with that, alternate path in Fig. 2.6(g)is also exerted on the same material. Later, circular/oval path was adopted by Kammerer,J.M. (2002) on Monterey No. 0/30 sand and Rutherford (2012) on Gulf of Mexico clay,with samples under CKM and CK conditions. Two nontrivial shearing paths, as shown inFig. 2.6(d,e), are termed by figure-8 type A and figure-8 type-B paths, respectively. Theyfirst came out in the experiments on Monterey No. 0/30 sand (Kammerer, J.M., 2002) andthen Gulf of Mexico clay (Rutherford, 2012). The path in Fig. 2.6(f) is termed as rotatedoval path and was used by Matsuda et al. (2011, 2016, 2013, 2004) and Nhan and Matsuda(2016) on a wide range of clays and sands with measurements of accumulation of only porepressure. Fig. 2.6(h) presents one case where the change only happens in the orientation ofthe horizontal total shear stress without change in the magnitude. This type of shearing hadbeen adopted by Rudolph et al. (2014) on North Sea sand and Nie et al. (2015) on Soft clay.Gulf of bzxixo xlvyCyclic multi-directional shear tests were performed by Rutherford (2012) on undisturbedGulf of Mexico clay with the PI for the tested samples in the range of 40-55, consistingof 51%-72% of clay particles. This database is the most comprehensive for clay among23FBFB Luvorutory ylymynt tysts (a) 1-D linear path (b) 2-D linear path (c) Circular/oval path (d) Figure-8 type A path (e) Figure-8 type B path (f) Rotated oval path (g) Alternate path (h) Sector path Figure 2. Cyclic shearing paths summarized in Yang et al. (2016) during cyclic shearing stage. Table 1. Experimental database for multi-directional cyclic shear tests on clays and sands. Shearing path Initial state Material Source 1-D linear path CKa Gulf of Mexico clay Rutherford (2012) 2-D linear path CKa Gulf of Mexico clay Rutherford (2012) circular/oval path CK0 Gulf of Mexico clay Rutherford (2012) CKa Gulf of Mexico clay Rutherford (2012) figure-8 type A path CK0 Gulf of Mexico clay Rutherford (2012) CKa Gulf of Mexico clay Rutherford (2012) figure-8 type B path CK0 Gulf of Mexico clay Rutherford (2012) CKa Gulf of Mexico clay Rutherford (2012) rotated oval path CK0 Kaolinite clay Matsuda et al. (2016), Nhan and Matsuda (2016) Tokyo bay clay Matsuda et al. (2016), Nhan and Matsuda (2016) Kitakyushu clay Matsuda et al. (2016), Nhan and Matsuda (2016) sector path CK0 Soft clay Nie et al. (2015) 1-D linear path CKa Sacramento River sand Boulanger et al. (1991) Monterey No. 0/30 sand Kammerer et al. (2002) 2-D linear path CKa Sacramento River sand Boulanger et al. (1991) Monterey No. 0/30 sand Kammerer et al. (2002) circular/oval path CI Fuji River sand Ishihara and Yamazaki (1980) CK0 Monterey No. 0/30 sand Kammerer et al. (2002) CKa Monterey No. 0/30 sand Kammerer et al. (2002) figure-8 type A path CK0 Monterey No. 0/30 sand Kammerer et al. (2002) CKa Monterey No. 0/30 sand Kammerer et al. (2002) figure-8 type B path CK0 Monterey No. 0/30 sand Kammerer et al. (2002) CKa Monterey No. 0/30 sand Kammerer et al. (2002) rotated oval path CK0 Toyoura sand Matsuda et al. (2004, 2011) GBFS Matsuda et al. (2004, 2011) alternate path CI Fuji river sand Ishihara and Yamazaki (1980) sector path CK0 North Sea sand Rudolph et al. (2014) irregular path CI Fuji river sand Ishihara and Nagase (1988) τx/σ′vcτy/σ′vcSSRx CSRxOCτx/σ′vcτy/σ′vcOCSSRxCSRyτx/σ′vcτy/σ′vcSSRx CSRxCSRyOCτx/σ′vcτy/σ′vcSSRx CSRxCSRyO Cτx/σ′vcτy/σ′vcSSRx CSRxCSRyO C γxγyπ/4C(O) τx/σ′vcτy/σ′vcCSRxCSRyC(O) τx/σ′vcτy/σ′vcψC(O)Figure 2.6: Cyclic shearing paths summarized in Yang et al. (2016) for cyclic multi-directionalshearingothers. The 1-D line r, 2-D linear, circular nd figure-8 type A shear stress paths with0:2Hz frequency were applied onto CKM and CK specimens. In the experiments the authormeasures both generation of pore pressure and shear strains. The following main observationswere mad from the experiments:• the largest accumulations of pore pressure was measured for the shearing paths appliedonto CKM specimens• figure-8 path accumulated permanent shear strains in fewer cycles than 1-D linear,2-D linear and circular tests due to the combination of the horizontal shear stressrotation, complete shear stress removal, and reversal of stresses as shown in Fig. 2.7for comparison between circular and figure-8 paths• for the tests on CK specimens, the shear strains migrated in the direction of initialshe r stress regardless of the orientation of a shear ng path, whereas the strains in theopposite direction were oscillating around 0 % (i.e., no accumulation) as demonstratedin Fig. 2.724FBFB Luvorutory ylymynt tystsNormalized Shear Stress, τx/σvc-0.4 -0.2 0 0.2 0.4Normalized Shear Stress, τy/σvc-0.4-0.200.20.4Strain γx (%)-20 -15 -10 -5 0 5 10 15 20Strain γy (%)-20-15-10-505101520Pore Pressure Ratio, ∆ u/ σvc00.20.40.60.81Number of Cycles, N012345678Shear Strain γ (%)-20 -15 -10 -5 0 5 10 15 20Number of Cycles, N012345678Normalized Shear Stress, τx/σvc-0.4 -0.2 0 0.2 0.4Normalized Shear Stress, τy/σvc-0.4-0.200.20.4Strain γx (%)-20 -15 -10 -5 0 5 10 15 20Strain γy (%)-20-15-10-505101520Pore Pressure Ratio, ∆ u/ σvc00.20.40.60.81Number of Cycles, N00.511.522.53Shear Strain γ (%)-20 -15 -10 -5 0 5 10 15 20Number of Cycles, N00.511.522.53Figure 2.7: Illustration of migration of shear strains in the direction of initial shear stressfor circular, GOM-11 (top), and figure-8 type A, GOM-14 (bottom), cyclic shear tests (afterRutherford, 2012)25FBFB Luvorutory ylymynt tystsTable 2.1: Experimental database for cyclic multi-directional shear tests on clayshhzvring pvthInitivl aovyingbvtzrivl hourxz“tvtz frzquznxz1-D linear CKM 0:2Hz Gulf of Mexico clay Rutherford (2012)CK 0:2Hz Gulf of Mexico clay Rutherford (2012)2-D linear CK 0:2Hz Gulf of Mexico clay Rutherford (2012)circular CKM 0:2Hz Gulf of Mexico clay Rutherford (2012)CK 0:2Hz Gulf of Mexico clay Rutherford (2012)figure-8 type A CKM 0:2Hz Gulf of Mexico clay Rutherford (2012)CK 0:2Hz Gulf of Mexico clay Rutherford (2012)figure-8 type B CKM 0:2Hz Gulf of Mexico clay Rutherford (2012)CK 0:2Hz Gulf of Mexico clay Rutherford (2012)rotated oval CKM 0:5Hz Kaolinite clay Matsuda et al. (2013),CKM0:5HzTokyo bay clayMatsuda et al. (2016),Nhan and Matsuda (2016)CKM0:5HzKitakyushu clayMatsuda et al. (2016),Nhan and Matsuda (2016)sector CKM 0:1Hz Soft clay Nie et al. (2015)KvolinitzA iokyo WvyA vny Kitvkyu“hu xlvy“Cyclic multi-directional shear tests were performed by Matsuda et al. (2016, 2013) and Nhanand Matsuda (2016) on Kaolinite clay with PI for tested specimens of 25:5, Tokyo bay claywith PI = 41:6, and Kitakyushu clay with PI = 63:8. The 1-D linear, and circular/oval shearstrain paths with 0:5Hz frequency were applied onto CKM and CK specimens. The two com-ponents of applied strain paths were having a phase shift thereby producing a circular pathfor 90◦ phase shift and a rotating oval path for larger phase shift values. In the experi-ments the authors measure only the generation of pore pressure without any measurementsof shear stresses. In this way, the experiments are limited only to the observations madefor pore pressure generation, which is known to be questionable in the device as mentionedpreviously. Nevertheless, the following main observations were reported by the authors:• the larger the shear strain amplitude and the number of cycles, the higher the porepressure ratio and the post-cyclic settlement• at the same shear strain amplitude and number of cycles, the excess pore pressureand the post-cyclic settlement induced by cyclic multi-directional shear are larger than26FBFB Luvorutory ylymynt tyststhose induced by the uni-directional one• for multi-directional shear, the excess pore water pressure and the post-cyclic set-tlement increase with the phase difference between applied horizontal shear straincomponents• the higher the plasticity index of the soil, the slower the rate of cyclic shear-inducedpore water pressure accumulation which leads to the lower level of pore water pressureaccumulation, irrespective of cyclic shearing conditionshoft xlvyCyclic multi-directional shear tests were performed by Nie et al. (2015) on Soft clay withthe PI for tested specimens of 35:3. Varying sector strain paths with 0:1Hz frequency wereapplied onto CKM and CK specimens. The sectors were rotated throughout a test untilspecified sector orientation angle and then returned back to their initial position where =0◦. In the experiments the author measures the generation of pore pressure and shear stresswith the results interpreted in terms of shear modulus G. The effect of cyclic shear directionand shear strain amplitude on the shear modulus G and the pore pressure u were investigatedduring the shear process and compression in the reconsolidation process. Measurements ofactual shear stresses are not provided by the author thereby stress-strain response cannotbe studied from the tests. The following main observations were reported by the authors:• the shear modulus G was found to be different in different directions of sector loadingpath due to soil anisotropy• the changing of directions of sector loading accelerated the decrease of G (1.51 timesof unidirectional attenuation of G); the shear tests with shear strain greater than acertain value had no such phenomenon• the changing of directions of sector loading promoted the development of pore pressurein the shear process (1.67 times of the increment of the unidirectional pore pressure);the shear tests with shear strain greater than a certain value had no such phenomenon27FBGB hhyorytiwul moxylingFrom the above observations primarily on Gulf of Mexico clay, it is clear that the cyclicmulti-directional shear path on clay induces larger strains than the 1-D linear cyclic shear.Rutherford (2012) also observes that among the multi-directional stress paths the figure-8shearing induces larger strains than both the circular and the 1-D linear paths. For CK specimens, the accumulation of the shear strains is always in the direction of initial staticshear stress.GCH ihzorztixvl moyzlingA constitutive model is required for numerical solution of a boundary value problem in orderto represent stress-strain response of a material. It is imperative that such representation ismade correctly. This is especially true when modeled material is subjected to cyclic responsewhere a number of loading/unloading/reloading sequences is typical. Below, a general re-view of plasticity theories and detailed review of classical plasticity are given first. Then ahistorical development of bounding surface formulation, commonly used in the constitutivemodeling research community for description of cyclic response of a material, is provided.This is followed by discussion on constitutive models that incorporate bounding surface for-mulation within classical plasticity theory for description of clay response subjected to cyclicloading. The revision of available constitutive models with the focus of their advantagesand disadvantages provides opportunity for development of a constitutive model that can beused for modeling clay response when subjected to cyclic loading.GCHCF dvzrvizfi of xon“titutivz moyzlingConstitutive models describing stress-strain response of geomaterials are predominantly de-veloped based on macroscopic observations made in experiments (e.g., triaxial or directsimple shear tests). Some of these models are purely empirical, and others are developedon the basis of certain theoretical frameworks, such as classical plasticity or hypoplasticity.Some models are quite simple and have material parameters (e.g., Poissons ratio, Shearmodulus, and friction angle) that are easy to comprehend and calibrate, while others aremore sophisticated and have larger number of parameters, including model specific parame-28FBGB hhyorytiwul moxylingters, that require availability of more advanced laboratory data for their calibration. Ideally,a model should have a balance between the simplicity of its formulation and its ability torepresent an intended material response. The choice of a proper constitutive model in anapplication largely depends on the type of loading and nature of governing aspects of ma-terial response during that loading. A number of constitutive models have been developedfor simulation of response of soil subjected to cyclic loading. According to type of their for-mulation, these models can be divided into plane strain and multi-axial models. The formermodels are developed only for plane strain conditions, and therefore cannot describe generalstress and strain. These models are typically developed based on DSS testing data. Someexamples of these models are UBCHYST (Naesgaard, 2011), UBCSAND (Beaty and Byrne,2011), PM4SAND (Boulanger and Ziotopoulou, 2013), and SimpleDSS (Pestana and Bis-contin, 2000). These models are often developed with the shear and volumetric stress-strainresponse coupled using certain assumptions. In the category of multi-axial constitutive mod-els, the formulation incorporates full tensors of stress and strain that can describe generalstress and strain, and, therefore, shear and volumetric response fully coupled. Many of thesemodels are developed as multi-axial extensions of simpler formulations initially developedfor triaxial stress-strain response. Although many of them show satisfactory performancein various triaxial loading conditions, their predictive capabilities are very rarely tested inmulti-axial loading. As the goal of this thesis is development of a constitutive model that canbe later used for describing generalized stress-strain response of clay subjected to cyclic load-ing, only constitutive models developed using multi-axial formulation are discussed. A briefoverview of available plasticity theories is provided next, which is by no means exhaustive,and is provided in order to demonstrate available approaches of plasticity theory.elv“tixity thzoriz“According to Batdford and Budiansky (1949), plasticity of materials historically has beendescribed using deformation theory of plasticity (Hencky, 1924; Ilyushinl, 1963; Nadai andWahl, 1931) and flow theory of plasticity (e.g. Hill, 1950; Prager, 1955). Deformation theoryof plasticity is based on the assumption that the state of stress is uniquely determined bythe state of strain. Flow theory of plasticity is based on the assumption that the increment29FBGB hhyorytiwul moxylingof stress is uniquely determined by the state of stress and increment of strain. Experimen-tal observations of soil behavior suggest that history of stress strongly influence materialresponse. Therefore, flow theory of plasticity is considered only and hereafter referred to asplasticity.A number of plasticity theories have been developed over decades. This includes classicalplasticity, hypoplasticity, and endochronic plasticity. These theories are developed based ondifferent assumptions and, to a certain extent, involve thermodynamic considerations usedfor development of their formulations. With the choice of plasticity theories one often facesa situation where the problem of identification of appropriate constitutive relation from testdata available does not have a unique solution (Bazˇant, 1978). Each of these theories hasadvantages and disadvantages. It is therefore important to acknowledge existence of differentplasticity frameworks and perhaps learn from them. In classical plasticity, plastic strainincrement is a function of state variables only and is independent of loading direction, whilein hypoplasticity plastic strain increment also depends on the loading direction (Dafalias,1986v). The latter is incrementally nonlinear as the loading direction set by the change instress is the unknown. Endochronic plasticity in this sense is similar to classical plasticityand considers plastic strain increment independent from the loading direction (Bazˇant, 1978).In endochronic plasticity, just like in hypoplasticity, plastic response is smooth, i.e., there isa smooth transition between loading, neutral loading, and unloading. In classical plasticity,however, this is not possible due to the consistency condition that has to be satisfied, whichinduces into artificial splitting between elastic response and elastoplastic response (Einav,2012).In addition to different classes of plasticity theories, constitutive models can also be clas-sified based on their formulation with internal variables defined in stress, and strain space.The stress space formulation is used in classical plasticity theory where yield criterion isdefined in the stress space. The yield surface limits purely elastic response from the elasto-plastic response. This includes soil models such as Modified Cam-Clay (Roscoe and Burland,1968) and its anisotropic extension SANICLAY model (Dafalias et al., 2006) with plasticpotential derived from work dissipation assumption. A number of hyperplasticity modelsare formulated in strain space. These constitutive models are developed from hypotheses30FBGB hhyorytiwul moxylingabout the form of an energy potential and a dissipation function (Collins and Houlsby, 1997;Houlsby and Puzrin, 2000; Likitlersuang and Houlsby, 2006). A number of hypoplasticityand endochronic models are formulated in the strain space. These models are worked out byconstruction of a single constitutive equation relating the stress rate as a function of strainrate (Kolymbas, 1988, 1991). However, recent works on hypoplasticity are formulated as amix between stress and strain spaces (Fuentes et al., 2017; Fuentes and Triantafyllidis, 2015)for better replication of soil response. Endochronic models deal with the plastic response ofmaterials by means of memory integrals expressed in terms of memory kernels which containthe essence of the material (Valanis and Peters, 1991; Valanis and Lee, 1984).From the above plasticity theories, classical plasticity with the yield criterion defined inthe stress space is well established and is primary plasticity theory used by the researchersin geomechanics (e.g. Dafalias, 1987; Roscoe and Burland, 1968; Wood, 1990). Certainlimitations in classical plasticity do exist as discussed above. The rest of the section isfocused on classical plasticity theory, and then followed by the presentation of the boundingsurface formulation for description of clay subjected to cyclic loading.Xlv““ixvl plv“tixityClassical elastoplastic models assume the existence of a yield surface in the generalized stressspace (e.g., Hill, 1950; Prager, 1955). Its general equation can be expressed asf (;qn) = 0 (2.1)where  is a stress tensor; and qn are n internal variables controlling the geometry of theyield surface. This yield surface encloses the region inside which any stress increment thatdoes not cross the yield surface induces only elastic strains. Once a stress increment crossesthe yield surface a plastic strain increment is induced in addition to the elastic one. A basickinematic assumption is a decomposition of a total strain tensor " into elastic "e and plastic"p parts. The rate of " can be decomposed accordingly:"˙ = "˙e + "˙p (2.2)31FBGB hhyorytiwul moxylingwhere a superposed dot stands for the material time derivative, i.e., rate.An input into a constitutive model could be either a strain increment or stress increment.General constitutive relations between stress increment and elastic strain increment are thengiven for the corresponding inputs, respectively,˙ = De : "˙e; (2.3)"˙e = Ce : ˙ (2.4)where operator : is an inner product of two adjacent tensors; De and Ce are fourth ordertensors of elastic modulus and compliance, respectively.Rates of plastic strain and internal variables are assumed to be given by Eqs. 2.5 and2.6."˙p = 〈a〉cp (2.5)q˙n = 〈a〉q¯n (2.6)where a is a plastic scalar-valued multiplier, or loading index to be defined in the sequel ofthis section; 〈〉 are Macaulay brackets such that 〈a〉 = a if a S 0, and 〈a〉 = 0 if a ≤ 0; cpand q¯n are tensorial and scalar/tensorial, respectively, functions of state variables  and qn.It is then common to definecp = @gR@, where @gR@ is a gradient of a plastic potentialgiven by:g (;qn) = 0 (2.7)An associative flow rule considers the yield surface and plastic potential coinciding witheach other, while a non-associative flow rule assumes non-coincident yield surface and plasticpotential.Then Eqs. 2.3 and 2.4 are re-written on the basis of Eqs. 2.2 and 2.5:˙ = De :("˙− 〈a〉 @g@)(2.8)32FBGB hhyorytiwul moxyling"˙ = Ce : ˙ + 〈a〉 @g@(2.9)The consistency condition requires that the stress point cannot lie outside the domainenclosed by the yield surface. Thus, once a stress increment crosses the yield surface, in-ternal variables are updated to maintain an updated stress point on the yield surface. Theconsistency condition is expressed in mathematical form asf˙ =@f@: ˙ +@f@qn: q˙n = 0 (2.10)Substituting Eq. 2.6 into Eq. 2.10 gives the loading indexa =1Kp(@f@: ˙)(2.11)where Kp = − (@fR@qn) : q¯n is a plastic modulus.The constitutive relationships in Eqs. 2.8 and 2.9 can be re-written on the basis of Eq. 2.11so that the the constitutive matrices are:˙ = Dep"˙ =[De −H (a) (De : @gR@)⊗ (@fR@) : DeKp + @fR@ : De : @gR@]: "˙ (2.12)"˙ = Cep : ˙ =[Ce +(@gR@)⊗ (@fR@)Kp]: ˙ (2.13)where H (a) is the Heaviside step function, H (a) = 1 for a S 0 and H (a) = 0 for a ≤ 0;Dep; Cep represent an elastoplastic modulus and compliance fourth order tensors; and ⊗ isan outer product of two adjacent tensors.Eqs. 2.12 and 2.13 are complete sets of constitutive relations for inputs of strain andstress increments, respectively, in a classical elastoplastic model. It is sufficient to providethe yield surface f = 0, plastic potential g = 0 or any other appropriate function for cp,and evolution laws for plastic strain rate and rate of internal variables shown generally inEqs. 2.5 and 2.633FBGB hhyorytiwul moxylingGCHCG Wounying “urfvxz formulvtionelv“tix “trvin“ in“iyz thz yizly “urfvxzThe observed experimental undrained cyclic response of clay in triaxial tests illustrated thatthe behavior of clay is not elastic within the yield surface. From what the question follows if itis necessary to account for inelastic strains within the yield surface in constitutive modeling.The answer depends on what type of loading is required to be simulated. For monotonicloading, concerned with peak and ultimate strength of clay or the stress states mainly on theyield surface, classical elastoplastic models with incorporated certain complexities are ableto predict soil response with fair accuracy. If monotonic loading is concerned with simulatedsoil response for the stress states within the yield surface, a model allowing inelastic strainswithin the yield surface should be considered. For earthquake loading, the imposed stresspath always involves loading and unloading processes. Even if the loading stress path ison the yield surface, unloading is directed towards the interior of the yield surface whichis then followed by reloading. The necessity to simulate plastic deformations within theyield surface becomes pronounced as the number of such loading-unloading-reloading cyclesincreases. Earthquake-induced deformations within the soil often involve a considerablenumber of these cycles, which does require simulation of inelastic soil response inside theyield surface for successful model predictions.As has been discussed in section 2.2.1, experimental testing requires that “slow” undrainedcyclic loading should be performed for accurate pore water pressure measurements. In “slow”undrained cyclic loading tests rate effect is still present, but to a much lower degree thanin “fast” undrained cyclic loading. Therefore, literature review is conducted below on rate-independent models with incorporated bounding surface formulation. The bounding surfaceframework is an attractive tool to model undrained cyclic loading imposed on clay used bymany researchers. The bounding surface formulation can be incorporated into any classicalelastoplastic model by adjusting a plastic modulus Kp and leaving all other relationshipsunchanged. First, development of the bounding surface formulations is discussed. Then,rate-independent elastoplastic models with incorporated bounding surface formulation formonotonic and cyclic loading simulations are reviewed.34FBGB hhyorytiwul moxylingNot all of the reviewed models are referred to by the authors proposing them as boundingsurface plasticity models; however, the concept adopted for plastic strains simulation withinthe yield surface is similar to that used in the bounding surface formulation. To keepa common definition of the yield surface, inside which only elastic strains are allowed, theterm bounding surface is used for the yield surface with inelastic strains allowed inside it. Asin various publications, the bounding surface is referred to in different ways, a conventionalyield surface term is to be used in such cases.Yzvzlopmznt of wounying “urfvxz formulvtionAttempts to simulate inelastic response within the conventional yield surface were madein the early 1950s; slip theory proposed by Batdford and Budiansky (1949) was adoptedby Koiter (1953) for this purpose. Further development of models describing soil behaviorin multiaxial stress space was conducted separately by Iwan (1967) and Mro´z (1967, 1969).They adopted an idea of family of surfaces of progressively larger size, one enclosing the other,with constant work-hardening moduli for each surface. The inner-most surface enclosed atruly elastic region, while the outer-most served as the conventional yield surface. Thechange of the work-hardening modulus was linear for the stress states on a single surfaceuntil a surface of larger size was reached; at this point the work-hardening modulus changedand corresponded to that of a larger surface at the current stress point until the next surfaceof larger size was attained. In this way, piece-wise constant work-hardening moduli allowedsimulation of inelastic strains inside the outer-most surface. The main disadvantage of thisapproach is that a number of surfaces have to be stored in the model memory.The need of a simpler approach inspired development of the bounding surface formulationproposed by Dafalias and Popov (1975). Rules of plastic modulus variation were proposedbased on uniaxial cyclic loading for metals and generalized to multiaxial stress space. Since itsintroduction, the bounding surface formulation has been used by various investigators. In aseries of publications by Dafalias (1986u), Dafalias and Herrmann (1986), and Anandarajahand Dafalias (1986), the bounding surface formulation is reviewed for simulations of soilresponse. Because the current work is dedicated to cyclic response of clay, the publicationsmentioned will be discussed in detail.35FBGB hhyorytiwul moxylingThe bounding surface formulation introduces inelastic strains for the stress states insidethe conventional yield surface through plastic modulus. All other relations assumed for thebounding surface are similar to those used in classical elastoplastic models. The conventionalyield surface is replaced by the bounding surface. Within the bounding surface F = 0, plasticstrains depend on the distance from the actual stress point ij to the corresponding imagepoint ¯ij on the bounding surface, where the latter is determined by a mapping rule. Thedegree of plasticity increases as this distance becomes smaller and reaches its limit oncethe stress point is on the bounding surface. A loading surface f = 0 passes through ijand is enclosed by the bounding surface so that both surfaces coincide once a current stresspoint is on the bounding surface. The bounding surface concept requires conformance of thefollowing:1. the mapping rule has to be such that a unique ¯ij is defined for any stress statewithin the bounding surface, and @fR@ij = @FR@¯ij, where @fR@ij and @FR@¯ij aregradients of the loading surface and the bounding surface,respectively, for ij = ¯ij toguarantee that the loading surface never intersects the bounding surface; the mappingrule is not invertible in a sense that an infinite number of the stress states within thebounding surface could correspond to a given ¯ij.2. plastic strains are allowed for the stress states within the bounding surface; thus, as¯ij must remain on the bounding surface, the plastic modulus for the stress states onthe bounding surface is obtained from the consistency condition, F˙ = 0:K¯p = − @F@q¯n: ˙¯qn (2.14)3. The actual plastic modulus Kp is related to K¯p through Euclidean distance: = [(¯ij − ij) (¯ij − ij)]N=2 (2.15)such that Kp S K¯p for  S 0, and Kp → K¯p for  → 0.Dafalias (1986v) reviewed two different mapping rules; each of the two mapping rulesdeserves to be discussed in more detail.36FBGB hhyorytiwul moxylingdriginvl wounying “urfvxz mvpping rulzThis mapping rule was originally proposed by Dafalias (1975). The loading and boundingsurfaces, having kinematic hardening rules that can be independent, were expressed respec-tively as follows:f (ij − ij; qn) = 0 (2.16)F (¯ij − ij; qn) = 0 (2.17)where ij and ij are the corresponding kinematic hardening internal variables. Adoptedkinematic and other hardening rules for the two surfaces should be such that surfaces neverintersect. Then the mapping rule, with the assumption that bounding and loading surfaceshad similar shapes, was given by:¯ij = b (ij − ij) + ij (2.18)where b is a similarity ratio between the two surfaces. The proposed plastic modulus for anystress state was:Kp = K¯p + h (|in|)  : n〈(in − ) : n〉 (2.19)where  = (¯ij − ij); in is equal to  at the moment of initiation of new loading process;and n is a loading direction. The in was updated when (in : n−  : n) ≤ 0. Notice thatthe first term was always larger than the second for a loading process and vice versa forunloading as the terms changed signs. The use of tensorial quantities was advocated by thecase of neutral loading simulations (see Dafalias, 1986v, for details).With the original bounding surface mapping rule, Dafalias (1986v) was aware that theproblem of “overshooting” exists. This was the case for short unloading with new in afterthe loading process, followed by reloading with updated in. This would result in a verysmall denominator in Eq. 2.19, i.e., a large plastic modulus, as opposed to the case where nosuch small perturbation had occurred and stress–strain curve in reloading would overshootthe one in loading. However, the overshooting was not infinite, as it was bounded by thebounding surface.37FBGB hhyorytiwul moxylinggvyivl mvpping rulzDafalias (1986v) referred to the work of Hashiguchi and Ueno (1977) as the pioneering workfor the radial mapping rule, where the origin had been used as a projection center. Theproposed radial mapping rule, shown in Fig. 2.8, required back-stress ratio , which wasthe projection center, to be always located within or on the bounding surface. The imagethe 8in (symbolized by dmax in their paper) to the value it had when a minimum 8 was achieved in the past, if this 5 is reached again. A dis-continuous Kp may occur though, since the same 8 will be related to two 8in • Another attempt to avoid overshooting was made by Tseng and Lee (37) using the elastic unloading chord, but it appears to have been re-stricted only to the cases where elastic deformation takes place upon unloading/reloading. Radial Mapping Plasticity Formulation.—This particular formulation is characterized by the "radial mapping rule" as proposed by Dafalias (13), and deserves a detailed investigation since only its main features were described in Ref. 13. In addition, many existing models with dif-ferent names are in essence radial mapping plasticity formulations and will be discussed subsequently. Fig. 1 shows the concept. The bounding surface is schematically shown as a circle of center p,-, (kinematic hard-ening) and can be described analytically by Eq. 25b. The novelty here is the introduction of a second back-stress, a,y, always within or on F = 0, not as the center of an enclosed yield surface as in the two-surface approach, but as the projection center used to radially project the cur-rent stress cr,y onto <r,y on F = 0. Thus, a radial mapping from cr,y to cr,-,- is introduced, and Eq. 19 takes the form b(<r,y 7) + «« (29) The b is s i and can be determined by inserting the cr,-, from Eq. 29 into the analytical expression of = 0. Observe that b is now variable, con-trary to the constant similarity ratio b entering Eq. 26. When er,y = <r,y the b = 1, i.e., the identity condition is satisfied, while when o^ = a,-, the b = 00 and & j is ind terminate. Eq. 29 together with the assumption L,-, = Ljj = dF/d&ij, efine indirectly a l ading surface / = 0 shown by a dashed line in Fig. 1, which is homologous to F = 0 with a,, being the center of ELASTIC NUCLEUS LOADING SURFACE Jki -BOUNDING SURFACE F ( £ f M , , „ ) = !> FIG. 1.—Schematic Diagram of Bounding Surface and Related Concepts 977 Downloaded 18 Mar 2011 to 128.189.245.242. Redistribution subject to ASCE license or copyright. Visit http://www.ascelibrary.orgFigure 2.8: Radial mapping rule (from Dafalias, 1986v, with permission from ASCE)point on the bounding surface was a projection from through the current stress point. Theprojection center also served as the center of homology between the loading and boundingsurfaces, such that @fR@ij points in the same direction as @FR@¯ij. This ensures that thetwo surfaces never intersect. The radial mapping rule was given by¯ij = b (ij − ij) + ij (2.20)38FBGB hhyorytiwul moxylingwhere the value of b was determined by plugging Eq. 2.20 into F = 0, 1 ≤ b ≤ ∞. Becauseof the two surfaces being homologous to each other no consistency condition was requiredfor the stress states within the bounding surface.The plastic modulus equation for any stress state was proposed as follows:Kp = K¯p + Hˆ〈r − s〉 = K¯p + Hˆ〈bb− 1 − s〉−N (2.21)where r is the distance from the projection center to the image point, Hˆ is a positive shapehardening function, s is an elastic factor, and rR = bR (b− 1). When  ≥ rRs, i.e., for thestress states within the elastic nucleus, Kp =∞, rendering purely elastic strains. For s = 1,the elastic domain degenerated into a single point, whereas for s → ∞, the elastic domaincoincided with the bounding surface, rendering a conventional elastoplastic model.A particular radial mapping rule for the bounding surface formulations was presented byDafalias and Herrmann (1986), as a continuation of previous work by Dafalias (1986v), forisotropic cohesive soils within the Critical State Soil Mechanics (CSSM) theory. In additionto the radial mapping, the authors included Lode angle dependency in the proposed model,which enhanced predictive capabilities for more general stress states. For the formulation,I was a first invariant of the stress tensor, and J was a second invariant of the deviatoricstress tensor. The proposed projection center in Fig. 2.9 was positioned on the hydrostaticline, represented by I-axis, and was related to the size of the bounding surface along thisaxis by:Ic = XpIM (2.22)where Ic is the position of the projection center on the I-axis, Xp is a new model constantin the range of 0 ≤ Xp Q 1, and IM is the size of the bounding surface. Similar to Eq. 2.20,image stress for a given projection center was:I¯ = b (I − XpIM) + XpIM (2.23)The composite bounding surface, consisting of two ellipses and one hyperbola with con-tinuous tangents, as shown in Fig. 2.9, was used for better predictions at high OCRs. In39FBGB hhyorytiwul moxylingaddition, the use of g S 1 parameter, which determined the position of point IN in Fig. 2.9and related two major axes of the ellipse 1 as IN = IMRg, was considered. Lode angle wasdenoted by .HYPERBOLA ELLIPSE 2 ELLIPSE 1 ELASTIC FIG. 1.—Schematic Illustration of Bounding Surface and Radial Mapping Rule in Stress Invariants Space it will be explained in the sequel. Eq. 30 has the following specific forms. For ellipse 1: F=(I-I0)[l+^I0)+(R-lfU) =0. For the hyperbola: 2 I- h N R\ N/ LJjL RJ \N R For ellipse 2: F = (7 - TI0)(I - (X + 2QI0] + p/2 = 0 T(Z + TF) T2 £ = - - = - — = — ; p = 0 . Z + 2TF' RA " N • (31) • (32) (33a) (33b) N Z(Z + 2TF') N i :; Z = - (1 + y - VT + 72) (33c) VT+f' R The dependence on ep occurs through loie7), and the dependence on a through the parameters N(a), R(<x), and A(a) according to Q(a) = g(a,c)Qc (34«) Qe c = ~ (34b) g(a,c) = — C > • , (34c) 1 + c - (1 - c) sin 3a 1270 Downloaded 18 May 2009 to 193.156.5.162. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyrightFigure 2.9: Radial mapping rule from (from Dafalias and Herrmann, 1986, with permissionfrom ASCE)The plastic modulus for any stress state was the same as the one in Eq. 2.21. Dafaliasand Herrmann (1986) elaborated on the shape hardening function Hˆ and considered thefollowing equation:Hˆ =1 + zin−  g∗2pa [zmh( ) + (1− zm)hM] (2.24)where zin is an initial void ratio;  and  are slopes of the normal compression and therebound lines, respectively, in the z-ln p plane; g∗ =((@FR@p¯)2 + (@FR@q¯)2)N:2was used toaccount for a jump of a bounding surface gradient in transition points between the threesurfaces comprising the bounding surface; pa is an atmospheric pressure; z = JRJN; m =0:02, rendering zm ≈ 1 to maintain dependency on h( ) rather than hM; h( ) is a shapehardening function of h, dependent on , and it required determination of hc and he fortriaxial compression and extension; hM = (hc + he) R2.Effects of hc and X parameters in triaxial compression were shown in detail. While theformer controlled the degree of plastic deformation for the stress states within the bounding40FBGB hhyorytiwul moxylingsurface, the latter was successfully used to control contracting or dialative responses foroverconsolidated stress states. Calibration of the parameters and model performance inundrained monotonic triaxial compression suggested that the projection center along thehydrostatic axis was an attractive feature of the model for monotonic loading simulationsfor high OCR states.Anandarajah and Dafalias (1986) further incorporated inherent and induced anisotropyinto the bounding surface soil plasticity model. The orientation of the bounding surface wascontrolled by evolving anisotropy, represented by the second-order tensor aij in Fig. 2.10,during the loading process. The projection center was determined as XpIaM , where Xp wasthe same model parameter proposed by Dafalias and Herrmann (1986) and IaM was a size ofthe bounding surface; the radial mapping rule was also identical, given the projection center.FIG. 1.—Schematic Illustration of Bounding Surface and Geometrical Interpreta-tion of Stress Variables was accounted for in a rather simple manner. Additionally, the concept of elastic nucleus was introduced in order to model the cyclic stabili-zation exhibited by clays. Some important modifications are incorporated in applying the con-cept of bounding surface for anisotropic clays. Referring to Fig. 1, the bounding surface is assumed to be oriented at any stage along a line (/"-axis) whose direction changes during the loading, representing the evo-lution of anisotropy. A symmetric second-order tensor 8°;, such that 8|8| = 3, defines the orientation of the bounding surface axis (/"-axis), in the stress space. According to the concept of bounding surface, every point in the stress space, such as point B in Fig. 1, is associated with a corresponding image point on the bounding surface via a suitable map-ping rule. The current theory, as in the isotropic formulation, employs a radial mapping rule (Fig. 1) where point C is obtained by radially pro-jecting point B from point A, which is assumed to be on the /"-axis. The projection center A is the counterpart of point Ic of the isotropic for-mulation, shown in figure 1 of Part II (11). The scalar distance between the origin and point.A is taken as CI"0 where C is a model parameter, referred to as the projection parameter. The gradient of the bounding surface at the image point C, defines the loading/unloading direction, as well as the direction of plastic strain rate for the actual stress point B. The plastic modulus Kp at the actual stress point a,y, is assumed to be a function of the plastic modulus Kp (bounding modulus) at the image point <r,y, and the scalar distance 8 between the image and the actual stress points. 1294 Downloaded 18 Mar 2011 to 128.189.245.242. Redistribution subject to ASCE license or copyright. Visit http://www.ascelibrary.orgFigure 2.10: Radial mapping rule (from Anandarajah and Dafalias, 1986, with permissionfrom ASCE)Due to the bounding surface orientation being controlled by evolved anisotropy, theplastic modulus and the shape-hardening function were adjusted accordingly, with the latter41FBGB hhyorytiwul moxylinggiven by:Hˆ =1 + zin−  pa [zmh( ) + (1− zm)hM][9(@FR@I¯a)2+ 3(@FR@J¯a)2](2.25)The model was tested in comparison with experimental data in undrained triaxial com-pression and extension tests on isotropically and KM-consolidated specimens. The use ofthe bounding surface with the projection center on evolving Ia-axis improved model perfor-mance during undrained shearing on KM-consolidated samples for the stress states within thebounding surface.In the radial mapping rule adopted by Dafalias and Herrmann (1986) and Anandarajahand Dafalias (1986) for isotropic and anisotropic cohesive soils, respectively, the projectioncenter imposed a restriction on plastic strains. Purely elastic strains occurred upon unloadinguntil loading index a = ((@FR@¯ij) ˙ij) RKp became positive as the definition of the plasticstrains increment is "˙ij = 〈a〉 @FR@¯ij, where Macaulay brackets rendered null for negative a.This restriction might be appropriate for monotonic loading simulations, while unsatisfactoryresults are expected for undrained cyclic loading simulations.GCHCH Wounying “urfvxz moyzl“ for xyxlix rz“pon“z of xlvyThe review of the bounding surface models that have been developed and evaluated for clayresponse in undrained cyclic loading is conducted below. Only some models are selectedand presented in this discussion, as they illustrate some important features for cyclic loadingsimulations. Other models available in the literature are either somehow related to theones discussed below or found to be irrelevant for elaboration in the current work (e.g.,viscoplasticity models as the time effect is not being considered).himplz moyzl for trvn“iznt “oil lovyingA simple model was proposed by Zienkiewicz et al. (1985) for transient soil loading in earth-quake analyses which incorporated the bounding surface formulation. The projection centerfixed at the origin was considered in the radial mapping rule. The authors distinguished theplastic modulus in unloading and reloading for the stress states within the bounding surface.42FBGB hhyorytiwul moxylingUnloading was defined for the stress increments pointing towards the interior of the loadingsurface, passing through the current stress point and the origin, and reloading for the stressincrements pointing outwards. Very simple plastic moduli for unloading and reloading wereconsidered. The former was given by:Hr = Hcsi[M]1(2.26)and the latter byHi = Hcsi[M](2.27)where Hcsi is a plastic modulus at the image stress, and N are model constants, anddefinitions of  and M are the same as in previously described models. In this way theplastic modulus was larger than Hcsi for the stress states inside the bounding surface andwas equal to Hcsi for the stress states on the bounding surface.The proposed model simulations were then compared with actual experimental data fromundrained cyclic triaxial tests. It was clear that the model predicts considerable overdamp-ingS as opposed to experimental data and a way to model plastic strains in unloading wasrequired. Based on experimental observations, the authors suggested the use of cumulativeplastic deviatoric strain, as the proposed model was not able to simulate accumulation ofplastic strains for clay once the stress path reached equilibrium in undrained cyclic triaxialloading. This was also done in the hope of achieving further reduction of stresses to simulateliquefaction for sands; however, it was still not accomplished with inclusion of cumulativeplastic deviatoric strains. The authors expressed the need to simulate plastic strains inunloading to avoid overdamping.Kinzmvtix hvryzning qwuwwlz7 moyzl“Al-Tabaa and Wood (1989) extended the Modified Cam-Clay model by adding a kinemati-cally hardening yield locus within the Cam-Clay locus. The adopted translation of the yieldlocus guaranteed that the outward normal of the inner yield locus was in the same direction6ovzrzstimvtion of yvmping rvtioP yvmping rvtio is yztzrminzy vs AL= =I.AT) × FEE fiith AL wzing thzvrzv of thz strzssBstrvin loop vny AT wzing thz zquivvlznt zlvstix znzrgy43FBGB hhyorytiwul moxylingas the outward normal of the Cam-Clay locus. This avoided the possibility that the twoloci would intersect. The response inside the yield locus was assumed to be elastic; once thestress point was on the yield locus inelastic strains took place. While the inner yield locuswas allowed translating within the Cam-Clay locus, the latter was allowed only expanding orshrinking, as in the Modified Cam-Clay model. Expansion/shrinkage of both loci occurredany time plastic volumetric strains took place. The degree of plasticity for the stress stateson the yield locus and inside the Cam-Clay locus was dependent on the distance from thecurrent stress point to its conjugate point on the Cam-Clay locus. This enabled smooth tran-sition of soil response from the initial stress point on the inner yield locus to the stress pointon the Cam-Clay locus. This approach of smooth transition is somewhat similar to thatproposed for soils by Dafalias (1986v); the difference is that, instead of keeping in memoryan additional memory parameter or having a projection center, the memory of the kinematicyield surface was required. However, the transition from purely elastic response for the stressstates inside the inner yield locus to elastoplastic response for the stress states on the inneryield locus was not smooth. In contrast, Dafalias (1986v) offers smooth transition betweenpurely elastic and elastoplastic responses.The model was attractive in that it required only 2 more parameters in addition to thoserequired for the Modified Cam Clay model. With the 6 model parameters close match ofvolumetric and shear strains accumulation was achieved during one-way undrained cyclictriaxial loading consisting of 6 cycles.Stallebrass and Taylor (1997) further incorporated an additional kinematic hardeningsurface, history surface, which was always inside the outer surface and enclosed the inneryield locus. The history surface was required to account for an effect of recent stress historyand yield at small strains. The translation laws, ensuring that the three surfaces neverintersect, required that the centers of the surfaces moved in the direction of a vector fromthe current stress state to its conjugate point on the next surface. The surfaces translationsoccurred anytime plastic deformations took place. The proposed 3-SKH model required 8model parameters. The model performance was illustrated for a single element simulationsin monotonic and two cycles of undrained cyclic triaxial loading with overpredicted stiffnessreduction as opposed to observed experiments. The authors implemented the proposed model44FBGB hhyorytiwul moxylinginto finite element code and simulated the available centrifuge test of a stresses distributionunderneath a foundation. The analysis showed relatively good model predictive capabilitiesunder a low level of loads, while for increased loads the model underpredicted displacementsunderneath the foundation.Hizrvrxhixvl “inglz “urfvxz moyzlThe Hierarchical Single Surface (HiSS) model approach by Desai et al. (1986) for isotropicmaterial with associative flow rule was used by Wathugala and Desai (1993) as a basis model;the bounding surface formulation was incorporated for plastic strains generation inside theconventional yield surface. The authors suggested that other complexity features (e.g., non-associative flow rule and anisotropy) could be included in the proposed model if desired.The model considered virgin loading, unloading, and reloading separately. Virgin loadingcorresponded to the stress states on the conventional yield surface and increments of a stressdirected outwards of the conventional yield surface. For virgin loading, the usual approachof finding the plastic modulus from the consistency condition was used.A reference surface, passing through the current stress point, was used to distinguishbetween unloading and reloading in the same way as the loading surface is used in othermodels. This surface was similar in shape to the conventional yield surface and also passedthrough the origin; both surfaces were homologous to each other with the center of homologylying at the origin. Based on laboratory tests, the authors proposed unloading to be essen-tially elastic, while for reloading the plastic modulus was a function of the plastic modulusfor virgin loading such that the former approached the latter:Hoi = HsiI1 +HsiI2rN(1− ps r)r2(2.28)where HsiI1 and HsiI2are plastic moduli at image stress and intersection of the conventionalyield surface with the hydrostatic axis, respectively, rN and r2 are model constants, ps and r are distances from the origin to the current stress state and to the image stress point,respectively. The psR r was used to impose distance dependency of the plastic modulus forthe reloading process.45FBGB hhyorytiwul moxylingThe model required calibration of 12 parameters based on isotropic consolidation andundrained triaxial compression/extension tests. Simulations of undrained cyclic triaxialtests, consisting of 4 cycles of two-way loading, suggested that the model was able to capturethe generation of plastic deviatoric strains within a considered number of cycles, while it over-estimated damping as unloading was simulated as elastic. The model was then implementedinto finite element code for pile installation, consolidation after installation, application ofload tests, and final cyclic loading simulation; the whole procedure of model implementation,calibration, and simulation was described by Desai et al. (1993). The results obtained gavea relatively good match to field observations of pile segment displacement, shear transfer,and horizontal stress.Kinzmvtix hvryzning tfioB“urfvxz plv“tixity moyzlLi and Meissner (2002) proposed a two-surface plasticity model with an associative flowrule and incorporated bounding surface formulation for simulation of clay cyclic undrainedbehavior. The bounding surface represented the consolidation history of clay, while theloading surface passing through current stress point was related to each loading process.The authors utilized the kinematic hardening rule of the bounding surface that allowed itstranslation in the stress space to simulate induced anisotropy. Expansion and contraction ofthe bounding surface were also permitted. The elastic domain was reduced to a single pointat the last stress reversal. The plastic modulus within the bounding surface was dependenton the relative configurations of the loading and bounding surfaces and varied from its localto prescribed value on the bounding surface as the stress point approached the latter.The authors extended Masing’s rule of stress reversal to a multiaxial case. They defineda memory center as the point where loading changed its direction so that it pointed towardsthe interior of the loading surface, i.e., the memory center was the point at the last stressreversal. The loading and bounding surfaces tangentially touched each other at the memorycenter and never intersected. The loading surface was homologous to the bounding surfaceand passed through the memory center and current stress point. Once the loading surfacecoincided with the bounding surface, the plastic modulus reached its prescribed value on thebounding surface, and the two surfaces evolved identically. If the stress path was reversed46FBGB hhyorytiwul moxylingand directed towards the interior of the loading surface, the memory center was updated. Anew loading surface evolved, starting from the point at the memory center, and the boundingsurface was translated to the same point with its size unchanged so that the two surfaceswere tangential. For the stress reversals occurring inside the bounding surface the latterindeed translated from its previous position to the memory center, whereas for the stressreversals on the bounding surface the latter stayed in the same position.To simulate cyclic degradation of clay stiffness, the authors used a varying power param-eter enabling reduction of the plastic modulus:hm = H¯m +(Hj − H¯m)( M)(2.29)where H¯m is a plastic modulus for the stress points on the bounding surface, Hj is a localvalue of the plastic modulus, and definitions of  and M are the same as in previouslydescribed models. The was constant for monotonic loading, while for cyclic loading itincreased hyperbolically with cumulative shear strain.It should be noted that the proposed Hj required specification of the applied dynamicshear stress amplitude qd and monotonic shear stress at failure qf, which must be knownprior to cyclic loading. This makes the model not applicable for simulation of irregularcyclic loading, where amplitude of shear stress cannot be specified in advance.The model required 12 parameters, including qd and qf, for cyclic loading simulations.The authors did not indicate if the clay tested in monotonic and undrained cyclic triaxialloading was the same and prepared under similar conditions; however, a different set ofparameters was used for monotonic and each cyclic loading. Monotonic loading simulationsclosely matched experimentally observed clay responses. One-way and two-way undrainedcyclic triaxial tests were well simulated in terms of reduction of p and accumulation of shearstrains.Xlvy vny “vny moyzlClay And Sand Model (CASM) proposed by Yu (1998), and Yu et al. (2007) is a unifiedmodel for simulations of soil behavior; CASM is based on the Modified Cam-Clay model with47FBGB hhyorytiwul moxylingtwo extra parameters required to control the shape of the conventional yield surface. Oneof the parameters controlled the position of the point where the conventional yield surfaceintersected with the CSL, while the other controlled a “drop-like” shape.Incorporation of the bounding surface formulation into CASM model required 5 extraparameters, rendering in total 12 parameters. The radial mapping rule assumed the projec-tion center at the origin of the stress space. The proposed model treated the plastic modulusfor virgin loading, unloading, and reloading separately. The description of the virgin loadingplastic modulus is omitted and interested readers are referred to the original paper. Theplastic modulus for unloading process was given by:H = Hr11− (2.30)where Hr is an unloading hardening parameter, representing the limit for H in unloading, = pRpj = qRqj is a non-dimensional ratio between current and image stresses. Thus, for anunloading process with initial stress state on the bounding surface, Hr ≤ H ≤ ∞.In reloading the plastic modulus was given by:H = Hj +Ho(1− )(1 + "pq)k(2.31)where Hj is equal to H at the image point, Ho is a reloading hardening parameter, "pq isa cumulative plastic deviatoric strain, and k is a model parameter controlling the rate of ashakedown. In reloading, H = ∞ for the stress states far away from the projected imagepoint, and H → Hj as → 1. The "pq allowed for stiffness degradation as plastic deviatoricstrains accumulated throughout loading.The model was further compared with bubble-type models proposed by Al-Tabaa andWood (1989); Stallebrass and Taylor (1997) in simulations of one-way and two-way undrainedcyclic triaxial loading with available experimental results. In terms of deviatoric strains, theproposed model illustrated better predictive capabilities and was satisfactory for volumetricstrains. For two-way cyclic loading simulations, the model predictions of deviatoric strainswere also satisfactory.48FBHB gummuryGCI hummvryA brief summary is provided on the trends experimentally observed in clay response whensubjected to cyclic loading and constitutive modeling of such response. The trends in con-stitutive modeling are then to be used in model development.GCICF Expzrimzntvlly ow“zrvzy rz“pon“zPresented in this chapter literature review on experimental undrained cyclic response of clayin triaxial test suggests the following• for normally consolidated specimens, the stress path migrates towards the origin ofthe stress space and eventually stops migrating before reaching the origin as shown inFigs. 2.1–2.4• deviatoric strain develop throughout applied cyclic loading in both directions (com-pression and extension) for two-way cyclic loading as shown in Figs. 2.2–2.5, and inthe direction of applied cyclic loading (compression) in Fig. 2.1• there exists a cyclic stress ratio threshold, below which deviatoric strain developmentslows down after a finite number of cycles as shown in Fig. 2.3(a,b) and Fig. 2.5From the review on undrained cyclic multi-directional response of normally consolidatedclay the following is observed• figure-8 type A path accumulated permanent shear strains, as shown in Fig. 2.7, infewer cycles than 1-D linear, 2-D linear and circular tests• for the tests on CK specimens, the shear strains migrated in the direction of initialshear stress regardless of the orientation of a shearing path as shown in Fig. 2.7GCICG Xon“titutivz moyzlingThe literature review on current trends in constitutive modeling of cyclic response of clay isalso conducted in this chapter. The bounding surface formulation is used by a number of49FBHB gummuryresearchers to simulate the cyclic response observed in the experiments. The following aretwo major findings from conducted literature review:• projection center updated at the point of last stress reversal produces more realisticsimulation results; this was initially mentioned by Zienkiewicz et al. (1985) who noticedthat with the projection center fixed at the origin simulated unloading exhibited purelyelastic strains, while plasticity was in effect suggested by experiments, thereby resultingin overdamping; Jung and Yune (2011) concluded, based on stress probe experiments,that the projection center at the point of the last stress reversal predicted plastic strainincrements closer to those observed in experiments; Li and Meissner (2002) adoptedthe projection center fixed at the last stress reversal and were able to obtain relativelygood predictions of one-way and two-way undrained cyclic triaxial loading• cumulative plastic deviatoric strain used to continuously reduce plastic modulus pro-duces continuous degradation of stiffness as observed in the experiments throughoutcyclic loading; this was initially proposed by Zienkiewicz et al. (1985) who noticed thatmodel simulations with the plastic modulus possessing only the distance dependencelacked the feature of simulating increased rate of deviatoric strain accumulation oncethe stress path stopped migrating towards the origin; a similar approach for reducingthe plastic modulus during simulations of undrained cyclic triaxial loading was usedby Yu (1998) for the isotropic model, and by Li and Meissner (2002) for the kinematichardening two-surface model; Huang et al. (2011) used the same idea for monotonicloading simulations in an anisotropic model with destructurationWhile the above findings were advantageous for the reviewed models, a number of fol-lowing disadvantages were present for their practical application in cyclic loading• focus of Jung and Yune (2011) was on application of the bounding surface formulationfor simulations at small strains in monotonic loading rather than cyclic loading• model proposed by Li and Meissner (2002) required the knowledge of the appliedstress amplitude prior to cyclic loading thereby making it impractical for irregularcyclic loading simulations50FBHB gummury• models proposed by Wathugala and Desai (1993); Yu (1998); Zienkiewicz et al. (1985)treated the plastic modulus in loading, unloading, and reloading separately therebyrequiring a large number of parameters• a special treatment of unloading involving only elastic strains was required for modelsproposed by (Wathugala and Desai, 1993) to artificially account for missing plasticdeformationsThe goal of this thesis is to incorporate the bounding surface formulation into an ex-isting model for better predictions of clay response in undrained cyclic loading imposed byearthquakes. In the development of the model, the position of the projection center andsimulation of cyclic softening will be addressed based on the observed experimental evidenceand the current trends in the reviewed constitutive models combined into the model. In theprocess of the development, the above mentioned disadvantages should be avoided to makesure that the model can be used in practical applications. The complexities incorporatedinto the developed model should be adequate to maintain its simplicity for further modelimplementation.51HC Yzvzlopmznt of hVcIXaVnBWboyzl for Xyxlix gz“pon“z of XlvyHCF IntroyuxtionThe modified cam-clay (MCC) model (Roscoe and Burland, 1968) is being applied in rela-tively advanced analyses in geotechnical engineering and is perhaps among the few modelsbesides the very simple ones like Mohr-Coulomb being widely used for this purpose. Basedon a work dissipation assumption with a contribution coupling the volumetric and deviatoricplastic strain rates, Dafalias (1986u) proposed an anisotropic extension of the MCC model,which has a simple and elegant formulation with clear physical interpretation. The result-ing plastic potential surface in the triaxial p–q space, which for associative plasticity servesalso as a yield surface, is a rotated and distorted ellipse. Employing the plastic potentialsurface proposed by Dafalias, a number of models developed subsequently (e.g., Korhonenand Lojander, 1987; Newson and Davies, 1996; Thevanayagam and Chameau, 1992; Wheeleret al., 2003). Later, by using a non-associated flow rule, Dafalias et al. (2006) extended theearlier work of Dafalias to allow the simulation of a softening response under undrainedcompression following oedometric consolidation. This model is named a Simple ANIsotropicCLAY (SANICLAY) plasticity model. Although the SANICLAY model is able to captureanisotropy as well as softening response in soft clays, it does not have enough means to cap-ture the response of sensitive clays. More recently, Taiebat et al. (2010b) proposed a versatileextension of the SANICLAY model by incorporating two distinct mechanisms for modelingof destructuration: isotropic and frictional. The former is a concept already presented inrelation to other models (e.g., Baudet and Stallebrass, 2004; Gajo and Wood, 2001; Kav-vadas and Amorosi, 2000; Nova, 1992) and in essence constitutes a mechanism of isotropic52GBEB Introxuwtionsoftening of the yield surface with destructuration. The latter is a novel proposition thatrefers to the reduction of the critical stress ratio reflecting the effect of destructuration onthe friction angle. They also generalized the SANICLAY model itself from its previous formby additional dependence of the yield surface on the third isotropic stress invariant. Sucha generalization allows the obtaining, as particular cases, simplified model versions of lowercomplexities, including one with a single surface and associative flow rule by simply settingaccordingly parameters of the generalized version. Hierarchical evolution of the SANICLAYfamily of models is illustrated in Fig. 3.1 with the SANICLAY-D referring to the model withisotropic destructuration mechanism.So far, all extensions of the SANICLAY model have included only elastic strains for stressstates within the yield surface. The bounding surface concept along with radial mapping rule(e.g., Anandarajah and Dafalias, 1986; Dafalias and Herrmann, 1986; Kaliakin and Dafalias,1989) can be employed to permit plastic strains for the stress states within the surface(contrary to classical plasticity) and thereby to simulate the stress-strain response. Thebounding surface formulation differs from classical elasto-plastic models in that the plasticmodulusKp also depends on the distance between the stress point and a so-called image pointon the bounding surface. This process requires an appropriate mapping rule for introducingthe image point. A projection center is to be used to map the actual stress point to theimage stress point on the bounding surface. The projection center of the bounding surfacein the above models was set along the induced hydrostatic line. Whittle and Kavvadas(1994) and Crouch et al. (1994), among others, expressed with the bounding surface theoryhaving the projection center at the origin of the stress space. Ling et al. (2002) employedan associated flow rule which uses a bounding surface that has a slightly modified shapefrom the one proposed by Dafalias (1986u), and specified the projection center along the KMline rather than at the origin of the stress space. This model was later enhanced to alsoinclude a non-associative flow rule (Jiang et al., 2012), however the same projection centeron the anisotropic line was used in the new model. The above models have been successfulin improving the stress-strain response in monotonic loading of clays, but they were notspecifically examined for addressing the response in cyclic loading.The model proposed in the present work incorporates the bounding surface plasticity53GBEB Introxuwtionp0 + p˙0pp0NNf = 0q=v)α+ α˙pαp0NNf = 0q=w)NNMcMeα∂f/∂σ∂g/∂σ(p, q)pqg = 0 p0f = 0=x)NNαpqp0p0df = 0=y)Figure 3.1: Evolution of the SANICLAY family of models: (a) MCC (Roscoe and Burland,1968), (b) anisotropic MCC (Dafalias, 1986u), (c) SANICLAY (Dafalias et al., 2006), and(d) SANICLAY with destructuration, SANICLAY-D (Taiebat et al., 2010b)54GBFB aoxyl zormulution in triufiiul spuwyformulation in the SANICLAY model in an efficient form aiming for cyclic loading. It isa continuation of the work initiated by Taiebat et al. (2010a) who extend the the originalSANICLAY model Dafalias et al. (2006) while preserving its attractive features. The modelincorporates a simplified non-associative flow rule with plastic potential sharing same devi-atoric back-stress ratio , and most importantly a bounding surface formulation simulatingthe plastic strains for the stress states inside the bounding surface. A proper radial mappingrule has been used in the present bounding surface model. This includes, following the trendsin the reviewed constitutive models (see Section 2.4.2), the projection center that needs tobe updated at any stress reversal. A novel feature is that the projection also evolves duringthe loading process in order to maintain its relative position within the evolving boundingsurface. Furthermore, the bounding surface hardening parameter is set to be a function ofa newly introduced damage parameter. The damage parameter is set, following the trendsin the reviewed constitutive models (see Section 2.4.2), to evolve with cumulative plasticdeviatoric strain. The resulting model is named SANICLAY-B referring to its associationwith the SANICLAY models. For simulating the structure loss in sensitive clays one canalso include the isotropiw destructuration mechanism of Taiebat et al. (2010b) as an efficientand practical version of that general model for destructuration. The destructuration or thebounding surface association features can simply be deactivated, if desired, by assigningcertain values to the corresponding model constants. The model requires calibration of 11model constants. These include 5 constants of the MCC, 1 for the non-associated flow rule, 2for the anisotropic hardening of the model (as in the SANICLAY model), 1 for the isotropicdestructuration mechanism (as in the SANICLAY model with destructuration), and 2 newconstants related to the bounding surface formulation.HCG boyzl formulvtion in trivxivl “pvxzThe formulation of the proposed model in the triaxial space is presented first in terms of tri-axial stress quantities. The essential elements of the model in triaxial space are schematicallypresented in Fig. 3.2. The model adopts the same hypoelastic relations, isotropic harden-ing, rotational hardening, and the isotropiw destructuration mechanism as in the original55GBFB aoxyl zormulution in triufiiul spuwyp0pαNcNeMcMeα∂F/∂σ¯∂g/∂σ¯(p˙, q˙)Bounding surfaceF = 0Loading surfacef = 0Plastic potentialg = 0(p, q)(pc, qc)pq(p¯, q¯)Figure 3.2: Schematic illustration of the loading, bounding, and plastic potential surfaces ofthe SANICLAY-B model in the triaxial stress spaceSANICLAY class of models. The elements of bounding surface formulation incorporatedinto the model are presented and discussed here. The bounding surface formulation includesthe projection center updated at each stress reversal and evolving with bounding surfaceevolution, and continuously reducing plastic modulus through the use of a newly introduceddamage parameter evolving with cumulative plastic deviatoric strain.HCGCF Wounying “urfvxz vny plv“tix potzntivl “urfvxzWhat originally was serving as the yield surface in the SANICLAY model is used as abounding surface in the SANICLAY-B model via the following expression of F = 0:F = (q¯ − p¯ )2 − (N2 − 2) p¯ (pM − p¯) = 0 (3.1)The bounding surface is the locus of the “image” stress (p¯,q¯). The actual stress (p,q) liesin or on the F = 0. A properly defined mapping rule associates a unique image stress tothe actual stress. The pM and constitute the two hardening internal variables, the formerrepresents isotropic hardening due to density changes and controls the size of the boundingsurface, and the latter represents rotational hardening introducing anisotropy in the model.The Lode angle dependent stress ratio N = Nc when the image stress ratio ¯ = q¯Rp¯ S , and56GBFB aoxyl zormulution in triufiiul spuwyN = Ne when the image stress ratio ¯ = q¯Rp¯ Q . The N serves as a bound for evolutionof the rotational variable and here is assumed to have the same value in compression andextension. The plastic potential function is presented via the expression g = 0 on the imagestress asg = (q¯ − p¯ )2 − (b2 − 2) p¯ (p − p¯) = 0 (3.2)where b is the critical stress ratio, and p is the the value of p¯ at q¯ = p¯ ; adjusted so thatEq. (3.2) is satisfied for a given pair of (p¯; q¯) values at the bounding surface. The criticalstress ratio b = bc when the image stress ratio ¯ = q¯Rp¯ S , and b = be when the imagestress ratio ¯ = q¯Rp¯ Q . Fig. 3.2 illustrates the bounding and plastic potential surfaces inthe triaxial stress space. Note that the bounding surface F = 0 and the plastic potentialsurface g = 0 share the same rotational variable , but different b and N parameters. Thesame rotational variable for both surfaces is for simplicity, as a particular case of whatwas originally used in the SANICLAY models. Such simplification was originally introducedby Jiang and Ling Jiang and Ling (2010), and Jiang et al. Jiang et al. (2012), whereby theyobtained a satisfactory agreement to fit a number of yield data on natural clays. Associativeflow rule follows by setting b = N ; however, a different value of N from b allows morerealistic prediction of experimentally determined yield points, at the expense of having anon-associative flow rule.HCGCG Imvgz “trz““ vny projzxtion xzntzrAccording to the bounding surface formulation, every point in the stress space is associatedwith a corresponding image point on the bounding surface via a suitable mapping rule. Thismapping rule must satisfy the integrability and identity conditions. The former impliesexistence of a “loading surface” through the actual stress (p,q), which does not have to beexplicitly defined. The latter guarantees that the actual stress (p,q) does not cross the F = 0,by identifying it with its image when it lies on the bounding surface.Here a “radial mapping” theory is employed where the image stress (p¯,q¯) is obtainedby radially projecting the stress state (p,q) from a “projection center” (pc; qc). This radial57GBFB aoxyl zormulution in triufiiul spuwymapping results in the following expression for finding the image stressp¯ = pc + b (p− pc) ; q¯ = qc + b (q − qc) (3.3)where 1 ≤ b ≤ ∞ is a similarity ratio between the loading surface f = 0 and the boundingsurface F = 0. The corresponding value of the similarity ratio b can then be obtained bysubstituting the image stress from Eq. 3.3 to the bounding surface expression in Eq. 3.1. Aloading surface f = 0, shown with dashed lines in Fig. 3.2, is an implicitly defined surfacehomologous to the bounding surface with the center of homology located at a projectioncenter (pc; qc). The projection center always lies inside the bounding surface and evolvesaccording to a proper rule, to be discussed.The rest of this subsection is about the choice of the projection center (PC). In theproposed model a moving PC is adopted which should be updated to the current stress stateat any “stress reversal” to better predict experimentally observed plastic strains during thecourse of cyclic loading. This idea follows the trends in the reviewed constitutive models(see Section 2.4.2). Analytically, the stress reversal occurs any time that loading indexa ≤ 0. The loading index a is obtained from the consistency condition F˙ = 0, resultingin a = [(@FR@p¯) p˙+ (@FR@q¯) q˙] RKp, where Kp is the plastic modulus that will be definedlater. The general concept of updating the projection center at the last stress reversal hasbeen discussed by Dafalias (1986v).To guarantee a unique image stress (p¯,q¯), the PC has to be always positioned withinthe bounding surface. If the PC established at the point of last stress reversal is keptfixed until the next stress reversal, it might be found outside the bounding surface as thelatter evolves (expands/contracts and rotates). A novel approach is introduced to avoidsuch scenarios. The idea is to maintain the PC within the bounding surface while it isexpanding/contracting and rotating. This will be achieved by allowing the PC to moveinside the bounding surface and such movement will be calculated by separating on the onehand the effect of expansion/contraction via the change of pM, and on the other the effect ofrotation via the change of and then add the two contributions.With reference to Fig. 3.3, consider (pc,qc) to be the PC at the instance of its creation58GBFB aoxyl zormulution in triufiiul spuwyNNαpcqcqαqbpqp0Bounding surfaceF = 0Figure 3.3: Schematic illustration of the moving projection center inside the bounding surfaceupon reversal of loading. If one assumes only a change of pM, this creates a proportional top˙M change of all points within the bounding surface; hence, the corresponding change of thePC will be given byp˙c =pcpMp˙M; q˙c =qcpMp˙M (3.4)The effect of change of is more difficult to determine. The idea is to maintain pro-portionality of the distance of the PC along the q–axis from the –axis, to say the distanceof the point on the bounding surface with same pc as the PC again from the –axis. Inother words it is a matter of distances along the q–axis only that will be affected by thechange of while the pM and pc will remain fixed. Let us call q = pc. The q coordinate ofthe point on the bounding surface with p = pc, namely the qb in Fig. 3.3, can be obtainedfrom the equation of the bounding surface, i.e., qb − q = [(N2 − 2) pc (pM − pc)]N=2. Theaforementioned distances are then given by qc − q and qb − q , respectively. Then, the ratiothat is needed to be kept constant readsqc − q qb − q = m (3.5)with m a fixed number obtained at the instance of stress reversal (creation of PC). Eq. 3.559GBFB aoxyl zormulution in triufiiul spuwyallows taking the rate of qc due to change of only, while keeping pc and pM fixed, since onlythe effect of rotational hardening is examined. On that basis it is straightforward to obtainthe corresponding change of the PC asq˙c =[pc −m pc (pM − pc) [(N2 − 2) pc (pM − pc)]N=2] ˙ (3.6)Finally the rate of change of the two components of the PC are given for pc from Eq. 3.4and for qc by the summution of Eqs. 3.4 and 3.6.It is interesting to analyze Eq. 3.6. The first term, i.e. the pc ˙ provides a “movement”of qc along the q–axis due only to the rate of change of (i.e, the qc moves as much asq = pc ) for fixed pc), as if the bounding surface only rotates but it does not “distort”along q. The second term involving the m, gives an additional movement along q due tothe distortion of the bounding surface in the q–axis, enough to maintain the aforementionedratio m fixed. This combination guarantees that the PC will not cross the bounding surfacebecause both the rotation and the distortion were accounted for. Observe also the product ˙ of the second term of Eq. 3.6; it acquires positive or negative values depending on thesigns of and ˙ and this means that the second term adds to the rotation-related first termif the bounding surface increases in thickness as it rotates back towards zero , or subtractsfrom the rotation-related first term if increases in absolute value which implies that thebounding surface becomes thinner.The projection center used for radially mapping the current stress onto the boundingsurface locating the image stress is introduced. The projection center is chosen to updateat the instance of each “stress reversal” following the trends in the reviewed constitutivemodels (see Section 2.4.2). In order to maintain a relative position of the projection centerwithin the bounding surface, a novel approach is introduced which allows for evolution ofthe projection center with the evolution of the bounding surface.HCGCH elv“tix moyulu“The essence of the bounding surface concept is the proposition that the plastic modulus Kpat the actual stress (p,q) is related to a bounding plastic modulus K¯p at the corresponding60GBFB aoxyl zormulution in triufiiul spuwyimage stress (p¯,q¯) by means of the Euclidean distance  between (p¯,q¯) and (p,q), such thatKp ≥ K¯p. The K¯p is defined by the consistency condition F˙ = 0, and for  = 0 one hasKp = K¯p. Thus the plastic states are not restricted only to those lying on F = 0 as inclassical yield surface plasticity, and this is the great advantage of the bounding surfaceconcept. In the proposed model, the Kp for any stress state (p; q) is related to K¯p by meansof the Euclidean distances  (between (p¯; q¯) and (p; q)), and r (between (p¯; q¯) and (pc; qc)),in the following form:Kp = K¯p +h p3M 〈r − s〉 = K¯p +h p3M〈bR(b− 1)− s〉 (3.7a)K¯p = −(@F@pMp¯M +@F@ ¯)(3.7b)where h is a positive shape hardening parameter, and s ≥ 1 is an indirect measure of thesize of the elastic nucleus that is a surface homologous to the bounding surface with theprojection center as the center of homology. The p3M in Eq. 3.7a is a desirable term in thisequation for better data simulations and also in order to make the h (or more specificallyhM that will be defined later on) a dimensionless model parameter. The values of p¯M and ¯in Eq. 3.7b are obtained from the hardening rules p˙M =Q a S p¯M and ˙ =Q a S ¯. Thehardening rules here are same as those in Taiebat et al. Taiebat et al. (2010b). Note thatrR = bR (b− 1).By setting a large number to s, the size of the elastic nucleus becomes so large that themodel behaves as if no bounding surface formulation exists. Hence, the model does not showany plastic deformation during the cycles of loading that fall within the bounding surface.On the other hand when setting s = 1, the size of the elastic nucleus is shrunk to a singlepoint, hence the bounding surface formulation becomes fully active and any change of stresswithin the bounding surface that is associated with a positive loading index a would beelastoplastic. The default value of the elastic nucleus is s = 1 for the current formulationand the elastic nucleus is shown as a single point at the projection center (pc; qc) in Fig. 3.2.The parameter s is mainly for the plastic modulus interpolation and not only for renderingthe elastic range to zero because of relocation of it upon loading reversal. Observe that61GBFB aoxyl zormulution in triufiiul spuwythe bounding surface formulation with b-dependent plastic modulus allows simulation of asmooth transition from an approximately elastic response, when the stress is at the onsetof stress reversal (b ≫ 1 and Kp ≫ K¯p), to the regular SANICLAY elastoplastic response,when the stress reaches the bounding surface (b→ 1 and Kp → K¯p).As it will be shown later in Sec. 3.4.1, the stress-strain loops in cyclic loading could stopevolving if a constant value of h is used in the model. In order to properly simulate thecontinuous evolution of stress-strain loops and the resulting degradation of stiffness duringcyclic loading, as observed in the experiments, the shape hardening parameter h that controlsthe plastic modulusKp is assumed to be a decaying function of plastic strain. This is achievedby the following simple expressionh =hM1 + y(3.8)where hM is a model constant representing the initial value of h, and y is a new model statevariable simulating the damage effect. Introduction of the damage parameter arises fromphenomenological observations made in laboratory experiments suggesting that clay stiffnesstend to reduce as cyclic loading progresses. This perhaps is attributed to damage of bondsbetween clay particles during cyclic loading. In the formulation, the damage state variabley is assumed to evolve linearly with the increment of plastic deviatoric strain, following thetrends in the reviewed constitutive models (see Section 2.4.2), asy˙ = vd |"˙pq| (3.9)with vd being a new model constant that controls the rate of evolution of y. Note thatusing the absolute value of the increment of plastic deviatoric strain in Eq. 3.9 means thatduring the process of loading the damage parameter y can only increase. From Eqs. 3.8and 3.9 it can be observed that with accumulation of the plastic deviatoric strains, thedamage state variable y increases, leading to reduction of the shape hardening parameterh, which in turn leads to reduction of plastic modulus Kp and simulation of smaller overallstiffness. It is recommended that the evolution of the damage parameter is activated onlyduring cyclic loading simulations as its introduction is based on the observations from cyclicloading tests. For simulations of monotonic loading, use of vd = 0 deactivates evolution62GBFB aoxyl zormulution in triufiiul spuwyof damage parameter y. The effect of the damage parameter in monotonic loading is outof scope of this research. Finally note that the effect of parameter h, with or without thedamage effect, is only while the actual stress point is inside the bounding surface. For thestress point on the bounding surface the similarity ratio b = 1 which eliminates the effect ofh.Plastic modulus is introduced here as a function of a newly introduced damage parameter.The damage parameter is chosen to accumulate with plastic deviatoric strain incrementsthereby allowing for continuous reduction of the plastic modulus.HCGCI Yz“truxturvtionFor lesser complexity, among the two distinct isotropic and frictional destructuration mecha-nisms introduced by Taiebat et al. (2010b), one may choose to only use the classical isotropicdestructuration for capturing the degradation of shear strength. To this end, the internalvariable pM is set to equal to hipMd, where hi ≥ 1 is the isotropic structuration factor and pMdis the destructured value of pM (or the value of pM at hi = 1). In Taiebat et al. (2010b) thenotation p∗M and pM instead of pM and pMd, was used, respectively. With h˙i =Q a S h¯i thehardening equation p˙M =Q a S p¯M is now expressed in the following form.p¯M = hip¯Md + h¯ipMd (3.10a)p¯Md =(1 + z− )pMd(@g@p¯)(3.10b)h¯i = ki(1 + z− )(hi − 1) "¯pd (3.10c)with z being the void ratio,  and  the slopes of the normal compression and rebound lines,respectively, and ki a material constant that controls the rate of destructuration. The de-structuration mechanism is active when ki S 0. By setting ki equal to 0, the destructurationmechanism is deactivated. The degradation of hi has been taken into account by means of"˙pd =Q a S "¯pd, which is the rate of an auxiliary internal variable called the destructuration63GBGB Gynyrulizution to multiufiiul spuwyplastic strain rate and is defined as"¯pd =√(1− V)(@g@p¯)2+ V(@g@q¯)2(3.11)where the parameter V is a material constant distributing the effects of volumetric anddeviatoric plastic rates to the value of "˙pd. A default value of 0.5 can be used for thisparameter.It is important to emphasize on the distinction between the isotropic destructurationmechanism and the damage effect in the proposed formulation. The former is aimed to servefor simulating loss of strength, while the latter to serve for simulating reduction of modulus(stiffness). Both of these features are required for successful model performance. Never-theless, there is a partial overlap between the two. As the isotropic destructuration takesplace, the size of the bounding surface shrinks thereby reducing the ratio b and consequentlythe plastic modulus Kp (i.e., modulus) on the basis of Eq. 3.7a. As the damage parameterevolves, the amount of plastic strains increases and as a result the rate of structure lossincreases on the basis of Eqs. 3.10a–3.10c.HCH Gznzrvlizvtion to multivxivl “pvxzThe multiaxial stress space generalization of different elements of the SANICLAY andSANICLAY-D models, such as the hardening rules for pM and and the isotopic destructura-tion mechanism, were presented in Dafalias et al. (2006) and Taiebat et al. (2010b) and willnot be repeated here. Only the generalization of the new equations related to the boundingsurface, i.e. Eqs. 3.1–3.9 are presented.HCHCF Wounying “urfvxz vny plv“tix potzntivl “urfvxzFollowing the above systematic rule for generalization of the deviatoric stress components,the bounding surface Eq. 3.1 and the plastic potential surface Eq. 3.2 can be generalized to64GBGB Gynyrulizution to multiufiiul spuwythe multiaxial stress space asF =32(“− p¯ ) : (“− p¯ )−(N2 − 32 : )p¯ (pM − p¯) = 0 (3.12a)g =32(“− p¯ ) : (“− p¯ )−(b2 − 32 : )p¯ (p − p¯) = 0 (3.12b)For the bounding surface F = 0, recall that a parameter N acquires different values Ncand Ne according to the sign of  − as explained after Eq. 3.1. Similarly for the plasticpotential surface g = 0, the critical state stress-ratio b acquires different values bx andbe according to the sign of  − as explained after Eq. 3.2. As in the other SANICLAYmodels, this is achieved by using the Lode angle dependent b() via the interpolationfunction proposed by Argyris et al. (1974). The Lode angle  is defined by cos 3 =√6trn3and is used to define the modification of b for various  tracing the critical state failuresurface in the stress ratio .-plane. Here n is a unit norm deviatoric tensor along r− , andr = “Rp¯ is a deviatoric stress ratio tensor at the image stress.Given image stress ¯, Eq. 3.12b yields the following solution for p :p =(3R2) (“¯− p¯ ) : (“¯− p¯ )(b2 − (3R2) : ) p¯ + p¯ (3.13)However, this equation fails to define p for image stress ¯ at the origin of the stress space.65GBGB Gynyrulizution to multiufiiul spuwyTo accommodate this case, a limiting value of p is obtained for r¯→ E:lim3r→0p = lim3r→0y (“¯; p¯)z (“¯; p¯)= lim3r→0@y (“¯; p¯) R@r¯@z (“¯; p¯) R@r¯= lim3r→0@y (“¯; p¯)@“¯@“¯@r¯+@y (“¯; p¯)@p¯@p¯@r¯@z (“¯; p¯)@“¯@“¯@r¯+@z (“¯; p¯)@p¯@p¯@r¯=(N2 − 32 : )(b2 − 32 : )pM (3.14)wherey (“¯; p¯) = (3R2) (“¯− p¯ ) : (“¯− p¯ ) +(b2 − 32 : )p¯2andz (“¯; p¯) =(b2 − 32 : )p¯so that it could be assured that p is in the range between lim3r→0 p and pM. In the aboveequation, N = min (Nc; Ne) and b = max (bc;be) are selected to avoid setting p = pMwhen N S b .The analytical expression in Eq. 3.12a for the bounding surface F = 0 represents arotated and distorted ellipsoid that is Lode angle dependent via N as illustrated in Fig. 3.4.The Eq. 3.12b for the plastic potential g = 0 also represents a rotated and distorted ellipsoidthat it is also Lode angle dependent via b . The above interpolation function for b formsa critical stress ratio envelope in the principal stress space, as illustrated in Fig. 3.4.HCHCG Imvgz “trz““ vny projzxtion xzntzrThe bounding surface formulation postulates that for any stress  a unique image stress ¯exists on the bounding surface (Dafalias, 1986v). The image stress ¯ is obtained by a radialmapping of stress  from a projection center c, to be discussed in the sequel, onto thebounding surface.66GBGB Gynyrulizution to multiufiiul spuwyFigure 3.4: Schematic illustrations of the model surfaces in the principal stress space:anisotropic bounding surface, anisotropic plastic potential surface, and non-uniform criti-cal stress ratio b()-coneGiven a projection center c, image stress is expressed in terms of its hydrostatic p¯I anddeviatoric “¯ counterpartsp¯ = pc + b (p− pc) (3.15)“ = “c + b (“− “c) (3.16)where b is a ratio between ‖¯ − c‖ and ‖ − c‖.The value of b is determined by plugging Eqs. 3.15 and 3.16 into Eq. 3.12a, which inmultiaxial space is solved term by term as follows:32(“− p¯ ) : (“− p¯ ) = b2VN + bBN + XN (3.17)67GBGB Gynyrulizution to multiufiiul spuwywhereVN =32(s− sc) : (s− sc)− 3 (p− pc) (s− sc) : + 32(p− pc)2 : (3.18)BN = 3 (s− sc) : sc − 3 (pcs+ (p− 2pc) sc) : + 3pc (p− pc) : (3.19)XN =32sc : sc − 3pcsc : + 32p2c : (3.20)Similarly,p¯(pM − p¯) = −b2V2 − bB2 − X2 (3.21)whereV2 = (p− pc)2 (3.22)B2 = (2pc − pM) (p− pc) (3.23)X2 = pc (pc − pM) (3.24)Then a quadratic equation is obtained from Eqs. 3.17 and 3.21Vb2 +Bb+ X = 0 (3.25)whereV = VNR(N2 − 2)+ V2 (3.26)B = BNR(N2 − 2)+B2 (3.27)X = XNR(N2 − 2)+ X2 (3.28)Ratio b is then obtained asb =b2 +√b2 − 4VX2V(3.29)Notice that maximum root is chosen as a solution of quadratic equation. For the projectioncenter inside the bounding surface, see Fig. 3.5(a), the maximum root yields a solution for theimage stress along the direction of −c as desired. For the projection center outside (due68GBGB Gynyrulizution to multiufiiul spuwyto allowable tolerance) the bounding surface, see Fig. 3.5(b), both minimum and maximumroots yield solutions for the image stress along the direction of −c. However, the maximumroot yields solution for the image stress on the opposite side of the bounding surface whichis desired. Therefore, maximum root is the only correct solution for b value for any positionof projection center c.F = 0 c¯ (bmax S 0)¯ (bmin Q 0)=v)F = 0F = TOL+ c¯ (bmax S 0)¯ (bmin S 0)=w)Figure 3.5: Illustration of correct solution for ratio b using maximum root for: (a) c insideF = 0, and (b) c outside F = 0 (due to allowable tolerance in drift correction)Determination of the image stress on the bounding surface requires a projection center.The projection center at the last point of stress reversal can be used to simulate plasticstrains in unloading, required for better undrained cyclic loading simulations. The issue ofsimulating plastic strains in unloading during cyclic loading was discussed by Zienkiewiczet al. (1985). Jung and Yune (2011) showed from stress probing tests that simulations withthe projection at the last stress reversal are favorable for monotonic loadings.The model proposed in this thesis, thus, considers a projection center c established atthe last point of stress reversal. Stress reversal is considered in the proposed model any timethe Loading index becomes negative a Q 0. The projection center c is updated at themoment of the stress reversal to the latest stress state . However, if the projection centerc is simply chosen to be a fixed point until the next stress reversal occurs, then there couldbe a case such that the projection center stays outside the bounding surface. This is true forthe model as it adopts isotropic and rotational hardening of the bounding surface relative tothe origin. The projection center outside of the bounding surface violates the concept of the69GBGB Gynyrulizution to multiufiiul spuwybounding surface, suggesting the uniqueness of the image stress for a given projection centerand stress state. Thus, the idea of the projection center at the last stress reversal shouldbe modified for the proposed model to maintain the desired effect of plastic strains uponunloading and, at the same time, to keep the well-established bounding surface concept. Thedetails of this novel feature are followed next, and in addition to the formulation specified inthe triaxial stress space, requires accounting for the Lode angle dependency of the boundingsurface in the multiaxial stress space.The proposed idea is to maintain the relative position of the projection center while thebounding surface evolves. Consider c as an updated projection center at the moment ofstress reversal. If one requires that the relative position of the projection center is keptunaltered owing only to a change of the size of the bounding surface, then the projectioncenter should evolve accordingly:p˙c =p˙MpMpc (3.30)“˙c =p˙MpM“c (3.31)The relative position of the projection center c should also be maintained upon rotationand distortion of the bounding surface. For this to be in effect, the following ratio should bekept constant [32(“c − “ ) : (“c − “ )]N=2[32(“b − “ ) : (“b − “ )]N=2 = m (3.32)where “b is a deviatoric stress tensor on the bounding surface along nc = (“c − “ ) R ‖“c − “ ‖(see Fig. 3.6) for a given pc, and “ = pc . The denominator can be obtained from the F = 0equation as[(N2 − 32 : )pc (pM − pc)]N=2.An important observation for the multiaxial formulation is that the Lode angle for theprojection center c must be kept unaltered during rotational hardening in order to keep therelative position of the projection center within the bounding surface. This is demonstratedin Fig. 3.6 for the initial bounding surface F = 0 and subsequently rotated bounding surface70GBGB Gynyrulizution to multiufiiul spuwysNs2 s3F = 0“ “cncF ′ = 0“′ “′cncFigure 3.6: Illustration of an unaltered Lode angle for the projection center evolution duringrotational hardening of the bounding surfaceF ′ = 0. As the Lode angle is kept unaltered the following simplification can be made:“c − “ = vcnc (3.33)where vc = ‖“c − “ ‖ is a scalar whose multiplication with unit tensor nc replaces “c− “ inEq. 3.32.Eq. 3.32 can be solved for vc asvc = m[23(N2 − 32 : )pc (pM − pc)]N=2(3.34)The rate of vc due to rotation of the bounding surface is given byv˙c = −m√23pc (pM − pc) 32 : ˙[(N2 − 32 : )pc (pM − pc)]N=2 (3.35)Observing that the rate of “c − “ is equal to “˙c − “˙ the rate of “c due to rotation and71GBGB Gynyrulizution to multiufiiul spuwydistortion is given by“˙c = pc ˙+ v˙cnc= pc ˙−m√23pc (pM − pc) 32 : ˙[(N2 − 32 : )pc (pM − pc)]N=2 nc (3.36)The complete evolution of the projection center in multiaxial stress space is then givenby Eq. 3.30 for pc and a combination of Eqs. 3.31 and 3.36 for “cp˙c =p˙MpMpc (3.37)“˙c =p˙MpM“c +pc ˙−m√23pc (pM − pc) 32 : ˙[(N2 − 32 : )pc (pM − pc)]N=2 nc (3.38)HCHCH elv“tix moyulu“The plastic modulus Kp in the multiaxial stress space has the same form as in Eq. 3.7a, butwith generalized scalar distances r = |−c| and  = |−|. The generalized form of thesimilarity ratio b is given and discussed in Eq. 3.29. The bounding plastic modulus in themultiaxial stress space becomesK¯p = −(@F@pMp¯M +@F@ : ¯)(3.39)with the bounding surface expression F = 0 from Eq. 3.12a, and the generalized forms of p¯Mand  as presented and discussed in the original SANICLAY and SANICLAY-D models.The last element of the model that needs generalization to multiaxial space is the rate ofdamage parameter y in Eq. 3.9. Following the systematic rule generalization of the deviatoricstrain components, Eq. 3.9 can be generalized asy˙ = vd [(2R3) tzp : tzp]N=2 (3.40)where z˙p is the multiaxial counterpart of the triaxial plastic deviatoric strain rate "˙pq.72GBHB Improvyx wonstitutivy zyuturys oz gAbIWLAmAVHCI Improvzy xon“titutivz fzvturz“ of hVcIXaVnBWThe incorporated bounding surface formulation significantly enhances the performance ofthe model in simulation of clay cyclic behavior. The improved model response is attributedto the projection center updated at each “stress reversal” and evolving with bounding surfaceevolution, and the new damage parameter continuously reducing plastic modulus with cumu-lative plastic deviatoric strain. In this section, a number of characteristic aspects of responsein undrained cyclic clay behavior have been simulated, and the effects of the new featuresof the model have been highlighted by sensitivity analyses on the new model parameters hMand vd entering Eqs. 3.7a and 3.39 in triaxial and multiaxial formulations, respectively. Inaddition, a concept of response envelopes is adopted in order to show the model’s improvedpredictive capabilities for strain increments acting in different directions of the triaxial space.The model parameters and initial internal variables adopted in these simulations are givenin Tables 5.4 and 3.2, respectively. A total of 13 model constants are listed in Table 5.4: theelasticity constants  and ,, and the critical state constants , bc and be are same as thosein the Modified Cam-Clay model (Roscoe and Burland, 1968); the constants Nc and Ne,which could be different from bc and be, respectively, control the shape of the boundingsurface; the rotational hardening constants X and x are from the original SANICLAY model;the destructuration constant ki is from the SANICLAY model with destructuration (Taiebatet al., 2010b); and the bounding surface constants hM and vd are the two new boundingsurface model constants for better simulation of cyclic loading, and s is a standard constantin the adopted bounding surface formulation. The constants Nc and Ne are often set equalto each other. The bounding surface constant s is often set to 1 by default to reduce purelyelastic region to a single point. The number of model constants reduces to 12 assumingthe default value of s. bc and be, and bc and be values are assumed to be equal, whichwould result into stress-strain loops evolving by approximately same amount of axial strainaccumulation in both triaxial compression and extension. For the presented simulations inthis section, the focus is only on the new bounding surface features; therefore for simplicitythe destructuration feature is deactivated by setting ki = 0 and initial structuration factorhi = 1. The presented stress quantities are normalized for cyclic loading simulations with73GBHB Improvyx wonstitutivy zyuturys oz gAbIWLAmAVTable 3.1: Model parameters used for illustration of improved constitutive featuresboyzl xon“tvnt xvtzgory Yz“ignvtion hzt 1 hzt 2Elasticity  0:03 0:037, 0:2 0:2Critical state bc 1 0:87be 1 0:86 0:15 0:121Bounding surface Nc 1 0:8Ne 1 0:8hM varied 50vd varied 7Rotational hardening X 5 3x 1:7 1:69Destructuration ki 0 0Table 3.2: Initial internal variables used for illustration of improved constitutive featuresboyzl vvrivwlz Yz“ignvtion hzt F hzt GSize of bounding surface pM 200 kPa 345 kPaStructuration factor hi 1 1Void ratio z 0:7 variedRotation of plastic potential and bounding surface 0 0Damage parameter y 0 0Stress state (p; q) (1; 0) variedStress projection center (pc; qc) (1; 0) (1; 0)either the consolidation stress c = 200 kPa or the undrained shear strength Su = 63:7 kPathat is measured from simulations of undrained triaxial compression.HCICF htrz““ pvth vny “trz““B“trvin vvrivtionFigure 3.7 shows results of the model simulation of stress-strain path for six cycles ofundrained triaxial loading under constant cyclic stress amplitude qcyRc = 0:35, to high-light the improved constitutive features of the SANICLAY-B model in terms of stress pathand stress-strain variation. In particular Figs. 3.7(a,b) show the results of simulation usingthe SANICLAY model which does not include the bounding surface feature and the damageeffect. For this purpose, the SANICLAY-B model has been employed but the boundingsurface feature and the damage effect have been deactivated by setting hM to a very large74GBHB Improvyx wonstitutivy zyuturys oz gAbIWLAmAVvalue and vd = 0. It can be observed that without the bounding surface feature the modelsimulates plastic strains mainly in the first cycle of loading for the stress states on the yieldsurface (the term yield surface is used here because the bounding surface feature has beendeactivated). In the subsequent five cycles the stress moves only inside the yield surface andno additional plastic strain is produced. Figures 3.7(c,d) show the simulated results whenthe bounding surface is active (hM = 50) but the damage effect is still inactive (vd = 0).Compared to the previous case, here plastic strains are produced even for the stress stateswithin the bounding surface, accompanied by reduction of p (contractive response), due tothe distance dependent plastic modulus Kp employed in the bounding surface formulation.Although the model can generate plastic strains in all of the loading cycles, it appears thatthe stress-strain loops do not evolve sufficiently. Additional investigation shows that thisphenomenon happens when the image stress (stemmed from the projection center that getsupdated at the last stress reversal) approaches the critical stress ratio. As such only plas-tic deviatoric strain rate is produced while plastic volumetric strain rate approaches zero.Recalling that isotropic and rotational hardening laws adopted from the SANICLAY arefunctions of the volumetric counterpart of gradient of the plastic potential, in this scenariothe bounding surface stops evolving and with no further plastic volumetric strain the meaneffective stress also stops changing from cycle to cycle. The almost locked state of the pro-jection center close to the critical stress ratio also stops evolution of the stress-strain loopsas shown in Fig. 3.7(c,d). The latter can be resolved by using the newly introduced damageeffect feature in the model which is activated by setting vd = 5 for the simulations resultsshown in Fig. 3.7(e,f). Notice that in this case, plastic deviatoric strains keep accumulatingeven when the mean effective stress stops changing; this is because the plastic modulus Kpnow also depends on the evolving damage parameter via the hardening function h. Theproduced stress-stain results in this case are more realistic and qualitatively similar to theones typically produced from experiments.HCICG kvrivtion“ of GszxRGmvfl vny yvmping rvtio fiith “hzvr “trvinIn engineering practice it is very common to use the modulus reduction and damping ratiocurves for conducting equivalent linear site-response evaluations and seismic microzonation.75GBHB Improvyx wonstitutivy zyuturys oz gAbIWLAmAV0 0.2 0.4 0.6 0.8 1 1.2−0.6−0.4−0.200.20.40.6p/σcq/σc(a)−4 −3 −2 −1 0 1 2 3 4−0.6−0.4−0.200.20.40.6εa (%)q/σcSANICLAY(b)0 0.2 0.4 0.6 0.8 1 1.2−0.6−0.4−0.200.20.40.6p/σcq/σc(c)−4 −3 −2 −1 0 1 2 3 4−0.6−0.4−0.200.20.40.6εa (%)q/σch0=100ad=0SANICLAY−B withoutdamage parameter(d)0 0.2 0.4 0.6 0.8 1 1.2−0.6−0.4−0.200.20.40.6p/σcq/σc(e)−4 −3 −2 −1 0 1 2 3 4−0.6−0.4−0.200.20.40.6εa (%)q/σch0=100ad=40SANICLAY−B withdamage parameter(f)Figure 3.7: Illustration of the bounding surface and the damage parameter effects in simu-lation of undrained cyclic triaxial loading: (a-b) SANICLAY, (c-d) SANICLAY-B withoutdamage effect, (e-f) SANICLAY-B with damage effect76GBHB Improvyx wonstitutivy zyuturys oz gAbIWLAmAVVucetic and Dobry (1991) compiled experimental data from various sources encompass-ing normally and overconsolidated clays (OCR = 1–15), and provided ready-to-use curves,showing the effect of plasticity index (PI) on the location of the modulus reduction curveGsecRGmax versus cyclic shear strain amplitude , and on the material damping ratio curve.Here Gsec and Gmax represent the secant shear modulus and maximum shear modulus, re-spectively. These typical curves are shown in Figs. 3.8(a) and 3.9(a), where the originalcurves of Vucetic and Dobry (1991) have also been extrapolated from about 1% to 10%shear strain as it is commonly done in practice, e.g. in the computer program ProShake.In order to evaluate the capability of the SANICLAY-B model in reproducing the typicalshape of the GsecRGmax and damping ratio curves, a number of plane strain simulations areperformed with different cyclic shear strain amplitudes ranging from 0:001% to 10% withthe constraints corresponding to an element in the middle of direct simple shear (DSS) de-vice. The results are shown in Figs. 3.8(b,c) and 3.9(b,c). Following Seed et al. (1986), thesecant shear modulus Gsec presented in these figures is measured from the extreme points ofthe 5th cycle stress-strain loop and the damping ratio (%) is determined as ViR (4.Vq)×100with Vi being the area of the stress-strain loop and Vq being the equivalent elastic energyor the area of a triangle with the vertices at the origin of the stress-strain space, the pointof maximum shear strain, and its projection on the shear strain axis. For reference, thedashed lines show the results of simulations using the SANICLAY model which does notinclude the bounding surface feature and the damage effect. Again for this purpose, theSANICLAY-B model has been employed but the bounding surface feature and the damageeffect have been deactivated by setting hM to a very large value and vd = 0. Figs. 3.8(b,c)show that without the bounding surface feature, the SANICLAY model might significantlyoverpredict the secant modulus. Similarly Figs. 3.9(b,c) show that without the boundingsurface feature, the SANICLAY model underpredicts the damping ratio in small strains andit might significantly over predict the damping ratio in larger strains. The solid lines inFigs. 3.8(b) and 3.9(b) represent the simulations with the SANICLAY-B, with various levelsof hM parameter to activate the bounding surface feature, but with vd = 0 to deactivate thedamage parameter effect. In Figs. 3.8(c) and 3.9(c), in addition to the bounding surfacefeature, the damage parameter effect is also activated by setting vd = 60. It can be observed77GBHB Improvyx wonstitutivy zyuturys oz gAbIWLAmAVthat with the aid of the bounding surface feature and the damage parameter, the perfor-mance of the model in controlling the variations of GsecRGmax and the damping ratio curvescan be significantly improved and the level of response can be better controlled in a widerrange of strains via parameters hM and vd.HCICH Vxxumulvtion of pzvk "a fiith numwzr of xyxlz“Development of the axial strain "a at peaks of cyclic stress with the number of cycles, duringundrained cyclic triaxial compression-extension loading to a cyclic stress level of qcyR2Su =0:69 is illustrated in a set of simulations shown in Figs. 3.10(b,c) along with a sample ofexperimental data on Cloverdale from Zergoun and Vaid (1994) shown in Fig. 3.10(a). Thecyclic stress level is expressed by the ratio of the maximum shear stress amplitude cy = qcyR2and the undrained strength in compression Su. This trend of response in development of "aat peak cyclic stresses for the Cloverdale clay is similar to that of other clays, e.g. Drammenclay under comparable loading conditions (Andersen et al., 1980). Simulations for a range ofvalues for hM and a damage effect parameter of vd = 40, shown in Fig. 3.10(b), suggest thatsmaller values of hM result in a faster accumulation of peak "a. For reference, the dashedlines show the results of simulations using the SANICLAY model which does not include thebounding surface feature and the damage effect. Again for this purpose, the SANICLAY-Bmodel has been employed but the bounding surface feature and the damage effect have beendeactivated by setting hM to a very large value and vd = 0. Note that the simulations usingthe SANICLAY model don’t capture such an accumulation of peak axial strain after the firstcycle. Similarly, simulations with hM = 50 and a range of value for vd, shown in Fig. 3.10(c),show that larger values of vd result in faster accumulation of peak "a. Observe also thatwhen no damage effect is considered, i.e. vd = 0, after a number of cycles difference betweenthe peak axial strain in compression and extension remains almost constant. The reason ofsuch response is the freezing of stress-strain loops as discussed for Fig. 3.7.78GBHB Improvyx wonstitutivy zyuturys oz gAbIWLAmAV10−3 10−2 10−1 100 10100.20.40.60.81Gsec/Gmax PI=200PI=100PI=50PI=30PI=15(a)Typicalexperimental data10−3 10−2 10−1 100 10100.20.40.60.81Gsec/Gmax ad=0h0=800h0=200h0=50SANICLAY(b)SANICLAY−Bwithout damageparameter10−3 10−2 10−1 100 10100.20.40.60.81Shear strain, γ, (%)Gsec/Gmax ad=60h0=800h0=200h0=50SANICLAY(c)SANICLAY−Bwith damageparameterFigure 3.8: Modulus reduction curves GsecRGmax versus cyclic shear strain amplitude : (a)typical curves of Vucetic and Dobry (1991) showing the effect of plasticity index (PI), and(b,c) model simulations of undrained cyclic direct simple shear (DSS) loading, illustratingthe effects of parameters hM and vy79GBHB Improvyx wonstitutivy zyuturys oz gAbIWLAmAV10−3 10−2 10−1 100 1010102030405060Damping ratio (%) PI=200PI=100PI=50PI=30PI=15(a)Typical experimental data10−3 10−2 10−1 100 1010102030405060Damping ratio (%) ad=0 h0=800h0=200h0=50SANICLAY(b)SANICLAY−B withoutdamage paramer10−3 10−2 10−1 100 1010102030405060Shear strain, γ, (%)Damping ratio (%) ad=60h0=800h0=200h0=50SANICLAY(c)SANICLAY−B withdamage parameterFigure 3.9: Damping ratio curves versus cyclic shear strain amplitude : (a) typical curvesof Vucetic and Dobry (1991) showing the effect of plasticity index (PI), and (b,c) modelsimulations of undrained cyclic direct simple shear (DSS) loading, illustrating the effects ofparameters hM and vy80GBHB Improvyx wonstitutivy zyuturys oz gAbIWLAmAV100 101 102−10−50510Peak εa (%) τcy/Su=0.69(a)Typical experimental data100 101 102−10−50510Peak εa (%) h0=100 200 400 800ad=40τcy/Su=0.69SANICLAYSANICLAY−B(b)100 101 102−10−50510Number of cyclesPeak εa (%)τcy/Su=0.69ad=40SANICLAY−B20 10 5 20h0=100(c)Figure 3.10: Development of the axial strain "a at peaks of cyclic stress with the numberof cycles: (a) sample experimental data on Cloverdale clay after Zergoun and Vaid (1994)and (b) model simulations of undrained cyclic triaxial loading, illustrating the effects ofparameters hM and vy81GBHB Improvyx wonstitutivy zyuturys oz gAbIWLAmAVHCICI kvrivtion“ of Xhg fiith numwzr of xyxlz“The last set of qualitative simulations presented in this section to highlight the new featuresof the SANICLAY-B model are related to a number of undrained cyclic triaxial loadingswith various constant cyclic stress ratio amplitudes CSR = qxyR2Su = xyRSu versus num-ber of cycles required to reach "a = 3%. Results of these simulations along with sampleexperimental data on Cloverdale clay adopted from Boulanger et al. (1991), to illustrate thetypical trend of response, are shown in Fig. 3.11. Again for reference, the dashed lines showthe results of simulations using the SANICLAY model which does not include the boundingsurface feature and the damage effect, by employing the SANICLAY-B model with a verylarge hM and vd = 0. It can be observed that for each of the simulations using the SANICLAYmodel, the "a = 3% can be reached in a finite number of cycles only for CSR values in therange of 0.9–1; but for CSR Q 0:9 the "a = 3% cannot be reached even with 100 loadingcycles, i.e. the required number of cycles to reach the "a = 3% approaches ∞. However,when the bounding surface feature is activated, e.g. with the a range of finite value for hMand vd = 5, the "a = 3% can be reached for CSR Q 0:9 as well, suggesting the more realisticsimulations using the SANICLAY-B in comparison with the experiments. Observe that toreach the "a = 3%, fewer number of cycles is required for the smaller hM values.HCIC5 gz“pon“z znvzlopz“Methods for generating stiffness locus (Bazˇant, 1978) and response envelopes (Gudehus,1979) were proposed for demonstration of the response of endochronic and hypoplasticityconstitutive models, respectively. Stress Response Envelopes (SRE) are generated in thissection for demonstration of improved features of the proposed SANICLAY-B model. In thesimulations, strain probes of equal total strain increment size "˙tot =√"˙2v + "˙2q are applied in"v-"q plane to generate corresponding SREs plotted in the triaxial stress plane. Initial voidratio z = 0:794 and isotropic normally consolidated stress state (p; q) = (345; 0) are usedas a reference state. For any other state, the reference state is used for computation of aninitial void ratio and a stress state. Modified overconsolidation ratio, defined as OCRm =‖¯‖ R ‖‖, where  is radially mapped from the origin of the stress space onto the bounding82GBHB Improvyx wonstitutivy zyuturys oz gAbIWLAmAV0 20 40 60 80 10000.20.40.60.811.2Cyclic stress ratio, τcy/Su Fitted curve(a)Typical experimental data0 20 40 60 80 10000.20.40.60.811.2Number of uniform cyclesCyclic stress ratio, τcy/Su ho=500ho=250ho=125SANICLAYad=5(b)SANICLAY−B with damage parameterFigure 3.11: Cyclic stress ratio amplitudes CSR = xyRSu versus number of cycles requiredto reach "a = 3%: (a) sample experimental data on Cloverdale clay adopted from Boulangeret al. (1991) and (b) model simulations of undrained cyclic triaxial loading, illustrating thecollective effects of parameters hM and vy83GBHB Improvyx wonstitutivy zyuturys oz gAbIWLAmAVsurface to obtain ¯, in the multiaxial formulation or as OCRm = p¯Rp or OCRm = q¯Rqin the triaxial quantities, is introduced here for convenience of computing the stress state(p; q) = (p¯ROCRm; q¯ROCRm).Initivl “trz““ “tvtz“ on thz wounying “urfvxzThe SREs produced by applying "˙tot = 1% starting from the initial stress states at OCRm = 1corresponding to the initial stress ratios M = 0:5bc, 1bc, 2bc are shown in Fig. 3.12 forSANICLAY (dash-dotted lines) and SANICLAY-B (solid lines) simulations. The SREs con-sists of top and bottom segments which is typical for classical plasticity and hyperplasticitymodels (Wu and Kolymbas, 1990). The top segments of SREs for the SANICLAY andSANICLAY-B coincide for the strain probes pushing the stress state outward of the yieldand the bounding surfaces, respectively. The top segments of the SRE for the SANICLAYsimulation that does not coincide with the one for the SANICLAY-B are for the strain probesthat at initial stress states are bringing the stress states inside the yield surface and crossit afterwards. The bottom segments (lying inside the initial yield surface) of SRE for theSANICLAY are elastic (hypoelasticity) SREs. The distance from the initial stress state tothe bottom segments of SREs for the SANICLAY-B is smaller than those of the SANICLAY,suggesting that the modulus is reduced for the former due to the updated projection centerproducing elastoplastic response within the bounding surface. In fact, for the SANICLAY-B simulations the projection center is initially at the origin and is updated for the strainprobes related to the bottom segments thereby providing the reduced modulus. Anotherobservation is that the initial stress state: stays inside SREs for M Q bc (green), lies onSREs for M = bc (blue), and lies outside SREs for M S bc (black) as expected for classicalplasticity models (Wu and Kolymbas, 1990).gzquirzmznt for limiting p The requirement for the limit of p introduced in Eq. 3.14 is demonstrated now. The SREsproduced for the initial stress state at OCRm = 1 and M = 0 are shown in Fig. 3.13 fortwo cases when: (a) the limit of p is not considered and, (b) the limit of p is considered.Observe that a kink in the SRE, when the limit of p is not considered, is present for the84GBHB Improvyx wonstitutivy zyuturys oz gAbIWLAmAVp/p00 0.2 0.4 0.6 0.8 1 1.2q/p 0-0.4-0.200.20.40.6Mcεv(%)-1 0 1εq(%)-101strain probes SANICLAY-BSANICLAYFigure 3.12: Stress response envelopes produced using "˙tot = 1% for the initial stress statedefined by OCRm = 1 and M = 0:5bc, 1bc, and 2bc demonstrate the response of theSANICLAY and SANICLAY-B models (initial yield and bounding surface shown in gray)strain probe corresponding to the isotropic unloading. This is a result of considering Eq. 3.13alone which cannot provide an adequate estimate of p for the image stress at the origin ofthe stress space. Consideration of p = lim p for such cases, eliminates the kink and theSRE becomes smooth.Initivl “trz““ “tvtz“ in“iyz thz wounying “urfvxzThe generation of SREs for the initial stress states inside the bounding surface are alsoinvestigated to demonstrate the effect of location of the initial projection center. The SREsproduced using the SANICLAY-B for the initial stress state at OCRm = 1:5 and M = 0,0:5bc, 1bc, 2bc are shown in Fig. 3.14. Here two scenarios are considered for the initialPC: (i) PC at the origin of the stress space, and PC at the initial stress states. Noticethat discontinuities in all SREs are present for the former scenario shown in Fig. 3.14(a),while SREs are smooth for the latter scenario shown in Fig. 3.14(b). The reason for suchdiscontinuities is the required update of PC to induce plasticity whenever response becomes85GBHB Improvyx wonstitutivy zyuturys oz gAbIWLAmAVp/p00 0.2 0.4 0.6 0.8 1 1.2q/p 0-0.6-0.4-0.200.20.40.6=v) thz limit of p is not xonsiyzrzyp/p00 0.2 0.4 0.6 0.8 1 1.2q/p 0-0.6-0.4-0.200.20.40.6=w) thz limit of p is xonsiyzrzyFigure 3.13: Stress response envelopes produced using "˙tot = 1% demonstrate the require-ment for limit of p given in Eq. 3.14elastic (i.e., stress reversal takes place). When the initial PC is at the initial stress states,any strain probe involves plasticity and no update is required thereby providing smoothSRE. Notice also that the SRE segments corresponding to the updated projection center inFig. 3.14(a) coincide with the portion of SREs in Fig. 3.14(b) corresponding to the samestrain probes. This is expected as, for these portions of SREs, the location of PC is identicalin both cases. These two scenarios demonstrate that identical stress and state variables butdifferent locations of PC result into the difference in the response and more importantly theupdated PC results into discontinuities. Such discontinuities are not observed for the initialstress states on the bounding surface because the initial PC has no effect on the strain probespushing the stress state outwards of the bounding surface.The success of the SANICLAY-B model in cyclic loading simulations lies in the updateof the projection center during stress reversal (e.g., conventional triaxial compression load-ing vs. unloading) where drastic changes of the modulus are observed in experiments. theSANICLAY-B identifies the instance of stress reversal whenever the loading index becomesnegative a Q 0 (Eq. 2.11). As a result, update of PC can take place when a = 0− (i.e.,close to neutral loading), while for a = 0+ the model does not update PC thereby producingdiscontinuities in SREs. It is, therefore, expected that any constitutive model accounting for86GBIB gummury oz proposyx gAbIWLAmAV moxylp/p00 0.2 0.4 0.6 0.8 1 1.2q/p 0-0.6-0.4-0.200.20.40.6Mcεv(%)-1 0 1εq(%)-101strain probes=v) hVcICaVnBBA initivl eC vt thz originp/p00 0.2 0.4 0.6 0.8 1 1.2q/p 0-0.6-0.4-0.200.20.40.6Mcεv(%)-1 0 1εq(%)-101strain probes=w) hVcICaVnBBA initivl eC vt initivl strzssFigure 3.14: Stress response envelopes produced using "˙tot = 1% demonstrate the effect ofinitial location of the projection center (initial bounding surface shown in gray)the change in the modulus during the stress reversal would result into the produced discon-tinuities. In the current version of SANICLAY-B model there is no solution for this problemand the model should be used with caution in boundary value problems. It is possible thatduring neutral loading (although its occurrence is limited) drastic change in modulus can re-sult in numerical difficulties. Such numerical difficulties could arise in finite element iterativesolution which relies on the modulus of a constitutive model. If such iterations take placewith the modulus of a constitutive model operating in the vicinity of such discontinuities,then finite element iterative solution can start diverging.HC5 hummvry of propo“zy hVcIXaVnBW moyzlThe bounding surface radial plasticity has been used to extend the applicability of the SAN-ICLAY family of models, with and without destructuration, to cyclic loading conditions, andthis extended version of the model is named SANICLAY-B. So far the SANICLAY modelshad a distorted elliptical yield surface in the triaxial p–q space, implying that changes ofstress state within this relatively large surface are assumed to cause only elastic strains. Bycontrast, in the formulation of the SANICLAY-B model the yield surface concept of the87GBIB gummury oz proposyx gAbIWLAmAV moxylSANICLAY models has been substituted by the bounding surface concept along with a ra-dial mapping rule as showing in Fig. 3.2 in the triaxial stress space. This feature enablesprediction of plastic strains for changes of stress state with the bounding surface for bothmonotonic and cyclic loadings, a feature lacking from the previous SANICLAY models, with-out losing the other well established predictive capabilities of the SANICLAY for simulatinganisotropy and destructuration as was demonstrated, for example, by Dafalias et al. (2006)and Taiebat et al. (2010b).Two simplifications in the formulation of the adopted SANICLAY model are (a) unifyingthe anisotropy tensors for the yield surface (or bounding surface in the SANICLAY-B model)and the plastic potential adopted from the suggestion of Jiang and Ling (2010), and Jianget al. (2012), and (b) incorporating only isotropic destructuration mechanism out of the twogeneral mechanics proposed by Taiebat et al. (2010b). Two of the novel elements of thenewly proposed SANICLAY-B model are the moving projection center (given by Eq. 3.38for multiaxial formulation) and the damage mechanism employed in the bounding surfaceformulation (given by Eqs. 3.7a, 3.8, 3.39 and 3.40 for multiaxial formulation) of the model.Following the trends in the reviewed constitutive models (see Section 2.4.2), the proposedprojection center is established at any stress reversal to avoid overdamping as plastic strainsresume shortly after stress reversal. The novel feature of the proposed projection center con-siders its evolution with the bounding surface evolution required to eliminate the possibilityof the projection center being found outside the bounding surface. The bounding surfacehardening function h (given by Eq. 3.8) that controls the plastic modulus Kp (given byEq. 3.7a) is proposed to be a decaying function of its initial value hM and the newly proposeddamage parameter y that evolves with deviatoric strain as specified by Eq. 3.40. The modelconstant vd regulates the rate of evolution of the damage parameter y allowing control ofthe rate of stiffness degradation during cyclic loading observed in the experiments.Detailed formulation of the new elements of the extended model, i.e. the SANICLAY-B,has been presented in both triaxial and multiaxial stress spaces. In order to illustrate theeffects of the new features of the model, comparisons have been made between simulationsusing the full SANICLAY-B model and simulations using the simplified version of it, inwhich the novel features have been deactivated by prescribing specific value to the new88GBIB gummury oz proposyx gAbIWLAmAV moxylmodel parameters hM and vd, which therefore can be referred to as the SANICLAY. Suchcomparison shows that the full model has significantly improved predictive capability incapturing the response of clays in undrained cyclic loading. This includes better simulationsof• stress and stress-strain paths shown in Fig. 3.7 with plastic strains progressively accu-mulating with number of cycles• modulus reductions and damping ratio curves versus cyclic shear strain amplitudesin undrained cyclic direct simple shear shown in Figs 3.8 and 3.9 suggesting thatthe dynamic properties of soil can be better replicated for the range of shear strains1:0× 10−2 ≤ ≥ 1:0× 10N• development of the axial strains at peaks of cyclic stress versus the number of cyclesshown in Fig. 3.10• variations of cyclic stress ratio amplitudes versus the number of cycles required to reachcertain axial strain shown in Fig. 3.11 in undrained cyclic triaxial loadingAll of the above add to the credibility of the model application in simulating responseof clay subjected to cyclic loading. It is important to point out that site effects can bereproduced only if a constitutive model is capable of realistically accounting for modulusreductions and damping ratio curves versus cyclic shear strain. As the main objective ofthis work is development of a constitutive model that can be used for seismic site responseanalysis, successful reproduction of these curves demonstrates model credibility. In addition,the improved features of the proposed formulation are demonstrated in producing responseenvelopes shown in Figs. 3.12 and 3.14.The SANICLAY-B model formulation is completed and its constitutive equations areto be numerically implemented in a constitutive driver prior to further assessment of itspredictive capabilities.89IC hVcIXaVnBW cumzrixvlImplzmzntvtionChapter 3 describes the formulation of developed SANICLAY-B model. The constitutiveequations of the model are highly nonlinear as they are intended for simulations of the re-sponse of clay exhibiting nonlinear behavior when subjected to cyclic loading. Constitutiveequations have to be integrated numerically for their further application in simulating lab-oratory element testsT and boundary value problems. In this chapter, adopted numericalintegration method and model specific integration details are presented within an imple-mentation of a constitutive driver considering multi-axial formulation of the model. Theconstitutive driver is designed for its usage within a finite difference or finite element codewhere it receives a strain increment and outputs corresponding stress. In order to simulatelaboratory element tests, a linearized loading constraints method is adopted and describedfor the tests to be used in this dissertation. Finally, implementation of the constitutive driverwithin OpenSEES finite element code is presented with the description of necessary elementsfor interaction between the constitutive driver and the numerical platform.ICF IntroyuxtionIn numerical solutions such as the finite element method, a domain is discretized into anumber of finite elements. Within each element, the displacement field is approximated usinginterpolation functions thereby providing estimates of strain increments that are passed toa number of Gauss points where a stress-strain response is computed using a constitutivedriver which solves a general constitutive relation given in Eq 2.12. For a domain with a7rzfzrring to thz tzsts fiithin fihixh strzss vny strvin zlys vrz uniformly yistriwutzy90HBEB Introxuwtionsingle finite element, loading acting on the element can be specified directly as the loadingis at the boundaries of the domain, thereby strain increments passed to a constitutive drivercan be controlled. In contrast, for a boundary value problem with the domain discretizedinto a number of finite elements, the loading acting on each element and, as a result, strainincrements passed to a constitutive driver cannot be controlled. Therefore, it is imperative toreassure that numerical implementation of a highly nonlinear model is so that it can handleany size of increments of strain passed from a finite element code. A simplified diagramillustrating interaction of finite element method with a constitutive driver representing stress-strain response of a material point is shown in Fig. 4.1. The diagram is split into global andconstitutive sectorsU with the former assembling loads, displacements, and stiffnesses fromall elements in the domain, and the latter representing only a single point field of stressesand strains. A global displacement increment u˙ is computed using a global stiffness K anda residual force g estimated as a difference between a global external Fext and an internalFint forces. An element displacement increment u˙el is extracted from the u˙ to pass a strainincrement "˙ for each Gauss point within an element. A constitutive driver describing stress-strain response at each Gauss point outputs stresses that are integrated to obtain elementinternal forces Fintel . The Fintel from all elements are then assembled into the Fint. To maintainequilibrium between the global external Fext and the global internal Fint forces, a Newton-Raphson type of algorithm is often used in finite element codes. During this procedure themagnitudes of strain increments "˙ passed into the constitutive driver cannot be controlledfor boundary value problems.Integration of the constitutive equation given in Eq. 2.12 follows a stress return algorithmthat applies a plastic corrector of an initially assumed elastic predictor which can be written8 nitz zlzmzntA vpproflimvting yisplvxzmznt zly vny xontvining v numwzr Gvuss points fiith v strzssBstrvin rzsponsz yzsxriwzy wy v xonstitutivz yrivzrA vrz not shofin on thz yivgrvm in FigC ICF vs this is notthz foxus of xurrznt numzrixvl implzmzntvtion91HBEB Introxuwtionglobalg˙ = FextM − FintMu˙ = K−Ng˙"˙ = Wi u˙elu˙el for an el-ement is ex-tracted fromglobal u˙""˙?"˙?constitutive driverat each Gauss pointwithin an elementFintel =∫kWiykFintel from all el-ements assembleinto global Fintg˙ = Fext − FintglobalNewton-Raphson alogrithm:iterate until∣∣∣g˙∣∣∣ ≤ TOLFigure 4.1: Illustration of the strain increment "˙ passed from a finite element code into aconstitutive driver (Newton-Raphson type algorithms are often used in finite element analysisto assure equilibrium between external and internal forces)as followsn+N = n + De : ("˙− "˙p)= (n + De : "˙)− De :(〈a〉 @g@)= pred − De :(〈a〉 @g@)(4.1)where n and n+N are the stress states at the beginning and at the end, respectively, ofstep; De is an elastic modulus; "˙p is a plastic strain increment; a is a loading index; andg = 0 is a plastic potential. The return algorithm requires the knowledge of the stressstate and the internal variables for the numerical integration. Researchers often use eitherexplicit or implicit integration techniques which differ in the consideration of the stress stateand the internal variables. The explicit integration is the most common and simple oneas it relies on the stress state and the internal variables at the beginning of the step (orat the onset of plasticity at the stress state intersecting the elastic domain). The plasticcorrection, in this case, is performed along the gradient of the plastic potential at the stressstate corresponding to the beginning of the onset of plasticity as shown in Fig. 4.2(a). Noticethat the stress state at the end of the integration is outside of the yield surface and in orderto satisfy the consistency condition, given in Eq. 2.10, either consistent and/or normal drift92HBEB Introxuwtioncorrections are often adopted (Crisfield, 1991; Potts and Gens, 1985; Sloan et al., 2001). Theimplicit integration is much more complicated with the plastic correction performed alongthe gradient of the plastic potential at the final stress state. As the final stress state isnever known in advance, this integration technique requires iterations until the solution isconverged with the stress state on the yield surface as shown in Fig. 4.2(b). Notice that in thiscase there is no need to locate the stress state intersecting the elastic domain as the procedurerelies on the final stress state. This integration technique has an important advantage overexplicit in that it offers quadratic rate of convergence in finite element iterative solutionthereby considerably reducing time in numerical solution of boundary value problems. Theimplicit integration technique was successfully used for soil plasticity models with Lode angleindependent plastic potential (e.g., Amorosi et al., 2008; Borja and Lee, 1990; Jeremic´ andSture, 1997). For Lode angle dependent constitutive models, the implicit integration requiresevaluation of the second order derivative of the plastic potential that is laborious to solveanalytically. Up to date, only Coombs and Crouch (2011) are known to have used theimplicit integration with an analytical solution for the second order derivative of the plasticpotential; however, their constitutive model adopted yield surface and plastic potential withisotropic hardening only. Finite difference approximation such as that proposed by Dennisand Schnabel (1983) can also be adopted for the solution of the second derivative of the plasticpotential. As an alternative, Macari et al. (1997) proposed a two level iterative procedurewith the additional iterative process for the Lode angle dependency of the plastic potential asthe means to avoid laborious analytical solution. Another approach used by researchers (e.g.,Alawaji et al., 1991; Ortiz and Simo, 1986; Simo and Taylor, 1985) considers an iterativesolution that does not require laborious computation of the second order derivative of theplastic potential and therefore can be used for Lode angle dependent plastic potential. In thisintegration technique, the stress state is updated in each iteration and the plastic correctionis performed along the gradient of the plastic potential passing through the stress state atthe beginning of each corresponding iteration. Although it is often referred to as implicitintegration, this method in fact relies on the gradient of the plastic potential at the stressstate at the beginning of a corresponding iteration and therefore is to be referred to assemi-implicit integration.93HBEB Introxuwtionnint@gR@int predn+Nfn = 0fn+N = 0Elastic domain=v) Zflplixitnpredfn = 0fkn+N = 0kn+N@gR@kn+NElastic domain=w) Implixit =xonvzrgzy in k itzrvtions)Figure 4.2: Geometric interpretation of the integration algorithms used in numerical imple-mentation of the general constitutive relation given in Eq. 2.12Among the reviewed above integration algorithms, the explicit integration is applicableonly for small strain increments which is not feasible for applications in boundary valueproblems. The implicit integration is believed to be the most robust one and it has beenshown that this method is much more reliable than semi-implicit integration for increas-ing size of the strain increment (e.g., Manzari and Prachathananukit, 2001). Both implicitand semi-implicit integration use a first order Taylor series expansion of the yield function(e.g., Jeremic´ and Sture, 1997; Manzari and Prachathananukit, 2001). For the semi-implicitintegration, this also results into drift-off error thereby requiring drift correction as in theexplicit integration method. In order to increase accuracy of the integration, extended ex-plicit integration schemes, namely modified Euler and high order Runge-Kuta formula, havebeen proposed by researchers. The modified Euler integration is a second order Taylor seriesexpansion of stress. In addition, to higher order accuracy the integration can also considersubstepping and error control performed automatically as described by Sloan (1987). AlsoPotts and Ganendra (1994) demonstrated that substepping results into accurate predictionsfor the loading conditions with the strain increments varying “proportionally” (i.e., when theratio between the strain increment components remains constant as in undrained conditions).Considering the above implications, the refined explicit integration with substepping and au-tomatic error control is to be used for the numerical implementation of the SANICLAY-Bmodel in this chapter.The chapter is organized in the following way. First, details of numerical implementa-94HBFB Wonstitutivy xrivyrtion of the SANICLAY-B model using the refined explicit integration with substepping andautomatic error control are presented within a constitutive driver in section 4.2. The stand-alone constitutive driver receives strain increment "˙ and outputs corresponding stress . Inorder to simulate a laboratory test, a method describing corresponding loading constraintsis presented in section 4.3 for laboratory tests simulated in this dissertation. The role ofconstitutive modeling in a finite element code is briefly described followed by description ofa finite element code interface with the implemented constitutive driver in section 4.4.ICG Xon“titutivz yrivzrThe details of the SANICLAY-B numerical implementation scheme are presented in thissection. The implementation scheme is presented with the description of important fea-tures and written pseudo-codes that could be followed for numerical implementation of theSANICLAY-B model as a constitutive driver. The inputs, outputs, and general descrip-tion for each pseudo-code are given in the corresponding captions in the following manner:[Outputs] = gznzrvl yz“xription (Inputs). Summary of all the pseudo-codes is given inTable 4.1. In the table, a reference to each pseudo-code, a brief description and a referenceto the pseudo-codes called by each pseudo-code are included. The first pseudo-code, given inTable 4.2, breaks a strain increment "˙ into smaller substepping increments i˙ "˙, if a maximumrelative error between modified Euler and forward Euler is greater than a specified toleranceSTOL. This pseudo-code is a core for controlling the accuracy of the obtained numericalsolution in the refined explicit integration. The modified Euler trial increments j = 1 andj = 2 are computed in Table 4.3 and returned to the preceding pseudo-code. The procedurein Table 4.3 also distinguishes between elastic and elastoplastic responses. In case of theelastic response, the procedure assigns same trial increments j = 1 and j = 2 to the statevariables so that when returned to Table 4.2 the estimated error is 0. In case of crossingelastic region, the procedure in Table 4.5 computing intersection state on the elastic nucleusis called first and then elastoplastic response is computed for the remaining increment. Incase of elastoplastic response, the procedure for computation of elastoplastic increments inTable 4.9 is called. The remaining pseudo-codes are required for elastoplastic computations.95HBFB Wonstitutivy xrivyrComputation of b ratio given in Table 4.13 is required for computation of the image stresson the bounding surface. Evolution of projection center scheme is in Table 4.16 is usedto compute evolution of c. A combination of a consistent and a normal drift correctionmethods are given in Table 4.17 required to reassure that the stress state is on the boundingsurface. Evolution "cum in Table A.1 of Appendix A is used as an additional measure toavoid spurious updates of the projection center.Table 4.1: Summary of pseudo-codes for implementation schemePseudo-code Description Required pseudo-codesTable 4.2 automatic substepping and error control Tables 4.3, 4.16, A.1, 4.17Table 4.3 modified Euler trials Tables 4.4, 4.9, 4.16Table 4.4 state at the intersection of the elastic nucleus Tables 4.5, 4.8Table 4.9 elastoplastic response Tables 4.10, 4.13Table 4.13 ratio b between the bounding and the loadingsurfacesTables 4.14, 4.15Table 4.16 evolution of the projection center c noneTable 4.17 combination of consistent and normal driftcorrectionsTable 4.16Table A.1 evolution of "cum for detection of stress reversal Tables A.2, A.3, A.4ICGCF gz nzy zxplixit intzgrvtionThe constitutive equations of the SANICLAY-B model are numerically integrated using sec-ond order accurate forward Euler solution (modified Euler). In addition, automatic errorcontrol where the errors between the first order (forward Euler) and the second order (mod-ified Euler) estimates of stress and internal variables are computed. If the maximum of thecomputed errors exceeds a specified tolerance, then substepping of the input strain incrementis employed.96HBFB Wonstitutivy xrivyrboyi zy Eulzr intzgrvtionModified Euler integration scheme used here is based on the second order Taylor seriesexpansion of stressn+N = n +@n@": "˙+12(@2n@"@": "˙): "˙= n + Dep;(N) : "˙+12(Dep;(2) − Dep;(N)) : "˙= n + ˙(N) +12(˙(2) − ˙(N))= n +12(˙(N) + ˙(2))(4.2)where subscripts n and n+1 denote state variables at the start and at the end of an increment"˙, respectively; and superscripts (1) and (2) denote trials j = 1 and j = 2 of modified Eulerintegration, respectively. More detailed explanation of j trials follows in conjunction withthe schematic representation of the modified Euler integration shown in Fig. 4.3. In thefirst trial denoted by j = 1, the stress increment ˙(N) is approximated by multiplicationof elastoplastic modulus Dep;(N), evaluated at the initial stress state n, and applied strainincrement "˙. At this point, the stress state and internal variables are temporarily updatedin order to evaluate Dep;(2). In the second trial denoted by j = 2, the stress increment ˙(2) isapproximated by multiplication of elastoplastic modulus Dep;(2) and applied strain increment"˙. The final stress state n+N is then computed following Eq. 4.2. Notice that in the equation@2n@"n@"n: "˙ = Dep;(2)−Dep;(N) is a change in Dep over a strain increment "˙, i.e., the differencebetween Dep at the beginning and at the end of the strain increment for j = 1 trial.Similar procedure of the second order approximation is also applied to the model internalvariablespM;n+N = pM;n +12(p˙(N)M + p˙(2)M)(4.3) n+N = n +12( ˙(N) + ˙(2))(4.4)hi;n+N = hi;n +12(h˙(N)i + h˙(2)i)(4.5)yn+N = yn +12(y˙(N) + y˙(2))(4.6)97HBFB Wonstitutivy xrivyr""nn"n+Nn+NDep;(N)˙(N)Dep;(2)Dep;(2)˙(2)"˙Figure 4.3: Schematic interpretation of the two-step modified Euler stress integration scheme(interpreted from Andrianopoulos et al., 2010)Vutomvtix “uw“tzpping vny zrror xontrolAn estimate of the difference between modified Euler (complete Eq. 4.2) and forward Euler(first two terms of Eq. 4.2), i.e.,E =12(˙(2) − ˙(N)) (4.7)is normalized by the magnitude of current stress, yielding a relative error as:g =‖E‖‖n‖ (4.8)where ‖◦‖ denotes a magnitude of ◦. Similar relative errors gp0 , g , gSi , and gy are specifiedfor the internal variables and the largest out of the five is selected asgmax = max (g; gp0 ; g ; gSi ; gy) (4.9)For substepping, a pseudo-time i lying in the range 0 Q i ≤ 1 is used. This variablemultiplies the input strain increment "˙ during substepping procedure with i = 1 meaningthat "˙ is fully considered in the iteration. The maximum error gmax is compared with98HBFB Wonstitutivy xrivyra specified tolerance STOL in each iteration. In case gmax S STOL, the modified Eulerestimate is not accepted and i˙ has to be reduced for a new iteration using an estimatedfactor  through i˙ ← i˙ such that = max(0:9√STOLRgmax; 0:1)(4.10)In case gmax ≤ STOL, the modified Euler estimate is accepted but i˙ has to be adjustedfor the next substep to accelerate the time of computation using an estimate of the factor given as = min(0:9√STOLRgmax; 1:1)(4.11)A procedure of refined explicit integration with substepping and automatic error control ispresented in pseudo-code shown in Table 4.2. In styp G of the procedure, the trial incrementsj = 1 and j = 2 of the modified Euler are computed using the procedure described inTable 4.3. The latter procedure distinguishes between purely elastic and elastoplastic trialincrements using the initial stress and the stress computed with the assumption of the purelyelastic response, i.e., multiplying secant elastic modulus Desec (e.g., Borja and Lee, 1990; Sloanet al., 2001) of the adopted hypoelasticity with a strain increment "˙ and summing up with theinitial stress. In the case of both stresses lying within the elastic nucleus Feln, the responseis considered elastic. Otherwise, two possibilities are evaluated in the procedure. The firstpossibility is that the response is partly elastic and for the remaining part elastoplastic.In this case, stress state and internal variables at the intersection of the elastic nucleusare computed first and then elastoplastic response is considered. The second possibility isthat the response is elastoplastic from the beginning of the increment. During elastoplasticresponse, the formulation of the model is such that the projection center c is updated duringstress reversals (see section 3.2.2) when a ≤ 0 in unloading. It is possible that unloadingcan take place with small magnitudes of strain increments resulting into updated of theprojection center. If shortly after such unloading, reloading takes place then the responsecan be unrealistically differ from the response during the last loading phase. In order toavoid update of projection center in such cases, a scalar valued variable "cum is accumulatedin unloading and stress reversal is set to occur only when the variable "cum exceeds an99HBFB Wonstitutivy xrivyrTable 4.2: [; q; z; hi; y;c; "cum] = vutomvtix “uw“tzpping vny zrror xontrol (; q; z; hi; y;c; "cum; "˙)Step Description1 Set i = 0, i˙ = 1, count = 0.2 Check for the end of increment: i Q 1?YES: EXIT;NO: GO TO styps G.3 Compute trial Modified Euler increments and corresponding "cum for detection of stress reversal:[˙(N;2); q˙(N;2); z˙; h˙(N;2)i ; y˙(N;2); "˙(N;2)cum]=moyi zy Eulzr inxrzmznt“(; q; z; hi; y;c; i˙ ; "˙);[el; zel;elc ; "trcum; i˙el]= “trz““ rzvzr“vl “tvtz(; z;c; "cum; "˙(N;2)cum ; i˙ ; "˙).4 Check for stress reversal: "trcum S "ˆ?YES: set[; z;c; "cum; i˙]=[el; zel;elc ; 0; i˙ − i˙ el]and GO TO styp G.NO: compute Modified Euler increments˙tr =0:5(˙(N) + ˙(2)); tr = + ˙tr; q˙tr =0:5(q˙(N) + q˙(2)); qtr =q + q˙tr;ztr = z− z˙; htri = hi + 0:5(h˙(N)i + h˙(2)i); ytr = y+ 0:5(y˙(N) + y˙(2));trc = projzxtion xzntzr zvolution(c; q; q˙tr).5 Compute relative errors between first and second Modified Euler trials:g = 0:5∥∥˙(2) − ˙(N)∥∥ R ‖tr‖; gq = 0:5 ∥∥q˙(2) − q˙(N)∥∥ R ‖qtr‖;gmax = max (g; gq).6 Check for allowable tolerance: gmax S STOL?YES: GO TO styps K and L ;NO: set count = 0 and GO TO styps M.7 Reduce pseudo-time increment i˙ : = max(0:9√STOLRgmax; 0:1);i˙ = max(i˙ ; i˙min); if i˙ = i˙min, set count = count + 1.8 Check if i˙min has been reached more than once: count S 1?YES: EXIT with error message “reduce i˙min”;NO: GO TO styp F.9 Check for stress drift from the bounding surface: F (tr; qtr) S 0?YES: set{[; q; hi; y;c] = yrift xorrzxtion(tr; qtr; ztr; htri ; ytr;trc)z = ztr; "cum = "trcum};NO: set [; q; z; hi; y;c; "cum] = [tr; qtr; ztr; htri ; ytr;trc ; "trcum].10 Update pseudo-time and pseudo-time increment for the next subincrement:i = i + i˙ ; = min(0:9√STOLRgmax; 1:1);i˙ = (i˙ − i˙ el); or, if i˙ el = i˙ : i˙ = i˙if(i + i˙)S 1, set i˙ = (1− i )GO TO styp F.input threshold value "ˆ as shown in styp H of Table 4.2. The complete procedure to updateprojection center during detected stress reversal using the "cum is given in Appendix A.3.1.The rest of the procedure in the table follows the logic of automatic substepping and errorcontrol described above. At the end of each accepted substep, a drift correction of a finalstress state onto the bounding surface is performed as shown in styp M. This step is necessaryto avoid accumulation of the drift-off.100HBFB Wonstitutivy xrivyrTable 4.3:[˙(N;2); q˙(N;2); z˙; S˙(N;2)i ; d˙(N;2); "˙(N;2)cum]=moyi zy Eulzr inxrzmznt“(; q; z; hi; y;c; i˙ ; "˙)Step Description1 Compute elastic trial and corresponding void ratio:˙el = Delsec(; z; i˙ "˙):(i˙ "˙); el =  + ˙el;z˙ = (1 + z) tr(i˙ "˙); ztr = z− z˙:2 Check for elastic shoot: Feln (; q;c) Q TOL and Feln(el; q;c)Q TOL?YES: set ˙(N;2) = ˙el, q˙(N;2) = E, h(N;2)i = 0, y˙(N;2) = 0, "˙(N;2)cum = 0 and EXIT;NO:GO TO styp G.3 Check for crossing elastic nucleus: Feln (; q;c) Q TOL?YES: set[int; zint; i˙ el; "˙ep]= zlv“tix nuxlzu“ intzr“zxtion “tvtz(; q; z;c; i˙ ; "˙; ˙el);NO: set[int; zint; i˙ el; "˙ep]= [; z; 0; "˙].4 Set j = 1 and initialize state for modified Euler elastoplastic increments:[(j); q(j); z(j); h(j)i ; y(j);(j)c ; "˙(j);]=(int; q; zint; hi; y;c; "˙ep).5 Perform modified Euler elastoplastic increments:[˙(j); q˙(j); z˙; S˙(j)i ; d˙(j); "˙(j)cum]= zlv“toplv“tix inxrzmznt((j); q(j); z(j); h(j)i ; y(j);(j)c ; "˙(j)).6 Check if j = 1?YES: set j = 2, and(j) = (j−N) + ˙(j−N);q(j) = q(j−N) + q˙(j−N);z(j) = z− z˙;h(j)i = h(j−N)i + h˙(j−N)i ;y(j) = y(j−N) − y˙(j−N);(j)c = projzxtion xzntzr zvolution((j−N)c ; q(j−N); q˙(j−N))and GO TO styp I.NO: EXIT;ICGCG Elv“tix rz“pon“zThe stress increment lying ˙ within the elastic nucleus yields purely elastic repose. However,the elastic nucleus is only implicitly defined in the model formulation by the location of theprojection center c and the radius of the elastic nucleus s. Below, an equation definingelastic nucleus Feln = 0 is obtained first, and then an iterative solution of intersection stressint on the elastic nucleus is provided. With the int, a strain intersection factor v" requiredfor splitting applied strain increment "˙ into elastic and elastoplastic parts is then obtained.Xomputing zxplixit zquvtion of thz zlv“tix nuxlzu“In the SANICLAY-B, the elastic nucleus is implicitly defined via a model parameter s whichenters the equation for the plastic modulusKp = K¯p +h p3M〈bR(b− 1)− s〉 (4.12)101HBFB Wonstitutivy xrivyrwhere h is a hardening function, and b is a ratio between the bounding F = 0 and loadingf = 0 surfaces. Whenever the expression within the Macaulay brackets is less or equalto zero, the denominator becomes zero and the second term of Eq. 4.12 is infinitely large,rendering fully elastic response. Thus, the elastic nucleus confines a region in the stress spacefor which the following holds truebb− 1 ≤ s (4.13)Equating the above equation renders beln that is a ratio between the bounding surface F = 0and the elastic nucleus Feln = 0beln =ss− 1 (4.14)The elastic nucleus, just like the loading surface, is homologous to the bounding surfacewith the projection center being the center of homology. Therefore, a radial mapping rulecan be exploited in order to define the elastic nucleus with the value of beln given in Eq. 4.14p¯ = pc + beln (p− pc) (4.15)“ = “c + beln (“− “c) (4.16)where p¯ and “ are the images of p and “ on the bounding surface, pc and “c are the p and “at the projection center.The origin of the elastic nucleus can now be evaluated using Eqs 4.15 and 4.16, givenp¯ = 0 and “ =#0 (where#0 denotes a tensor with all zero components) are the image p and“ at the origin of the stress spacepor = pc(1− 1beln)(4.17)“or = “c(1− 1beln)(4.18)To obtain the size of the elastic nucleus, first the p on Feln = 0 that corresponds to theimage stress at the tip of the bounding surface, where p¯ = pM, is computed asp =pM − pcbeln+ pc (4.19)102HBFB Wonstitutivy xrivyrGiven values of p at the origin of the elastic nucleus and at the tip of the elastic nucleus inEqs 4.17 and 4.19, respectively, the size of the Elastic nucleus is obtained aspM;eln = p− por=(pM − pcbeln+ pc)− pc(1− 1beln)=pMbeln(4.20)Notice that beln is the ratio between the size pM of the bounding surface and the size pM;eln ofthe elastic nucleus.With the help of Eqs 4.17, 4.18 and 4.20 the equation for elastic nucleus is defined asFeln =32((“− “or)− (p− por) ) : ((“− “or)− (p− por) )−(N2 − 32 : )(p− por)(pMbeln− (p− por))= 0 (4.21)which is similar to the equation of the bounding surface with the adjustments required toaccount for the shifted origin and reduced size of the elastic nucleus.In Eq. 4.21, the N is interpolated between its values in triaxial compression Nc andtriaxial extension Ne by means of the Lode angle Θ as follows:N =2n(1 + n)− (1− n) cos 3Θ; n =NeNc(4.22a)cos 3Θ =√6 trn3; n =r′ − [(r′ − ) : (r′ − )]N=2(4.22b)where the Lode angle is defined along r′ − with the stress-ratio r′ = rc + (r− rc) belndenoting an image stress ratio that lies inside the bounding surface when r is inside theelastic nucleus, and on the bounding surface when r is on the elastic nucleus. Due to thehomology between the elastic nucleus and the bounding surfaces, the Lode angle of theformer for the stress-ratio r coincides with the Lode angle of the latter for the correspondingr′ as illustrated in Fig. 4.4.103HBFB Wonstitutivy xrivyrrNr2 r3F = 0Feln = 0 rcrr′nnΘΘFigure 4.4: Illustration of the Lode angle for the elastic nucleus coinciding with the Lodeangle for the bounding surface in the deviatoric stress ratio planeFinying intzr“zxtion “trz““ of thz zlv“tix nuxlzu“In each step of the numerical integration it is checked if the initial stress  and the predictedstress pred for a given strain increment "˙ are within the elastic nucleus. If the stressesare within the elastic nucleus, then the increment is accepted as elastic. If not, then anintersection stress int on the elastic nucleus is located with the procedure described belowand schematically illustrated in Fig. 4.5.For any stress state  within the elastic nucleus and the predicted stress pred lyingoutside the elastic nucleus, the stress state int at the intersection of the elastic nucleuscan be obtained using the projection center pc that, in fact, is the center of homologybetween the bounding surface and the elastic nucleus. For convenience, two stress tensorsare introduced. The first one is the difference between the current stress state and theprojection center˙a =  − c (4.23)and the second one is the difference between the stress state at the intersection of the elastic104HBFB Wonstitutivy xrivyrF = 0¯cFeln = 0intpred˙pred˙b˙aFigure 4.5: Schematic diagram for finding intnucleus and the projection center˙b = int − c (4.24)To solve for the int, the ˙b can be expressed as a combination of the ˙a and an increment˙pred used to obtain the pred˙b = ˙a + v˙pred (4.25)where v is unknown stress intersection factor in int =  + v˙pred.The image stress radially mapped from int is given by¯ = c +(int − c)beln (4.26)where beln = sR (s− 1) and s S 1 is an indirect measure of elastic nucleus. Noting that theparenthesis could be replaced by Eq. 4.24 which in turn can be substitute by Eq. 4.25, imagestress in Eq. 4.26 is expressed as¯ = c +(˙a + v˙pred)beln (4.27)105HBFB Wonstitutivy xrivyrwhich can be decomposed into its hydrostatic p¯I and deviatoric counterparts “¯“¯ = “c +(“˙a + v“˙pred)belnp¯ = pc +(p˙a + vp˙pred)beln(4.28)The image stress counterparts are plugged into the bounding surface equation and thev is obtained from a quadratic equationVv2 +Bv + X = 0 (4.29)whereV = VNR(N2 − 32 : )+ V2;B = BNR(N2 − 32 : )+B2;X = XNR(N2 − 32 : )+ X2;(4.30)withVN =32b2eln[“˙pred − p˙pred ] : [“˙pred − p˙pred ] ;BN = 3 beln [(“c + “˙abeln)− (pc + p˙abeln) ] :[“˙pred − p˙pred ] ;XN =32[(“c + “˙abeln)− (pc + p˙abeln) ] : [(“c + “˙abeln)− (pc + p˙abeln) ] ;V2 = b2eln(p˙pred)2;B2 = beln [2 (pc + p˙abeln)− pM] p˙pred;X2 = − (pc + p˙abeln) [pM − (pc + p˙abeln)] :(4.31)The maximum root of Eq. 4.29 is of interestv =−B +√B2 + 4VX2V(4.32)which is discussed for two possible cases. In the first case, the current stress  is inside theelastic nucleus as shown in Fig. 4.6(a). For this case, one of the roots is positive and theother one is negative. Clearly, the int is along the ˙pred, i.e., the scalar multiple of ˙pred106HBFB Wonstitutivy xrivyrmust be positive to obtain int thereby the maximum root of the equation is the solution.In the second case, the current stress  is outside the elastic nucleus as shown in Fig. 4.6(b)and the ˙pred crosses the elastic nucleus at two stress points. For this case, both roots of theequation are positive with the minimum root corresponding to the first instance of crossingthe elastic nucleus while the maximum root corresponding to the second instance. As theonset of plastic response corresponds to the state when the ˙pred is pointing outside of theelastic nucleus, the second instance is of interest here. It should be noted that as the elasticFeln = 0int (v;max S 0)int (v;min Q 0)pred˙pred=v)Feln = 0int (v;max S 0)int (v;min S 0)pred˙pred=w)Figure 4.6: Illustration of correct solution for v using maximum root for: (a)  insideFeln = 0, and (b)  outside Feln = 0nucleus is homologous to the bounding surface, the Lode angle dependency applies to theformer as well. Therefore, iterative solution for solving value of intis required.At the last step, int from Eq. 4.24 is obtained asint = c +(˙a + vpred): (4.33)A similar approach could be used for computation of int of other SANICLAY family ofmodels. A center of homology in this case would be the origin of the stress space.107HBFB Wonstitutivy xrivyrsNs2 s3F = 0“ “c““¯ncFigure 4.7: Illustration of the Lode angle dependency of the bounding surface and, con-sequently, the elastic nucleus in the deviatoric stress plane requiring iterative solution forintItzrvtivz “olution for intzr“zxtion “trz““ of thz zlv“tix nuxlzu“Solution of the intersection stress int on the elastic nucleus requires locating the imagestress ¯ on the bounding surface according to Eq. 4.26. The image stress cannot be solvedanalytically as it requires the image stress to be known in advance in order to define Lodeangle along nc = (“¯− “ ) R ‖“¯− “ ‖ as shown in Fig. 4.7. Therefore, an iterative solutionV isproposed next which is then followed by its numerical implementation with a minimizationof an error function using the Pegasus method presented by Dowell and Jarratt (1972).As the image stress ¯ is not known in advance, its important to set the boundaries withinwhich the image stress could lie. Such boundaries are shown in Fig. 4.8 in the deviatoricstress plane along the red line crossing the bounding surface. These boundaries are givenby the predefined Ne ≤ N ≤ Nc range. Selecting a trial value N tr, a corresponding trialintersection stress int,tr can be computed using Eqs. 4.23–4.33. For a trial image stress ¯trcorresponding to the int,tr, a trial Lode angle Θtr can be established along the correspondingunit tensor ntrc = (“¯tr − “ ) R ‖“¯tr − “ ‖. The trial Lode angle Θtr is then used to computethe corresponding value of N . And the Pegasus method is then adopted for the minimizationof the difference between trial N tr and computed Nfc = N tr −N (4.34)9this solution is vlso follofizy for v wounying surfvxz intzrszxtion fvxtor acrocc vny thz rvtio b108HBFB Wonstitutivy xrivyrsNs2 s3F (N (Θ)) = 0“ “c““¯ (?)nc (?)Figure 4.8: Illustration of the trial N values established along the trial nc for the iterativesolution of the intersection stress intWhen such minimization is achieved, the intersection stress int is set to its latest trial int,tr.The complete procedure implemented for finding the intersection stress int on the elasticnucleus is given in Tables 4.4, 4.5, 4.6, and 4.7. In styp E of Table 4.4, the int is obtainedusing the iterative procedure described in Table 4.5. The initial values of N are first setfor trials i − 1 and i, and the corresponding inti−N and inti are obtained using Eqs. 4.23–4.33 as described in the procedure given in Table 4.6. Then the obtained inti−N and intiare used in Table 4.7 to compute the differences fc between the trial N tr values, Ni−N andNi, and the N values corresponding to the Lode angles at the computed inti−N and inti ,respectively. For the trial i+1, the value of N tr is computed using interpolation function ofits initial trial values, Ni−N and Ni, and the corresponding differences fc . A difference fci+Nis then obtained and checked to be within a specified tolerance TOL. If within the toleranceor exceeding a specified maximum number of iterations MAXITS, the inti+N correspondingto Ni+N is accepted as the final value. If not, then further iterations are performed withcertain adjustments made following the Pegasus method. For the obtained int, the strainintersection factor v" is computed in styp F of Table 4.4 using the procedure described below.Given the v", the corresponding pseudo-time i˙e, void ratio zint at the intersection stress, andthe strain increment "˙ep required for elastoplastic computations are also computed in stypsGAI of Table 4.4.109HBFB Wonstitutivy xrivyrTable 4.4:[int; zint; i˙ e; "˙ep]= zlv“tix nuxlzu“ intzr“zxtion “tvtz(; q; z;c; i˙ ; "˙; ˙pred)Step Description1 Compute stress at the intersection of elastic nucleus and elastic stress increment:int = intzr“zxtion “trz““(; q;c; ˙pred)2 Compute intersection factor:v" = “trvin intzr“zxtion fvxtor(; "˙;int).3 Compute elastic pseudo-time increment i˙ pred :i˙ pred = v"i˙4 Compute void ratio corresponding to int :zint = z− (1 + z) tr(i˙ pred"˙)5 Compute elastoplastic strain increment:"˙ep = (1− v") "˙Table 4.5: int = intzr“zxtion “trz““(; q;c; ˙pred)Step Description1 Set ncount = 0, set Ni−N = Ne, Ni = Nc and compute corresponding fc values:inti−N = trivl intzr“zxtion “trz““(Ni−N;; q;c; ˙pred)inti = trivl intzr“zxtion “trz““(Ni;; q;c; ˙pred)fci−N = N yiffzrznxz for zlv“tix nuxlzu“ intzr“zxtion(Ni−N;inti−N;c)fci = N yiffzrznxz for zlv“tix nuxlzu“ intzr“zxtion(Ni;inti ;c)2 For Ni+N =(Nifci−N −Ni−Nfci)R(fci−N − fci), compute fci+N:inti+N = trivl intzr“zxtion “trz““(Ni+N;; q;c; ˙pred)fci+N = rvtio N yiffzrznxz for zlv“tix nuxlzu“ intzr“zxtion(Ni+N;inti+N;c)3 Check if: fci+N Q TOL?YES: EXIT,NO: go to styp H.4 Perform the following operations:If fci+N fci Q 0:set Ni−N = Ni, and fci−N = fci .If fci+N fci S 0:set fci−N = fci−Nfcifci + fci+N.Set Ni = Ni+N, fci = fci+N and Ni+N =(Nifci−N −Ni−Nfci)R(fci−N − fci).5 Set ncount = ncount + 1 and compute new values of inti+N and fci+N as in styp F.6 Check if: fci+N S TOL or ncount ≤ MAXITS?YES: return to styp H,NO: set int = inti+N and EXIT.110HBFB Wonstitutivy xrivyrTable 4.6: int,tr = trivl intzr“zxtion “trz““(N tr;; q;c; ˙pred)Step Description1 Compute stress difference between current stress and projection center, and ratio beln:˙a =  − c, beln = sR (s− 1).2 Compute coefficients of a quadratic equation VN (v)2 +BNv + X = 0:VN =32b2eln[“˙pred − p˙pred ] : [“˙pred − p˙pred ] ;BN = 3 beln [(“c + “˙abeln)− (pc + p˙abeln) ] :[“˙pred − p˙pred ] ;XN =32[(“c + “˙abeln)− (pc + p˙abeln) ] : [(“c + “˙abeln)− (pc + p˙abeln) ] ;3 Compute V2, B2, and X2:V2 = b2eln(p˙pred)2, B2 = beln [2 (pc + p˙abeln)− pM] p˙pred, andX2 = − (pc + p˙abeln) [pM − (pc + p˙abeln)] :4 Compute V, B, and X:V = VNR((N tr)2 − 32 : )+ V2, B = BNR((N tr)2 − 32 : )+B2, andX = XNR((N tr)2 − 32 : )+ X2.5 Compute the v as a maximum root of the quadratic equation:v =−B +√B2 − 4VX2V6 Compute int,tr:int,tr = c +(˙a + v˙pred):Table 4.7: fc = rvtio N yiffzrznxz for zlv“tix nuxlzu“ intzr“zxtion(N tr;int,tr;c)Step Description1 Compute image stress ¯tr for a given int,tr value:¯tr = c +(int,tr − c)beln2 Given “ = pc , compute corresponding unit tensor ntrc :ntrc =“¯tr − “ ‖“¯tr − “ ‖3 For the unit tensor ntrc , compute corresponding value of N :cos (3Θtr) =√6tr (ntrc )3N =2n(1 + n)− (1− n) cos (3Θtr) ; n =NeNc4 Compute the difference:fc = N tr −NFinying “trvin intzr“zxtion fvxtorGiven the int, a purely elastic stress increment with the secant elastic modulus isint −  = Desec : v""˙ (4.35)where v" is the strain intersection factor that needs to be found.111HBFB Wonstitutivy xrivyrIn Desec, the secant bulk modulus Ksec can be factored out as followsDesec = Ksec1 + 4fR3 1− 2fR3 1− 2fR3 0 0 01− 2fR3 1 + 4fR3 1− 2fR3 0 0 01− 2fR3 1− 2fR3 1 + 4fR3 0 0 00 0 0 2f 0 00 0 0 0 2f 00 0 0 0 0 2f(4.36)where Ksec =pv""˙v[exp((1 + z) v""˙v)− 1]and f =3 (1− 2,)2 (1 + ,).As the Ksec includes the intersection factor v" in the denominator, Eq. 4.35 can be re-written, replacing Ksec outside of the parenthesisint −  = p"˙v[exp((1 + z) v""˙v)− 1](1KsecDesec : "˙)(4.37)The strain intersection factor v" is then obtained from Eq. 4.37 asv" =(1 + z) "˙vln ‖int − ‖‖ 1KsecDesec : "˙‖"˙vp+ 1 (4.38)Notice that Eq. 4.38 cannot explicitly define value of intersection factor v" when volumetricstrain increment "˙v = 0. In order to define a limiting value of v", l’Hospital’s rule can be112HBFB Wonstitutivy xrivyrused yielding the following:lim_"v→Mv" = lim_"v→My ("˙v)z ("˙v)= lim_"v→M@y ("˙v) R@"˙v@z ("˙v) R@"˙v= lim_"v→M (1 + z)‖int − ‖‖ 1KsecDesec : "˙‖1p R ‖int − ‖‖ 1KsecDesec : "˙‖"˙vp+ 1=(1 + z)‖int − ‖‖ 1KsecDesec : "˙‖1p(4.39)where y ("˙v) =  ln ‖int − ‖‖ 1KsecDesec : "˙‖"˙vp+ 1 and z ("˙v) = (1 + z) "˙v.Complete procedure implemented for computing the v" is summarized in Table 4.8.Table 4.8: v" = “trvin intzr“zxtion fvxtor(; "˙;int)Step Description1 Set matrix1KsecDesec:1KsecDesec =1 + 4fR3 1− 2fR3 1− 2fR3 0 0 01− 2fR3 1 + 4fR3 1− 2fR3 0 0 01− 2fR3 1− 2fR3 1 + 4fR3 0 0 00 0 0 2f 0 00 0 0 0 2f 00 0 0 0 0 2fwith f =3 (1− 2,)2 (1 + ,)2 Compute volumetric strain increment:"˙v = tr ("˙)3 Check if "˙v ≤ EPS (machine epsilon)?YES: compute limiting value v" =(1 + z)‖int − ‖‖ 1KsecDesec : "˙‖1p;NO: v" =(1 + z) "˙vln ‖int − ‖‖ 1KsecDesec : "˙‖"˙vp+ 1.113HBFB Wonstitutivy xrivyrICGCH Elv“toplv“tix rz“pon“zNumerical integration of SANICLAY-B constitutive equations follow the return algorithmgiven in Eq. 4.1 that applies a plastic corrector of an initially assumed elastic predictor.The algorithm is implemented with the details given in Table 4.9. The ratio b is computedstyp E for the initial stress state . In styp F, assumed elastic response is estimated withthe computed pred. In order to avoid non-uniqueness of the image stress when the initialstress state  coincides with the projection center c, additional measures are undertakenin styps G to 5. In styp G, the detection of the case is first confirmed so that the step is tobe simulated as elastic. In styp H, it is checked if the assumed elastic response crosses thebounding surface. If so, then the cross stress cross on the bounding surface is located usingan estimated bounding surface intersection factor vcross whose solution is given in Table4.10. If not, then the step is assumed elastic with the state variable increments updatedaccordingly in styp L. Following the computation of the vcross and the cross in styp H, itis further checked in styp I if the initial stress state is on the bounding surface within agiven tolerance TOL of the computed increment ˙cross. If so, then the response is assumedelastoplastic with the ratio b set to 1:0 as it cannot be computed otherwise for  = c.If not, then the state variables are updated in accordance with the cross assuming elasticresponse for "˙e = vcross"˙ and elastoplastic response for "˙p = (1− vcross) "˙ stain increments.The rest of the pseudo-code follows the computational procedure described in the modelformulation.114HBFB Wonstitutivy xrivyrTable 4.9:[˙; q˙; z˙; S˙i; d˙; "˙cum]= zlv“toplv“tix inxrzmznt (; q; z; hi; y;c; "˙)Step Description1 Compute the ratio b for the initial stress state:b = rvtio b (; q;c).2 Set ˙cross = E, and compute predicted stress pred assuming elastic response (prior to the plasticcorrection):˙pred = De (; z) : "˙, pred =  + ˙pred.3 Check if the stress state coincides with the projection center: ‖ − c‖ Q TOL?NO: GO TO styp J.4 Check for crossing the bounding surface: ‖F () ‖ Q TOL and F (pred) S 0?YES: compute vcross = wounying “urfvxz intzr“zxtion fvxtor(; q; ˙pred), and˙cross = vcross˙pred;NO: set a = 0, ("˙p) = E and GO TO styp L.5 Check if the stress state is on the bounding surface: ‖˙cross‖ Q TOL?YES: set b = 1;NO: set  =  + ˙cross, z = z− vcross (1 + z) tr ("), "˙ = (1− vcross) "˙and recompute b = rvtio b (; q;c).6 Compute the image stress:¯ = c + ( − c) b.7 Compute plastic modulus, loading index and plastic strain increment:h = hMR (1 + y), Kp = −@F@q: q¯ + h (pM)3 R〈bR (b− 1)− s〉;a =@F@¯: De (; z) : "˙R(@F@¯: De (; z) :@g@¯+Kp);("˙p) = a@g@¯.8 Record trial increments for all state variables:˙ = De (; z) : ("˙− ("˙p)) + ˙cross;q˙ = aq¯; h˙i = ah¯i; y˙ = vd [(2R3) ( tzp) : ( tzp)]N=2 ;"˙cum = ‖"˙− ("˙p) ‖(− @f@: ˙predR(‖ @f@‖‖˙pred‖)).Itzrvtivz “olution for wounying “urfvxz intzr“zxtion fvxtorThe iterative procedure given in Table 4.10 for solution of the intersection factor vcross followsthe proposed solution for the computation of the int in section 4.2.2. Computation of atrial intersection factor vcross,tr using trial N tr value is performed in Table 4.11. The solutionis obtained by substituting “¯ = “ + vcross,tr(“˙pred), p¯ = p + vcross,tr(p˙pred)and N = N tr inEq. 3.12a. The difference between trial N tr value and the N value corresponding to the Lodeangle at the computed trial cross,tr is estimated in Table 4.12.115HBFB Wonstitutivy xrivyrTable 4.10: vcross = wounying “urfvxz intzr“zxtion fvxtor(; q; ˙pred)Step Description1 Set ncount = 0, set Ni−N = Ne, Ni = Nc and compute corresponding fc values:vcrossi−N = trivl wounying “urfvxz intzr“zxtion fvxtor(Ni−N;; q; ˙pred)vcrossi = trivl wounying “urfvxz intzr“zxtion fvxtor(Ni;; q; ˙pred)fci−N = rvtio N yiffzrznxz for wounying “urfvxz xro““(Ni−N; + vcrossi−N ˙pred)fci = rvtio N yiffzrznxz for wounying “urfvxz xro““(Ni; + vcrossi ˙pred)2 For Ni+N =(Nifci−N −Ni−Nfci)R(fci−N − fci), compute fci+N:vcrossi+N = trivl wounying “urfvxz intzr“zxtion fvxtor(Ni+N;; q; ˙pred)fci+N = rvtio N yiffzrznxz for wounying “urfvxz xro““(Ni+N; + vcrossi+N ˙pred)3 Check if: fci+N Q TOL?YES: EXIT,NO: go to styp H.4 Perform the following operations:If fci+N fci Q 0:set Ni−N = Ni, and fci−N = fci .If fci+N fci S 0:set fci−N = fci−Nfcifci + fci+N.Set Ni = Ni+N, fci = fci+N and Ni+N =(Nifci−N −Ni−Nfci)R(fci−N − fci).5 Set ncount = ncount + 1 and compute new values of vcrossi+N and fci+N as in styp F.6 Check if: fci+N S TOL or ncount ≤ MAXITS?YES: return to styp H,NO: set vcross = vcrossi+N and EXIT.Table 4.11: vcross,tr = trivl wounying “urfvxz intzr“zxtion fvxtor(N tr;; q; ˙pred)Step Description1 Compute coefficients of a quadratic equation V (vcross)2 +Bvcross + X = 0:V = 32(“˙pred − p˙pred ) : (“˙pred − p˙pred )+ (p˙pred)2 ((N tr)2 − 32 : )B = 3 (“− p ) : (“˙pred − p˙pred )− p˙pred (pM − 2 p)((N tr)2 − 32 : )X = 32(“− p ) : (“− p )− p (pM − p)((N tr)2 − 32 : )2 Compute vcross,tr as a maximum root of the quadratic equation:vcross,tr =−B +√B2 − 4VXV2Table 4.12: fc = rvtio N yiffzrznxz for wounying “urfvxz xro““ (N tr;cross,tr)Step Description1 Given “ = “cross,tr and “ = pc , compute corresponding unit tensor ntrc :ntrc =“− “ ‖“− “ ‖2 For the unit tensor ntrc , compute corresponding value of N :cos (3Θtr) =√6tr (ntrc )3N =2n(1 + n)− (1− n) cos (3Θtr) ; n =NeNc3 Compute the difference:fc = N tr −N116HBFB Wonstitutivy xrivyrItzrvtivz “olution for rvtio bThe iterative procedure given in Table 4.13 for solution of the ratio b follows the proposedsolution for the computation of the int in section 4.2.2. Computation of a trial ratio btr usingtrial N tr value is performed using Eqs. 3.15–3.29 as followed in Table 4.14. The differencebetween trial N tr value and the N value corresponding to the Lode angle at the trial imagestress ¯tr corresponding to the trial ratio btr is estimated in Table 4.15.Table 4.13: b = rvtio b (; q;c)Step Description1 Set ncount = 0 and set trial N values:Ni−N = Ne, Ni = Nc2 Compute corresponding b values:bi−N = trivl rvtio b (Ni−N;; q;c), andbi = trivl rvtio b (Ni;; q;c)3 Compute corresponding differences fc :fci−N = rvtio N yiffzrznxz for imvgz “trz““ (Ni−N; bi−N;;c), andfci = rvtio N yiffzrznxz for imvgz “trz““ (Ni; bi;;c)4 For linearly interpolated Ni+N = (Nifi−N −Ni−Nfi) R (fi−N − fi), compute b and fc :bi+i = trivl rvtio b (Ni+N;; q;c),fci+i = rvtio N yiffzrznxz for imvgz “trz““ (Ni+i; bi+i;;c).5 Check if: fci+N Q TOL?YES: set b = bi+N and EXIT,NO: go to styp J.6 Perform the following operations:If fci+N fci Q 0:set Ni−N = Ni, and fc;i−N = fc;i.If fci+N fci S 0:set fci−N = fci−Nfcifci + fci+N.Set Ni = Ni+N, fci = fci+N and Ni+N =(Nifci−N −Ni−Nfci)R(fci−N − fci).7 Set ncount = ncount + 1 and compute new values for bi+N, and fci+N as in styp H.8 Check if: fci+N S TOL?YES: return to styp J,NO: set b = bi+N and EXIT.117HBFB Wonstitutivy xrivyrTable 4.14: btr = trivl rvtio b (N tr;; q;c)Step Description1 Compute VN, BN, and XN:VN =32(s− sc) : (s− sc)− 3 (p− pc) (s− sc) : + 32(p− pc)2 : ;BN = 3 (s− sc) : sc − 3 (pcs+ (p− 2pc) sc) : + 3pc (p− pc) : ;XN =32sc : sc − 3pcsc : + 32p2c : :2 Compute V2, B2, and X2:V2 = (p− pc)2, B2 = (2pc − pM) (p− pc), X2 = pc (pc − pM).3 Compute coefficients for a quadratic equation V (btr)2+Bbtr + X = 0:V = VNR((N tr)2 − 32 : )+ V2, B = BNR((N tr)2 − 32 : )+B2, andX = XNR((N tr)2 − 32 : )+ X2.4 Compute the ratio btr as a maximum root of the quadratic equation:btr =−B +√B2 − 4VX2V.Table 4.15: fc = rvtio N yiffzrznxz for imvgz “trz““ (N tr; btr;;c)Step Description1 Compute trial image stress ¯tr for a given btr value:¯tr = c + ( − c) btr2 Given “ , compute corresponding unit tensor ntrc :ntrc =“¯tr − “ ‖“¯tr − “ ‖3 For the unit tensor ntrc , compute corresponding value of N :cos (3Θtr) =√6tr (ntrc )3N =2n(1 + n)− (1− n) cos (3Θtr) ; n =NeNc4 Compute the difference:fc = N tr −NEvolution of projzxtion xzntzrEvolution of the projection center given by its hydrostatic and deviatoric counterparts inEq. 3.37 and 3.38 of the model formulation, respectively, is implemented with the straight-forward procedure shown in Table 4.16.118HBFB Wonstitutivy xrivyrTable 4.16: c = projzxtion xzntzr zvolution (c; q; q˙)Step Description1 Compute projection of c onto :“ = pc :2 Compute unit tensor along “c − “ :nc =“c − “ ‖“c − “ ‖ :3 Compute a ratio of the projection center relative position:m =[32(“c − “ ) : (“c − “ )]N=2[32(“b − “ ) : (“b − “ )]N=24 Compute complete evolution of the projection center hydrostatic and deviatoric counter-parts:pc = pc +p˙MpMpc;“c = “c +p˙MpM“c +pc ˙−m√23pc (pM − pc) 32 : ˙[(N2 − 32 : )pc (pM − pc)]N=2 nc :Yrift xorrzxtionThe drift correction adopted from Sloan et al. (2001) considers a combination of a consistentand a normal drift correction. For the consistent drift correction, the stress state is correctedalong with the internal variables. For the normal drift correction, the stress state is enforcedalong the normal to the bounding surface without the correction of the internal variables.Within each iteration, the former drift correction is used first and whenever it diverges, thelatter is used. Such a combination of the consistent and the normal drift correction methodsis adopted and implemented with the procedure shown in Table 4.17.119HBGB Linyurizyx wonstruints oz luvorutory tystsTable 4.17: [; q; hi; y;c] = yrift xorrzxtion (; q; z; hi; y;c)Step Description1 Set count = 0.2 Check if: (‖F (; q)‖ S TOL) OR (count ≤ MAXITS) ?YES: GO TO styp G,NO: EXIT3 Compute a plastic multiplier and corresponding plastic strain increment:@l = F (; q) R(@F@: D (; z) :@g@− @F@q: q¯)."˙p = @l@g@4 Compute trial stress state and internal variables:tr =  − De (; z) : "˙p; qtr = q + @l q¯;htri = hi + @l h¯i; ytr = y− vd [(2R3) tzp : tzp]N=2.5 Check if ‖F (tr; qtr)‖ S ‖F (; q)‖?YES: GO TO styps J and L,NO: GO TO styps K and L.6 Perform normal drift correction:@l = F (; q) R(@F@:@g@) =  − @l @F@7 Update projection center and accept trial stress state and internal state variables fromconsistent drift correction:c = projzxtion xzntzr zvolution (c; q; @l q¯); = tr; q = qtr; hi = htri ; y = ytr.8 Set count = count + 1 and GO TO styp F.ICH ainzvrizzy xon“trvint“ of lvworvtory tz“t“The constitutive driver presented in section 4.2 numerically integrates SANICLAY-B con-stitutive equations for any input strain increment "˙. To simulate laboratory element tests,however, appropriate loading constraints must be specified, thereby providing input strainincrement "˙ that can be passed to the constitutive driver. One approach to obtain loadingconstraints is the use of the constitutive driver in finite element or finite difference code. Inthe approach by Alawaji et al. (1992), constitutive equations are assembled into a set of lin-earized algebraic equations and solved in an iterative manner for prescribed mixed controlledconstraints. As current constitutive driver is written for its implementation into a numericalplatform so that it outputs stresses for given strain increments, this method would requireforming another constitutive driver compatible with it. Another approach employed in thissection follows the numerical technique described by Bardet and Choucair (1991). This tech-120HBGB Linyurizyx wonstruints oz luvorutory tystsnique allows the user to input any type of loading increments (stress, strain, or combinationsof stress and strain) and convert them into the increments of strain passed to the consti-tutive driver. It linearizes the loading constraints of laboratory experiments, links them toa constitutive relation, and forms a linear system of ordinary differential equations. Theemployed method does not compete with the efficient stress calculation algorithms of finiteelement or finite difference methods but rather provides bench-mark solutions to calibratethem (Bardet and Choucair, 1991). This approach pertains to most laboratory experimentsof geomechanics and is very useful for the research on material modeling, engineering practiceand computational mechanics.ICHCF ainzvrizzy xon“trvint“ vny yriving vvrivwlz“The principle of linearized loading constraints are described by Bardet and Choucair (1991).Here the procedure is followed with the difference that more than one driving variable isallowed thereby allowing simulation of more complex laboratory tests (e.g., multi-directionalshear test). There are twelve unknowns - six for ˙ and six for "˙. Six linear relations areobtained by linearizing the loading constraints of laboratory experiments. These relationsare assembled into six linear relations and form a system of twelve linear equations thatdetermine twelve unknowns.The constraints during a loading increment are described by 6− n linear relationsS∑j=N(hij˙j + Eij˙j) = 0; i = 1; :::; 6− n (4.40)where hij and Eij are constant during a loading increment but may vary from one loadingincrement to another in order to describe non-linear constraints, and n is a number ofloading variables m˙N; ::m˙n: that control the evolution of a loading process. The remaining nrelationships are linear combinations of stress and strain incrementsS∑j=N(hij˙j + Eij˙j) = m˙i; i = (6− n) + 1; :::; 6 (4.41)121HBGB Linyurizyx wonstruints oz luvorutory tystsCombining Eqs. 4.40 and 4.41 yieldsS∑j=N(hij˙j + Eij˙j) = n˙i; i = 1; :::; 6 (4.42)or in abbreviated notationh t + E t" = tn (4.43)where tn =[0 ::: 0 m˙N ::: m˙n]qis a six component vector.Bardet and Choucair (1991) proposes two solution techniques: one for stress- and an-other for strain-controlled testing. As the SANICLAY-B constitutive driver is devised toaccept strains as an input, a solution technique for strain-controlled testing is adopted whichsubstitutes t = D : t" in Eq. 4.43 yielding the following solution fort" = (hD+ E)−N tn (4.44)that is passed to the SANICLAY-B constitutive driver.ICHCG Vpplixvtion to “omz lvworvtory zlzmznt tz“t“An example of how to define h, E and tn is given next for a particular test and then asimilar procedure is followed for conventional and complex laboratory element tests used inthis dissertation. Consider the case of strain-controlled shearing under constant p triaxialloading on an isotropically consolidated specimen. This means that the confining pressurep = (NN + 22 + 22) R3 is kept constant, i.e., ˙NN + ˙22 + ˙22 = 0, while the axial strain "NNis gradually increased with a small increment of a single input m˙N until the specimen fails.The triaxial loading means that 22 = 33, i.e., ˙22 = ˙33, and shear stresses are absent, i.e.,˙N2 = 0, ˙N3 = 0, and ˙23 = 0. Assuming that the loading direction is 1, and that there isno shear stress within the sample, the loading constraints are summarized as˙NN + ˙22 + ˙22 = 0; ˙22 = ˙33 = 0; ˙N2 = 0; ˙N3 = 0 ˙23 = 0 (4.45)122HBGB Linyurizyx wonstruints oz luvorutory tystswhich assemble into h ˙ that together with E "˙, indicating driving strain increment, yieldthe following set of linear equations that are to be solved for "˙ (using Eq. 4.44):1 1 1 0 0 00 1 −1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 0 0 0 0˙NN˙22˙33˙N2˙N3˙23+0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 01 0 0 0 0 0"˙NN"˙22"˙33"˙N2"˙N3"˙23=00000m˙N(4.46)which is a test specific representation of h ˙ + E "˙ = n˙.htrvinBxontrollzy lvworvtory tz“t“Below, constraints for a number of conventional laboratory tests are given demonstratingversatility of the method. The tests include drained and undrained triaxial loading con-straints. In all tests considered here m˙N is a single driving strain increment applied until aspecimen fails.Isotropiw wonsolixution tyst. Driving variable: m˙N = "˙NN. Constraints: ˙NN − ˙22 = 0,˙22 − ˙33 = 0, ˙N2 = 0, ˙N3 = 0, ˙23 = 0. Therefore,h =1 −1 0 0 0 00 1 −1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 0 0 0 0; E =0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 01 0 0 0 0 0; tn =00000m˙N(4.47)KM wonsolixution tyst. Driving variable: m˙N = "˙NN. Constraints: "˙22 = 0, "˙33 = 0, ˙N2 = 0,123HBGB Linyurizyx wonstruints oz luvorutory tysts˙N3 = 0, ˙23 = 0. Therefore,h =0 0 0 0 0 00 0 0 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 0 0 0 0; E =0 1 0 0 0 00 0 1 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 01 0 0 0 0 0; tn =00000m˙N(4.48)Wonstunt{p triufiiul tyst. Driving variable: m˙N = "˙NN. Constraints: ˙NN + ˙22 + ˙33 = 0,˙22 − ˙33 = 0, ˙N2 = 0, ˙N3 = 0, ˙23 = 0. Therefore,h =1 1 1 0 0 00 1 −1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 0 0 0 0; E =0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 01 0 0 0 0 0; tn =00000m˙N(4.49)Wonvyntionul xruinyx triufiiul tyst. Driving variable: m˙N = "˙NN. Constraints: ˙22 = 0,˙33 = 0, ˙N2 = 0, ˙N3 = 0, ˙23 = 0. Therefore,h =0 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 0 0 0 0; E =0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 01 0 0 0 0 0; tn =00000m˙N(4.50)Wonvyntionul unxruinyx triufiiul tyst. Driving variable: m˙N = "˙NN. Constraints: "˙NN+ "˙22+124HBGB Linyurizyx wonstruints oz luvorutory tysts"˙33 = 0, ˙22 − ˙33 = 0, ˙N2 = 0, ˙N3 = 0, ˙23 = 0. Therefore,h =0 0 0 0 0 00 1 −1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 0 0 0 0; E =1 1 1 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 01 0 0 0 0 0; tn =00000m˙N(4.51)Druinyx simply shyur tyst. Driving variable: m˙N = "˙N2. Constraints: ˙NN = 0, ˙22 = 0,˙33 = 0, ˙N2 = 0, ˙N3 = 0. Therefore,h =1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 0 1 00 0 0 0 0 10 0 0 0 0 0; E =0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 1 0 0; tn =00000m˙N(4.52)inxruinyx simply shyur tyst. Driving variable: m˙N = "˙N2. Constraints: "˙NN = 0, "˙22 = 0,"˙33 = 0, ˙N2 = 0, ˙N3 = 0. Therefore,h =0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 0 0 0 0; E =1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 1 0 0; tn =00000m˙N(4.53)htrz““Bxontrollzy lvworvtory tz“t“In experiments and simulations it is preferred to apply displacements/strains as drivingvariables in order to capture softening response of a material. However, specific loading125HBGB Linyurizyx wonstruints oz luvorutory tystspaths are sometimes required to study material response subjected to certain stress paths.All tests considered here are driven by applying loading increments required to follow aspecified stress path.K wonsolixution tyst. Driving variables: m˙N = ˙NN, m˙2 = ˙N2. Constraints: "˙22 = 0,"˙33 = 0, "˙N3 = 0, "˙23 = 0. Therefore,h =0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 01 0 0 0 0 00 0 0 1 0 0; E =0 1 0 0 0 00 0 1 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 0 0 0 0; tn =0000m˙Nm˙2(4.54)In K consolidation test it is important to maintain desired value of K = N2RNN anddue to nonlinearity of constitutive equations applying increments ˙NN and ˙N2 alone cannotguarantee constant K . Therefore, additional measures that are generalized for all stress-controlled tests are to be described at the end of this section.inxruinyx multiAxirywtionul shyur tyst. Driving variables: m˙N = ˙N2 and m˙2 = ˙N3.Constraints: "˙NN = 0, "˙22 = 0, "˙33 = 0, "˙23 = 0. Therefore,h =0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 1 0 00 0 0 0 1 0; E =1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0; tn =0000m˙Nm˙2(4.55)Druinyx multiAxirywtionul shyur tyst. Driving variables: m˙N = ˙N2 and m˙2 = ˙N3. Con-126HBGB Linyurizyx wonstruints oz luvorutory tystsstraints: ˙NN = 0, "˙22 = 0, "˙33 = 0, "˙23 = 0. Therefore,h =1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 1 0 00 0 0 0 1 0; E =0 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0; tn =0000m˙Nm˙2(4.56)For stress-controlled multi-directional shear tests, shear stress histories N2 and N2 areassumed as sinusoidal functions:N2 = VN2 sin (vN2 ( + N2)) (4.57a)N3 = VN3 sin (vN3 ( + N3)) (4.57b)where VN2, VN3 are stress amplitudes; vN2, vN3 are frequencies; N2 is a phase angle forthe respective shear stress; and  is driving input angle. Various shear stress paths can beachieved using the equations above. Below are a few examples on variation in shear stresspath that can be achieved by proper adjustment of the input constants. y = N3 x = N2+VN2−VN2+VN2−VN2vN2 = 1, vN3 = 1N2 =.2 y = N3 x = N2+VN2−VN2+VN2−VN2vN2 = 1, vN3 = 2N2 = 0 y = N3 x = N2+VN2−VN2+VN2−VN2vN2 = 2, vN3 = 1N2 = 0Figure 4.9: Examples of multi-directional shear stress paths using a combination of Eq. 4.57In order to maintain the desired loading path, additional measures are to be taken.This applies to all stress-controlled tests considering nonlinear constitutive equations. The127HBHB Finity ylymynt implymyntutionidea is to use the Newton-Raphson iterative procedure to ensure that intended stress andconstitutive stress, for this purpose referred to as ext and int, respectively, are withinallowable tolerance NRTOL. Implemented iterative procedure for one step of loading isshown in Table 4.18. This procedure ensures that an intended loading is correctly solved ina constitutive model despite highly nonlinear constitutive equations and is called for eachloading step until final load is attained.Table 4.18: czfitongvph“on (; q; z; hi; y;c; "cum;ext)Step Description1 Set i = 0 and record initial stress state i and internal variables qi; zi; hi;i; yi;c;i; "cum;ifrom the last successful step.2 Set initial strain increment "˙i = E and initial internal stress int;i = i.3 Update i = i+ 1 and compute new ith stress state and internal variables:[int;i; qi; zi; hi;i; yi;c;i; "cum;i] = vutomvtix “uw“tzpping vny zrror xontrol(M; qM; zM; hi;M; yM;c;M; "cum;M; "˙i)4 Compute residual gNo = [(ext − int;i) : (ext − int;i)]N=2 and check if gNo S NRTOL?YES: GO TO styp I,NO: set [; q; z; hi; y;c; "cum] = [int;i; qi; zi; hi;i; yi;c;i; "cum;i] for next step and EXIT.5 Compute tangent modulus Di, corresponding to ith stress state and internal variables.6 Compute driving loading vector n˙ with loading variables from components of ext −int,i.7 Compute new increment t"i+N = t"i + (hDi + E)−N tn and GO TO styp G.ICI Finitz zlzmznt implzmzntvtionIn this section, a brief description of a finite element code used as an example of SANICLAY-B model numerical implementation is given. Then the procedure followed for interac-tion of the constitutive driver with Open System for Earthquake Engineering simulations(OpenSEES) finite element program (also applicable for other finite element platforms) ispresented.ICICF Gznzrvl yz“xription of dpznhEEh nitz zlzmznt xoyzNumerical solutions for complex boundary value problems are important tools in today’s en-gineering design. Finite element programs are one of the most attractive numerical solutionapproaches. With rapidly changing hardware, improving numerical computation algorithms& solvers, advancements in types of elements and development of more representative consti-tutive models for materials, it is essential for a finite element program to be able to adopt to128HBHB Finity ylymynt implymyntutionthese changes. The constitutive driver described in section 4.2 is implemented in OpenSEESfinite element program. The program is an object-oriented, open source software that al-lows the user to develop finite element computer applications for simulating the responseof structural and geotechnical systems subjected to earthquake excitation and other haz-ards. In OpenSEES, an object class defines specification of methods and internal data ofan object, and an abstract class defines specification of abstract methods and interface forits subclasses. The advantages of such object-oriented structure are in that it allows ananalyst to introduce new subclasses which greatly increases flexibility of the computationalcapabilities of the software. More details of OpenSees structure can be found in McKenna(1997). For description of porous media, OpenSEES elements adopt u-p version of finiteelement that can be used for simulating dynamic response of solid–pore fluid fully coupledmaterial, based on Biots theory of porous medium (Zienkiewicz and Shiomi, 1984).ICICG dpznhEEh intzrvxtion fiith xon“titutivz yrivzrKey methods required for OpenSEES interaction with the SANICLAY-B model are givennext. The SANICLAY-B model is implemented using NDMaterial abstraction which rep-resents multi-dimensional constitutive response. NDMaterial class interface that must beincluded in SANICLAY-B implementation is listed below in Listing 4.1 with the explanationof the interface methods described by Scott and Fenves (2001) as follows.The method “ztirivlhtrvin(tr") takes trial strain as an argument. Method instancewith two arguments is for strain rate dependent materials. As SANICLAY-B is rate-independent material the method with one argument is used for its implementation whichcomputes and passes tr"˙ = tr" − com" to “ztirivlhtrvinInxr(tr"˙). Similar to “ztirivlBhtrvin(tr"), “ztirivlhtrvinInxr(tr"˙) with one input argument is used for SANICLAY-Bimplementation. This method invokes SANICLAY-B numerical integration by passing tr"˙ tovutomvtix “uw“tzpping vny zrror xontrol in Table 4.2 computing trial stress tr for agiven trial strain increment tr"˙ and the last committed state. The current material tangentD corresponding to tr is returned by the method gztivngznt() required for constructionof global stiffness K. In SANICLAY-B implementation depending on tr and trial statevariables, D can be either elastic or elastoplastic. As can be noted gztInitivlivngznt() is129HBHB Finity ylymynt implymyntutionredirected to gztivngznt() and therefore does nothing other then that. The next methodis gzthtrz““() returning trial stress tr which is a function of the trial strain increment tr"˙and last committed stress com. The gzthtrvin() method retrieves last committed straincom".All Material objects in OpenSEES are responsible for keeping track of their own historyvariables. First the method xommithtvtz() is invoked to inform the NDMaterial objectthat its current state is on the converged solution path, and its internal history variablesshould be updated accordingly. Next the method rzvzrtioav“tXommit() is provided tolet an NDMaterial object know that it should return to its last committed state. Finallyrzvzrtiohtvrt() informs the NDMaterial object to revert to its initial state, i.e., at thestart of the analysis.The gztXopy() method is declared so a calling object, e.g., Element, can obtain an exactcopy of this NDMaterial object. A pointer to the new object is returned by this functionand the calling object is responsible for deleting this dynamically allocated memory. Thegztiypz() method is returning either “Plane strain” or “Three dimensional” for 2D and 3Danalysis, respectively, in SANICLAY-B implementation. The gztdryzr() method returns2nd or 3rd order for 2D and 3D analysis, respectively, in SANICLAY-B implementation.The final two methods “ztgz“pon“z() and gztgz“pon“z() are declared for recordingNDMaterial state information. In SANICLAY-B implementation, these methods are usedto invoke gzthtrz““(), gzthtrvin() and gzthtvtz() methods when setting string argument“Stress”, “Strain”, and “State”, respectively. Also notice that in SANICLAY-B, the gztBhtvtz() method is implemented to retrieve trial state variables.130HBIB gummury oz numyriwul implymyntutionListing 4.1: NDMaterial interface methods (McKenna et al., 2000)8inx luyz "bvtzr iv l C h"class bvtrifl Pclass IY Pclass kzxtor Pclass In formvt ion Pclass gzsponsz Pclass cYbvtzrivl O public bvtzr iv l{public OcYbvtzrivl = int tvg A int x lvssivg ) PcYbvtzrivl = ) Pvirtual ~cYbvtzrivl = ) Pvirtual int s z ti r i v l h t r v i n = const kzxtor ;v ) Pvirtual int s z ti r i v l h t r v i n = const kzxtor ;v A const kzxtor ;r ) Pvirtual int s z ti r i v l h t r v i n I n x r = const kzxtor ;v ) Pvirtual int s z ti r i v l h t r v i n I n x r = const kzxtor ;v A const kzxtor ;r ) Pvirtual const bvtrifl ;gztivngznt = void ) Pvirtual const bvtrifl ;gz t In i t i v livngzn t = void ) {return this−>gztivngznt = ) P } Pvirtual const kzxtor ;g z t h t r z s s = void ) Pvirtual const kzxtor ;gz th t r v i n = void ) Pvirtual int xommithtvtz = void ) R E Pvirtual int rzvzrtioavstCommit = void ) R E Pvirtual int rzvzr tiohtvr t = void ) R E Pvirtual cYbvtzrivl ∗gztCopy = void ) R E Pvirtual cYbvtzrivl ∗gztCopy = const char ∗ xoyz ) Pvirtual const char ∗gztiypz = void ) const R E Pvirtual int gztdryzr = void ) const {return E P } Pvirtual gzsponsz ∗ sztgzsponsz = const char ∗∗vrgv A int vrgx Adeh htrzvm ;s ) Pvirtual int gztgzsponsz = int rzsponszIY A In formvt ion ;mvtInformvtion ) Pprotected Oprivate O} PIC5 hummvry of numzrixvl implzmzntvtionIn this chapter, the SANICLAY-B model formulation described in chapter 3 is translated intoa numerical code that can be implemented for its use in numerical simulations of laboratorytests or boundary value problems. The adopted refined explicit integration with automaticsubstepping and error control is chosen for the implementation to make sure that the con-131HBIB gummury oz numyriwul implymyntutionstitutive driver can handle any input strain increments without diverging from the intendedmodel response. Then the constitutive driver implementation for simulations of laboratorytests using linearized constraints is illustrated. For model application in numerical solutionof complex boundary value problems, OpenSEES finite element code interaction with theconstitutive driver is described.The procedure followed for model implementation is presented in section 4.2 along withthe pseudo-codes illustrating details that can be followed for generation of the constitutivedriver. The procedure also includes model specific details of integration. The elastic nucleuswhich is implicitly stated in the model formulation is defined in order to allow separatingthe elastic response from the elastoplastic response. Due to Lode angle dependency of thebounding surface adopted in the formulation, the iterative solution is proposed which usesminimization of an error function using Pegasus iterative method for computation of stressstates at the crossing of the elastic nucleus, the crossing of the bounding surface, and com-putation of the image stress. The linearized constraints of laboratory element tests adoptedfor model simulations are presented in section 4.3. First the concept of linearized constraintsis explained and details of the constraints for laboratory element tests used in this disserta-tion are given. These tests include isotropic- and KM- consolidation, constant-p, conventionaldrained and undrained triaxial, drained and undrained direct simple shear tests which arechosen as strain-controlled to allow simulation of strain softening response. In addition,K -consolidation and multi-directional shear tests are included as stress-controlled to allowsimulation of specified stress paths. Newton-Raphson iteration is adopted to ensure thata specified stress path is followed for stress-controlled laboratory element tests. Implemen-tation of the constitutive driver in OpenSEES finite element code is described in section4.4. The procedure for interaction of OpenSEES (also applicable for other finite elementplatforms) with the constitutive driver is presented. The constitutive driver can also be im-plemented in other finite element (e.g., PLAXIS) or finite difference (e.g., FLAC) platforms.The changes that the constitutive driver may undergo are not related to the model numer-ical implementation but are related to platform specific communication of the constitutivedriver.The above summarizes SANICLAY-B model adopted integration within the generated132HBIB gummury oz numyriwul implymyntutionconstitutive driver and its implementation in linearized constraints technique for simulationsof laboratory tests and finite element method for simulations of boundary value problems.The next chapter focuses on verification and validation of the SANICLAY-B model forbuilding the confidence in translation of the constitutive equations into the constitutivedriver and in model predictive capabilities prior to its application in simulating complexcyclic laboratory element tests.1335C kzri xvtion of cumzrixvlImplzmzntvtion vny kvliyvtion ofboyzl ezrformvnxz in Xyxlix hhzvrThe constitutive driver for the developed SANICLAY-B model is used in this chapter forverification of its implementation and model validation against laboratory cyclic test data. Inthe introductory part the difference between the validation and the verification processes, andtheir importance are explained. Then the verification of the implemented constitutive driveris performed to demonstrate that it can reproduce its theoretically intended response, andthat the implementation of the model is robust and can handle a large range of magnitudesof input strain increments. The model is then validated in comparison with experimentaldata on three clays subjected to undrained cyclic triaxial loading to gain confidence in itspredictive capabilities in cyclic testing. In this chapter, laboratory constraints are simulatedusing the method specified in 4.3.5CF IntroyuxtionPrediction of mechanical behavior comprises the use of computational model to foretell thestate of a physical system under consideration with conditions for which the computationalmodel has not been validated (Oberkampf et al., 2004). Confidence in computational simu-lations relies heavily on proper Verification and Validation (V&V) processes.jyri wution is the process of determining that a model implementation accurately rep-resents developer’s conceptual description and specification. It is a Mathematics issue. Ver-ification provides evidence that a conceptual model is solved correctly in comparison with134IBEB Introxuwtionits highly accurate solutions. Verification is also meant to identify and remove errors incomputer coding and verify numerical algorithms and is desirable in quantifying numericalerrors in a computed solution.julixution is the process of determining the degree to which a conceptual model is anaccurate representation of the real world from the perspective of the intended uses of themodel. It is a Physics issue. Validation provides evidence that the correct model is solvedin comparison with experimental data. Validation serves as a strategic goal to increaseconfidence in the quantitative predictive capability of the computational model.V&V procedures are the primary means of assessing accuracy in modeling and compu-tational simulations. The V&V procedures are essential tools in building confidence andcredibility in modeling and computational simulations. Verification procedure is the firststep and it builds confidence in computerized model. Validation procedure is the next stepand it builds confidence in model predictive capabilities. The issue of V&V for modelingsoil response in cyclic loading has been extensively discussed by Taiebat (2008); Tasiopoulouet al. (2015u,v). It is believed that the V&V process is an essential attribute for developmentand numerical implementation of the SANICLAY-B model in this dissertation.Verification of the SANICLAY-B model implementation is performed first against ana-lytical solution of the constitutive equations reduced, by setting appropriate values to modelparameters, to Modified Cam-Clay (MCC) proposed by Burland (1965), with the additionalfeature of Lode angle dependency, for a given loading condition. Further verification of thenumerical implementation is conducted using ISO-error maps. These maps are constructedusing different sizes of strain increments thereby evaluating the adopted integration techniqueof the model.Validation of SANICLAY-B model is then performed based on the experimental (physi-cal) data on Georgia kaolin clay (Sheu, 1984), Cloverdale clay (Zergoun and Vaid, 1994), andAriake clay (Yasuhara et al., 1992). Step by step procedure for calibration of model param-eters is first given for each set of simulations. An emphasis is made on the model validationagainst “slow” undrained cyclic data. Cyclic loading in experiments on Georgia kaolin andCloverdale clay is conducted “slow” allowing uniform distribution of pore water pressureand thereby reliable quantification of the measured stresses. Cyclic loading in experiments135IBFB jyri wution oz numyriwul implymyntutionon Ariake clay is conducted “fast” and cannot be fully relied on. More discussion on defi-nition of “slow” and “fast” cyclic loading along with their advantages and disadvantages ispresented in section 2.2.1.5CG kzri xvtion of numzrixvl implzmzntvtionIn this section, the verification of the model implementation is performed using an exactsolution for the SANICLAY-B model reduced to the MCC model by setting appropriate val-ues to model parameters, and ISO-error maps. Model parameters, initial internal variables,and numerical integration parameters used in the simulations (unless otherwise specified)are given in Tables 5.1, 5.2, and 5.3, respectively. Set 1 in these tables is used for modelsimulations in verification of reduced model against the exact solution. The reduced modelis SANICLAY-B with deactivated features of the rotational hardening, the destructuration,and the bounding surface formulation. The plastic potential and the bounding surface areset to coincide, thereby adopting associative flow rule in the model. The surfaces are set todepend on the Lode angle. Set 2 is used for model simulations in the verification processusing ISO-error maps. In this set, only the destructuration feature is deactivated. Initialvoid ratio z, initial triaxal stress state (p; q), and substepping relative error tolerance STOLare varied with the details given prior to the corresponding simulations.Table 5.1: Model parameters used in verification of model implementationboyzl xon“tvnt Yz“ignvtion hzt F hzt GElasticity  0:037 0:037, 0:2 0:2Critical state bc 0:87 0:87be 0:8 0:86 0:121 0:121Bounding surface Nc 0:87 0:8Ne 0:8 0:8hM 50 50vd 0 7Rotational hardening X 0 3x 1:69 1:69Destructuration ki 0 0136IBFB jyri wution oz numyriwul implymyntutionTable 5.2: Initial internal variables used in verification of model implementationboyzl intzrnvl vvrivwlz Yz“ignvtion hzt F hzt GSize of the bounding surface pM 345 kPa 345 kPaStructuration factor hi 1 1Void ratio z 0:794 variedOrientation of plastic potential and bounding surface 0 0Initial stress state (p; q) (345; 0) variedInitial projection center (pc; qc) (0; 0) (0; 0)Table 5.3: Numerical integration parameters used in verification of model implementationIntzgrvtion pvrvmztzr“ Yz“ignvtion hzt F hzt GSubstepping relative error tolerance STOL 1:0× 10−3 variedMinimum pseudo-time increment i˙min 1:0× 10−4 1:0× 10−NMBounding surface and elastic nucleus tolerance TOL 1:0× 10−T 1:0× 10−TMaximum number of iterations for drift correction ISMAX 40 40Threshold for detection of stress reversal "ˆ 0 05CGCF kzri xvtion of rzyuxzy moyzl vgvin“t zxvxt “olutionThe SANICLAY-B model describes highly nonlinear response of clay and therefore it is notfeasible to obtain exact solution using its constitutive equations and some simplifications arerequired. MCC is a basis model in the hierarchical development of the SANICLAY family ofmodels, and the SANICLAY-B model can be reduced to MCC by adjusting its parameters.In order to deactivate the rotational hardening, the rate of rotational hardening X is set to0. To deactivate the destructuration mechanism, the rate of destructuration ki is set to 0.To deactivate the bounding surface formulation, the indirect measure of the elastic nucleuss is set to 100. In fact, simulations considered here start from normally consolidated statethereby the bounding surface formulation is not in effect; thus, any values for boundingsurface parameters hM and vd can be used. The model adopts Lode angle dependency,however, which is not a feature of MCC model. Nevertheless, solution of the MCC model fortriaxial compression and extension can be considered withb set tobc andbe, respectively,in Table 5.6 for comparison with the reduced SANICLAY-B model. Analytical solutions forthe stress path of undrained triaxial loading conditions using MCC constitutive equationshas been previously provided by Wood (1990), and for the stress-strain path by Potts and137IBFB jyri wution oz numyriwul implymyntutionGanendra (1994), and Peric´ and Ayari (2002).The stress path for undrained triaxial loading is estimated below. The estimated path isobtained for an initially rotated bounding surface, which is an essential difference from thestress path estimate proposed by Wood (1990). For this purpose, triaxial form of Eq. 3.12aexpressed as F = (q¯ − p¯ )2 − (N2 − 2) p¯ (pM − p¯) = 0 is first used with the substitution ofq¯ = p¯ yieldingp¯ =(N2 − 2) pM[( − )2 + (N2 − 2)] (5.1)which together with the q¯ = p¯ expresses the stress state on the bounding surface for agiven  along unloading-reloading line (i.e., with constant pM and ). Given the undrainedconditions, Eq. 5.1 has to be reduced to bring the void ratio z from the unloading-reloadingline to the initial void ratio zin at isotropically normally consolidated state thereby accountingfor change in pM. Then an undrained mean stress pu is obtained aspu =((N2 − 2in)( − in)2 + (N2 − 2in))−p(=)M;in (5.2a)qu = pu (5.2b)where Λ = (− ) R, and pM;in and in are initial values of pM and , respectively. The aboveequation describes the undrained triaxial stress path for an input  and initial rotation ofthe bounding surface given by in. Setting in = 0, reduced the provided stress path solutionto that of MCC.Peric´ and Ayari (2002) integrates "˙q over input  and obtains the following equation:"q =Λbvin[ln(b + ) (b − in)(b − ) (b + in) + 2(b23 − 1)(arctanb− arctan inb)]++ (1− 2Λ)3 (1 + zin)( − in) (5.3)where an initial specific volume vin = (1 + zin), =3 (1− 2,)2 (1 + ,). The above equation describesundrained triaxial stress-strain path for an input  and does not consider the initial rotationof the bounding surface.138IBFB jyri wution oz numyriwul implymyntutionp/p0,in0 0.2 0.4 0.6 0.8 1 1.2q/p 0,in-0.6-0.4-0.200.20.40.6Exact solution10000 steps1000 steps100 steps10 steps1 stepsa| (%)0 2 4 6 8 10 12q/p 0,in-0.6-0.4-0.200.20.40.6Figure 5.1: The analytical solution and reduced SANICLAY-B model simulation resultsfor undrained triaxial loading on normally consolidated clay (Note: the symbols are plot-ted equally spaced along horizontal axis of the stress and stress-strain paths to avoid plotoverloading for illustration purposes)Set of Eqs. 5.2 and 5.3 providing the exact solution of MCC constitutive equations forthe undrained triaxial loading are plotted (red line) in Fig. 5.1 in terms of the stress andthe stress-strain paths. SANICLAY-B model simulations are performed strain-controlledwith five different number of steps varying from 10000 steps to 1 step. Notice that all of thesimulation results fall along the exact solution for both the stress and the stress-strain paths.Potts and Ganendra (1994) suggests that undrained (no volume change) nature results ina constant ratio between radial to axial strain components throughout the whole loadingprocess and strains are therefore varying in a “proportional” manner and consequently thesubstepping algorithm, using substeps of proportional strains, will be correct and indepen-dent of the size of the strain increment applied. This observation has been also supportedin numerical simulations by other researchers (e.g., Andrianopoulos et al., 2010).It should be mentioned again that the rotational hardening and bounding surface formu-lation cannot be verified using an analytical solution due to additional nonlinearities thesefeatures bring into the constitutive equations. Despite such inability to provide full verifica-tion of all model features, successful comparison of model simulation results with the exactsolution demonstrated above provides a basis for further model verification.139IBFB jyri wution oz numyriwul implymyntution5CGCG IhdBzrror mvp“Further verification of the accuracy of the adopted refined explicit integration, namely mod-ified Euler with automatic substepping and error control, is presented here. The ISO-errormaps are often developed for this purpose based on either the stress- or the strain-controlledhomogeneous problem. The procedure has been used by many researchers for general plas-ticity models by (e.g., Iwan and Yoder, 1983; Krieg and Krieg, 1977; Ortiz and Popov, 1985;Ortiz and Simo, 1986; Schreyer et al., 1979; Simo and Hughes, 1998); and for soil models by(e.g., Amorosi et al., 2008; Andrianopoulos et al., 2010; Jeremic´ and Sture, 1997; Manzariand Prachathananukit, 2001).Strain probes ("˙a; "˙r) of different sizes up to (10; 10)% in magnitude are passed to theconstitutive driver and final stress  are used in the comparison with the stress ∗ fromthe “exact” solution for generation of ISO-error maps using the following equation for eachstrain probeError =√( − ∗) : ( − ∗)∗ : ∗100% (5.4)The “exact” solution is obtained using 500 steps and STOL = 1× 10−R. This number ofsteps along with STOL are considered to be sufficient for the “exact” solution from a pre-liminary analysis and was previously adopted by Amorosi et al. (2008) who also appliedlarge magnitude strain increments. Small value of i˙min = 1:0× 10−NM shown in Table 5.3is required for larger strains used in ISO-error maps generation. The error estimates areconsidered for 100 × 100 mesh strain probes to construct detailed ISO-error maps in threedifferent zones shown in Fig. 5.2(a) spanning ("˙a; "˙r) = (−10 : 10;−10 : 10)% strain range.The region, not considered for the generation of ISO-error maps, contains strain incrementsthat are inadmissible by the model as there is a limit of strain increment that can be ap-pliedNM. Initial stress states are considered at a number of initial stress ratios M shown inFig. 5.2(b). For computation of the corresponding stress states (p¯; q¯) on the initial bound-ing surface, p¯ from Eq. 5.1 and q¯ = p¯M are used. Modified overconsolidation ratio OCRmdescribed in section 3.4.5 is used for convenience in defining initial stress state. The voidratio, corresponding to the computed p¯ on the unloading-reloading line in z–ln p space and10thz most owvious zflvmplz is v strvin inxrzmznt thvt xvn wz vpplizy in isotropix unlovying140IBFB jyri wution oz numyriwul implymyntutionε˙a(%)-10 0 10ε˙r(%)-10010Not considered Zone 1Zone 2Zone 3=v) oonzs spvnning = t"a; t"r) R =−FE O FE;−FE O FE):p/p00 0.2 0.4 0.6 0.8 1 1.2q/p 0-0.6-0.4-0.200.20.40.60.5Mc1Mc2McB1C1D1A8 A4 A2 A1=w) Consiyzrzy initivl strzss stvtzs =initivl wounyingsurfvxz shofin in grvy)Figure 5.2: Zones and initial stress states for generation of ISO-error mapson the bounding surface in q–p space, is then estimated as z = z(p=p0) −  ln (pMRp) with avoid ratio on the isotropic normal consolidation line z(p=p0) = 0:794. A similar procedure isfollowed for computing the initial stress states for response envelopes in section 3.4.5.Effzxt of tolzrvnxz hida on gznzrvtion of IhdBzrror mvp“The first set of ISO-error maps shown in Fig. 5.3 is generated for the initial stress state atOCRm = 1:0 and stress ratio M = 0, corresponding to the point AN in Fig. 5.2(b). The effectof the tolerance STOL used in the adopted refined explicit integration is illustrated. Thethree columns of ISO-error maps represent the inputs of the strain increments correspondingto: (a) Zone 1, (b) Zone 2, and (c) Zone 3 with each row corresponding to a specifiedSTOL. The amount of error directly depends on the tolerance STOL used in the integration,i.e., for smaller tolerances the error is reduced. The smallest errors in Zones 1 and 3 areassociated with the strain paths that result into loading along gradient @gR@ throughoutthe substepping process. They form zigzag pattern contours in Zone 1 and slightly curvedcontours in Zone 3 that are more pronounced with the reduced tolerance STOL. Negligiblediscontinuities in the contours due to the update of the projection center from its initial141IBFB jyri wution oz numyriwul implymyntutionlocation at the origin to the point of stress reversal are noticeable for the strain paths in thevicinity of "˙a = −2"˙r in Zones 1 and 3. Also observe oscillating contours in the ISO-errormaps. The oscillations are due to allowable tolerance STOL in the refined explicit integrationand therefore are proportional to the adopted STOL values. Hereafter, all ISO-error mapsare to be generated using STOL = 1:0× 10−3 as an optimal value providing errors within0:2% for the considered strain paths.ε˙a(%)0 2 4 6 8 10ε˙r(%)-10-8-6-4-20VNAhidaRF×FE−2ε˙a(%)0 2 4 6 8 10ε˙r(%)0246810ε˙a(%)-10 -8 -6 -4 -2 0ε˙r(%)024681000.20.40.60.811.21.4Error (%)ε˙a(%)0 2 4 6 8 10ε˙r(%)-10-8-6-4-20VNAhidaRF×FE−3ε˙a(%)0 2 4 6 8 10ε˙r(%)0246810ε˙a(%)-10 -8 -6 -4 -2 0ε˙r(%)024681000.20.40.60.811.21.4Error (%)ε˙a(%)0 2 4 6 8 10ε˙r(%)-10-8-6-4-20VNAhidaRF×FE−4oonz Fε˙a(%)0 2 4 6 8 10ε˙r(%)0246810oonz Gε˙a(%)-10 -8 -6 -4 -2 0ε˙r(%)024681000.20.40.60.811.21.4Error (%)oonz HFigure 5.3: ISO-error maps for the initial stress state at point VN showing the reductionof the Error with the reduced tolerance STOL (each row corresponds to an Error and eachcolumn corresponds to a Zone)In Fig. 5.3 the discontinuities are present within the ISO-error maps generated for "˙a = "˙r142IBFB jyri wution oz numyriwul implymyntutionin such a way that the maximum error is estimated along the isotropic strain path. In orderto explain the discontinuities, Zone 2 is re-plotted with the contour lines included in Fig. 5.4and the ISO-error map with deactivated (by setting parameter X = 0) rotational hardeningis generated for comparison. Although the maximum error is still estimated along "˙a = "˙rstrain path in the latter case, no sharp changes in the contours observed in the formercase are noticeable. Recall from section 4.2.1 that the relative error gmax of the numericalintegration is computed as the maximum out of the relative errors g, g , gp0 , and gycorresponding to the computations of , , pM, and y, respectively. In the generation of theISO-error map with no rotational hardening along isotropic loading, gmax is computed outof only g, gp0 , and gy as g = 0 for all strain probes. In the generation of the ISO-errormap with activated rotational hardening, gmax is computed out of g, gp0 and gy only forthe strain path "˙a = "˙r, while for any other strain path g is also effective. As computationof gmax based on all four relative errors is more demanding than that based only on g, gp0and gy for the numerical integration, more substeps are associated with the "˙a ̸= "˙r strainpath thereby resulting in the abrupt changes in the contours of ISO-error corresponding tothe transition from the "˙a = "˙r to the "˙a ̸= "˙r strain paths.ε˙a(%)0 2 4 6 8 10ε˙r(%)0246810VNAhidaRF×FE−3fiith rotvtionvl hvryzningε˙a(%)0 2 4 6 8 10ε˙r(%)024681000.020.040.060.080.10.120.140.160.180.20Error (%)fiithout rotvtionvl hvryzningFigure 5.4: The effect of the rotational hardening on the contours of ISO-error maps in thevicinity of "˙a = "˙r strain pathThe effect of the tolerance STOL on CPU time spent for simulations using strain probes,corresponding to Zone 2, are shown in Fig. 5.5. The results are shown for the simulations143IBFB jyri wution oz numyriwul implymyntutionstarting with the initial state at point AN in Fig. 5.2(b). The three columns of the ISO-errormaps represent different values of the tolerance STOL. As expected the largest tolerancevalue STOL = 1 × 10−2 requires least amount of CPU time, with maximum of 0:0191s,by the adopted numerical integration. With the tolerance increased to STOL = 1 × 10−3and 1 × 10−4, the maximum CPU time increases to 0:0426s and 0:1182s, respectively. Asexplained in the previous paragraph, more substeps are associated with the "˙a ̸= "˙r strainpath thereby resulting in the abrupt changes in the contours of CPU time corresponding tothe transition from the "˙a = "˙r to the "˙a ̸= "˙r strain paths. There appear to be patches oflarger CPU time that are randomly positioned throughout the contours plots. Consideringthat the CPU time is relatively small for strain probes as large as ("˙a; "˙r) = (10; 10)%, andthe difference between these patches and CPU time values within the contours around it aresmall, numerical integration is considered to be efficient in terms of its computational cost.ε˙a(%)0 2 4 6 8 10ε˙r(%)0246810hida R F× FE−2ε˙a(%)0 2 4 6 8 10ε˙r(%)0246810hida R F× FE−3ε˙a(%)0 2 4 6 8 10ε˙r(%)024681000.020.040.060.080.10.12time (s)hida R F× FE−4Figure 5.5: CPU time (in seconds) spent for simulations of Zone 2 strain increments startingfrom the initial stress state at point VN, showing CPU time for different STOL valuesIhdBzrror mvp“ for initivl “trz““ “tvtz“ in“iyz wounying “urfvxzThe ISO-error maps in Figs. 5.6 are generated for the initial overconsolidation ratios OCRm =1, 2, 4, and 8, corresponding to the points AN, A2, A4, and AU in Fig. 5.2(b), respectively.These points correspond to initially isotropic state and are intended to test the numericalimplementation of the model at the initial stress states inside the bounding surface. Thesmallest error contours in Zones 1 and 3 change their shape as different initial stress statesresult into different image stresses on the bounding surface. This results into different gradi-144IBGB julixution in unxruinyx wywliw triufiiul tystents @gR@ and therefore different strain paths along the gradients producing the contours.Discontinuity in Zone 2 is present for all OCRm due to the reasons discussed above as imagestresses are always located on the bounding surface such that rotational hardening is inac-tive. Overall, the largest errors associated with different initial OCRs are very similar andproduced for isotropic "˙a = "˙r strain path.IhdBzrror mvp“ for initivl “trz““ “tvtz“ vt yiffzrznt “trz““ rvtio“The ISO-error maps shown in Figs. 5.7 are generated for the initial stress ratios M = 0,0:5bc, 1bc, and 2bc, corresponding to the points AN, BN, CN, and DN in Fig. 5.2(b), re-spectively. These points are intended to test the numerical implementation of the model atthe initial stress states corresponding to different stress ratios. The smallest error contoursin Zones 1 and 3 change their shape as different initial stress states on the bounding surfacecorresponding to M result into different gradients @gR@ and therefore different strain pathsalong these gradients. The discontinuity in Zone 2 for M = 0, due to inactive rotationalhardening as discussed above, is replaced by the continuous contours for M S 0. The reasonbehind it is that gmax is computed out of all g, g , gp0 and gy with the strain paths inthe vicinity of "˙a = "˙r for M S 0. Moreover, the strain paths just above "˙a = "˙r produce thesmallest errors for M S 0 as these strain paths induce loading along the gradient @gR@. InZone 3 for M = 2bc, the largest error is associated with the strain path inducing the stressstate in the vicinity of the origin of the stress space. Eq. 5.4 inevitably results into com-putations of the error that are very sensitive for small stresses in the vicinity of the origin,thereby producing larger error estimates. Nevertheless, the stress states in the vicinity ofthe origin are handled well by the adopted model numerical integration scheme consideringthat large strains (up to 10%) are applied.5CH kvliyvtion in unyrvinzy xyxlix trivxivl tz“tThe SANICLAY-B model is validated in this section in comparison with laboratory data onthree clays: Georgia kaolin, Cloverdale, and Ariake clays. The specimens of Georgia kaolinclay are prepared from a slurry of commercially available clay powder. The specimens of145IBGB julixution in unxruinyx wywliw triufiiul tystε˙a(%)0 2 4 6 8 10ε˙r(%)-10-8-6-4-20VN=dCgmRF)ε˙a(%)0 2 4 6 8 10ε˙r(%)0246810ε˙a(%)-10 -8 -6 -4 -2 0ε˙r(%)024681000.020.040.060.080.10.120.140.160.180.2Error (%)ε˙a(%)0 2 4 6 8 10ε˙r(%)-10-8-6-4-20V2=dCgmRG)ε˙a(%)0 2 4 6 8 10ε˙r(%)0246810ε˙a(%)-10 -8 -6 -4 -2 0ε˙r(%)024681000.020.040.060.080.10.120.140.160.180.2Error (%)ε˙a(%)0 2 4 6 8 10ε˙r(%)-10-8-6-4-20V4=dCgmRI)ε˙a(%)0 2 4 6 8 10ε˙r(%)0246810ε˙a(%)-10 -8 -6 -4 -2 0ε˙r(%)024681000.020.040.060.080.10.120.140.160.180.2Error (%)ε˙a(%)0 2 4 6 8 10ε˙r(%)-10-8-6-4-20VU=dCgmRM)oonz Fε˙a(%)0 2 4 6 8 10ε˙r(%)0246810oonz Gε˙a(%)-10 -8 -6 -4 -2 0ε˙r(%)024681000.020.040.060.080.10.120.140.160.180.2Error (%)oonz HFigure 5.6: ISO-error maps for different OCRm values showing the robustness of the nu-merical implementation for the initial stress states inside the bounding surface (each rowcorresponds to an OCRmand each column corresponds to a Zone)146IBGB julixution in unxruinyx wywliw triufiiul tystε˙a(%)0 2 4 6 8 10ε˙r(%)-10-8-6-4-20VN=MRE)ε˙a(%)0 2 4 6 8 10ε˙r(%)0246810ε˙a(%)-10 -8 -6 -4 -2 0ε˙r(%)024681000.050.10.150.20.250.3Error (%)ε˙a(%)0 2 4 6 8 10ε˙r(%)-10-8-6-4-20BN=MRE:5bc)ε˙a(%)0 2 4 6 8 10ε˙r(%)0246810ε˙a(%)-10 -8 -6 -4 -2 0ε˙r(%)024681000.050.10.150.20.250.3Error (%)ε˙a(%)0 2 4 6 8 10ε˙r(%)-10-8-6-4-20CN=MRFbc)ε˙a(%)0 2 4 6 8 10ε˙r(%)0246810ε˙a(%)-10 -8 -6 -4 -2 0ε˙r(%)024681000.050.10.150.20.250.3Error (%)ε˙a(%)0 2 4 6 8 10ε˙r(%)-10-8-6-4-20YN=MRGbc)oonz Fε˙a(%)0 2 4 6 8 10ε˙r(%)0246810oonz Gε˙a(%)-10 -8 -6 -4 -2 0ε˙r(%)024681000.050.10.150.20.250.3Error (%)oonz HFigure 5.7: ISO-error maps showing effect of initial stress ratio M147IBGB julixution in unxruinyx wywliw triufiiul tystCloverdale clay are prepared from undisturbed samples of natural structured marine clay.The specimens of Ariake clay are prepared from reconstituted samples of natural marineclay.Required experimental data for model calibration are listed first. A calibration process ofthe model constants for each clay is illustrated then prior to the validation process. Numericalintegration parameters used in all simulations correspond to Set 1 of Table 5.3.5CHCF boyzl xvliwrvtion proxzyurzThe calibration procedure refers to the following list of experiments:XI isotropically consolidationXKM KM consolidationXIjX undrained triaxial compression test on isotropically consolidated clayXIjE undrained triaxial extension test on isotropically consolidated clayXKMjX undrained triaxial compression test on KM consolidated clayXKMjE undrained triaxial extension test on KM consolidated clayjcy undrained cyclic triaxial or direct simple shear testsThe proposed SANICLAY-B model requires 12 model constants, whose roles are sum-marized in Table 5.4. Initial internal variables required for the proposed model with theirdefinition are shown in Table 5.5. The sequence of the calibration procedure coincides withthe hierarchical evolution of SANICLAY family of models adding on successive developmentsto enhance model predictive capabilities as described in section 3.1. The reader is referredto the works by Dafalias et al. (2006) and Taiebat et al. (2010b) for comprehensive descrip-tion of anisotropy and destructuration related parameters calibration, respectively. A briefdescription of the calibration procedure for the required constants is given as follows:• Calibration of  and  is based on, preferably, CI tests. The use of data from the CKMtest is also acceptable.148IBGB julixution in unxruinyx wywliw triufiiul tyst• Calibration of , is based on initial KM-unloading stress path preceded by CKM test.Typical range of , = 0:2–0:3 for clays can be assumed, if the required stress path isnot available.• Calibration of bc and be is based on CIUC and CIUE, or CKMUC and CKMUE tests.• Calibration of Nc is based on CKMUC test on normally consolidated clay. If not avail-able, Nc can be calibrated along with X from CIUC or CIUE test on normally consol-idated clay.• Ne = Nc is assumed in all simulations considered in this dissertation. The value canbe adjusted for matching monotonic triaxial extension tests.• Calibration of X is based on the stress path from CKMUE test. If not available, X canbe calibrated along with Nc from CIUC or CIUE test on normally consolidated clay.X = 3–20 is the typical range for most clays (Dafalias et al., 2006).• Calibration of x is based on the closed-form relation given by Eq. 27 in Dafalias et al.(2006).• Calibration of ki for structured clays is based on CI or CKM tests to stresses, ensuringcomplete loss of structure. ki = 0 should be used for unstructured clays.• Calibration of hM is based on stress and stress–strain paths from initial cycles of triaxialor direct simple shear Ucy test, preferably on normally consolidated clay. If monotonicloading is of interest, the calibration can be performed in comparison with undrainedmonotonic loading test. For calibration of hM in cyclic loading, vd = 0 should be used.The calibration is performed in trial-and-error simulations of initial few cycles to makesure that when damage parameter is in effect the results of simulations are not alteredsignificantly by the damage parameter starting to evolve.• Calibration of vd is based on cyclic test data used for hM calibration, but using cyclesfollowing initial few cycles used for hM calibration. vd = 0 should be used for monotonicloading, and set to its calibrated value prior to cyclic loading simulations.149IBGB julixution in unxruinyx wywliw triufiiul tystTable 5.4: Model constants for the proposed SANICLAY-B modelboyzl xon“tvnt Yz“ignvtion Yz“xription of it“ rolzElasticity  Slope of unloading-reloading line in z–ln p plane, Poisson’s ratioCritical state bc Critical stress ratio in triaxial compressionbe Critical stress ratio in triaxial extensionNc Stress ratio at peak q on the bounding surface intriaxial compressionNe Stress ratio at peak q on the bounding surface intriaxial extension Slope of compression line in z–ln p planeRotational hardening X Rate of anisotropic hardening evolutionx for anisotropic hardening saturation during KM con-solidationDestructuration ki Rate of destructurationBounding surface hM Initial value of hardening parameter hvd Rate of damage evolutionTable 5.5: Initial internal variables for the proposed SANICLAY-B modelYz“ignvtion Yz“xription of it“ rolzpM Size of the bounding surfacehi Structuration factorz Initial void ratio Rotation of plastic potential and bounding surfacey Initial damage parameter(p; q) Initial stress state in the triaxial stress plane(pc; qc) Initial projection center in the triaxial stress plane5CHCG gzxon“titutzy Gzorgiv kvolin xlvyExperimental data after Sheu (1984) on Georgia kaolin clay is used for model calibration.The soil tested was composed of 62% clay and 38% silt. The plasticity index for this claywas 20%.Kaolin clay was mixed with salty water for preparation of Georgia kaolin clay in order toreduce the effect of anisotropy through formation of a flocculated structure. The mixture waspoured into the consolidometer with two-way drainage and incremental vertical pressureswere applied with the final pressure of 205 kPa. Then a block sample was trimmed intotriaxial specimens.150IBGB julixution in unxruinyx wywliw triufiiul tystboyzl xvliwrvtionModel calibration is based on available experimental results. Parameters for destructurationare set to the values corresponding to already destructured clay as the clay considered wasreconstituted. Details of the calibration of model parameters for this clay are presentedbelow.durumytyrs  unx B In Fig. 5.8 the response of Georgia kaolin clay in isotropic con-solidation with three unloading-reloading sequences is presented in the z– log p plane. Theslopes of the normal compression and the rebound lines are Xc and Xr, respectively, in thez– log p plane, whereas the required  and  are measured in the z– ln p plane. The relation-ships between the slopes are as follows:  = XcR ln 10 and  = XrR ln 10. Then the valuesXc = 0:28 and Xr = 0:085 measured in Fig. 5.8 render  = 0:121 and  = 0:037.101 102 1030.50.60.70.80.91p (kPa)e experimentCc=0.28Cr=0.085Figure 5.8: Calibration of Xc and Xr in z– log p plane based on isotropic consolidation testafter Sheu (1984)durumytyrs bc unx beB CIUC and CIUE stress paths on initially normally consolidatedclay to a range of preconsolidation stresses x = 276, 345,414, and 552 kPa are shownin Fig. 5.9. From direct measurements at the ultimate stress states, critical state ratiosare estimated as bc = 0:87 and be = 0:86. For later calibration of x, friction angle in151IBGB julixution in unxruinyx wywliw triufiiul tystcompression ϕc might be required to come up with the KM value; in this case, bc is relatedto ϕc bybc =6 sinϕc3− sinϕc (5.5)0 200 400 600 800−400−2000200400p (kPa)q (kPa) Mc=0.87Me=0.86414345276Compression σc (kPa):552414345276Extension σc (kPa):Figure 5.9: Calibration of bc and be in the triaxial plane based on undrained triaxialcompression and extension tests starting from a normally consolidated state, experimentalresults after Sheu (1984)durumytyr N B Due to the unavailability of CKMUC test, the value for N must be assumed.Looking ahead, N = bc or N = be would require parameter X to be very small relative toits typical values for clay in the range of X = 3–20 mentioned previously. In addition, cyclicloading simulations would suggest that the value should be considerably smaller for bettersimulations, especially for high cyclic stress amplitudes. For these reasons, the decision ismade to adopt N = 0:8.durumytyr ,B Due to the unavailability of KM-unloading preceded by KM-consolidation152IBGB julixution in unxruinyx wywliw triufiiul tysttest for Georgia kaolin clay, parameter , = 0:2 is assumed. This value has been calibratedfor other clays by Taiebat et al. (2010b), and Dafalias et al. (2006).durumytyr xB The parameter x for successful KM-loading simulation can be obtainedusing the following equation proposed by Dafalias et al. (2006):x =2K0ϵ(1− R)Bϵ3K0 + 3K0+ [2 (1− R)−Bb2c ] ϵK0 −b2c; B = − 2 (1 + ,)9 (1− 2,) (5.6)where ϵ = 3R2 for KM-loading path; KM is a measured value of the earth pressure coefficientat rest; and h0 = 3 (1−KM) R (1 + 2KM).If KM is not available, Jacky’s equation, KM = 1− sinϕ, can be used. For Georgia kaolinclay, ϕ = ϕc = 22:3◦ from Eq. 5.5 yields KM = 0:62, h0 = 0:51 and x = 1:69.durumytyr XB Calibration of X requires all other constants calibrated in advance (Dafaliaset al., 2006). In the proposed model, the novel bounding surface parameters hM and vd arenot required to be known for calibration of X. This is true since parameters hM and vdare in effect for overconsolidated states, while X calibration is based on the stress pathcorresponding to the normally consolidated state. From the available data, a CIUC testis the most appropriate for X calibration, ensuring rotation of the bounding surface. TheX = 3 is chosen as an optimum value from trial-and-error simulations shown in Fig. 5.10.Vounxing surzuwy purumytyrs hM unx vdB Calibration of hM and vd requires experimentaldata involving the stress path corresponding to overconsolidated states. As cyclic loadingis of concern, the parameters should be calibrated based on a cyclic loading test. To beconsistent, calibration of the two parameters should be performed from the same test results.Parameter hM is responsible for the initial part of the cyclic loading, when accumulation ofdamage parameter y is still not significant to induce considerable reduction of h. Therefore,initial cycles from available undrained cyclic triaxial test are used for hM calibration. hM = 50is calibrated from trial-and-error simulations with vd = 0 shown in Fig. 5.11.Once hM is known, vd calibration is performed based on the same undrained cyclic triaxialtest used for hM calibration, but considering all 15 available cycles. For Georgia kaolin clay,vd = 7 is calibrated in a trial-and-error procedure.Calibrated constants and initial internal variables for Georgia kaolin clay are shown in153IBGB julixution in unxruinyx wywliw triufiiul tyst0 50 100 150 200 250 300 350 4000100200300p (kPa)q (kPa) experimentC=9C=6C=3Figure 5.10: Calibration of X based on undrained triaxial compression test on normallyconsolidated clay, experimental results after Sheu (1984)Tables 5.6 and 5.7, respectively. In Table 5.7, z = 1:5 and p = 1 kPa are used for simulationsof undrained cyclic triaxial loading. For simulations of consolidation, CIUC and CIUE tests,different values of z and p are used, which are to be specified for the corresponding testsimulations.boyzl pzrformvnxzModel constants calibrated previously (Table 5.6) are used in this section to simulate isotropicconsolidation, CIUC, CIUE, and undrained cyclic triaxial tests on normally consolidatedGeorgia kaolin clay. The initial internal variables specified in Table 5.7 are used in thesimulations, unless otherwise specified.Isotropiw wonsolixution tystB The results of the isotropic consolidation test simulationin comparison with experimental data are shown in Fig. 5.12. pM = 75 kPa and z = 0:88are adopted for the simulation. Notice that plastic volumetric strains are involved uponreloading, even before the past maximum preconsolidation stress is reached. The magnitudeof the plastic volumetric strains in reloading is in agreement with the experimental results,suggesting that hM = 50 calibrated based on cyclic loading can be successfully used formonotonic loading simulations.154IBGB julixution in unxruinyx wywliw triufiiul tyst0 100 200 300 400 500−200−1000100200p (kPa)q (kPa) qcy=140.7 kPa experimentsimulation=v)−3 −2 −1 0 1 2 3−200−1000100200εa (%)q (kPa) h0=50experimentsimulation=w)Figure 5.11: Initial hardening parameter calibrated to hM = 50 based on undrained cyclictriaxial loading, experimental results after Sheu (1984)inxruinyx triufiiul wompryssion unx yfitynsion tystsB Simulated stress and stress–strainpaths of CIUC and CIUE loading on clay initially normally consolidated to a range ofpreconsolidation stresses c = 276, 345, 414 and 552 kPa are shown in Fig. 5.13 againstavailable experimental data. For the stress path in compression, simulations closely matchexperimental results, while in extension the “hook-type” stress path cannot be reproduced bythe model because of the evolution of anisotropic variable being a function of tr (@FR@),i.e., the stress path freezes as the critical stress ratio is reached (Dafalias and Taiebat, 2012;Taiebat and Dafalias, 2013). In terms of the stress–strain response, the simulated pathultimately matches experimental data, although experimental results suggest larger plasticaxial strains at earlier stages.Note that there is no effect from parameters hM and vd for CIUC and CIUE on normallyconsolidated clay.inxruinyx wywliw triufiiul louxingB The main focus of the current study is on simulationsof cyclic loading. Such simulations are performed on Georgia kaolin clay first isotropciallyconsolidated to normally consolidated state at p = 345kPa. For convenience, stress andstress–strain paths are shown separately in Figs. 5.14 and 5.15, respectively, with the digi-tized experimental data on the left and simulations on the right.The results of 22 cycles for qcy = 121:4 kPa, 15 cycles for qcy = 136 and 140:7 kPa,155IBGB julixution in unxruinyx wywliw triufiiul tystTable 5.6: Model parameters for Georgia kaolin clayboyzl xon“tvnt Yz“ignvtion kvluzElasticity  0:037, 0:2Critical state bc 0:87be 0:86 0:121Bounding surface Nc 0:8Ne 0:8hM 50vd 7Rotational hardening X 3x 1:69Destructuration (no effect) ki 0Table 5.7: Initial internal variables for Georgia kaolin clayboyzl intzrnvl vvrivwlz Yz“ignvtion kvluzSize of the bounding surface pM 1 kPaStructuration factor hi 1Void ratio z 1:5Rotation of plastic potential and bounding surface 0Damage parameter y 0Initial stress state (p; q) (1; 0) kPaInitial projection center (pc; qc) (0; 0) kPaand 3 cycles for qcy = 165:5 kPa are available from the experiments. Simulations are notexhibiting the same amount of p reduction as suggested in experiments owing to the balancereached by the simulated stress path at OCR = 2. In the experiments the stress path fallswell beyond the critical stress ratio. A similar response was observed by Takahashi et al.(1980). They found that if the drainage was allowed after the undrained loading, then thestress ratio finally reduced to the critical stress ratio. This suggests that unreliable porewater pressure measurements were the reason of the stress ratio exceeding the critical stressratio. Shear response however is not affected by unreliable pore pressure measurements andtherefore these results are considered for model validation.For the stress–strain path, the model is able to reproduce cyclic softening observed in theexperiments for the whole range of cyclic stress amplitudes. It is no surprise that simulations156IBGB julixution in unxruinyx wywliw triufiiul tyst101 102 1030.50.60.70.80.91p (kPa)e experimentsimulationFigure 5.12: Simulation of isotropic consolidation test, experimental results after Sheu (1984)for qcy = 140:7 kPa closely match experiments as hM and vd calibration is based on thisparticular cyclic stress amplitude. For a slightly lower stress amplitude, qcy = 136 kPa, thesimulated stress–strain path is also matching the experiment well. For stress amplitudesmuch different from 140:7 kPa, simulations slightly deviate from experiments, although theresults can be considered satisfactory for simulations with the single set of model constants.Observe that the simulated strain levels are less than experimentally observed ones for 165:5kPa cyclic stress amplitude and larger for qcy = 121:4 kPa.5CHCH htruxturzy Xlovzryvlz xlvyExperimental data after Zergoun (1991), Zergoun and Vaid (1994) on Cloverdale clay areused for model calibration. Soil tested was sensitive undisturbed marine clay consisting of49% clay and 45% silt with PI = 24%. The sensitivity was a result of surface infiltrationfollowing uplift above the sea level.The clay was block sampled from a depth of (2:0–2:6) m below ground level. Then thetriaxial specimens were trimmed from the same horizon and hydrostatically loaded under aneffective stress of 200 kPa for 48 h. The consolidation stress beyond apparent preconsolida-157IBGB julixution in unxruinyx wywliw triufiiul tyst0 200 400 600 800−400−2000200400p (kPa)q (kPa) CIUC/E on NC clay414345276Compression σc (kPa):552414345276Extension sigmac (kPa):=v)−20 −10 0 10 20−400−2000200400εa (%)q (kPa) 414345276Compression σc (kPa):552414345276Extension σc (kPa):=w)Figure 5.13: Simulation of CIUC and CIUE tests on normally consolidated clay, experimentalresults after Sheu (1984)158IBGB julixution in unxruinyx wywliw triufiiul tyst0 100 200 300 400 500−200−1000100200p (kPa)q (kPa) qcy=121.4 kPa experiment (6 cycles)=v)0 100 200 300 400 500−200−1000100200p (kPa)q (kPa) qcy=121.4 kPa simulation (22 cycles)=w)0 100 200 300 400 500−200−1000100200p (kPa)q (kPa) qcy=136 kPa experiment (1−6,15 cycles)=x)0 100 200 300 400 500−200−1000100200p (kPa)q (kPa) qcy=136 kPa simulation (15 cycles)=y)0 100 200 300 400 500−200−1000100200p (kPa)q (kPa) qcy=140.7 kPa experiment (1−6,15 cycles)=z)0 100 200 300 400 500−200−1000100200p (kPa)q (kPa) qcy=140.7 kPa simulation (15 cycles)=f)0 100 200 300 400 500−200−1000100200p (kPa)q (kPa) qcy=165.5 kPa experiment (3 cycles)=g)0 100 200 300 400 500−200−1000100200p (kPa)q (kPa) qcy=165.5 kPa simulation (3 cycles)=h)Figure 5.14: Undrained cyclic triaxial loading stress paths with the digitized experimentalresults after Sheu (1984) (left) and model simulations (right) for cyclic stress amplitudes:(a–b) qcy = 121:4 kPa, (c–d) qcy = 136 kPa, (e–f) qcy = 140:7 kPa, and (g–h) qcy = 165:5kPa159IBGB julixution in unxruinyx wywliw triufiiul tyst−15 −10 −5 0 5 10−200−1000100200εa (%)q (kPa) experiment (1−6,19−22 cycles)=v)−15 −10 −5 0 5 10−200−1000100200εa (%)q (kPa) qcy=121.4 kPasimulation (22 cycles)=w)−15 −10 −5 0 5 10−200−1000100200εa (%)q (kPa) experiment (15 cycles)=x)−15 −10 −5 0 5 10−200−1000100200εa (%)q (kPa) qcy=136 kPasimulation (15 cycles)=y)−15 −10 −5 0 5 10−200−1000100200εa (%)q (kPa) qcy=140.7 kPaexperiment (1−7,13−15 cycles)=z)−15 −10 −5 0 5 10−200−1000100200εa (%)q (kPa) qcy=140.7 kPasimulation (15 cycles)=f)−15 −10 −5 0 5 10−200−1000100200εa (%)q (kPa) experiment (3 cycles)=g)−15 −10 −5 0 5 10−200−1000100200εa (%)q (kPa) qcy=165.5 kPasimulation (3 cycles)=h)Figure 5.15: Undrained cyclic triaxial loading stress–strain paths with the digitized ex-perimental results after Sheu (1984) (left) and model simulations (right) for cyclic stressamplitudes: (a–b) qcy = 121:4 kPa, (c–d) qcy = 136 kPa, (e–f) qcy = 140:7 kPa, and (g–h)qcy = 165:5 kPa160IBGB julixution in unxruinyx wywliw triufiiul tysttion stress destructured the clay to some extent, inducing reduced sensitivity. Prior to theseexperiments, the specimens were left undrained under constant total stress for 24 h to allowpore water pressure generation due to arrest of secondary consolidation.boyzl xvliwrvtionCalibration of the model parameters given below is based on available experimental data.The clay considered requires calibration of the parameters ki and hi.durumytyrs  unx B The response of Cloverdale clay in an oedometer test during one-dimensional compression and swelling is shown in Fig. 5.16. Lacking lateral stress mea-surements in this test, the data are plotted in the z– log a plane, where a is an axial(vertical) stress. Xc and Xr measured in Fig. 5.16 are related to  and  as before, giving: = XcR ln 10 = 0:21 and  = XrR ln 10 = 0:03. Note that while this indirect estimationof  is accurate, determination of  is only approximated, given the relative increase of thepRa ratio in swelling due to the increase of KM for the overconsolidated state (Wood, 1990),i.e., rate of p and a change is no longer constant as in the loading process. Therefore, calibration based on isotropic consolidation test, if available, is preferred (Papadimitriouet al., 2005).durumytyrs bc unx beB The response of Cloverdale clay in CIUC tests on normallyconsolidated and overconsolidated specimens is shown in Fig. 5.17. The data is presented inthe MIT stress plane, normalized by the value of the equivalent consolidation stress e, whichis an equivalent stress on the normal consolidation line in the z– log c plane corresponding tothe same void ratio as the overconsolidated clay specimen (Zergoun, 1991). Given these tests,the friction angle in compression is estimated as ϕc = 32◦, rendering bc = 1:29 according toEq. 5.5. be = 1:27 is chosen for better undrained cyclic triaxial loading simulations.durumytyr N B Due to the unavailability of a CKMUC test the value of N = 1 is assumed.Similar to the N preference for Georgia kaolin clay in section 5.3.2, this value is chosen owingto its appropriateness for calibration of the constant X and better undrained cyclic triaxialloading simulations.durumytyr ,B Due to unavailability of KM-unloading stress path following KM-consolidationfor Cloverdale clay the parameter , = 0:2 is assumed.161IBGB julixution in unxruinyx wywliw triufiiul tyst100 101 102 1030.60.811.21.41.6σa (kPa)e experimentCc=0.48Cr=0.07Figure 5.16: Calibration of Xc and Xr in z– log p plane based on oedometer test after Zergoun(1991)durumytyr xB Utilizing Eq. 5.6 along with Jacky’s equation, ϕ = 32◦ yields KM = 0:47,h0 = 0:82 and x = 1:73.durumytyrs hi unx kiB The best experiment for calibration of the initial internal variablehi and the value of destructuration parameter ki is an isotropic compression test on anintact sample until complete loss of structure occurs (Taiebat et al., 2010b). The onlyavailable consolidation test is oedometric consolidation, which is used for the calibration ofthe parameters.Parameter hi can be estimated from the compression plane as the ratio of the axial stress(or mean stress, in the case of isotropic loading) at the end of the initial stiff regime andthe corresponding stress on the Intrinsic Compression Curve (ICC) at the same void ratio(Taiebat et al., 2010b). The location of the ICC can be approximated by the dotted line inFig. 5.18(a) and the ratio yields hi = 90R24 = 3:7 for Cloverdale clay.The parameter ki is calibrated to ki = 2 from trial-and-error simulations shown inFig. 5.18(b). The initial value of pM = 94 kPa has been used to successfully simulate theinitial stiff (elastic) part of the response. New model constants that still require calibration162IBGB julixution in unxruinyx wywliw triufiiul tyst0 0.2 0.4 0.6 0.8 1 1.200.20.40.6(σa + σr) /2 σe(σ a − σr) /2 σe φc=32°1.01.32.03.8Compression, OCR:Figure 5.17: Calibration of ϕc in the triaxial plane based on undrained triaxial compressiontest on normally consolidated and overconsolidated specimens, experimental results afterZergoun (1991)are set to hM = 1z10 and vd = 0 to simulate stiff response for overconsolidated state, sincethe calibration of ki involves a normally consolidated state.durumytyr XB CIUC is the most appropriate test from the available data for calibrationof X, involving rotation of the bounding surface. From trial-and-error calibration procedureX = 3 is selected as an optimum value. Simulation results with different X values for theCIUC test on normally consolidated clay to p = 200 kPa are shown in Fig. 5.19.Vounxing surzuwy purumytyrs hM unx vdB Model parameters hM = 550 and vd = 68 arecalibrated based on the available experimental cyclic loading with cyRSuc = 0:75 cyclic stressamplitude.The 11 calibrated model constants are shown in Table 5.8, and initial internal variablesin Table 5.9. These parameters are to be used in simulations of the available experiments.boyzl pzrformvnxzModel performance for Cloverdale clay is illustrated in this section. Simulations are con-ducted for one-dimensional consolidation, CIUC on normally consolidated and overconsol-idated clay, and undrained cyclic triaxial tests on normally consolidated specimens. Theinitial internal variables specified in Table 5.9 are used in the simulations, unless otherwise163IBGB julixution in unxruinyx wywliw triufiiul tyst100 101 102 1030.60.811.21.41.6σa (kPa)e experimentICC (approx.)9024Si=90/24=3.7=v)100 101 102 1030.60.811.21.41.6σa (kPa)e experimentki=2.0ki=2.4ki=1.6=w)Figure 5.18: Calibration of: (a) hi, and (b) ki based on oedometer test after Zergoun (1991)164IBGB julixution in unxruinyx wywliw triufiiul tyst0 0.2 0.4 0.6 0.8 100.20.40.6(σa + σr) /2 σe(σ a − σr) /2 σe experimentC=3C=9C=6Figure 5.19: Calibration of X based on CIUC test on initially normally consolidated clay,experimental data after Zergoun (1991)specified.cyxomytyr tystB In Fig. 5.20 experimental results and model simulations are shown foran oedometer test on Cloverdale clay. Destructuration observed in laboratory testing can bewell simulated by the model. In the reloading following initial unloading, plastic volumetricstrains for the stress states before the preconsolidation stress is attained are predicted by themodel in agreement with the experiment. For unloading, simulated response closely matchesexperiment for a ≥ 20 kPa; however, the dialative nature of clay in KM-unloading cannotbe predicted for a Q 20 kPa, resulting in slightly underpredicted void ratio at the end ofunloading.One might anticipate a bounding surface providing the desired dialation as the imagepoint upon KM-unloading should be located so that the negative plastic volumetric strains arepredicted. It is indeed so, and the proposed model better simulates dialative response thanthe previous SANICLAY family of models, but the degree of plastic strains with calibrated hMand vd does not allow reaching dialation observed in the experiment. During the calibrationof the parameters, it was noticed that the parameter X is the factor that can control thedegree of dialation through rotational hardening. For lower values of X, less rotation of165IBGB julixution in unxruinyx wywliw triufiiul tystTable 5.8: Model parameters for Cloverdale clayboyzl xon“tvnt Yz“ignvtion kvluzElasticity  0:03, 0:2Critical state bc 1:29be 1:27 0:21Bounding surface Nc 1Ne 1hM 550vd 68Rotational hardening X 3x 1:73Destructuration ki 2Table 5.9: Initial internal variables for Cloverdale clayboyzl intzrnvl vvrivwlz Yz“ignvtion kvluzSize of the bounding surface pM 94 kPaStructuration factor hi 3:7Void ratio z 1:48Rotation of plastic potential and bounding surface 0Damage parameter y 0Initial stress state (p; q) (1; 0) kPaInitial projection center (pc; qc) (0; 0) kPathe bounding surface renders more dialative response simulated for KM-unloading. Thus,adjusted rotational hardening in conjunction with the bounding surface formulation canpotentially enhance model predictions in KM-loading.inxruinyx triufiiul wompryssion unx yfitynsion tystsB Isotropic compression to p = 200kPa is imposed prior to simulations of undrained triaxial compression and extension tests.Simulations of CIUC and CIUE tests on normally consolidated and overconsolidated speci-mens are performed with the results shown in Fig. 5.21. Simulated response approximatelyfollows the experimental results in terms of the stress path for OCR ≤ 2. As OCR increasesthe stress path deviates from experiment in that the dialative response observed in actualtests is not captured. Similarly, the stress–strain path is closely matching for OCR ≤ 2throughout the whole process of undrained loading. For OCR = 3:8, initial stress–strain166IBGB julixution in unxruinyx wywliw triufiiul tyst100 101 102 1030.60.811.21.41.6σa (kPa)e experimentsimulationFigure 5.20: Simulation of oedometer test on Cloverdale clay, experimental results afterZergoun (1991)simulation exhibits larger plastic strains than desired and then falls close to the experiment.inxruinyx wywliw triufiiul louxingB Experimental data, where “slow” undrained cyclictriaxial loading was performed by Zergoun (1991) to maintain the uniformity of pore waterpressure, is used in this section. Prior to simulations of undrained cyclic triaxial loading,isotropic consolidation to p = 200 kPa is simulated. The stress and stress–strain path inundrained cyclic triaxial loading are shown separately in Figs. 5.22 and 5.23, respectively,with the experimental results on the left and model simulations on the right.Unlike undrained cyclic triaxial loading tests on Georgia kaolin clay, the “slow” cyclicloading imposed on Cloverdale clay renders the stress path that does not travel beyond thecritical stress ratio, as shown in Fig.5.22. The simulated stress path that freezes at OCR = 2still deviates from the observed experiments, but to a smaller degree than the simulatedstress path for Georgia kaolin clay. In terms of the stress–strain simulations, cyRSuc = 0:75cyclic stress amplitude that is used for hM and vd calibration is apparently well simulated.For cyRSuc = 0:70, the stress–strain response closely matches the experiment in terms of167IBGB julixution in unxruinyx wywliw triufiiul tyst0 0.2 0.4 0.6 0.8 1 1.200.20.40.6(σa + σr) /2 σe(σ a − σr) /2 σe 1.01.32.03.8Compression, OCR:=v)0 2 4 6 8 1000.20.40.6εa (%)(σ a − σr) /2 σe 1.01.32.03.8Compression, OCR:=w)Figure 5.21: Simulations of CIUC and CIUE tests on normally and overconsolidated clay,experimental results after Zergoun (1991)168IBGB julixution in unxruinyx wywliw triufiiul tyst0 0.2 0.4 0.6 0.8 1 1.2−0.4−0.200.20.4(σa + σr) /2 σe(σ a − σr) /2 σe τcy/Suc=0.70 experiment (4 cycles)=v)0 0.2 0.4 0.6 0.8 1 1.2−0.4−0.200.20.4(σa + σr) /2 σe(σ a − σr) /2 σe τcy/Suc=0.70 simulation (5 cycles)=w)0 0.2 0.4 0.6 0.8 1 1.2−0.4−0.200.20.4(σa + σr) /2 σe(σ a − σr) /2 σe τcy/Suc=0.75 experiment (13 cycles)=x)0 0.2 0.4 0.6 0.8 1 1.2−0.4−0.200.20.4(σa + σr) /2 σe(σ a − σr) /2 σe τcy/Suc=0.75 simulation (13 cycles)=y)0 0.2 0.4 0.6 0.8 1 1.2−0.4−0.200.20.4(σa + σr) /2 σe(σ a − σr) /2 σe τcy/Suc=0.75 experiment (4 cycles)=z)0 0.2 0.4 0.6 0.8 1 1.2−0.4−0.200.20.4(σa + σr) /2 σe(σ a − σr) /2 σe τcy/Suc=0.79 simulation (4 cycles)=f)Figure 5.22: Undrained cyclic triaxial loading stress path with the digitized experimentalresults after Zergoun (1991); Zergoun and Vaid (1994) (left), and model simulations (right)for cyclic stress amplitudes: (a-b) cyRSuc = 0:70, (c-d) cyRSuc = 0:75, and (e-f) cyRSuc =0:79169IBGB julixution in unxruinyx wywliw triufiiul tyst−3 −2 −1 0 1 2 3−0.4−0.200.20.4εa (%)(σ a + σr) /2 σe τcy/Suc=0.70experiment (4 cycles)=v)−3 −2 −1 0 1 2 3−0.4−0.200.20.4εa (%)(σ a − σr) /2 σe τcy/Suc=0.70simulation (5 cycles)=w)−10 −8 −6 −4 −2 0 2 4 6 8 10−0.4−0.200.20.4εa (%)(σ a + σr) /2 σe experiment (13 cycles)=x)−10 −8 −6 −4 −2 0 2 4 6 8 10−0.4−0.200.20.4εa (%)(σ a − σr) /2 σe τcy/Suc=0.75simulation (13 cycles)h0=550ad=68=y)−10 −8 −6 −4 −2 0 2 4 6 8 10−0.4−0.200.20.4εa (%)(σ a + σr) /2 σe τcy/Suc=0.75experiment (4 cycles)=z)−10 −8 −6 −4 −2 0 2 4 6 8 10−0.4−0.200.20.4εa (%)(σ a − σr) /2 σe τcy/Suc=0.79simulation (4 cycles)=f)Figure 5.23: Undrained cyclic triaxial loading stress path with the digitized experimentalresults after Zergoun (1991); Zergoun and Vaid (1994) (left), and model simulations (right)for cyclic stress amplitudes: (a-b) cyRSuc = 0:70, (c-d) cyRSuc = 0:75, and (e-f) cyRSuc =0:79170IBGB julixution in unxruinyx wywliw triufiiul tystattained peak strains. Simulation of cyRSuc = 0:79 cyclic loading provides strains that aremuch smaller than the ones observed during the experiments.Finally, simulations in terms of peak "a versus number of cycles for the range of availablecyclic stress amplitudes in undrained cyclic triaxial loading are shown in Fig. 5.24(b). Ex-perimental data is digitized and is shown in Fig. 5.24(a) for comparison . Considering thatearthquake loading is of interest, a M7.5 earthquake corresponds to approximately 30 equiv-alent number of cycles imposed on fine-grained material according to Idriss and Boulanger(2008). Therefore, the results of simulations and experimental data are plotted within 30cycles. Evolution of peak "a with number of cycles is closely simulated for cyRSuc = 0:75and 0:69. Simulated cyRSuc = 0:79 cyclic loading requires about 5 extra cycles to reachthe experimentally observed peak "a = 10%. For lower cyclic stress amplitudes, the picturebecomes less satisfactory, simulating a softer clay response than desired. Cyclic stress am-plitude of cyRSuc = 0:62 reaches approximately "a = 10% in 30 cycles as simulated, whilein experiments only "a = 2% is attained in 30 cycles. With a cyRSuc = 0:57, peak "a = 2%is simulated in 30 cycles, while less than 1% is reached in the experiments. A decreased rateof peak "a with a number of cycles is observed in the tests for cyRSuc = 0:54 approachinga plateau, whereas strains keep accumulating in simulation. Such a plateau, however, isattained for lower stress amplitudes, although at a strain levels above that observed in theexperiments. It should be noted that the data is plotted in logarithmic scale for the verticalaxis and therefore deviation from experimental data is somewhat exaggerated for lower cyclicstress amplitudes.Observe that at lower cyclic stress amplitudes the simulated curves (Fig.5.24b) lie abovethe ones experimentally obtained (Fig.5.24a) from initiation of cyclic loading. This suggeststhat either too large plastic strains are simulated at low strain levels or elasticity modelconstants/relation used is/are not appropriate at low level of strains. Simulations results,shown in Fig. 5.25, obtained with the original SANICLAY model reveal that approximatelythe same peak "a are attained during initial cycles for the lowest cyclic stress amplitude.This eliminates the possibility of too large plastic strains induced by the model, implyingthat either adopted elasticity constants  and , or elasticity relation in the proposed modelmight be not appropriate. Changing parameter  would render inappropriate simulation of171IBGB julixution in unxruinyx wywliw triufiiul tystthe consolidation process and would affect all other simulations, thus requiring re-calibrationof some parameters. Parameter , can potentially enhance the simulated elastic response, butthis would require its reduction to values far below the typical range for clays. Simulationswith reduced , = 0:05 are shown in Fig. 5.24(c). Note that the overall simulated responsedoes not change noticeably, except for the lowest cyRSuc, where elastic strains are governingstrain during initial cycles. Thus, only simulations of very low cyRSuc can be controlled tosome extent by elastic parameters. The results for simulations using , = 0:05 are shownonly for demonstration purpose.Undrained cyclic triaxial loading tests are also simulated with the original SANICLAYmodel for the whole range of available cyclic stress amplitudes with the results shown inFig. 5.25. Model parameters and internal variables are the same as those used for theproposed SANICLAY-B model simulations. Parameters hM and vd are not required for theoriginal SANICLAY model. Clearly, no accumulation of peak "a can be simulated as thestress path falls inside the conventional yield surface, rendering purely elastic strains afterthe first half of cycle for all cyRSuc.From the above simulated response of structured clay, the proposed SANICLAY-B modelis considered to give adequate results within 30 cycles of applied cyclic loading, which isequivalent to a M7.5 earthquake (Idriss and Boulanger, 2008). Larger number of cycles mightbe considered with the reduced degree of satisfaction of the proposed model performance forlow cyRSuc.172IBGB julixution in unxruinyx wywliw triufiiul tyst0 5 10 15 20 25 3010−210−1100101Number of cyclesPeak |ε a| (%) compressionextension0.540.570.620.690.790.750.47 τcy/Suc=0.270.36Experiment=v)0 5 10 15 20 25 3010−210−1100101 Number of cyclesPeak |ε a| (%) compessionextensionτcy/Suc=0.270.360.470.540.570.620.690.750.79SANICLAY−B=w)0 5 10 15 20 25 3010−210−1100101Number of cyclesPeak |ε a| (%) compressionextension0.36 τcy/Suc=0.270.570.620.69 0.540.750.790.47SANICLAY−B (ν=0.05)=x)Figure 5.24: Simulations of peak "a versus number of cycles in undrained cyclic triaxialloading for the range of available cyclic stress amplitudes: (a) experimental results fromZergoun and Vaid (1994), and model simulations with (b) calibrated model parameters, (c)calibrated model parameters and , = 0:05173IBGB julixution in unxruinyx wywliw triufiiul tyst0 5 10 15 20 25 3010−210−1100101Number of cyclesPeak |εa| (%) compressionextensionτcy/Suc=0.270.47 0.360.79 0.750.69 0.62 0.57 0.54SANICLAYFigure 5.25: Simulations of peak "a versus number of cycles in undrained cyclic triaxialloading for the range of available cyclic stress amplitudes with the original SANICLAYmodel5CHCI gzxon“titutzy Vrivkz xlvyAvailability of reliable undrained cyclic experiments on clays is very limited and therefore inorder to build a database for model performance less reliable set of experiments is consideredin this section. In order to illustrate if the model is able to reproduce the clay responsein “fast” cyclic loading, although with less reliable experimental data (see section 2.2.1 fordetails), model parameters are calibrated approximately based on available data by Yasuharaet al. (1992). Two reconstituted marine clays, namely two Ariake clays, were used with thefollowing index properties: PI = 69, Gs = 2:65; and PI = 72, Gs = 2:58.A clay slurry was consolidated under 55 kPa of vertical stress in a large consolidationvessel. Triaxial specimens were cut from the block samples formed in the consolidationvessel. The specimens were isotropically consolidated to c = 200 kPa with the drainageallowed through filter paper surrounding the specimen in the triaxial cell. Then isotropicallyconsolidated specimens were subjected to undrained cyclic triaxial loading with constantstress amplitude.boyzl xvliwrvtionBecause few experimental results are available, most of the model parameters are eitherapproximated from known empirical relations or assumed as typical values for clays. A brief174IBGB julixution in unxruinyx wywliw triufiiul tystdescription of an approximate calibration procedure is given below.durumytyrs  unx B Given average PI ∼ 71% and Gs ∼ 2:62, approximation  ≈ 0:6PIfrom Wood (1990) yields  = 0:41. The  = 0:05 is chosen simply as 1R8.durumytyrs bc unx beB Yasuhara et al. (1992) obtained bc = 1:68, correspondingto ϕ = ϕc = 41◦ from Eq. 5.5, based on monotonic undrained triaxial compression testwith a strain rate of 0:05%Rmin. Such values for critical stress ratio b and friction angleϕ are questionable and perhaps are erroneous owing to the high rate of applied monotonicloading resulting in non-equalized pore water pressure distribution within the tested specimen(see section 2.2.1). However, all undrained cyclic triaxial tests were performed with highfrequency loading of 1 Hz. Therefore, it was decided to adopt the reported bc = 1:68 andassumed value of be = 1:65 for simulations of high frequency cyclic loading.durumytyr N B For Ariake clay, N = 1:68 is assumed.durumytyr xB Based on Eq. 5.6 and the adopted value of bc, x = 1:76 is determined.durumytyr ,B For Ariake clay, , = 0:2 is chosen as this value is representative for mostclays.durumytyr XB X = 15 is calibrated from available CIUC test.Vounxing surzuwy purumytyrs hM unx vdB For calibration of parameters hM and vd, plots ofpeak "a versus number of cycles generated for qcyR2c = 0:3 and qcyR2c = 0:25 in undrainedcyclic triaxial loading tests are used. This is an example of the parameters calibration basedon the history of peak "a accumulation with number of cycles rather than already shown forGeorgia kaolin and Cloverdale clay calibration of the parameters based on stress and stress–strain paths. In trial-and-error simulations the most appropriate parameters are hM = 1600and vd = 80 with the final simulation shown in Fig. 5.26.Calibrated model parameters are shown in Tables 5.10 and 5.11.boyzl pzrformvnxzWith the calibration of parameters completed, it is of interest to see how well the model canreplicate the experimental data in cyclic stress ratio CSR versus number of cycles requiredto reach double amplitude axial strain "aA = 5%. The comparison of the experimental datawith the simulated results is shown in Fig. 5.27. CSR = qcyR2c is adopted in accordance175IBGB julixution in unxruinyx wywliw triufiiul tyst0 5 10 15 20 25 30 35 40−5−4−3−2−1012345Number of cyclesPeak ε a (%) experimentsimulation0.3qcy/2σc=0.225Figure 5.26: Calibration of hM and vd based on peak "a versus number of cycles in undrainedcyclic triaxial loading, experimental results after Yasuhara et al. (1992)with the experiments. For CSR = 0:3, model simulates "aA = 5% reached in 10 cycles, whilein experiments the same strain level is reached in 6 cycles. As CSR decreases, simulatedresults fall closer to experiments. The two fitted curves for simulation and experimental datapoints intersect at CSR = 0:24. For further reduced CSR, approximately the same numberof cycles is required to reach "aA = 5%. In general, reasonable results are obtained with thecalibrated model parameters.It is pertinent to say that the use of CSR versus number of cycles for calibration purposesis not suggested as it considers only one particular strain level that a given CSR reaches ina number of cycles and neglects the history of strains from before and after that particularstrain level.176IBHB gummury oz jyri wution unx julixutionTable 5.10: Model parameters for Ariake clayboyzl xon“tvnt Yz“ignvtion kvluzElasticity  0:05, 0:2Critical state bc 1:68be 1:65 0:41Bounding surface Nc 1:68Ne 1:68hM 1600vd 80Rotational hardening X 15x 1:76Destructuration (no effect) ki 0Table 5.11: Initial internal variables for Ariake clayboyzl intzrnvl vvrivwlz Yz“ignvtion kvluzSize of the bounding surface pM 37 kPaStructuration factor hi 1Void ratio z 2Rotation of plastic potential and bounding surface 0Damage parameter y 0Initial stress state (p; q) (1; 0) kPaInitial projection center (pc; qc) (0; 0) kPa5CI hummvry of kzri xvtion vny kvliyvtionThis chapter is focused on verification of SANICLAY-B numerical implementation and val-idation of its performance. A clarification in the difference between Verification and Valida-tion (V&V) processes and their importance is made first. Combination of V&V processes isoften neglected by researchers and instead their models are directly validated against physi-cal data. In order to minimize potential errors, verification of the numerical implementationis performed. In the first step of the verification, a comparison is made between the exactsolution and the numerical simulations using the reduced model (SANICLAY-B without ro-tational hardening, destructuration and bounding surface formulation) in undrained triaxialloading. This step provides confidence in the model implementation for the model featuresthat correspond to the MCC model because it is not possible to obtain an exact solution177IBHB gummury oz jyri wution unx julixution0 5 10 15 20 25 30 35 4000.10.20.30.40.5Number of cyclesCyclic stress ratio, qcy/2 σc experimentsimulationFitted curvesFigure 5.27: Simulation of cyclic stress ratio CSR versus number of cycles required to reach"aA = 5%, experimental results after Yasuhara et al. (1992) and fitted power functiony = vCSR−bfor the full SANICLAY-B model due to its highly nonlinear constitutive equations resultingfrom the rotational hardening, the destructuration, and the bounding surface formulation.In the next step of the verification, numerical simulations of the loads that are different fromtypical conventional triaxial tests are performed employing ISO-error maps. The ISO-errormaps are constructed in the strain space by measuring the error in the computed stressesobtained using small strain increments with small error tolerance vs. the computed stressesobtained using single input strain increments. The results suggest that the adopted numer-ical integration scheme can provide solutions with small estimates of the error when a smallerror tolerance is used. The model is then used for validation of its performance againstexperimental (physical) data in order to gain confidence in its predictive capabilities. Theemphasis is stressed on the validation of the model performance in reliable “slow” undrainedcyclic triaxial tests conducted on specimens of Georgia kaolin and Cloverdale clays. Cali-brated bounding surface parameters hM and vd are order of magnitude different in these clays.This is due to different nature of these two clays. Georgia kaolin clay specimens are pre-pared from slurry making it softer than Cloverdale clay specimens prepared from samples ofnatural structured clay. Simulation results suggest that model can predict a relatively widerange of applied cyclic stress ratios with a single set of model parameters as demonstrated178IBHB gummury oz jyri wution unx julixutionin Figs 5.15, 5.24, and 5.26. For Georgia kaolin clay, experimentally observed pore pressuremeasurements produce unrealistic stress path with the stress ratio well above the criticalstate. This could be attributed to the fact that the test was performed in stress controlledmanner and therefore increased strain rate. In addition, “fast” undrained cyclic triaxialtests on specimens of Ariake clay are also used for the model validation with satisfactoryperformance. It should be mentioned that “fast” undrained cyclic tests are less desirable asmeasured response is less reliable due to the reasons explained in section 2.2.1. The use ofthese data sets is included due to unavailability of reliable test results.The verification of the numerical implementation of the SANICLAY-B model and thevalidation of its performance are perceived successful. The success is qualitatively assessed inmodel’s ability to accumulate plastic strains development and modulus reduction throughoutcyclic loading simulations. The model is next to be used for simulations of clay subjected tomulti-directional cyclic shear.179KC hVcIXaVnBW gz“pon“z in XyxlixbultiByirzxtionvl hhzvrKCF IntroyuxtionIn the preceding chapters, SANICLAY-B model formulation is developed and numericallyimplemented, which is then followed by verification of the implementation and validationof the model performance. The validation is made in comparison against undrained cyclictriaxial test data. It is typical for researchers to fully rely on the limited validation ofconstitutive models for clays subjected to cyclic loading against either triaxial data (e.g.,Kimoto et al., 2015; Maleki and Cambou, 2009; Ni et al., 2015) and simple shear (e.g., Pestanaand Biscontin, 2000) or directly apply a constitutive model for solution of a boundary valueproblem without the validation process (e.g., Huang and Liu, 2015). A very limited numberof work has been focusing on validation of a constitutive model in more advanced experiments(e.g., Wood, 1974, in true triaxial cyclic tests). It is well-known that loads imposed in-siturarely resemble the triaxial conditions, while simple shear tests are more representative of in-situ loading conditions. Biscontin and Pestana (2006) developed a constitutive model basedon simple shear tests and successfully used in 1-D seismic wave propagation. However, for theapplication in modeling actual seismic wave propagation the multi-directionality of seismicloading also has to be accounted (Li and Shen, 1995). In recent years cyclic multi-directionalshear tests become available allowing researchers to test capabilities of their constitutivemodels (e.g., Nouri, 2013). In this chapter, SANICLAY-B response in cyclic multi-directionalshear is explored in order to examine predictive capabilities of the model under complexloading conditions. First, model simulations are performed in comparison with availableexperimental data from 1D linear, 2D linear, circular and figure-8 cyclic shear paths. Then1806BEB Introxuwtionthe results are explored in an attempt to reveal the elements of model formulation preventingsuccessful simulations in some of the cyclic shear paths.The cyclic multi-directional shear tests used for exploration of the model response isperformed in a manner introducing some uncertainties that do not allow the utilization ofthe data for validation of the model. This includes:• non-uniform distribution of stresses,• un-equalized pore pressures, and• rate effect.The first two items are related to the uncertainties in the experiments, while the last itemis related to the model formulation. The first item is discussed in more details in section2.2.1; this issue is minimized with the reduced height to diameter ratio of a specimen in thecyclic multi-directional shear data. The second item is discussed in more details in section2.2.1; this issue is particularly pronounced for clays and is also minimized with reducedheight to diameter ratio of a specimen. The available multi-directional shear tests are allperformed with the application of high frequency cyclic loading which is perceived to notallow for sufficient time required for equalization of pore pressure within a specimen; thusthe pore pressure measured at the boundaries of a specimen are not representative of theelement test. The third item is related to the rate-independent model formulation. Certainassumptions have to be made in the model to account for the rate effect, to be describednext. The issues discussed above make the experimental data inapplicable for the validationof the model performance. Instead, the available data is used to explore the response ofthe SANICLAY-B model in cyclic multi-directional shear tests after certain assumptions aremade for the model.Yi“xlvimzrC It is well known that clay is a rate-dependent material, i.e. its responseis largely affected by the applied rate of strain. When subjected to cyclic loading, such asthose induced by seismic and/or ocean waves, clays experience a very wide range of strainrates. It is therefore of importance to make sure that, in simulating the response of clays,the rate effect is accounted for. The SANICLAY-B model is developed based on the data1816BFB Dyswription oz Gulz oz ayfiiwo wluyfor “slow” undrained cyclic loading to reassure the uniform pore pressure and strains dis-tribution within tested specimens (Zergoun and Vaid, 1994). Thus, the current version ofSANICLAY-B is not meant to account for the strain rate effect, although the rate effect hasbeen recently added into a SANICLAY family model by Rezania et al. (2016). This could bedone in the future for the SANICLAY-B model, once the cyclic loading predictive capabilitiesof the model are well established, which is the goal of this dissertation. The experimentaldata on cyclic multi-directional shearing of clay summarized in Table 2.1 is limited to “fast”cyclic loading tests with the range of 0:1 to 0:5 Hz frequency of applied loading. As theSANICLAY-B model cannot account for the strain rate effect, some measures have to be un-dertaken without further complications of the model. In strain rate dependent constitutivemodels, researchers developed different methods such as nonstationary flow surface (Naghdiand Murch, 1962), Perzyna’s overstress (Perzyna, 1966), and isotach (Leroueil et al., 1985)theories that account for the rate effect. In all of these methods increased strain rate, ingeneral, results into increasing size of a so-called “dynamic yield surface” and consequentlystrength of soil. This methods have been successfully incorporated into classical elastoplas-ticity; in particular, Perzyna’s overstress theory is the most popular one as it requires onlytwo additional parameters. Perzyna’s overstress theory has been also combined with thebounding surface formulation; however, it further complicates the bounding surface formu-lation by the introduction of three additional parameters (e.g., Jiang et al., 2016; Kaliakin,2005). As the current work aims to the development of a “simple” constitutive model alongwith its validation and verification against cyclic loading, it is important to avoid introduc-ing additional features and focus on the cyclic aspects of the model. In this section, forsimulating the rate effect in “fast” undrained cyclic multi-directional shearing, the size ofthe bounding surface is simply increased without implementation of additional features.KCG Yz“xription of Gulf of bzxixo xlvyGulf of Mexico clay is tested in cyclic multi-directional loading with the detailed resultspresented by Rutherford (2012). The geology is described by Rutherford (2012) as follows.The Southern Green Canyon, where Gulf of Mexico samples were taken, lies along the Sigsbee1826BGB Wulivrution oz moxyl purumytyrsEscarpment in the Gulf of Mexico. The lower continental slope consists of uniform surficialdeposits of highly plastic Holocene clays overlying older, stiffer, less plastic, Pleistocene clays.The Sigsbee Escarpment consists of stiff clays that have been exposed by slides and slumps.These sediments have also experienced uplift by the underlying salt and erosion caused bybottom currents. The upper continental rise consists of uniform deposits of highly plasticclays with a thin drape of Holocene clays.Samples were retrieved from the water depth of 1000 m to 1300 m at 30 different locationswith varying degrees of seabed sloping. As the shallower samples were too soft with thenatural water content well above the liquid limit, the testing program was performed onsamples obtained at approximately 11 m depth underneath the seabed. The plasticity indexfor these samples was in the range of 40-55, consisting of 84 % to 99 % of fine particles ofwhich 51 % to 72 % are clay size. Minivane shear strength measured in-situ at this depth isapproximately 20 kPa (indicating normally consolidated clay deposit). The preconsolidationstress p = 72 kPa is measured during consolidation for the samples retrieved from 11 mdepth beneath the seabed. The diameter to height ratio DRH = 4 of the tested specimensis chosen in order to minimize the effects of non-uniform stress distribution along the topand bottom of the sample due to the lack of the complementary shear stress on the lateralboundaries of the tested specimens (as discussed in section 2.2.3).KCH Xvliwrvtion of moyzl pvrvmztzr“durumytyrs  unx B In Fig. 6.1(a), the response of Gulf of Mexico clay in two constantrate of strain consolidation tests (Rutherford, 2012) with an unloading-reloading sequencepresented in the z-log a space. The parameters  and , however, are measured in z-ln p space. Taiebat et al. (2010b) proposed a procedure for plotting the results from z-log aspace in the z-log p space. In the procedure, p˙ = ˙a (1 + 2KM) R3 for the loading part with theassumption of constant KM loading, and p˙ = (˙aR3) (1 + ,) R (1− ,) for the unloading partwith the assumption of elastic response are computed. The results are plotted in Fig. 6.1(b)in the z-log p space with the slope of the normal compression Xc = 0:569 and the slope of therebound line Xr = 0:138 that corresponds to  = XcR ln 10 = 0:247 and  = XrR ln 10 = :06,1836BGB Wulivrution oz moxyl purumytyrsσa (kPa)100 101 102 103 104e0.60.811.21.41.61.8experiment, CRS-GOM7experiment, CRS-GOM6=v)p (kPa)100 101 102 103 104e0.60.811.21.41.61.8 Cc=0.569Cr=0.276=w)Figure 6.1: Calibration of Xc and Xr from oedometer testrespectively.durumytyrs bc unx beB The response of Gulf of Mexico clay in CKMUC and CKMUEtests, performed by Murali (2011), on normally consolidated and overconsolidated specimensis shown in Fig. 6.2. The data is presented in the MIT stress space, normalized by thevalue of maximum vertical stress v;max = 137:9 kPa. Given these results, the critical statefriction angle (from stress state achieved at large strains) is estimated as ϕ = 27:7◦ with thecorresponding critical state ratios shown by the solid line on the plot. The friction anglerenders critical state line slopes in triaxial compression as bc = 6 sinϕR (3− sinϕ) = 1:1and extension as bc = 6 sinϕR (3 + sinϕ) = 0:81.durumytyr N B Parameter Nc = 1:1 is calibrated for Gulf of Mexico clay from the CKMUCtest on a normally consolidated specimen shown in Fig. 6.3. The value corresponds to thestress ratio at the peak q for the provided stress path and coincides with bc. The parameterNe is chosen to be equal to Nc for simplicity.durumytyr ,B Due to the unavailability of KM-unloading stress path required for , cali-bration, the value of , = 0:2 is assumed.durumytyr xB The parameter x for a successful KM-loading simulation can be obtained1846BGB Wulivrution oz moxyl purumytyrs(σv+σh)/(2σv,max)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1(σv-σh)/(2σv,max)-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5φ=27.7°Figure 6.2: Calibration of bc and be (experimental data from Murali, 2011)p/σv,max0 0.2 0.4 0.6 0.8 1q/σv,max00.20.40.60.81Nc=1.1Figure 6.3: Calibration of Nc (experimental data from Murali, 2011)1856BGB Wulivrution oz moxyl purumytyrsusing the following equation proposed by Dafalias et al. (2006):x =2K0ϵ(1− R)Bϵ3K0 + 2K0+ [2 (1− R)−Bb2c ] ϵK0 −b2c; B = − 2 (1 + ,)9 (1− 2,) (6.1)where ϵ = 3R2 for KM-loading path; KM is a measured value of the earth pressure coefficient atrest; and h0 = 3 (1−KM) R (1 + 2KM). The h0 = 0:414 is estimated from the initial stresssate of CKMUC and CKMUE tests on normally consolidated specimens shown in Fig. 6.2. Toget to h0 = 0:414, unreasonable value of x = −13:6 is estimated using Eq. 6.1 suggestingthat during h0-loading the bounding surface should rotate in the direction opposite to thedirection of loading. This is a deficiency in the model that could not be resolved with thecurrent formulation. In order to get a more reasonable model parameter x, Jacky’s equationKM = 1 − sinϕ is used yielding KM = 0:535, h0 = 0:674 and x = 1:79 which is adopted forGulf of Mexico clay.durumytyrs hi unx kiB The destructuration effect as defined for the model by Taiebatet al. (2010b), where the normal consolidation line is changing its slope, cannot be observedfrom Fig. 6.1 with constant rate of strain consolidation tests. Therefore, parameters hi = 1and ki = 0 are assigned to deactivate destructuration mechanism of the model.durumytyr XB For calibration of the parameter X, controlling the rate of anisotropicevolution, the CKMUE test on normally consolidated specimen (Dafalias et al., 2006) is used.Model simulations using different values of X along with the experimental data are shownin Fig. 6.4. At this point, the bounding surface parameters are not calibrated yet and areset to hM = 10U and vd = 0 which eliminate the effect of the bounding surface formulationin the simulations. In comparison to experimental data, simulation using X = 4 yields the“best” fit for the stress state at the end of the test.Vounxing surzuwy purumytyrs hM unx vdB Calibration of hM and vd requires experimentaldata involving the cyclic stress path within the bounding surface. The parameter hM controlsthe response during the earlier cycles of cyclic loading, when the accumulation of the damageparameter y is still not significant to induce considerable reduction of h. The parameter vdcontrols the response during the remaining cycles of cyclic loading. Experimental data shownin Fig. 6.5 corresponds to 1-D linear cyclic shear path on CKM-consolidated specimens. The1866BGB Wulivrution oz moxyl purumytyrs(σv+σh)/(2σv,max)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1(σv-σh)/(2σv,max)-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5C=3, 7, and 11Figure 6.4: Calibration of X (experimental data from Murali, 2011)cyclic loading experiments are performed in a truly undrained fashion with the measurementsof pore pressure at the top and the bottom of the tested specimens. The shear stress , thevertical effective stress v, and the excess pore pressure ∆u are normalized by a verticalconsolidation stress vc. For the model simulations, excess pore pressure is estimated as thedifference between the normal effective stress at the beginning of the test and its variationthroughout the test. Based on the experimental data, the bounding surface parameters arecalibrated to hM = 66 and vd = 3 with the simulations also shown in Fig. 6.5.Inwryusyx si–y oz thy vounxing surzuwy prior to wywliw louxingBFast cyclic shearing of clay leads to apparent increase of its stiffness and strength. Themodel is rate independent and therefore does not have a mechanism to account for suchincrease. Without that, the shear strength of the model may not be sufficient to sustainthe intended shear stress amplitudes in the stress-controlled cyclic shearing simulations, andtherefore the simulations may quickly lead to the critical state and therefore infinite shearstrains without even reaching the desired shear stress amplitudes.To prevent this problem, a virtual increase of the size of the bounding surface is adoptedin all simulations in order to partially account for this shortcoming of the model. Through1876BGB Wulivrution oz moxyl purumytyrsTable 6.1: Model parameters for Gulf of Mexico clayboyzl xon“tvnt xvtzgory Yz“ignvtion Gulf of bzxixo XlvyElasticity  0:06, 0:2Critical state  0:247bc 1:1be 0:81Bounding surface Nc 1:1Ne 1:1hM 66vd 3Rotational hardening X 4x 1:79Destructuration ki 0Table 6.2: Initial internal variables for Gulf of Mexico clayboyzl vvrivwlz Yz“ignvtion Gulf of bzxixo XlvySize of the bounding surface (kPa) pM 45:73Structuration factor hi 1:0Void ratio z 1:59Rotation of bounding surface [0:22;−0:11;−0:11; 0; 0; 0]Damage parameter y 0Stress state  [5; 14; 14; 0; 0; 0]Projection center c [0; 0; 0; 0; 0; 0]iterative procedure it is found that for these cyclic loading simulations the initial pM shouldbe increased at least by a factor of 1.5, and therefore this factor is adopted for all of thepresent cyclic loading simulationModel constants and initial internal variables calibrated for Gulf of Mexico clay are shownin Tables 6.1 and 6.2, respectively. These parameters are to be used in all the followingcyclic loading simulations. The initial internal variables are obtained from an attemptedsimulation of depositional history of clay and its extrusion in the following way. For thispurposed p = pM = 1 kPa and z = 2:45 are used at the beginning of KM-consolidation tov = 72 kPa, as reported pre-consolidation stress by Rutherford (2012). This is then followedby KM-unloading to v = 5 kPa which is considered as an initial state for all simulations.One such simulation is shown in Fig. 6.6 for constant strain rate consolidation tests.1886BGB Wulivrution oz moxyl purumytyrs 119 Fig. 6.4. GOM-5 CKo cyclic test results (a) stress path, (b) stress-strain curve, (c) pore pressure ratio, and (d) shear strain with cycles (a) (b) (c) (d) Normalized vertical stress, σv/σvc Shear strain, 훾 (%)Normalized shear stress, 휏/σ vcNumber of cycles, NNormalized shear stress, 휏/σvcNumber of cycles, N Pore pressure ratio, Δu/σvc Shear strain, 훾 (%)Normalized vertical stress, σv/σvc0 0.2 0.4 0.6 0.8 1Normalized shear stress, τ/σvc-0.4-0.200.20.4Shear strain, γ (%)-15 -10 -5 0 5 10 15 Normalized shear stress, τ/σvc-0.4-0.200.20.4Pore pressure ratio, ∆u/σvc00.20.40.60.81Number of cycles, N0255075100Shear strain, γ (%)-15 -10 -5 0 5 10 15Number of cycles, N0255075100Figure 6.5: Experiment (top) vs. simulation (bottom) results of 1-D linear cyclic shear test,GOM-5, used for hM and vd calibration (experimental data from Rutherford, 2012)1896BHB gimulution oz wywliw multiAxirywtionul shyurσa (kPa)100 101 102 103e0.60.811.21.41.6experiment, CRS-GOM7experiment, CRS-GOM6simulationFigure 6.6: Constant strain rate consolidation test simulation (experimental data fromRutherford, 2012)KCI himulvtion of xyxlix multiByirzxtionvl “hzvrAll the cyclic multi-directional shear tests that are simulated in this paper are listed in Table6.3. The input variables are explained schematically in Fig. 6.7. In the consolidation stage,the soil specimen is consolidated starting from the initial state given in Table 6.2, with thelateral normal strain constrained, to the vertical consolidation stress denoted by vc, and thenormalized magnitude of initial consolidation shear stress c;x along x-direction denoted bya static stress ratio SSRx = c;xRvc and the normalized magnitude of initial consolidationshear stress c;yNN along y-direction denoted by a static stress ratio SSRy = c;yRvc. For thecyclic shearing stage, the cyclic shear stress paths are applied with the quantities of a cyclicstress ratio CSRx = cy,xRvc along the x-direction, CSRy = cy,yRvc along the y-direction,and the number of loading cycles N given in the table. Prior to the circular loading stresspath application, the shear stress is first applied along x-direction to the circular loading11initivl xonsoliyvtion shzvr strzss vlong yByirzxtion is not mzvnt to wz vpplizy yuring thz xonsoliyvtionstvgzA wut is mzvsurzy yuring somz of thz zflpzrimznts1906BHB gimulution oz wywliw multiAxirywtionul shyurpath. It should be noted that the initial shear stress path applied along x-direction is anassumption as there is no indication in the experimental data in which direction the initialshear stress is applied. The references for the figures with the experimental and simulationresults are also provided. It should be noted that as the experimental data is not availabledigitally, the figures from Rutherford (2012) are used which appear in low quality.Table 6.3: Summary of cyclic multi-directional shear tests for Gulf of Mexico clayiz“t typz iz“t 8 zinXon“oliyvtion “tvgz hhzvring “tvgzFigurzvc[kPa] SSRx SSRy CSRx CSRy N1-D linear GOM-5* 2.23 83.6 0.00 0.00 0.20 0.00 90 Fig. 6.5GOM-6 2.23 83.6 0.20 0.00 0.20 0.00 5 Fig. 6.8GOM-7 2.23 83.6 0.20 0.00 0.15 0.00 26 Fig. 6.92-D linear GOM-8 2.23 83.6 0.20 0.00 0.00 0.20 14 Fig. 6.10Circular GOM-10 2.23 83.6 0.00 0.00 0.20 0.20 13 Fig. 6.12GOM-11 2.23 83.6 0.20 -0.06 0.20 0.20 7 Fig. 6.13GOM-12 2.23 83.6 0.20 -0.03 0.15 0.15 28 Fig. 6.14Figure-8 type A GOM-13 2.23 83.6 0.02 -0.04 0.20 0.20 12 Fig. 6.17GOM-14 2.23 83.6 0.20 0.00 0.20 0.20 3 Fig. 6.18GOM-15 2.23 83.6 0.20 0.00 0.15 0.15 11 Fig. 6.19* test used for calibration of the bounding surface parameters hM and vdKCICF FBY vny GBY linzvr xyxlix “hzvrThe first test results are shown in Fig. 6.8 (GOM-6) for clay anisotropically consolidatedwith SSRx = 0:20 and sheared using the 1-D linear cyclic stress path. In 1-D linear cyclicshear, total shear stress and strain are composed of only x and x components, re-spectively. Accumulation of the pore pressure in the simulation follows a similar trend asin the experiment with increasingly larger amplitudes as the number of cycles increases forthe experiment. Shear strain also follows a very similar trend in the simulation and theexperiment. The results shown in Fig. 6.9 (GOM-7) are for clay anisotropically consolidatedto SSRx = 0:15 and sheared using the 1-D linear cyclic shear path. Simulated pore waterpressure ratio follows similar trend as in the experiments. However, simulated shear responseis softer in the simulation: it takes around 16 cycles to reach = 15% in the simulationwhile in the experiment it takes 25 cycles.1916BHB gimulution oz wywliw multiAxirywtionul shyurDHxyzvc cy,x cy,y c=v) htrzsszs vpplizy on v spzximzn=Y=H R I) yRvc xRvcO CCSRxSSRx=w) FBY linzvr yRvc xRvcO CSSRxCSRy=x) GBY linzvr yRvc xRvcO CSSRxCSRyCSRx=y) Cirxulvr yRvc xRvcO CSSRxCSRyCSRx=z) FigurzBM iypz VFigure 6.7: Schematic illustration of multi-directional loading for Gulf of Mexico clay1926BHB gimulution oz wywliw multiAxirywtionul shyur 165 Fig. C.8. GOM-6 CKα cyclic x-direction, 0.1 Hz, τcyc/σ′p = 0.2, δ = 0°. Normalized shear stress, 휏/σ vcNumber of cycles, NPore pressure ratio, Δu/σvc Shear strain, 훾 (%)Normalized shear stress, 휏/σvcNumber of cycles, NNormalized vertical stress, σv/σvc Shear strain, 훾 (%) Normalized vertical stress, σv/σvc0 0.2 0.4 0.6 0.8 1Normalized shear stress, τ/σvc-0.4-0.200.20.4Shear strain, γ (%)-15 -10 -5 0 5 10 15 Normalized shear stress, τ/σvc-0.4-0.200.20.4Pore pressure ratio, ∆u/σvc00.20.40.60.81Number of cycles, N012345Shear strain, γ (%)-15 -10 -5 0 5 10 15Number of cycles, N012345Figure 6.8: Experiment (top) vs. simulation (bottom) results of 1-D linear cyclic shear test,GOM-6 (experimental data from Rutherford, 2012)1936BHB gimulution oz wywliw multiAxirywtionul shyur 166 Fig. C.9. GOM-7 CKα cyclic x-direction , 0.1 Hz, τcyc/σ′p = 0.15, δ = 0°. Normalized shear stress, 휏/σ vcNumber of cycles, NPore pressure ratio, Δu/σvcNormalized shear stress, 휏/σvcNormalized vertical stress, σv/σvc Shear strain, 훾 (%)Shear strain, 훾 (%)Number of cycles, N Normalized vertical stress, σv/σvc0 0.2 0.4 0.6 0.8 1Normalized shear stress, τ/σvc-0.4-0.200.20.4Shear strain, γ (%)-15 -10 -5 0 5 10 15 Normalized shear stress, τ/σvc-0.4-0.200.20.4Pore pressure ratio, ∆u/σvc00.20.40.60.81Number of cycles, N0612182430Shear strain, γ (%)-15 -10 -5 0 5 10 15Number of cycles, N0612182430Figure 6.9: Experiment (top) vs. simulation (bottom) results of 1-D linear cyclic shear test,GOM-7 (experimental data from Rutherford, 2012)1946BHB gimulution oz wywliw multiAxirywtionul shyurThe results for clay anisotropically consolidated with SSRx = 0:20 and sheared using the2-D linear cyclic shear path are shown in Fig. 6.10 (GOM-8). In the figure, the total shearstress and strain are defined as =√ 2x + 2y and =√2x + 2y , respectively. The porepressure in the simulation develops much faster at the beginning of cyclic loading and shortlystarts accumulating very slowly with the number of cycles. In the experiment, however, theaccumulation of pore pressure is gradual and continues accumulating up to about 14 cycles,after which the pore pressure starts reducingN2. At the end of the 16th cycle, the modelpredicts pore pressures that are about half of those in the experiment. The trend followedby the shear strain accumulation in the simulations is similar to that in the experiment withminor differences in the rate of accumulation.A summary of the results for the experiments and the simulations in 1-D and 2-D linearcyclic stress paths is shown in Fig. 6.11. The colors used in the results shown for thesimulations are in accordance with those for the experiments. While the pore pressurepredictions are poor, accumulation of the shear strain with the number of cycles N issimulated relatively well for all considered tests. Note that the test data shown in red isused for the calibration of the bounding surface model parameters, while the other tests areused to evaluate the model performance.KCICG Xirxulvr xyxlix “hzvrThe results of circular cyclic shear on clay anisotropically consolidated with SSRx = 0 (i.e.,under KM condition) and sheared with the cyclic stress ratios CSRx = CSRy = 0:20 areshown in Fig. 6.12 (GOM-10). It is clear that the model fails to replicate the experimentalresults. The pore pressure ratio is largely underestimated in the simulation although thetrend of the positive pore pressure accumulation is followed. The amplitude of the shearstrain variation in both x and y directions is also largely underestimated as the number ofcycles increases. The reason for such poor performance of the model is possibly rooted inits formulation that produces an elastic response for loading directions along neutral loadingpath (investigated in Section 6.5.2). Moreover, the trend of the continuous shear strainsaccumulation along x axis is not captured.12this is pzrxzivzy vs vn inyixvtion of unrzlivwlz porz przssurz mzvsurzmznts1956BHB gimulution oz wywliw multiAxirywtionul shyur 167 Fig. C.10. GOM-8 CKα cyclic x-direction, 0.1 Hz, τcyc/σ′p = 0.2, δ = 90°. Normalized shear stress, 휏/σ vcNumber of cycles, NPore pressure ratio, Δu/σvcNormalized shear stress, 휏/σvcNormalized vertical stress, σv/σvc Shear strain, 훾 (%)Shear strain, 훾 (%)Number of cycles, N Normalized vertical stress, σv/σvc0 0.2 0.4 0.6 0.8 1Normalized shear stress, τ/σvc-0.4-0.200.20.4Shear strain, γ (%)-15 -10 -5 0 5 10 15 Normalized shear stress, τ/σvc-0.4-0.200.20.4Pore pressure ratio, ∆u/σvc00.20.40.60.81Number of cycles, N0481216Shear strain, γ (%)-15 -10 -5 0 5 10 15Number of cycles, N0481216Figure 6.10: Experiment (top) vs. simulation (bottom) results of 2-D linear cyclic shear test,GOM-8 (experimental data from Rutherford, 2012)1966BHB gimulution oz wywliw multiAxirywtionul shyur 120 Fig. 6.5. Cyclic tests (a) development of pore pressures, (b) accumulation of plastic strains (a) (b) 120 Fig. 6.5. Cyclic tests (a) development of pore pressures, (b) accumulation of plastic strains (a) (b) Number of cycles, NNumber of cycles, NPore pressure ratio, Δu/σ vcShear strain, 훾x (%) Number of cycles, N0 20 40 60 80 100Pore pressure ratio, ∆u/σvc00.250.50.751Number of cycles, N0 20 40 60 80 100Shear strain, γx (%)-40-2002040Figure 6.11: S marized experiments (left) and simulations (right) of 1-D and 2-D linearcyclic shear tests, GOM-5, GOM-6, GOM-7, GOM-8 (experimental data from Rutherford,2012)1976BHB gimulution oz wywliw multiAxirywtionul shyur 169 Fig. C.12. GOM-10 CKR circular 0.1Hz, τcyc/σ′p = 0.2. Number of cycles, NShear strain, 훾 (%)Number of cycles, N 169 Normalized shear stress, 휏y/Shear strain, 훾y (%)Normalized shear stress, 휏x/σvc Shear strain, 훾x (%)Pore pressure ratio, Δu/σvcNormalized Shear Stress, τx/σvc-0.3 -0.2 -0.1 0 0.1 0.2 0.3Normalized Shear Stress, τy/σvc-0.3-0.2-0.100.10.20.3Shear Strain, γx (%)-20 -15 -10 -5 0 5 10 15 20Shear Strain, γy (%)-20-15-10-505101520Pore Pressure Ratio, ∆u/σvc00.20.40.60.81Number of Cycles, N02468101214Shear Strain, γ (%)-20 -15 -10 -5 0 5 10 15 20Number of Cycles, N02468101214Figure 6.12: Experiment (top) vs. simulation (bottom) results of circular cyclic shear test,GOM-10 (experimental data from Rutherford, 2012)1986BHB gimulution oz wywliw multiAxirywtionul shyurThe results of circular cyclic shear on clay anisotropically consolidated with SSRx = 0:20and sheared with the cyclic stress ratios CSRx = CSRy = 0:20 and CSRx = CSRy = 0:15are shown in Figs. 6.13 (GOM-11) and 6.14 (GOM-12), respectively. The accumulation ofthe pore pressure for both cyclic stress ratios is positive in the experiments, while the simu-lations predict negative pore pressures that are outside of the plotting limits and are plottedseparately in Fig. 6.15. Shear strains simulated by the model are following a slightly differenttrend with well captured accumulation of the x and an overestimation of accumulation ofthe y.The results of the experiments and simulations for circular cyclic shear tests are summa-rized in Fig. 6.16. The model predictions of the pore pressure are not captured by the modelin all of the simulations. The variation of the shear strains is also not captured by the modelfor GOM-10 test, and captured relatively well for GOM-11 and GOM-12 tests.KCICH FigurzBM xyxlix “hzvrThe results of the figure-8 cyclic shear loading on clay anisotropically consolidated withSSRx = 0:02 and SSRy = −0:04 (i.e., under approximately KM condition) are shown inFig. 6.17 (GOM-13). Most of the pore pressure is accumulated within the 1st cycle inthe simulation, while gradual increase of the pore pressure is observed in the experimentthroughout the whole course of loading. The shear strains are simulated to be accumulatingin the negative y direction with the amplitudes of x and y strains increasing with thenumber of cycles. The experiment, however, exhibits shear strain accumulating in positivex and negative y directions simultaneously and to a much smaller degree in y-direction.The amplitudes of of x and y strains increase to a much larger degree in the experiment.The results of figure-8 cyclic shear loading on clay anisotropically consolidated withSSRx = 0:20 and sheared in cyclic loading with the cyclic stress ratios CSRx = CSRy =0:20 and CSRx = CSRy = 0:15 are shown in Figs. 6.18 (GOM-14) and 6.19 (GOM-15),respectively. The pore pressure variation for the cyclic stress ratio CSRx = CSRy = 0:20follows the same trend in the simulation as in the experiment exhibiting the smaller amplitudewith the number of cycles in the simulation. For the cyclic stress ratio CSRx = CSRy = 0:15,the pore pressure variation follows the trend similar to that in the experiment but with the1996BHB gimulution oz wywliw multiAxirywtionul shyur 170 Fig. C.13. GOM-11 CKα circular 0.1Hz, τcyc/σ′p = 0.2. 170 . . . . , ′ . . Number of cycles, NShear strain, 훾 (%)Number of cycles, NShear strain, 훾y (%)Shear strain, 훾x (%)Pore pressure ratio, Δu/σvc Normalized shear stress, 휏x/σvcNormalized shear stress, 휏y/σ vcNormalized Shear Stress, τx/σvc-0.4 -0.2 0 0.2 0.4Normalized Shear Stress, τy/σvc-0.4-0.200.20.4Shear Strain, γx (%)-20 -15 -10 -5 0 5 10 15 20Shear Strain, γy (%)-20-15-10-505101520Pore Pressure Ratio, ∆u/σvc00.20.40.60.81Number of Cycles, N012345678Shear Strain, γ (%)-20 -15 -10 -5 0 5 10 15 20Number of Cycles, N012345678Figure 6.13: Experiment (top) vs. simulation (bottom) results of circular cyclic shear test,GOM-11 (experimental data from Rutherford, 2012)2006BHB gimulution oz wywliw multiAxirywtionul shyur 171 Fig. C.14. GOM-12 CKα circular 0.1Hz, τcyc/σ′p = 0.15. 171 Number of cycles, NShear strain, 훾 (%)Number of cycles, NShear strain, 훾y (%)Pore pressure ratio, Δu/σvc Normalized shear stress, 휏x/σvcNormalized shear stress, 휏y/σ vc Shear strain, 훾x (%)Normalized Shear Stress, τx/σvc-0.4 -0.2 0 0.2 0.4Normalized Shear Stress, τy/σvc-0.4-0.200.20.4Shear Strain, γx (%)-20 -15 -10 -5 0 5 10 15 20Shear Strain, γy (%)-20-15-10-505101520Pore Pressure Ratio, ∆u/σvc00.20.40.60.81Number of Cycles, N051015202530Shear Strain, γ (%)-20 -15 -10 -5 0 5 10 15 20Number of Cycles, N051015202530Figure 6.14: Experiment (top) vs. simulation (bottom) results of circular cyclic shear test,GOM-12 (experimental data from Rutherford, 2012)2016BHB gimulution oz wywliw multiAxirywtionul shyurPore Pressure Ratio, ∆u/σvc00.20.40.60.81Number of Cycles, N051015202530GOM-10GOM-11GOM-12Figure 6.15: Initially negative accumulation of simulated pore pressure for circular cyclicshear tests GOM-11 and GOM-12pore pressure accumulation in simulations about half of that in the experiment at the endof the loading. The variation of the shear strains for both cyclic stress ratios is abouty = 0 value and accumulates along the x axis. The simulations however deviate from theexperiments with the accumulation of the shear strains in both positive x and negative ydirections. Such deviation is reduced noticeably for the larger CSRx = CSRy = 0:20 cyclicstress ratio, although it is not fully eliminated.The experimental and simulation results for figure-8 type A multi-directional test aresummarized in Fig. 6.20. Although the model predicts positive pore pressures similarly tothe experiments, quantitatively simulated results provide smaller values for GOM-13 andGOM-15 tests. The pore pressure accumulation is predicted well by the model for GOM-14test. In terms of the shear strains, the simulation for GOM-13 test exhibits much smallerincrease of the amplitude of x and much smaller increase of the amplitude along with theincorrect direction of accumulation of y with the number of cycles. Simulations resultsare comparable qualitatively and quantitatively well with the experimental results in termsof the accumulation of x for both larger and smaller cyclic stress ratios in GOM-14 andGOM-15 tests, respectively. Satisfactory predictions of accumulation of y are observed forthe larger cyclic stress ratio in GOM-14 test and unsatisfactory for the smaller cyclic stressratio in GOM-15 test as the negative accumulation is simulated.2026BHB gimulution oz wywliw multiAxirywtionul shyur 124 Fig. 6.7. Circular tests (a) development of pore pressures, (b) accumulation of plastic strains x-direction, (c) accumulation of plastic strain y-direction (a) (b) (c) Number of cycles, NNumber of cycles, NNumber of cycles, NPore pressure ratio, Δu/σ vcShear strain, 훾x (%)Shear strain, 훾y (%) Number of Cycles, N0 5 10 15 20 25 30Pore Pressure Ratio, ∆u/σvc00.250.50.751Number of Cycles, N0 5 10 15 20 25 30Shear Strain, γx (%)-40-2002040Number of Cycles, N0 5 10 15 20 25 30Shear Strain, γy (%)-20-1001020Figure 6.16: Summarized experiments (left) and simulations (right) of circular cyclic sheartests, GOM-10, GOM-11, GOM-12 (experimental data from Rutherford, 2012)2036BHB gimulution oz wywliw multiAxirywtionul shyur 172 Fig. C.15. GOM-13 CKR figure-8, 0.1 Hz x-direction, τcyc/σ′p = 0.2, δ = 0°. 172 . . . , . , ′ . , . Number of cycles, NShear strain, 훾 (%)Number of cycles, NPore pressure ratio, Δu/σvc Normalized shear stress, 휏x/σvcNormalized shear stress, 휏y/σ vc Shear strain, 훾x (%)Shear strain, 훾y (%) Normalized Shear Stress, τx/σvc-0.3 -0.2 -0.1 0 0.1 0.2 0.3Normalized Shear Stress, τy/σvc-0.3-0.2-0.100.10.20.3Shear Strain, γx (%)-20 -15 -10 -5 0 5 10 15 20Shear Strain, γy (%)-20-15-10-505101520Pore Pressure Ratio, ∆u/σvc00.20.40.60.81Number of Cycles, N024681012Shear Strain, γ (%)-20 -15 -10 -5 0 5 10 15 20Number of Cycles, N024681012Figure 6.17: Experiment (top) vs. simulation (bottom) results of figure-8 cyclic shear test,GOM-13 (experimental data from Rutherford, 2012)2046BHB gimulution oz wywliw multiAxirywtionul shyur 173 Fig. C.16. GOM-14 CKα figure-8, 0.1 Hz x-direction, τcyc/σ′p = 0.2, δ = 0°. 173 ′ Number of cycles, NShear strain, 훾 (%)Number of cycles, NPore pressure ratio, Δu/σvc Normalized shear stress, 휏x/σvcNormalized shear stress, 휏y/σ vc Shear strain, 훾x (%)Shear strain, 훾y (%) Normalized Shear Stress, τx/σvc-0.4 -0.2 0 0.2 0.4Normalized Shear Stress, τy/σvc-0.4-0.200.20.4Shear Strain, γx (%)-20 -15 -10 -5 0 5 10 15 20Shear Strain, γy (%)-20-15-10-505101520Pore Pressure Ratio, ∆u/σvc00.20.40.60.81Number of Cycles, N00.511.522.53Shear Strain, γ (%)-20 -15 -10 -5 0 5 10 15 20Number of Cycles, N00.511.522.53Figure 6.18: Experiment (top) vs. simulation (bottom) results of figure-8 cyclic shear test,GOM-14 (experimental data from Rutherford, 2012)2056BHB gimulution oz wywliw multiAxirywtionul shyur 174 Fig. C.17. GOM-15 CKα figure-8, 0.1 Hz x-direction, τcyc/σ′p = 0.15, δ = 0°. 174 Number of cycles, NShear strain, 훾 (%)Number of cycles, NPore pressure ratio, Δu/σvc Normalized shear stress, 휏x/σvcNormalized shear stress, 휏y/σ vc Shear strain, 훾x (%)Shear strain, 훾y (%) Normalized Shear Stress, τx/σvc-0.4 -0.2 0 0.2 0.4Normalized Shear Stress, τy/σvc-0.4-0.200.20.4Shear Strain, γx (%)-20 -15 -10 -5 0 5 10 15 20Shear Strain, γy (%)-20-15-10-505101520Pore Pressure Ratio, ∆u/σvc00.20.40.60.81Number of Cycles, N024681012Shear Strain, γ (%)-20 -15 -10 -5 0 5 10 15 20Number of Cycles, N024681012Figure 6.19: Experiment (top) vs. simulation (bottom) results of figure-8 cyclic shear test,GOM-15 (experimental data from Rutherford, 2012)2066BHB gimulution oz wywliw multiAxirywtionul shyur 127 Fig. 6.9. Figure-8 tests (a) development of pore pressures, (b) accumulation of plastic strains in x-direction, and (c) accumulation of plastic strain in y-direction (a) (b) (c) Number of cycles, NPore pressure ratio, Δu/σ vcShear strain, 훾x (%)Shear strain, 훾y (%) Number of cycles, N Number of cycles, N Number of Cycles, N0 5 10 15 20Pore Pressure Ratio, ∆u/σvc00.250.50.751Number of Cycles, N0 5 10 15 20Shear Strain, γx (%)-40-2002040Number of Cycles, N0 5 10 15 20Shear Strain, γy (%)-40-2002040Figure 6.20: Summarized experiments (left) and simulations (right) of figure-8 cyclic sheartests, GOM-13, GOM-14, GOM-15 (experimental data from Rutherford, 2012)2076BHB gimulution oz wywliw multiAxirywtionul shyurKCICI Vxxumulvtzy porz prz““urz rvtio vny “hzvr “trvin“The performance of the SANICLAY-B model is summarized in Fig. 6.21 where the values ofthe pore pressure and shear strains at the end of cycles indicated in the legend of the plot areincluded for the experimental and simulation results. The lines along the plotted diagonalsof the plots are indicative of the simulation results coinciding with the experimental data. Itshould be noted that accumulated shear strains for 2-D linear case are not provided becausethey could not be interpreted from the available experimental data.In most of the simulations, the pore pressure is underestimated with the largest deviationsobserved for the specimens without initial consolidation shear stress, i.e., SSRx ≈ 0 (i.e.,under approximately KM condition). This could be attributed to the increased size of thebounding surface, bringing the initial stress state to high OCR prior to cyclic loading, inorder to indirectly account for the rate-effect induced by the “fast” cyclic loading used in theexperiments. The model performance in terms of the accumulated shear strains is satisfactorywith enough room for further refinements of its predictive capabilities, in particular forthe specimens anisotropically consolidated with SSRx ≈ 0 (i.e., under approximately KMcondition) prior to cyclic multi-directional shear (GOM-10 and GOM-13), where the initialconsolidation shear stress is nearly absent and the response is governed by the fabric. Alsosimulation results overpredict shear strains for the specimen anisotropically consolidatedwith SSRx = 0:2 and sheared using the 1-D linear stress path with the smallest cyclic stressratio CSRx = 0:15 (GOM-7). The accumulated x values are simulated satisfactory bythe model in most tests as suggested by the points lying within the 5% difference range.Considerable deviations are observed for the simulations of GOM-7 and GOM-10 tests. Theaccumulated y values are lying within the 5% difference for circular and figure-8 cyclic shearstress paths for the specimens anisotropically consolidated with SSRx = 0:2 and shearedwith CSRx = CSRy = 0:20 (GOM-11 and GOM-14). For the smaller cyclic stress ratioCSRx = CSRy = 0:15 the accumulated y values are slightly outside of the range. Theaccumulated y value for figure-8 cyclic shear stress path for the specimens anisotropicallyconsolidated with SSRx ≈ 0 (GOM-13) is largely deviating between the simulation and theexperiment.2086BIB Yfiploring moxyl simulution rysultsExperimental pore pressure ratio ∆u/σvc-0.2 0 0.2 0.4 0.6 0.8 1Simulated pore pressure ratio ∆u/σvc-0.200.20.40.60.811-D linear - GOM-5 (cycle 90)1-D linear - GOM-6 (cycle 4.5)1-D linear - GOM-7 (cycle 24)2-D linear - GOM-8 (cycle 13.5)circular - GOM-10 (cycle 13)circular - GOM-11 (cycle 7)circular - GOM-12 (cycle 24)figure-8 - GOM-13 (cycle 5)figure-8 - GOM-14 (cycle 2)figure-8 - GOM-15 (cycle 8)Experimental pore pressure ratio ∆u/σvc-0.2 0 0.2 0.4 0.6 0.8 1Simulated pore pressure ratio ∆u/σvc-0.200.20.40.60.81Experimental shear strain γx (%)-30 -20 -10 0 10 20 30Simulated shear strain γx (%)-30-20-100102030Experimental shear strain γy (%)-30 -20 -10 0 10 20 30Simulated shear strain γy (%)-30-20-100102030Figure 6.21: Simulations vs. experiments for pore pressure ratio, and shear strains at theend of the cycles indicated in the legend for cyclic shear tests on Gulf of Mexico clayKC5 Exploring moyzl “imulvtion rz“ult“In the previous section, SANICLAY-B model simulation results of 1-D and 2-D linear, circu-lar and figure-8 cyclic shear tests are presented. The model predictions for the cyclic sheartests performed on the specimens anisotropically consolidated with SSRx = 0:2 (i.e., withthe presence of initial consolidation shear stress) are perceived as satisfactory. However, con-siderable deviations are observed in all the tests performed on the specimens anisotropicallyconsolidated with SSRx ≈ 0 (i.e., under approximately KM condition, without the presence ofinitial consolidation shear stress). In particular, the shear strains in the circular cyclic shearstress path (GOM-10) are much smaller in the simulation than in the experiment through-out the test. The shear strains simulated for figure-8 cyclic shear stress path (GOM-13) are2096BIB Yfiploring moxyl simulution rysultsmuch smaller in terms of the amplitudes of the strains in each cycle and accumulate in thedifferent direction than observed in the experiment. In this section, potential SANICLAY-Bmodel features that can influence the unsatisfactory response are explored. This includesexploring the influence of the hypoelastic formulation adopted for the elastic response of themodel and the resemblance of the cyclic multi-directional shear stress path to the neutralloading.KC5CF Hypozlv“tixityThe hypoelastic relation adopted in all SANICLAY family of models is given as"˙e =s˙2G+p˙3KI (6.2)where z is a current void ratio, K and G are bulk and shear moduli. The bulk modulus Kis expressed as a function of the void ratio z, the mean effective stress pK =p (1 + z); G =3K (1 + z)2 (1 + ,)(6.3)where  is a slope of the rebound line in the z– ln p space, and , is a Poisson’s ratio.Significant underestimation of the accumulated shear strains for the circular (GOM-10),and the accumulation of the shear strains in the wrong direction for figure-8 (GOM-13) cyclicshear stress paths on the specimens anisotropically consolidated with SSRx ≈ 0 (i.e., underapproximately KM condition) are explored on the influence of the adopted elasticity. Forcomparison, simulations are also performed with the z and p values set to be constant in thehypoelastic formulation (constant elasticity) and equal to their initial values at the beginningof cyclic loading. The results for the two most deviating cyclic stress paths are shown interms of the simulated shear strain paths in Fig. 6.22. It can be observed that the results ofsimulations obtained using the original hypoelastic formulation and the constant elasticityare similar. Neither the amplitudes nor the directivity of the simulated shear strains areinfluenced by the constant elasticity. This suggests that the observed deficiencies in themodel response are attributed to the plastic formulation of the model.2106BIB Yfiploring moxyl simulution rysultsShear strain, γx (%)-3 -2 -1 0 1 2Shear strain, γy (%)-3-2-1012hypoelasticityconstant elasticity=v) xirxulvr B GdbBFEO hhgx R EShear strain, γx (%)-20 -15 -10 -5 0 5 10 15 20Shear strain, γy (%)-20-15-10-505101520hypoelasticityconstant elasticity=w) gurzBM B GdbBFHO hhgx R E:EGFigure 6.22: Comparison between SANICLAY-B simulation results with the model hypoe-lasticity and constant elasticityKC5CG czutrvl lovyingThe classical plasticity formulation requires that the stress point stays on the yield surfaceduring the loading inducing plastic response. In order to ensure the requirement is satisfied,the consistency condition applies which results into purely elastic response for the loadingdirections tangential to the yield surface which is also known as a neutral loading. Thebounding surface formulation adopted for the SANICLAY-B model, although is intended toreduce the elastic region confined by the yield surface in the classical plasticity, inherits theneutral loading deficiency from the classical plasticity. Thus, the neutral loading can takeplace even within the bounding surface. In such cases, the neutral loading is defined as theloading path tangential to the loading surface. The loading surface within the boundingsurface formulation is defined by the current stress state  and the projection center cshown in Fig. 6.23. The projection center serves as the center of homology between theloading surface f = 0 and the bounding surface F = 0. When the stress increment ˙ isorthogonal to the loading surface gradient@f@as shown in the figure, the loading directionis tangential to the loading surface and thus would produce purely elastic response.In order to study how close the cyclic shear stress paths simulated in the previous section2116BIB Yfiploring moxyl simulution rysultsF = 0¯cf = 0@fR@˙Figure 6.23: Schematic illustration of neutral loading tangential to the loading surface f = 0for SANICLAY-B modelare to the neutral loading, an angle Θ = arccos(@f@: ˙)is used. When Θ = 90◦, theneutral loading takes place and the stress increment ˙ produces purely elastic response.In the SANICLAY-B formulation, whenever Θ S 90◦ in the stress-controlled simulationsstudied here, the stress reversal takes place and the projection center is updated to the pointof the stress reversal that reduced the Θ. In this way, Θ ≤ 90◦ is expected to be the case inall of the simulations. The variations of Θ for the 1-D linear cyclic shear paths in GOM-5and GOM-6, the circular cyclic shear paths in GOM-10 and GOM-11, and the figure-8 cyclicshear paths in GOM-13 and GOM-14 are plotted in Fig. 6.24 for the first 7 cycles of theloadings. The red circles in the plots indicate the moment of projection center updated tothe point of stress reversal.The results for the 1-D linear cyclic shear paths are provided in Figs. 6.24(a,b) as referencesimulation results to illustrate how, with the proposed projection center updated at stressreversals, the model produces the desired reduction of Θ during cyclic loading. Notice thatfor the 1-D linear shear on the specimens anisotropically consolidated with SSRx = 0 (GOM-5), the Θ is kept approximately at 20◦ shortly after the commencement of the loading. Thisindicates that the stress increments ˙ are acting at a similar angle Θ in relation to theloading surfaces produced by updating projection center in unloading instances. For the2126BIB Yfiploring moxyl simulution rysults1-D linear shear on the specimen anisotropically consolidated with SSRx = 0:2 (GOM-6),the Θ is oscillating around 20◦. The update of the projection center at the stress reversalssuccessfully keeps Θ = 20◦ in both cases which allows for simulations of the loading awayfrom the neutral loading that would produce a purely elastic response.It is expected that for the cyclic multi-directional shear paths, the variation of Θ is sig-nificant. However, the variation should be such that the cyclic shear paths are sufficientlyaway from the neutral loading to produce observed in the experiments accumulation ofshear strains with the number of cycles. The results for circular cyclic shear on the speci-mens anisotropically consolidated with SSRx = 0 (GOM-10) and SSRx = 0:2 (GOM-11) areshown in Figs. 6.24(c,d). Observe that the Θ in the simulations of GOM-10 test is varyingbetween 45◦ and 90◦. This variation of Θ is insufficient to reproduce the strains observed inthe experiment as illustrated in the previous section. For GOM-11 test, Θ is varying between22◦ and 90◦. A significant portion of GOM-11 test is simulated with Θ varying between 25◦and 35◦ thereby producing shear strains that are comparable with those observed in the ex-periment. The results for figure-8 cyclic shear on the specimens anisotropically consolidatedwith SSRx = 0:02 (GOM-13) and SSRx = 0:2 (GOM-14) are shown in Figs. 6.24(e,f). Inboth cases, the variation of Θ is between 17◦ and 90◦ thereby producing large plastic shearstrains. However, the accumulation of shear strains for GOM-13 test is in the directiondifferent from that in the experiment.The comparison of the angle Θ, between the gradient@f@of the loading surface and theapplied stress increment ˙, variation in 1-D linear, circular and figure-8 cyclic shear pathsis provided. While the Θ is kept around 20◦ for 1-D linear cyclic shear paths, it is varyingfor circular and figure-8 cyclic shear paths. The Θ ≈ 20◦ for 1-D linear path is a result ofthe proposed update of the projection center during stress reversals and is one of the mainfeatures for successful simulations in cyclic loading. The update of the projection centeris also demonstrated to provide the variation of Θ Q 90◦ for cyclic multi-directional shearpaths that enables plastic shear strains simulations. However, in the circular cyclic shearpath on the specimen anisotropcially consolidated with SSRx = 0 (GOM-10), the variationof Θ demonstrates that the simulated loading is the closest to the neutral loading out of allthe tests considered here. As a result accumulated plastic shear strains are insufficient to2136BIB Yfiploring moxyl simulution rysultsθ (deg.)0 10 20 30 40 50 60 70 80 90Number of cycles, N01234567=v) FBY linzvr B GdbB5O hhgx R EΘ (deg.)0 10 20 30 40 50 60 70 80 90Number of Cycles, N01234567=w) FBY linzvr B GdbBKO hhgx R E:GΘ (deg.)0 10 20 30 40 50 60 70 80 90Number of Cycles, N01234567=x) xirxulvr B GdbBFEO hhgx R EΘ (deg.)0 10 20 30 40 50 60 70 80 90Number of Cycles, N01234567=y) xirxulvr B GdbBFFO hhgx R E:GΘ (deg.)0 10 20 30 40 50 60 70 80 90Number of Cycles, N01234567=z) gurzBM B GdbBFHO hhgx R E:EGΘ (deg.)0 10 20 30 40 50 60 70 80 90Number of Cycles, N01234567=f) gurzBM B GdbBFIO hhgx R E:GFigure 6.24: The angle Θ between the applied stress increment ˙ and the gradient to theloading surface @fR@ for cyclic tests simulations. Red circles indicate stress reversal occur-ring during the corresponding increment.2146B6B gummury oz simulutyx wywliw multiAxirywtionul shyurreplicate the observed response in the experiment.KCK hummvry of “imulvtzy xyxlix multiByirzxtionvl“hzvrSimulations of cyclic multi-directional shear tests using the SANICLAY-B model are ex-plored in this chapter. Available experimental data on Gulf of Mexico clay is used for thispurpose. The data includes 1-D linear, 2-D linear, circular and figure-8 cyclic shear paths.Calibration of the model parameters in section 6.3 follows similar procedure used in section5.3 for triaxial test data. The calibration of the newly proposed bounding surface param-eters hM and vd is performed against 1-D cyclic shear test (GOM-5). The model is thenused to simulate all of the other available tests in section 6.4. As illustrated in section 6.4.4,the model significantly underpredicts pore pressure ratios observed in the experiments. Itshould be noted however that pore pressure measurements during cyclic loading in cyclicmulti-directional shear device, inheriting its shortcomings from the direct simple shear de-vice, is difficult to perform for clays and it is suggested to focus on strains accumulationinstead (Andersen, 2009). A satisfactory model performance is illustrated in terms of theshear strains accumulation. Significant deviations from the experiments are observed forcircular (GOM-10) and figure-8 (GOM-13) cyclic shear tests on the specimens anisotropi-cally consolidated to SSRx ≈ 0 (i.e., approximately under KM condition). In the attempt tounderstand the source of such deviations, the hypoelasticity formulation used for the elasticresponse of the model and closeness of the simulated shear paths to the neutral loading areexplored in section 6.5. The hypoelasticity appears to have no effect when compared tothe cyclic loading simulations using constant elasticity. The study on the simulated shearpaths suggests that the circular loading path GOM-10 is the closest to the neutral loadingpath and therefore produces insufficient plastic shear strains. The source of the deviation forGOM-13 could not be established. One of the potential sources of such deviation is that theintended figure-8 stress path could not be maintained in the experiment as can be observedin Fig. 6.17, while in the simulations such shear path is maintained throughout the test. The2156B6B gummury oz simulutyx wywliw multiAxirywtionul shyurinability to maintain the desired shear path is also observed for experiments of the circularloadings GOM-10 and GOM-12, but to a much smaller extent.The goal of this chapter is to examine model response in the explored cyclic shear pathsand explain potential shortcomings in the model formulation. The findings of this chaptersuggest that current model formulation inherits from classical plasticity its inability to inducesufficient plasticity for circular shear path when compared to experimental data. While suchshear path is perhaps not commonly encountered, it could potentially lead to underestimationof clay response when encountered. The findings provide further opportunity to enhancemodel formulation for future research.216LC hummvryA Xonxlu“ionA vnygzxommznyvtion“ for Futurz lorkLCF hummvryIn this dissertation, the SANICLAY-B constitutive model is developed for the description ofthe response of clay subjected to cyclic loading. The model is prepared for future researchand applications in geotechnical earthquake engineering for clay deposits as a numericaltool package. The package is framed for numerical solution of boundary value problemsthrough the process of the development of model formulation, its numerical implementation,verification of the implementation, validation of its performance against laboratory elementtesting, and simulations of cyclic multi-directional shear tests.The bounding surface formulation is incorporated as an extension of the SANICLAYfamily of models to simulate plastic strains within the conventional yield surface duringundrained cyclic loading. The formulation of the proposed SANICLAY-B model in chapter3 is developed based on the experimental observations and study from other constitutivemodels that were successful to some extent. To this end, in addition to the well-establishedbounding surface formulation, a projection center updated at the instance of “stress rever-sal” and degradation of plastic modulus with cumulative plastic deviatoric strain are adoptedfollowing the trends in the reviewed constitutive models (see Section 2.4.2). This allows toincrease plastic strains as the cyclic loading proceeds according to the experiments. Anevolving projection center is newly proposed in a way that allows it to keep its relative posi-tion within the bounding surface established at the moment of stress reversal. This featureis required in order to satisfy uniqueness of the image stress within the bounding surface for-mulation. A damage parameter is newly proposed which is accumulated with accumulation2177BEB gummuryof plastic deviatoric strain thereby allowing for continuous degradation of plastic modulusrequired to model the observed response of clay subjected to cyclic loading. In order tohighlight the effects of the new features of the model, comparisons are made between thesimulations using the full SANICLAY-B model and the simulations using its simplified ver-sion in which the novel features have been deactivated by prescribing specific values to thenew model parameters hM and vd, which therefore can be referred to as SANICLAY. Sucha comparison shows that the full model has significantly improved predictive capabilitiesin capturing the response of clays in undrained cyclic loading. A number of characteristicaspects of response in undrained cyclic clay behavior have been qualitatively simulated bythe model. In particular, model’s ability to realistically account for shear modulus reductionand damping ration evolution with shear strain amplitude gives credibility to the model forits use in seismic site response analysis.The constitutive equations of the SANICLAY-B model are numerically implementedwithin a constitutive driver provided in chapter 4 that can be used in numerical platformssuch as finite element or finite difference for solution of boundary value problems. Themethod chosen for the integration of the equations is refined explicit integration with auto-matic supstepping and error control which reassures that the intended stress-strain responseis integrated correctly for any input strain increment. This method reassures that intendedstress-strain response is simulated for a wide range of input strain increments thereby pro-viding confidence in application of the model for solution of boundary value problems wherethere is no control over the size of the input strain increment. Details of the steps under-taken for the development of the constitutive driver are presented along with accompanyingpseudo-codes. A number of the model specific details of the integration are included. Inaddition, the Bardet’s constraints method that is used to simulate laboratory element testsfor the model validation process are presented. This approach is chosen to concentrate onthe verification of the model numerical integration and validation of the model constitutivefeatures in the following chapter. The constitutive driver interaction with OpenSEES finiteelement platform is also provided as an example of the model implementation in a finiteelement code.The verification of the implementation and the validation of the model performance in2187BEB gummuryundrained cyclic triaxial loading are provided in chapter 5. The verification of the implemen-tation is first performed for the reduced model against the exact solution of the constitutiveequations for undrained triaxial compression and extension tests. The results suggest thatthe considered exact solution can be closely replicated by the implementation of the modeleven with the strains applied in one single step. The next step of the verification processis conducted using ISO-error maps illustrating the robustness of chosen integration algo-rithm. The results suggest that the implementation of the model can provide solutions withsmall estimates of the error. The validation of the model is then performed in compari-son with undrained cyclic triaxial loading tests on reconstitutive Georgia kaolin, structuredCloverdale, and reconstitutive Ariake clays. For reconstituted Georgia kaolin clay, simula-tions of undrained cyclic triaxial loading closely match experimentally observed strains forhigher cyclic stress amplitudes, whereas for lower cyclic stress amplitudes the model slightlyoverpredicts the observed strains. Reasonable simulation results are obtained for structuredCloverdale clay in undrained cyclic triaxial loading with a wide range of available cyclic stressamplitudes, although for lower cyclic stress amplitudes the strains are also overpredicted.Finally, model simulation of Ariake clay in “fast” undrained cyclic triaxial loading illustratesthat the new model parameters approximately calibrated based on sufficient experimentaldata can be successfully used to reproduce experimentally obtained cyclic stress ratio CSRversus number of cycles required to reach a certain strain level. Overestimation of strainsfor lower cyclic stress amplitudes is attributed to model hypoelasticity resulting into elasticshear modulus that is smaller than those observed in the experiments.Simulations of cyclic multi-directional shear tests on Gulf of Mexico clay are performedin chapter 6. It should be noted that the experiments on Gulf of Mexico clay are performedin “fast” undrained cyclic loading; therefore, pore pressure measurements in truly undrainedtests are not reliable, providing effective stresses that cannot be fully relied on. However,the cyclic shear response is independent of pore pressure measurements (e.g., Andersen,2009) and can be used for exploring model performance in multi-direction cyclic shear.Simulated shear paths include 1-D linear, 2-D linear, circular, and figure-8 cyclic loadingtests. The model is able to capture the observed cyclic response of Gulf of Mexico clay onthe specimens anisotropically consolidated with the initial consolidation shear stress. The2197BFB Wonwlusionresults of simulations, however, do not compare well to the experiments on the specimensanisotropically consolidated without the initial consolidation shear stress. The main findingfrom the results reveals that the model formulation cannot produce sufficient plasticity forthe loading close to neutral loading which is not in agreement with the observed response.LCG Xonxlu“ionThe proposed model has an adequate level of complexity for capturing a number of impor-tant characteristic features of clay response in cyclic loading. The most important modelfeatures enabling successful simulations are the projection center, updated at the instance of“stress reversal” and evolving with the bounding surface evolution, and the degradation ofplastic modulus through the introduced damage parameter evolving with cumulative plasticstrain increments. The evolving projection center and damage parameters are novel features,while the update of the projection center at the instance of “stress reversal” and degradationof plastic modulus with cumulative plastic strain increments are adopted following reviewedconstitutive models (see Section 2.4.2). However, none of the reviewed rate-independentbounding surface models demonstrated the ability to predict well a wide range of cyclicstress amplitudes. Some models were missing features of the damage effect proposed in thiswork, others were not able to adequately simulate the stress-strain response, and yet othersused model parameters that were not feasible to be calibrated for complex loading such asthose imposed by seismic, wind and tidal waves (e.g., cyclic stress amplitude parameter).In addition, none of the models were used for simulations of cyclic multi-directional shear,which in some cases is more representative to the loading conditions in situ. Robust imple-mentation of the model, verification of the implementation, validation of its performance,and simulations of multi-direction cyclic shear provide a package of the model that can beused in numerical solution of boundary value problems. Its application is primarily aimed,but not limited, to numerical solution of boundary value problems requiring appropriaterepresentation of the response of clay subjected to cyclic loading. The package provides aframework that can be used by future researchers and, when necessary, further refinementsof the model can be incorporated.2207BGB fywommynxutions zor zutury rysyurwhLCH gzxommznyvtion“ for futurz rz“zvrxhThe proposed SANICLAY-B model has been designed with particular attention on main-taining the simplicity of the formulation and the model calibration. However, if necessary,one may address some of the shortcomings of the proposed model as discussed below. Thiswould perhaps require a higher level of sophistication in the formulation and subsequently inthe calibration process of the model. Although the model has significantly improved from itspredecessors, it still may not sufficiently reproduce variations of shear modulus and dampingratio observed in the experiments at very small strains. This shortcoming may be due tothe very simple elastic relation (hypoelasticity) adopted in the model and may be resolvedby adopting a more nonlinear elasticity formulation that can provide hysteretic stress–strainrelationships at small strain levels (e.g., Jung and Yune, 2011; Nouri, 2013) which wouldrequire additional parameters. Also the present model may overpredict the plastic strains atsmall cyclic stress amplitudes with larger number of cycles (S100). The hardening functionh may be the key issue for addressing some of the shortcomings of the model in a widerange of cyclic stress amplitudes and a more extensive study on this function may provide afurther enhanced formulation, if needed. In addition, as more reliable data become availableit would help to establish range of model constants (in particular, new model constants hMand vd). Finally the time dependency and creep have not been addressed in the presentedformulation due to the focus on the cyclic aspects of the model and may be considered in thefuture. Rate-dependent SANICLAY model has been previously presented by Rezania et al.(2016) and can be followed for the SANICLAY-B model which would require additional con-siderations and adoption of additional model parameters for bounding surface fromulation(e.g., Jiang et al., 2016; Kaliakin, 2005). 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I: Basic model and its application.’, Intyrnutionul Journul zorbumyriwul unx Unulytiwul aythoxs in Gyomywhuniws N(5), 453–476.237VC cumzrixvl Implzmzntvtion indpznhEEhNumerical solutions for complex boundary value problems are important tools in today’s en-gineering design. Finite element programs are one of the most attractive numerical solutionapproaches. With rapidly changing hardware, improving numerical computation algorithms& solvers, advancements in types of elements and development of more representative con-stitutive models for materials, it is essential for a finite element program to be able to adoptto these changes. Open System for Earthquake Engineering simulations (OpenSEES) pro-gram is an object-oriented, open source software that allows user to develop finite elementcomputer applications for simulating the response of structural and geotechnical systemssubjected to earthquake excitation and other hazards. In OpenSEES, an object class de-fines specification of methods and internal data of an object, and an abstract class definesspecification of abstract methods and interface for its subclasses. The advantages of suchobject-oriented structure are in that it allows an analyst to introduce new subclasses whichgreatly increases flexibility of the computational capabilities of the software. More detailsof OpenSees structure can be found in McKenna (1997) and brief description for options ofmodel implementation into OpenSees are described in Appendix A.Below a brief outline is given on the requirements for generation of a subclass of NDMa-terial abstract class.VCF gzquirzmznt“ for “uwxlv““z“ of cYbvtzrivl xlv““Any n-dimensional material implemented into OpenSees must meet the interface of NDMa-terial abstract class to be recognized as an object of OpenSEES domain. The abstract class238ABFB Gynyrution oz xynumiwully linkyx livrurycontains virtual methods that can be part of a subclass (in this case SANICLAY-B is asubclass of NDMaterial). The virtual methods that are called purely virtual methods (set tobe equal to zero in NDMaterial abstract class) are engaged by other objects and, therefore,must be over-written by a subclass. As an example setTrialStrain (") is a purely virtualmethod of NDMaterial abstract class that passes " to NDMaterial subclass from an Elementobject.A NDMaterial subclass could be either a part of built-in OpenSEES materials or gen-erated as a dynamically linked library (dll). In current implementation SANICLAY-B isgenerated as a dll file.VCG Gznzrvtion of yynvmixvlly linkzy liwrvryInstructions that cover steps required on adding a DLL file for a new material in WindowsOS are given within the source files of OpenSees. The instructions are focused on procedureof project generation within Visual Studio. In the next paragraph a main requirement for adll file is described.During a run OpenSees looks for a material dll file. The name of a dll file should besame as the name of the material in an input file. Once inside a dll file, OpenSees looks fora procedure with the name OPS SANICLAY B. Within the procedure, OpenSees parses theinput parameters it stores in integer and double arrays to make sure that number of param-eters are sufficient for SANICLAY B constructor. If so, OpenSees creates a new instanceof SANICLAY B material by calling a constructor with the input parameters assigned fromthe storage arrays.When a material instance is assigned to an element a copy of the material instanceis passed to that element using a getCopy () method which is a purely virtual method ofNDMaterial abstract class and SANICLAY B subclass. Depending on the element type acopy of either plane strain or 3-dimensional SANICLAY B is assigned.239ABGB aoxyl rylutyx uxxitionul zyuturysVCH boyzl rzlvtzy vyyitionvl fzvturz“In this section, additional options implemented for SANICLAY-B material are described.This includes evolution of stress reversal variable to avoid spurious update of projectioncenter during stress reversal, linear elastic stage, update of material stage and model pa-rameters, and initialization of model variables. The first two features can be called from anOpenSEES script, and the last feature is called automatically either while material initial-ization or switching from linear elastic to elastoplastic stages.VCHCF jpyvtz of projzxtion xzntzr yuring “trz““ rzvzr“vlModel formulation defines stress reversal at anytime loading index becomes negative, i.e., a Q0. This approach, however, does not recognize the magnitude of an input strain incrementand direction of loading it results into. As a result, it can impose superficial update ofc to the latest stress state thereby directly impacting elastic/elastoplastic modulus. For aboundary value problem such abrupt changes of modulus result into abrupt changes in globalstiffness. When Newton-Raphson algorithm is used in finite element code, variation of globalstiffness controls convergence of the algorithm. Therefore, it is of interest to make sure thatstress reversal is identified correctly. Below a new approach is introduced for identification ofstress reversal. This approach relies on accumulation of strain in unloading which dependson the the magnitude of elastic strain and the direction of loading.XonxzptA scalar valued cumulative strain "cum is introduced which initially is set to zero andevolves according to the following equation:"˙cum = ‖"˙e‖ (− cos ) ; (A.1)where cos =@f@: ˙R(‖ @f@‖‖˙‖)and "˙e is an elastic strain increment, and is anangle between gradient of loading surface@f@and elastic stress increment ˙. The "˙e ="˙ (i.e., elastic response) for unloading, and "˙e = "˙ − "˙p (i.e., elastoplastic response) forloading/reloading. The negative sign is required to accumulate "cum in unloading, i.e., when240ABGB aoxyl rylutyx uxxitionul zyuturysF = 0cf = 0@fR@˙˙Figure A.1: Illustration of stress increments: shown in blue, tangential to the loading surface(neutral loading) , and, shown in red, opposite in direction to @fR@−1 ≤ cos Q 0. In loading/reloading case, 0 Q cos Q 1, and, therefore, "cum reduces inmagnitude. To avoid negative "cum, a restriction "cum ≥ 0 is introduced.Notice that, for neutral loading (shown in blue in Fig. A.1), cos = 0, and, no ‖"˙e‖ isaccumulated to "cum. For unloading case, when stress increment is opposite to the gradient ofloading surface (shown in red in Fig. A.1), cos = −1, and, ‖"˙e‖ is completely accumulatedto "cum. For intermediate loading directions, a portion of ‖"˙e‖ accumulated to "cum dependson cos z value. Similarly, a reduction of "cum in loading/reloading depends on cos value,and once the "cum = 0 no further reduction is recorded.The above cumulative strain "cum is used to identify an instance of stress reversal whichtakes place whenever it is equal to a new input parameter, threshold strain "ˆ, i.e., whenever"cum = "ˆ. At this instance, sr is updated to the latest stress state , and "cum is set to zero.The next step will be simulated using updated sr. In case of "cum S "ˆ, it is necessary to finda pseudo-time increment i˙ (0 Q i˙ Q 1) for i˙ "˙ that results into "cum = "ˆ, update sr, set"cum to zero, and simulate(1− i˙)"˙ using the updated variables. As the increase of "cum ispossible only in unloading (which, therefore, is simulated as an elastic response), finding "˙resulting into "cum = "ˆ for a forward euler integration scheme, with tangent elastic modulus,is a trivial matter. However, for the modified Euler integration scheme this requires extrasteps to be undertaken which is to be discussed next.241ABGB aoxyl rylutyx uxxitionul zyuturysVpplixvtion to moyi zy Eulzr intzgrvtion “xhzmzModified Euler integration scheme uses two trials j = 1 and 2 whose combination forms asecond order approximation. The j = 1 trial is based on the initial stress state and internalvariables, while the j = 2 trial uses stress state and internal variables calculated at the endof the j = 1 trial. Therefore, Eq. A.1 for the modified Euler trials is written in the followingform:"˙(N)cum =‖ ("˙e)(N) ‖(− cos (N)) ; cos (N) = @f@(N): ˙R(‖ @f@(N)‖‖˙‖)"˙(2)cum =‖ ("˙e)(2) ‖(− cos (2)) ; cos (2) = @f@(2): ˙R(‖ @f@(2)‖‖˙‖) (A.2)where superscripts refer to the modified Euler trials j = 1 and 2. As in Eq. A.1, ("˙e)(N;2) = "˙(i.e., elastic response) for unloading, and ("˙e)(N;2) = "˙− ("˙p)(N;2) (i.e., elastoplastic response)for loading/reloading. Also notice that for both modified Euler trials the same elastic stressincrement ˙ = De((N); z(N)): "˙ is used because it is required only to know the direction ofstress increment which is unaltered for the two trials.Modified Euler "˙cum is then computed using the trials as"˙cum =12["˙(N)cum + "˙(2)cum](A.3)which is used for accumulation of "cum.At the end of the modified Euler trials, the cummulative "cum is checked for the stressreversal in a similar manner as discussed in the previous section. For the case of "cum S "ˆ,finding a scalar i˙ sr requires extra computations. As mentioned in the previous section theincrease of "cum is possible only in unloading which is simulated as an elastic increment andthereby ("˙e)(N;2) = "˙. Based on Eqs A.2 and A.3, this case requires solution of the followingequation for i˙ sr:f(i˙ sr)= ("ˆ− "cum)− 12i˙ sr‖"˙‖(− cos (N) − cos ( _i sr))= 0 (A.4)where cos (_i sr) =@f@(_i sr): ˙R(‖ @f@(_i sr)‖‖˙‖)and (_i sr) is the stress state at the end ofj = 1 resulting from the strain increment i˙ sr"˙.242ABGB aoxyl rylutyx uxxitionul zyuturysSubstituting  + i˙ sr˙ for  in@f@, the following equation is obtained:@f@(_i sr)= V+Wi˙ sr +X1b(_i sr)(A.5)whereV =3 ((“− “sr)− (p− psr) )−[((“− “sr)− (p− psr) ) : − 13(N2 − 32 : )2 (p− psr)]IW =3 (t“− p˙ )−[(t“− p˙ ) : − 13(N2 − 32 : )2p˙]IX =3 (“sr − psr )−[(“sr − psr ) : + 13(N2 − 32 : )(pM + 2psr)]I(A.6)andb(_i sr) =(−B +√B2 − 4VX)R (2V) (A.7)which is obtained by plugging ¯ = ¯sr+ b(_i sr)((_i sr) − sr)into a bounding surface equa-tion F = 0 resulting into a quadratic equation withVb =32(v : v)− 3 (v : ) v− (N2 − 3 : ) v2Bb =3 (“sr : v)− 3 [(“sr : ) v+ (v : ) psr] +[3 ( : ) psr −(N2 − 32 : )pM]vXb =32(“sr − psr ) : (“sr − psr )−(N2 − 32 : )psr (pM − psr)(A.8)and v = (“− “sr) + i˙ sr“˙ and v = (p− psr) + i˙ srp˙.The solution of i˙ sr in Eq. A.4 can be obtained numerically using, for example, bisectionmethod. The value of i˙ sr corresponds to the amount of strain increment at the end of which"cum = "ˆ and sr is updated to the latest .It is possible that, during unloading, a strain increment is so large that j = 1 trial iselastic and j = 2 trial is elastoplastic. This can result into "cum Q "ˆ which cannot trulyeliminate the possibility of "cum S "ˆ throughout the strain increment. To make sure suchpossibility is accounted for, it is needed to separate elastic response from elastoplastic forthe given strain increment. The point of such separation is the stress state (a=M) at which243ABGB aoxyl rylutyx uxxitionul zyuturys@f@(a=M): ˙ = 0. The gradient of loading surface at (a=M) can be obtained by substituting + i˙(a=M)˙ for  in loading surface gradient equation which yields the following:@f@(a=M)= V(a=M) +W(a=M)i˙(a=M) +X(a=M)1b(a=M)(A.9)whereV(a=M) =3 [(“− “sr)− (p− psr) ]−[((“− “sr)− (p− psr) ) : − 13(N2 − 32 : )2 (p− psr)]IW(a=M) =3 [“˙− p˙ ]−[(“˙− p˙ ) : − 13(N2 − 32 : )2 p˙]IX(a=M) =3 [“sr − psr ]−[(“sr − psr ) : − 13(N2 − 32 : )2 (pM − 2 psr)]Iand the value of b(a=M) is obtained by solving@f@(a=M): ˙ = 0 which yieldsb(a=M) = − X(a=M) : V(a=M) :  +(W(a=M) : )i˙(a=M)= − X(a=M)V(a=M) +B(a=M)i˙(a=M)(A.10)In Eq. A.9, the unknown is i˙(a=M) whose solution is obtained by setting¯ = sr + b(a=M)( + i˙(a=M)˙ − sr)into F = 0:i˙(a=M) =√−32Xi : Xi − Xi32Vi : Vi − Vi (A.11)244ABGB aoxyl rylutyx uxxitionul zyuturyswithVi =[(B(a=M)“sr − X(a=M)“˙)− (B(a=M)psr − X(a=M)p˙) ]Vi =(N2 − 32 : )(B(a=M)psr − X(a=M)p˙) (B(a=M)pM −(B(a=M)psr − X(a=M)p˙))Xi =[(V(a=M)“sr − X(a=M) (“− “sr))− (V(a=M)psr − X(a=M) (p− psr)) ]Xi =(N2 − 32 : )(V(a=M)psr − X(a=M) (p− psr)) (V(a=M)pM −(V(a=M)psr − X(a=M) (p− psr))):The obtained value of i˙(a=M) is substituted for i˙sr in Eq. A.4, and in case if f(i˙(a=M))Q=0 stress reversal takes place within i˙(a=M). The actual i˙sr is then solved for using theprocedure described before and sr is updated.Detailed procedure implemented for solution of "cum for the modified Euler integration isshown in Tables A.1, A.2, A.3, and A.4. This includes computation of pseudo-time incrementi˙(a=M) at the moment of neutral loading, required to isolate elastic and elastoplastic parts ofunloading process, and i˙ sr required to establish instance of "cum = "ˆ.Table A.1:[sr; zsr;src ; "srcum; i˙sr]= “trz““ rzvzr“vl “tvtz(; z;c; "cum; "˙(N;2)cum ; i˙ ; "˙)Step Description1 Set "srcum = "cum + 0:5("˙(N)cum + "˙(2)cum), and, if "srcum Q 0, set "srcum = 02 Check if: "srcum S "ˆ?YES: GO TO styp INO: GO TO styp G3 Check for elastic (j = 1) trial followed by elastoplastic (j = 2) trial: "˙(N)cum S 0 and "˙(2)cum Q 0?YES: compute i˙(a=M) = p“zuyoBtimz inxrzmznt vt nzutrvl lovying (; q; z;c; "˙),f_i = "˙cum yiffzrznxz(; q; z;c; "˙; i˙(a=M))and GO TO styp HNO: set[sr; zsr;src ; i˙sr]=[; z;c; i˙]and EXIT4 Check if: f_i Q 0?YES: set i˙ = i˙(a=M) and GO TO styp INO: set[sr; zsr;src ; i˙sr]=[; z;c; i˙]and EXIT5 Update variables for stress reversal:i˙ sr = p“zuyoBtimz inxrzmznt vt “trz““ rzvzr“vl(; q; z;c; "˙; i˙);"˙sr = i˙ sr"˙, zsr = z− (1 + z) tr ("˙sr);˙srN = Desec (; z; "˙sr) : "˙sr; ˙sr2 = Desec ( + ˙srN ; z; "˙sr) : "˙sr; sr =  + 0:5 (˙srN + ˙sr2 );"srcum = 0and EXIT.245ABGB aoxyl rylutyx uxxitionul zyuturysTable A.2: i˙(a=M) = p“zuyoBtimz inxrzmznt vt nzutrvl lovying (; q; z;c; "˙)Step Description1 Set ncount = 0, set i˙i−N = 0, i˙i = i˙ and, given ˙ = De (; z) : "˙, compute correspondingf(a=M) values:set b(a=M)(i−N) = rvtio b( + i˙i−N˙; q;c), b(a=M)(i) = rvtio b( + i˙i˙; q;c),and compute corresponding corresponding @fR@(a=M)(i−N), @fR@(a=M)(i);compute f(a=M)(i) = @fR@(a=M)(i) : .2 For i˙i+N =(i˙if(a=M)(i−N) − i˙i−Nf(a=M)(i))R(f(a=M)(i−N) − f(a=M)(i)), compute f(a=M)(i+N):set b(a=M)(i+N) = rvtio b( + i˙i+N˙; q;c)and compute corresponding @fR@(a=M)(i+N),f(a=M)(i+N) = @fR@(a=M)(i+N) : .3 Check if: f(a=M)(i+N) Q TOL?YES: EXIT,NO: go to styp J.4 Perform the following operations:If f(a=M)(i+N) f(a=M)(i) Q 0:set i˙i−N = i˙i, and f(a=M)(i−N) = f(a=M)(i).If f(a=M)(i+N) f(a=M)(i) S 0:set f(a=M)(i−N) = f(a=M)(i−N)f(a=M)(i)f(a=M)(i) + f(a=M)(i+N).Set i˙i = i˙i+N, f(a=M)(i) = f(a=M)(i+N) andi˙i+N =(i˙if(a=M)(i−N) − i˙i−Nf(a=M)(i))R(f(a=M)(i−N) − f(a=M)(i)).5 Set ncount = ncount + 1 and compute new value for f(a=M)(i+N) as before.6 Check if: f(a=M)(i+N) S TOL or ncount ≤ MAXITS?YES: return to styp H,NO: set i˙ (a=M) = i˙i+N and EXIT.246ABGB aoxyl rylutyx uxxitionul zyuturysTable A.3: i˙ sr = p“zuyoBtimz inxrzmznt vt “trz““ rzvzr“vl(; q; z;c; "˙; i˙)Step Description1 Set ncount = 0, set i˙i−N = 0, i˙i = i˙ and compute corresponding f _i values:f( _i)(i−N) = "˙cum yiffzrznxz(; q; z;c; "˙; i˙i−N)f( _i)(i) = "˙cum yiffzrznxz(; q; z;c; "˙; i˙i)2 For i˙i+N =(i˙if( _i)(i−N) − i˙i−Nf( _i)(i))R(f( _i)(i−N) − f( _i)(i)), compute f( _i)(i+N):f( _i)(i+N) = "˙cum yiffzrznxz(; q; z;c; "˙; i˙i+N).3 Check if: f( _i)(i+N) Q TOL?YES: EXIT,NO: go to styp J.4 Perform the following operations:If f( _i)(i+N) f( _i)(i) Q 0:set i˙i−N = i˙i, and f( _i)(i−N) = f( _i)(i).If f( _i)(i+N) f( _i)(i) S 0:set f( _i)(i−N) = f( _i)(i−N)f( _i)(i)f( _i)(i) + f( _i)(i+N).Set i˙i = i˙i+N, f( _i)(i) = f( _i)(i+N) andi˙i+N =(i˙if( _i)(i−N) − i˙i−Nf( _i)(i))R(f( _i)(i−N) − f( _i)(i)).5 Set ncount = ncount + 1 and compute new value for f( _i)(i+N) as before.6 Check if: f( _i)(i+N) S TOL or ncount ≤ MAXITS?YES: return to styp H,NO: set i˙ el = i˙i+N and EXIT.Table A.4: f tr_i = "˙cum yiffzrznxz(; q; z;c; "˙; i˙tr)Step Description1 Given ˙ = De (; z) : "˙, compute cos (N):cos (N) =(@f@: ˙)R(‖ @f@‖‖˙‖)2 For [b _i ; N ] = rvtio b( + i˙ tr˙; q;c), compute @fR@ _i :V = 3 ((“− “c)− (p− pc) )−[((“− “c)− (p− pc) ) : − N3(N2 − 32 : )2 (p− pc)]IW = 3 (“˙− p˙ )− [(“˙− p˙ ) : − N3(N2 − 32 : )2p˙]IX = 3 (“c − pc )−[(“c − pc ) : + N3(N2 − 32 : )(pM + 2pc)]I@f@ _i= V+Wi˙ tr +X1b _i3 Compute cos _i :cos _i =(@f@ _i: ˙)R(‖ @f@ _i‖‖˙‖)4 Compute f tr_i :f tr_i = ("ˆ− "cum)− N2 i˙ tr‖"˙‖(− cos (N) − cos _i )247ABGB aoxyl rylutyx uxxitionul zyuturysVCHCG ainzvr zlv“tix “tvgzA linear elastic stage is an option for quick application of self-weight loading. However, itshould be noted that this is valid only for one-dimensional case with KM in-situ stress condi-tions. Linear elasticity requires much lesser number of steps and is more stable than eithernonlinear elastic or elastoplastic analysis that are computationally much more expensive.Stresses from linear elastic stage then can be passed to SANICLAY-B model.According to linear elasticity stress-strain relationship is expressed as = KelKst"vI+ 2GelKstz (A.12)where KelKst = 3GelKst (1 + ,elKst) R (2 (1− 2,elKst)) a linear elastic bulk modulus, ,elKst is aninput linear elastic Poisson’s ratio, GelKst is an input linear elastic shear modulus, "v is avolumetric strain, z is a deviatoric strain tensor, and I is an identity tensor.In vectors and matrix form Eq. A.12 can be expressed as followsNN2233N2N323=KelKst + 4GelKstR3 KelKst − 2GelKstR3 KelKst − 2GelKstR3 0 0 0KelKst − 2GelKstR3 KelKst + 4GelKstR3 KelKst − 2GelKstR3 0 0 0KelKst − 2GelKstR3 KelKst − 2GelKstR3 KelKst + 4GelKstR3 0 0 00 0 0 2GelKst 0 00 0 0 0 2GelKst 00 0 0 0 0 2GelKst"NN"22"33"N2"N3"23(A.13)The linear elastic analysis requires specification of two input parameters ,elKst and GelKst.,elKst could be assigned based on assumption of KM in-situ stress conditions for which "22 ="33 = 0 and 22 = 33 (and all shear components are equal to 0). With these conditions,in-situ stress ratio KM is expressed asKM =33NN=(KelKst − 2GelKst3)(KelKst +4GelKst3) (A.14)248ABGB aoxyl rylutyx uxxitionul zyuturysWith the previously given definition of KelKst, Eq. A.14 can be rearranged to express ,elKst interms of the desired KM,elKst =KM1 +KM(A.15)Input parameter GelKst controls only deformations induced during linear elastic stage. Avery large value should be assigned to make sure that initial geometry is not changed.VCHCH jpyvtz of mvtzrivl “tvgz vny othzr moyzl pvrvmztzr“In order to update a SANICLAY-B model parameter, two methods in SANICLAY B.cppare used. The first one sets an ID for a parameter and the second updates the parameter.The list of parameters that could be updated include loadStage parameter used for switchingfrom linear elastic to elastoplastic stage.VCHCI Initivlizvtion of hVcIXaVnBW “tvtz vvrivwlz“Once material is either initialized or a loading stage is updated from linear elastic to elasto-plastic, projection center c, size of the Bounding surface pM, deviatoric back-stress ratio representing anisotropy, and void ratio z are initialized.Projection center is simply set toc =  (A.16)where  is either an initial stress state or a stress state from linear elastic stage. Notethat with such initialization  cannot be radially mapped onto the bounding surface; and,therefore, image stress ¯, required for plasticity, cannot be obtained. As a measure ofmitigation, it has been described previously that such cases are treated as elastic therebynot requiring obtaining the image stress.The next step of initialization is determination of stress tensor Bp that is a stress stateon the Bounding surfaceBp = OCRm  (A.17)where OCRm is a modified overconsolidation ratio.249ABGB aoxyl rylutyx uxxitionul zyuturysThe is initialized as = (f ) rRx (A.18)where f is either 0 (no initial anisotropy) or 1 (conventional SANICLAY anisotropy), r =“Rp with “ =  − pI and p = tr () R3, and x model’s saturation parameter required forsaturation of .Finally pM can be initialized using equation for Bounding surface F = 0pM =32(“Bp − pBp ) : (“Bp − pBp )(N2 − 32 : ) + pBp (A.19)where “Bp = Bp − pBpI and pBp = tr (Bp) R3In the first step of z initialization, void ratio at pM is first determinedzp0 = zincl −  ln (pM) (A.20)where zincl is a void ratio on the isotropic normally consolidation line at p = 1,  is a slopeof compression line in z–ln p plane. Then z for the current stress state is initialized asz = zp0 +  ln (pMRp) (A.21)where  is a slope of unloading-reloading line in z–ln p plane.250WC eropvgvtion of Hvrmonix hzi“mixlvvz“ through hvturvtzy hoft XlvyFollowing the general approach described by McGann and Arduino (2011), effective stresssite response analysis is carried out in OpenSEES for a 20 m deep deposit of normally con-solidated clay characterized by SANICLAY and SANICLAY-B models with the parameterslisted in Table B.1. The profile is assumed to be on an infinite slope with a 2% grade. Thegroundwater table is assumed to be located at a depth of 1 m. A 2D plane strain column ofsoil is modeled using 20 nine-node quadrilateral elements. Nine-node quadrilateral elementswith both displacement and pore pressure degrees of freedom enable the model to trackchanges in pore pressure and effective stress during the earthquake excitation. The nodes atthe base of the soil column are fixed against vertical translation and the pore pressure nodesabove the groundwater table are fixed in the pore pressure degree of freedom (representingan open drainage condition). Periodic boundary conditions are assumed for the purposes ofmimicking a 1D analysis. A Lysmer and Kuhlemeyer (1969) dashpot is utilized to accountfor the finite rigidity of the underlying elastic medium such as bedrock. At its base the soilcolumn is subject to a harmonic excitation with frequency of 2 Hz and maximum accelerationof 0.1g (Fig. B.1). The analysis consists of a gravity stage followed by a dynamic excitation0 1 2 3 4−0.12−0.08−0.0400.040.080.12time (s)acceleration (g)Figure B.1: Time history of the input acceleration at the base.251Appynxifi VB dropugution oz Hurmoniw gyismiw kuvys through guturutyx gozt Wluystage.Table B.1: Model parameters used for the analysisboyzl xon“tvnt Yz“ignvtion kvluzElasticity  0:03, 0:2Critical state bc (be) 1:0 (1:0) 0:15Bounding surface Nc (Ne ) 1:0 (1:0)hM 200vd 60s 1Rotational hardening X 5x 1:7Destructuration ki 0Figs. B.2 and B.3 summarize the time e histories of shear stress and shear strain atdepth of 5 m for the site response analysis using the SANICLAY and SANICLAY-B models,respectively. The comparison between the results at this depth shows that the SANICLAY-B model generates larger amplitudes of shear strains and larger residual shear strains. Forbetter comparison of the soil response in this example, the corresponding plots of shearstress vs. shear strain are presented in Fig. B.4. The figure clearly shows the advantageof the SANICLAY-B model in simulation of the stress-strain loops; something that in thisproblem is primarily simulated as elastic response with the SANICLAY model. It should benoted that depending on the characteristics of the motion and properties of the soil column,the SANICLAY can under-damp or over-damp the stress-strain loops, leading to differentsite response analysis results.252Appynxifi VB dropugution oz Hurmoniw gyismiw kuvys through guturutyx gozt Wluy1 2 3 4−10−50510time (s)τ (kPa)0 1 2 3 4−1−0.500.51time (s)γ (%)Figure B.2: Simulation results for time histories of shear stress and shear strain atdepth of 5 m in the soil column using the SANICLAY model.1 2 3 4−10−50510time (s)τ (kPa)0 1 2 3 4−1−0.500.51time (s)γ (%)Figure B.3: Simulation results for time histories of shear stress and shear strain atdepth of 5 m in the soil column using the SANICLAY-B model.253Appynxifi VB dropugution oz Hurmoniw gyismiw kuvys through guturutyx gozt Wluy−1.2 −0.8 −0.4 0 0.4−10−50510γ (%)τ (kPa)−1.2 −0.8 −0.4 0 0.4γ (%)Figure B.4: Simulation results for shear stress-strain response at depth of 5 m in the soilcolumn using the SANICLAY (left) and SANICLAY-B (right) models.254XC eropvgvtion of Evrthquvkzinyuxzy hzi“mix lvvz“ throughhvturvtzy hoft XlvyEffective stress site response analysis is carried out in OpenSEES following the general ap-proach described by (McGann and Arduino, 2011). A 20-m soil profile composed of uniformnormally consolidated clay material is characterized by SANICLAY and SANICLAY-B mod-els with the parameters listed in Table B.1. The profile is assumed to be on an infinite slopewith a 2% grade and a groundwater table located at a depth of 1 m. Total of 20 plane strainnine-node quadrilateral u–p elements are used to model soil column. In addition to twodegrees of freedom for displacement (u), the corner nodes of the elements have an additionaldegree of freedom for pore pressure (p). This enables the models to track changes in porepressure and effective stress during the earthquake excitation. The nodes at the base ofthe soil column are fixed against vertical translation and the pore pressure nodes above thegroundwater table are fixed in the pore pressure degree of freedom (representing an opendrainage condition). Periodic boundary conditions are assumed for the purposes of mimick-ing a 1D analysis. A Lysmer and Kuhlemeyer (1969) dashpot is utilized to account for thefinite rigidity of the underlying elastic medium such as bedrock. Prior to seismic loading, agravity load is applied, and, then, at the base the soil column is subject to an earthquakeexcitation from Imperial Valley record scaled to PGA of 0.35g shown in Fig. C.1.For comparison of simulated responses, stress and strain time histories recorded at 5.5mdepth are shown in Figs. C.2 and C.3 for the SANICLAY and SANICLAY-B models, re-spectively. Amplitudes of shear stresses for SANICLAY model are considerably smallerthan those of SANICLAY-B, suggesting that the latter dissipates more energy before shear255Appynxifi WB dropugution oz Yurthquuky inxuwyx gyismiw kuvys through guturutyx gozt Wluy0 2 4 6 8 10 12 14 16 18 20-0.4-0.3-0.2-0.100.10.20.30.4time (s)Acceleration (g)Figure C.1: Imperial Valley record scaled to PGA of 0.35g2 4 6 8 10 12 14 16 18−16−80816time (s)τ (kPa)0 2 4 6 8 10 12 14 16 18 20−6−4−202time (s)γ (%)Figure C.2: Simulation results for time histories of shear stress and shear strain atdepth of 5.5 m in the soil column using the SANICLAY model.waves reach 5.5m depth. Notice that major earthquake excitation is in the range of 4–6 seconds. SANICLAY model simulates accumulation of residual strains (in the directionof sloping) within this range, and afterwards no more accumulation is observed. This re-sponse is expected as the yield surface expands considerably during the major excitationand, afterwards, the stress state is mainly contained within the yield surface not inducingsignificant strains. SANICLAY-B model, in contrast, clearly shows accumulation of resid-ual strains up to about 12 seconds. This demonstrates that bounding surface formulationenables larger accumulation of residual strains in the direction of sloping. The profiles ofresidual horizontal displacements are shown in Fig. C.4, suggesting that the SANICLAY-B256Appynxifi WB dropugution oz Yurthquuky inxuwyx gyismiw kuvys through guturutyx gozt Wluy2 4 6 8 10 12 14 16 18−16−80816time (s)τ (kPa)0 2 4 6 8 10 12 14 16 18 20−6−4−202time (s)γ (%)Figure C.3: Simulation results for time histories of shear stress and shear strain atdepth of 5.5 m in the soil column using the SANICLAY-B model.predicts about 50% more displacement at the surface compared to the SANICLAY. Fig. C.5shows acceleration response spectra for the input base motion and the predicted responseat the surface. Both SANICLAY&SANICLAY-B models predict considerable amplificationfor periods below 0.6sec and little amplification for larger periods. SANICLAY predictslarger amplification than SANICLAY-B for periods in the range between 0.2 and 0.7sec. Forperiods above 2sec, SANICLAY-B shows slightly larger amplification as simulated responseof column involves larger degree of plasticity making soil behave as a softer material. It−0.5 −0.4 −0.3 −0.2 −0.1 005101520Residual horizontal displacement (m)Depth (m) SANICLAYSANICLAY−BFigure C.4: Residual displacement with depth at the end of seismic loading.257Appynxifi WB dropugution oz Yurthquuky inxuwyx gyismiw kuvys through guturutyx gozt Wluy10−3 10−2 10−1 100 1010246S a (g)Period, T (sec) base motionground surface(SANICLAY)ground surface(SANICLAY−B)Figure C.5: Acceleration response spectra of the base input motion and the predictedground surface response (SANICLAY & SANICLAY-B).should be noted that depending on the characteristics of the motion and properties of thesoil column, the SANICLAY can under-damp or over-damp the stress-strain loops, leadingto different seismic site response analysis results.258
Thesis/Dissertation
eng
Civil Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Constitutive and numerical modeling of clay subjected to cyclic loading
Text