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UBC Theses and Dissertations

A critical assessment of a constitutive theory for soils Wong, Colin L. Y. 1983

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A CRITICAL ASSESSMENT OF A CONSTITUTIVE THEORY FOR SOILS by COLIN L.Y. WONG B.Eng., McGill University , Montreal, 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1983 © Colin L.Y. Wong, 1983 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C i v i l Engineering The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date A p r i l 27, 1983 ABSTRACT A cons t i t u t i v e theory for s o i l proposed by J.H. Prevost is explained in d e t a i l and examined for i t s l i m i t a t i o n s , appropriateness of assumptions, and c a p a b i l i t i e s . Three models using t h i s theory are appraised: an undrained t o t a l stress formulation, an e f f e c t i v e stress model for cohesionless s o i l , and a general e f f e c t i v e stress model for any type of s o i l . The necessary equations are derived and the consequences of the im p l i c i t assumptions discussed. Methods for determining the parameters are presented and comparisons of the model predictions with actual test data are also made. It was found that the undrained t o t a l stress model i s remarkably accurate for predicting the behavior of a k a o l i n i t e clay under a complex monotonic stress path. However, caution must be exercised when applying the model to heavily overconsolidated clays. Problems may also be encountered when applying the model to s t r a i n softening s o i l s subjected to complex load paths. The e f f e c t i v e stress models do not give good comparisons between predictions and test data. Some of the problems include a r e s t r i c t i o n of the models to monotonic loading and a mathematical inconsistency in formulating the general e f f e c t i v e stress model. In their present form, the e f f e c t i v e stress models are not suitable for predicting the e f f e c t i v e stress behavior of s o i l s . i i i TABLE OF CONTENTS L i s t of Tables iv L i s t of Figures v Acknowledgements ix Nomenclature x Introduction 1 Chapter 1 - The Stress and Strain Tensors 3 Chapter 2 - General Theory 8 Chapter 3 - Theory for the Undrained Total Stress Analysis of Clays 51 Chapter 4 - Application of the Total Stress Model to Undrained Clays 64 Chapter 5 - Predictions Using the Undrained Total Stress Model- 79 Chapter 6 - A p p l i c a b i l i t y of the Model to C y c l i c Loading .. 95 Chapter 7 - Some Limitations of the Total Stress Model ....107 Chapter 8 - Theory for the E f f e c t i v e Stress Analysis of S o i l s 114 Chapter 9 - A Formulation For Cohesionless S o i l s 126 Chapter 10- A General Formulation For So i l s 145 Chapter 11- A Discussion of the E f f e c t i v e Stress Models ...170 Conclusions 183 References 185 Appendix 188 iv LIST OF TABLES Table 1 - Model Parameters for Drammen Clay 80 Table 2 - Model Parameters for Kao l i n i t e 84 Table 3 - Total Stress Path - Test 2 86 Table 4 - Total Stress Path - Test 3 87 Table 5 - Total Stress Path - Test 5 88 Table 6 - Total Stress Path - Test 7 89 Table 7 - Model Parameters for Santa Barbara S i l t 99 Table 8 - Model Parameters for Cook's Bayou Sand 136 Table 9 - Model Parameters for Ottawa Sand 143 Table 10 - Model Parameters for Haney Clay 160 Table 11 - Model Parameters for Weald Clay 165 V LIST OF FIGURES Figure 2.1 - Stress Strain Curve of a Uniaxial System 12 Figure 2.2 - Isotropic Hardening 13 Figure 2.3 - Effe c t of Hardening Type on the Stress Strain Curve of a Uniaxial System 14 Figure 2.4 - Kinematic Hardening 15 Figure 2.5 - Decomposition of the Stress Increment 18 Figure 2.6 - Example Showing that S o i l Under Drained Conditions may not Satisfy Drucker's Work Hardening Hypothesis 24 Figure 2.7 - F i e l d of Hardening Moduli Surfaces in Stress Space 26 Figure 2.8 - The Advantages of Using Multiple Y i e l d Surfaces 28 Figure 2.9 - Consequences of Intersecting Y i e l d Surfaces .. 33 Figure 2.10 - Translation of f m Due to Hardening 35 Figure 2.11 - Translation of f m Requires Many Small Stress Increments be Used 46 Figure 2.12 - Position of f m After a Small Stress Increment has been Applied 47 Figure 2.13 - Special Case in which a Large Stress Increment may be Applied 49 Figure 3.1 - Y i e l d Surfaces in Stress Space A l l Centered About the Origin 62 Figure 3.2 - Yi e l d Surfaces in Stress Space with Y i e l d Surface B Offset from Origin 62 Figure 4.1 - Comparison Between T r i a x i a l Tests and True Tests for Anisotropy 66 Figure 4.2 - Stress Paths Comparing T r i a x i a l Tests with True Tests for Anisotropy 67 Figure 4.3 - T r i a x i a l Stress-Strain Curve and Corres-ponding Representation in Stress Space 69 v i Figure 4.4 - Simple Shear Stress Path and Corresponding Stress-Strain Curve 77 Figure 5.1 - Prediction of Drammen Clay Simple Shear Behavior 81 Figure 5.2 - T r i a x i a l Stress-Strain Curve for Kaolinite ... 83 Figure 5.3 - D e f i n i t i o n of CJ for Kaolinite Predictions .... 84 Figure 5.4 - Kaolinite Y i e l d Surfaces and Stress Paths for Prediction in S,-S3 Plane 90 Figure 5.5 - Stress-Strain Comparisons for Kaolinite -Test 2 91 Figure 5.6 - Stress-Strain Comparisons for Ka o l i n i t e -Test 3 92 Figure 5.7 - Stress-Strain Comparisons for Ka o l i n i t e -Test 5 93 Figure 5.8 - Stress-Strain Comparisons for Ka o l i n i t e -Test 7 94 Figure 6.1 - Simple Shear Stress Path for Monotonic Loading 96 Figure 6.2 - Simple Shear Stress Path for C y c l i c Loading 96 Figure 6.3 - T r i a x i a l Stress-Strain Curve - Santa Barbara S i l t 98 Figure 6.4 - Simple Shear Stress-Strain Comparison for Santa Barbara S i l t 100 Figure 6.5 - Decrease of Stress Ratio During a Cyclic Simple Shear Test 101 Figure 6.6 - Softening of Stress-Strain Curve with Cycling at a Strain Amplitude of 1% 104 Figure 7.1 - T r i a x i a l Stress-Strain Curve df K 0 Normally Consolidated, Resedimented Boston Blue Clay 110 Figure 7.2 - Comparison of Actual and Pseudo-Hardened Stress-Strain Curves 111 Figure 7.3 - Behavior Predicted by Model as Compared to "Actual" Behavior 113 Figure 8.1 - Y i e l d Surfaces in T r i a x i a l Plane ....121 v i i Figure 9.1 - Yi e l d Surfaces During Isotropic Compression to p = P i 129 Figure 9.2 - Experimental, Isotropic Compression/Rebound Curve for Cook's Bayou Sand 133 Figure 9.3 - Experimental, Drained T r i a x i a l Compression Curves for Cook's Bayou Sand 134 Figure 9.4 - Y i e l d Surfaces in the T r i a x i a l Plane for Cook's Bayou Sand 138 Figure 9.5 - Comparison of Model F i t and Test Data from which Model Parameters were Derived for Cook's Bayou Sand 141 Figure 9.6 - Comparison of Model F i t Using 8 and 19 Y i e l d Surfaces 1 42 Figure 9.7 - Predictions; Ottawa Sand 144 Figure 10.1 - The C r i t i c a l State Line 147 Figure 10.2 - Normalizing Relationships for Normally Consolidated Clay 148 Figure 10.3 - D e f i n i t i o n of d 150 Figure 10.4 - Haney Clay. Stress Difference Versus Axial Strain Curves 159 Figure 10.5 - Haney Clay. Volumetric Strain Versus Axial Strain Curves 161 Figure 10.6 - Haney Clay. O r i g i n a l Positions of Y i e l d Surfaces and Stress Paths 162 Figure 10.7 - Haney Clay. Stress Difference Versus Axial Strain Predictions 163 Figure 10.8 - Haney Clay. Volumetric Strain Versus Ax i a l Strain Predictions 164 Figure 10.9 - Weald Clay. Stress Difference Versus Strain Difference 166 Figure 10.10- Weald Clay. Volumetric Strain Versus Strain Difference 167 Figure 10.11- Weald Clay. Predictions 168 v i i i Figure 10.12- Pore Water Pressure Predictions for Kaolinite 169 Figure 11.1 - Position of Yi e l d Surfaces Upon Translation 172 Figure 11.2 - When Stress Point F i r s t Reaches f 2 173 Figure 11.3 - When Stress Point is on f 2 After Unloading and Reloading E l a s t i c a l l y 173 Figure 11.4 - Model F i t Using Prevost's Method and the Averaging Method .177 Figure 11.5 - Discontinuity of Volumetric Strain when 5=90° 181 Figure D1 - T r i a x i a l Plane in Stress Space 195 ix ACKNOWLEDGEMENTS The author wishes to thank Professor W.D. Liam Finn for being his advisor and primary reviewer during the preparation of t h i s thesis. The discussions we had were a valuable learning experience. Professor D.L. Anderson also reviewed the text and his comments are appreciated. Many thanks are also due to Professor R.N.Yong and Professor W.D.L. Finn for having o r i g i n a l l y stimulated the author's interest in the application of p l a s t i c i t y theory to geotechnical engineering. The author sincerely appreciates the postgraduate scholarship awarded to him by the Natural Sciences and Engineering Research Council of Canada. This scholarship program did much to encourage the author to undertake postgraduate studies. To my family, friends and fellow students in Vancouver who have tolerated my non-technical discussions and bad jokes, you are not forgotten. And to the reader who takes i t upon him/herself to wade through t h i s d i s s e r t i o n , be forewarned that the text contains no intentional humour and a pillow might come in handy. X NOMENCLATURE A - a material parameter associated with shear induced volume change B - e l a s t i c bulk modulus B, - e l a s t i c bulk modulus at p=p, B' - a parameter associated with the p l a s t i c bulk modulus c - a constant in the e f f e c t i v e stress y i e l d function dju - a scalar parameter associated with M to describe the kinematic hardening of the y i e l d surfaces e~ - t o t a l deviatoric s t r a i n tensor e,j - p l a s t i c deviatoric s t r a i n tensor f - y i e l d surface or y i e l d function F - loading function g - p l a s t i c potential function G - e l a s t i c shear modulus G, - e l a s t i c shear modulus at p=Pi h' - p l a s t i c shear modulus H - e l a s t o p l a s t i c modulus H' - p l a s t i c modulus k - a measure of the size of the y i e l d surface k 0 - o r i g i n a l size of the largest y i e l d surface N - number of cycles applied beyond i n i t i a l loading curve plus 1 p=am - mean normal or hydrostatic stress PT - a reference mean normal stress Pjj - normal to the p l a s t i c potential surface g Pg' - projection of the normal to the p l a s t i c potential surface onto the deviatoric plane xi P" - projection of the normal to the p l a s t i c potential surface onto the hydrostatic axis (a scalar) q - stress difference: a¥ - a, - normal to the y i e l d surface Q' - projection of the normal to the y i e l d surface onto the deviatoric plane Q" - projection of the normal to the y i e l d surface onto the hydrostatic axis (a scalar) |Q| - scalar magnitude of QT. s,. - deviatoric stress S; - stress in the 1,2,3 coordinate system a(j - stress coordinates of the center of the y i e l d surface in the deviatoric subspace P - stress coordinate of the center of the y i e l d surface along the hydrostatic axis (a scalar) 5;. - Kronecker delta; the hydrostatic axis ef. - s t r a i n tensor ev - volumetric s t r a i n (a scalar) e - s t r a i n difference: e y - e x 7 - shear s t r a i n = 2e x y X - a measure of the t o t a l accumilated p l a s t i c shear s t r a i n - stress coordinates of the center of the y i e l d surface J in stress space HJJ ~ a tensor in the d i r e c t i o n of a l i n e joining the stress point on f m and a point on f m, ( with the same outward normal Q - Lame's constant ( e l a s t i c i t y ) p - d(ln p)/de„ in a consolidation test r - shear stress 1 INTRODUCTION It i s a well known fact that the p a r t i c u l a t e nature of s o i l s causes i t s s t r e s s - s t r a i n behavior to be non-linear, hysteretic, and stress path dependent. Cycling the stresses may cause the strength and s t i f f n e s s to increase or decrease. In addition, the mode of deposition and stress history can cause the s o i l to behave d i f f e r e n t l y for d i f f e r e n t orientations of a set of stresses, resulting in pronounced anisotropy. Another p e c u l i a r i t y of s o i l i s the phenomenon of coupling between the d i f f e r e n t components of stress or s t r a i n . For example, under drained conditions, dense sands d i l a t e when subjected to shear s t r a i n s . This shear-induced volume change is an important consideration when analysing the behavior of s o i l . Any r e a l i s t i c s o i l model should include the above behavioral c h a r a c t e r i s t i c s . However, most present day models are able to describe only certain aspects with reasonable accuracy. Recently, J.H. Prevost (18,19) has introduced a t o t a l stress model to describe the non-linear, hysteretic, stress path dependent behavior of anisotropic, undrained clays subjected to monotonic or quasi-static c y c l i c loading conditions. He has also extended the concept to describe the e f f e c t i v e stress behavior of s o i l s (20-22). Published comparisons with experimental data have shown the undrained t o t a l stress model to be remarkably accurate and i t s future in 2 geotechnical applications appears to be very promising. However, l i t t l e work has been done on the model by people other than Prevost. To examine the l i m i t a t i o n s , assumptions, and c a p a b i l i t i e s of the Prevost model, a study was undertaken by the author and the results are the subject of t h i s thesis. The f i r s t chapter reviews the concepts of stress and s t r a i n tensors so that the presentation can be self-contained. The second deals with the fundamental theoret i c a l considerations which form the basis of the model. Chapters 3 to 7 and 8 to 11 present an undrained t o t a l stress and e f f e c t i v e stress formulation, respectively. Methods for determining the necessary parameters and a comparative study of predictions and experimental results are also presented. 3 CHAPTER 1  THE STRESS AND STRAIN TENSORS Tensor Notation Tensor notation i s used to define stresses, stra i n s , and y i e l d c r i t e r i a discussed in t h i s thesis. Therefore, a brief description of tensor notation, and in p a r t i c u l a r the stress and s t r a i n tensors, w i l l be presented here. A tensor i s a set of numbers or functions which can undergo coordinate transformations according to cer t a i n mathematical laws. Only the notation of tensors, and not their transformation laws, w i l l be discussed here. If we wish to write the vector {x 1 f x 2, x 3} in tensor notation, i t would appear as X ; where the subscript i (a l a t i n or other non-numeric symbol) can take on the values 1, 2, and 3. X j i s a tensor of order one since i t has only one subscript. Stresses however, require two subscripts and are second order tensors: one to describe the d i r e c t i o n of the associated force and the other to specify the d i r e c t i o n of the normal to the face of the material element i t acts upon. There are then nine components of stress or which are shown in eq.1.1: 4 r„ r X / xz T a r yx y yz T T a . zx *y z C l 1 °\ 2 C 1 3 o2 1 ^ 2 2 0 2 3 = °<i = a (1.1) 3^ 1 a 3 2 <>3 3 Just as i took on values of 1, 2, and 3 in the f i r s t example, both i and j take on these values here. More generally, when a non-numeric subscript appears only once in a term, the subscript takes on the values 1, 2, and 3. In p r a c t i c a l terms, the indices may be thought of as representing the x, y, and z axes, respectively. S i m i l a r l y for the st r a i n s : e 1 1 e 1 2 « 1 3 e 2 1 e 2 2 * 2 3 « 3 i e 3 2 £ 3 3 e{. = e (1.2) where: 7- i * 3 If a subscript i s repeated in a term, the range of the subscript i s then summed: on = a, , + a2 2 + 03 3 a,. de;j = a 1 1 d e 1 1 + a 1 2 d e 1 2 + . . . . + a 2 i c l e 2 i + a 3 3de33 5 No s u b s c r i p t may be r e p e a t e d more t h a n o n c e . I f t h e s u b s c r i p t s a r e i n t e r c h a n g e a b l e w i t h o u t a l t e r i n g t h e c o m p o n e n t s , t h e n t h e s y s t e m i s s a i d t o be s y m m e t r i c . E x c e p t i n v e r y s p e c i a l c a s e s , t h e s t r e s s a n d s t r a i n t e n s o r s a r e s y m m e t r i c : o-. = a,, % -A s p e c i a l s e c o n d o r d e r t e n s o r , c a l l e d t h e K r o n e c k e r d e l t a 5,. , i s d e f i n e d a s : 6 G « 8 1 0 0 0 1 0 0 0 . 1 r i f i f (1.3) Some s p e c i a l p r o p e r t i e s o f t h i s t e n s o r a r e : 3 6 Deviatoric Stresses and Strains It i s often convenient in p l a s t i c i t y theory and s o i l mechanics to separate the stress tensor into two parts. One i s c a l l e d the spherical stress tensor and i s defined as: P 0 0 0 0 0 P (1.4) where: P =  +°Y  +°* ) = 3^kk = om (1 -5 ) The quantity p i s also c a l l e d the mean normal stress or hydrostatic stress. It i s the same for a l l possible orientations of axes and i s thus an invariant. The second part of the stress tensor i s c a l l e d the deviatoric stress tensor s;- and i s defined as: s;j = o;- - am 6(j (1.6) Obviously, the subtraction of the hydrostatic stress from the stress tensor does not change the d i r e c t i o n of the p r i n c i p a l deviator stresses. The three invariants of the stress deviator tensor are: = 0 7 J 2 ~ 2 S;: S;; 'J i S i k =>!<,• J 3 = aS;; S...S,., The second invariant J 2 i s es p e c i a l l y s i g n i f i c a n t in p l a s t i c i t y because i t forms part of the Von Mises y i e l d c r i t e r i o n , which w i l l be discussed l a t e r . Strains can be treated in much the same way as stresses. The spherical s t r a i n tensor i s given by: 1 1 3 3 e v 0 0 0 e v 0 0 0 e„ (1.7) where: = volumetric str a i n The deviatoric s t r a i n tensor i s defined as: (1.8) (1.9) The deviatoric s t r a i n invariants are: J 1 J ' 2 = 0 _ 1 5 e - e-1 'j J = i e o e i k e f e i 8 CHAPTER 2 GENERAL THEORY The theory of p l a s t i c i t y provides a powerful basis upon which to construct models to describe the s t r e s s - s t r a i n behavior of s o i l s . The simplest formulation would be to assume ideal or perfect p l a s t i c i t y . A sample with t h i s property subjected to a uniaxial load would deform p l a s t i c a l l y at constant load when a certain stress, c a l l e d the y i e l d stress, i s reached. However, s o i l s often exhibit hardening; that i s , they have load-deformation curves which monotonically increase during p l a s t i c deformation. The con s t i t u t i v e model described herein uses p l a s t i c i t y theory to model t h i s phenomenon. A general formulation w i l l now be presented. P a r t i c u l a r i z a t i o n s to describe the undrained and drained behavior of s o i l s w i l l be presented in l a t e r chapters. Components of Strain The incremental theory of p l a s t i c i t y i s based on the assumption that a given s t r a i n increment can be separated into an e l a s t i c and a p l a s t i c component: Where: e l a s t i c s t r a i n increment tensor = p l a s t i c s t r a i n increment tensor 9 For convenience, the e l a s t i c i t y of the s o i l is assumed to be i s o t r o p i c . The e l a s t i c s t r a i n increments are then related to the stress increments by generalized Hooke's law expressed in terms of the shear modulus G and the bulk modulus B. These moduli may be assumed to be constant or functions of the e f f e c t i v e stress and/or void r a t i o . To describe the p l a s t i c i t y of the s o i l , a y i e l d c r i t e r i o n , flow rule, and hardening rule are required. The y i e l d c r i t e r i o n or y i e l d function s p e c i f i e s the stress states for which p l a s t i c deformation may take place. It i s represented by a scalar function f (cn. , e/j ,k) which forms a closed surface in stress space. As such, f i s also c a l l e d a y i e l d surface. Stress states on f may cause p l a s t i c deformations whereas only e l a s t i c behavior occurs for states within the surface. Mathematically, i t i s defined as: f = F (cr. ) - k ( ee. ) = 0 (2.1) where: F describes the shape of the y i e l d surface and i s a function of only the stresses k i s a measure of the size of the y i e l d surface and may be a function of the p l a s t i c strains and ; history of loading. If f<0, i . e . F<k, then the stress point l i e s inside the 10 y i e l d surface and only e l a s t i c deformations occur. P l a s t i c deformations may occur i f f=0. Now suppose- f>0; then F>k and the stress point would be outside the y i e l d surface. Since loading from a p l a s t i c state must lead to another p l a s t i c state, f=0 at a l l times during p l a s t i c deformation. This requirement i s known as the consistency condition (16) and demands that the stress point always be on the current y i e l d surface during loading. f>0 i s therefore inadmissable and has no meaning. To determine whether p l a s t i c deformation occurs for a perfectly p l a s t i c or hardening material when the stress point is on the y i e l d surface f=0, the scalar (dot) product between a r the outward normal to the y i e l d surface ; T and the stress increment da, i s evaluated: a) For loading, do, > 0 and hardening occurs. do;- J b) For neutral loading -4^— do; = 0. P l a s t i c 3 d \T:, 'J deformation may occur for perfectly p l a s t i c materials, but no hardening may take place. c) For unloading do. < 0 and only e l a s t i c strains occur. Notice that dF/do;. = bt/ bo,- since k i s not a function of a... F i s often c a l l e d the "loading function" because i t i s only a function of the stresses and defines the the loading condition. 11 The quantity ~~|-£r c a n t»e thought of as being a measure of the components of the stress increment tensor in the d i r e c t i o n of the outward normal. Thus for loading to occur, the stress "vector" 1 must be directed outwards from the y i e l d surface. The flow rule relates the p l a s t i c s t r a i n increment tensor to the stress tensor. The most common rule i s the normality rule, which assumes that the s t r a i n increments are normal to a surface in stress space c a l l e d the p l a s t i c potential surface, g. Now l e t us assume that the p l a s t i c s t r a i n increment and stress axes coincide in stress space. If f=g, the y i e l d surface also plays the role of the potential surface and the flow rule i s known as an associated flow rule ( i . e . the flow rule i s associated with the y i e l d function f ) . A non-associated flow rule results i f f*g, and the s t r a i n increment tensor i s not related to the y i e l d surface. Hardening rules are a consequence of the consistency condition. Figure 2.1 shows how p l a s t i c hardening i s achieved in a uniaxial t e s t . Let aA be the o r i g i n a l y i e l d stress. Now suppose a stress increment da i s applied causing p l a s t i c deformation. The consistency condition requires that the y i e l d point be shifted to point B. S i m i l a r i l y , the y i e l d surface in stress space must assume a new position consistent with the new stress state. The modification of the y i e l d surface's position 1 Although stress i s a tensor, i t can be v i s u a l i z e d as being a "vector" in stress space. F i g . 2 . 1 . Stress-strain curve of a un i a x i a l system and stress space representation of stress state. a) I n i t i a l ; b) After load increment has been applied. 13 in stress space as p l a s t i c flow occurs i s described by the hardening rul e . Isotropic and kinematic hardening are two fundamental types of hardening. Isotropic hardening assumes that the y i e l d surface expands proportionately outwards while the center of the y i e l d surface remains stationary. Figure 2.2 gives two examples of t h i s . F i g . 2.2. Isotropic hardening. The dashed l i n e s show the y i e l d surface's size before p l a s t i c deformation. Now suppose that the material in Fig.2.3 i s loaded u n i a x i a l l y and has i t s y i e l d surface centered at the o r i g i n (a,j=0). Once the applied stress exceeds the y i e l d stress oyo , the material hardens. If the load i s reversed from oc' , iso t r o p i c hardening would predict that p l a s t i c deformation w i l l not occur u n t i l a stress of aE', where | oc' | = | ffE' | , i s reached t r Kinematic Isotropic Kinematic Isotropic F i g . 2.3. E f f e c t of hardening type on the s t r e s s -s t r a i n curve of a u n i a x i a l system. Dashed y i e l d s u r f a c e l i n e s show y i e l d s u r f a c e ' s p o s i t i o n a f t e r hardening. 15 F i g . 2.4. Kinematic hardening. in extension. S o i l s , however, often exhibit the Bauschinger e f f e c t ; that i s , p l a s t i c deformations occur well before the stress has been f u l l y reversed. Thus is o t r o p i c hardening, which i s a useful concept for monotonic loading s i t u a t i o n s , i s not convenient to use when loading reversal occurs. Kinematic hardening (17) provides a simple means of modelling the Bauschinger e f f e c t . It assumes that the y i e l d surface's size and orientation i s retained but i t s center i s allowed to translate in stress space (Fig.2.4). If we use kinematic hardening on the material in Fig.2.3, i t can be seen that ay in extension occurs well before a£' . 16 General P l a s t i c Stress-Strain Relation A general form of the p l a s t i c s t r e s s - s t r a i n r e l a t i o n for hardening materials can be developed using three assumptions: 1) The stress state l i e s on the y i e l d surface f and further p l a s t i c deformation occurs only for: a f da,j > 0 (2.2) 2) There exists a surface g in stress space such that the normal to g gives the d i r e c t i o n of the p l a s t i c s t r a i n increments. 3) The r e l a t i o n between the i n f i n i t e s i m a l s of stress and st r a i n i s l i n e a r : de,p = C5jkl dokl (2.3) where Cjjk, may be a function of stress, s t r a i n , or history of loading but i s independent of de?- or dak, . Using assumption 2, C,jkl can be written: 1 7 69 C i j k i = — rk, < 2 - 3 a > do-,. w h e r e : a g = o u t w a r d n o r m a l t o t h e p l a s t i c p o t e n t i a l ; t h i s do;-. t e n s o r i s c o - d i r e c t i o n a l w i t h de ; i . r k | = an a s y e t u n d e f i n e d t e n s o r w i t h s u b s c r i p t s " k l " The t e n s o r r k l c a n be o b t a i n e d by n o t i n g t h a t a s s u m p t i o n 3 a l l o w s t h e p r i n c i p l e o f s u p e r p o s i t i o n t o a p p l y . T h u s : da k l = daj, + da;, ( 2 . 4 ) w h e r e da k' p r o d u c e s no p l a s t i c d e f o r m a t i o n a n d da k" i s p r o p o r t i o n a l t o t h e g r a d i e n t o f f ( F i g . 2 . 5 ) : a t daj; = r ( 2 . 5 ) r > 0 Eq.2.4 i n t o eq.2.2 g i v e s : a f — ^ + d < ; ) > ° ( 2 . 6 ) 18 F i g . 2.5. Decomposition of the stress increment da k ) dak" i s perpendicular and dak', i s tangent to the y i e l d surface at the stress point. But dak',, by d e f i n i t i o n , produces no p l a s t i c flow; therefore: 3f do' = 0 (2.7) Combining equations 2.5, 2.6, and 2.7 gives: af df af df do k , doki dakl doM or: 19 df 7 d a k i d o k \ • , (2.8) af af Comparing eq.2.8 with eq.2.3a, i t can be concluded that a f 1 dok | r k I = (2.9) H ' af af d<>mn aa m n where 1 / H ' i s a factor of proportionality and H ' i s c a l l e d the p l a s t i c modulus. The most general form of the p l a s t i c s t r e s s - s t r a i n r e l a t i o n for hardening materials can therefore be written: d e : af 1 d g ^ oki do, kl H * do., dt dt (2.10) This can be expressed more succinctly as: 20 d e ' P I Q I 2 (2.11) w h e r e : P = p.. a_g_ do. Q = Q 5 n o r m a l t o t h e p l a s t i c p o t e n t i a l s u r f a c e ( i . e . i n t h e d i r e c t i o n o f t h e p l a s t i c s t r a i n i n c r e m e n t s ) dt do-. 'j = n o r m a l t o t h e y i e l d s u r f a c e IQI af at do„„ &o. mn vtnn = — Q-da = H' 1 a f H' do. do. kl <L> = i f L > 0 i f L < 0 D r u c k e r (3) f u r t h e r r e s t r i c t s t h e f o r m o f t h e s t r e s s -21 s t r a i n r e l a t i o n by defining that a work hardening material be in equilibrium when a s e l f - e q u i l i b r a t i n g set of stresses i s added to the material and further that: a) p o s i t i v e work is done by the application of a set of s e l f - e q u i l i b r a t i n g forces b) the net work performed over a complete cycle of application and removal i s zero or p o s i t i v e . Consider a volume of material under a homogeneous state of stress a. and s t r a i n . If a s e l f - e q u i l i b r a t i n g set of forces applies small surface tractions r e s u l t i n g in a stress change at each point of do,. , the resulting s t r a i n increments are de^ . Now i f these small surface tractions are removed, the e l a s t i c s t r a i n increments w i l l be released. Thus condition "a" becomes: (2.12) Condition "b" results i n : d o . ( d e 0 - de,!: ) > o d a 0 de j > 0 (2.13) 22 where: da. de.? =0 i f def! = 0 Equations 2.12 and 2.13 are a mathematical d e f i n i t i o n of Drucker's work hardening hypothesis. Recalling from Fig.2.5 that the stress increment can be decomposed, eq.2.13 may be written as: da^defj = (da,; + da0" ) def! > 0 (2.14) But da,.' produces no p l a s t i c flow and therefore the stress increment da,. = C da' + da,]' w i l l produce the same p l a s t i c s t r a i n increment de/! regardless of the value of C. However, da.'def- must vanish, otherwise C could be chosen to be a large negative number such that eq.2.14 i s vi o l a t e d . Therefore: da.dej = 0 or from eq.2.11: dg <L> da' = 0 (2.15) J doj. |Q|2 P l a s t i c loading requires that <L> > 0. Eq.2.15 therefore implies that <}g/<3q. i s perpendicular to da,' and thus also to 23 f at the stress point during y i e l d i n g . This indicates that f=g and therefore equation 2.10 becomes: 3f 1 a f bok. de,.; = - (2.16) J H' GO,- of bf This proves that a work hardening material, in the Drucker sense, must have an associated flow rule. Drucker (3) has pointed out that the d e f i n i t i o n of work hardening defined by equations 2.12 and 2.13 implies s t a b i l i t y of the system; that i s , no deformations may occur without an input of work. However, the behavior of s o i l s under drained conditions ( i . e . when the behavior i s a function of the mean normal stress) does not always s a t i s f y Drucker's d e f i n i t i o n of work hardening. For example, consider a specimen of dry sand at the c r i t i c a l void r a t i o , subjected to a shear stress r and a mean normal stress p (Fig.2.6). If the stress point for the system l i e s on the y i e l d surface, then a cycle of adding and then removing a hydrostatic stress decrement dp would cause an increment in p l a s t i c shear s t r a i n . Assuming that no volume change occurs, the work done by the added stress dp would be zero. This behavior i s in v i o l a t i o n of eq.2.13 and therefore s o i l s under drained conditions are not, in general, work hardening materials. Materials which may show the c h a r a c t e r i s t i c s of hardening ( i . e . a monotonically increasing 24 F i g . 2.6. Example showing that s o i l under drained conditions may not s a t i s f y Drucker's work hardening hypothesis. s t r e s s - s t r a i n curve) but do not necessarily s a t i s f y Drucker's requirement for s t a b i l i t y are c a l l e d hardening materials and the flow rule i s given by eq.2.10. It should be recognized that work hardening materials are a subset of hardening materials. General Y i e l d Function It w i l l be assumed herein that the y i e l d surface can be represented by an expression of the form: 0 (2.17) 25 where: $. = offset of the y i e l d surface from the o r i g i n in stress space. k(efj) = a measure of the size of the y i e l d surface, n = an integer exponent; n > 1 For s o i l s , F i s conveniently chosen to be a homogeneous function of degree n in (a.. - ) (see Appendix A). Thus, for J ' J quadratic F, n=2 . For convenience (19) the size of the y i e l d surface k may be a constant or a function of: =p = jde,' and/or A = I ^ A a * where: efv = p l a s t i c volumetric s t r a i n = p l a s t i c deviatoric s t r a i n tensor To allow for the anisotropic hardening of a material subjected to any path in stress space, the concept of a f i e l d of hardening moduli surfaces, f 1 f f 2 , , f P with sizes k l 1 ,< k < 2 )< < k (P', respectively, was proposed by Iwan (8) and Mroz (12) and adapted for modelling s o i l behavior by Prevost (18-22). These surfaces represent stress l o c i at which the p l a s t i c modulus H' (or other p l a s t i c parameters) acquires a new value. For example, when the stress point M in Fig.2.7 touches f m , a new p l a s t i c modulus (or other p l a s t i c parameters) 26 i s used i n the s t r e s s - s t r a i n r e l a t i o n . A l though the f u n c t i o n s d e s c r i b i n g each hardening sur face need not be the same, f 2 , f 3 , , f p are chosen to be of an i d e n t i c a l form as the y i e l d f u n c t i o n f , fo r s i m p l i c i t y . fr = F r (o,. - $.! r>) - ( k " > ) n = 0 (2.18) fo r a l l r When the s t r e s s p o i n t reaches the hardening s u r f a c e f m du r ing l o a d i n g , f m a c t s as a y i e l d s u r f a c e and equat ion 2.10 becomes: 27 dfm 1 a 9 m ^ da. H ' a a , df df. (2.19) 3o r i 3 a,, Prevost c a l l s the hardening surfaces " y i e l d surfaces". This is not e n t i r e l y correct i f y i e l d i n g is defined as the " l i m i t of e l a s t i c i t y under any possible combination of stresses" (7). For example, i f f, i s a true y i e l d surface in Fig.2.7 ( i . e . purely e l a s t i c behavior within i t ) then f m cannot also be a y i e l d surface since p l a s t i c deformations can occur within i t . Nevertheless, when the stress point l i e s on the hardening surface fm , t h i s function acts l i k e a y i e l d surface. In p a r t i c u l a r , i f a stress increment i s applied tangent to f m : bt — d o k ) = 0 and no p l a s t i c s t r a i n occurs. If stress increments are continuously applied in t h i s manner, the hardening surface may t h e o r e t i c a l l y be circumscribed by the stress point without causing p l a s t i c s t r a i n s . This i s a property of a y i e l d surface. To simplify the terminology, Prevost's convention of c a l l i n g hardening surfaces " y i e l d surfaces" w i l l be used herein. Now l e t us q u a l i t a t i v e l y examine one of the advantages of (a) F i g . 2.8. (b) Example showing how multiple y i e l d surfaces can be used to simulate the dif f e r e n t behavior under compression and extension, and also anisotropy. a) Yi e l d surfaces in stress space. b) Corresponding s t r e s s - s t r a i n curve. 29 a multiple y i e l d surface approach using a uniaxial system. Let f 1 f f 2 , and f 3 in Fig.2.8a be y i e l d surfaces having constant p l a s t i c moduli of H,, H 2, H 3 and constant sizes k 1<k 2<k 3, respectively. Assuming the e l a s t i c strains are zero, the corresponding s t r e s s - s t r a i n curves are i l l u s t r a t e d in Fig.2.8b. Notice the d i f f e r e n t behavior in compression and extension i f the y i e l d surfaces are offset with respect to the o r i g i n 0. The multiple y i e l d surface approach would thus be a convenient method of simulating the d i f f e r e n t behavior under compression and extension, and also anisotropy. One of the most well known y i e l d c r i t e r i o n in p l a s t i c i t y i s the Von Mises y i e l d c r i t e r i o n : J 2 = |s f j s-. = k where: k = a measure of the size of the y i e l d surface, which may vary due to str a i n hardening or softening. If the above re l a t i o n i s s a t i s f i e d , then p l a s t i c deformation may occur. Kinematic Hardening Rule The translation and change in size of a y i e l d surface due to p l a s t i c deformation is described by the kinematic and isotropic hardening rules respectively. Isotropic hardening i s the simplest of the two and i s b a s i c a l l y a c o r r e l a t i o n between the observed behavior of a material and the changes in the sizes of the y i e l d surfaces ( i . e . dk). An example of an 30 isotropic hardening rule i s given in Chapter 6 . However, kinematic hardening can be quite complex for a m u l t i - y i e l d surface model because the y i e l d surfaces interact with one another. The mathematics which describes these movements w i l l now be explained. If a material hardens, then the consistency condition must be s a t i s f i e d . Applying the consistency condition to: F ( a ; j - 5 . . ) = F ( 0 ; j ) = h(k) = k" ( 2 . 2 0 ) requires that: F ( a t j + da0- - - d$;j) = F ( 0 l j + 6QSJ) = h(k + dk) = (k + dk)' ( 2 . 2 1 ) Expanding F(0,-j + d0,j ) into a Taylor series we get: 1 3 2 F F ( 0 ; j + d e ( j ) = F(efj ) ' + d 0 ; . + -2! a 0 f j 2 + ( 2 . 2 2 ) S i m i l a r i l y : 31 dh 1 a 2h h(k + dk) = h(k) + — dk + (dk) 2 + dk 2! ak 2 (2.23) Substituting eq.2.22 and eq.2.23 into eq.2.21 and since F(0 ( j ) = h(k) : a F 1 s 2F a h d©,- + (d©,,)2 + = — dk + a©,. 2> a©,.2 ak 1 a 2h (dk) 2 + (2.24) 21 ak 2 From the d e f i n i t i o n of 0-: 3F 3F a~* ' ~ °Qi' Equation 2.24 then becomes: Qfj <doy - d$ ) + - — i <dok| - d$ k |) 2 + 2! do-,-ah 1 aah _ dk + (dk) 2 + ..... (2.25) ak • 2! ak 2 32 or: 1 3Q. Q i j ( d 0 i j - d 5 {., + - - 1 (da k| - d$ k l) 2 + 2! boi: n kn"' dk + n _ ( n - l ) k"'1 (dk) 2 + . . 2! (2.26) This i s a mathematical statement of the consistency condition. If (do(j - d$ r ) and dk are small, then the second order terms can be neglected: Qfj (do. - d§ ; j) = n kh"' dk (2.27) This simpler form of the consistency condition w i l l be used in subsequent derivations unless s p e c i f i e d otherwise. Since F i s a homogeneous function, Euler's theorem (see Appendix A) can be used on eq.2.27 to give : U 0 - Q l j = n F = n k" (2.28) or: 33 n k""' dk = dk C-fo, - ^ ) (2.29) k It i s imperative that the y i e l d surfaces never overlap. A physical reason for t h i s can be seen in Fig.2.9. .Fig. 2.9. Intersection of two y i e l d surfaces gives two d i f f e r e n t outward normals at A and thus the s t r a i n increment would not be unique. If the stress point reaches point A, two d i f f e r e n t outward normals, corresponding to f, and t2, would e x i s t . From eq.2.9, i t can be seen that the s t r a i n increment would not be unique at t h i s point and thus overlappings cannot be permitted. A point where dt/do^ i s not uniquely defined i s c a l l e d a singular point. To prevent the y i e l d surfaces which are smaller than fm from intersecting one another, Mroz (12) proposed that f 1 f f 2 , f m_, be made tangent to f m at the stress point M. Since f 1 r f 2 , f m., d i f f e r only in size but not form: 34 ( 1 ) ( rn ) 0 k ( r n ) ( 2 . 3 0 ) The geometric meaning of t h i s r e l a t i o n i s that the y i e l d surfaces are tangent at the stress point and have their centers lying on the "vector" (o-^  - ^j""') as shown in Fig.2.7. It should be noted that eq.2.30 i s not a hardening rule since p l a s t i c deformation need not occur for i t to be true. When the stress point i s on f m and when p l a s t i c hardening occurs, the y i e l d surfaces f 1 f f 2 , , tm are translated together and remain tangent at the stress point as i t moves along i t s stress path towards f m M . Thus the stress point must never be outside any y i e l d surface. To avoid the overlapping of f m and f„,+l , these y i e l d surfaces may touch only at points with the same outward normal. Let M be the stress point lying on f m and l e t R be the point on f„,+l which has an outward normal in the same d i r e c t i o n as at M in Fig.2.10. For i l l u s t r a t i v e purposes, the y i e l d surfaces are shown as " c i r c l e s " in stress space. We s h a l l now demand that points M and R move towards each other during kinematic and isotropic hardening of the y i e l d surfaces. Points A and B denote the centers of the y i e l d surfaces f m and f m + i respectively. Since similar triangles are formed by AEBM and ARMB, |BM| = |MB j and MB = -(o ~ $ ( m + 1 > ) . Also, |BR| = k ( m + 1 ) and |AM| = k ( m ) . But from the s i m i l a r i t y of triangles, |EM| = |BR| = k ( m + 1 ) . Therefore MR i s given by : F i g . 2 . 1 0 . T r a n s l a t i o n of f m due t o h a r d e n i n g . 36 M = MR = EB = EM + MB k if* * 1 ) = AM + MB k ( « n k (m + 1 ) M = (a - $ ( m>) - (a - $<m + , >) k ( m ' (2.31) This result was f i r s t obtained by Mroz (12). Following an analysis by Prevost (20), we s h a l l now find the movements of f m and f m + 1 with respect to each other such that no overlapping may occur ( i . e . we seek an expression for d £ ( m ) - d $ ( m + 1 > ) . Let 5a and 6a' be the movements of M and R respectively due to kinematic (d$ ( n n ) and d $ ( m + 1 ) ) and iso t r o p i c (dk""' and d k ( m + 1 ) ) hardening. The stress increment 5a should not be confused with da. Its purpose i s to track the change in position of M due to hardening. The new position of M after hardening w i l l not necessarily coincide with the position of the stress point. The consistency condition for point M can be written: Q.(m>(5o.. - d^"") = n k<m)"~] dk"*" But d$.(.n,l)) i s in the same di r e c t i o n as ' since t h i s 37 q u a n t i t y o n l y c o n t a i n s i s o t r o p i c h a r d e n i n g . T h i s c a n be m a t h e m a t i c a l l y s t a t e d a s : Q' m > = T ? ( 6 a , - d $ , ( . " > > ) ( 2 . 3 2 ) w h e r e : T J = a s c a l a r c o n s t a n t To f i n d T J : (*) I = n )""' d k " * * |Q.(."" | |Qi<w> | = n ( k ^ ' f " 1 dk<™> T h e r e f o r e : | Q ( « , | 2 n ( k " * ' ) 0 " ' dk' 0 1' ( 2 . 3 3 ) E q u a t i o n 2.33 i n t o 2.32 g i v e s : n ( k l m ) ) n " ' dk'™' Son = d$.< m > + Q'"" ( 2 . 3 4 ) J 'J | Q (m ) | 2 J F o r p o i n t R, (80^ - d$.p+,)) w i l l be i n t h e same d i r e c t i o n a s Q--m) s i n c e t h e o u t w a r d n o r m a l s a t M a n d R a r e i n t h e same 'j d i r e c t i o n : 38 Q.(.m» = <M So-1- - d$.(mtl > ) Where I|I i s a scalar constant. To get an expression for |Q' m + l ' I I S a i ' " dS i ( m* , ,| = n (k ( m t , ))' >' 1 dk(n, ,+ " | Q .<."»» | n |Q(m+i)| '1 . _ n (k («)* l ) ) n " ak ( m + l ) | Q i - m t l > I lQ|jm) I n ( k ^ " ) " " ' dk< m t | ) Therefore: Note that although Q.<.n'1' and Q['n + 1 ' are in the same d i r e c t i o n , their magnitudes are d i f f e r e n t . In accordance with our demand that M and R move towards each other: 5a - 5a' = dn ju (2.36) The scalar parameter da can be obtained from eq.2.36 : Q ( m » -5a - Q ( w » .8a ' = dn Q( m ' • M n-1 0 f j n ( k ( m + l >) dk ( m + l ) IQ ( nn ) |Q(n,+ I ) (2.35) 39 Q ( m ) • ( 5 a - 6 a ' ) dju = (2.37) Q<m>.£ Since the consistency condition s p e c i f i e s that point M and the stress point must both remain on the y i e l d surface after the stress increment da has been applied, for point M : F(a + 6a - £ ( m ) - d $ ( m ) ) = k < m >" + d_F . ( 5 a - d $ ( m ) ) do = k'"1'" + Q - ( 5 a - dtlm) ) (2.38) For the stress point : F(a + da - t ( m ' " d $ ( m ) ) = 0 = k"">n + Q • (da - 6tim) ) (2.39) Equating eq.2.38 and eq.2.39 gives : Q< m > . 5 a = Q l m )-da (2.40) Letting a' be the stress coordinates for R, the consistency condition requires: 40 F(a' + 6a' - ^C"* 1) - c U < m + l ) ) - = 0 Q ( m* n-6a' - Q ( m + I' • d l ( n i + " - n (k""*")"" 1 dk(n,,+ l ) = 0 |Q(m + |) I |Q<m + l ) I {Q^i-So') - ( Q ( m ' •d| ( m* l ) ) | Q ( m , | |Q ( m ) - n ( k ( m + l ' )""' dk ( m + l > = 0 j ( n r , ).6a' = Q"* > • d$(n,, +l' + n ( k ( m + l ) ) n " ' d k l m + l ) | Q ( m ) | | Q ( ^ l ) | (2.41 ) Substituting eq.2.40 and eq.2.41 into eq.2.37 y i e l d s : d M = Q ( m ' - da - Q ( m ) • df ( m * n - n k""*" dk ( m*" l Q ( m ' Q ( m , - M (2.42) Combining equations 2.34, 2.35, and 2.36 gives the rela t i o n between d $ ( m ) and dflfn + 1 ' : 41 d " £ < m ) _ d ^ ( m + 1 ) = K ( 0 1 + 1) n-l n-1 n d k ( m * l ) k ( m ) d k ( m ) Q ( m) Q(m) |Q |Q(«, I (2.43) Note that dju i s also a function of d $ ( n r ) + 1 > . The y i e l d surfaces f m and f m t , translate in stress space in a manner which prevents their i n t e r s e c t i o n . But eq.2.43 does not describe the kinematic movement of f m or f m + , ; i t only s p e c i f i e s t h e i r motions r e l a t i v e to each other. Thus no r e s t r i c t i o n s are placed on how any one of the y i e l d surfaces translates. In the following, several kinematic hardening rules for f m ( i . e . expressions for dtitn)) are considered. d £ " n + 1 ) can then be found from eq.2.43. Prager's Kinematic Rule According to (17), d t ( m ) = du Q ( w ) and the instantaneous translation of f m occurs in the d i r e c t i o n of the l o c a l outward normal. The analogy of a f r i c t i o n l e s s ring (representing the y i e l d surface) being dragged by a stick (the stress point) i s often used. From the s i m p l i f i e d consistency condition (eq.2.27) Q ( m )-da - n k^""' dk ( f n' = Q ( m ) - d t l m ) 42 dv Q<m) . d a _ n k ( « ) d k ( n n ) |Q(« ) I 2 and: ( n r > ) Q(nr,) -da - n k"" > " dk(m > IQ ( m > I 2 Q(rn) (2.44) Ziegler's Kinematic Rule According to (3.2), d $ ( m ) = dv (a - tlrn)) and the instantaneous t r a n s l a t i o n of f M occurs in the d i r e c t i o n of the 'vector' connecting the center of the y i e l d surface to the stress point. From the consistency condition (eq.2.27) : r v - l Q(m,-da - n k ( m ) dk""' = Q<m > • d$lm' = Q< m> • (a - t i m ) ) dv and: d$ ( m ) n-1 Q l r n ) -da - n k'™' dk ( r , , ) i ( m ) (a - S(m)) (a - Stn,))) (2.45) Using equation 2.28, eq.2.45 becomes Am) do - n k'"1'""' dk d$ (m ) (o - $<">) (2.46) 43 P h i l l i p ' s Kinematic Rule From (15), d$ ( n f , > = vdo and the instantaneous translation i s in the di r e c t i o n of the stress 'vector' at the stress point. From the consistency condition (eq.2.27) : n-l Q < n f , )-da - n kirn) dkim) = Qim > • d$ i n )' = Q(m) • da v and: as ( m ) n - l n k l m ' dk(nn> 1 -da da (2.47) Mroz's Kinematic Rule According to (12), d $ ( m ) = dva . The consistency condition (eq.2.27) gives : . n-l Q ( m > - d a - n k < m ) d k < m ) = Q ' ^ ' - M df (m ) n-l Q ( r n ) -da - n k ( m> dk^ > (2.48) A New Sim p l i f i e d Kinematic Rule The tensors d $ ( m ) , given by equations 2.44 through 2.48 would a l l result in a tra n s l a t i o n of f m + i unless some 44 a d d i t i o n a l r e s t r i c t i o n s a r e p l a c e d on d $ ( f y , ) a n d / o r d $ ( m + 1 ) . H o w e v e r , e q . 2 . 4 3 i s q u i t e c o m p l e x t o s o l v e s i n c e d $ ( m + 1 ) a l s o o c c u r s i n a s c a l a r p r o d u c t w i t h Q ( M ) . In a d d i t i o n , any movement o f f m t ) w o u l d r e q u i r e a s i m i l a r a n a l y s i s t o p r e v e n t i t s i n t e r s e c t i o n w i t h f m + i . T h i s p r o c e d u r e must be done f o r a l l f r i n w h i c h m<r<p . The r e s u l t i n g mass o f e q u a t i o n s w o u l d be q u i t e cumbersome t o h a n d l e u n l e s s some s i m p l i f y i n g a s s u m p t i o n s were made. In o r d e r t o r e t a i n g e n e r a l i t y w i t h o u t h a v i n g t o c o m p u t e t h e t r a n s l a t i o n o f s u c c e s s i v e y i e l d s u r f a c e s , d f ( m + 1 ) may be s e t e q u a l t o z e r o . I f i t i s a l s o a s sumed t h a t d k ( r ) < d k ( r + 1 > f o r r^m+1, d ? ( r ) c a n be c o m p u t e d f r o m e q . 2 . 4 3 : d £ < m + 1 > = 0 as ( m ) IQ (m ) n - l n - 1 k ( m * i ) d k ( m + 1 ) k ( m > d k ( m ) ln ( ">« M I | Q < M > I n - 1 da - n k(m +1 ' dk""*1 > |Q(<Y1) IQ(m * ' I ( 2 . 4 9 ) The c o n d i t i o n s t h a t dS(r*'> = 0 a n d d k ( r ) < d k ( r + 1 ) f o r a l l r^m+1 a s s u r e s t h a t no o v e r l a p p i n g s w i l l o c c u r f o r r^m+1. 45 P r a c t i c a l Considerations in Applying Kinematic Hardening In general, the kinematic rules given by equations 2.44, 2.46, 2.47, 2.48, and 2.49 only describe the instantaneous tra n s l a t i o n of the y i e l d surfaces f m and fm<., when the stress point i s at M in Fig.2.10. This means that only small and t h e o r e t i c a l l y i n f i n i t e s i m a l stress increments may be applied when following a stress path. The reason for this can be demonstrated using Fig.2.11. For c l a r i t y , i t i s assumed that Mroz's kinematic rule is applicable, d$ ( r n + 1 )=0, and no isotropic hardening occurs ( i . e . dk ( r )=0 for a l l r ) . If a large stress increment Ao is applied to the stress point M such that the new stress state i s at point A, then Mroz's kinematic rule would demand that point M translate in the d i r e c t i o n of MR. This raises the problem of how f m can translate along MR and yet s t i l l have point A on f m , as the consistency condition demands, after Ao has been applied. It is evident that the choice of a large value of ACT i s responsible for thi s d i f f i c u l t y . If a small stress increment dc; i s applied instead such that the new stress state i s at point B, then the situation becomes manageable. f m translates towards f m + ) in the di r e c t i o n of MR but only for a short distance because of the small value of do. Point B can therefore remain on f m while simultaneously s a t i s f y i n g the kinematic hardening rule. Fig.2.12 i l l u s t r a t e s the positions of the y i e l d surfaces after do has been applied. Note that the corresponding outward normal to point B (on f m ) on y i e l d surface f m + , i s point R' and thi s point l i e s between R 46 F i g . 2 . 1 1 . T r a n s l a t i o n of f m r e q u i r e s tha t many smal l s t r e s s increments be used , such as the i n i t i a l one to p o i n t B, i n s t e a d of one l a r g e one, such as a s i n g l e one to po in t A . 47 F i g . 2 . 1 2 . P o s i t i o n of f m a f t e r a s m a l l s t r e s s increment has been a p p l i e d . 48 and the intersection of the stress path with f m + l (point G). When another stress increment i s applied along the indicated stress path, f m translates in the direction of BR' which is d i f f e r e n t , in general, to MR. Clearly, the application of many more small stress increments would cause the corresponding outward normal on f m t l to s h i f t along the surface of f m., towards point G, with f m ultimately being tangent to fm+., at t h i s point. It i s evident that only "small" stress increments can give consistent r e s u l t s . Of course, this raises the problem of defining what a small stress increment i s . Unfortunately, there i s no single answer and the maximum stress increment which can be used depends on the desired accuracy of the computations and the positions of the y i e l d surfaces with respect to one another. For example, many stress increments would be required for the y i e l d surfaces shown in Fig.2.11 for reasons discussed e a r l i e r . In contrast, a single stress increment between y i e l d surfaces would be adequate i f the centers of f m and f m t ) l i e along an extension to a linear stress path, as i l l u s t r a t e d in Fig.2.13, since n,- would have the same value at intermediate points ( i . e . using small stress increments) as the i n i t i a l value. It should be noted that the s p e c i f i c a t i o n of small (dcr;j-d$;. ) and dk in the shortened form of the consistency condition (eq.2.27) i s not related to the problem just discussed. The consistency condition, whether i t be eq.2.26 or 2.27 only influences the magnitude of the kinematic movement and not the d i r e c t i o n . The d i r e c t i o n i s specified by the p a r t i c u l a r 49 stress path F i g . 2 . 1 3 . A l a r g e s t r e s s increment may be a p p l i e d when the c e n t e r s of f m and f m t i l i e a long an ex tens ion to a l i n e a r s t r e s s path because would not change even i f s m a l l s t r e s s increments were used . hardening r u l e used. 51 CHAPTER 3 THEORY FOR THE UNDRAINED TOTAL STRESS ANALYSIS OF CLAY Total stress type analyses have been widely used in the past for determining the s t a b i l i t y of slopes and for stress-deformation problems. Although i t i s widely acknowledged that e f f e c t i v e stresses govern the behavior of s o i l s , the t o t a l stress approach i s s t i l l widely used because of i t s s i m p l i c i t y and i t s successful application to many geotechnical problems in the past. This i s p a r t i c u l a r i l y true for saturated clays subjected to loadings of short duration. Under these conditions, no volume change occurs and any increase in the t o t a l mean normal stress has no eff e c t on the e f f e c t i v e mean normal stress. The behavior of the s o i l i s therefore independent of the t o t a l mean normal stress and only the deviatoric stresses need to be considered in any con s t i t u t i v e r e l a t i o n . Changes in the s t r e s s - s t r a i n c h a r a c t e r i s t i c s due to pore pressure generation during deformation must be i n d i r e c t l y accounted for by varying the y i e l d stress and strength of the s o i l . The behavior of p l a s t i c a l l y deformed, work hardening metals i s also considered to be independent of the mean normal stress. Mathematical models for describing the hysteresis and Bauschinger e f f e c t during p l a s t i c deformation of metals were proposed by Iwan (8) and Mroz (12) in 1967. But i t was not u n t i l Prevost published his work in 1977 (18) that these ideas 52 were u t i l i z e d to construct a p r a c t i c a l model for clays. This chapter p a r t i c u l a r i z e s the theory of Chapter 2 to model the undrained t o t a l stress behavior of clay. To describe the s t r e s s - s t r a i n behavior of a s o i l subjected to undrained . loading, the stra i n s are separated into e l a s t i c and p l a s t i c deformations. The e l a s t i c i t y i s assumed to be constant, l i n e a r , and i s o t r o p i c . The r e l a t i o n between the stress and e l a s t i c s t r a i n i s then given by: = 2Ge« + (3.1 ) where: 0 = Lame's constant e = e l a s t i c s t r a i n = e l a s t i c volumetric s t r a i n = e l a s t i c deviatoric s t r a i n om = mean normal stress Multiplying eq.3.1 by 8r gives: 3om = 2Ge*60 + 8;j 6(. J2e« Thus: 53 m = 2Ge.J + 5 ; jfie, e - 8-t] (^Ce? = 2 G e J - ^Ge: 5, = 2G<e« - ± e ! 5. ) = 2Ge^ o r : ( 3 . 2 ) 2G N o n - l i n e a r i t y i s a s sumed t o be a r e s u l t o f t h e p l a s t i c i t y . The y i e l d c r i t e r i o n f o r u n d r a i n e d s a t u r a t e d c l a y s , a s m e n t i o n e d p r e v i o u s l y , i s u s u a l l y c o n s i d e r e d t o be i n d e p e n d e n t o f t h e t o t a l mean n o r m a l s t r e s s . The y i e l d f u n c t i o n t h e n r e q u i r e s o n l y d e v i a t o r i c s t r e s s e s f o r i t s a r g u m e n t s . F o r m a t h e m a t i c a l s i m p l i c i t y , a Von M i s e s t y p e y i e l d s u r f a c e i s a s s u m e d . E q u a t i o n 2.18 t h e n b e c o m e s : The a b s e n c e o f t h e t o t a l mean n o r m a l s t r e s s f r o m e q . 3 . 3 i n d i c a t e s t h a t t h e r e i s no p l a s t i c v o l u m e c h a n g e . T h i s i s i n c o n t r a s t t o t h e common a s s u m p t i o n i n c o n s t i t u t i v e t h e o r i e s f o r s o i l w h i c h a s sumes t h a t t h e t o t a l v o l u m e c h a n g e i s z e r o w i t h t h e p l a s t i c vo l ume c h a n g e e q u a l a n d o p p o s i t e t o t h e e l a s t i c v o l u m e c h a n g e . A l t h o u g h i t i s p o s s i b l e t o c a l c u l a t e an e l a s t i c f m = 0 = $ ( s i j - a < m l ) ( s i . - a $ m > ) - (k<">>)2 ( 3 . 3 ) 54 volume change, t h i s quantity i s o r d i n a r i l y set to zero in thi s model for consistency with the assumption of zero t o t a l volume change in clays during undrained loading. The at" 1' are a subset of the defined in eq.2.17 and represent the coordinates of the center of the y i e l d surface, f m , in the deviatoric stress subspace. The k^' are a measure of the size of the y i e l d surfaces. To conceptualize eq.3.3, l e t us assume that there are only two deviatoric stress dimensions: 12 (one-two) and 13 (one-three), with the remaining deviatoric stresses and offs e t s (a;-) being zero. Expanding eq.3.3 we have: ( s l 2 - a 1 2 ) 2 + ( s 1 3 - a l 3 ) 2 = -|k2 . (3.4) This i s the equation of a c i r c l e in s 1 2 - s 1 3 space with of f s e t s of a, 2 and a 1 3 from the o r i g i n along the 12 and 13 axes respectively. Thus, y i e l d surfaces can be thought of as being 2 dimensional c i r c l e s having r a d i i of j^k in a 2 dimensional deviatoric stress space. In a three dimensional deviatoric stress space, the y i e l d surfaces would appear as spheres. To obtain the p l a s t i c s t r a i n s , a flow rule i s required. For mathematical s i m p l i c i t y , an associated flow rule i s adopted. Although some researchers have questioned i t s v a l i d i t y (4,27), the associated flow rule has enjoyed some success in other s o i l p l a s t i c i t y models (24) and i t i s widely used. Equation 2.19 then becomes: 55 p 1 ( Q k r M d s k l ) Q d e ' = 1 (3.5) H; |QCrr», , | 2 w h e r e : Q -( rvi ) i _ dsr 3s . J J = o u t w a r d n o r m a l t o y i e l d s u r f a c e fm | Q ( m > - | = ( Q ( « ) • Qt*)•) d s ^ = d e v i a t o r i c s t r e s s i n c r e m e n t de? = p l a s t i c d e v i a t o r i c s t r a i n i n c r e m e n t T o o b t a i n a p r a c t i c a l s t r e s s - s t r a i n r e l a t i o n , e q . 3 . 5 i s s i m p l i f i e d by r e c a l l i n g t h a t : <3fm Q ( m , , = = 3 ( S ( _ _ a , m ) ) { 3 6 ) OS--| Q ( m ) . | = J6W»>- (3.7) T h e r e f o r e : 1 [3(s ; j - a ^ > ) ] 3 ( ( m ) ki ) d s kl H: 6 k ( m ) 2 (3.8) T h e c o m p l e t e c o n s t i t u t i v e r e l a t i o n t h e n b e c o m e s : 56 ds. 3 (s. de; a'"" ) 2G 2Hr k. ( m > 2 ( s - a ( m ' ) d s (3.9) This can be inverted (Appendix B) to give the stress increment in terms of the s t r a i n increment: ds;j = 2Gde;j |(2G-Hm)(s,. - a<»>>) k ( r- ) 2 (s k l " (3.10) where; 1 H„ HI 2G (3.11) Now l e t us examine the form of HI. By forming a scalar product of both sides of eq.3.5 with i t s e l f we have: def. de' 1 H I ( Q ^ M ds i 3) - 2 |Q(m ) I 2 O ( m ) » r» ( m ) » w k l v k l The square root of th i s becomes: HI (Q.^" dsy) |Q ( m" | |dep | where: |de p| = (de^de",)1 57 The s t r e s s increment ds. can be s p l i t i n t o two p o r t i o n s : ds- = ds-! + ds.V where: ds;V i s i n the d i r e c t i o n of the outward normal t o the y i e l d s u r f a c e ; ds,j i s tangent t o the y i e l d s u r f a c e and produces no p l a s t i c f l o w . T h e r e f o r e : Q.(m) 'da. = Q ( m ) 'dsV = |Q(,T" ' | |ds" | and: |ds" | H ; = ( 3 . 1 2 ) Idep I T h i s shows t h a t the p l a s t i c modulus i s not a f u n c t i o n of Q'. and thus may assume a c o n s t a n t v a l u e a l o n g the y i e l d s u r f a c e . Both i s o t r o p i c and k i n e m a t i c h a r d e n i n g a r e i n c o r p o r a t e d i n t h i s model. K i n e m a t i c h a r d e n i n g i s adopted t o model th e 58 B a u s c h i n g e r e f f e c t w h i l e i s o t r o p i c h a r d e n i n g i s used f o r i n c o r p o r a t i n g the e f f e c t s of s t r e n g t h and s t i f f n e s s l o s s d u r i n g c y c l i c l o a d i n g . The k i n e m a t i c h a r d e n i n g of the y i e l d s u r f a c e s i s d e s c r i b e d by Mroz's k i n e m a t i c r u l e ( 1 2 ) : da ;< m ) = d/i M,J (3.13) i n which (eq.2.31): Mi; = k ( m+ 1 } ( s . . . - ( s ; i - a t " ^ 1 > ) (3.14) 1 k ( nr, ) •> J J J where s, may be s u b s t i t u t e d f o r av s i n c e o n l y d e v i a t o r i c s t r e s s e s a r e c o n s i d e r e d . The s c a l a r du can be found from eq.2.48 (assuming s m a l l ds,-da, and d k ) : Q\.m ' ds, - 2 k ( f M ) d k ( m ) dM = — 3(s i ; j - alim) )ds, - 2k ( f" 'dk*"1 3 ( s k ( - akl">>) Mk, (3.15) I f i t i s f u r t h e r assumed t h a t d k ( r ) = d k ( r + 1> f o r a l l r and Q f . m ) d a J m + 1 >=0 ( 2 0 ) , then eq.2.43 g i v e s : 59 da.<m + 1 ' = 0 and the y i e l d surfaces larger than the one which the stress point i s currently on do not translate ( i . e . the unengaged surfaces do not move). This resulting computational s i m p l i c i t y i s the primary reason for using Mroz's kinematic rule over Prager's, Ziegler's, or P h i l l i p ' s rule. A l t e r n a t i v e l y , eq.3.15 could have been derived by using the hardening rule proposed in eq.2.49 with d k ( r ' = d k ( r + 1 1 for a l l r. A more exact expression for d/u without assuming small (dSij-dciij) and dk can be derived by f i r s t assuming dk ( r ' =dk ( r + 1 ' for a l l r and da ;] m + 1 ,=0 (20). Eq.2.26 for n=2 becomes: Q... (ds 1— dajj) + - (ds da k,) 2 = 2kdk + dk 2 2 ds(j kl 2 ds 1 3Q kl kl ds,? + Q 5d S i j 2kdk 0 L e t t i n g : Q = 3 f _ = 3(s ; -then: 60 d a 2 ( f ) + d a i j[-3dSj.-3 ( s i j - a l j ) ] + [-|ds.,2 + 3 ( s ( j - a , . )ds ; j-2kdk-dk 2 ] = 0 But da,-. = dix Mjj. Therefore: dn = -B-yBz-4AC 2A (3.16) where: A = B = -[3/x..dsij+3(sij-aij ] C = [%ds? + 3 ( s 5 - a ^ )dst. - 2kdk - dk 2 ] For isotropic hardening/softening, the size of the y i e l d surface k, i s conveniently taken to be a function of (19): X = j^dejjdef. )* (3.17) The parameter X then becomes a measure of the t o t a l p l a s t i c s t r a i n which has taken place. For s i m p l i c i t y , the surfaces may retain their o r i g i n a l size u n t i l the f i r s t load reversal occurs. By varying the size of the y i e l d surfaces, the behavior of 61 a s o i l subjected to a small number of load cycles can be adequately modelled. For a large number of cycles, both k and H' w i l l vary for each y i e l d surface. This w i l l be explained further in the chapter dealing with c y c l i c loading. As mentioned previously, the <~;\r ) represent the coordinates of the center of the y i e l d surface f r , in the deviatoric stress subspace. The significance of the a- can best be examined by f i r s t assuming that they are a l l zero (Fig.3.1). Any c i r c l e A with i t s center at 0 would represent an invariant measure of the deviatoric stress state of the system. From eq.3.12, i t can be seen that |dep | would be the same for any monotonic stress path which starts at the o r i g i n O and terminates on c i r c l e A. Since |dep | i s an invariant measure of the p l a s t i c s t r a i n and the e l a s t i c i t y i s assumed to be i s o t r o p i c , the s o i l would behave i s o t r o p i c a l l y . Thus i f a;j=0 , the material i s i s o t r o p i c . I f , on the other hand, we offset y i e l d surface B as in Fig.3.2, the |ds"| (stress increment component perpendicular to the y i e l d surface) to c i r c l e A would be d i f f e r e n t and hence |dep | would depend on the loading path. The s o i l would then behave a n i s o t r o p i c a l l y . In general, there are six deviatoric axes (of which five are independent) and thus six components of a must be evaluated for each material type. However, i f the physical coordinate axes coincide with the p r i n c i p a l axes of material anisotropy, a x y r ' » a y 2 r ' a n d a z X r ' a r e t n e n i n i t i a l l y a l l equal to zero for 62 F i g . 3 . 2 . Y i e l d s u r f a c e s i n s t r e s s space w i t h y i e l d s u r f a c e B o f f s e t from o r i g i n . A n i s o t r o p i c behav ior would r e s u l t . 63 a l l r . F u r t h e r , i f a{xr) = a< r ' f o r a l l r, then the m a t e r i a l i n i t i a l l y e x h i b i t s c r o s s - a n i s o t r o p y about - the y a x i s ( i . e . the behavior i s the same on a g iven p lane which i s p e r p e n d i c u l a r to the y a x i s ) . Th is i s p a r t i c u l a r i l y r e l e v a n t to g e o t e c h n i c a l eng ineer ing because f i n e - g r a i n e d s o i l s are o f t e n cons idered to be d e p o s i t e d i n h o r i z o n t a l l a y e r s . 64 CHAPTER 4 APPLICATION OF THE TOTAL STRESS MODEL  TO UNDRAINED CLAYS The application of the model requires the determination o f : 1. The e l a s t i c shear modulus G 2. The i n i t i a l positions (a 0) and sizes (k 0) of the y i e l d surfaces and t h e i r associated p l a s t i c moduli (H') 3. The changes in the p l a s t i c moduli or size of the y i e l d surfaces as loading occurs. To describe the undrained behavior of clays which maintain their s t i f f n e s s and do not soften requires only the f i r s t two sets of parameters. Under these conditions, H' and k are constant. This section w i l l be confined to the application of the model to such s o i l s . Clays which soften when subjected to c y c l i c loads require that the changes in H' and k be determined. A simple example i l l u s t r a t i n g how the functional change in k (and H') can be determined w i l l be described in Chapter 6. Although the model i s capable of analyzing a s o i l which i s completely anisotropic, only a procedure to determine the parameters for the case of cross-anisotropy (or rotational symmetry) about the v e r t i c a l y axis w i l l be outlined here. In 65 t h i s case, both the experimental t r i a x i a l (ax=oz ) compression and extension s t r e s s - s t r a i n curves are required. Theoretically, i t is immaterial whether the t r i a x i a l compression test i s conducted by holding ax and az constant and increasing oY , or by holding aY constant and decreasing the confining pressure since both loading patterns would have i d e n t i c a l stress paths in the deviatoric plane. S i m i l a r i l y , the method of imposing the stress difference (ay-a„ ) i s immaterial for t r i a x i a l extension. Test data to support t h i s finding w i l l be presented further on. The t r i a x i a l compression and extension tests described above are not true tests for anisotropy. This i s demonstrated in Fig.4.1 where the equal and opposite stresses necessary for equilibrium have been removed for c l a r i t y . However, i f the material i s cross-anisotropic about the v e r t i c a l axis, an associated flow rule i s assumed, and a Von Mises type y i e l d surface i s adopted, then the behavior of the s o i l under t r i a x i a l compression and extension tests i s also a r e f l e c t i o n of i t s anisotropy. To i l l u s t r a t e t h i s , stress paths corresponding to the stress states I, II, and III in Fig.4.1 have been shown in Fig.4.2 in the deviatoric stress space. Cross-anisotropy about the y axis and an associated flow rule require that the center of each y i e l d surface l i e on the (positive or negative) s y a x i s . If a y i e l d point i s selected along the positive s y axis, say at point A, then only one other corresponding y i e l d point in the deviatoric plane, c a l l i t B, i s needed to completely a) Tr iax ia l Tests Compression b) True Tests for Anisotropy cr H II °3 I I I T r i a x i a l Compression F i g . 4 . 1 . a) The t r i a x i a l compression and e x t e n s i o n t e s t s sub jec t the s o i l to a d i f f e r e n t set of s t r e s s e s i n each case whereas b) the same set of s t r e s s e s at d i f f e r e n t o r i e n t a t i o n s i s r e q u i r e d fo r a t r u e t e s t f o r a n i s o t r o p y . 67 " I A Extension F i g . 4.2. Stress paths for true tests for anisotropy (II and III) compared with the stress path for the extension test (IV). describe the position of that y i e l d surface. Thus i f a test corresponding to stress state II or III (which i s t r u l y i n d i c a t i v e of anisotropy) i s performed, the corresponding y i e l d point in t r i a x i a l extension can be interpolated from the y i e l d function. S i m i l a r i l y , i f B were chosen from a t r i a x i a l extension t e s t , the corresponding y i e l d points for II and III may be determined. Therefore the extension test i s a r e f l e c t i o n of the s o i l ' s anisotropy in t h i s p a r t i c u l a r case. If the axes of material anisotropy coincide with the p r i n c i p a l axes of loading in the test apparatus, then for ro t a t i o n a l symmetry about the y a x i s : 68 a x y = a y z = a 2 X = 0 (4.1) a x = az Since the a;j are deviatoric stresses, then a;; =0 and: a x = - ay (4.2) 2 A procedure to find G and the i n i t i a l values of H', k, and a i s as follows: 1. Choose a piecewise linear approximation of the stress difference (ay-ax ) versus a x i a l s t r a i n (ey ) curve in compression as shown in Fig.4.3a. A corresponding representation in stress space i s shown with i t for explanatory purposes in Fig.4.3b. Appendix D derives the scale factor /3/2 which i s necessary to est a b l i s h equivalence between the stress axes of Fig.4.3a and Fig.4.3b. 2. Points where the piecewise linear curve changes slope are referred to here as y i e l d points. An e l a s t o - p l a s t i c modulus Hf i s associated with the l i n e a r segment above y i e l d point r: Fig. 4 . 3 . a) Piecewise linear approximation of the st r e s s - s t r a i n curve for an axisymmetric t r i a x i a l compression and extension test, b) Corresponding representation of y i e l d surfaces in stress space. 70 3 3 s l o p e o f segment r on s t r e s s —H r = = d i f f e r e n c e v s . a x i a l s t r a i n 2 2 J_ c u r v e H; G ( 4 . 3 ) T h i s i s d e r i v e d i n A p p e n d i x C. F i n d c o r r e s p o n d i n g y i e l d p o i n t s on t h e e x t e n s i o n s i d e . As m e n t i o n e d p r e v i o u s l y , t h e p l a s t i c s h e a r m o d u l u s HI i s a c o n s t a n t f o r e a c h y i e l d s u r f a c e m. Thus t h e c o r r e s p o n d i n g y i e l d s t r e s s e s i n e x t e n s i o n r e p r e s e n t p o i n t s where t h e s l o p e s . o f t h e i d e a l i z e d , p i e c e w i s e l i n e a r , s t r e s s - s t r a i n c u r v e s a r e t h e same a s i n c o m p r e s s i o n . C a l c u l a t e k ( m ) f o r e a c h m: 2 k ( m ' = d i f f e r e n c e b e t w e e n t h e y i e l d s t r e s s i n c o m p r e s s i o n a n d e x t e n s i o n o f s u r f a c e m. = (oy -a„ ) ( m ) - (oy -ax )<?> 7 * Corn- f * ey(\ C a l c u l a t e a l ( m ) f o r e a c h m w h e r e : a t m ) _ o f f s e t o f c e n t e r o f y i e l d s u r f a c e m f r o m o r i g i n . 71 k(m> = f a y m ) = -3a y ( r n ) = - 3 a J m ) These parameters together with equations 3.9 and 3.13-3.15 give the complete set of equations necessary to describe the s t r e s s - s t r a i n behavior of non-softening s o i l s . It should be noted that the parameters obtained by the above method are a function of the selection of the piecewise linear approximation of the s t r e s s - s t r a i n curve in compression. Thus, there are an i n f i n i t e number of possible parameters for a given stress-s t r a i n curve. For many common problems, some components of a- and the stress tensor vanish. For example, in situations for which either axisymmetric or plane s t r a i n loading conditions occur ( i . e . two dimensional a n a l y s i s ) , T = T X Z = 0. The following stress coordinates become convenient for this case(!9): S , = (4.4a) = ' l U z - O = V l ( s z - s x ) (4.4b) S 3 = /3 r (4.4c) These stress coordinates and th e i r associated strains w i l l be referred to as 1,2,3 coordinates. The distance from the o r i g i n to the stress point in the orthogonal stress space 72 S i , S 2 , S 3 is: / S , 2 + S 2 2 + S 3 2 = /3J7 = /|s,.s,. By analogy with eq .3 .3 , we have: (L(S. - a , ( m > ) 2 ) - k ( m' a= 0 i(s,-ai\n»>)(s;.-oi{.-)) - k"">* ( 4 . 5 ) in which the coordinates of the center of the y i e l d surface m are: 3 a ( m ) = _ a ( n n ) i y 2 /3 a ( m ) = — ( a ^ ' - a ^ ' ) 2 a j m ) = y3a^> The str a i n s E,, E 2 , and E 3 associated with the stresses S 1 f S 2 , and S 3 respectively are defined so that an increment of work would be the same in either x,y,z or 1,2,3 coordinates: dW = s;j de.j = ?S, dE,. ' ( 4 . 6 ) Combining equations 4 . 4 and 4 . 6 gives: 73 sx de„ + s yde y + s, de, +2sXv de 3sydE, + ^ S ( s 2 - s x ) d E 2 + /3sxy dE 3 (4.7) To get an expression for E 3 , note that the last term on the l e f t hand side of eq.4.7 corresponds to the last term of the right hand side since these are the only terms involving s x y . Therefore: 2s x ye x y = j3s X ydE 3 Integrating the l e f t hand side between 0 and E 3 and the right hand side between 0 and 2fXy gives: E 3 = ^ * X y (4.9) E, can be found from the terms in eq.4.7 which do not have any xy terms: sx dex + sy de y + s z de 2 = |s ydE, + J£L (S z-s„ )dE 2 (4.10) Recall that: 7 4 and i f Poisson's ratio=.5: ex + e y + e- = 0 then: 2 s x d e x + 2 s yde y + s yde x + s x de y = l s y d E , - - s y + 2 s x )dE 2 ( 4 . 1 1 ) 2 s x (dex +{dey ) + 2 s y d e y + s y dex = i s y d E , --§•( s y + 2 s x )dE 2 ( 4 . 1 2 ) Taking terms containing only s x : - j 3 s,<dE 2 = 2 s x (de x + {dey ) = 2 s x (de x - {dex - {de. ) = s x (de x - de 2) Therefore: dE 2 = ^L(de 2 - de x) Integrating, we get: E 2 = 7^(e 2 - e, ) (4.13) F i n a l l y E, can be determined by substituting the expression for E 2 into eq.4.12 and integrating the resu l t i n g t o t a l d i f f e r e n t i a l to give: 75 E i = e , (4.14) The s t r e s s - s t r a i n r e l a t i o n i s obtained by substituting eq.4.5 into eq.3.5 giving: dE : dS, 2 (S; - tti(m> ) + ?(Sj -a.'-')dSj 3G 3H1 (k<">>)2 (4.15) in which the inverse i s : 3 \ S; - a}m> dS; = 3GdE| - 3G - -Hm 2 J ( k ( m > ) 2 J Z(Si - aj(nr'>)dSi (4.16) The kinematic hardening rule follows from equations 2.31 and 2.48: da, = dju k ( m + I ) k ( m ) (Sj - > ) - (S, - a ; ( m t 1 ' ) (4.17) in which: dM = -B- JBZ-4AC 2A A = Mi M; B = -2M; (S; + dS; - a\» > ) 76 C = 2(Sj - o, ( m l)dS; + dS, dS; -2k ( f r i )dk (' r , ) - ( d k ( f r ) ) ) 2 or for small (dS; -da, ) and dk: d M = ; < m ) )dS; - k'^'dk"1"1 (4.18) ?(S. - a-'"*' > ) (S-, - a!*1) The S 1,S 2,S 3 stress space i s p a r t i c u l a r i l y useful when interpreting t r i a x i a l , simple shear, or plane st r a i n loading conditions. During axisymmetric t r i a x i a l tests, a„ = o z and Txy=0. Thus S2=S3=0 and the stress point moves along the S, axis only. Simple shear s o i l tests demand that dex =dey =dez =0. For a s o i l i n i t i a l l y subjected to equal horizontal normal stresses: E2=0, S2=0, and a 2 =0. Applying equations 4.16 and 4.17 gives dS2=0 and da2=0 respectively. The stress point therefore remains in the S,,S3 plane at a l l times. If the e l a s t i c strains are assumed to be zero, def^O and dE^O. The stress point then follows a stress path such that the outward normal to the y i e l d surface which i t i s on, has no component in the S, dir e c t i o n (Fig.4.4). Plane s t r a i n tests require rxy= ryx= rxi=0 and thus S 3 = 0 at a l l times. Conventional plane s t r a i n tests constrain de2=0 and thus: de x + de y + de 2 = 0 77 F i g . 4.4. a) Stress path for simple shear test in S,-S3 plane. Resulting b) shear stress vs. shear s t r a i n and c) stress difference vs. shear s t r a i n curves. [from Prevost (18)] de x = -de y - /3dE2 = -dE, dE 2 = -jLdB, In s i t u pressuremeter te s t s , which are a type of plane st r a i n t e s t , i d e a l l y have dey =0 and thus dE,=0. If the e l a s t i c strains are n e g l i g i b l e , then the stress point l i e s in the S 1-S 2 plane and follows a path such that the normal to the y i e l d s u r f a c e which i t i s on has no component i n the S , d i r e c t i o n . 79 CHAPTER 5 PREDICTIONS USING THE UNDRAINED TOTAL STRESS MODEL To evaluate the accuracy of the model proposed by Prevost, the s t r e s s - s t r a i n behavior of a clay subjected to four d i f f e r e n t undrained, monotonic stress paths was predicted from the results of undrained t r i a x i a l compression and extension t e s t s . The predictions were subsequently compared with stress-s t r a i n curves obtained from actual test data. Computer Program Check In order to v e r i f y that the computer program developed for t h i s study was able to duplicate the c a l c u l a t i o n procedures as outlined in Prevost's papers (18,19), a prediction of the simple shear behavior of Drammen clay K 0 overconsolidated to a r a t i o of 4, was made using the model parameters published in (18) and repeated here in Table 1. Figure 5.1 shows the s t r e s s - s t r a i n curves as calculated by the computer program. The curves match those published in (18) very well, thus v e r i f y i n g the accuracy of the computer program used. Test Data The tests were conducted as part of the May 1980 NSF/NSERC North American Workshop on P l a s t i c i t y Theory and Generalized Stress-Strain Modelling of S o i l s (29). F u l l d e t a i l s of the 80 TABLE 1 Model Parameters for Drammen Clay I from Prevost (18)] m k a H o* a' a' psi psi psi 1 0.300 0. 100 266.67 2 0.350 0.150 133.33 3 0.600 0.300 100.00 4 0.700 0.400 73.33 5 0.775 0.475 54.67 6 0.875 0.525 40.00 7 0.950 0.550 31 .00 8 1 .025 0.575 24.33 9 1 .050 0.600 17.33 10 1 . 1 25 0.575 13.33 1 1 1 .200 0.550 1 0.00 12 1 .250 0.550 6.67 13 1 .275 0.525 3.33 14 1 .373 0.467 0.00 G = = 200a' testing procedure are available in the Proceedings of the above conference and only a summary w i l l be given here. A laboratory prepared k a o l i n i t e clay with a l i q u i d l i m i t of 62.5%, a p l a s t i c l i m i t of 39.0%, and an average undrained water content of 38.8% was used for the tests. The samples 81 X .8 t vc o . o (%) o 2 4 6 8 F i g . 5.1 Prediction of Drammen clay simple shear behavior. were t y p i c a l l y K 0 consolidated with an e f f e c t i v e confining pressure of 40 psi and an additional a x i a l load s u f f i c i e n t for a K0=.48 condition to be maintained. After consolidation was completed, the additional (K 0) a x i a l load was removed and the sample was allowed to rebound under an isotropic e f f e c t i v e stress of 40 p s i . The samples were then subjected to stress-controlled tests in an undrained condition. Four basic tests were provided upon which model parameters were to be derived: Test 1 - A compression test in which the a x i a l stress was increased while the confining pressure was held constant. 82 Test 4 - A compression test in which the a x i a l stress was increased and the l a t e r a l stress decreased such that the t o t a l mean normal stress remained constant. Test 10 - An extension test in which the a x i a l stress i s decreased while the confining pressure was held constant. Test 13 - An extension test in which the a x i a l stress was decreased and the l a t e r a l stress increased such that the t o t a l mean normal stress remained constant. The stress difference versus a x i a l s t r a i n curves are plotted in Fig.5.2. Note that tests 1 and 4 in compression have almost i d e n t i c a l behavior. This i s to be expected since these tests have i d e n t i c a l stress paths in the deviatoric plane and the changes in the t o t a l mean normal stress do not a f f e c t the e f f e c t i v e mean normal stress (because of undrained conditions). A similar arguement holds for tests 10 and 13 in extension. Table 2 l i s t s a set of parameters for t h i s s o i l using the procedure outlined in Chapter 4 while Tables 3, 4, 5, and 6 tabulate the stress paths for prediction. Note that tests 2 and 5 have i d e n t i c a l stress paths in the deviatoric plane. Fig.5.3 shows how the angle u i s defined for the stresses paths to be predicted. A representation of the y i e l d surfaces in the S 1 f S 3 plane i s given in Fig.5.4 together with the stress paths F i g . 5.2. T r i a x i a l s t r e s s - s t r a i n curve for Ka o l i n i t e . Note the almost i d e n t i c a l behavior of the conventional t r i a x i a l test and the constant t o t a l mean normal stress test, [data from (29)] p> CO 84 TABLE 2 Model Parameters for Kaol i n i t e m k a H psi psi psi 1 10.0 10.0 2963.0 2 20.0 8.0 1436.0 3 27.0 15.0 800.0 4 30.0 18.0 444.4 5 34.4 17.8 288.0 6 38. 1 14.2 138.4 7 44.3 12.3 98.8 8 49.3 9.3 .01 G = 3030. psi F i g . 5.3. D e f i n i t i o n of u. 85 to be predicted. Predictions Figures 5.5, 5.6, 5.7 and 5.8 compare the predicted s t r e s s - s t r a i n curves for tests 2, 3, 5,and 7 respectively with actual test data. While tests 2, 3, and 7 agree very well, the results of test 5 appear to be somewhat disappointing. Because the stress paths in the deviatoric plane of samples 2 and 5 are the same, one would expect the correspondence between the predicted and experimental s t r e s s - s t r a i n curves to be the same for both tests. One explanation for thi s discrepancy could be that test 5 had a water content almost 1% higher than the average of the other t e s t s . In e f f e c t , t h i s sample was less overconsolidated than the others and thus one would expect i t s strength to be lower than that predicted from parameters obtained from specimens with lower water contents. With these considerations in mind, i t appears that the model was able to predict the behavior of t h i s clay very well. These comparisons further demonstrate the a p p l i c a b i l i t y of the model to the undrained, monotonic loading of clays as previously shown by Prevost (19). TABLE 3 Test 2 Total P r i n c i p a l Stresses (psi ) o2 o3 58.00 58.00 58.00 78.90 58.00 56.50 86.22 58.00 55.98 92.69 58.00 55.51 96.78 58.00 55.22 100.01 58.00 54.99 102.28 58.00 54.82 104.32 58.00 54.68 105.72 58.00 54.58 107.88 58.00 54.42 109.60 58.00 54.30 110.79 58.00 54.21 1 1 1 .76 58.00 54. 15 112.40 58.00 54. 10 112.94 58.00 54.06 113.48 58.00 54.02 113.91 58.00 53.99 114.13 58.00 53.97 114.34 58.00 53.96 water content = 38.8% co = 15° TABLE 4 Test 3 Total P r i n c i p a l Stresses (psi) o2 o3 58.00 58.00 58.00 74. 18 58.00 51 .82 81.14 58.00 49. 1 6 88. 10 58.00 46.50 91 .98 58.00 45.02 94.08 58.00 44.22 96. 19 58.00 43.41 97.64 58.00 42.86 98.45 58.00 42.55 99.58 58.00 42. 12 100.39 58.00 41 .81 101.20 58.00 41 .50 101.85 58.00 41 .25 102. 17 58.00 41.13 102.33 58.00 41.07 water content = w = 38.8% 37.5° TABLE 5 Test 5 Total P r i n c i p a l Stresses (psi ) a2 58.00 58.00 58.00 72.36 51.57 50.08 73.62 51.00 49.38 77.34 49.33 47.32 79.87 48.20 45.93 81 .58 47.43 44.98 82.92 46.83 44.25 83.74 46.47 43.79 84.41 46. 17 43.42 84.86 45.97 43. 18 45.67 45.60 42.73 86.27 45.33 42.40 86.72 45.13 42. 1 5 87.01 45.00 41 .99 87.31 44.87 41 .82 87.46 44.80 41 .74 87.53 44.77 41 .70 water content = 39.7% co = 15° TABLE 6 Test 7 Total P r i n c i p a l Stresses (psi) o^ a2 o3 58.00 58.00 58.00 64. 1 0 58.00 51 .90 68. 10 58.00 47.90 70. 10 58.00 45.90 72. 10 58.00 43.90 74. 10 58.00 41 .90 77.90 58.00 38.20 79.40 58.00 36.60 81 .00 58.00 35.00 81 .80 58.00 34.20 82.50 58.00 33.50 83.40 58.00 32.60 84. 10 58.00 31 .90 84.20 58.00 31 .80 84.40 58.00 31 .60 water content = 38.8% CJ = 45° 90 S3 ( p s i ) I6 0 P r o j e c t i o n o f T e s t 3 o n S1-S3 p l a n e + -60 F i g . 5.4. K a o l i n i t e y i e l d surfaces and stress paths for prediction in S,-S3 plane. 91 12ot 40 I , , , , , H 0 1 2 3 ^ 5 6 I STRAIN i (%) F i g . 5.5. Str e s s - s t r a i n comparisons for Kaolinite - Test 2. [data form (29)3 92 F i g . 5.6. Str e s s - s t r a i n comparisons for Kaolinite - Test 3. [data from (29)] 93 1 0 0 + i 1 1 1 1 1 1 • 0 1 2 3 ^ 5 6 I STRAIN I (%) F i g . 5.7. St r e s s - s t r a i n comparisons for Ka o l i n i t e - Test 5. [data from (29)] 94 F i g . 5.8. Str e s s - s t r a i n comparisons for Kaolinite - Test 7. [data from (29)] 95 CHAPTER 6 APPLICABILITY OF THE MODEL TO CYCLIC LOADING One of the most a t t r a c t i v e features of t h i s model i s i t s a b i l i t y to simulate some aspects of c y c l i c loading. Foremost amongst these i s the a b i l i t y to exhibit Masing behavior. Figures 6.1 and 6.2 i l l u s t r a t e how the model generates hysteresis loops in a simple shear test . The y i e l d surfaces and stress paths are plotted in S,,S3 space whereby S, = J3 sy /2 (eq.4.4a) and S3= j3 T x y (eq.4.4c). Let us f i r s t assume that the s o i l i s cross-anisotropic about the v e r t i c a l axis and that the e l a s t i c s t r a i n s are n e g l i g i b l e . Then the JT r X y required to reach any y i e l d surface m upon i n i t i a l loading i s equal to k ( m > since the stress path goes through the apex of f m (Fig.6.1). If no softening occurs, then the change in rxy to reach y i e l d surface tm upon unloading i s 2 k ( m ) as shown in Fig.6.2. This i s p r e c i s e l y the behavior predicted by applying Masing's rules. S o i l s which soften when subjected to c y c l i c loads require that the y i e l d surfaces decrease in s i z e . A s i m p l i s t i c method of doing t h i s would be to u t i l i z e an isotropic softening rule whereby a l l the y i e l d surfaces contract in size simultaneously and by the same amount ( i . e . d k ' r } = d k ( r + 1 ) for a l l r) without changing th e i r positions ( a ; j r ) ) . While this may be appropriate i f only a few load cycles are applied, the remolding effect of large cumulative p l a s t i c s t r a i n i n g eventually causes a loss of 96 (a) (b) F i g . 6.1. a) Y i e l d surfaces before loading and i n i t i a l monotonic stress path to point N. b) Shear stress - s t r a i n curve re s u l t i n g from monotonic stress path. [from Prevost (18)] (a) (b) F i g . 6.2. a) Y i e l d surfaces upon reaching point N and loading reversal stress path. )b Hysteretic loop formed by stress cycle, [from Prevost (18)] 97 the s o i l ' s inherent anisotropy. To account for t h i s , Prevost (18) suggests that, in addition to contracting the size of a l l the y i e l d surfaces by the same amount, the offsets along the S, axis be made to decrease ( i . e . a kinematic change) such that a,=a,C\) u n t i l a 1 ( r >=0 for a l l y i e l d surfaces, f r , which have not yet been translated. Inspection of the experimental c y c l i c load test data for loss of inherent anisotropy ( i . e . a tendency towards isotropic behavior) can determine whether this additional sophistication i s warranted. To demonstrate the ef f e c t of decreasing the sizes of the y i e l d surfaces, the isotropic softening rule w i l l be applied to model the behavior of Santa Barbara s i l t when subjected to a st r a i n controlled, c y c l i c simple shear test with an amplitude of 7X =1%. Fig.6.3 shows the re s u l t s of a monotonic 'max t r i a x i a l compression and an extension test as reported in reference 23 and Table 7 l i s t s a set of parameters derived from th i s data. The predicted simple shear s t r e s s - s t r a i n curve together with the experimental test results are compared in Fig.6.4. Good correspondence between the predicted curve and experimental points was achieved for the T x / versus yx^ p l o t . To model the softening behavior during c y c l i c loading, the change in the r e s i s t i n g stress when the s o i l i s subjected to a constant c y c l i c s t r a i n amplitude must be found experimentally. Fig.6.5 plots t h i s change versus the number of cycles for a st r a i n amplitude of 1% as reported by Prevost (23). The quantity N i s defined to be the number of cycles applied beyond the i n i t i a l loading curve plus one. Thus the end of the F i g . 6.3. T r i a x i a l t e s t r e s u l t s used f o r determining the parameters f o r Santa Barbara s i l t , [data from (23)] CO 99 TABLE 7 S a n t a B a r b a r a S i l t m k a' a a' H a' 1 . 1 50 .500 52.630 2 .225 .425 27.780 3 .298 .403 16.667 4 .340 .390 13.333 5 .375 .375 8.888 6 .405 .365 4.444 7 .450 .350 3.030 8 .550 .300 1 .587 9 .615 .335 0.001 G = 8 3 . 3 3 a ' m o n o t o n i c s t r e s s - s t r a i n c u r v e w o u l d be c y c l e 1 a n d t h e c o m p l e t i o n t h e r e a f t e r o f a h y s t e r e s i s l o o p w o u l d be c y c l e 2. N e g l e c t i n g e l a s t i c s t r a i n s ( a n d t h e r e f o r e t h e c u m u l a t i v e p l a s t i c s t r a i n i n v a r i a n t X a l s o e q u a l s t h e i n v a r i a n t m e a s u r e o f t h e t o t a l c u m u l a t i v e s t r a i n ) t h e i n c r e m e n t o f p l a s t i c s t r a i n i s e q u a l t o dX: 100 F i g . 6 . 4 . Simple shear s t r e s s — s t r a i n c o m parison f o r Santa B a r b a r a s i l t . dX = ( i d ^ d e j ) * - (2«§de 2) i = £de*y = U s i n g the above d e f i n i t i o n f o r N, the t o t a l c u m u l a t i v e p l a s t i c s t r a i n f o r a c y c l i c s t r a i n c o n t r o l l e d s i m p l e shear t e s t f o r one c y c l e would be: 101 0.4 0 . 1 0 . 0 I I M i l I M i l 1 M i l 1 1 0 1 0 0 1 0 0 0 N = number of c y c l e s F i g . 6.5. Decrease of s t r e s s r a t i o d u r i n g c y c l i n g i n a s i m p l e shear s t r a i n - c o n t r o l l e d t e s t a t a shear s t r a i n a m p l i t u d e of 1%. [ d a t a from ( 2 3 ) ] X or a f t e r 2 c y c l e s : or a f t e r N c y c l e s : •xy X = -ll [ i + 4 ] 73 X = — J l [1 + 4(N-1)] N*1 (6.1) 73 102 For s i m p l i c i t y in presentation, only the s t r e s s - s t r a i n traces during fiv e cycles with 7 =1% w i l l be demonstrated here. From the experimentally obtained curves in Fig.6.5, the rela t i o n between the r e s i s t i n g stress r V u and the number of cycles N for a stra i n amplitude of 1% can be approximated to be (for N<10) : r h = -0.0767 log(N) + 0.230 (6.2) Alog N Equation 6.1 may be rewritten: where: -0.0767 1 N = -4 J3\ + 3 N > 1 Substituting this expression into eq. 6.2 for 7 =1% gives: /3 0.0767 log /3 — X + 0.75 4 + 0.230 (6.3) 103 for a l l r, where X i s in percent. Eq.6.3 describes k as a function of X for a c y c l i c s t r a i n - c o n t r o l l e d simple shear test with a s t r a i n amplitude of 1% and 10^N>1. Fig.6.6 shows the effect of the decrease in size of the y i e l d surfaces on i t s s t r e s s - s t r a i n behavior. A reduction in the shear resistance at a given s t r a i n and softening of the s o i l are a result of t h i s degradation. The dotted l i n e shows the hysteresis loop generated by cycl i n g at a constant s t r a i n amplitude of 1% without changing the sizes of the y i e l d surfaces. The above analysis i s a very s i m p l i s t i c one and i s meant only to demonstrate the mechanics of the model. A more sophisticated approach for describing the changes in k i s required i f the model i s to describe the behavior under a general c y c l i c stress path. Prevost has developed a set of equations for completely describing the degradation of Drammen clay at an overconsolidation r a t i o of 4 (18). These equations are e s s e n t i a l l y a curve f i t t i n g approach but are quite complex. A simpler method for describing the degradation of the y i e l d surfaces would be desirable i f possible. If a s o i l s t r a i n softens enough, i t i s apparent that the smaller y i e l d surfaces would disappear as isotropic softening occurs. This could lead to u n r e a l i s t i c a l l y " f l a t " stress-s t r a i n curves as only the larger, and consequently softer, y i e l d surfaces remain. To overcome t h i s , Prevost (18) suggests a minimum size k' r > for each y i e l d surface " r " so that: F i g . 6 . 6 . Softening of s t r e s s - s t r a i n curve with c y c l i n g at a s t r a i n amplitude of 1%. Dotted l i n e shows the hysteretic loop formed i f no softening occured. 105 where: k < r ) = W } (6.4) X= k< P > k 0 (P > k ( P > = size of largest y i e l d surface ( i . e . the f a i l u r e surface) after very large cumulative p l a s t i c s t r a i n i n g or remolding. kj P ' = i n i t i a l size of largest y i e l d surface. When the surfaces reach their l i m i t i n g sizes (usually i n d i v i d u a l l y ) their associated p l a s t i c shear moduli HJ. then start to vary. k ( r ) then remains constant while HJ. becomes a function of X. A procedure similar to finding k(X) must then be used for H'(X) (18). The change in slope of the stress-s t r a i n curve ( e l a s t o p l a s t i c modulus H r) can be plotted versus the number of load cycles in a manner similar to Fig.6.5 at the various stress levels corresponding to the individual y i e l d surfaces. Using eq.3.11, a plot of Hr' with the number of load cycles can be computed. A function can then be found equating with N, much l i k e eq.6.2 for the r e s i s t i n g stress versus N. Equation 6.1 can be used to eliminate N such that H'r becomes a function of only X. Unlike the expression for the change in size of the y i e l d surfaces, which assumes that they a l l change by the same amount, H,!(X) i s l i k e l y to be d i f f e r e n t for each y i e l d surface r (18). As such, individual expressions of H,!(X) 106 are required for each y i e l d surface. It should be noted that HJ. is a constant for each y i e l d surface r u n t i l the y i e l d surfaces have reached their l i m i t i n g s i z e s . Only after y i e l d surface r has reached i t s l i m i t i n g size k[r ' does HJ. become a function of X. 1 07 CHAPTER 7 SOME LIMITATIONS OF THE UNDRAINED TOTAL STRESS MODEL Although the comparisons between the test data and the model predictions given in Chapter 5 together with those reported by Prevost (19) show that the model can be quite accurate, many more such comparisons with a wide variety of clays must be made before the model can be used with greater confidence. The results of a study of t h i s nature would act u a l l y be an assessment of the v a l i d i t y of the assumptions made in formulating the model. To achieve a workable model, simplifying assumptions must necessarily be made. However, some of these assumptions need to be reassessed to allow the model to describe a wider range of behavior. In the following, some of the li m i t a t i o n s of the model in i t s present form w i l l be discussed from a speculative point of view. The assumption of a Von Mises type y i e l d surface may not always be v a l i d . Yong and McKyes (30) showed that a Von Mises y i e l d surface existed in a clay they tested provided the applied stress was less than one half the peak stress. Beyond t h i s l i m i t , the y i e l d surface gradually changed in shape u n t i l a Mohr Coulomb surface (31) developed at f a i l u r e . This change was due to the development of a l o c a l i z e d region of high disturbance which ultimately developed into a f a i l u r e plane. Thus, a Mohr Coulomb type y i e l d surface may be more appropriate for s o i l s once a l o c a l i z e d s l i p zone develops. Heavily 108 overconsolidated clays are p a r t i c u l a r i l y prone to f a i l i n g along a s l i p plane and therefore a Von Mises type y i e l d surface may not give very good results for such s o i l s . If the s t r e s s - s t r a i n curve of a material decreases beyond the peak stress with continued straining and without rupturing, the material i s said to be s t r a i n softening in t h i s portion of the curve. The application of the model to such s o i l s subjected to a complicated loading path presents the problem of defining when p l a s t i c deformations occur. Since the condition -j^j- do;. > 0 i s not applicable in t h i s case, only two conditions e x i s t : df A) dc^ = 0 . No p l a s t i c flow occurs, neutral loading, df B) dc^ < 0 . P l a s t i c flow may occur. a a,. If the stress point l i e s on the y i e l d surface, e l a s t i c behavior i s also possible and da. <0 in t h i s case as well. By multiplying the outward normal to f with the stress increment, i t i s not possible to d i s t i n g u i s h between p l a s t i c and e l a s t i c loading states. Therefore the model may not be suitable for describing the behavior of s t r a i n softening s o i l s subjected to a complicated loading path. H i l l (7) suggested that eq.3.17 i s applicable to i n i t i a l l y i sotropic metals. This expression may not be e n t i r e l y suitable 109 for anisotropic s o i l s . For example, consider the monotonic, t r i a x i a l , s t r e s s - s t r a i n - curve for K 0 normally consolidated, resedimented, Boston Blue clay (19) i l l u s t r a t e d in Fig.7.1. If e l a s t i c strains are neglected, the horizontal axis |e y |, could also be denoted as X since: /|de~de~ = /$[de/ + 2de.»] = y§[de» + 2(-idey )2] Notice that at a compressive s t r a i n of | ey | = 0.25%, the s o i l strain-softens and therefore the y i e l d surfaces contract in size beyond t h i s point. In extension however, hardening continues to occur up to a s t r a i n of 3.0%. Thus the hardening-softening behavior of the s o i l i s d i f f e r e n t . One approach to solving t h i s problem would be to pseudo-harden the extension curve so that when the function k(X) in compression i s applied to i t , the resulting softening would give the actual s t r e s s - s t r a i n curve in extension. Fig.7.2 i l l u s t r a t e s t h i s concept. This procedure i s workable for monotonic loading conditions but i t s application to c y c l i c loads i s questionable. To i l l u s t r a t e t h i s , consider a s t r a i n controlled, c y c l i c 110 .8 .6 .4 0~y- o~, .2 0.0 -.2 -.4 0 1 2 3 4 | V e r t i c a l S t r a i n | (*) F i g . 7.1. Stress-strain curve of K 0 normally consolidated, resedimented Boston Blue clay. [from Prevost (19)] t r i a x i a l test of the s o i l in Fig.7.1. If the s t r a i n i s cycled between +0.5% and 0.0% in compression, one would expect a r e l a t i v e l y rapid reduction in compressive strength since the applied s t r a i n exceeds the s t r a i n at peak strength (+0.25%) and the o r i g i n a l structure i s destroyed. However, i f the s t r a i n i s cycled between -0.5% and 0.0% in extension, the s o i l would s t i l l be in the s t r a i n hardening portion of the curve and the rate of degradation in strength would be s i g n i f i c a n t l y l e s s . Nevertheless, the pseudo-hardening approach would in c o r r e c t l y predict i d e n t i c a l rates of degradation for both loading conditions. T Compression 111 F i g . 7.2. Comparison of actual and pseudo-hardened s t r e s s - s t r a i n curves. A more consistent approach would be to use a method which incorporates the anisotropic s t r a i n softening rates d i r e c t l y without having to modify the experimentally obtained curves. To the author's knowledge, such a method has yet to be developed. Cl e a r l y , further research in t h i s area must be undertaken. The s i m p l i f i c a t i o n that the p l a s t i c shear moduli vary only afte r the y i e l d surfaces reach their l i m i t i n g sizes during c y c l i c loading imposes a r e s t r i c t i o n on the type of behavior which the model i s capable of handling. S o i l s which undergo l i t t l e loss in peak strength during cycling but experience a 1 1 2 s i g n i f i c a n t reduction in s t i f f n e s s cannot be modelled using the existing procedure. Figure 7.3 i l l u s t r a t e s t h i s problem for a hypothetical clay. If the isotropic softening rule i s applied to a y i e l d surface m corresponding to point B on the i n i t i a l loading curve, then a f t e r , say 10 cycles, the model would predict the dotted curve in Figure 7.3 with the slope at point B being the same as at point A. Since the remolded strength i s less than the strength aft e r 10 cycles, y i e l d surface m would not have reached i t s l i m i t i n g size (although the smaller y i e l d surfaces may have). As such, the p l a s t i c modulus for f m cannot vary and thus a softer response than the dotted curve in Fig.7.3 cannot be modelled. It i s therefore not possible to obtain the "actual" stress-strain, curve after 10 cycles i f the p l a s t i c moduli are not permitted to vary u n t i l the l i m i t i n g y i e l d surface's size i s reached. 113 • F i g . 7.3. Behavior predicted by model as compared to "actual" behavior for a hypothetical clay. 1 14 CHAPTER 8 THEORY FOR THE EFFECTIVE STRESS ANALYSIS OF SOILS The behavior of s o i l s i s governed to a large extent by i t s void r a t i o and the imposed e f f e c t i v e mean normal stress. Other p e c u l i a r i t i e s of s o i l behavior include p l a s t i c volume changes and a coupling between volume change and shear s t r a i n under drained loading conditions. Consequently, i t is desireable to have the e f f e c t s of these variables in a constitutive r e l a t i o n . Using the general theory of Chapter 2, Prevost has extended his isotropic/kinematic p l a s t i c i t y model to describe the behavior of s o i l , in terms of e f f e c t i v e stresses under undrained or drained conditions. The t h e o r e t i c a l considerations of the model he has proposed w i l l be discussed in t h i s chapter. A l l stresses are e f f e c t i v e stresses unless stated otherwise. Theory  E l a s t i c i t y If the e l a s t i c i t y i s assumed to be i s o t r o p i c , two parameters are required to describe the e l a s t i c behavior: the shear modulus G and the bulk modulus B. The deviatoric e l a s t i c s t r a i n i s then given by: 1 1 5 (8.1) 2G and the volumetric e l a s t i c s t r a i n i s defined to be: For s i m p l i c i t y , the bulk modulus may be taken to be a function of the mean normal e f f e c t i v e stress while the shear modulus i s held constant (21). Janbu (9) has proposed the following r e l a t i o n for B: (8.3) where B, = bulk modulus at p = p, n = a constant depending on the s o i l type p, = a normalizing (reference) stress A more sophisticated approach would be to also make G a function of the mean normal e f f e c t i v e stress. The shear modulus could then assume a form similar to B: 116 c n P. G G (8.4) P i Hardin and Drenevich (6) have suggested that n=0.5 in equation 8.4 and show that G should also be a function of the void r a t i o and the overconsolidation r a t i o . The l a t t e r variable however is only meaningful for undisturbed clays since straining destroys the o r i g i n a l fabric of the s o i l . Plast ic i t y The description of the behavior of s o i l s in terms of e f f e c t i v e stresses requires that the y i e l d c r i t e r i o n include some measure of the e f f e c t i v e mean normal stress. In addition, i f hysteresis during hydrostatic loading and unloading i s to be modelled, then a parameter to track the kinematic change of the y i e l d surface along the hydrostatic axis should be incorporated. To s a t i s f y these requirements, Prevost (20) has proposed the following y i e l d c r i t e r i o n : f m = 0 = l ( s ; j - a<"")( S i j - a£«">) + c 2 ( p - 0"">)2 - (k'">) 2 (8.5) where: p = the e f f e c t i v e mean normal stress = ^ai; 1 17 0 = coordinate along the p axis of the center of the y i e l d surface c = a constant Let: Q = Q' + Q"5 Q' = projection of Q onto the deviatoric plane = BF_ = df_ ds, dSjj 3Q" = projection of Q along hydrostatic axis p = Q-5 = (Q' • 6+Q"6.5) .-= 3 df_ = dt do-- dp Then: | Q ( m ) | 2 _ ^{ m ) . Q ( m ) = Q'-Q' + (Q") 26-5 = 9(s ; j-a i j)(s ; j-a i j) +3(^c«(p-/3) 2) = 6 [ 4(s , . - a i j)(s i j - a i j ) + |c«(p-/3)2] For s i m i l a r i t y with eq.3.7, Prevost has set |Q|2 = 6k 2, which gives c = 3/J2. Other values of c may be used to give a better f i t with experimental data, but i t i s not mathematically 118 convenient to do so since |Q| must then be computed from s^, a , p, and /?. The influence of c can be seen by rewriting eq.8.5 as: [|(s;j -a,-){s,. - a f j)] + [c(p - &)V = (k<">>)2 (8.5a) The f i r s t term contains only deviatoric stresses while the second term involves only the mean normal stress. The y i e l d surfaces can then be thought of as being c i r c l e s in the jjs^ s:-versus cp stress space. A l t e r n a t i v e l y , the y i e l d surfaces would appear as e l l i p s o i d a l surfaces i f plotted in the Jfs, s;. versus p space with c being the r a t i o between the major and minor axes. A value of c greater than 1 would mean that the y i e l d surface would be elongated in the p d i r e c t i o n . If the physical axes of anisotropy coincide with the p r i n c i p l e axes of applied stress, then a ( r >=0 for i * j and for a l l r. Further, i f the s o i l i s cross-anisotropic about the y axis ( i . e . rotational symmetry) then ax =a2 . These assumptions w i l l be i m p l i c i t in a l l the following analyses. To conceptualize eq.8.5a, l e t us examine the special case of the axisymmetric t r i a x i a l test whereby ax =az . Recalling that |s y = (a y-a x ) in t h i s case, then eq.8.5 gives: [q-a,] 2 + c 2(p-|3) 2 = k 2 (8.6) 119 where: a, = § a y (8.7) p = ^ ( a y +2a x ) (8.8) q = ( a y - a, ) (8.9) Figure 8.1 shows how the y i e l d surfaces appear on the t r i a x i a l plane in stress space. The l i n e OC i s c a l l e d the C r i t i c a l State l i n e for compression and represents the locus of stress points for which large shear deformations occur at constant stress and sample volume. It does not necessarily indicate the stress conditions at which the peak strength is reached. Cohesion or d i l a t i o n of the s o i l could allow the stress point to cross above t h i s l i n e temporarily. However, i f large deformations occur so that these e f f e c t s are overcome, then the stress point w i l l ultimately l i e on the C r i t i c a l State l i n e . Thus, i t represents the upperbound for the stress point after large deformations occur. The slope M can be determined from the stress conditions in the t r i a x i a l compression test when the C r i t i c a l State has been reached: q 3 ( 0 , - 0 3 ) M = (8.10) p 2a, + a3 Using the Mohr Coulomb f a i l u r e c r i t e r i o n , for an axissymmetric state of stress, to represent the C r i t i c a l State l i n e ( 2 4 ) : 1 20 a, 1 + s i n 0 ' — = (8.11) a 3 1 - s i n 0 ' where: 0 ' = f r i c t i o n angle at the c r i t i c a l state An expression for Mc can be found in terms of 0 ' : 6 sintf>' Mc = (8.12) 3 - sintf>' S i m i l a r i l y the C r i t i c a l State l i n e can be defined for extension ( oy < ox ) : - 6 s i n 0 ' ME = (8.13) 3 + s i n ^ ' The p l a s t i c strains are obtained from the flow rule. As Palmer et. a l . (13) have pointed out, f r i c t i o n a l materials do not exhibit normality. So, for generality, a non-associated flow rule (eq.2.11) i s adopted. P can be decomposed into: p = P' + p"5 (8.14) where: P = normal to the p l a s t i c p o t e n t i a l surface g P' = projection of the p l a s t i c potential surface onto the deviatoric plane • M _ dS, 3P" = projection of the p l a s t i c potential surface onto 121 the hydrostatic axis. Note that t h i s i s a scalar quantity in which 6 i s the associated tensor. P-6 = a_g dp d«5 = de". + -de; 6,. (8.15) 8.1. Yie l d surfaces in t r i a x i a l plane. [from Prevost(20) ] 122 d e f i = <L> Pr' (8.16) J ToT j = p l a s t i c deviatoric s t r a i n increment tensor de¥f = de.p 6;i = 3<L> P" ' (8.17) J J TOT = p l a s t i c volumetric s t r a i n increment (a scalar) where: <L> = J_ Q-do = J_[Q' • ds + 3Q"dp] H' H' IQI = IQ-QI1 Prevost has chosen a p l a s t i c potential qm associated with y i e l d surface f m such that the p l a s t i c deviatoric s t r a i n increment vector is normal to the projection of f m onto the deviatoric subspace: dq = a_f_ ds-. ds-. P.:' •j However, PVQ" in general. Instead P" i s related to Q" by the following r e l a t i o n : P" = Q" + Am|Q'| (8.18) 123 where: A m = a material parameter to be determined from s o i l t e s t s. Equations 8.16 and 8.17 then become: de f = 1 (Q'-ds + 3Q"dp) Q' (8.19) H' |Q|2 del = 1 3(Q'-ds + 3Q"dp) [Q"+AjQ'|J (8.20) H' [OP The motivating factor for choosing P'=Q' was the good predictive c a p a b i l i t y of the undrained t o t a l stress model. Since the modelling c a p a b i l i t i e s were reasonably good, i t follows that the assumptions of a Von Mises type y i e l d function given by eq.3.3 and the use of an associated flow rule were acceptable for undrained, t o t a l stress loading. The l a t t e r i s es p e c i a l l y important since i t implies that the r e l a t i v e magnitudes of the d i f f e r e n t p l a s t i c deviatoric s t r a i n increment components ( i . e . the d i r e c t i o n of the p l a s t i c s t r a i n increment vector) were compatible with experimental observations. Thus a l o g i c a l extension to these findings would be to also have P'=Q' for the e f f e c t i v e stress model. However, only experimental comparisons with the model predictions can determine the true v a l i d i t y of t h i s assumption. The p l a s t i c modulus H' serves as a factor of proportionality for the p l a s t i c s t r a i n increment. One special 1 24 case for H' i s when Q'=0 ( i . e . the y i e l d surface has no component of i t s outward normal at the stress point in the deviatoric plane). Equation 8.20 then becomes: H' then becomes the p l a s t i c bulk modulus. If Q"=0, the result i s given by eq.3.12 in which case H' becomes the p l a s t i c shear modulus. The kinematic hardening rule describes the changes in the y i e l d surface(s) as p l a s t i c deformation occurs. For drained loading, Mroz's kinematic rule may be used: d^" 1' = dun where n i s given by eq.2.31 and d$ = da + d/35. If (da -df ) and dk are small, the scalar dn can be found from eq.2.42: Recalling that Q = Q' + Q"8 and rearranging: (8.21 ) Q < « n . da - n(k ( r n' )""' dk ( r n ) dn = (8.22) If i t i s further assumed that d k ( r ) = d k ( r + 1 ) for a l l r and Q<"»> . djen + i ) = o then eq.2.43 gives dt(m + 1 ' = 0.. With these assumptions, Mroz's kinematic rule s i m p l i f i e s to.a special case 1 25 of the hardening rule proposed in eq.2.49. 1 26 CHAPTER 9  A FORMULATION FOR COHESIONLESS SOIL The theory presented in Chapter 8 i s not usable by i t s e l f . Before i t can be put to use, a method for obtaining the necessary parameters must be formulated. Additional simplifying assumptions may also be required in order that the model be p r a c t i c a l . The f i r s t and simplest model proposed by Prevost (21) requires only the results of an isotropic compression/rebound test and an axisymmetric t r i a x i a l test but i s applicable only to cohesionless s o i l s . The primary assumptions used were: 1. "For cohesionless s o i l specimens prepared in the laboratory: o , t m ) = 0 and /3(nri> = j2klrn)/3 i n i t i a l l y for a l l m and o;j = 0 for a l l i , j . The y i e l d surfaces are thus i n i t i a l l y a l l centered along the hydrostatic axis and tangent to each other at the o r i g i n " . Note that this implies isotropic material behavior and no i n i t i a l l y imposed stresses. The expression for (3 comes from the y i e l d function, eq.8.6, with ay =ax =a, =p=0 . 2. k ( r ) i s a d i f f e r e n t constant for each y i e l d surface r. 3. The shear modulus G i s a constant. 4. The p l a s t i c modulus i s given by: 1 27 2B: H i (1-t) ( 9 . 1 ) 2B' (1+t) - t where: 3 ( p - 0 ) t = -1 < t < 1 J2 klrn) Note the following l i m i t i n g conditions: a) t = 1 and Q'=0; i . e . the s o i l i s being i s o t r o p i c a l l y compressed; and H'=©o. The strains are then purely e l a s t i c . b) t = 0 ; i . e . p=0 . The stress path then goes through the apex of the y i e l d surface and Q"=0. Then the result i s i d e n t i c a l to eq.3.12: HI = the p l a s t i c shear modulus = hi . c) t = -1 ; i . e . the isotropic stress is being released. Since Q'=0 for t h i s case, then i t follows from eq.8.21 that H^ = p l a s t i c bulk modulus = B l . Cases "a" and "c" appear to be an anomaly since the usual convention i s to have the p l a s t i c strains occur during compression (loading). However, the choice of whether to record the p l a s t i c strains during compression or rebound during computations i s largely a matter of bookeeping. What i s 1 28 s i g n i f i c a n t i s the permanent (p l a s t i c ) deformation during a cycle of loading and unloading. Whether t h i s quantity i s recorded during compression or during rebound i s of minor importance in the following. As a result of assumption 4 and equations 8.2 and 8.20, the volumetric str a i n increment during isotropic compression i s : dev = deve = d£ (9.2) B with B given by eq.8.3. When the compressive stress reaches a maximum value of pc and l i e s on y i e l d surface c, 0<m)+ jS.k<m>=pc for m<c since a l l the y i e l d surfaces which have been translated are tangent to each other at pc (Fig.9.1). During isotropic rebound, equations 8.2 and 8.20 give: BI = (9.3) dev 1 dp B The value for /3 ( m ) can be found by noting from Figure 9.1 that: pimi = i ( P c + p j ( 9 # 4 ) 1 2 9 F i g . 9 . 1 . Y i e l d s u r f a c e s d u r i n g compression to p=p c . i s o t r o p i c 130 The size of the y i e l d surface can be determined s i m i l a r i l y : k ( m > = 3 (p. - p m) (9.5) ~~& 2 Thus B,, Br', /3 ( r ), and k ( r ) for y i e l d surfaces f r , r<c can be determined from an is o t r o p i c compression/rebound test. The other parameters, such as hr' , AP and G cannot be determined from t h i s t e s t . It should, be noted that pe during testing should be at least as large as the p expected to be attained in the problem at hand since only j3 ( f , ) and k ( r ) (but not BP') for r>c can be determined from a t r i a x i a l t e s t . If a cohesionless s o i l i s i s o t r o p i c a l l y compressed to a confining pressure of p,, and then a conventional axisymmetric t r i a x i a l compression test i s performed, then: dq = 3dp where: q = (a y - ax ) p = \ ( oY + 2 ax ) When the stress point i n i t i a l l y reaches f m , equations 8.1, 8.6, and 8.19 and 8.2, 8.6, and 8.20 give: d(e y - ex ) 1 3q |q + 3 (p - 0<m > ) dp/dq de dq 2G 2H; (k (™' ) 2 dq (9.6) 131 dev 1 3 9(p - /3 ( m ) ) '+ j6A mq = - + ( 2 q + 3 ( p _ p i * ) ) dp/dq) dp B 2 H ; ( k ( m ' ) 2 (9.7) in which: J2 0<m> = k(m> ( g > 8 ) 3 m > c since the y i e l d surfaces are tangent at p=0, q=0. Therefore from eq.8.6: 3 lp2 + q 2 k' m' = (9.9) J2 9p The shear modulus G i s obtained from the steepest slope of the st r e s s - s t r a i n curve: G = 1 L. dq , i (9.10) steepest 2 l d ( e y - e „ ) ; Hi can then be found from eq.9.6: 1 32 H: q (q + i ( p - 0 ) ) ( k ( n n ) ) 2 d( ) dq 1 2G (9.11) while A m i s obtained from eq.9.7: J6A dev 1 dp B p(p-i3) d(e y - ex ) 1 dq 2G (9.12) Lastly, the p l a s t i c shear modulus h' can be determined from : 2B^(1+t) 2BZ H l O - t ) (9.13) + t It should be noted that while G, , /3 ( n n ), k ( m', and A m are obtainable solely from a t r i a x i a l test, where m i s greater than the largest y i e l d surface reached in isotropic compression under p, ( i . e . f p ), B^ , cannot be determined from t h i s t e s t . These equations w i l l now be used to determine the parameters for Cook's Bayou Sand from the data given in Figures 9.2 and 9.3. 133 F i g . 9.2. Experimental, isotropic compression/ rebound curve for Cook's Bayou sand, [from Prevost (21) ] 4 h F i g . 9 . 3 . Experimental, drained t r i a x i a l compression curves for Cook's Bayou sand, [from Prevost ( 2 1 ) ] 1 35 Upon i n i t i a l c o m p r e s s i o n , the s t r a i n s a r e p u r e l y e l a s t i c ; t h u s : B = dp_ • (9.14) de„ A p p l y i n g eq.8.3 t o eq.9.14 w i t h n=0.5 f o r sands g i v e s : del = dp B^p_y5 (9.15) where a v t = p, = a n o r m a l i z i n g ( r e f e r e n c e ) s t r e s s . L e t t i n g p 0 and e v o be the i n i t i a l s t r e s s and s t r a i n s t a t e s r e s p e c t i v e l y : de v = p dp /Po T h e r e f o r e d u r i n g c ompression 2 fp~ fPo" (9.16) B T J avc J o„, 136 Using the data in Figure 9.2, B,/avc = 472. , Upon isotropic rebound, values for p are selected to correspond to int e r v a l s at which the y i e l d surfaces are desired. Values for k < m ) and are then calculated from eq.9.5 and eq.9.3 respectively. Table 8 Model l Parameters For Cook's Bayou Sand m k_ 1_ BJ_ A ovc ovc 1 1 .061 0.500 2 1 .432 0.675 -8123. 338. 1 -2.328 3 1 .945 0.917 -6860. 205. 1 -1.508 4 2.663 1 .255 -6222. 1 25. 1 -1.004 5 3. 182 1 .500 -4996. 77.6 -0.814 6 3.450 1 .626 -4657. 56.4 -0.773 7 3.722 1 .755 -4581. 24.8 -0.637 8 4.561 2. 150 G = 400 avc B, = 472 avc a v c = 50 psi Turning to the t r i a x i a l test on a new but i d e n t i c a l sample which has been i s o t r o p i c a l l y compressed to a value of p/a V 6 = 1, 137 eq.9.8 gives values for j 3 ( r n ) , m>2. Note that for m=1, f, was translated during isotropic compression of the t r i a x i a l sample and thus j 3 < 1 ) i s given by eq.9.4. Values for p and q corresponding to y i e l d surfaces m>2 can be determined by r e c a l l i n g that 3dp=dq or p=1+iq, q>0 in the case of the t r i a x i a l compression test. Equation 9.9 then becomes: ( m ) 3 [{( 9 d + i q j (9.17) and the y i e l d stresses q m (and pm=1+3qm) associated with y i e l d surface f m are eas i l y determined. The parameters G, , A m, and h^ for m>2 can then be found from equations 9.10, 9.11, 9.12, and 9.13 respectively. It must be emphasized that H 1 f A,, and h, are indeterminate since the t r i a x i a l test was not conducted in i t s stress range ( i . e . the stress point had q=0 when i t f i r s t reached f i ) . Table 8 tabulates a complete set of parameters for the s o i l based on a piecewise linear approximation of the s t r e s s - s t r a i n curve and Figure 9 .4 shows the y i e l d surfaces in stress space. Other values for the parameters are possible depending opon the choice for the y i e l d stresses. The parameters obtained from the above procedure are only appropriate for the instant the stress point f i r s t touches the y i e l d surface. From eq . 9 . 6 and 9 . 7 , i t can be seen that any 138 F i g . 9 . 4 . Y i e l d surfaces in the t r i a x i a l plane for Cook's Bayou sand. 1 39 change in the stress point which causes a change in p-/3 and q-a would a l t e r the value of de/dq and de v/dp, assuming G, h', B 1, and A remain constant. To prevent any changes in de/dq and d e v / d p during a change in the stress point, Prevost computes these values when the stress point f i r s t touches f m and these remain constant u n t i l the next largest y i e l d surface i s reached. In comparison, the kinematic movement of the y i e l d surfaces does not impose any such r e s t r i c t i o n on the undrained t o t a l stress model for v e r t i c a l l y cross-anisotropic s o i l s . In th i s case, de;-/dsk| i s constant during the t r i a x i a l tests to determine the parameters, by virtu e of the y i e l d surfaces being centered along the /3/2 ay axis (Fig.4.3b). Since (s^-a-) remains constant during any stress increment in the t r i a x i a l test, de;./dskl also remains constant. Thus the de-/ds k | determined when the stress point f i r s t touches y i e l d surface f m remains the same u n t i l f m t 1 i s reached. It should be noted that the parameters obtained from equations 9.11, 9.12, and 9.13 do not necessarily imply that a piecewise linear approach to matching the experimental stress-s t r a i n curve must be used. These equations are also applicable for instantaneous values of the parameters at given values of q, p, a, and /?, whereby de/dq and d e v / d p are the tangents on the respective s t r e s s - s t r a i n curves at the stress point of inte r e s t . Using the parameters given in Table 8, a comparison between actual test data, the model f i t assuming constant de/dq 140 and d e v /dp between y i e l d surfaces, and the model f i t using the same parameters but permitting d e/dq and d e v /dp to vary as the kinematic movement of the y i e l d surfaces demands is shown in Figure 9.5. Clearly, the parameters of Table 8 obtained from the i n i t i a l ("original") positions of the y i e l d surfaces, using the previously described method, are not compatible with kinematic hardening in this case. If a larger number of y i e l d surfaces had been used, the model f i t for the non-constant de/dq and d e v /dp curves would have been better because q-a and p-/3 would not change as much during kinematic hardening. As an example, Fig.9.6 compares the model f i t using 8 and 19 y i e l d surfaces. It i s evident that, although using 19 y i e l d surfaces gives a r e l a t i v e l y better f i t , many more y i e l d surfaces are required i f the test data i s to be matched more c l o s e l y . To evaluate the predictive c a p a b i l i t i e s of the model the parameters for dry Ottawa sand at a r e l a t i v e density of 87% and an i n i t i a l , i s o t ropic confining pressure of 5 psi were determined from data given in Reference 29. The s t r e s s - s t r a i n curves were then predicted for a t r i a x i a l compression and an extension test in which the mean normal stress was held constant at 5 p s i . Table 9 gives a set of parameters for t h i s sand. To give a r e a l i s t i c ultimate strength for the sand, the Mohr Coulomb f a i l u r e c r i t e r i o n was used. A comparison between the predictions and the actual test data i s shown in Fig.9.7. Except for the shear stress - s t r a i n curve at low s t r a i n s , the performance of the model i s not good. Model F i t F i g . 9.5. Cook's Bayou sand. Comparison of model f i t and test data. 142 I b l Movement of y i e l d s u r -faces not considered. 8 y i e l d surfaces w i t h kinematic movement considered. 19 y i e l d surfaces w i t h kinematic movement considered. F i g . 9 . 6 . Comparison of model f i t u s i n g 8 and 19 y i e l d s u r f a c e s u s i n g " o r i g i n a l " parameters and c o n s i d e r i n g the k inemat ic movement of the y i e l d s u r f a c e s . 143 T a b l e 9 O t t a w a Sand P a r a m e t e r s m k_ 11 A -Oo o0 o 0 o0 1 1 .061 0 .500 2 1 . 1 40 0 .537 - 1 8 3 6 2 . 15.00 -7.036 3 1 .591 0 .750 - 1 6 9 0 6 . 249.72 -1.276 4 1 .871 0 .882 - 2 1 5 9 1 . 260.00 -1.354 5 2.334 1 .100 - 2 1 6 5 4 . 209.08 -1.130 6 2.834 1 .336 - 1 9 7 2 6 . 165.58 -1.044 7 3.182 1 .500 - 1 7 2 9 6 . 145.53 -0.993 8 3.540 1 .669 - 1 6 6 4 5 . 93.79 -0.998 9 4.091 1 .929 - 1 1 4 6 6 . 54. 10 -0.929 10 4.846 2 .285 -5281 . 19.46 -0.909 1 1 5.619 2 .649 G = 7 7 0 a o B, = 1724 a 0 a0 = 5 p s i 144 2 + COMPRESSION b* I 1 +. 0 + O Compression - experimental A Extension - experimental Predicted EXTENSION -1 A 0 1 2 3 4 1 STRAIN 1 (%) a O Extension 0 1 2 3 4 I STRAIN i (%) Fi g . 9.7. Predictions. Ottawa sand. [data from (29)] 145 CHAPTER 10 A MORE GENERAL FORMULATION FOR SOILS The formulation given in Chapter 9 was r e s t r i c t e d to a par t i c u l a r class of s o i l s , namely i s o t r o p i c , cohesionless s o i l s . Further simplifying assumptions include a constant shear modulus and a constant y i e l d surface size k during i n i t i a l loading. In a c t u a l i t y , the shear modulus i s known to be a function of the e f f e c t i v e mean normal stress (6) and k ( n n ) i s related to the volumetric s t r a i n as w i l l be demonstrated l a t e r . In l i g h t of these d e f i c i e n c i e s , Prevost proposed a model to characterize the behavior of any given s o i l under monotonic loads (22) in which the parameters can be obtained solely from a t r i a x i a l compression and an extension t e s t . However, the following presentation w i l l be concerned mainly with modelling the behavior of cohesive s o i l s . The a p p l i c a b i l i t y of the model to cohesionless s o i l s w i l l be discussed in Chapter 11. The primary assumptions of thi s method are that: (10.1a) (10.1b) (10.1c) 1 46 (10.1d) k ( m ' ( 1 0.2a) (10.2b) (m) (10.2c) Hi + J3 Q" " Q(m) (10.3) where p = d(ln p) in the normally consolidated range of the e„-ln(p) isotropic consolidation curve. Since the slope of the normal consolidation l i n e for one-dimensional compression i s p a r a l l e l to the slope for isotropic compresssion (1), either test may be used to obtain p. While equations 10.1a and 10.1b, as previously demonstrated, are j u s t i f i a b l e from a s o i l mechanics point of view, evidence to prove the a p p l i c a b i l i t y of 10.1c and 10.1d i s scarce at the present time. The prime motivation for having the l a t t e r relations appears to be the mathematical s i m p l i c i t y i t affords in subsequent derivations. The basis for equation 10.2a l i e s in the experimental observation that for a given void r a t i o , there exists a unique value for p and q at which the C r i t i c a l State l i n e i s reached. Thus there i s a unique rela t i o n s h i p between the size of the largest y i e l d surface at large strains and the volumetric s t r a i n ev . Further,it has been shown by Roscoe and Burland (24) that the projection of the C r i t i c a l State l i n e onto the de„ 147 volume-ln(p') plane i s p a r a l l e l to the normal consolidation l i n e (Figure 10.1). In l i g h t of these observations, the re l a t i o n k(P> = k^P^^'' i s acceptable and a l o g i c a l extension would be eq.10.2a. F i g . 10.1. The C r i t i c a l State l i n e in v -ln(p') space. [from Atkinson and Bransby (1)] It i s an experimental observation that for normally consolidated clays, the s t r e s s - s t r a i n curve for samples consolidated under d i f f e r e n t consolidation pressures would appear to be the same i f the stress difference i s normalized by the consolidation pressure p e (Fig.10.2). Analogously, the y i e l d surfaces for a normally consolidated clay should also be the same i f the stresses are normalized with respect to the consolidation pressure. It then follows that a and 0 should have the same form as k; thus giving eq.l0.2b and c. Note that 148 2a 0 5 10 ea (per cent) F i g . 10.2. a) Relationship between stress difference q and a x i a l s t r a i n e„ in undrained t r i a x i a l tests on samples normally consolidated to p e = a, 2a, 3a. b) Relationship between normalized stress difference q/pe and a x i a l s t r a i n ea . [from Atkinson and Bransby (1)] 1 49 eq.10.2 i s a "quasi-hardening" rule since e l a s t i c volume changes can modify the y i e l d surfaces. True hardening rules, as defined in p l a s t i c i t y theory, are caused by p l a s t i c strains only. The r e l a t i o n for the p l a s t i c modulus (eq.10.3) was proposed by Prevost for i t s s i m p l i c i t y . Note the three l i m i t i n g values for H I : a) Q" = 0 ; H^ , = h i = p l a s t i c shear modulus b) Q' = 0 , Q" = |Q| ; HI = hi + BI = p l a s t i c bulk modulus in loading c) Q' = 0 , Q" = -|Q| ; Ul = hi - BI = p l a s t i c bulk modulus in unloading ' Let us now consider a sample, which i s cross-anisotropic about the v e r t i c a l y axis, subjected to a stress path in .the axisymmetric t r i a x i a l plane ( i . e . q-p plane). Equation 8.6 i s then applicable. Defining e? to be as in Figure 10.3: q " a ( rn ) sine?"" > (10.4) k ( m ) c(p - / 3 ( m ' ) cose?"1"' (10.5) k(m) Eq.8.19 then becomes: 150 q > cp F i g . 10.3. D e f i n i t i o n of 0. de 1 1 — = — + — sint? (sine? + cTcost?) (10.6) dq 2G HI For t h i s formulation, Prevost (22) has modified the expression for P" such that: Q" P" = Q" + A|Q' |Q"I Substituting t h i s expression into eq.8.17 gives: dev 1 1 1 = - + — (2c cost? + km /6 cos0| tant?| ) — (sine? + cTcost?) dp B H ; 3T (10.7) 151 in which: T = dp dq Let 0 c l r n ) and 0e(nri> be the values of 9 when the stress point f i r s t reaches y i e l d surface fm in t r i a x i a l compression and extension respectively under drained conditions. Dividing the p l a s t i c component of the volumetric st r a i n (eq.10.7) by the deviatoric component (eq.10.6): dp (2c cos0 + A m y6cos0|tan0| ) — (sin0 + cTcos0) 3T dq — (sin0) (sine + cTcos0) H: (10.8) d e y f Subtracting of the compression side from the extension side: del defr de de -p 1 3 2c .tan 6L tanac tan ft. 2c tan0 c tana* tan0 c 2c / 1 1 \ d e l de p 3 (tan0 c tandtJ defve de + i f : tan ft. tan0 c < 0 tanftc tana E > 0 But: de v f = de v - de ¥ de v dp = dp - — dp B S i m i l a r i l y for d e p : de dq d e p = dq — - — dq 2G Therefore: 1 53 del de dp /de1( vdp de dq — \dq V B> 1 2GJ ' P 'P vP v dp 'P \" de v v P i / dp v P i / B dq 'P \" de v P i / dq P i / 2G dp 'P V de v v P i / dp 1 dq 'P \ de v P i / dq 1 2G, x T -y (10.10) T h e r e f o r e : T c — ± T £ — 2c / 1 3 \tant? c tant? e (10.11) w h e r e : T = dp dq (10.12) 154 'p\ de 1 - — - (10.13) kp,/ dq 2G, 1 /p \" de, 1 y \ P i / dp B, (10.14) where the subscripts c and E refer to compression and extension loading, respectively. To get another r e l a t i o n involving 6C and d£ , r e c a l l that: k, = ke'P&v (10.15) a, = ae~pe" (10.16) 0, = /3e~p€v (10.17) Therefore: c a. E a. (q c - k sin0 c )e'p€" = q ce pe" - k, sinfl c (q - k sin0 e )e" f > e v = q.e"^6' - k, sin0 e 0, = (pc - k cos0 c )e~fi£" = pc e pe" - k, cos0 c c c c 0i = (p e - k cos0 £ )e" f e" = p£ e p e* - k, cos0 E c c Then: 155 cos0c - cos0 E c(pc, - $\) - c(p£, - 0, ) sin0 c - sin0 E (qc, - a,) - (q£, - a\ ) p - p e'"'- 6'> c — ' W £ c - £ f ) = R (10.18) Using algebraic manipulations, eq.10.18 can be rewritten: 1 1 \ 2R / 1 1 + + — 1) = 0 (10.19) ,tan0c tan0 e/ 1 - R 2\tan0 c tan0 E The smooth s t r e s s - s t r a i n curves are approximated by piecewise linear segments along which the tangent modulus i s a constant. To find the y i e l d surfaces, y i e l d points are chosen along the compression s t r e s s - s t r a i n curve. The corresponding y i e l d point in extension i s determined by specifying that the slope dq/de i s to be the same in both compression and extension once the stress'point has f i r s t reached that p a r t i c u l a r y i e l d surface. Values for 0C and 0E for each y i e l d surface can then 156 be determined from the two simultaneous equations eq.10.11 and 10.19. The parameters associated with each y i e l d surface can be found by rearranging eq.10.6 and 10.7: B' x csin0 cTC - xE sin0 E TE cos0 c - COS0 E (10.20) h; = x c s i n 0 c T C - B„;cos0c (10.21) K J6 = ( m ) 1 |tan0 c | pel 3 T C — tan0 c - 2c <3ce ~ q Ee •pe. sin0 r - sin0 e (10.22) (10.23) , ( m ) (fe,) _ k,(r"»sin0c ) = p e- (f e„> k,(m 'cos0. (10.24) (10.25) where: TC = sin0 c + cT ccos0 c TE = sin0 e + cT ecos0 e (10.26) (10.27) The s p e c i f i c a t i o n that the smooth s t r e s s - s t r a i n curves be approximated by linear segments demands that d e / d q and d e v / d p be constant. For th i s to be true, the value of (de/dq) and ( d e v / d p ) must be calculated when the stress point f i r s t reaches f m and cannot change, despite the tr a n s l a t i o n and 1 57 expansion of the y i e l d surfaces according to eq.10.2, u n t i l f,,,,, i s reached. As with the model for cohesionless s o i l , kinematic and isotropic hardening does not influence the s t r a i n increments. Let us now use this model to predict the drained behavior of two normally consolidated clays, one natural and the other remolded, subjected to two d i f f e r e n t axisymmetric t r i a x i a l stress paths: active compression (AC) whereby ox =o. i s decreased while a y i s held constant, and passive extension (PE) whereby the confining stress is increased while the v e r t i c a l stress i s held constant. The two stress paths required for the determination of the model parameters are: passive compression (PC) whereby the v e r t i c a l stress i s increased with the confining stress constant, and active extension (AE) in which oY i s decreased with ox=a^ held constant. The natural clay used for the study i s known as Haney clay. It has a l i q u i d l i m i t of 46%, a p l a s t i c l i m i t of 26%, and a natural water content of 41-44%. Its s e n s i t i v i t y is of the order of 6-10. A complete description of the s o i l properties and testing program can be found in Reference 28. The samples were K 0 consolidated to a v e r t i c a l e f f e c t i v e stress of 5.95 kg/cm2 and a horizontal e f f e c t i v e stress of 3.30 kg/cm2 giving a K 0 value of 0.56. The specimens were then loaded at a constant rate of str a i n in a drained condition. Figures 10.4 and 10.5 show stress difference versus a x i a l s t r a i n and volumetric . s t r a i n versus a x i a l s t r a i n curves respectively for the passive and active extension tests. Also 1 58 shown are the predictions which the model made using the parameters in Table 10. The good correspondence i s to be expected since the parameters were derived from t h i s set of data. The stress paths of these two tests as well as the two for prediction are shown along with the i n i t i a l sizes and positions of the y i e l d surfaces in Figure 10.6. A comparison between the predicted s t r e s s - s t r a i n curves and the actual test data for AC and PE are shown in Figures 10.7 and 10.8. The predicted behavior bears no resemblence to the actual data. Obviously, the model was of l i t t l e value in predicting the e f f e c t i v e stress behavior of thi s s o i l . Remolded Weald clay was the second s o i l used in thi s evaluation. It has a l i q u i d l i m i t of 20% and a p l a s t i c l i m i t of 30%. The samples were i s o t r o p i c a l l y consolidated to an ef f e c t i v e mean normal stress of 30 psi and then sheared in a drained condition. Details of the test program are given in Reference 14. The test data for stress paths PC and AE used ' to derive the parameters in Table 11 i s shown in Figures 10.9 and 10.10. The comparison between the predicted s t r e s s - s t r a i n curves and the actual test data for AC and PE i s shown in Fig.10.11. The predictions for thi s clay are not very good either. The Prevost e f f e c t i v e stress model can be used to predict pore pressures under undrained conditions by setting de v= 0 and rearranging eq.10.7 to express dp as a function of dq. Although the t o t a l volume change i s zero, the e l a s t i c and p l a s t i c volume changes are non-zero, and equal and opposite. _12 _8 -4 0 4 8 12 AXIAL STRAIN (%) F i g . 10.4. Haney c l a y . S t r e s s d i f f e r e n c e v e r s u s a x i a l s t r a i n c u r v e s . [from V a i d ( 2 8 ) ] 160 Table 10  Parameters For Haney Clay m a. 01 h' B' A/6 1 .6723 2 .1316 3.9129 192.094 -54.9757 -2.2646 2 2.1338 1 .4605 3.5315 41.569 -14.7826 -2.1709 3 2.4957 1 .1329 3.4464 15.609 -5.1436 -2.4666 4 2.9147 1 .1357 3.2100 12.055 -4.5119 -2. 1659 5 3.0040 1 .2339 3.2061 5.280 -2.0401 -1.9167 6 3.1094 1 .0528 3.2363 .001 -.0001 0.0000 G, = 400 kg/cm2 B, = 201 kg/cm2 p = 3.612 n = 0.5 Note: k, a, 0, h \ and B' a l l have units of kg/cm2. A J6 i s dimensionless. Prevost (22) has used th i s method, to predict the pore pressures of a clay during undrained shear, with only mixed success. Fig.10.12 shows two comparisons between predicted and measured pore pressures as reported in reference 22 for the Kaolinite clay discussed in Chapter 5. 1 2 •• ACTIVE EXTENSION PASSIVE COMPRESSION • Experimental -f Experimental — Model P i t — Model P i t ' AXIAL STRAIN (%) F i g . 10.5. Haney c l a y . Volumetric s t r a i n v ersus a x i a l s t r a i n c urves. [from V a i d (28)] q (kg/cm2) 4 -3 1 Fig. 10.6. Haney Clay. O r i g i n a l positions of y i e l d surfaces and stress paths used for study. to o °° o f q (kg/cm 2) o 34> A & ^ PASSIVE EXTENSION O E x p e r i m e n t a l P r e d i c t e d \ \ \ A A A / / / / \ / \ / \ / \/ / / / ACTIVE COMPRESSION / A E x p e r i m e n t a l P r e d i c t e d -12 -8 _4 0 4 AXIAL STRAIN (%) 12 16 F i g . 10.7. Haney clay. Stress difference q versus ax i a l strain e predictions. [data from (29)] €V (50 o o 8 t P A S S I V E EXTENSION O Experimental P r e d i c t e d - 1 2 i - 8 - 4 / A C T I V E COMPRESSION A Experimental / P r e d i c t e d / / / A A 8 1 2 A X I A L S T R A I N (%) F i g . 1 0 . 8 . Haney c l a y . V o l u m e t r i c s t r a i n ev v e r s u s a x i a l s t r a i n ey p r e d i c t i o n s . [ d a t a f r o m ( 2 9 ) ] 165 Table 11  Parameters For Weald Clay m k, a, 0, h' B' A J6 1 1 .574 -1 . 0591 29.4514 1 1 23 .71 -309.95 -2 .461 6 2 6.945 -0. 6027 28.8374 571 .44 -120.37 -2 .2525 3 12.254 2. 5129 28.8502 264 .30 -65.40 -1 .9343 4 16.411 3. 1065 28.0681 159 .32 -43.54 -1 .8564 5 19.759 5. 5773 27.8338 81 .74 -27.78 -1 .5626 6 21 .727 6. 0652 27.4839 28 .82 -10.96 -1 .5154 7 22.783 5. 5821 27.0053 .01 -.01 0 .0000 G = 833.33 psi B = 8333.3 psi n = 0.5 p = 7.443 Note: k, a, |3, h', and B' a l l have units of p s i . Ay6~ is dimensionless. q ( p s i ) 4 0 T - 2 0 1 F i g . 10.9. Normally consolidated, remolded Weald clay. Drained t r i a x i a l data used to get the parameters. Stress difference q versus s t r a i n difference e = ey-eK . [data from (14)] ACTIVE EXTENSION PASSIVE COMPRESSION F i g . 10.10. Normally consolidated, remolded Weald clay. Drained t r i a x i a l data used to get the parameters. Volumetric s t r a i n ev versus str a i n difference e . [data from (14)] 168 ( p s i ) 20 10 A o •10 o A E x p e r i m e n t a l - P a s s i v e E x t e n s i o n E x p e r i m e n t a l - A c t i v e Comp. P r e d i c t e d - P a s s i v e E x t e n s i o n P r e d i c t e d - A c t i v e Comp. -20 •30 •40 10 15 20 (%) E 5 V Ui Ui QJ S-CL E o c o AXIAL STRAIN (%) r— Q ~r- "5 10 15 20 AXIAL STRAIN (%) F i g . 10.11. Normally consolidated, remolded Weald clay. Predictions. [data from (14)] Fig. 10.12. Pore water pressure predictions for Kaolinite. [data from (22,29)] cn vo 170 CHAPTER 11 A DISCUSSION OF THE EFFECTIVE STRESS MODEL The comparisons shown in Figures 9.7, 10.7, 10.8 and 10.11 demonstrate quite c l e a r l y that the e f f e c t i v e stress formulation of the theory of Chapter 2 i s not very good for predicting the behavior of s o i l under drained conditions. This implies that some of the assumptions used to formulate the model are not v a l i d . An assessment of the propriety of each assumption would be very d i f f i c u l t to make. For example, the value for "c" in the y i e l d c r i t e r i o n (eq.8.5a), the normal to the p l a s t i c potential component for volumetric s t r a i n (eq.8.18), and the expression for H' (eq.9.1 and eq.10.3) were chosen largely for mathematical convenience. It could then be said that one, two, or possibly a l l three are not good assumptions; but to i s o l a t e which combination i s responsible for the poor predictions would be extremely d i f f i c u l t . In some cases however, c r i t i c i s m can be made on a more rat i o n a l basis. The assumption of i n i t i a l isotropy for cohesionless s o i l s prepared in the laboratory, which the model in Chapter 9 makes, i s one of these. Ladd et. a l . (10) have shown that anisotropy can be quite pronounced in laboratory samples prepared using standard methods. The aV} * 0 in t h i s case. V e r i f i c a t i o n of anisotropy or isotropy requires at least one other test, under the same stress path, which shears the sample in a d i f f e r e n t d i r e c t i o n . Thus the two tests prescribed 171 for obtaining the parameters for the cohesionless s o i l model are not adequate. Fixing the value of d e/dq and d e v / d p from when the stress point f i r s t touches f m u n t i l f m H is reached may only be suitable for monotonic loading unless a very large number of y i e l d surfaces are used. Under more complex stress paths, the appropriate value of d e/dq and d e v /dp would be d i f f i c u l t to determine. To i l l u s t r a t e t h i s point, consider the monotonic stress path of Fig.11.1 from point A to point B. If we now follow a new stress path, say to point C, the problem of obtaining an appropriate value of d e/dq and d e v /dp becomes apparent. Obviously, using the i n i t i a l values of d e/dq computed under stress path AB would be incorrect. From eq.9.6, de/dq=1/2G in t h i s case since q=0 i n i t i a l l y for a l l the y i e l d surfaces. Thus no p l a s t i c deviatoric strains would be predicted, which i s incorrect. It could be argued that the appropriate values for d e/dq and d e v/dp should be computed from the new orientation of the y i e l d surfaces. However, an inconsistency can arise whereby the p l a s t i c strains can be d i f f e r e n t under the same p l a s t i c stress path ( i . e . That portion of the stress path which causes p l a s t i c s t r a i n s ) . For example, l e t f, enclose an e n t i r e l y e l a s t i c region and l e t f 2 be the y i e l d surface which the stress point has just reached (Fig.11.2). de/dq and d e v / d p can then be computed and these values remain constant u n t i l f 3 i s reached. By moving the stress point to B, (p-0im)) and (q-a ( n r") F i g . 1 1 . 1 . Position of y i e l d surfaces upon tran s l a t i o n from A to B. F i g . 11.3. When s t r e s s p o i n t i s on f 2 a f t e r un load ing and r e l o a d i n g e l a s t i c a l l y . 174 change due to the kinematic movement of the y i e l d surfaces. We s h a l l now unload e l a s t i c a l l y to point C and then reload e l a s t i c a l l y to B again (Fig.11.3). A new value for de/dq and de„ /dp must now be computed i f p l a s t i c loading occurs; but these quantities would be d i f f e r e n t from those computed e a r l i e r . If loading is now continued along the same stress path to point D, the t o t a l deviatoric p l a s t i c s t r a i n which occurs from A to D would be: e p = e p + Jf ABD AB BD m (qA-q.) + felj <q.-q0> (11.D AB V /&D If the loading from A to D increased monotonically instead, the t o t a l p l a s t i c deviatoric s t r a i n would be: = (q A-q 8) (.11.2) But eAp0 when, in fact, they should be equal since the unloading and reloading to point C was purely e l a s t i c . This shows that f i x i n g the value of de/dq and de„/dp i s only suitable for monotonic loading. Of course, special rules can always be created to deal with these sit u a t i o n s . But such rules would complicate the 175 c o m p u t a t i o n a l p r o c e s s . The s i m p l e s t means f o r m o d e l l i n g the b e h a v i o r under any s t r e s s p a t h would be t o c a l c u l a t e a s e t of parameters which would be c o m p a t i b l e w i t h c h a n g i n g v a l u e s of de/dq and de v/dp d u r i n g each s t r e s s i n c r e m e n t . One method f o r d o i n g t h i s would be t o c a l c u l a t e h i , H I and hm u s i n g an average v a l u e f o r (p - /3 ( r n )) and (q - a ( n r ) >) o b t a i n e d i n the t r i a x i a l t e s t . L e t ( p 0 - and ( q 0 - a<m)) denote o r i g i n a l v a l u e s when the s t r e s s p o i n t f i r s t touches fm. However,as the y i e l d s u r f a c e t r a n s l a t e s , (p-j3) and (q-a) change and, a c c o r d i n g t o eq.2.30, r e a c h v a l u e s o f : k ( m ) (p - 0 ' " 1 ' ) , = (p - / 3 < , n + 1 ) ) o (11.3) k (m + 1 ) k ( m ) (q - a ( n i ) ) f = (q - a ( m + 1 ' ) 0 (11.4) k ( m * 1 ' when the next l a r g e s t y i e l d s u r f a c e i s reached ( s u b s c r i p t " f " denotes t h e i r f i n a l v a l u e upon r e a c h i n g f m <., ). Average v a l u e s can then be found from: (q - a ( m ' ) 0 + (q ~ <~ i m ) ) f ( q - a ' ^ U = ~ (11.5) 3 2 (p " / 3 " n ) ) 0 + (p " 0 ( m ) h ( p - 0 ( r r , , ) a M = ~ (11.6) 3 2 176 and these can be used in equations 9.11, 9.12, and 9.13. The computation can then proceed in the same manner as for the undrained t o t a l stress model, whereby updating of de/ds and detf/dp can be done after each stress increment. Figure 11.4 compares the response predicted for the t r i a x i a l compression test on Cook's Bayou sand by a) using parameters determined by th i s averaging method together with kinematically moving y i e l d surfaces, and b) using Prevost's method whereby " o r i g i n a l " parameters are used without considering the kinematic movement of the y i e l d surfaces. Clearly, the averaging method can give a very good model f i t while permitting f u l l consideration of the kinematic hardening which has occured. The assertion that the model discussed in Chapter 10 i s applicable to any given s o i l i s questionable. For example, eq.10.2a i s j u s t i f i a b l e i f the C r i t i c a l State l i n e also defines the state at which the peak stress i s achieved. This coincidence i s true for normally and l i g h t l y overconsolidated clays and for very loose sands. However, heavily overconsolidated clays and medium to very dense sands have peak strengths considerably above the C r i t i c a l State. After reaching the peak stress, these s o i l s s t r a i n soften towards the C r i t i c a l State condition. Unlike the C r i t i c a l State condition, t h i s peak strength cannot be uniquely determined for a given void r a t i o . Thus the size of the largest y i e l d surface, as given by eq.l0.2a in which k i s so l e l y a function of the volumetric s t r a i n , cannot also define f a i l u r e at peak stress 177 F i g . 11.4. Model f i t using Prevost's method and the averaging method. 178 for these s o i l s . As explained e a r l i e r , equations 10.2b and 10.2c are j u s t i f i a b l e i f the "strength" of a normally consolidated s o i l can be normalized by the consolidation pressure. While t h i s i s true for clays, i t is unreasonable in general for sands. Even the loosest sands w i l l show a peak stress and then subsequent s t r a i n softening behavior i f the confining stress i s low enough. On the other hand, i f thi s loose sand were subjected to a very high confining pressure, "normally consolidated" behavior would occur. Clearly, a normalization procedure based on a consolidation pressure cannot be used for sands in general. These arguements show that the model discussed in Chapter 10, while not being applicable to certain types of s o i l s , uses assumptions which are v a l i d for normally consolidated clays. A closer examination of eq.10.7 shows a possible reason why the model does not give good predictions for thi s s o i l e ither. Recall that: as. = 3(s -d: ) (11.7) Q.V ds- 3f_ ds.. = 3(s f j -oy )ds-= 3[(q- a i)^dq + 2(~\(q-a,))(-$dq)] = 2(q-a 1)dq (11.8) Q T! = u ^c2(p-/3) = 3(p-0) (11.9) dp 179 IQ'I = y 3 - 3 ( s ; j - 0 i j ) ( s ; j - a ; j ) = 3 y ( i ( q - a , ) ) 2 + 2 ( - ^ ( q - a 1 ) ) 2 = | y g ( q - a , ) | ( 1 1 . 1 0 ) Thus e q u a t i o n 8.20 becomes: d e ' 1 ( 2 ( q - a , ) d q + 2 c 2 ( p - 0 ) d p ) [ 2 c 2 ( p - 0 ) + 3 j6Am | q - a , | ] d q H; 6 k 2 ( 1 1 . 1 1 ) de y p 1 [2cc(p-/3) + 3 y§A m |q-a, | = L- L. ( ( q - a , ) + c T c ( p - 0 ) ) dp 3H; T k 2 ( 1 1 . 1 2 ) S u b s t i t u t i n g e q u a t i o n s 10.4 a n d 10.5 i n t o t h e a b o v e g i v e s : de v p 1 [2c cosd + 3/6A m | sine? 11 = — {sind + c T c o s f l ) dp H : 3T ( 1 1 . 1 3 ) N o t i c e t h a t t h e s e c o n d t e r m i n s q u a r e b r a c k e t s h a s a n u m e r i c a l f a c t o r o f 3/6 c o m p a r e d w i t h /6 f o r e q . 1 0 . 7 . The d i f f e r e n c e l i e s i n t h e d e f i n i t i o n o f A m . E q u a t i o n 10.7 i n c l u d e s t h e f a c t o r 3 i n i t w h e r e a s eq.11.13 s e p a r a t e s i t f r o m A m . Thus t h e A m o f e q u a t i o n 10.7 i s t h r e e t i m e s l a r g e r t h a n t h e A m d e f i n e d by eq.8.20 o r 11.13. To a v o i d c o n f u s i o n , o n l y eq.10.7 w i l l be u s e d h e r e i n . A l s o s i g n i f i c a n t i s t h a t t h e s e c o n d t e r m i n s q u a r e 180 brackets has cos0|tanc?| for eq.10.7 whereas eq .11.13 has | sine? | . Note that: cosf?|tanc?| = cost? |sin0| * | sine? j (11.14) | cost? | If eq.11.13 i s used instead of eq.10.7 to generate a second equation, instead of eq.10.11, to be used together with eq.10.19 to solve for c9 c ( m ) and d£lm), no solution can be obtained for either Weald clay or Haney clay. This implies that a solution for c9c(ni) and 6Elrn) using eq.11.13 is not, in general, obtainable and thus an alt e r n a t i v e equation (eq.10.7) i s necessary. Although eq.10.7 allows a solution to be obtained, i t has certain implications as to the type of behavior the model w i l l describe. Figure 11.5 and the following text describes one of these. Assume that the stress point l i e s i n f i n i t e s i m a l l y to the right of f m and that the stress increment points outward to f m (vector A in Fig.11.5). Thus: cose? = an i n f i n i t e s i m a l l y small positive number sine? = 1 = h'm = a constant A^ = a constant de, = dp + 1_ (0+Am /6)j_(dq) . (11.15) 181 q F i g . 11.5. Discontinuity of volumetric s t r a i n when 0=90°. Let us now examine what happens i f the stress point l i e s i n f i n i t e s i m a l l y to the l e f t of f m and the stress increment i s in the same di r e c t i o n as in the case above (vector B in Figure 11.5): cos0 = an i n f i n i t e s i m a l l y small negative number sine? = 1 H^ , = h i = a constant A m = a constant de v = dp + J _ (0-A y6)j_(dq) (11.16) B Hi 3 For c l a r i t y , l e t us assume that dp=0. Stress vector A gives: 182 A m J6 dev + dq 3HV while stress vector B gives: i . e . compression (11.17) A M / 6 dq i . e . expansion (11.18) 3 H I Thus two stress points which are i n f i n i t e s i m a l l y apart can give r a d i c a l l y d i f f e r e n t volume changes. This i s not consistent with the observed behavior of s o i l s in which i n f i n i t e s i m a l changes in stress give only i n f i n i t e s i m a l changes in s t r a i n ( i . e . continuity) at stress states below f a i l u r e . Given the above inconsistency, one would not expect t h i s model to give good predictions. d e v = -183 CONCLUSIONS From the results of the thesis, the following conclusions can be drawn: 1. P l a s t i c i t y theory u t i l i z i n g multiple y i e l d surfaces and kinematic hardening provides a convenient means of modelling the nonlinear, hysteretic, stress path dependent behavior of anisotropic s o i l s . 2. The t o t a l stress model proposed by Prevost appears to be quite accurate for predicting the monotonic behavior of undrained clays under complex stress paths. However, many more such comparisons between the s t r e s s - s t r a i n curves predicted by the model and actual test data must be made before the model can be used with confidence. 3. The reduction in strength and s t i f f n e s s of a s o i l during c y c l i c loading can be simulated by incorporating isotropic hardening/softening into the model. However, simpler expressions for describing the degradation of the y i e l d surfaces than what have been proposed would be desireable. 4. Neither of the e f f e c t i v e stress models i s capable of accurately predicting the s t r e s s - s t r a i n behavior of the s o i l s examined. 5. The general formulation for s o i l s has a mathematical inconsistency in the derivation of the equations for the parameters. This i s one of the reasons why t h i s e f f e c t i v e stress model does not give good predictions. 184 Unless a very la rge number of y i e l d su r faces are used, the p r a c t i c e of m a i n t a i n i n g de/ds and de^/dp constant in the e f f e c t i v e s t r e s s model between y i e l d su r faces i m p l i e s that only monotonic loads can be c o n s i d e r e d . 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Prevost, J.H., " P l a s t i c i t y Theory for S o i l Stress-Strain Behavior", Journal of the Engineering Mechanics Divis i o n , A.S.C.E., Vol.104, No.EM5, October, 1978, pp. 1 1 7,7-1 1 94. 21. Prevost, J.H., "Mathematical Modeling of S o i l Stress-Strain-Strength Behavior", Proceedings, Third International Conference on Numerical Methods in Geomechanics, Aachen, Germany, Vol.1, 1979, pp.347-361 . 22. Prevost, J.H., "Constitutive Theory for S o i l " , Proceedings of the NSF/NSERC North American Workshop on P l a s t i c i t y Theory and Generalized Stress-Strain Modelling of S o i l s , Montreal, May 1980. 23. Prevost, J.H.,et. a l . , "Offshore Gravity Structures: Analysis", Journal of the Geotechnical Engineering Divis i o n , American Society of C i v i l Engineers, Vol.107, NO.GT2, Feb., 1981, pp.143-165. 187 24. Roscoe, K.H., and Burland, J.B., "On the Generalized Stress-Strain Behavior of Wet Clay", Engineering  P l a s t i c i t y , Heyman and Leckis, eds., Cambridge University Press, Cambridge, England, 1968, pp.535-609. 25. Slater, R.A.C., Engineering P l a s t i c i t y , The McMillan Press Ltd., Great B r i t a i n , 1977. 26. Spiegel, M., Advanced Calculus , Schaum's Outline Series, McGraw H i l l , New York, 1963. 27. Tsotsos, St. S., and Hatzigogos, Th. N., discussion of "Anisotropic Undrained Stress-Strain Behavior of Clays", by J.H. Prevost, Journal of the Geotechnical Engineering D i v i s i o n , A.S.C.E., Vol.105, No.G.T.8, Aug., 1979, pp.991-993. 28. Vaid, Y.P., "Comparative Behavior of an Undisturbed Clay Under T r i a x i a l and Plane Strain Conditions", thesis presented in p a r t i a l f u l f i l l m e n t of the degree of Doctor of Philosophy, University of B r i t i s h Columbia, Vancouver, Canada, 1971. 29. Yong, R.N., and Ko, H.Y., Information Package for Predictions, Proceedings . of the NSF/NSERC North American Workshop on P l a s t i c i t y Theory and Generalized Stress-Strain Modelling of S o i l s , Montreal, May 1980. 30. Yong, R.N., and McKyes, E., "Yield and Failure of a Clay Under T r i a x i a l Stresses", Journal of the S o i l Mechanics and Foundations D i v i s i o n , American Society of C i v i l Engineers, Vol.97, No.SMI, January, 1971, pp.159-176. 31. Yong, R.N., and Warkentin, B.P., " S o i l Properties and Behavior", Els e v i e r S c i e n t i f i c , Amsterdam, 1975. 32. Ziegler, H. , "A Modification of Prager's Hardening Rule", Quarterly of Applied Mathematics, Vol.17, 1959, pp.55-65. 188 APPENDIX A  Homogeneous Functions A function f ( x x 2 x r ) i s c a l l e d a homogeneous function of degree n i f : f(ux 1 f u x 2 u x r ) = y nf(x,,x 2,....,x r) where v i s an a r b i t r a r y parameter. For example, the invariants of the stress deviation, J 2 and J 3 , are homogeneous functions of degree 2 and 3 respectively. The following i l l u s t r a t e s t h i s for J 2 : j 2 = s.. i(l>Sj. ) (us,. ) = iv2si} sl} Therefore J 2 i s a homogeneous function of degree 2. Euler's theorem on homogeneous functions states that i f f(x,,x 2,....,x r) i s homogeneous of degree n , then: x , df + x2df + dx, d x 2 + xr a f_ 189 o r : APPENDIX B Inversion of Equation 3.9 Let defj = y(s^-a-) ; where: 3 (s,, - a 0) v = ds l ; = a scalar 2H' k: 2 'J Recalling that ds,j = 2Gde* , equation 3.9 can be rewritten de 0 = de* + - I - ^  ^ (s k | - a k |)2G(de k l - depkl) 2H k z 3G Sjj - a-. d 6 ' j + T TT^ ( S * " " a * > ) d e k i H' k z 3G S ; j - ai} — — (ski - ak|)(Ski " a k , ) v H k 3G s r - a(. 2G ^ + H^ " k ^ ( S k' " a k , ) d e k ' " IV (S'J " a | j ) U = de?j. + del Therefore: 191 de- = v (s- - a ) 3 G s(j - a0-( s k i _ a k . ) d e k i 2 G (s ; H' a y ) v Since: 1 1 1 H H' 2 G then: 2 G N v i s , - a-.) I 1 + — 3 G s,: - a . 'j H' - (s k l - a k l)de k l 2 G (B1) Equation B1 becomes: \ 1 \ (s ; j - a ) = 3 - - — — J- (s k | - afc|)de \H 2 G / k 2 def:: vis,- - a ; j ) 1 - -2 \ 2 G H \ (s p j - a , . ) ; < ski _ a w > d e k i = de; de e 'j = de.. ds. 2G Therefore: H \ (s.3 - afj) = de;. - - 1 - — ( s M - a k l)de k l ds;j a t 2G J 2 \ 2G/ k 2 3 (s, - a r ) ds;j = 2Gde.- - ( 2 G - H) - — — (s k l - ak,)dekl } 2 k 2 193 APPENDIX C Derivation of Expression for E l a s t o p l a s t i c Modulus H For a t r i a x i a l test in which ax =oz , equation 3.9 gives: ds y 3 (s - a y) de = + [ ( s y - a y)ds Y + 2(s„ - a, )ds x ] 2H' 2H' k 2 ( C D But: 1 s, = - -s 2 y 1 2 1 ds = - —ds x 2 Therefore equation C1 becomes: ds 3 ( s y - a y ) 2 de y = + (ds y - ds y ) (C2) 2G 2H' k 2 Recall that: 3 - ( s r a , 3 ) ( s r S ) - k 2 = 0 194 which for a t r i a x i a l test becomes: 3 - [ ( s y - a y ) 2 + 2 ( s x - a x ) 2 ] - k 2 = 0 2 or: ^(Sy-ciy) 2 = k 2 Equation C2 i s then dSy 2 de = + ( d s y - d s x ) 2G 3H' ds y 2 + d ( oy - ax) 2G . 3H' 2 But e y = ey since ev = 0 and s y = -(oy-ax). Therefore: d ( ay - ax ) 2 d ev = + d ( 0 y - ax ) 3G 3H' de^ + de; And: d ( oy - ax ) 1 de y 1 2 3G 3H' 3 - H 2 195 APPENDIX D  The Scale Factor /3/2 of Figure 4.3 A stress space representation of Fig.4.3 from a d i f f e r e n t perspective i s shown in Fig.DI. Note that j ^ a y i s the ( a ) ( b ) Fig.DI. a) Oblique view of axisymmetric t r i a x i a l plane (shaded) in stress space. b) View perpendicular to t r i a x i a l plane. projection of oy onto a plane perpendicular to the space diagonal. This vector l i e s in the deviatoric plane and i s equal to sy ; j \ a y = s y . But since we are comparing Figure 4.3a with Figure 4.3b and r e c a l l i n g that ay-ax = \sy for a t r i a x i a l t e st, the axis j\oy must be multiplied by 3/2 to achieve correspondence: 196 l i t 0 ' " i f * SIG FONZ 

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