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A coupled non-orthogonal hypoelastic constitutive model for simulation of woven fabrics Haghi Kashani, Masoud 2017

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   A COUPLED NON-ORTHOGONAL HYPOELASTIC CONSTITUTIVE MODEL FOR SIMULATION OF WOVEN FABRICS By Masoud Haghi Kashani  M.Sc., Amirkabir University of Technology, 2013 B.Sc., Amirkabir University of Technology, 2011  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  Doctor of Philosophy  in  THE COLLEGE OF GRADUATE STUDIES  (Mechanical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA  (Okanagan)  October 2017  © Masoud Haghi Kashani, 2017ii  The following individuals certify that they have read and recommend to the College of Graduate Studies for acceptance, a thesis/dissertation entitled: A COUPLED NON‐ORTHOGONAL HYPOELASTIC CONSTITUTIVE MODEL FOR SIMULATION OF WOVEN FABRICS  Submitted by     Masoud Haghi Kashani   in partial fulfillment of the requirements of  the degree of   Doctor of Philosophy.  Dr. Abbas. S. Milani, School of Engineering Supervisor Dr. M. Shahria Alam, School of Engineering Supervisory Committee Member Dr. Rudolf Seethaler, School of Engineering Supervisory Committee Member  Dr. Sumi Siddiqua, School of Engineering University Examiner  Dr. Mehdi Hojjati, Concordia University External Examiner  October 21st, 2017 Date submitted to College of Graduate Studies     iii  Abstract Woven fabrics offer a number of advantages compared to their unidirectional counterpart, such as their superior formability and higher out-of-plane stiffness, making this class of materials a decent alternative in leading composite industries such as aerospace and automotive. While their performance merits originate from the interlacing architecture of yarns, this architecture causes some complications toward their reliable analyses; namely the presence of inherent couplings between families of yarns under different deformation modes. Theoretically, the inherent coupling means an arbitrary macro deformation in a given fabric direction can affect the individual effective properties in other directions.  This study aims to provide an enhanced understanding of the role of such couplings in the mechanical behavior of woven fabrics. More specifically, the study attempts to identify the underlying multi-scale sources responsible for the coupled mechanical response of woven fabrics. Subsequently, the work introduces a new non-orthogonal hypoelastic constitutive model to reflect the observed coupled deformation mechanisms in woven fabrics. Different modes of coupling are defined and distinguished from the general coupling scheme presented by the Hook’s law. In order to parameterize the model, a comprehensive characterization framework under tension-shear, tension-tension, and shear-tension coupling modes is developed, via a multi-scale analysis. The results show that the first two modes should be closely taken into account in the analyses of fabrics to achieve more accurate predictions of the material response. The attained macro-level characterization results are interpreted at micro and meso levels. Finally, the coupled non-orthogonal model is augmented with a new wrinkling criterion to precisely predict not only the stress-strain response, but also the wrinkling onset of a plain weave in the presence of inherent coupling. The comparison between the experimental and prediction results validates the capabilities of the proposed model. The main practical advantages of including couplings in the analyses of woven fabrics are considered to be (a) more reliable design of the fabric-reinforced composites, (b) better anticipating the shape of deformed fabrics in general, and the wrinkles in particular, and (c) determination of the required tension levels to prevent wrinkling during forming process of fabrics.  iv  Preface This PhD thesis has led to publication of the following works: Chapter 2: The presented discussions in chapter 2 which are in fact the main/original idea of this study; parts of the chapter has been published in the book chapter below:  M. Haghi Kashani, A. S. Milani. “Damage prediction in woven and non-woven fabric composites”. In: “Non-woven Fabrics", Editor: Han-Yong Jeon, InTech Publisher, ISBN 978- 953-51-4586-8, 2015:233-62. Chapter 3: Within the second year of my PhD, I initiated a close collaboration with one of the PhD students of Professor Frank Ko at UBC Vancouver, namely Mr. Abbas Hosseini who I met at the CANCOM Conference held in Edmonton in August 2015. I noticed his PhD research is complementary to my work. Thereafter, through the Composites Research Network at the Vancouver and Okanagan nodes, we started working collaboratively under joint supervisions of Professor Milani, Professor Ko, and Professor Sassani- my colleague’s co-supervisor. Eventually, we developed a new shear model for plain woven fabrics and employed this model to predict the fiber locking onset in these materials. Thereafter, based on the established model, we could derive a new wrinkling criterion for the bias extension and picture frame characterization modes. The new shear theory and expressions for the locking angles (Chapter 3) along with a new coupled fabric constitutive model (introduced in chapter 5) have been primarily undertaken by me, while Mr. Hosseini primarily focused on development of the novel wrinkling model. Eventually, findings of the two works was combined and led to introducing a new material model for woven fabrics which predicted not only the stress-strain response, but also the critical end point (wrinkling). The studies presented in Chapter 3 were published in 3 journal papers and 2 conference papers as follows:  M. Haghi Kashani, A. Hosseini, F. Sassani, F.K. Ko, A.S. Milani.  “The role of intra-yarn shear in integrated multi-scale deformation analyses of woven fabrics: A critical review”. Critical Reviews in Solid State and Materials Sciences, 2017 (In press, Impact Factor: 6.45).  A. Hosseini, M. Haghi Kashani, F. Sassani, A.S. Milani, F.K. Ko. “A Mesoscopic Analytical Model to Predict the Onset of Wrinkling in Plain Woven Preforms under v  Bias Extension Shear Deformation”. Materials, 2017; 10(10):1184 (Impact Factor: 2.65).  A. Hosseini, M. Haghi Kashani, F. Sassani, A.S. Milani, F.K. Ko. “Understanding Different Shear Wrinkling Behavior of Woven Composite Preforms under Bias Extension and Picture Frame Tests”. Submitted to Composite Structures, 2017 (received minor revisions; Impact Factor: 3.85).  M. Haghi Kashani, A. Hosseini, F. Sassani, F.K. Ko, A.S. Milani. “Bias extension test or picture frame test? Toward a best-practice for characterizing the true shear behavior of woven fabrics for forming simulations”.  The 21st Annual Pacific Centre for Advanced Materials and Microstructures (PCAMM 2016) Conference, December 2016, Vancouver, Canada.  A. Hosseini, M. Haghi Kashani, F. Sassani, A.S. Milani, F.K. Ko. “A mechanism-based analytical model for predicting the onset of wrinkling in plain-woven composite preforms – A bridge from micro-scale to macro-scale”. The 21st annual Pacific Centre for Advanced Materials and Microstructures (PCAMM 2016) Conference, December 2016, Vancouver, Canada. Chapter 4: The employed test device in this chapter, upon the onset of my PhD, had been preliminary designed and manufactured in the Composite Research Network (CRN) laboratory at the Okanagan node. The device, however, was suffering from some mechanical problems. Thus, I started resolving these issues in cooperation with Mr. Armin Rashidi Mehrabadi, a Master’s student at the time (and currently a PhD student) at CRN Okanagan, and Mr. Bryn Crawford as the node research engineer. We redesigned and modified the device under supervision of Prof. Milani. Subsequently, as I postulated some roots of observed conflicts in the literature regarding the characterization results of woven fabrics, we worked to develop a new analytical procedure to prove parts of my work hypotheses. All the steps of the analytical procedures were discussed and checked by Prof. Milani. This study was published as follows:  M. Haghi Kashani, A. Rashidi, B. Crawford, A.S. Milani. “Analysis of a two-way tension-shear coupling in woven fabrics under combined loading tests: Global to vi  local transformation of non-orthogonal normalized forces and displacements”. Composite Part A, 2016;88:272-85 (Impact Factor: 4.07). Moreover, a comprehensive experimental plan was introduced and conducted by myself to characterize the fabric coupling modes. The results were analyzed using the above-mentioned analytical procedure and test fixture. Mr. Hosseini closely helped me in the analysis discussions, while I enormously benefited from the advice of Prof. Milani, Prof. Sassani and Pro. Ko. We published the following journal article:  M. Haghi Kashani, A. Hosseini, F. Sassani, F.K. Ko, A.S. Milani. “Understanding Different Types of Coupling in Mechanical Behaviour of Woven Fabric Reinforcements: A Critical Review and Analysis”. Composite Structures, 2017;179:558-567 (Impact Factor: 3.85). Chapter 5: I developed the coupled hypoelastic constitutive model and examined its validity via experiments. The results have been submitted to the following journal, where I closely benefited from the advice of Prof. Milani, Prof. Ko, Prof. Sassani, and Mr. Abbas Hosseini.  M. Haghi Kashani, A. Hosseini, F. Sassani, F.K. Ko, A.S. Milani. “A coupled non-orthogonal hypoelastic model for woven fabrics”. Submitted. Finally, in order to implement the introduced material model in ABAQUS finite element software, the collaboration between the two nodes of CRN was extended where Mr. Masoud Hejazi, a Master’s student in the research group of Prof. Sassani, was involved to help the team with writing a User Material Subroutine (UMAT) code. The written UMAT code was submitted to the 2017 American Society of Composites conference in Purdue University to participate in the material model code competition, which was selected as one of the scholarship winners:  M. Haghi Kashani, M. Hejazi, A. Hosseini, F. Sassani, F.K. Ko, A.S. Milani. “Toward enhanced forming simulation of woven fabrics using a coupled non-orthogonal hypoelastic constitutive model, integrated with a new wrinkling onset criterion”. The 32nd American Society of Composites (ASC) Technical Conference, October 2017, Purdue University, USA.  vii  Table of Contents Abstract ................................................................................................................................... iii Preface ..................................................................................................................................... iv Table of Contents .................................................................................................................. vii List of Tables ............................................................................................................................x List of Figures ......................................................................................................................... xi List of Symbols ..................................................................................................................... xix List of Abbreviations .......................................................................................................... xxii Acknowledgements ............................................................................................................ xxiii Chapter 1: Introduction ..........................................................................................................1 1.1 General overview .............................................................................................................1 1.2 Motivation and objectives ................................................................................................2 1.3 Thesis framework .............................................................................................................3  Chapter 2: Background Review .............................................................................................6 2.1 Mechanical behavior of woven fabrics ............................................................................6 2.1.1 Tensile behavior of woven fabrics .............................................................................6 2.1.2 Shear behavior of woven fabrics ...............................................................................6 2.2 Inherent coupling ..............................................................................................................8 2.2.1 Primary definition of coupling ..................................................................................8 2.2.2 Coupling in the principal material coordinate system ...............................................9 2.3 Numerical modelling of woven fabrics ..........................................................................15 2.3.1 Kinematics models ..................................................................................................15 2.3.2 Discrete models .......................................................................................................15 2.3.3 Continuum shell models ..........................................................................................15 2.4 Summary ........................................................................................................................17  Chapter 3: Re-theorizing the True Shear Behavior of Woven Fabrics under a Multi-scale Framework ....................................................................................................................19 3.1 Overview ........................................................................................................................19 3.2 Shear characterization of woven fabrics ........................................................................20 viii  3.3 Understanding the general shear behavior of woven fabrics .........................................21 3.3.1 Pre-locking shear behavior of fabrics ......................................................................22 3.3.2 Post-locking shear behavior of woven fabrics .........................................................41 3.4 Summary of findings ......................................................................................................59  Chapter 4:  Developing a New Characterization Framework to Capture Different Modes of Inherent Coupling in Woven Fabrics ..................................................................61 4.1 Overview ........................................................................................................................61 4.2 Experimental study .........................................................................................................61 4.2.1 Material ....................................................................................................................61 4.2.2 Custom biaxial-picture frame test fixture ................................................................62 4.2.3 Sample shape and clamping ....................................................................................63 4.2.4 Loading modes and friction consideration ..............................................................64 4.2.5 Experimental plan ....................................................................................................65 4.3 Transformation of global force-displacement datasets to normalized local non-orthogonal values .................................................................................................................67 4.4 Results and discussion ....................................................................................................74 4.4.1 Tension-tension coupling ........................................................................................74 4.4.2 Tension-shear coupling ............................................................................................79 4.4.3 Shear-tension coupling ............................................................................................86 4.5 Summary of findings ......................................................................................................91  Chapter 5: Proposing a Coupled Non-Orthogonal Constitutive Model for Woven Fabrics .....................................................................................................................................93 5.1 Overview ........................................................................................................................93 5.2 The meaning of coupling in the course of constitutive modeling ..................................93 5.3 The proposed coupled non-orthogonal constitutive model ............................................94 5.4 Model identification .......................................................................................................96 5.5 Considering the load history dependency of woven fabrics ..........................................96 5.6 Determination of the stiffness functions ........................................................................99 5.7 Validity of the model ....................................................................................................101 5.7.1 The capability of the model to predict the effect of tension-tension coupling ......103 5.7.2 The effectiveness of the model to predict tension-shear coupling ........................104 5.8 A coupled numerical model for woven fabrics ............................................................104 ix  5.8.1. Implementation of the suggested simulation framework in ABAQUS ................105 5.8.2. Validity of the numerical model ...........................................................................105 5.8.3. A remark on low bending rigidity of fabrics ........................................................107 5.9 Summary of findings ....................................................................................................108  Chapter 6: Conclusion ........................................................................................................109 6.1 Summary ......................................................................................................................109 6.2 Contribution to knowledge ...........................................................................................110 6.3 Future work recommendations .....................................................................................112 Bibliography .........................................................................................................................114 Appendices ............................................................................................................................122 Appendix A: Details of the obtained stiffness functions ....................................................122 Appendix B: UMAT code for the developed material model for woven fabrics ...............123      x  List of Tables Table 3.1. Comparison between different underlying deformation mechanisms in shear                  deformation of woven fabrics …………………. …………...……………………41 Table 3.2. Geometric and mechanical characteristics of the tested carbon fabric (the                   parameters are defined in Figure 3.13(a)).………........................ ……………......54 Table 4.1. Specifications and geometrical properties of the tested PP/glass plain weave ……62 Table 5.1. The identification strategy proposed to determine the stiffness functions in                    the constitutive model and validate the model ..…........………………………….96 Table 5.2. Different biaxial stress states at an identical strain state, showing the presence                   of loading path dependency in the fabric behavior under individual                    deformation modes…………………………………………………………..…...98 Table 5.3. Different shear stress states at an identical biaxial strain state, showing the        presence of loading path dependency in the fabric behavior under combined       loading modes…………………………………………………………………....98   xi  List of Figures  Figure 1.1 Comparison between the architecture of a UD and woven fabric; a) a dry Prepreg UD, b) a dry woven fabric ..................................................................................... 1 Figure 1.2 Organization of the thesis ........................................................................................ 5 Figure 2.1 Uniaxial tensile behavior of a typical woven fabric (TWINTEX®TPP60N22P-060) including two stages [1]: straightening and stretching .................................. 6 Figure 2.2 The significance of shear modulus on the forming ability of woven preforms [4], a) Shear modulus is zero, the shear angle is close to 90 degree; b) Shear modulus is very low (non-zero, similar to woven fabrics), there are a number of wrinkles; c) The modeled material is isotropic in which the shear modulus is comparable with Young’s modulus, there are quite a few wrinkles observed in the formed shape. ...................................................................................................................... 7 Figure 2.3 A typical shear force-shear angle response of woven fabrics at macro-level, showing four phases: 1- shear with static friction, 2- shear with dynamic friction, 3- locking, and 4- wrinkling ................................................................................... 8 Figure 2.4 Effect of inherent coupling on the shear behavior of woven fabrics over four deformation regions: 1- shear with static friction, 2- shear with dynamic friction, 3- locking, and 4- wrinkling ................................................................................. 10 Figure 2.5 a) A typical material behaviour without inherent coupling under sequential tensile loading in which firstly pre-tension is applied in the transverse direction (direction 2), followed by subsequent tension in the longitudinal direction (direction 1), b) The expected tensile behavior from a linear elastic material, c) The expected tensile behavior from a non-linear elastic material ....................... 12 Figure 3.1 Two different test set-ups commonly used for the shear characterization of woven fabrics: (a) Bias extension/BE testing, (b) Picture frame/PF testing; in both tests the yarns are initially oriented at ±45o with respected to the vertical loading direction; In the BE test, three different deformation regions are formed, whereas in the PF test, the deformation in the mid-region is uniform ............................... 21 Figure 3.2 Different types of shear deformation modes in a plain woven fabric. (a)  Pure shear deformation (intra-yarn shear); (b) Trellising shear deformation (rigid xii  rotation of yarns based on pin joint theory); (c) translational sliding of yarns at crossovers ............................................................................................................. 22 Figure 3.3 Kinematics of trellising shear deformation: (a) Rotation of yarns about their respective center of rotations;  (b) different velocities of the points at the same location from two distinct yarns ........................................................................... 24 Figure 3.4 In-detail meso-level comparisons between different deformation mechanisms subjected to the same extent of shear angle, γ. (a) Before deformation; (b) Trellising shear deformation (only inter-yarn shear i.e. relative rigid rotation of yarns and hence, ߛ ൌ ߛݎ; (c) “Pure” shear deformation (intra-yarn shear, ߛ ൌߛݕሻ; (d) Combination of the pure and trellising shear modes (ߛ ൌ ߛݕ ൅ ߛݎሻ. Notice that bending is not shown in these images to simplify the visual comparisons ......................................................................................................... 25 Figure 3.5 Schematic of a unit cell of a typical plain weave (TWINTEX® TPP60N22P-060) modeled in TexGen. (a) Top view; (b) 3D presentation of a single yarn in the unit cell subject to in-plane shear force; (c) a simplified structural beam representation. Notice that the in-plane shear force is along with the width of yarns, which is generally not substantially less than the length of a yarn between two crossovers and hence, should not be considered under a long beam theory . 26 Figure 3.6 (a) Yarns before deformation; (b) Deformed yarns in the first phase of response under trellising shear mode and considering bending (note that the lines perpendicular to the longitudinal axis remain at 90o); (c) Deformed yarns in the second phase of trellising response where yarns now undergo relative rotation with respect to each other, while bending stops; (d) The actual meso-level deformation of the yarns under a typical picture frame test (showing combination of trellising and pure shear deformation modes). No fiber misalignment was assumed in these images ...................................................................................... 27 Figure 3.7 Comparison between experimental data and numerical results with two distinct boundary conditions on yarns [49], implying the significance of intra-yarn shear .............................................................................................................................. 28 Figure 3.8 Comparison between intra-yarn shear deformation and the applied frame shear angle [50]. RR1, RR2, and RR3 are three different fabrics, the stiffest of which is xiii  RR3; A and B represent the shear angle within yarns spacing and the yarn itself .............................................................................................................................. 30 Figure 3.9 (a) Local bending of yarns close to the clamped boundary conditions in a typical picture frame test, which causes a deviation between the global shear angle and that within the yarn; (b) also its significance on the deviation of actual yarns width reduction from the theoretical formula [56]; ݓ0, ݓ, and ݓ′ are the initial width of yarns, instantaneous width in ideal PF test, and actual yarns width in typical PF test, respectively. Also, the ideal slip distance between AD and BC is denoted by ∆, whereas the actual one is presented by ∆′ ..................................... 30 Figure 3.10 Coordinate systems to demonstrate the undeformed and deformed shape of a fabric sample subject to an ideal picture frame test ............................................. 33 Figure 3.11 (a) multi-scale nature of woven materials [18]; (b) the micro-scale mechanisms contributing to meso-level intra-yarn shear phenomenon: i) intra-filament shear (i.e., there is static friction between filaments and the external shear force has not reached the critical force value, ii) inter-filament slippage (the imposed shear force on the yarn is now higher than the static friction limit, and a kinematic friction state is active) .......................................................................................... 36 Figure 3.12 Perpendicular lines to the axis of yarns remain at 90o over shear deformation under a bias extension test, implying the trellising shear mode in such test ........ 38 Figure 3.13 a) Proposed unit cell model for plain woven fabrics based on the pin joint theory in order to find the fabric locking angle; (b) Demonstration of the geometrical relationship between width and length of yarns ................................................... 43 Figure 3.14 Delay in locking initiation under pure shear deformation in comparison with trellising shear; (a) The representative fabric element before deformation; (b) Two parallel yarns of the deformed unit cell (black and red yarns present trellising and pure shear modes, respectively, as in a BE versus as ideal PF test). The red lines will reach other sooner during fabric shearing ............................... 44 Figure 3.15 a) Numerical simulation showing that a yarn from a given family of yarns is sandwiched between two yarns of the other family, at the locking point; b) Demonstration of the critical distance and the instantaneous thickness at locking point ..................................................................................................................... 45 xiv  Figure 3.16 Compressive forces induced between adjacent yarns during in-plane shear deformation of plain woven fabrics; (a) simulation, (b) experimental investigation of the yarn’s strain field in large shear deformation [3] ...................................... 48 Figure 3.17 Mechanism of wrinkle formation in a plain woven fabric representative element due to large in-plane shear deformation (T and Ms denote the tension along yarn and in-plane shear coupling, respectively) ........................................................... 49 Figure 3.18 Equivalent structure (a) before equilibrium deviation, (b) after equilibrium deviation ............................................................................................................... 49 Figure 3.19 Equivalent torsional spring Kt, mimicking the bending effect of interlacing yarn Y-3 on the out of plane deformation of Y-2, and vice versa ............................... 50 Figure 3.20 a) Modeling of yarn as a bending beam (made of an elastic, transversely isotropic, integrated, homogeneous material), and determining the equivalent torsional spring coefficient, Kt. Note that the width of yarns is considered as Eq. 3.13, meaning the obtained stiffness is for trellising deformation; b) The experimental set-up for measuring effective bending rigidity of the yarn ........... 52 Figure 3.21 The equivalent structure in an imaginary deviated mode; note that the P forces remain in the original directions .......................................................................... 52 Figure 3.22 The geometric characteristics of the tested carbon woven fabric measured by a Nikon optical microscope, a) the top view of a unit cell of the tested woven fabric, b) the elliptical cross section of a single yarn impregnated with PDMS resin ...................................................................................................................... 54 Figure 3.23 The carbon fabric specimen under BE test (a) before shear deformation, (b) at the onset of locking, (c) at the onset of wrinkling ..................................................... 55 Figure 3.24 The picture frame test set-up used in the experimental study ............................. 56 Figure 3.25 a) Using needles in the boundary conditions, b) a folded sample in grips to increase the interaction between fabrics and grips ............................................... 57 Figure 3.26 a) wrinkling initiation in arm regions due to the presence of transverse yarns b) removing the transverse yarns c) shear deformation of the fabric at 64 degrees . 58 Figure 3.27 Experimental measurements of locking and wrinkling angles versus the predictions proposed by the analytical model (Eqs. 3.11, 3.12, 3.18, and 3.19) . 59 Figure 4.1 Woven architecture of the tested fabric, TWINTEX® TPP60N22P-060 ............. 61 xv  Figure 4.2 Customized biaxial-picture frame test fixture employed for applying simultaneous and sequential combined loading mode ............................................................... 63 Figure 4.3 (a) Using needles instead of the conventional plate-bolts clamping to allow yarns rotate freely without local bending, (b) the resulting uniform shear deformation within the sample (the upper plates in these images have been removed after the test to be better visualize the effect of the needles in inducing homogenous deformation in the jaw’s neighborhood) .............................................................. 64 Figure 4.4 Different deformation mode configurations: (a) the original state, (b) picture frame mode, (c) biaxial tensile loading, (d) simultaneous biaxial-shear loading. Note that in (a), (b) and (c), the local fabric (material covariant) coordinate system (݂1 െ 	݂2ሻ is aligned with the fixture covariant coordinate system (1-2); however,  ݂1, and	݂2 directions are not parallel to 1 and 2 under the simultaneous mode in Figure 4.4 (d) .......................................................................................... 65 Figure 4.5 The biaxial-picture frame test fixture employed in the study; displacements in directions 1 and 2 cause biaxial loading, while displacement in direction 3 imposes in-plane shear deformation .................................................................... 66 Figure 4.6 Shear deformation of the picture frame fixture ..................................................... 68 Figure 4.7 The angle ߙ1	between the yarn in direction ݂1	and the fixture arm in direction 1, due to the tensile loading of motors in direction 1. Note that 1 and 2 are the bases of fixture covariant coordinate system, and ݂1	and	݂2 the bases of the material covariant (local) coordinate system ..................................................................... 69 Figure 4.8 Decomposition of the global external forces of the biaxial and shear motors into their normalized shear and longitudinal components imposed on the fixture frame (i.e., in 1-2 directions), in order to measure the net shear force (ܾܰݏ ൅ ܰ3ݏ), denoting a kinematic coupling between the global forces due the mechanism of the combined loading fixture. The normalized force vectors are shown as dashed lines and are related to the external motor forces via Eqs. 4.11-4.13 .................. 71 Figure 4.9 Schematic of the normalized net global forces and the resolved normalized local forces along yarns (ܰݏ	and	݈ܰሻ within the inner region of interest. Note that ܨݏ	and	ܾ݈ܰ	are parallel to the fixture arms (directions 1 and 2), whereas ܰݏ	ܽ݊݀	݈ܰ are aligned with the yarn directions (݂1	and	݂2ሻ ............................. 73 xvi  Figure 4.10 Free body diagram of the normalized net global and local forces on a triangular material element within the fabric based on Figure 4.9 ....................................... 73 Figure 4.11 Comparison between the tensile behavior of the fabric under different sequential tensile tests; a) The normalized force in the direction 1 (ܰ1ሻ	versus strain in the same direction (ߝ1ሻ, b) The normalized force in the second direction (ܰ2ሻ	against strain in the first direction (ߝ1ሻ, c) The adjusted normalized force in the direction 1 (ܰ′1ሻ	versus strain in the direction 1 (ߝ1ሻ, d) The adjusted normalized force in the second direction (ܰ′2ሻ	against strain in the first direction (ߝ1ሻ .................... 75 Figure 4.12 The tension-tension coupling factor and its effect on (a) the fabric longitudinal stiffness (using Eq. 4.20) and (b) the effective Poisson’s ratio (using Eq. 4.22) . 79 Figure 4.13 (a) Comparison between normalized global shear forces resulting from the shear motor for two picture frame tests with different pre-tension levels, and (b) the same comparison using the normalized net local shear force applied to the fabric. Notice the significance of considering the global shear force component resulting from the biaxial motors (coupling effect) in the resulting trends between the two curves with 0.8 and 3.1 N/mm pre-tensions, especially at higher shear angles. More specifically, by performing the force analysis in the local coordinate, the effect of pre-tension is consistent (the distance between the curves remain consistent) across the fabric shear angles ............................................................ 80 Figure 4.14 Variation of the global force of biaxial motors during picture frame test with 3.1 N/mm (total of 280 N) pre-tension; from the origin to point A (stage 1): applying pre-tension up to 380 N; from point A to point B (stage 2): the relaxation of the sample and causing reduction in the force from 380 N to 280 N; after point B (stage 3): applying the shear deformation ............................................................ 81 Figure 4.15 Normalized global force components of the biaxial motors during shear frame testing with 3.1 N/mm pre-tension; (a) the longitudinal (tensile) component, (b) the shear component. Note that as outlined in section 4.3, during the analysis ݈ܰ	is		assumed	to	be	alomost	equal	to	ܾ݈ܰ in the shear tests with a small yarn pre-tension level ................................................................................................... 83 Figure 4.16 Comparison between the shear behavior of the tested fabric under different pre-tension conditions. a) The shear force from the picture frame tests with uniaxial xvii  pre-tension, b) The shear force from the picture frame tests with biaxial pre-tension, c) Variation of the coupling factor for shear tests under uniaxial pre-tension, d) Variation of the coupling factor for shear tests under biaxial pre-tension .................................................................................................................. 84 Figure 4.17 A typical shear response of woven fabric at macro-level, showing four stages: 1- shear with static friction, 2- shear with dynamic friction, 3- locking, and 4- wrinkling. Notice that the applying yarn tension not only increases the effective shear rigidity, but also changes the range of each shear stage ............................. 85 Figure 4.18 (a) Change in the required load of the shear motor over time in the simultaneous loading test (showing the kinematic coupling effect between the biaxial and shear motors), (b) the global force of the biaxial motors under this mode, which reaches to a considerably high load magnitude ................................................................ 87 Figure 4.19 Deviation of the shear angle within the sample under simultaneous loading from the shear angle of the picture frame fixture based on the analytical procedure. The deviation between the curve and the dashed line represents the difference between the shear angle of the frame and fabric (2ߙ). ........................................ 88 Figure 4.20 (a) Comparison of the fabric’s tensile behavior under biaxial and simultaneous biaxial-shear loading modes, implying the effect of fabric shear on its tensile behavior; b) The calculated shear-tension coupling factor .................................. 89 Figure 4.21 Idealized change in the contact area at crossover points during (a) biaxial and (b) shear deformation; Notice that the right side rhomboid area is smaller, assuming identical yarn width in both cases ........................................................................ 89 Figure 4.22 Comparison between the elongation of fiber yarns during biaxial and simultaneous loading modes ................................................................................ 90 Figure 5.1  Using the non-orthogonal coordinate system to trace the fibers rotation under large shear. ߠ is the angle between warp/weft yarns in the deformed configuration, and ߙ is half of the engineering strain (ߛ12) calculated for each element during finite element implementations using a regular shell element .... 94 Figure 5.2 Different response of the woven fabric to different loading paths with the same final deformation condition, proving the presence of load history dependency in woven fabric; (a) Comparison between biaxial tensile test and uniaxial tensile test xviii  with 2% transverse pre-tension, (b) Comparison between simultaneous shear-tension test and picture frame test with 4% biaxial pre-tension .......................... 98 Figure 5.3 Concept map for different types of coupling and their effects on the mechanical behavior of woven fabrics .................................................................................. 100 Figure 5.4 Examining the accuracy of the model for datasets employed in the model identification stage; a) under sequential tensile test with 2% transverse pre-tension, , b) under sequential tensile test with 2% transverse pre-tension, c) under picture frame test with 3% biaxial pre-tension .................................................. 102 Figure 5.5 The validity of the proposed model via comparison between the accuracy of the coupled and uncoupled models using independent datasets, a) for biaxial tensile test, b) for simultaneous biaxial tensile-shear test ............................................. 103 Figure 5.6 The suggested simulation framework for woven fabrics .................................... 105 Figure 5.7 Validation of the UMAT code implemented in ABAQUS in terms of stress-strain behavior (a, b & c) and wrinkling prediction (d); a) Accurate prediction of longitudinal stress under sequential tensile test with 2% transverse pre-tension, b) The transverse stress under the sequential tensile test with 2% transverse pre-tension c) Schematic of stress in direction 1 of the specimen under sequential tensile test with 2% transverse pre-tension, d) Correct prediction of wrinkling onset in the woven fabric under bias extension test using the developed material model .................................................................................................................. 106 Figure 5.8 Comparison between two modes of wrinkles in a woven fabric under a 3D forming test. Compression wrinkle: A global phenomenon; Shear wrinkle: A local phenomenon. This trial illustrative draping test was done a TWINTEX® TPP60N22P-060 fabric with hemispherical punch with diameter of 120mm ... 108    xix  List of Symbols Symbols Definitions ߪ Stress components in the global orthogonal coordinate system ߝ Strain components in the global orthogonal coordinate system C Stiffness components in the global orthogonal coordinate system Dij Reduced stiffness components ݓ Instantaneous width of yarns ݓ଴ Initial width of yarns ߛ Shear angle ݓ′ Instantaneous width of yarns E Young’s modulus I Bending moment of inertia G Shear modulus J Polar moment of inertia A Cross-section area ߦ Cross-sectional shape coefficient ܯ௬ Resultant bending moment ௫ܶ, Torsional torque ௭ܸ Shear force X Undeformed position vector of material point ݔ Deformed position vector of material point Fij Deformation gradient tensor ܿ௚ Correction factor representing the impact of gaps between yarn filaments Cij Right Cauchy–Green tensor Eij Green–Lagrange strain tensor ߚ Factor representing the extent of intra-yarn shear ݈ Length between crossover center of yarns ߛ௟௢௖௞௜௡௚ Locking angle xx  ݐ଴ Initial thickness of yarns ௌܻ/ଶ Deflection at middle of yarns ܨ Transverse force on yarns ܵ Length of a unit cell T Applied tension on a single yarn ܳ௕ Flexural rigidity of yarn ܭ௧ Equivalent stiffness of torsional spring ܼ Lumped parameter ܭ Lateral stiffness of yarns Δ Compaction of yarns  The angle of bars with respect to the initial configuration P The lateral compressive forces on yarns ௖ܲ௥ Critical compressive force ߂௖௥ Critical compaction of yarns ߛௐ௥௜௡௞௟௜௡௚ି஻ா Wrinkling angle under bias extension test ߛௐ௥௜௡௞௟௜௡௚ି௉ி Wrinkling angle under picture frame test ܦ Distance of the opposing corners of region III in bias extension tests ݀ Relative displacement of the grips 2∅ Angle of the picture frame arms ݑ௦ Displacement in the direction of the shear motor ߠ௉ி Shear angle of the sample directly resulting from the picture frame deformation ܮ Length of the fixture arms ܮଵᇱ  Instantaneous length of yarns under simultaneous loading mode in direction 1 ܮଶᇱ  Instantaneous length of yarns under simultaneous loading mode in direction 2 ݑଵ Displacement of biaxial motor 1 ݑଶ Displacement of biaxial motor 2 ߙଵ,ߙଶ Angle between the fixture arm and the corresponding family of yarns while undergoing simultaneous loading (for a balanced case: αଵ=αଶ=α) a Yarn width in warp direction xxi  b Yarn width in weft direction W Jaw’s width ܨ௦ Normalized net global shear force ܰ௠௢௧௢௥	ଵ, ܰ௠௢௧௢௥	ଶ Global forces of biaxial motors (for a balanced material and symmetric loading: N୫୭୲୭୰	ଵ ൌ N୫୭୲୭୰ ଶ ൌ Nୠ) ଷܰ Global force of shear motor ଷܰ௦ Normalized global shear force of shear motor ௕ܰ௟ Normalized longitudinal force component of biaxial motors along the arms direction ௕ܰ௦ Normalized shear force component of biaxial motors ௟ܰ Normalized local longitudinal force applied to yarns within the region of interest ߝଵ, ߝଶ Longitudinal strains in yarn directions ߠ Net shear angle 1-2 Fixture covariant coordinate system ௦ܰ Normalized local net shear force applied to yarns within the region of interest ݓ Width of fixture jaws ଵ݂ െ ଶ݂ Material covariant (local) coordinate system ଵܰ, ଶܰ Normalized tensile forces along yarns in directions 1 and 2 ܰ′ଵ, ܰ′ଶ Adjusted normalized forces in directions 1 and 2 CF Coupling factor ݃௜ Local covariant base vector ௜ܲ௝ Transformation matrix of bases ݃௜ Local contravarient matrix ߪ෤ Contravariant components of the stress tensor in the covariant coordinate system ߝ̃ Covariant components of strain tensor TR Transformation matrix ܦ෩଴ Pure modulus in the non-orthogonal coordinate system ܦ෩௖ Coupling induced modulus in the non-orthogonal coordinate system    xxii  List of Abbreviations  Abbreviation  Deviation BE   Bias extension CF   Coupling factor ES    Equivalent structure PF    Picture frame test PJT    Pin-joint theory UD       Unidirectional composites   xxiii  Acknowledgements  First, I would like to extent my sincere gratitude to my supervisor, Professor Milani, for all his solid support, precious advice, and warm friendship throughout this journey. He taught me how to look at the bigger picture and develop new understandings in this magnificent field of science and engineering. He will always mean to me more than a supportive supervisor; he is truly like an elder, wiser and well-experienced mentor and caring family member, not only to studetns in his lab, but also to every single student who he interacts with. I would like to thank other faculty members at UBC, particularly Prof. Ko and Prof. Sassani, who generously spent their times with me and provided their constructive advice at different stages of my work. The constructive comments of my committee members in the proposal defense stage, Prof. Vaziri, Prof. Seethaler, and Prof. Alam were immensely useful and are highly recognized. Moreover, I am so grateful for the assistance of my CRN Okanagan labmates, namely Mr. Armin Rashidi, Mr. Bryn Crawford, Dr. Mohammad Alemi, Mr. Juan David Torres Ferrero and Dr. Mohammad Nouroz Islam. Furthermore, I would like to extend my gratitude to my colleagues and friends at UBC Vancouver, Mr. Abbas Hosseini and Mr. Masoud Hejazi who made considerable contributions to evolve this project. Above all, my especial thanks belong to my lovely parents and my brother who unconditionally and wholeheartedly supported me throughout my life by providing endless love, tireless encouragement and lifelong dedication.    xxiv  “Parents were the only ones obligated to love you; from the rest of the world you had to earn it.” Ann Brashares.  …In dedication to my beloved parents…       1  Chapter 1: Introduction 1.1 General overview Since the emergence of composite materials (1937), unidirectional (UD) fiber reinforced composites have caught the attention of most designers, mainly due to their high specific stiffness and strength properties as well as the simplicity in their analysis and design. Nevertheless, woven fiber reinforced composites (Figure 1.1) gradually became an acceptable alternative to the traditional UD composites in a wide range of industries [1]. A woven fabric reinforcement is defined as interlaced warp and weft fibers in a repetitive weave pattern such as plain, twill, and satin. Woven composites possess many advantages over UDs owing to their interlaced fibrous structures. To exemplify, they have higher impact resistance, superior resistance to crack growth, and better formability [1].    (a) (b) Figure 1.1 Comparison between the architecture of a UD and woven fabric; a) a dry Prepreg UD, b) a dry woven fabric Woven fabrics are today utilized at both at the consolidated level in composite structures and at the dry level in a range of technologies such as inflatable beams, flexible electronics, microfluidics, and artificial muscle fibers [2]. However, to date, there has been far more research conducted in the area of UD composites when compared to the woven fabric composites. As an example, according to the World Wide Failure Exercise [3], there exist nearly 20 failure theories derived for the UD composite materials, while no explicit failure 2  criterion has been derived specifically for cured woven composites. In essence, the interweaved structure of woven fabrics leads to a number of intrinsic complexities in the analyses of such hierarchical materials (from thousands of filaments in a yarn, to hundreds of yarns in the meso-scale fabric, to macro-scale composite part). In particular, there exists out-of-plane waviness (crimp) of yarns within a woven fabric, whereas UD tows are comprised of straight yarns. Moreover, in the consolidated form, matrix cracking can be confined in woven-reinforced composites, thanks to the cellular structure of the reinforcement. Another complication is that the interlacement of yarns can cause local stress concentrations at the meso level. Finally, there appear severe “inherent” couplings between the response of family of yarns in woven fabrics under different deformation modes, which is the main focus of this thesis and will be elaborated on in the following chapters. Theoretically, the coupling in this thesis is defined when a macro-level deformation in a given fabric direction causes variations in effective individual properties in other directions. 1.2 Motivation and objectives Although one of the practical outcomes of inherent mechanical coupling has started to be used by composite designers in real forming processes of woven fabrics - namely delaying the wrinkles initiation in forming processes through applying tension on fabric yarns - the nature of this phenomenon and the underlying mechanisms have not been fully understood for a systematic implementation in numerical simulation tools. More specifically, neither the significance nor the trend of the influence of different modes of coupling on the mechanical behavior of woven fabrics has been thoroughly investigated.  From a mechanical point of view, the inherent coupling has not been distinguished from the coupling concept expressed by the general Hook’s law, and has not been incorporated in a formal material constitutive model.  Recognizing the above gaps between knowledge and practice of woven fabric composites field, and pursuant to the more detailed literature review that will be presented in the next chapters, the ultimate goal of this PhD thesis is to develop an enhanced coupled non-orthogonal constitutive model for situation of dry woven fabrics. It is aimed that the model should be able to consider not only non-linearities in the response of fabrics, but also the influential coupling modes under complex combined biaxial-shear modes as experienced  by 3  the material during industrial forming processes. More specific objectives of the thesis are as follows: 1) Develop an enhanced knowledge of multi-scale tensile and shear behavior of woven fabrics Deliverable: This objective will provide a fundamental understanding of the contributing mechanisms in the mechanical behavior of woven fabrics, based on which the coupling phenomenon can be further explained. 2) Propose a new analytical framework to transform the global fabric response parameters to local non-orthogonal parameters, under combined loading conditions. Deliverable: This framework will allow analyzing the fabric characterization results in the context of couplings at local yarn levels. 3) Perform full characterization of different modes of inherent couplings in a typical woven fabric under combined tension-shear modes;   Deliverable: The results will determine the significance and trend of the effect of each coupling mode on the mechanical behavior of woven fabrics. In fact, the obtained results will be the basis to propose a new constitutive model for woven fabrics. 4) Propose a new coupled non-orthogonal hypoelastic fabric model, integrated with a wrinkle initiation criterion. Deliverable: The model may be used in fabric forming simulations to more accurately predict the shape of the deformed preforms in general, and the location of forming-induced wrinkles in particular. 1.3 Thesis framework This thesis is comprised of six chapters. Chapter 2 is a general background review as related to the inherent coupling concept. It will demonstrate a distinction between the inherent coupling and that presented by the Hook’s law in anisotropic materials. Next, the chapter presents a critical review on the experimentally, analytically, and numerically undertaken studies in the area of the mechanical behavior of woven fabrics in terms of the coupling. Chapters 3, 4 and 5 constitute the main, interconnected body of the research, each having its own methodology, results, and discussion. More specifically, Chapter 3 starts with developing a new multi-scale pure shear theory for woven fabrics by challenging the validity of the earlier 4  Pin-Joint Theory for Picture Frame tests. Under this theory, basic understandings about different response of woven fabrics under common shear characterization tests in the literature (namely the bias extension and picture frame tests) have been enhanced and linked to the underlying multi-scale deformation mechanisms as pertinent to the analyses of coupling. In Chapter 4, a new experimental characterization strategy including various combined loading tests is designed such that different coupling modes can be parameterized. In doing so, an analytical approach is developed and employed to transform the global forces and displacements to local stresses and strains along the yarns direction. The results in the local non-orthogonal coordinate system are analyzed to determine the role of each coupling mode in the mechanical behavior of a glass-polypropylenes woven fabric. In accordance with the results of Chapter 3 and 4, in Chapter 5 the coupled non-orthogonal hypoelastic model is developed, reflecting a more realistic of behavior of woven fabrics at macro-level under presence of tension-tension, and tension-shear couplings. The validity of the model is assessed by comparing the predictions with independent datasets, as well as its simulation model via a user-defined code in ABAQUS. For the latter, the constitutive model is augmented with a shear wrinkle initiation criterion. Finally, chapter 6 summaries the findings of the thesis and recommends some potential future work directions. Figure 1.2 illustrates the organization of the thesis and demonstrates how chapters accomplish the proposed objectives in Section 1.2.         5                     Figure 1.2 Organization of the thesis  Chapter 4: Developing a new characterization framework to capture different modes of inherent coupling in woven fabrics (Objectives 2 and 3) Chapter 3: Re-theorizing the shear behavior of fabrics (Objective 1) Chapter 2: Background and Review Chapter 1: Introduction Chapter 5: Proposing a coupled constitutive model for woven fabrics (Objective 4) Chapter 6: Conclusion  6  Chapter 2: Background Review 2.1 Mechanical behavior of woven fabrics As stated in Chapter 1, regardless of inherent coupling which shows its effect predominantly in combined loading conditions, the behavior of woven fabrics even under simple loading conditions such as uniaxial tensile and pure shear loadings can be more intricate when compared to the UD composites. Hence, prior to discussing the coupling concept, a review of general behavior of woven fabric materials is deemed essential.  2.1.1 Tensile behavior of woven fabrics The tensile behavior of woven fabrics in general is non-linear, including two phases of fiber straightening and fiber stretching, as shown in Figure 2.1. There exist a number of meso-scale mechanisms causing such response. In the straightening phase, the yarns under tension lose their waviness – become straighter, whereas the level of crimping in the crosswise yarns increases because of contact of yarns at crossovers. The crimping exchange actually happens through unbending and bending of the longitudinal and transverse yarns, respectively. Such reduction in the waviness of the yarns is up to a limit where yarns are virtually locked to one another. Subsequently, the yarns need to stretch to accommodate further global deformation. Thus, a stiffer behavior is observed at the macro-level in the second phase.   Figure 2.1 Uniaxial tensile behavior of a typical woven fabric (TWINTEX®TPP60N22P-060) including two stages [1]: straightening and stretching 2.1.2 Shear behavior of woven fabrics The shear behavior of woven fabrics has received more attention in the literature in that this deformation is the most dominant mode during the forming process of woven fabrics [4]. 7  In other words, a better formability of woven fabrics compared to UD composites is due to the substantially lower rigidity in fabrics. More specifically, the shear resistance of woven fabrics is much smaller than their tensile stiffness. Hence, based on the minimum total potential energy principle [5], when a woven preform is formed onto to a specific mould shape, it tends to deform through the way with least energy, which is shear deformation in the case of fabrics. The numerical study [4] clearly showed the influence of change in the fabrics shear rigidity on their formability, as shown in Figure 2.2.    (a) (b) (c) Figure 2.2 The significance of shear modulus on the forming ability of woven preforms [4], a) Shear modulus is zero, the shear angle is close to 90 degree; b) Shear modulus is very low (non-zero, similar to woven fabrics), there are a number of wrinkles; c) The modeled material is isotropic in which the shear modulus is comparable with Young’s modulus, there are quite a few wrinkles observed in the formed shape A typical shear response of woven fabrics is illustrated in Figure 2.3. According to this figure, the fabric can experience four different regimes during forming, which are referred to as shear with static friction, shear with dynamic friction, locking, and wrinkling. In the first stage, because the shear force is lower than the critical static friction force within fabrics, a stiff behavior is observed. Once the shear force exceeds the critical static friction force, the static friction is converted to dynamic friction, reducing the resistance of fabrics against shear deformation. Locking is defined as the moment the adjacent (warp/warp and/or weft/weft) yarns start to have side contact, upon which the wrinkling or out-of-plane deformation of fabrics may begin.  8   Figure 2.3 A typical shear force-shear angle response of woven fabrics at macro-level, showing four phases: 1- shear with static friction, 2- shear with dynamic friction, 3- locking, and 4- wrinkling 2.2 Inherent coupling 2.2.1 Primary definition of coupling    In accordance with the generalized Hook’s law, for anisotropic materials, there are two types of couplings: extension-extension and shear-extension coupling [6]. To elucidate, these forms of coupling mean that, for example, applying tensile deformation in one direction causes transverse contraction and/or in-plane shear, as shown in Eq. 2.1 where ߪ, ߝ, and C represent stress, strain and stiffness components, respectively. In fact, extension-extension coupling is a representative of the Poisson’s ratio effect. However, if the Hook’s law is written in the principal material coordinate system, e.g., in a plane stress domain such as plane fabrics along the yarns directions, the shear-extension coupling vanishes as shown in Eq. 2.2, and the original stiffness coefficients are converted to the reduced stiffness elements denoted by Dij.  9   (2.1) ൥ߪଵߪଶߪଵଶ൩ ൌ ൥ܦଵଵ ܦଵଶ 0ܦଵଶ ܦଶଶ 00 0 ܦ଺଺൩ ൥ߝଵߝଶߛଵଶ൩ (2.2) 2.2.2 Coupling in the principal material coordinate system 2.2.2.1 Definition of inherent coupling From a mechanistic point of view, this class of coupling would imply that “any macro deformation in a given material direction causes joint variations in meso-scale structure of the fabric, hence changing the individual effective macro-scale properties in other directions. To explain, in a macro-level presentation, this means e.g. the axial stiffness D11 can be a function of ߝଶ, as well as possibly other strain components. Note that such coupling can appear not only in the general coordinate systems but also in the principal material coordinates. Also under this category of coupling, applying load in one direction does not necessarily lead to normal deformation or shear in other directions. As the evidence to such inherent coupling, it has been experimentally reported that applying tension on woven yarns notably increases the effective shear rigidity of woven fabrics and postpone wrinkling [7]. Figure 2.4 schematically illustrates the role of coupling in the latter observation, i.e., its effect on both stress-strain path and the critical end point – wrinkling. Note that for a typical material without coupling the curves presented in Figure 2.4 have to be exactly the same. 10   Figure 2.4 Effect of inherent coupling on the shear behavior of woven fabrics over four deformation regions: 1- shear with static friction, 2- shear with dynamic friction, 3- locking, and 4- wrinkling 2.2.2.2 The importance of inherent coupling in both design and manufacturing Before further analyzing such coupling, it would be worthwhile to substantiate the importance of coupling in practice of woven composites. Based on the definition of section 2.2.2.1, inherent coupling would predominantly reveal its effect under combined (multi-directional) loading conditions, either sequential or simultaneous. In practice, the applied service loads are also often in the form of combined loadings; for instance, the Kevlar woven fabrics used in the inflatable heat shields for descending spacecraft, undergo a loading condition analogous to the simultaneous biaxial tensile test [2]. Hence, the coupling in woven fabric composites should be carefully considered to predict the design safety levels. More importantly, since the forming of dry woven fabrics (to produce near-net 3D shapes) often involves simultaneous loading modes including shear, tension, and compression in different regions of the reinforcement, the inherent coupling can become an influential factor to avoid defects during manufacturing. In fact, it has been shown that wrinkling - one of the most common defects in composites manufacturing - can be delayed by applying tension on fabrics during draping [7]. Accordingly, the advantages of assessment of coupling in woven fabrics at dry level is summarized as follows:  More reliable design of the textile fabrics over service time;  Anticipating the shape of a deformed fabric in general and wrinkles shape in particular over forming processes;  Determination of the required tension to prevent wrinkling with higher precisions; 11   More accurate prediction of the mechanical properties of the final consolidated composite parts such as stiffness and residual stresses;  Better estimation of the required force and facilities for draping processes. 2.2.2.3 Different types of inherent coupling in woven fabrics There can be at least three types of inherent coupling in woven fabrics, namely tension-tension, tension-shear, and shear-tension coupling, each of which is reviewed and discussed as follows. There exits also tension-bending coupling, i.e. the effect of yarn tension on bending rigidity of fabric, which is out of scope of this thesis; however, it is partially discussed in Chapter 3 and Chapter 5. a) Tension-tension coupling In a typical multi-directional material (e.g., a UD lamina) in which there is no coupling between the material properties in the longitudinal and transverse directions, the only difference between the response of the material under longitudinal tension with and without a transverse pre-tension is the initial jump (intercept) in the stress-strain curve, owing to the Poisson’s ratio effect. However, the slope of the curves – the stiffness - remains unchanged, as shown in Figure 2.5(b). Similarly, for a nonlinear material but without inherent coupling, Figure 2.5 (c) demonstrates that although the slope varies over tensile loading, its instantaneous value at each strain remains the same over the two curves, and the only difference is still in the initial jump.   12     (a) (b) (c) Figure 2.5 a) A typical material behaviour without inherent coupling under sequential tensile loading in which firstly pre-tension is applied in the transverse direction (direction 2), followed by subsequent tension in the longitudinal direction (direction 1), b) The expected tensile behavior from a linear elastic material, c) The expected tensile behavior from a non-linear elastic material On the other hand, due to the interweaved architecture of woven fabrics, there are inherent meso-level interactions between the warp and weft yarns such that applying tension in one direction, causes a change in e.g., crimp and hence the effective stiffness of the other direction. Graphically, it means that although there is still initial jump; the slopes of the curves would not be identical as in Figure 2.5 (c). Most of the earlier efforts to characterize the tensile behavior of woven fabrics have demonstrated a fair agreement between analytical/numerical models with uniaxial and biaxial tensile tests [8-11]. These studies, however, did not examine the validity of the proposed fabric models under various complex loading conditions. As an example, in the study by Sogar et al., results revealed that there are significant discrepancies between the predicted and experimental data in some specific loading cases, whereas the analytical and experimental data for the uniaxial extension mode were highly comparable [12]. Other numerical studies such as [13], in which an accurate 3D geometrical model of the woven material was implemented, showed improved predictions under combined loading modes. In particular, the study [13] pointed that the stiffness of the fabric in a given direction under biaxial loading has direct correlation with the ratio of axial strain to the transverse one. However, no further discrimination between the contributions of the inherent coupling and the Poisson’s ration effect was presented while both 13  make effect on the stress-strain curves. Theoretically, the influence of extension-extension coupling due to the Poisson’s effect on the material behavior in fractional biaxial loading dictates that, a higher fraction (i.e., ratio of  ఌభఌమ ) should yield a more compliant behavior in direction-1 (lower force), while a stiffer behavior (higher force) was noticed for woven fabrics in [13]. The latter implies that the inherent coupling can be so significant as to not only overcome the compliant behaviour resulting from the Poisson’s ratio effect, but also to cause a stiffer response in the yarns. In another work by Shahkarimi and Vaziri [14], with a main objective of introducing a shell finite element for impact simulation of woven fabrics, some levels of pre-tension was considered in weft direction. Although the model demonstrated a slight compliance in the warp direction (see Figure 9 of [14] which is consistent with [13]), the observed decrease was not significant in their numerical simulation. Accordingly, the coupling was ignored in the final shell element formulation [14]. Thereafter, Komeili [15] simulated dry woven fabrics at meso-level and applied sequential biaxial tensile loadings – i.e. first applying tension in one direction followed by tension in the other direction - on a representative unit cell. Again no actual tension-tension interaction was recognized via his numerical results [15]. On the other hand, another numerical simulation performed by Lee et al. [16] showed that the stiffness of the fabric in direction 1 is a function of applied ሺఌమఌభሻఈ where ߙ is a positive value, inferring that the latter observation is totally opposite to that in [13]. As a consequence, it appears that there is a conflict in the current literature regarding understanding the effect of the transverse yarn tension on the mechanical behavior of woven fabrics in the longitudinal direction. Moreover, the effect of the extension-extension coupling (Poisson’s ratio) and the meso-level tension-tension coupling has not been distinguished. What is more to be scrutinized is a distinction between different levels of transverse pre-tension as corresponding to the decrimping or stretching phase of woven fibers as shown in Figure 2.1 (i.e., a two-phase model may be required for capturing actual coupling in fabrics).   14  b) Tension-shear coupling In a conventional UD composite, applying tension on fibers does not change the in-plane shear resistance of the lamina, and vice versa. However, on account of the interlaced fiber architecture of woven fabrics, the coupling between the in-plane tensile and shear responses of the reinforcement is inevitable [7]. Despite the fact that the dependence of the shear rigidity of woven fabrics on yarns’ pre-tension has been reported in the recent literature, some conflicts regarding the trend of this effect are perceived. Namely, experimental results have shown that a higher biaxial pre-tension level can yield a higher fabric shear resistance at low shear angles, whereas surprisingly an opposite trend was noticed at high shear angles in [17]. On the contrary, the experimental work by Nezami et al. [7] and the numerical results by Komeili [18], indicated that applying pre-tension causes the shear rigidity to rise at any shear angle. This conflict should be addressed and its source should be identified. Moreover, the significance of tension-shear coupling, or the quantified correlation between the extent of tension and the shear behavior of fabrics has not been studied yet. c) Shear-tension coupling While the effect of yarn tensioning on the fabric shear rigidity has been addressed in a few studies as reviewed above, the effect of fabric shear on yarns tensile behavior has not received comparable attention. Buet-Gautier and Boisse [9] reported a slight decrease in the tensile stiffness of woven fabrics when the angle between weft and warp yarns was set to a value other than 90o. No meso-analyses regarding the extent and source of the noticed difference were presented. Komeili’s numerical results under simultaneous loading showed that the magnitude of the effect of shear-tension coupling may be ignored relative to that of tension-shear [18], while the individual properties may still be important for final structural properties after consolidation. No experimental evidence was, however, reported in [18]. Furthermore, although the roots of tension-shear coupling are known, the underlying sources of shear-tension coupling have not been offered yet. Additionally, this mode of coupling has not been quantified and compared to other types of couplings reviewed above.    15  2.3 Numerical modelling of woven fabrics There have been a number of numerical models employed in the past to simulate the mechanical behavior of woven fabrics. These approaches are briefly reviewed as follows. 2.3.1 Kinematics models In spite of the computational effectiveness of the well-known kinematics models [19-23], which are based on the Pin-Joint Network (PJN) theory, they do not take the material and processing parameters into consideration. In other words, in the PJN approach, fibers are assumed inextensible with negligible shear and bending stiffness, and with no consideration of tool-part interactions. Hence, kinematics models can only provide a first-order estimation of the fabric deformation by overlooking the strain energy. 2.3.2 Discrete models Despite the kinematic models, discrete meso-scale numerical simulations such as [24] can offer much more detailed information about the stress distribution within the fabric, change in meso-scale configurations of yarns, and yarns slippage. However, the discrete modeling of fabrics is computationally expensive for large-scale forming simulations. On account of reduction in computational time, instead of solid elements, trusses and membranes were employed to measure the energy resulting from the tension in yarns and the shear of fabrics [25-30]. In fact, yarns were presented by trusses, while the shear resistance of fabrics was modeled by membranes [27] or two diagonal trusses [25]. Yet, this approach is not computationally cost effective. 2.3.3 Continuum shell models Given the hieratical fibrous structure of woven fabrics, any micro/meso-scale material knowledge should be interpreted at macro-scale in order to propose a truly equivalent continuum, which in turn can act in accordance with micro/meso-scale physics of the fabric. As a result, despite the discontinuity in woven fabrics, the continuum modeling of woven fabrics using shell elements would be the most efficient method. To do so, in a number of studies, one unit cell of fabrics is presumed to derive the nominal macroscopic stresses on the cell boundaries [2, 18]. For instance, Erol et al. proposed a macro-scale non-orthogonal material model for plain woven fabrics based on the meso-scale unit cell of these materials 16  [18]. Nonlinear tensile behavior of woven fabrics, fibers rotation and shear behavior of woven fabrics were accounted for in that study. Boisse et al. investigated a unit cell from virtual energy point of view and introduced a specific element formulations in three following articles [4, 31-32]. In [4], it was demonstrated that wrinkling occurrence depends on all the terms of energy including tension, shear and bending. In addition to the above-mentioned studies in the literature, there are two other approaches known as hyper-elastic and hypo-elastic constitutive models. In an hyper-elastic model, the strain energy of the woven fabric is decomposed into the tensile energy in yarns and the shear energy at crossovers, each of which to be determined using curve fitting with e.g. uniaxial tensile and shear frame tests [33-35]. Notice that summing the individual tensile, shear and bending deformation energies under a total stain energy does not constitute the consideration of coupling. To take the latter issue into account, each energy term must be a function of all strain components. For instance, the shear deformation energy should be a function of longitudinal strains. Instead of postulating energy terms, constitutive models which explicitly relate the shell strain to stress components can be employed to equivalently reflect the meso-level behavior of the fabric. In this approach, the macro constitutive models introduced earlier in the literature have been in orthogonal coordinate systems (e.g. [36] and [37]) which were not able to take the relative weft/warp yarns rotations into consideration during draping. Subsequently, a non-orthogonal constitutive equation was introduced by Yu et al. [38] to account for yarns reorientation during large shear. More recently, a non-orthogonal constitutive model was introduced in [39] to trace the fibers direction and update the material stiffness matrix for finite element implementations. This model was developed by Peng et al. in [40, 41]. In [40], a comparison between the results of the orthogonal and non-orthogonal material models was drawn, and it was perceived that the accuracy of non-orthogonal model is substantially higher. However, no interaction effect (coupling) was considered in these past studies.  For the first time, the coupled effect of yarns tension on the shear rigidity of woven fabrics was incorporated in the fabric constitutive modeling in [42], by considering the shear modulus as G× G’; where G is the shear modulus of the material under a pure shear test and Gʹ is the tensile induced shear modulus (coupling factor). Both of these modulus functions 17  were fitted based on test simulations. Subsequently, the result of a macro-level forming simulation showed a higher punch load using the coupled model compared to the uncoupled model, as theoretically expected. However, no further comparison with experimental results was drawn to substantiate the validity of the model. In another investigation by Lee et al [16], in addition to the effect of tension on shear, the coupling between warp and weft directions was implemented in a non-orthogonal constitutive model. However, as discussed in Section 2.2.2.3, the assumed effect for tension-tension coupling was in opposite to some experimental results. Moreover, Poisson’s ratio was assumed zero which would seem similar to reality because woven fabrics are not integrated so that elongation in one direction can cause considerable contraction in other directions. However, tension-tension coupling may magnify Poisson’s ratio of fabrics. Thus, this issue needs to be studied.  Although the above past studies considered the influence of coupling on the shear stiffness modulus of fabrics, its effect on the wrinkling angle, the ultimate goal of forming modeling and simulation, has not been considered. Similarly, in the studies [4, 43] on wrinkling formation of fabrics, the dependency of wrinkling angle on the applied tension was not explicitly implemented in simulations. 2.4 Summary Chapter 2 critically reviewed the reported studies in the literature on the role of coupling in the mechanical behavior of woven fabrics. This chapter demonstrated that while the application of coupling to postpone wrinkling in woven fabrics is being established and employed in industries, its mechanics has not been intensely and clearly understood, let alone its underlying multi-scale sources and effects on the mechanical properties of fabrics. Furthermore, there are a number of conflicts observed between the results of conducted studies in the literature regarding the effect of coupling. Recognizing the above need in the current theory and practice of woven composites, the ultimate goal of this thesis in the subsequent chapters will be to fundamentally understand and characterise the influence of coupling on the mechanical behavior of woven fabrics, and in turn mathematically present them at macro-scale in the form of a new coupled material model. Such constitutive model which relates strains and 18  stresses of the material may then be employed in macro-level numerical simulation of woven fabrics, both during manufacturing and design.    19  Chapter 3: Re-theorizing the True Shear Behavior of Woven Fabrics under a Multi-scale Framework 3.1 Overview To study the coupling behaviour in woven fabrics, fundamental understanding of the shear behavior of this type of fabric reinforcements and the underlying multi-scale deformation mechanisms is required. This chapter attempts to bring an enhanced insight into the analysis of in-plane shear behavior of woven fabrics. Two common methods have been used in the literature to characterize the shear behavior of woven preforms, namely the Bias Extension (BE) and Picture Frame (PF) tests. In spite of the identical macro-scale shear deformation of fabrics in these two characterization procedures, the current chapter demonstrates that their underlying micro and meso-scale deformation mechanisms are quite distinct. Trellising mechanism, which is based on the well-known Pin-Joint Theory (PJT), has been regarded for a long time as the sole model to describe the meso-scale shear deformation of woven fabrics in both the BE and PF tests. Through this chapter, this mechanism is challenged for the PF test by undertaking a multi-scale analysis along with a critical review on the previous experimental, analytical and numerical studies. Intra-yarn shear, which has not been fully understood yet, is substantiated as a potential meso-level deformation mechanism in the PF test. Accordingly, a new meso-level deformation mode is proposed for PF tests and compared with the trellising shear pattern in the BE test. Afterward, the comparison is extended from meso-level to macro-level in order to provide more in-depth hypotheses for the differences reported in the literature between the shear characteristics of woven fabrics under BE and PF tests without firm explanations. To illustrate, it is hypothesized that locking and wrinkling, the two critical points in the shear behavior of fabrics, arise later in PF test compared to BE test. This postulation is examined through an experimental-analytical evaluation. A good agreement between the analytical and experimental results is observed. Moreover, to determine the effect of coupling on the aforementioned critical points, the influence of tension is taken into account in the proposed analytical procedure. Eventually, a guideline will be recommended to select more reliable characterization method in the context of real forming processes and optimizing their boundary conditions. 20  3.2 Shear characterization of woven fabrics As addressed, in order to characterize the shear behavior of woven fabric preforms, the Bias Extension (BE) and Picture Frame (PF) tests are frequently employed in the literature, each of which has its advantages and disadvantages. Figure 3.1 illustrates the general set-ups of the BE and PF tests. In addition to offering a simpler set-up, the BE test is known to be more analogous to real stamping process of fabrics with respect to the possibility of inducing both wrinkling and translational slippage defects, depending on the chosen dimensions of samples [43]. On the other hand, this test suffers from a heterogeneous deformation field within the sample, whereas the PF test is able to offer more uniform deformation fields without translational slippage of yarns at crossovers; hence simplifying the test characterization. As a disadvantage, the sample installation and fixture boundary effect in the PF test can cause very large deviations from true and repetitive characterization results. A few studies compared the shear force-shear angle response of woven fabrics under BE and PF tests, and the observed difference was attributed to unintentionally induced tension due to the clamped boundary condition in the PF test [17, 44, 45]. After altering the boundary condition to prevent such tension on yarns, still higher shear forces in the PF test were observed compared to the BE test [17]. Moreover, considerable discrepancies between the BE and PF tests in terms of the locking and wrinkling angels were noted [46, 47].  However, no further in-detail multi-scale analysis has been undertaken to unveil the facts behind such observations. This study hypothesizes that these macro-scale differences stem from an inherent sub-scale distinction between the BE and PF tests. After identifying and discussing the underlying roots, an ideal shear characterization technique based on real forming processes can be recommended.  21   (a) (b) Figure 3.1 Two different test set-ups commonly used for the shear characterization of woven fabrics: (a) Bias extension/BE testing, (b) Picture frame/PF testing; in both tests the yarns are initially oriented at ±45o with respected to the vertical loading direction; In the BE test, three different deformation regions are formed, whereas in the PF test, the deformation in the mid-region is uniform 3.3 Understanding the general shear behavior of woven fabrics The typical shear behavior of woven fabrics was discussed and illustrated in Figure 2.3 of Chapter 2, and four phases of shear deformation were identified. In order to explore the root of shear force trend over these four phases, possible contributing meso-scale deformation mechanisms during should be first recalled. There are several such mechanisms reported:  shear within yarns (known as intra-yarn shear) [48-50],   yarns rotation (known as trellising shear) [43, 50-56],   translational slippage at crossovers [57]   lateral (in-plane) yarn compression [43, 56].  Figure 3.2 shows a preliminary comparison between the three first mentioned mechanisms through illustration of their influence on the shear deformation of representative fabric element. The figure suggests the fact that the shear response of woven fabrics can be probably variant based on the induced shear deformation mode (e.g., BE versus PF), implying the importance of digging more into the meso-scale behaviour of such materials. Because lateral compaction arises after locking initiation, its effect has not been demonstrated in Figure 3.2. This effect will be comprehensively discussed in section 3.3.2 after achieving fundamental understanding about the pre-locking behavior of woven fabrics. It is worth mentioning that the 22  occurrence of each mechanism at any local point of a deformed fabric significantly depends on the shape of the preform, forming boundary conditions, etc. For instance, the translational slippage at crossovers is a common phenomenon seen in inflatable woven fabric structures [57]. The existence of a deformation mode, however, does not necessarily imply that this type of deformation makes a considerable contribution into the shear resistance of a given fabric. To elucidate this point, for instance, the energy dissipated through rotational friction at crossovers was found to be substantially less than that of the yarn lateral compaction [10], inferring that computational time of the wrinkling analysis will be probably sacrificed by taking each and every mechanism into account.   (a) (b) (c) Figure 3.2 Different types of shear deformation modes in a plain woven fabric. (a)  Pure shear deformation (intra-yarn shear); (b) Trellising shear deformation (rigid rotation of yarns based on pin joint theory); (c) translational sliding of yarns at crossovers 3.3.1 Pre-locking shear behavior of fabrics 3.3.1.1 Trellising shear In general, from relative force magnitude point of view, the shear response of woven fabrics can be divided into two main regions: below and above the locking angle, in that the shear force after reaching locking angle is significantly higher as compared to that of before locking. However, considering the initially stiff slope of the force response (Figure 2.3) over a small range of shear angle (typically 0 to 5 degrees) helps to acquire an in-depth understanding of the deformation mechanisms participating in the pre-locking region. This linear stiff behavior at initial degrees of fabric shear has been observed in quite a few experimental studies [2, 7, 54]. In the extent literature, the common justification of this initial stiff response followed 23  by a more compliant behavior prior to locking (Figure 2.3) is reported to be the static and subsequently dynamic rotational friction at crossovers. This explanation is based on the fact that the shear resistance of fabrics before locking has been attributed to the rotational friction resistance at crossovers, resulting from the most obvious deformation mechanism during fabric forming—yarns rotation (change in angle between yarns) as shown in Figure 3.2 (b). The yarns rotation has been described predominantly based on the Pin-Joint Theory (PJT) introduced by Mack and Taylor [58]. According to this kinematic model, yarns are analogous to trusses joined to each other at crossovers to create a trellis structure, as shown in Figure 3.3. Shear deformation of fabrics, which means change in the angle between yarns, can be accommodated by the displacement of pins A, E, F and G in the presented directions in Figure 3.3 (a). In fact, the displacement of pins leads to rotation of yarns with respect to each other about the interlacing pins. For instance, yarn 1 experiences relative rigid rotation to yarn 2 about pin A. However, the actual center of rotation of each yarn is indicated in Figure 3.3 (a) as Ci. Using the concept of dynamics of mechanisms, the center of rotation for each yarn can be identified through a geometrical approach by drawing perpendicular lines to the velocity vectors of two pins of each yarn. In order to evidently explain the reason of introduced relative displacement at crossovers, Figure 3.3(b) is presented. This figure depicts the different velocities of the points at the same location but within two distinct yarns. To illustrate, Points ܤଵ and ܤଶ which belong to the yarns 1 and 2, respectively, have the same velocity magnitude, but in different directions. On the other hand, the direction of the velocity vector of Points ܦଵ and ܦଶ is identical, but the magnitude is different because of different distances from the corresponding centers of rotation. The exclusive location at which two points from two distinct yarns have the same velocity is the pin’s location, implying that points ܣଵ and ܣଶ remain on each other over shear deformation. Figure 3.4(b) illustrates the resultant relative displacement between yarns 1 and 2 at crossover A. This relative displacement dissipates some energy through friction at crossovers, which is regarded as the source of shear resistance of the fabric. In the initial shear angle rage (e.g. 0 to 5 degree), there is static friction between the warp and weft yarns, inferring that there is no relative displacement at crossovers, and yarns are almost joined, creating a stiff fibrous network and causing the initially high slope of the shear force response. On the other hand, once the external shear force, which induces an external torque at the yarns contact areas, overcomes the resultant torque from the static fiction, relative rotational 24  movement of yarns begins, leading to a more compliant fibrous network and eventually the second phase of shear response curve.  (a) (b)  Figure 3.3 Kinematics of trellising shear deformation: (a) Rotation of yarns about their respective center of rotations;  (b) different velocities of the points at the same location from two distinct yarns   (a) (b) 25    (c) (d) Figure 3.4 In-detail meso-level comparisons between different deformation mechanisms subjected to the same extent of shear angle, γ. (a) Before deformation; (b) Trellising shear deformation (only inter-yarn shear i.e. relative rigid rotation of yarns and hence, ࢽ ൌ ࢽ࢘; (c) “Pure” shear deformation (intra-yarn shear, ࢽ ൌ ࢽ࢟ሻ; (d) Combination of the pure and trellising shear modes (ࢽ ൌ ࢽ࢟ ൅ ࢽ࢘ሻ. Notice that bending is not shown in these images to simplify the visual comparisons The implementation of such possible presumption about the first and second regimes in the shear analysis of fabrics is found diverse. In [54, 55], a simplified approach was adopted in which a non-zero term, as the representative of the initial friction, was considered. Subsequently, the force response curve did not explicitly demonstrate the first high-stiffness region. Instead, it started from a non-zero magnitude of shear force at zero angle. Grosberg et al. presented a more sophisticated modeling approach to capture the initial stiff behavior of woven fabrics [51], which was subsequently utilized in two follow-up studies [52, 53]. The assumption in these studies was that during the first phase of shear deformation, the yarns at joints are virtually welded, and hence the woven architecture is similar to a trellis structure until the applied shear force surpasses a critical value and initiates the second phase, in which rotational slippage can take place at crossovers. Hence, during the first stage of shear loading, the yarns in a representative fabric unit cell were modeled as cantilever beams subjected to the external force, as illustrated in Figure 3.5 (c). The angle of each beam (yarn) at its tip, also 26  regarded as the fabric shear angle, was derived from Castigliano Beam Theory based on the stored bending and torsional energies. Simply put, in the first stage because the shear force is not high enough to overcome the static friction at crossovers, there is no relative rotational movement at crossovers, and yarns have to bend to accommodate the imposed shear deformation. According to Figure 3.5(c), the resultant bending moment is about Y-direction. Once the shear force reaches the critical value, the relative rotation at crossovers is initiated, and the bending deformation stops. Consequently, it should be emphasized that Figure 3.4(b) does not completely demonstrate the actual meso-level “trellising shear” deformation of woven fabrics since the bending has been ignored. Instead, Figure 3.6 (a, b, and c) is presented to clarify different deformation mechanisms involved in the first and second stages of trellising shear of woven fabrics. Notice that the total shear angle of fabrics before locking is the summation of the angle resulting from first-stage bending and the subsequent change in the angle between yarns due to the rotation. However, in reality, the bending deformation contribution in shear angle is not sizable (when compared to the second region); hence Figure 3.6 is exaggerated for illustration purposes. In fact, Figure 3.5(c) indicates high magnitude of ܫ௬௬ that causes the bending deformation not to be substantial.   (a) (b) (c) Figure 3.5 Schematic of a unit cell of a typical plain weave (TWINTEX® TPP60N22P-060) modeled in TexGen. (a) Top view; (b) 3D presentation of a single yarn in the unit cell subject to in-plane shear force; (c) a simplified structural beam representation. Notice that the in-plane shear force is along with the width of yarns, which is generally not substantially less than the length of a yarn between two crossovers and hence, should not be considered under a long beam theory  27    (a) (b)   (c) (d) Figure 3.6 (a) Yarns before deformation; (b) Deformed yarns in the first phase of response under trellising shear mode and considering bending (note that the lines perpendicular to the longitudinal axis remain at 90o); (c) Deformed yarns in the second phase of trellising response where yarns now undergo relative rotation with respect to each other, while bending stops; (d) The actual meso-level deformation of the yarns under a typical picture frame test (showing combination of trellising and pure shear deformation modes). No fiber misalignment was assumed in these images 3.3.1.2 Pure shear on account of intra-yarn shear One missing point in the above analyses has been ignoring the transverse shear energy through intra-yarn shear deformation. This type of deformation has been experimentally and numerically substantiated more recently, and it appears to be an effective contributing mechanism in the shear deformation of fabrics [48-50]. In PF tests, the intra-yarn shear is highly possible to arise, compared to the BE tests, because this phenomenon originates from the boundary condition at yarn ends. The effect of boundary condition to create intra-yarn shear was first introduced numerically by Lin [49]. In the preliminary numerical simulations where yarns rotation was merely taken into account [18, 49, 59], the first stage of shear response was not captured, which probably challenges the validity of trellising mode as the true shear behavior of woven fabrics. The modified numerical simulation [49], however, indicated higher accuracy of the finite element modeling by taking account of intra-yarn shear (Figure 3.7). In order to incorporate this effect, an adjusted boundary condition, which would be more comparable to that of the picture frame tests, was implemented [49]. In fact, a geometrical 28  explanation - continuity at boundaries of the fabric sample subjected to a picture frame mode – followed by its implementation in the finite element modeling to lead more precise results has been presented as the evidence of intra-yarn shear existence. Likewise, another geometrical reasoning with different perspective is provided in chapter 4 of the current thesis, while transforming the global fabric forces to normalized local forces under combined tension-shear loading modes. In summary, looking at the edges of yarns in Figure 3.4 (b) does not meet the true boundary condition of PF tests. Based on the latter work, without considering the intra-yarn shear, a correct characterization of the woven fabric cannot be achieved.   Figure 3.7 Comparison between experimental data and numerical results with two distinct boundary conditions on yarns [49], implying the significance of intra-yarn shear Regarding experimental studies, the performed micro-level optical measurements of strain field within yarns of a plain-woven fabric under picture frame test [50] revealed the existence of shear deformation within the yarns. Additionally, Figure 3.8 (adapted from [50]) brought a new point into understanding these materials; that is, the shear angle within yarns for all tested fabric types was less than that of the picture frame (or global angle between yarns). A firm reason for such observation will be offered in a later section, by further digging into the interaction between intra-yarn shear and other mechanisms at meso-level shear deformation of woven fabrics. Another experimental investigation implicitly highlighted the occurrence of intra-yarn shear mechanism [56], by reporting the width of yarns over shear deformation. If only the trellising shear happens in PF tests, the width of yarns should remain constant before reaching the locking point. However, results in [56] informed that the width of yarns is in fact reduced during shear deformation even before reaching locking angle. Furthermore, the experimental change in yarns’ width before locking followed approximately a theoretical 29  cosine formula, which is actually derived by taking intra-yarn shear into account. As a matter of fact, when a yarn is sheared, its effective width declines as shown in Figure 3.4(c). Thus, the instantaneous width of the yarn at a given shear angle before locking can be calculated as: ݓ ൌ ݓ଴ cos ߛ (3.1) Where ݓ଴	corresponds to the initial width of yarns, while ݓ represents the instantaneous yarn width at a pre-locking shear angle, ߛ. Thus, it can be said that the change in the width of yarns based on this equation is theoretically a direct consequence of shear angle extent within yarns. In [56], the local bending of yarns due to the imposed PF boundary conditions was regarded as the reason of the deviation between the experimentally measured and theoretical values of yarns width, as demonstrated in Figure 3.9. Relying on this point, the reason of observing smaller shear angles within yarns in comparison with that of frame angle in [50] (Figure 3.8) can also be hypothesized. On account of the local bending close to jaws, the actual width of yarns is greater than the theoretical value, as shown in Figure 3.9, causing smaller shear angle within yarns according to Eq. 3.1. One point which substantiates the presented hypothesis is that the intra-yarn shear angle within a stiffer fabric, e.g. RR3 in Figure 3.8, is closer to that of the global angle, which can be interpreted in a way that the stiffer the fabric, the less the amount of bending deformation, the less the deviation from theoretical width decrease, and eventually the less the difference between meso and macro shear angles. This also implies that in an ideal picture frame test, no local bending of yarns close to jaws should be induced, which in turn necessitates some novel boundary conditions to put into practice in design of future PF fixtures. A combination of needles and jaws is recommended in the next chapter as a first-step toward this solution. 30   Figure 3.8 Comparison between intra-yarn shear deformation and the applied frame shear angle [50]. RR1, RR2, and RR3 are three different fabrics, the stiffest of which is RR3; A and B represent the shear angle within yarns spacing and the yarn itself  (a) (b) Figure 3.9 (a) Local bending of yarns close to the clamped boundary conditions in a typical picture frame test, which causes a deviation between the global shear angle and that within the yarn; (b) also its significance on the deviation of actual yarns width reduction from the theoretical formula [56]; ࢝૙, ࢝, and ࢝′ are the initial width of yarns, instantaneous width in ideal PF test, and actual yarns width in typical PF test, respectively. Also, the ideal slip distance between AD and BC is denoted by ∆, whereas the actual one is presented by ∆′ As far as theoretical evidence for the existence of shear deformation within yarns, although brief explanations using fabric deformation continuity at fixture boundaries were provided earlier in this section, a more detailed theoretical explanation from other mechanistic points of view is offered in the following section.  3.3.1.3 A theoretical representation of yarns based on high-order beam theory As stated earlier, the yarns were regarded in [51] as arc cantilever beams in which intra-yarn shear has not been accounted, meaning that the cross section at any point of the beam is perpendicular to the longitudinal axis. As a result, only bending and torsion energies were 31  calculated in the analytical derivation of the shear force-shear angle curve. Considering a beam subjected to a force along its thickness direction, it is common to overlook the contribution of transverse shear energy in the total strain energy of the beam, providing its length to thickness ratio is greater than 16 (ASTM D7264). However, the ignorance of the transverse shear energy, in the yarns of woven fabric structures subjected to in-plane shear loading is highly questionable, in that the ratio of the length to the thickness of an equivalent beam representing a typical yarn unit cell is only slightly greater than 2. To clarify, Figure 3.5(a) demonstrates a typical woven fabric with commercial name of TWINTEX® TPP60N22P-060 modeled in TexGen. According to this figure, the length of the yarn is the length of the modelled cantilever beam while the thickness of the cantilever beam is the width of yarn, rather than the thickness of yarn, because the in-plane shear force is along with the width of yarn.  Consequently, for related meso-level deformation analysis purposes, the ratio of length to the thickness of the proposed equivalent beam is the ratio of the length to the width of yarn in a representative fabric element, whose value is nearly 2 given the fact that length of a yarn in a unit cell of a balanced woven fabric is around two times of the width of yarn plus the yarns spacing as shown in Figure 3.5(a). Hence, the presumption of bending deformation in the first stage of shear deformation based on trellising shear is not fairly reasonable. Furthermore, even another assumption used in trellising model reduces the effective length of the equivalent beam, violating the presumption of only bending deformation in the initial stiff response. To clarify, in accordance with the trellising mechanism, there is a static friction between yarns at crossovers in the first phase of shear behavior, leading to presumed clamped boundary condition in contact areas at crossovers. As a result, the part of yarn that undergoes bending at this stage is the yarns spacing section region between lines BB and DD depicted in Figure 3.5(a), which is substantially less than the total length of yarn in the representative element. Therefore, the effective ratio of the length to thickness in the equivalent beam is even less than 2 in the first stage of shear response. Accordingly, instead of Euler-Bernoulli beam theory, the High-Order beam theory or the First-Order Beam Theory (Timoshenko Beam Theory) should be employed. Under such theories, the stored shear energy in an arc beam shown in Figure 3.5(c) should be added to the bending and torsional deformation energies as follows: 32  ܷ ൌ 	න ܯ௬ଶ2ܧܫ௬௬ ݀ݔ ൅	න௫ܶଶܩܬ ݀ݔ ൅ න௭ܸሺݔሻଶ2ܩܣ/ߦ ݀ݔ (3.2) Where E, I, G, J, A, and ߦ represent the Young’s modulus, bending moment of inertia, shear modulus, polar moment of inertia, cross-section area, and cross-section shape coefficient of the equivalent beam, respectively. Also, the resultant bending moment, torsional torque, and shear force at each point of the beam are regarded as ܯ௬, ௫ܶ, and ௭ܸ, respectively. It should be noted that based on a given fabrics geometry, mechanical properties of yarns, and friction between yarns which affects the strictness of boundary conditions at crossovers, the contribution percentage of the energy terms in Eq. (3.2) can be varied. 3.3.1.4 A continuum mechanics based evidence for presence of intra-yarn shear: Effect of filament gaps Another theoretical argument to substantiate the existence of intra-yarn shear in fabric shearing, herein namely the PF test, can be offered in regards to kinemics of such tests from a continuum mechanics viewpoint. Milani et al. using a continuum mechanics based model proposed an explicit predictive equation for the shear response of woven fabrics under fiber misalignment [60]. If X and ݔ vectors denote the undeformed and deformed vector of a material point within the fabric subjected to PF testing (Figure 3.10), the relationship between components of these vectors under a fully uniform PF test without local bending close to grips—ideal PF test—can be attained as: ൞ݔଵ ൌ 	 ଵܺ cos ߛ2 ൅ ܺଶ sinߛ2ݔଶ ൌ 	 ଵܺ sin ߛ2 ൅ ܺଶ cosߛ2(3.3) Thereafter, the deformation gradient tensor can be calculated as: 33  ܨ ൌ 	 ቈ߲ݔ௜߲ ௝ܺ቉ ൌ 	ۏێێێێۍcos ߛ2 sinߛ2 0sin ߛ2 cosߛ2 00 0 ߲ݔଷ߲ܺଷےۑۑۑۑې (3.4) (a) Before deformation (b) After deformation Figure 3.10 Coordinate systems to demonstrate the undeformed and deformed shape of a fabric sample subject to an ideal picture frame test Given the incompressibility assumption, det[F]=1, it leads to determination of the unknown component డ௫యడ௑య as ଵୡ୭ୱఊ. Existence of gaps between filaments within yarns, however, challenges the incompressible behavior assumption for woven fabrics. Comparison between the above presented expression for change in thickness ሺ	 ଵୡ୭ୱఊሻ with the experimental results observed in [56] suggests that yarns are slightly thickened during shear deformation, but not as significantly as the proposed formula by the continuum mechanics based model in [60]. Accordingly, ܿ௚ a correction factor representing the impact of gaps between filaments within a dry fabric yarn on the incompressibility assumption is introduced here. Subsequently, the right Cauchy–Green tensor and the Green–Lagrange strain tensor, known as C and E, respectively, can be calculated as: 34  ܥ ൌ ሾܨሿ்ሾܨሿ ൌ ቎1 sin ߛ 0sin ߛ 1 00 0 ሺܿ௚ sec ߛሻଶ቏ (3.5) ܧ ൌ 12 ሾܥ െ ܫሿ ൌ12 ቎0 sin ߛ 0sin ߛ 0 00 0 ሺܿ௚ sec ߛሻଶ െ 1቏ (Eq. 6) Eq. 3.6 clearly reveals that the fabric sample subjected to uniform picture frame loading must experience pure shear deformation at each and every single point. To elucidate more, thanks to arbitrarily chosen point P, it can be located within yarns spacing regions as well as within yarns. Hence, shear deformation field is created not only between yarns, but also inside the yarns. 3.3.1.5 Contributing mechanisms via intra-yarn shear deformation Thus far, experimental, numerical and theoretical evidence were discussed in support of the critical role of intra-yarn shear in the shear response of fabrics. Taking this meso-level deformation mode into account, however, brings us to potential sources of inconsistencies with respect to previous works in this area, which are attempted to review and address in this section. The first point of discussion is to propose a new meso-level deformation model based on the pure shear deformation field formulated in section 3.3.1.4. Figure 3.4(c) displays this proposed “pure shear” model. In addition to the comparison between Figure 3.3(a) and (b) which clarifies the role of intra-yarn shear in meso-level deformation of yarns, the comparison between Figs. 3.4 (b) and (c), which illustrate a close-up of the crossovers of yarns with and without intra-yarn shear, discloses more significance of this deformation mechanism. Namely, when the yarns undergo pure shear (Figure 3.4(c)), each pair of initially overlapping points from two distinct (interlacing warp/weft) yarns follow the same movement (Eq. 3.3). Thus, it can be inferred that there is no relative displacement between yarns at crossovers, disputing the idea of static and dynamic rotational friction as the main reason for the change between the force response slopes in Phase 1 and 2 (Figure 2.3). In other words, in spite of the existence of friction between yarns, under pure shear, there would be no relative displacement that can effectively dissipate energy at meso-level. This fact raises this question that, “What is then the 35  actual source of difference between Phases 1 and 2 and how does the friction effect connect to it?”. The answer to this question can be found by moving the analysis to the next material level: micro-level (i.e., deformation mechanisms of filaments in the yarn). A single yarn is a bundle of thousands of filaments which are in contact with each other (Figure 3.11(a)). Assume pure shear is imposed on a short single yarn, the trend of the obtained shear force-shear angle curve should be comparable with that of a woven fabric before locking, i.e. embracing an initially stiff behavior phase followed by a more compliant behavior. This trend can be justified given the fact that the pure shear behavior of a single yarn (intra-yarn shear) can be divided into intra-filament shear mechanism followed by inter-filament slippage, as depicted in Figure 3.11(b). In fact, in the first step of shear loading, there is static friction between filaments, meaning that filaments are virtually joined to each other, and they are sheared together. In that phase, the shear angle of the single yarn is the same as that inside the filaments. By gradually increasing the external shear force, the shear force reaches the level of the static friction force between filaments in each yarn, which then initiates the slippage of filaments over one another to accommodate the applied shear deformation. In this regime, the shear angle of the yarn increases through filaments slippage (relative movement), whereas the shear angle within filaments remains constant, as illustrated in Figure 3.11(b). As a result of this shift in micro-level deformation mechanism, the yarn (meso-level) shear rigidity drops from a higher value in the first stage to a lower value in the second stage. To consider the influence of such micro-level mechanism on meso-level numerical simulations, the shear modulus of yarns should be a function of the shear angle, instead of a constant value in order to obtain more precise results. In fact, recalling Figure 3.7 from the study [49], suggests a clue to substantiate this variable shear modulus of yarns. This figure reveals that by implementing the adjusted boundary condition, the first phase of force response can be captured. The predicted first phase reported in [49], however, is from 0o to 20o, which is not in agreement with actual tests where the range of this phase is about 5o. In order to resolve this disagreement, either a coupled micro-meso numerical simulation (in which filaments are modeled separately) is recommended, or an equivalent meso-level model can be used in which the shear modulus of yarns is dependent on the forming shear angle. It should be mentioned that some fabrics are also comprised of twisted 36  yarns which make the micro-scale geometry of the yarn more complicated, yet the static and dynamic frictions would play a major role.                          (a)  (b) Figure 3.11 (a) multi-scale nature of woven materials [18]; (b) the micro-scale mechanisms contributing to meso-level intra-yarn shear phenomenon: i) intra-filament shear (i.e., there is static friction between filaments and the external shear force has not reached the critical force value, ii) inter-filament slippage (the imposed shear force on the yarn is now higher than the static friction limit, and a kinematic friction state is active) 3.3.1.6 A hierarchical definition of ‘pure’ shear mode  Summarizing the above analyses and discussions, we hypothesize that in the first and second stages of shear response (i.e., before locking) under an ideal “pure” picture frame test of woven fabric, at the micro level, inter-filament shear (due to the existence of static friction) and intra-filament slippage (due to change of friction to kinematic mode) take place, respectively. This, in turn, translates to the intra-yarn shear with a high and low modulus level, respectively, for meso-level analyses. For phenomenological-based macro-models (e.g. [4, 16]), this translates to high and low ply shear modulus, respectively (Figure 2.3). Although 37  micro/meso-scale deformation effects (and potential forming defects raising from them) cannot be fully captured in the macro-models, they are proven to be very useful for fast and large scale forming simulations [60, 61]. It should be added that under the above contributing deformation mechanisms, yarn bending still can occur in the first phase of force response, in that theoretically, there is no relative displacement between yarns at crossovers, mimicking a clamped boundary condition of the yarn at crossovers under loading. However, given the dimensions of the short beam clamped between two unit-cell crossovers, the contribution of bending and torsional energies in Eq. 3.2 would be negligible compared to that of transverse shear, also depending on the level of static friction between yarns as well as the given fabric crimp angle. In addition, during the second stage, because the filaments of yarns start moving, their contact at crossovers would not be as strict as in the first stage. As a result, the yarn bending would stop gradually when the second phase begins. It should be noted that the latter hypothesis is only valid when there is no major initial misalignment in the fabric. Otherwise other mechanisms such as slippage at crossovers (which again will depend on factors such as the friction coefficient), local buckling due to excessive length, etc can occur through deformation in phases 2, and they would continue to grow gradually. The effective friction coefficient itself would depend on the material and surface condition of filaments, as well as presence or absence of twist among the yarns, along with other factors such as processing temperature. 3.3.1.7 Understanding new differences between the bias extension and picture frame tests (before locking initiation) In order to precisely characterise the shear behavior of woven fabrics, it is vital to recognize which test condition/set-up would lead to the occurrence of which deformation mechanisms. It is our belief that in the bias extension tests in which most of the yarns are free either at their both ends or at least on one end, there is a slight possibility of intra-yarn shear deformation. That is to say, the trellising shear illustrated in Figure 3.4(b) takes place in bias extension tests, and energy is dissipated mainly through rotational friction at crossovers. Also, another known deformation in bias extension tests is the slippage (separation) of crossover centers as depicted in Figure 3.2(c). In order to further examine the validity of this hypothesis, a bias extension experiment was carried out where a simple inspection method was employed 38  through the new understanding of shear deformation at multi-scale. Case in point, if intra-yarn shear happens, a line that is initially perpendicular to the longitudinal axis of the yarn should not stay perpendicular over shear deformation. Accordingly, the sample was marked to monitor those lines perpendicular to the longitudinal axis of yarns. A close-up photos taken after deformation, Figure 3.12, divulges that the lines remain at their original alignment –– which indicates the absence of intra-yarn shear in BE tests.  Figure 3.12 Perpendicular lines to the axis of yarns remain at 90o over shear deformation under a bias extension test, implying the trellising shear mode in such test On the other hand, in an ideal picture frame test, which possesses more advanced (controlled) setup at the sample boundaries, “pure” shear deformation is more prone to exist. In such a ‘hypothetical’ test, the shear angle within yarns must be ‘exactly’ the same as the frame shear angle, as depicted in Figure 3.4(c). The extent of shear angle within yarns in actual tests, however, is usually less than the shear angle of fabric (the angle between yarns) [50, 56], owing to the inevitable local non-uniformities within the fabric and/or jaws’ boundary conditions. Regarding the occurrence of different shear deformation mechanisms in BE and PF tests, some other proofs will be provided in section 3.3.2 for their effect on fabric locking and wrinkling onsets. The next natural question would be, “whether or not the distinct shear deformation modes in bias extension and picture frame tests would cause considerable difference between the achieved shear force-shear angle response of a given fabric?” It has been reported that the shear resistance of woven fabrics before locking under the picture frame test is higher than that attained from a bias extension test [17, 44, 45]. The provided explanations in these works 39  mainly included unintentionally induced tension in yarns resulting from the clamped boundary condition, considering the fact that applying tension on woven fabrics increases their shear rigidity [7, 17]. Some efforts were made in [17, 44, 45] to prevent tension from generating and developing within fabric samples. By doing so, the woven fabric specimens can experience exclusively shear deformation without tension, and hence hypothetically delivering more similar results to the bias extension test. However, the observed shear response in the latter modified picture frame tests was still higher. Such unavoidable difference in the response probably points to the necessity of fundamentally distinguishing underlying shear deformation modes in BE and PF tests. It should be noted that the executed picture frame tests in these earlier studies [17, 44, 45] were not the hypothetical (ideal) picture frame test.  Employing typical clamped boundary conditions in those tests caused the local bending of yarns close to grips and, subsequently, the lower shear angle within yarns. As a result, in order to accommodate the total applied shear deformation, a woven fabric under the typical picture frame test has to experience some extent of inter-yarn (trellising) shear, in addition to some intra-yarn shear. Table 3.1 and Figure 3.4 illuminate these points by drawing a comparison between the resultant meso-scale deformation mode in trellising shear, pure shear, and typical PF tests. For instance, considering 40o as the total applied shear angle (frame shear angle), the globally observed angle change between weft/warp yarns has to be ideally 40o providing the absence of misalignment. In trellising shear, each warp and weft yarn family rigidly rotates 20o to accommodate the total 40o, with zero shear within the yarns. On the other hand, in a pure shear deformation, both warp and weft undergo 40o shear within their yarns, with no relative rigid rotation at crossovers. In a typical picture frame test, assume that yarns are sheared 22o, instead of 40o owing to local bending close to grips. Thus, each warp and weft yarn groups has to rotate 9o, so as to create the total 40o shear. Contrasting Figure 3.4(d) with Figure 3.4(b) and (c) as well as the last two columns of Table 3.1 represents these differences in a symbolic manner. In summary, despite the same extent of macro-scale shear angle, three distinct meso-scale deformation modes can be possible as presented in Figure 3.4 (b, c, and d). It should be mentioned that bending has been removed in Figure 3.4 to more clearly compare the effect of pure shear and trellising shear models. However, Figure 3.6(d) has also been provided to visualize the actual meso-level shear deformation including bending. In summary: 40   A typical PF test (with clamps) would induce a deformation between a BE and an ideal pure shear mode.   Yarns subjected to PF tests can dissipate energy through both intra-yarn shear and inter-yarn shear, whereas energy is dissipated only through inter-yarn shear in bias extension tests.   However, dissipated energy through the above two mechanisms in PF tests does not necessarily imply higher total required energy, and hence the effective rigidity of the fabric in PF deformation may not be necessarily larger than that in BE deformation.  In typical picture frame tests, the deviation of results from BE test would depend on the contribution percentage of intra-yarn shear depending on the boundary condition applied through PF setup.  As an example, the total energy in a typical PF test with 40o shear at the fixture (global) level, would be the summation of the required energy for 22o intra-yarn shear and 18o inter-yarn shear; while in an ideal bias extension test (assuming no transverse slippage) the entire energy would arise from the 40o inter-yarn shear. Although both intra-yarn shear and inter-yarn shear absorb a slight amount of elastic energy through intra-filament shear and bending of yarns (the first phase of shear deformation), the most energy needed would be through friction coming from inter-filament slippage and relative displacement at crossovers (the second phase of shear). Note that the effective contact area between filaments within yarns are substantially more than that between yarns at crossovers. However, the friction coefficient between filaments should not be very different from that between filaments of two distinct yarns at crossovers. Consequently, the required energy to induce intra-yarn shear would be higher than inter-yarn shear. This conclusion is in agreement with Figure 3.7 obtained from the numerical simulation in [49] which took intra-yarn shear into account. This figure shows that the shear rigidity of woven fabric under pure shear deformation is higher than that under trellising shear.   41  Table 3.1. Comparison between different underlying deformation mechanisms in shear deformation of woven fabrics Mode First phase Second phase Intra-yarn shear angle Inter-yarn shear angle  Trellising shear (close to bias extension test) In-plane bending (elastic energy) Rotational friction (dissipative energy) 0 ߛ Pure shear (ideal picture frame test) Intra-filament shear and transverse bending (elastic energy) Inter-filament slippage (dissipative energy) ߛ 0 Combination of trellising and pure shear (typical picture frame tests using clamped boundary conditions) Intra-filament shear and transverse bending (elastic energy) Rotational friction and inter-filament slippage (dissipative energy) ߚߛ where 0 ൏ ߚ ൏ 1. ߚ is dependent on boundary conditions, fabric geometry, mechanical properties of yarns, and friction coefficient  ሺ1 െ ߚሻߛ  3.3.2 Post-locking shear behavior of woven fabrics In large shear deformation, woven fabrics experience the third and fourth phases (Figure 2.3), known as locking and wrinkling, respectively. As a matter of fact, over shear deformation from the early stages, the distance between yarns decreases gradually, and at some point, yarns of the same family (weft or warp) start to have side-to-side contact and apply a lateral force to each other, which initiates the ‘locking’ [34]. In the locking region, lateral compaction of yarns arises to accommodate the applied additional shear angle [56, 62], and this deformation mode proceeds to a certain limit known as the onset of wrinkling, which is regarded a major manufacturing defect [63-65]. To determine theoretical locking point along with the onset of wrinkling in the course of shear deformation, a considerable amount of research has been dedicated and will be briefly discussed below. Moreover, an analytical approach is present to predict the locking and wrinkling angles of plain woven fabrics. These points are critical in the shear behavior of woven fabrics; hence, to propose an accurate constitutive model which anticipates the stress-strain curves, these points should be identified. Furthermore, because the focus of this thesis is on the effect of coupling on the mechanical behavior of woven fabrics, the most known and practical of which is delay in wrinkling by 42  applying tension, the need in analytical expression to predict locking and wrinkling angles is highlighted. Practically, the analytical study can help to estimate the required level of tension in real forming processes to prevent wrinkling. On top of these points, in light of the noticed distinct meso-scale deformation modes of BE and PF tests, the analytical method to predict locking and wrinkling is adjusted to both meso-level deformation modes to introduce distinct equations for these two points for each BE and PF tests, separately. By doing so, section 3.3.2 and its sub-sections substantiates the asynchronous onset of locking and wrinkling in BE test with those in PF test. In practice, however, it has been widely presumed that the obtained locking and wrinkling angles from BE and PF tests should be the same, if the unintentionally induced tension in PF tests could be prevented. 3.3.2.1 Locking in woven fabrics Prodromou and Chen [66] are among pioneers who realized the importance of a model to predict the locking angle and the onset of wrinkling in plain woven fabrics. Prior to their research, there were no analytical means for computing the locking angle, and the only practical approach, at the time, was experimental methods [67, 68]. In the model introduced by Prodromou and Chen, locking angle was defined as the shear angle at which the distance between the yarns is zero, as shown in Figure 3.13 [66]. In order to develop their model, they modified the pin-joint theory, which is merely based on the kinematics of the fabric structure, and included fabric construction parameters such as yarn spacing and yarn size in the calculations. However, intra-yarn shear was not taken into account in their model. Also, wrinkling initiation in their model was assumed to be coincident with the locking angle, while in practice (mechanically) the fabric structure can undergo further shear deformation after locking region through lateral compaction of yarns. Hence, it provides more room for additional shear deformation and delaying the onset of wrinkling. In light of this fact, it may be argued that Prodromou and Chen’s model does not represent precisely the behavior of the woven fabrics after locking. Furthermore, ignoring the material properties of yarns in the PJT models dismisses their versatility when different types of fiber material and yarn architecture are used [56].  43    (a) (b) Figure 3.13 a) Proposed unit cell model for plain woven fabrics based on the pin joint theory in order to find the fabric locking angle; (b) Demonstration of the geometrical relationship between width and length of yarns 3.3.2.2 Prediction of locking in woven fabrics According to Figure 3.13, the first expression of locking in terms of fabric geometries was offered based on the Pin-Joint Theory, as presented in Eq. 3.7 [36]. The constant length between the shown pins in Figure 3.13(a) – the crossover center of yarns in a representative unit cell - during shear loading, the yarns width and the shear angle at locking are denoted by ݈, ݓ଴ and ߛ௟௢௖௞௜௡௚, respectively. It should be mentioned that the width of yarns in this model was considered as constant. ݓ଴ ൌ ݈ cos ߛ௟௢௖௞௜௡௚ (3.7) The validity of this equation can be assessed by comparing the predicted values with the experimental results of a tested fabric in the literature with known fabric geometry. The study presented in [56] was opted for this comparison as all the geometrical properties of the tested fabric were given in [56]. The predicted value using Eq. 3.7 for the given fabric dimensions in [56] is found to be 29.76o, whereas the experimental locking angle has been reported to be nearly 50o. Such considerable deviation shows the inaccuracy of the Pin Joint Theory (PJT) for PF tests. Accounting for intra-yarn shear alters the deformation kinematics of PJT as illustrated in Figure 3.14 and, in turn, it changes the left side of Eq. 3.7 in that the width of yarns does not remain constant over shear deformation and is gradually reduced. 44  Figure 3.14 clearly indicates that considering intra-yarn shear postpones the locking point and consequently the wrinkling. Thus, in order to modify the left side of Eq. 3.7, the relationship between instantaneous width and shear angle in a “pure” shear deformation (Eq. 3.1) is utilized, leading to: ݓ଴ cos ߛ௟௢௖௞௜௡௚ ൌ ݈ cos ߛ௟௢௖௞௜௡௚ (3.8)  (a) (b) Figure 3.14 Delay in locking initiation under pure shear deformation in comparison with trellising shear; (a) The representative fabric element before deformation; (b) Two parallel yarns of the deformed unit cell (black and red yarns present trellising and pure shear modes, respectively, as in a BE versus as ideal PF test). The red lines will reach other sooner during fabric shearing As can be concluded from Eq. 3.8, there is no solution for this equation unless ݓ଴ and ݈ are equal, geometrically meaning that there is no yarns spacing in the given fabric, which is not practical. In other words, it seems that locking is not geometrically possible to predict in 2D under “pure” shear deformation, which is in conflict with reality. The key point resolving this dispute is that the critical distance between two parallel yarns at which locking begins was considered zero in Prodromou and Chen model [66]. Their erroneous assumption stems from looking at a unit cell in a 2D schematic from top view. Instead, at locking, a 3D unit cell model similar to Figure 3.15 can be assumed where each yarn of a given family is sandwiched between two adjacent yarns of the other family. This in turn implies that the critical distance between two parallel yarns to initiate locking is the ‘effective’ thickness of the other family of yarn, rather than zero. Mathematically, this point adds one more term to Eq. 3.8, leading to Eqs. 3.9 and 3.10 for the bias extension and picture frame modes, respectively. Note that in the bias extension mode, per discussion in the previous sections, it is considered that the intra-yarn shear is absent; hence there is no change in yarn width. Regarding the use of effective 45  thickness, it should be mentioned that the thickness of yarns increases over shear deformation as explained in section 3.3.1.4. However, the extent of growth in thickness is not as high as theoretical prediction due to the existing gaps in woven fabrics. Moreover, Figure 3.15(b) illustrates that the distance has a sinusoidal relationship with the instantaneous thickness of yarns at each shear angle, which is greater than the initial thickness. As a result, the critical distance is presented as ݂ሺݐ଴ሻ in Eqs. 3.9 and 3.10. In accordance with the provided explanations, initial thickness can be regarded as a simple, but fairly accurate approximation of  ݂ሺݐ଴ሻ, resulting in Eqs. 3.11 and 3.12, the simplified form of Eqs. 3.9 and 3.10. The precision of this estimation is examined in the following sections of this thesis.  ݂ሺݐ଴ሻ ൌ ݈ cos ߛ௟௢௖௞௜௡௚ െ ݓ଴ (3.9) (locking angle for a plain woven fabric under bias extension test) ݂ሺݐ଴ሻ ൌ ሺ݈ െ ݓ଴ሻ cos ߛ௟௢௖௞௜௡௚ (3.10) (locking angle for a plain woven fabric under for ideal picture frame test) ݐ଴ ൌ ݈ cos ߛ௟௢௖௞௜௡௚ െ ݓ଴ (3.11) (locking angle for a plain woven fabric under for bias extension test) ݐ଴ ൌ ሺ݈ െ ݓ଴ሻ cos ߛ௟௢௖௞௜௡௚ (3.12) (locking angle for a plain woven fabric under for ideal picture frame test)   (a) (b) Figure 3.15 a) Numerical simulation showing that a yarn from a given family of yarns is sandwiched between two yarns of the other family, at the locking point; b) Demonstration of the critical distance and the instantaneous thickness at locking point As to the influence of coupling on locking angle, applying tension may slightly change the locking angle, given that the width of yarns is reduced by applying tension. Hence, according to Eqs. 3.9-3.12, the locking angle should be increased. The extent of growth, however, would not be considerable in that the Poisson’s ratio of a single yarn is not high due 46  to the discrete nature of yarns made of thousands of filaments. Hence, in Chapter 5 where the constitutive model is proposed the locking angle is assumed independent of coupling, while it is found that wrinkling affected by coupling – the level of applied tension.   3.3.2.3 Wrinkling in woven fabrics Wrinkling is the last stage of large shear behavior of woven fabrics (Figure 2.3). When locking arises, the adjacent yarns begin to press each other laterally. Further shear deformation of plain woven fabrics in this stage is dominantly by yarns lateral compaction. By growing the shear deformation, the pressure between adjacent parallel yarns rises, triggering the formation of wrinkles. The first attempt to anticipate the wrinkling onset was undertook by Prodromou and Chen [66]. In that work, as mentioned in the previous sections, it was assumed that locking and wrinkling onsets are coincident, which is not in agreement with reality. Lightfoot et al. investigated in-plane waviness and out-of-plane tow wrinkles of the composite preforms in the course of draping process [69]. Through their study, they elucidated the mechanism behind the formation of these two defects. In a research conducted by Launay et al. [17], the effect of tension on yarns was experimentally studied. They demonstrated that the onset of wrinkling is delayed by increasing the tension on yarns. In spite of providing reasonable experimental explanations about the wrinkle formation and affecting parameters such as yarn tension, theoretical interpretation was not presented [17, 69]. An energy-based model to predict the wrinkle initiation in plain woven fabrics has been proposed by Zhu [43]. The study developed an analytical criterion based on the assumption that wrinkles initiate whenever the energy required for in-plane shear deformation mode becomes greater than that of the out-of-plane deformation mode. This assumption could be challenged given the fact that the comparison was made between two terms of energy that are both internal, and their total always equals the external work done by shear forces. Regarding the difference between the obtained wrinkling angle of a given fabric under BE and PF tests, there exist a few studies. Lanuany et al. [17] justified the distinct response of the fabric in the PF and BE tests – particularly asynchronous onsets of wrinkling – based on the existence of some spurious tension in the yarns during picture frame test. They explained that tension along the yarns, which would be a result of misalignment of the yarns, noticeably 47  delays the onset of wrinkling in the PF test. Hivet and Duong also tackled this issue using the same assumption and correlated the different responses of the fabric to the tension induced in the yarns [70]. However, in addition to misalignment, increase in the crimping of the yarns can be regarded as one of the sources of tension generation. The numerical results in [49] and the analytical study using in continuum mechanics in Section 3.3.1.4 indicated that the thickness of yarns and in turn the crimping of yarns, see Figure 3.15(b), rises over the shear deformation. Next to the experimental report on the increase in thickness over shear deformation [56], this issue was experimentally observed in the BE test in which the yarns are free at their ends [46]. In simple words, on one hand, the effective length of the yarns – the length of the projection of the yarn on a plane that contains the fabric – over shear deformation in the PF test must remain constant; on the other hand, the growth in crimping tends to reduce the effective length. Accordingly, some tension has to be produced in the yarns to maintain the effective length unchanged. 3.3.2.4 Prediction of wrinkling in woven fabrics  Contrary to the pre-locking stage, the behavior of woven fabrics in the post-locking stage is dominantly governed by the material properties of the yarns (e.g., its transverse compaction resistance and flexural rigidity). Hence, the onset of wrinkling depends not only on test kinematics and geometry of the fabric elements, but also the material properties of the yarns. Figure 3.16 illustrates compressive forces establishing between adjacent parallel yarns during the course of lock-up, and Figure 3.17 depicts that at a certain instant the fabric structure alters its deformation pattern from the in-plane mode to the out-of-plane mode. In mechanical structures under compressive forces, there are discrete values of the load at which secondary equilibrium configurations may appear in the neighborhood of the initial equilibrium position. The nature of wrinkling in woven fabrics can be analogous to a deviation in deformation mode of the material structure. This analogy opens up a window toward theoretical analysis of wrinkling from an instability point of view.  48   Figure 3.16 Compressive forces induced between adjacent yarns during in-plane shear deformation of plain woven fabrics; (a) simulation, (b) experimental investigation of the yarn’s strain field in large shear deformation [4] Given this definition, a simple four-link equivalent structure mimicking the behavior of a representative volume element in the post-locking stage is proposed as shown in Figure 3.18. This Equivalent Structure (ES) comprises of four rigid bars. The length of bars are in fact the instantaneous width of yarns. For BE and PF tests, the instantaneous width can be presented as: ݓ ൌ ݓ଴ െ ∆ (3.13) (For BE test) ݓ ൌ ݓ଴ cos ߛ െ ∆ (3.14) (For PF test) Where w, w0 and Δ are, in turn, the instantaneous width of yarn, initial width of yarn and compaction of the yarn due to lateral compressive force at each shear angle. First, the analytical model and the equivalent structure are presented for the simpler condition which is BE test (Trellising shear) where the width is constant before locking. Subsequently, the model is modified to take into account intra yarn shear. These bars are clamped elastically at one end against rotation – about the dash-lined axes illustrated in Figure 3.18 – using four torsion springs of stiffness Kt, and linked to ball-joints at the other end (Point O). In other words, these torsion springs represent the equivalent resistance of the representative fabric volume against out-of-plane deformation. Four sliding springs with the stiffness coefficient K are placed around the bars to capture the lateral compaction stiffness of yarns. The bars are treated as the 49  cores of the sliding springs. The quasi-static compressive forces P act on the system as illustrated in Figure 3.18. These forces are to model the compaction forces between adjacent parallel yarns in the real fabric. If the compressive forces P reach a critical value termed as Pcr, the equivalent structure will become unstable. At the onset of instability, the meso-level structure will change its deformation mode from in-plane to out-of-plane (Figure 3.18). Accordingly, obtaining Pcr is a significant step to predict the onset of wrinkling in the fabric structure.  Figure 3.17 Mechanism of wrinkle formation in a plain woven fabric representative element due to large in-plane shear deformation (T and Ms denote the tension along yarn and in-plane shear coupling, respectively)  Figure 3.18 Equivalent structure (a) before equilibrium deviation, (b) after equilibrium deviation The stiffness of the sliding yarns can be determined using a lateral compression test on a single yarn. On the other hand, torsional springs which imitate the role of interlacing yarns 50  in the fabric (Figure 3.19) can be characterized analytically for a given fabric. To elucidate, yarn Y-2 at the onset of wrinkling tends to rotate about its longitudinal outer edge (i.e., about JJʹ axis in Figure 3.19) and form a wrinkle. In order to accommodate such deformation, the bending of the interlacing yarn Y-3 is necessitated. That is, the torsional springs represent the bending resistance of the interlacing yarns.  Figure 3.19 Equivalent torsional spring Kt, mimicking the bending effect of interlacing yarn Y-3 on the out of plane deformation of Y-2, and vice versa To derive the equivalent torsion spring coefficients, it is presumed that yarn Y-03 is a straight beam (Figure 3.20(a)) made of a linear, transversely isotropic, integrated, homogeneous material with an effective bending rigidity in the longitudinal direction ܳ௕ [71]. The Euler-Bernoulli theory was used to analyze such hypothesized beam [56, 71]. Moreover, as the proposed model takes the effect of tension on yarns into account, instead of a first order analysis which is based on superposition principle and considers the deflection of the beam independent of applied tension, the second-order analysis [72] was undertaken which accounts for the effect of tension on the overall flexural rigidity of yarns, ி	௒ೄ/మ. Considering the assumptions made in Figure 3.20, the deflection of the yarn as a beam at its mid-span and under arbitrary vertical force F can be written as [72]: ௌܻ/ଶ ൌ ܨ	2ܳ௕ 	ቀ ܶܳ௕ቁଷଶ ۉۇܵට ܶܳ௕2 െ tanhۉۇܵට ܶܳ௕2یۊیۊ (3.15) where T, ܳ௕, and S are tension along the yarn, the flexural rigidity of yarn, and the length of yarn respectively. 51  Note that the above second order analysis determines the extent of increase in bending stiffness due to tension for an integrated (continuum) beam. However, dry yarns are comprised of thousands of filaments which can slide over each other. Accordingly, the effective bending rigidity of a yarn, ܳ௕, is much less than that of the beam where filaments are perfectly joined and make an integrated beam. To exemplify, consider two beams, one an integrated large beam with the same total cross-section area as that of a second beam with a number of sub-beams stacked on top of each other with a minimal friction between them. The inherent difference between such two beams can be addressed by measuring the flexural rigidity of the fabric beam, ܳ௕,	 using the setup shown in Figure 3.20(b), which indirectly takes the discrete nature of yarns into account.  In essence, the second-order bending behaviour of dry fabrics relates to the fact that applying tension increases the overall flexural rigidity of yarns, ி	௒ೄ/మ, in two ways:  The resultant bending moment from tension (second-order analysis) reduces the magnitude of ௌܻ/ଶ and in turn it raises the overall flexural rigidity. This issue is incorporated in Eq. 3.15.  An increase in the internal bending rigidity of yarns can occur due to tension; e.g. by considering more interaction between filaments. This fact can be accounted for when ܳ௕ in Eq. 3.15 becomes a function of tension. Finally, the tensile behavior of yarns is typically non-linear as shown in Figure 2.1, while Eq. 3.15 is based on a linear behaviour assumption. In this phase of thesis, however, since the main goal is to analytically prove the pure shear theory through proposing two distinct equations for wrinkling of plain woven fabrics under BE and PF tests without yarn tension, the above-mentioned assumption will not violate the subsequent analyses. The second-order analysis would be deemed as the first step towards a realistic prediction of wrinkling angle when influence of tension is considered (please see [73, 74] for more detailed discussions and analytical examples).  52  (a) (b) Figure 3.20 a) Modeling of yarn as a bending beam (made of an elastic, transversely isotropic, integrated, homogeneous material), and determining the equivalent torsional spring coefficient, Kt. Note that the width of yarns is considered as Eq. 3.13, meaning the obtained stiffness is for trellising deformation; b) The experimental set-up for measuring effective bending rigidity of the yarn The instability approach using the presented methods in [75, 76] can be employed to determine the critical P and in turn the critical compaction of yarns based on the proposed mechanical model. The details of this approach is presented in the joint works [73, 74].  Figure 3.21 The equivalent structure in an imaginary deviated mode; note that the P forces remain in the original directions ௖ܲ௥ ൌ ܼܭݓ଴ܼ ൅ ܭ (3.16) ߂௖௥ ൌ ௖ܲ௥ܭ ൌܼݓ଴ܼ ൅ ܭ ൌ2ܳ௕ ቀ ܶܳ௕ቁଷଶ ݓ଴2ܳ௕ ቀ ܶܳ௕ቁଷଶ ൅ ܭۉۈۇۉۇܵටܶܳ௕2یۊ െ tanhۉۇܵටܶܳ௕2یۊیۋۊ (3.17) 53  Finally, referring to the discussion about locking prediction, the shear angle at which wrinkling occurs – critical shear angle – for plain woven fabrics under BE test is: ߛௐ௥௜௡௞௟௜௡௚ି஻ா ൌ cosିଵ ൬ݓ଴ ൅ ݐ െ ∆௖௥݈ ൰ (3.18) The same approach can be employed to determine the wrinkling angle in PF tests. The only difference with the BE test case is that the width of the yarns, which was assumed unchanged in the BE test, is now updated (ݓ ൌ ݓ଴ܿ݋ݏߛ) to take the intra-yarn shear into account. Consequently, the wrinkling onset for plain woven fabrics subjected to PF test can be expressed as: ߛ௪௥௜௡௞௟௜௡௚ି௉ி ൌ ܿ݋ݏିଵ ቌ ݐܮ଴ െ 2ݓ଴ ൅ 2ܣݓ଴ܣ ൅ ܭቍ (3.19) Where ܮ଴ is the length between the outer edges of two adjacent yarns. 3.3.2.5 Experimental evaluation  A commercially available plain woven fabrics made of carbon fiber was acquired from APC Composite Inc. The geometric specifications of the fabric (Figure 3.22) were obtained using a Nikon optical microscope. Figure 3.22(a) shows the top view of the unit cell of the studied woven fabric, and Figure 3.22(b) shows the elliptical cross-section and the thickness of a single yarn. To facilitate the visualization of yarn thickness, a piece of fabric was impregnated with PDMS resin-- a clear transparent and very thick resin-- and cured at room temperature.   54   (a)                                              (b) Figure 3.22 The geometric characteristics of the tested carbon woven fabric measured by a Nikon optical microscope, a) the top view of a unit cell of the tested woven fabric, b) the elliptical cross section of a single yarn impregnated with PDMS resin Moreover, Table 3.2 shows the relevant geometric and mechanical characteristics of the carbon fabric used in the test. Table 3.2. Geometrical and mechanical characteristics of the tested carbon fabric (the parameters are defined in Figure 3.13(a)). Yarn’s Material Geometric Characteristics Mechanical Characteristics w0 (mm) S (mm) l (mm)t (mm) Flexural Rigidity - ܳ௕ (N.mm2) Yarn Lateral Stiffness - K (N/mm) Carbon fiber 1.4 4.1 2.05 0.355 2.75 11.06 Now, by substituting the geometric and mechanical properties of the fabric – presented in Table 3.2 – into Eqs. 3.18 and 3.19, the critical shear angle of the woven fabric subjected to BE and PF tests can be determined. To evaluate the effectiveness of the proposed analytical approach for predicting wrinkling onset, actual bias extension and picture frame tests were carried out. The samples for the bias extension test were prepared from the carbon fabric with the aspect ratio of 2 (14 cm × 7 cm). Considering the gripped area of the sample, rectangular fabric specimens were cut into 24 cm × 7 cm to provide sufficient interaction between the fabric samples and the grips. Instron 5969 was used to apply the tensile load to the prepared samples 55  (Figure 3.23). The shear angle of the region III (Figure 3.23(a)) is calculated using the equation introduced in [9]: ߛ ൌ ߨ2 െ 2 arccos ൬ܦ ൅ ݀√2ܦ ൰ (3.20) where D and d are the distance of the opposing corners of region III (Figure 3.23) and the relative displacement of the grips, respectively.  Figure 3.23 The carbon fabric specimen under BE test (a) before shear deformation, (b) at the onset of locking, (c) at the onset of wrinkling It is worth mentioning that Eq. (3.20) is valid if there is no slippage between the yarns. To ensure the accuracy of Eq. (3.20) for our tests, the analytically obtained shear angles at a number of displacements were compared with the experimentally measured angles. To experimentally determine the shear angles, a camera was installed in front of the sample and synchronized with the Instron software in a manner to correlate each image with its corresponding displacement. After image processing and comparing the measured shear angles with those obtained from the analytical equation, it was observed that there is an acceptable agreement between the analytical and experimental results, proving the validity of Eq. (3.20) for our further experimental investigation. Three bias extension tests were conducted to ensure the repeatability of the results.  In spite of the simplicity in conducting the BE test, the picture frame test requires more sophisticated equipment, in particular if the yarn tensioning needs to be controlled/monitored. A customized biaxial tensile-picture frame fixture was designed and fabricated to be vertically 56  installed on Instron 5969. The fixture was comprised of four steel bars with equal length (23.5 cm) joint to each other with pins, as demonstrated in Figure 3.24. The tensile force was applied along the diagonal of the square frame, generating shear deformation. In addition to the Instron motor to provide the shear load, four servomotors were perpendicularly attached to each of four arms to provide tensile/compressive loading on the warp and weft yarns of fabrics. The device took advantage of five load cells, four of which to monitor the tension in the yarns over shear loading, and the remaining one to measure the global shear force supplied by the Instron machine. The load cell data were transferred to a data acquisition system, and the LABVIEW software was used to maintain the tension in the yarns at zero level.  Figure 3.24 The picture frame test set-up used in the experimental study Using appropriate boundary conditions to generate a uniform deformation field not only in the region of interest but also close to the grips is regarded as one of the challenges of the picture frame test. Non-uniformity in the deformation field, which is a type of misalignment, can cause fabric samples to experience tension, leading to a discrepancy in the results. This issue will be proven further in Chapter 4 through an analytical framework to transform the global force-displacement curves to local non-orthogonal stress-strain curves. To cope with the stated experimental challenge, needles were used to facilitate the yarns settlement at the grips, resulting in more uniform deformation fields. The reason of the helpfulness of two jaws with needles and slight pressure is analytically introduced and experimentally proved in the next chapter. Such boundary condition employed in this study can be regarded a modified boundary condition for PF test, which is a combination of a perfectly clamped boundary condition and free boundary condition. The implementation of the 57  proposed boundary conditions causes uniform deformation, and in turn, tension in the yarns during the shear deformation becomes substantially less than that using the simply clamped boundary conditions. Selecting the size of the samples was another critical aspect of the experiments. The proper width of each strip of the samples was found to be 12 cm. Although the effective distance between the grips is 23.5 cm, the length of the samples was chosen 30 cm to provide some room for folding the ends of the sample within the grips and creating good hold between the fabrics and the grips (Figure 3.25(b)).  Figure 3.25 a) Using needles in the boundary conditions, b) a folded sample in grips to increase the interaction between fabrics and grips It has been recommended that the transverse yarns in the arm regions of the sample (this region is indicated in Figure 3.1) should be removed; otherwise, wrinkling would initiate in the arm regions rather than in the region of interest. The same observation is made in this study as shown in Figure 3.26 (a). This observation is, in fact, another proof of the proposed “Pure Shear” model. To explain, the meso-scale deformation field in the region of interest follows the pure shear deformation, whereas, in the arm regions of the samples, the meso-level deformation pattern is something between trellising and pure shear because a family of yarns are free at their both ends. As a result, locking and wrinkling appear in the arm regions earlier than in the region of interest. To alleviate the chance of wrinkling in the arm regions, the transverse yarns were removed from the sample as illustrated in Figure 3.26 (b). Figure 3.26 (c) shows the sample without transverse yarns in the arm regions at 64 degrees, while no wrinkling is initiated neither in the arm regions nor in the region of interest. 58   Figure 3.26 a) wrinkling initiation in arm regions due to the presence of transverse yarns b) removing the transverse yarns c) shear deformation of the fabric at 64 degrees Using the modified boundary conditions, a slight increase in tension was noticed by the load cells. This increase in tension is due to the increase in the crimping of yarns during the shear deformation which reduces the effective length of the yarns. To remove the induced tension, the LABVIEW was programmed in a way that once the load cells show nonzero values, the LABVIEW turns the biaxial motors on in the compressive direction for a few seconds, keeping the tension within fabric samples zero. Doing so eliminates the possibility of the tension along yarns. 3.3.2.6 Validation of the model The incorporation of intra-yarn shear in the shear mechanism of woven fabrics led to propose two distinct meso-scale deformation modes for BE and PF tests. In accordance with the introduced deformation modes, it was postulated that locking and wrinkling in a woven fabric arise later in PF test compared to BE test. The experimental results presented in Figures 3.23, 3.26, and 3.27 substantiate this hypothesis. Moreover, Figure 3.27 shows a fair agreement between the experimentally measured and analytically predicted locking and wrinkling angles in the BE as well as PF tests.  59   Figure 3.27 Experimental measurements of locking and wrinkling angles versus the predictions proposed by the analytical model (Eqs. 3.11, 3.12, 3.18, and 3.19) 3.4 Summary of findings The presented critical review in the early part of this chapter undertook a comparison between different possible deformation mechanisms contributing to the in-plane shear behavior of woven fabrics. Intra-yarn shear, inter-yarn shear, translational slippage, and lateral compaction of yarns were discussed in general. Moreover, a coupled analytical, experimental and numerical discussion was presented to elaborate specifically on the intra-yarn shear effect. In fact, the validity of trellising shear for the shear behavior of fabrics subjected to PF test was challenged. With respect to the introduced influence of intra-yarn shear, a new deformation mechanism, “pure” shear mode was suggested in the context of multi-scale nature of woven fabrics. Accordingly, fundamental differences between bias extension tests and picture frame tests in terms of contributing micro-meso-macro level deformation mechanisms were illustrated.  Such differences were interpreted at macro-scale, leading to a number of hypotheses, the most important of which is delay in locking and wrinkling of woven fabrics under PF test. Locking and wrinkling of fabrics under both BE and PF tests were determined in terms of the mechanical and geometrical characteristics of fabrics. Furthermore, the effect of tension on these two angles were obtained. The conducted experimental study proved not only the hypotheses established based on “Pure Shear” model, but also the effectiveness and 60  accuracy of the offered analytical model to predict locking and wrinkling of woven fabrics under both bias extension and picture frame tests.  The aforementioned distinctions between the two tests raise this question that, which test is more reliable to undertake material characterization for a given manufacturing process. On account of the existence of blank holders in most forming processes [77-79], yarns are not perfectly free ended in practice. The applied force on blank holders, however, is not normally high in draping set-ups. As a result, there would exist a compromise between the trellising shear and pure shear deformation modes in the 3D forming of fabrics. Picture frame testing offers a promising experiment to correctly characterize shear modulus of woven fabrics for numerical simulation purposes, while there remain improvements to be achieved regarding imposing “ideal” boundary conditions on the specimens under this test. On the other hand, bias extension tests remain of high interest in the literature due to the simplicity of test set-up and presence of translational slippage that can be useful for mimicking some potential forming defects.    61  Chapter 4: Developing a New Characterization Framework to Capture Different Modes of Inherent Coupling in Woven Fabrics 4.1 Overview In this chapter, a new characterization framework is introduced where (a) a given woven fabric can be subjected to a designed set of combined loading tests, and subsequently (b) an analytical procedure can be employed to transform the global force-displacement data to the normalized local non-orthogonal stress-strain curves. Thereafter, the obtained results are compared with each other to assess the significance of coupling. Results will reveal the highly dominant influence of tension-shear coupling on the effective mechanical properties of the fabric, followed by the tension-tension and shear-tension couplings. Discussions are made as to how these macro-level observations are linked to the underlying deformation mechanisms at lower material scales. The main outcome of this chapter is the identified meso-scale sources which will be incorporated in the proposed constitutive model in Chapter 5.    4.2 Experimental study 4.2.1 Material The material used in the experiments (Figure 4.1) was a comingled polypropylene/glass plain weave, which was composed of 60% E-glass reinforcement fibers and 40% Polypropylene (PP) thermoplastic fibers. Specifications and geometrical properties of the material are shown in Table 4.1.  Figure 4.1 Woven architecture of the tested fabric, TWINTEX® TPP60N22P-060  62  Table 4.1. Specifications and geometrical properties of the tested PP/glass plain weave Commercial code TWINTEX® TPP60N22P-060 Weight fraction (glass) 60% Area density, g/ m2 745 Yarn linear density, tex 1870 Nominal fiber diameter, μm 18.5 Nominal thickness, mm 0.6 Nominal yarn spacing, warp,  mm 4.9 Nominal yarn spacing, weft,  mm 5.2 Nominal yarn width, warp, mm 3.5 Nominal yarn width, weft,  mm 3.8  4.2.2 Custom biaxial-picture frame test fixture Shown in Figure 4.2, the employed fixture embraced eight aluminum arms with equal lengths of 300 mm, hinged to each other using tubular bearings. Four servomotors were mounted perpendicular to each arm to apply axial tension along the warp and weft yarns. Another synchronous motor was oriented in the bias direction, by which a tensile force is applied across diagonally opposing corners of the picture frame and causing the arms to move from a primarily square shape into a rhomboid. The device orientation is horizontal to facilitate the sample mounting and to avoid local misalignments due to mishandling of the material. Besides picture frame testing, the sliding shear mode is also accessible using the device, though it was out of the scope of the present thesis. Each biaxial and shear motor had a load cell and linear variable differential transformer (LVDT) to monitor the global forces and displacements over time. The load cells were connected to a data acquisition unit, to program the device for a variety of simultaneous loading modes. Moreover, it could be set to run a sequence of loading modes on the sample to investigate, e.g., the effect of loading history on the material response. 63   Figure 4.2 Customized biaxial-picture frame test fixture employed for applying simultaneous and sequential combined loading mode The clamping mechanism of picture frame test is known to be important in obtaining reliable results. Tight plate clamping condition can be responsible for generating local bending of yarns as well as their spurious tensions during the test [46]; in fact it is deemed that different clamping conditions have been one of the main sources of large discrepancy seen in the benchmark investigation [80] in comparing results of different shear fixtures on the same material.  To induce a deformation close to pure shear, following [7], in this work needles were employed in the clamp jaws as described follows. 4.2.3 Sample shape and clamping A cross shaped sample (Figure 4.2) was mounted on the device for all the experiments. On one hand, because tensile forces are to be applied to the sample, a higher width may bring about more homogenous deformation. On the other hand, to avoid misalignments and unintentional errors—among obstacles of having uniform shear deformation in the gauge section— the specimen size should be small enough to provide an easy mounting process. As the fixture’s effective arm length was 300 mm, using 18 yarns, around 90mm, were found to give most repeatable tests. Each strip of the cross-shape sample was cut to 450 mm in length so that the sample ends could be folded two times, when mounting on the spaces between needles (see Figure 4.3(a)). The latter condition provided a very good gripping as the fiber slippage during the biaxial loading was fully hindered. No extra pressure (e.g., bolts) was applied to the upper light plate. The upper plate’s small weight was just sufficient to keep the yarns at the same level within the jaw, causing fiber yarns to deform uniformly together. Two layers of separate fabric cut-offs were stacked below and above the needled sample to engage the most effective load distribution of the upper plate weight while preventing it from crushing 64  the yarns. It should be mentioned that in spite of the same number of yarns in both warp and weft directions, due to the slight inherent unbalencedness in the material (see yarn widths and spacing in Table 4.1), the width of the sample strips in these directions was cut to 87.6 mm and 92.3 mm, respectively. Although the studied plain weave is nominally manufactured to be balanced along warp and weft, the source of the above difference in the yarn widths is related to the weaving process. That is, warp and weft yarns undergo various trends of tension and bending while being interweaved [81]. The statistical significance of such inherent irregularities in the response of fabrics has been reported, e.g., in [18, 82].   (a) (b) Figure 4.3 (a) Using needles instead of the conventional plate-bolts clamping to allow yarns rotate freely without local bending, (b) the resulting uniform shear deformation within the sample (the upper plates in these images have been removed after the test to be better visualize the effect of the needles in inducing homogenous deformation in the jaw’s neighborhood) 4.2.4 Loading modes and friction consideration Figure 4.4 schematically shows three fundamental deformation modes induced by the biaxial-shear test fixture; namely the picture frame, biaxial tension, and combined loading modes. The displacement rate for the biaxial and shear motors was 4.5 and 2.6 mm/min (i.e., quasi-static), respectively. Three replications were performed for each experiment and force-displacement data were recorded. To account for mechanical friction in the fixture, prior to actual testing, the friction force was recorded without the material mounted. The repeatability of this effect was observed through three replications and an average value was regarded as the effective friction force in the device. This force was then subtracted from the output curves for each subsequent experiment with the material mounted. The friction effect between the fabric and needles was neglected.  65    (a) (b)   (c) (d) Figure 4.4 Different deformation mode configurations: (a) the original state, (b) picture frame mode, (c) biaxial tensile loading, (d) simultaneous biaxial-shear loading. Note that in (a), (b) and (c), the local fabric (material covariant) coordinate system (ࢌ૚ െ	ࢌ૛ሻ is aligned with the fixture covariant coordinate system (1-2); however,  ࢌ૚, ܉ܖ܌	ࢌ૛ directions are not parallel to 1 and 2 under the simultaneous mode in Figure 4.4 (d) 4.2.5 Experimental plan  In order to assess three different inherent coupling types (per section 2.2.2.3), three sets of experiments were conducted, with each repeated at least three times. First, the tension-tension coupling was examined by performing a series of sequential biaxial tensile tests. The sequential biaxial tensile loading implies that tension is first applied and fixed in the second (transverse) direction (shown in Figure 4.5) up to a certain point, and then the tensile test in the first direction is carried out. Given that the uniaxial tensile behavior of woven fabrics is non-linear, including two regions of yarn straightening and stretching (Figure 2.1), the level of transverse pre-tensions in the tests was carefully chosen to cover both regions. More 66  specifically, a preliminary uniaxial tensile test revealed that the transition point between the first and second phases, schematically defined in Figure 2.1, for the selected fabric occurs at about 2.5% elongation. Accordingly, 1, 2, 3 and 4% transverse strain levels are selected for sequential biaxial loading tests, where the first and second levels would correspond to the yarn straightening phase, and the latter two to the yarn stretching phase.   Figure 4.5 The biaxial-picture frame test fixture employed in the study; displacements in directions 1 and 2 cause biaxial loading, while displacement in direction 3 imposes in-plane shear deformation Second, to capture the effect of tension-shear coupling, a series of picture frame tests with an initial biaxial tension on yarns were performed, again taking the two phases of fiber elongation phase by applying 1, 2, 3, and 4% pre-tensions. It is important to note that the meso-scale nature of the coupling modes may not follow the superposition rule. For instance, in the course of tensioning effect on the shear rigidity of the fabric, superposition would imply that the shear modulus under a biaxial pre-tension of (4%, 4%) should be twice that under a uniaxial pre-tension of (4%, 0%). To evaluate this important phenomenon, picture frame tests under uniaxial pre-tensions were also conducted and compared.  Finally, to capture the effect of shear-tension coupling, applying shear up to a certain shear angle followed by imposing biaxial tension directly along the warp and weft directions would be regarded as a natural choice. This type of sequential experimentation, however, was not be feasible with the current version of the device due to the presence of dynamic rotational coupling between the frame angle and axial loading guides. This issue will be illustrated later on in this chapter. Alternatively, a simultaneous shear-tension testing mode was opted in which all the motors were turned on from the beginning of the experiment. Then, using the global to 67  local force transformation framework (outlined in the next section), the comparison between this simultaneous shear-tension loading with biaxial tensile test was attained and elucidate the effect of shear-tension coupling. 4.3 Transformation of global force-displacement datasets to normalized local non-orthogonal values The customized device provides the raw force-displacement data measured by five load cells and five LVDTs. Despite the use of these datasets in the literature to characterize the inherent coupling in fabrics, the actual local force and displacements which warp and weft yarns experience over the loading should be analyzed. An analytical approach is presented here to derive explicit expressions for local stresses and strains based on the global forces and displacements. First, a geometrical analysis is performed to find the relationship between the deformation output parameters (yarn longitudinal strain and shear angle) and pertinent inputs such as global biaxial and shear displacement (ݑଵ, ݑଶ, and	ݑ௦ሻ. Figure 4.4 illustrates these three independent global deformation parameters. As depicted in Figure 4.4, the deformation of specimens under combined loading is most complicated in which the angle between yarns in the central gauge section is not the same as the angle between the arms of picture frame. It should be noted that tension-shear combined loading can be conducted under a simultaneous or a sequential regime (e.g., first pre-tension and then shear). The subsequent deformation analysis of the cross-shape sample is followed under simultaneous regime, through which general expressions can be obtained and then applied to other simpler modes including the sequential loading. In the first step of the solution, the angle of the rhombus-shape fixture at a given time, which is dependent on the shear displacement (ݑ௦), should be obtained. According to Figure 4.6, this angle	ሺ2∅) can be defined in terms of ݑ௦ as [39]:   2∅ ൌ 2 arccos ሺݑ௦ܮ ൅1√2ሻ (4.1) where, L is the length of fixture arms. If the biaxial motors are off while the shear motor is on, a pure picture frame test is conducted; thus the warp and weft yarns remain theoretically 68  parallel to the arms. The shear angle of the sample over time can be determined under the picture frame mode by: ߠ௉ி ൌ 90 െ 2∅            (4.2)   Figure 4.6 Shear deformation of the picture frame fixture On the other hand, if both the biaxial and shear motors are turned on simultaneously, the angle between the warp and weft yarns will be different from that of the fixture arms, as shown in Figure 4.4 (d). Accordingly, in the picture frame and pure biaxial tensile tests, the ଵ݂ െ ଶ݂	local	coordinate system remains parallel to the fixture coordinate system 1-2, while in the simultaneous loading, the local fabric coordinate bases do not remain parallel to the bases of fixture coordinate (Figure 4.4(d)). Figure 4.7 demonstrates the position of a yarn in direction	 ଵ݂	under simultaneous loading, which leads to the angle ߙଵ between the yarn and the arm in the corresponding direction due to loading motor 1. To calculate this angle with respect to the dimensions of fixture and the input displacement parameter (ݑଵሻ, the following trigonometric equations are employed: ሺܮଵᇱ2 ሻଶ ൌ ݑଵଶ ൅ ሺܮ2ሻଶ െ 2ݑଵܮcosሺ90 ൅ 2∅ሻ (4.3) sin ߙଵݑଵ ൌsinሺ90 ൅ 2∅ሻܮଵᇱ  (4.4) where ܮଵᇱ  corresponds to the instantaneous length of yarn under simultaneous loading. By substituting Eq. 4.3 into Eq. 4.4 and rearranging, we obtain: 69  ߙଵ ൌ arcsin	ۏێێۍ cosሺ2∅ሻݑଵ2ටݑଵଶ ൅ ሺܮ2ሻଶ ൅ 2 ∗ ݑଵ ∗ ܮ ∗ sinሺ2∅ሻےۑۑې             (4.5) Similarly, ܮଶᇱ  and ߙଶ	arising from the axial tension in direction-2, can be obtained as: ሺܮଶᇱ2 ሻଶ ൌ ݑଶଶ ൅ ሺܮ2ሻଶ െ 2 ∗ ݑଵ ∗ ܮ ∗ cosሺ90 ൅ 2∅ሻ (4.6) ߙଶ ൌ arcsin	ۏێێۍ cosሺ2∅ሻݑଶ2ටݑଶଶ ൅ ሺܮ2ሻଶ ൅ 2 ∗ ݑଶ ∗ ܮ ∗ sinሺ2∅ሻےۑۑې (4.7) Consequently, the net shear angle is calculated as: ߠ ൌ 90 െ 2∅ െ ߙଵ െ ߙଶ (4.8)  Figure 4.7 The angle ࢻ૚	between the yarn in direction ࢌ૚	and the fixture arm in direction 1, due to the tensile loading of motors in direction 1. Note that 1 and 2 are the bases of fixture covariant coordinate system, and ࢌ૚	܉ܖ܌	ࢌ૛ the bases of the material covariant (local) coordinate system The difference between ܮଵᇱ  or ܮଶᇱ  and L represents the increase in the yarn length in direction 1 or 2. Thus, assuming infinitesimal deformation in the fibers directions, given their high normal moduli, the strain along the fabric principal directions is obtained by: ߝଵ ൌ ܮଵᇱ െ ܮܮ  (4.9) 70  ߝଶ ൌ ܮଶᇱ െ ܮܮ  (4.10) A noteworthy point regarding the deformation of sample under simultaneous loading is that the same displacement rate of biaxial motors (ݑଵ ൌ ݑଶሻ causes ߙଵ and ߙଶ to be the same according to Eqs. (5) and (7), as was also observed in the test replications. Accordingly, ߙଵ and ߙଶ are hereafter referred to as ߙ. The next step of solution is to develop expressions for local stresses in terms of the applied global forces and displacements. Due to the difficulties with precise measurement of the thickness of fabrics, normalized forces (force per length) are typically studied instead of stresses for characterization of dry fabrics [83], as also employed in this study. As mentioned in section 4.2.3, a slight unbalance was noticed in the tested fabric material (Table 4.1), hypothetically causing differences in the required forces in warp and weft directions for the same amount of extension. However, monitoring the load history of both biaxial motors showed less than 10% for such difference. Consequently, the average value of ܰ௠௢௧௢௥	ଵ and ܰ௠௢௧௢௥	ଶ is reassigned as ௕ܰ for both directions in subsequent calculations. Also, to obtain the normalized force values, the average width of warp and weft strips were used. The global force from the shear motor can cause the specimen to experience shear loading, whereas the global pulling forces from the biaxial motors, as a result of being always perpendicular to the arms in the fixture mechanism, produce two components on the frame, as shown in Figure 4.8.  One component induces extra force in addition to the shear motor to apply picture frame shear deformation and the other one results in pure tension in the direction of arms. In other words, in the given setup the global force of the biaxial motors (ܰ௠௢௧௢௥	ଵ	and	ܰ௠௢௧௢௥	ଶሻ	and that of the shear motor ( ଷܰሻ are not independent of each other. Figure 4.8 represents the coupling between these forces. Note that the direction of forces were chosen based on the mechanical design of the device. For instance, the shear motor applies the shear force directly on the frame which is transferred and applied on specimens, whereas the biaxial motors are part of the frame connected to fabric specimens. The normalized resultant shear force of the shear motor ( ଷܰ௦ሻ can be found based on its global value, as shown in Eq. 4.11, where ܹ refers to the clamping width. Furthermore, Eqs. 4.12 and 4.13 express the 71  normalized longitudinal and shear force components of the biaxial motors. It is noted that there are two terms of sin 2∅ in Eq. 4.13, one of which is incorporated to transfer ௕ܰ to the direction of yarn. The second one comes from considering the intra-yarn shear which has been proved in chapter 3. Subsequently, instead of jaw width, its projection should be used to calculate the normalized longitudinal force component of the biaxial motors, showing the importance of intra-yarn shear in correct characterization of woven fabrics. ଷܰ௦ ൌ ଷܰ2ܹܿ݋ݏ ∅ (4.11) ௕ܰ௦ ൌ 	 ௕ܹܰݐܽ݊ 2∅ (4.12) ௕ܰ௟ ൌ ௕ܹܰݏ݅݊ଶ2∅ (4.13)  Figure 4.8 Decomposition of the global external forces of the biaxial and shear motors into their normalized shear and longitudinal components imposed on the fixture frame (i.e., in 1-2 directions), in order to measure the net shear force (ࡺ࢈࢙ ൅ ࡺ૜࢙), denoting a kinematic coupling between the global forces due the mechanism of the combined loading fixture. The normalized force vectors are shown as dashed lines and are related to the external motor forces via Eqs. 4.11-4.13  Experimental evidence of the above coupling between global forces of biaxial and shear motors will be provided in section 4.4.3. The main outcome drawn from Figure 4.8 is that the net shear force applied to the fabric sample should be obtained by: ܨ௦ ൌ ଷܰ௦ ൅ ௕ܰ௦     (4.14) 72  Next, given the inherent coupling between global biaxial and picture frame deformations, it is essential to reemphasize that the angle between yarns during combined loading does not follow the frame shear angle (Figure 4.9). As a result, the obtained net normalized forces ܰ ௕௟	& ܨ௦ in Figure 4.8 cannot be deemed as in the actual local fabric direction (i.e., parallel to yarns) in the middle region of interest. For the picture frame testing with small pre-tension, however, the yarns nearly remain parallel to the arms during shearing. Hence, for the latter tests, the normalized local tensile force applied to yarns ( ௟ܰሻ may be regarded the same as the normalized global force component of the biaxial motors ( ௕ܰ௟ ൌ ௟ܰሻ. For the simultaneous loading, to complete the transformation analysis from the frame of non-orthogonal fixture coordinate to the frame of non-orthogonal fabric coordinate, an approach analogous to [84] was implemented. Namely, Figure 4.9 displays a rhombic element within the sample region of interest, on which the normalized global forces are applied, and another inside rhomboid whose sides are parallel to the warp and weft directions. To acquire the relationships for the normalized local forces, a triangular material element was extracted from Figure 4.9 and shown in Figure 4.10. Considering force equilibrium conditions on this element, Eqs. 4.15 and 4.16 were arbitrarily formulated in the directions 1’ and 2’—perpendicular to the original directions 1 and 2, respectively, which were merely chosen to attain more straightforward formulation. The terms sin	ሺ2∅ሻ and sin	ሺ2∅ ൅ 2ߙሻ stem from the fact that ௕ܰ௟ and ௟ܰ, respectively, are not perpendicular to the sides of the triangle. Also, ୱ୧୬ఈୱ୧୬ሺଶ∅ାఈሻ and ୱ୧୬ ଶ∅ୱ୧୬ሺଶ∅ାఈሻ are utilized to consider different length of material element edges in Figure 4.10, on which the normalized forces are applied. Equilibrium in Direction 1’: ሾ ௕ܰ௟ ൈ sin 2∅ሿ ൈ ୱ୧୬ఈୱ୧୬ሺଶ∅ାఈሻ ൈ sin 2∅ ൅	ܨ௦ ൈ sin 2∅ ൌ 	 ௦ܰ ൈୱ୧୬ଶ∅ୱ୧୬	ሺଶ∅ାఈሻ ൈ sinሺ2∅ ൅ ߙሻ െ	ሾ ௟ܰ ൈ sin	ሺ2∅ ൅ 2ߙሻሿ ൈୱ୧୬ଶ∅ୱ୧୬ሺଶ∅ାఈሻ ൈ sin	ߙ                                                                (4.15) Equilibrium in Direction 2’: [ ௕ܰ௟ ൈ sin	2∅ሿ ൈ sin		2∅ ൅	ܨ௦ ൈ ୱ୧୬ఈୱ୧୬ሺଶ∅ାఈሻ ൈ sin	2∅ ൌ 	െ ௦ܰ ൈୱ୧୬ଶ∅ୱ୧୬ሺଶ∅ାఈሻ ൈ sin	α ൅ ሾ ௟ܰ ൈ sinሺ2∅ ൅ 2ߙሻሿ ൈୱ୧୬ଶ∅ୱ୧୬ሺଶ∅ାఈሻ ൈ sin	ሺ2∅ ൅ ߙሻ                                                                 (4.16)     73   Figure 4.9 Schematic of the normalized net global forces and the resolved normalized local forces along yarns (ࡺ࢙	܉ܖ܌	ࡺ࢒ሻ within the inner region of interest. Note that ࡲ࢙	܉ܖ܌	ࡺ࢈࢒	are parallel to the fixture arms (directions 1 and 2), whereas ࡺ࢙	ࢇ࢔ࢊ	ࡺ࢒ are aligned with the yarn directions (ࢌ૚	܉ܖ܌	ࢌ૛ሻ  Figure 4.10 Free body diagram of the normalized net global and local forces on a triangular material element within the fabric based on Figure 4.9 Finally, rearranging and solving Eqs. 4.15 and 4.16, the normalized local longitudinal and shear forces were calculated as in Eqs. 4.17 and 4.18. These equations in combination with Eqs. 4.11-4.14 give the on-axis net normalized forces in the most general form as a function of global forces applied to the balanced woven fabric. As a check point, by setting the value of 2∅ ൌ 90° as well as ߙ=0° in Eq. 17 and ߙ=0 in Eq. 4.18, relationships for a conventional biaxial and pure picture frame mode can be obtained, respectively. ௟ܰ ൌ ୱ୧୬ሺଶ∅ାఈሻୱ୧୬ሺଶ∅ାଶఈሻሺୱ୧୬మሺଶ∅ାఈሻିୱ୧୬మ ఈሻ ሾ ௕ܰ௟ሺୱ୧୬	ଶ∅		 ୱ୧୬మ ఈୱ୧୬ሺଶ∅ାఈሻ ൅ ݏ݅݊ 2∅	sin	ሺ2∅ ൅ ߙሻሻ ൅ 2ܨ௦sinߙሿ         (4.17) ௦ܰ ൌ ୱ୧୬ሺଶ∅ାఈሻୱ୧୬మሺଶ∅ାఈሻିୱ୧୬మ ఈ ሾ ௕ܰ௟ sin α ሺ1 ൅ sin 2∅ሻ൅	ܨ௦ ሺsinሺ2∅ ൅ ߙሻ ൅ୱ୧୬మ ఈୱ୧୬ሺଶ∅ାఈሻሻሿ                           (4.18)  74  4.4 Results and discussion 4.4.1 Tension-tension coupling As stated in Chapter 2, these are disagreements about the tension-tension coupling between the experimental results and numerical results. Moreover, numerical results conducted by different groups were not consistent with each other. Hence, this section of the thesis attempts to comprehensively investigate the influence of tension-tension coupling on the mechanical behavior of fabrics at different levels, namely micro, meso and macro levels. More importantly, the underlying micro and meso-scale sources of each observation are identified. By doing so, the missing issues in the conducted numerical simulations will be revealed, resolving the existing conflict in the literature. Figure 4.11 demonstrates the behavior of the tested fabric under tension in the longitudinal direction, 1, with various levels of pre-tension in the transverse direction 2. To clarify, first tension was applied in the second direction where the sample was fixed in the first direction; subsequently, tension in the longitudinal direction, 1, was imposed while maintaining the transverse strain constant. The response of the fabric in these tests are depicted in Figures 4.11 (a & b).  If no inherent coupling existed, all the response curves would be parallel, analogous to Figure 2.5(c). By contrast, as noticed in Figure 4.11(a), the force response curves in direction 1 cross each other, inferring that there is some extent of inherent coupling embedded in the stiffness ܦଵଵ (see  also Eq. 2.2). In fact the data trend shows that ܦଵଵ is a function of ߝଶ as well as ߝଵ (with the latter making the individual curves non-linear). Also per Figure 4.11(a), the initial jumps (i.e., when ߝଵଵ	 ൌ 0ሻ	for different transverse pre-tensions refer to the extension-extension coupling (i.e., the initial Poisson’s ratio represented by ܦଵଶ	in Eq. 2 for when ߝଵଵ	 ൌ0). In other words, the initially applied tension in the transverse direction has generated some residual force in direction 1 owing to the constrained yarns. The negligible intercepts in Figure 4.11(a) indicate a small value of initial ܦଵଶ	in woven fabrics, as also assumed zero in some numerical studies such as [16]; however the difference between the curves is relatively amplified at higher strains. Figure 4.11(a) also demonstrates that applying tension in the transverse direction results in a more compliant response in the longitudinal direction, providing that the applied pre-tension does not exceed the yarn straightening (decrimping) 75  phase. Within this phase, by comparing 1% and 2% elongation curves, it can be understood that the higher the transverse yarn elongation, the more compliant behavior of the longitudinal yarn. On the contrary, if the applied transverse elongation is beyond the straightening phase (i.e., in stretching phase for 3% and 4% curves), a much stiffer response is seen in the beginning of the test.     (a) (b)   (c) (d) Figure 4.11 Comparison between the tensile behavior of the fabric under different sequential tensile tests; a) The normalized force in the direction 1 (ࡺ૚ሻ	versus strain in the same direction (ࢿ૚ሻ, b) The normalized force in the second direction (ࡺ૛ሻ	against strain in the first direction (ࢿ૚ሻ, c) The adjusted normalized force in the direction 1 (ࡺ′૚ሻ	versus strain in the direction 1 (ࢿ૚ሻ, d) The adjusted normalized force in the second direction (ࡺ′૛ሻ	against strain in the first direction (ࢿ૚ሻ Figure 4.11(b) illustrates the change in the transverse force while applying the longitudinal force; mathematically, this figure represents	ܦଶଵ ∗ ߝଵ.  Due to the initial pre-tension in the transverse direction, there exists a jump for all the corresponding loading cases, the extent of which directly depends on the level of applied pre-tension. Also in this figure, although the curves do not cross each other, they have different slopes during test, pointing to the presence of coupling effect on the effective Poisson’s ratio through ܦଶଵ in Eq. 2.2; in fact, the difference between these curves implies that ܦଶଵ is a function of ߝଶ and it is also a function 76  of ߝଵ due to the non-linear curves observed during the test. Note that for a balanced fabric ܦଵଶ	and ܦଶଵ should be comparable under a biaxial extension mode with the same loading rate in both directions. The above macro-level coupling observations may be contemplated at meso-level via trade-offs between three determining mechanisms: variation in the crimp of yarns, variation in the lateral contact forces at yarn crossovers, and variation in the bending stiffness of yarns explained in Chapter 3. When the applied transverse strain is within the uncrimping (straightening) phase, the initial waviness of yarns in direction 1 is increased (with a low lateral reaction forces/resistance at crossovers from the transverse yarns). Given that the yarns in the fabric are similar to sinusoidal beams, this higher level of initial crimping yields more curvature and hence a more compliant tensile response during the test in direction 1, as shown in the 1 and 2% pre-tension curves in Figure 4.11(a). However, the figure shows an opposite trend (i.e. a stiffer behavior of the fabric in the straightening phase of longitudinal yarns) under the high transverse pre-tension levels (3 and 4% curves). To explain, when the applied transverse pre-tension exceeds the transition point and reaches the second phase, the yarns in direction 2 undergo stretching. This meso-scale transition would result in both higher lateral contact forces at crossovers [22] and also a higher effective bending resistance of each transverse yarn. Accordingly, the increased lateral contact forces at crossovers and the bending stiffness of the transverse yarns under high transverse pre-tensions, overcome the effect of increase in crimp of longitudinal yarns, causing initially a much stiffer behavior in direction 1, which is in agreement with the observations in Figure 4.11(a).  Interestingly, although a stiffer behavior is initially observed in the straightening phase of curves in Figure 4.11(a) under high levels of transverse pre-tension (3% and 4% curves), this response is transited to a more compliant response (lower slope) in the stretching phase of these curves. To explain, as the transition from straightening to stretching phase begins in direction 1, the yarns are virtually locked, and there is slight or no further change in the amplitude of crimps of yarns. Hence, the increase in the lateral contact forces and the bending rigidity of transverse yarns, which resist the unbending/decrimping of longitudinal yarns, may no further affect the stiffening of tensile behavior in the longitudinal direction. Instead, the locked high amplitude of crimping in the main direction causes more compliant response for 77  the rest of deformation, given that yarns are analogous to sinusoidal beams whose effective stiffness is directly function of the beam curvature. As such, the main contributing mechanism to the deformation becomes the stretching of yarns under a curved beam with solid reaction points at cross-overs. Multi-scale numerical simulations are underway to provide better assessment of these deformation hypotheses in future studies. The observed conflicts in the literature roots from two facts, first of which is that mostly biaxial tensile test with different loading ratios were utilized where both inherent coupling and extension-extension coupling affect the stress-strain curves. The sequential biaxial tensile test along with the separate analysis of forces in the longitudinal and transverse directions could help to distinguish these two couplings. Moreover, paying attention to the two distinct phases of tensile behavior of woven fabrics was helpful to comprehensively assess the effect of tension-tension coupling. 4.4.1.1 Quantification of tension-tension coupling factor For quantification of the tension-tension coupling in woven fabrics, a limited mathematical definition is found in the literature. In [7], the tension-shear coupling factor was expressed as the shear force of the fabric under biaxial pre-tension divided by that under the pure shear with no pre-tension. In the current study, the latter definition is adapted to the tension-tension coupling with a minor modification: the tension-tension coupling factor is measured as the ratio of an adjusted normalized force in the fabric under transverse pre-tension to that under uniaxial tension. Namely, for the adjusted curves (shown in Figures 4.11(c & d)), the initial jumps in Figures 4.11(a & b) should be removed, hence excluding the effect of initial Poisson’s ratio from the quantification. Accordingly the tension-tension coupling factors are defined as: ܥܨ௧௘௡௦௜௢௡ି௧௘௡௦௜௢௡	ሺ௘௙௙௘௖௧	௢௡ ஽೔೔ሻ ൌே೔ᇲ|ఌೕୀఌೕ_௣௥௘௧௘௡௦௜௢௡ே೔ᇲ|ఌೕୀ଴     (i, j=1,2) (4.19) ܥܨ௧௘௡௦௜௢௡ି௧௘௡௦௜௢௡	ሺ௘௙௙௘௖௧	௢௡ ஽ೕ೔ሻ ൌேೕᇲ|ఌೕୀఌೕ_೛ೝ೐೟೐೙ೞ೔೚೙ሻேೕᇲ|ఌೕୀ଴    (i, j=1,2,  i്j) (4.20) Where ௜ܰᇱ represents the adjusted normalized force in direction i. It should be noted that Eq. 4.19 determines the effect of tension-tension coupling on ܦଵଵand ܦଶଶ, while Eq. 4.20 measures 78  the influence of this coupling on ܦଶଵand ܦଵଶ as in Eq. 2.2. Based on the conducted experiments in which the pre-tension is applied in direction 2, the coupling factors of interest can be rewritten as:  ܥܨ௧௘௡௦௜௢௡ି௧௘௡௦௜௢௡ ሺ௘௙௙௘௖௧ ௢௡ ஽భభሻ ൌேభᇲ|ఌమୀఌమ_೛ೝ೐೟೐೙ೞ೔೚೙ேభᇲ|ఌమୀ଴  (4.21) ܥܨ௧௘௡௦௜௢௡ି௧௘௡௦௜௢௡ ሺ௘௙௙௘௖௧ ௢௡ ஽మభሻ ൌேమᇲ|ఌమୀఌమ_೛ೝ೐೟೐೙ೞ೔೚೙ேమᇲ|ఌమୀ଴  (4.22) Figures 4.12 (a & b) show the variation of coupling factors in Eqs. (4.21) and (4.22), indicating the significance of tension-tension coupling on ܦଵଵ and ܦଶଵ. Theoretically, a ܥܨ of 1 implies no inherent coupling. Moreover, a ܥܨ greater or lower than 1 indicates a stiffer or a more compliant behavior, respectively. As observed in Figure 4.12(a), the coupling factor for 1% and 2% transverse pre-tensions is lower than 1, meaning a more compliant behavior of yarns in direction 1, as also observed in Figure 4.11(a & c). Comparison between the curves of 1 and 2% pre-tension discloses that the higher the level of the transverse pre-tension, the lower the magnitude of the coupling, and subsequently the more the compliant longitudinal behavior. However, when the transverse pre-tension exceeds 2.5% that is about the nominal transition point, the magnitude of coupling factor has been raised to 3.5. This suggests, when the transverse pre-tension is within the stretching region, the fabric in the main direction becomes much stiffer. While the coupling factor is nearly constant for the pre-tensions within the straightening region, a varying trend of the curves is seen for the pre-tensions within the stretching region, which is due to the interaction between three different mechanisms explained above. Figure 4.12(b) shows another aspect of the tension-tension coupling. For all the curves in this figure, ܥܨ is greater than one, meaning that the coupling always increases the interaction between the warp and weft yarns and subsequently the effective Poisson’s ratio. To put in numbers, the ratio of ܦଶଵto ܦଵଵ for the studied fabric without pre-tension was nearly 0.03, whereas in the case of 4% transverse pre-tension, the ratio of the effective ܦଶଵto ܦଵଵwas elevated to 0.25. Hence, ignoring ܦଶଵ and ܦଵଶ in the fabric numerical simulations (such as [16]) may be disputed at high strain levels due to the magnified coupling effect. In fact when both families of yarns fall in the stretching region, due to high yarn stiffness values and 79  established contact forces, the fabric integrity as a whole becomes high and makes it more like a solid shell with a sizable Poisson’s ratio.    (a) (b) Figure 4.12 The tension-tension coupling factor and its effect on (a) the fabric longitudinal stiffness (using Eq. 4.20) and (b) the effective Poisson’s ratio (using Eq. 4.22) 4.4.2 Tension-shear coupling Prior to thoroughly discussing the effect of tension-shear coupling, identifying the underlying sources and determination of its significance, it is necessitated to disclose the roots of the observed conflict in the literature regarding the shear force-shear angle response of fabrics under tension. As stated in Chapter 2, although it is expected the higher pre-tension, the higher shear force at both low and high shear angles, the experimental study in [17] showed lower shear force at high angles. Hence, this section of the thesis attempts to firstly resolve this conflict, and then investigate the role of tension in the shear rigidity of fabrics. In this section, local parameters were determined employing the proposed analytical procedure. It is worth adding that the accuracy of the analytical prediction of the local shear angle and longitudinal displacements was verified by comparing to experimental observations. To experimentally determine the shear angle, a camera was installed above the samples and synchronized with LabView software in a manner to correlate each image with its corresponding displacement. 4.4.2.1 Identification of the causes of the conflict in the literature Figure 4.13(a) demonstrates the normalized global force of the shear motor versus shear angle of the test fabric at two different pre-tension levels (1 and 2% biaxial pre-tension). The global results in this figure are in full agreement with the results of Launay et al. (Figure 15 of [17]). Namely, the observed significant increase in the magnitude of global force of picture frame shear motor at initial stages of the loading is an evidence of the yarn tension 80  effect on the shear response of the fabric. Despite the substantial deviation between the two curves with different levels of pre-tension (0.8 and 3.1N/mm corresponding to 1 and 2% elongation) in the small shear angles, they become closer to one another in the larger shear angle region. This convergence may mistakenly imply that the effect of tension on the fabric shear behaviour fades at the higher forming angles. Even in the study [17], the shear force at some low pre-tension levels exceeded that of higher pre-tension. On the other hand, the study conducted by Nezami et al. [7] showed a more significant deviation between the shear responses of tensioned samples at higher shear angles.  (a) (b) Figure 4.13 (a) Comparison between normalized global shear forces resulting from the shear motor for two picture frame tests with different pre-tension levels, and (b) the same comparison using the normalized net local shear force applied to the fabric. Notice the significance of considering the global shear force component resulting from the biaxial motors (coupling effect) in the resulting trends between the two curves with 0.8 and 3.1 N/mm pre-tensions, especially at higher shear angles. More specifically, by performing the force analysis in the local coordinate, the effect of pre-tension is consistent (the distance between the curves remain consistent) across the fabric shear angles This conflict may be rooted in two points. The first and main point is that neither of the past studies performed the coupling characterization in the local fabric coordinate (the effect to be further elucidated in the next section). The second point is that Launay et al. [17] applied pre-tension by fixing the yarns at a given displacement; hence the tensile force level varied over shear loading, whereas the device in [7] employed pneumatic cylinders to keep the pre-tension force constant. The boundary condition of fabric samples in the current study was also displacement-control in which the displacement could be controlled while the force varies. Figure 4.14 depicts the measured force of the biaxial motors against time in the picture frame test when a 2%, or 3.1N/mm, pre-tension level is applied. In fact, this sequential experiment 81  was comprised of three stages, the first of which corresponded to turning on the biaxial motors to reach a certain level of pre-tension. The second stage referred to the interval between turning off the biaxial motors and turning on the shear motor (relaxation/settling time), and at the third stage the picture frame/shear loading was applied. The allocated time span between applying the pre-tension and starting the shear test was to perform the experiment in a fashion comparable to practice. That is, during the stamping process of fabrics, normally there is a delay between the clamping of the fabric via blank holders and the initiation of draping. By performing the test in a sequential manner, the effect of viscoelasticity—the reason of reduction in force value in the second stage in Figure 4.14—could be accurately identified and removed from the third stage so that the perceived quasi-static behavior under the shear mode regime became intact.   Figure 4.14 Variation of the global force of biaxial motors during picture frame test with 3.1 N/mm (total of 280 N) pre-tension; from the origin to point A (stage 1): applying pre-tension up to 380 N; from point A to point B (stage 2): the relaxation of the sample and causing reduction in the force from 380 N to 280 N; after point B (stage 3): applying the shear deformation Recalling the third stage of response in Figure 4.14, it is noted that that the global biaxial force illustrates a reduction in its magnitude. This can be justified in that, by applying shear deformation, a non-zero angle is created between the yarn direction and the corresponding pulling direction of the biaxial motor, resulting in a decline in the stiffness against fiber pulling in that direction. For the same reason, during the shear test, the pre-tension effect may abate, and the global force response curves get closer at higher shear deformation ranges (Figure 4.13(a)).  Remark 4.1: Another notable observation was that by performing additional picture frame tests with a strictly clamped boundary, the effect of clamping condition was confirmed. Namely, with full-plate clamping the third stage of the load history did not mimic the same 82  behavior as in Figure 4.14. In some cases, a decreasing trend was followed by an increasing trend, or vice versa; and in some other repeats, the load in both directions continuously rose. Similar random (uncontrollable) behavior using strict clamping condition was reported in [17]. In a following study, some techniques such as accurately positioning the sample and using aluminum plates with silicon glue were suggested to reduce the level of the resultant tension within yarns during picture frame tests with clamped boundary conditions [70]. However, there was still non-homogenous deformations close the clamp regions, causing error in the characterization data. On the contrary, using needles in the fixture jaws highly assisted in inducing a homogenous deformation and consistent test results (Figures 4.3(a) and 4.13). Moreover, it was noticed that using needle clamps, acceptable test repeatability can be attained even without pre-conditioning of the fabric (as seen from the small range of error bars in Figure 4.13, and also previously reported in [7]). Pre-conditioning of fabrics is normally employed to get repeatable results between tests within a laboratory and among laboratories [80]. The aforementioned observation in the present study, though on one type of fabric, suggests that modifying boundary conditions of tooling to needle clamping may rectify the need for pre-conditioning. While the precise ‘statistical’ effect of boundary condition on forming repeatability of different types of fabrics remains the subject of a future study via ‘hypothesis testing’ tools, it is believed that the main reason for this effect is as follows. Using needle clamps prevents from local bending close to the jaws as discussed in Chapter 3. The local bending causes two sources of discrepancy in the results, one of which is the difference between the actual yarns shear angle and the frame shear angle, while the second consequence is the induced tension in fabrics due to the local bending. According to the presented discussions in this section, the trend of tensile biaxial forces confirmed that the global tension level declines during the shear frame test, which may be deemed as an explanation of the convergence between the response curves in Figure 4.13(a) at higher shear angles. However, it is more accurate to assess the normalized local tensile force in the fabric coordinate	ሺ ௟ܰሻ, rather than the global force. Based on Eq. 4.13, since both numerator and denominator decrease over the shear loading, it cannot be readily presumed that the axial force level also diminishes in the fabric as the shear angle increases. To assess this effect and reveal the yarn’s axial force variation, the normalized local tensile force is plotted 83  using Eq. 4.13, and shown in Figure 4.15(a). According to this figure, the yarns pertinently experience lower tensile forces as the shear angle proceeds.   (a) (b) Figure 4.15 Normalized global force components of the biaxial motors during shear frame testing with 3.1 N/mm pre-tension; (a) the longitudinal (tensile) component, (b) the shear component. Note that as outlined in section 4.3, during the analysis ࡺ࢒	ܑܛ		܉ܛܛܝܕ܍܌	ܜܗ	܊܍	܉ܔܗܕܗܛܜ	܍ܙܝ܉ܔ	ܜܗ	ࡺ࢈࢒ in the shear tests with a small yarn pre-tension level The main underlying point helping to eliminate the conflict is that the comparison between the global shear loads measured by the shear motor load cell in two tests with distinct pre-tension levels is not a good representation of the coupling effect of yarn tension on fabric shear. As substantiated in the analytical part (Section 4.3), the biaxial motors due to their alignment relative to the frame arms not only exert yarns elongation, but also an extra shear force. Consequently, an accurate collation on the coupling effect was accomplished by comparing the normalized local shear force applied to the yarns ( ௦ܰሻ. To obtain the variation of this force, first the contributed shear component of the biaxial load was obtained via Eq. 4.12 and results were shown in Figure 4.15(b). Comparison of Figures 4.15(a&b) illustrates that in spite of the reduction in the longitudinal force component, the shear force component increases at the frame coordinate, owing to their dependence on the angle between the arms. Finally, the normalized local (fabric) net shear force can be obtained using the procedure presented in section 4.3 (results shown in Figure 4.13(b)). In fact, this figure discloses that the deviation between the two curves corresponding to pre-tensioned samples is maintained by the material, and applying yarn tension increases the shear rigidity of the fabric at both low and high shear angles. This macro-level behaviour may be explained at meso-level by the fact that the shear resistance of woven fabrics stems from the friction between yarns and within yarns (between filaments), which is increased by applying tension. 84  4.4.2.2 Effect of tension on shear resistance of fabrics As proved in section 4.4.2.1, the effect of tension on the shear behavior of the fabric should be assessed by comparing the local shear force-shear angle curves, rather than the global ones, of the sample under different levels and modes of pre-tensions. Figures 4.16 (a & b) depict the shear behavior of the fabric under uniaxial and biaxial pre-tension modes, respectively. The data trends show that in both modes, a higher level of pre-tension leads to a higher shear rigidity. The dissipation of energy through friction is regarded as the main sources of shear resistance in fabrics.    (a) (b)   (c)  (d) Figure 4.16 Comparison between the shear behavior of the tested fabric under different pre-tension conditions. a) The shear force from the picture frame tests with uniaxial pre-tension, b) The shear force from the picture frame tests with biaxial pre-tension, c) Variation of the coupling factor for shear tests under uniaxial pre-tension, d) Variation of the coupling factor for shear tests under biaxial pre-tension From Figure 4.16(a & b), it is also seen that the increase in the shear rigidity is more considerable when the applied pre-tension is beyond the straightening region (curves of 3% and 4 % pre-tension). In addition, comparison between Figures 4.16 (a & b) reveals that the tension-shear coupling does not follow the superposition rule. As an illustration, the shear force of the sample subjected to the 4% biaxial pre-tension is larger than twice that under 4% uniaxial  pre-tension. That is, the meso-level features of yarns such as crimping as well as the contact 85  force level at yarn crossovers would be very distinct  between (4%, 0%) and (4%, 4%) strain states. This issue, violation of superposition rule, should be taken into consideration in the proposal of a new constitutive model.  Not only does the tension-shear coupling increase the fabric shear resistance, but also it changes the range (length) of each region of shear response. To elaborate, it is known that the shear behavior of woven fabrics encompasses four regions: a short stiff behavior, followed by a long compliant response, subsequently yarns locking, and finally wrinkling, as shown in Figure 4.17. It can be expected that applying tension slightly reduces the width of the yarns, postponing the locking initiation similar to delay in wrinkling onset. A close-up of the curves in Figures 4.16(a & b) for the pure shear deformation reveals that the initially stiff behavior of the material can continue up to about 4 degrees. Applying yarn pre-tension shortens this range to about 1 degree; a similar observation was reported in [7]. Theoretically, the transition from the first to the second shear region would occur when the external shear force exceeds the critical static friction force between and within the yarns, as discussed in the previous chapter. Yarn tension increases both the external shear force and the critical static friction force, but perhaps to different extends—a notion that requires more investigations through future multi-scale numerical simulations.   Figure 4.17 A typical shear response of woven fabric at macro-level, showing four stages: 1- shear with static friction, 2- shear with dynamic friction, 3- locking, and 4- wrinkling. Notice that the applying yarn tension not only increases the effective shear rigidity, but also changes the range of each shear stage 86  4.4.2.3 Quantification of the tension-shear coupling  In order to quantify the influence of tension-shear coupling on the mechanical behavior of the fabric, the coupling factor was calculated using Eq. 4.23 [7] for each pre-tension mode, the results of which are depicted in Figures 4.16 (c & d).  ܥܨ௧௘௡௦௜௢௡ି௦௛௘௔௥	ሺ௘௙௙௘௖௧	௢௡ ொలలሻ ൌேೞ|ఌభୀఌభ_೛ೝ೐೟೐೙ೞ೔೚೙; ఌమୀఌమ_೛ೝ೐೟೐೙ೞ೔೚೙ேೞ|ఌభୀఌమୀ଴     (4.23) For each pre-tension level, a similar trend, including a sharp reduction in the coupling magnitude followed by a relatively constant trend is observed. Considering the first and second regions of the fabrics shear response (Figure 4.17), such behaviour in Figures 4.16 (c & d) can be expected. In the first region, the contributing mechanisms to shear deformation are the bending and shear of yarns and high static friction, whereas relative movements of yarns (i.e., low dynamic friction) is considered to be the dominant mechanism in the second region. The sharp contrast between the underlying meso-level mechanisms in the two regions is reflected in the trend of curves in Figures 4.16(c & d). In terms of the magnitude, a higher pre-tension has consistently yielded a higher tension-shear coupling factor, though amplifying nonlinearly. Furthermore, by making a comparison between the curves of the same pre-tension (for instance 4%) for uniaxial and biaxial pre-tension modes, it is again understood that superposition cannot be utilized in the coupling analysis of fabrics (e.g., the initial value of 450 in Figure 4.16(d) is not twice the initial value of 150 in Figure 7(c)). 4.4.3 Shear-tension coupling This mode of coupling has not received considerable amount of attention by researchers compared to other applications. However, from safety level point of view, the importance of this mode is highlighted. First, the global results are analyzed to attain the local results, based on which the influence of shear on the axial tensile behavior of woven fabrics is assessed, and subsequently, the observed effect is quantified for further comparison with other coupling modes. 87  4.4.3.1 The analysis of the global results of simultaneous loading Under a simultaneous biaxial extension-picture frame loading mode, the presence of kinematic coupling between the global axial and shear forces first needed to be confirmed, as it was a cornerstone of the analytical framework in section 4.4.3. Figure 4.18 reveals that the required global force of the shear motor under this test is negative (opposite to direction 3 in Figure 4.5), meaning that to induce shear deformation in the frame diagonal direction, the only external force responsible to supply the necessary positive net load is the shear component from the biaxial motors. In other words, it confirms that biaxial motors contribute to shear deformation for the current combined loading setup. To further scrutinize this observation, an extra experiment was undertaken in which the shear motor was turned off after imposing 10 degrees of shear and then detached from the device, resulting in a complete degree of freedom in the shear direction 3 in Figure 4.5. It should be mentioned that during this trial the fixture and the sample did not come back to their original state, owing to the existing friction. Thereafter, the biaxial motors were turned on to start supplying tension. It was observed that the angle between the arms of the picture frame declined from the initial 80° (10° of shear) to 70° (20° of shear); hence, reconfirming that the biaxial motors via the kinematic coupling within the setups can help to apply a fraction of the shear on the material under simultaneous loading modes.    (a) (b) Figure 4.18 (a) Change in the required load of the shear motor over time in the simultaneous loading test (showing the kinematic coupling effect between the biaxial and shear motors), (b) the global force of the biaxial motors under this mode, which reaches to a considerably high load magnitude Under this loading mode, however, based on the geometrical analysis in section 4.3 (Figure 4.7) the local angle between the warp and weft yarns does not follow the frame angle. 88  According to Eq. 4.8, this deviation for a symmetric loading condition can be calculated over deformation through Eqs. 4.3-4.7 and measured data of the global displacements. The result has been plotted in Figure 4.19. Next to the analytical solution, the local angle between adjacent yarns was measured by employing image analysis (taking photos at the end of each test), confirming the validity of the analytical solution for estimating α (error less than 2%). Figure 4.19 illustrates that this deviation between the rhomboid sample shape and the rhomboid fixture is nonlinear and it becomes larger at higher deformation ranges. At the end of the simultaneous test, the angle in the fabric was 25° while the angle of the main fixture was 30.8°.  Figure 4.19 Deviation of the shear angle within the sample under simultaneous loading from the shear angle of the picture frame fixture based on the analytical procedure. The deviation between the curve and the dashed line represents the difference between the shear angle of the frame and fabric (૛ࢻ). 4.4.3.2 Influence of shear deformation on the axial tensile behavior of woven fabrics In order to analyze the shear-tension coupling, the longitudinal normalized forces under the biaxial tensile test and the simultaneous biaxial tensile-shear test should be obtained and compared. The global measured values in the simultaneous biaxial tensile-shear loading were transferred based on the approach presented in section 4.3, to arrive at the normalized local forces and strains along the yarns. Figure 4.20 (a) compares the local tensile behavior of the fabric under the two tests. Under the simultaneous test, as the normal strain reaches to 4%, the shear angle is raised to 12 degrees. Data in Figure 4.20(a) demonstrate that the shear deformation causes a more compliant tensile response of the yarns. More compliance means that not only the slope of the force response is smaller, but also the transition from straightening to stretching phase is postponed. This phenomenon may mean that designers in defining the 89  final mechanical properties of the part should take into account the two-way tension-shearing coupling imbedded during the stamping, especially at tight curvatures.  The primary justification for these observations would be the increase in the thickness and crimping as the fabric undergoes shear deformation. This point was substantiated in chapter 3 through continuum mechanics approach. Numerical simulations have similarly shown an increase in the crimping of yarns during shear deformation [49]. Another possible justification is that the fabric shear would yield less contact areas at crossovers and subsequently cause less interactions between the warp and weft yarns. Given the similarity of woven fabrics to a network of curved beams supporting each other [84], the less interaction between yarns can result in more complaint material medium.    (a) (b) Figure 4.20 (a) Comparison of the fabric’s tensile behavior under biaxial and simultaneous biaxial-shear loading modes, implying the effect of fabric shear on its tensile behavior; b) The calculated shear-tension coupling factor                                              (a)                                                           (b)  Figure 4.21 Idealized change in the contact area at crossover points during (a) biaxial and (b) shear deformation; Notice that the right side rhomboid area is smaller, assuming identical yarn width in both cases 90  To accurately compare the tensile behavior of fabrics under different deformation configurations, the elongation rate must also be checked, as it has been well-established that polymer composites are rate/time-dependent materials [85], even in the dry form as was seen in Figure 4.14. These composites in general behave stiffer at high deformation rates (e.g., 10/s), when compared to their behaviour at a quasi-static deformation rate (e.g., 0.01/s). On the other hand, based on the obtained geometrical equations (Eqs. 4.3 and 4.6), the elongation of tows under the simultaneous loading is not exactly the same as that of the biaxial motors (ݑ௕ሻ	at a given time. Hence, the deformation rates may not be exactly identical in the conducted tests, though they are all in quasi-static regime. The fiber-direction elongation over time for both biaxial and simultaneous loading experiments is shown in Figure 4.22. According to this figure, the elongation in the first stages of loading is almost identical for both modes, whereas there is a deviation between them in the later stages of loading. The slopes at each point of these curves were furthermore calculated and results confirmed that the rate of elongation in the conducted biaxial test remain constant (about 4.5 mm/min). On the other hand, the initial strain rate for the simultaneous loading mode was about 4.5 mm/min and diminished to about 2.9 mm/min as the fabric shearing proceeded. However, there is little probability that the dissimilarity seen between the two force responses in Figure 4.20(a) is due to this minor strain rate difference in Figure 4.22. Nevertheless, to decouple the effect of deformation rate and the loading mode on the tension-shear coupling characterization of fabrics, series of new tests with more sophisticated combined loading devices are worthwhile in future studies.  Figure 4.22 Comparison between the elongation of fiber yarns during biaxial and simultaneous loading modes 91  4.4.3.3 Quantification of the shear-tension coupling factor To investigate, the coupling factor in the shear-tension mode was defined as: ܥܨ௦௛௘௔௥ି௧௘௡௦௜௢௡	ሺ௘௙௙௘௖௧	௢௡ ொభభ,ொమమ,ொభమ,ொమభሻ ൌ ଵܰሺݏ݅݉ݑ݈ݐܽ݊݁݋ݑݏ	ݐ݁ݏݐሻଵܰሺܾ݅ܽݔ݈݅ܽ ݐ݁݊ݏ݈݅݁	ݐ݁ݏݐሻ (4.24) Since the shear deformation affects the tensile behavior of fabrics, the above-mentioned coupling factor changes the magnitude of ܳଵଵ, ܳଶଶ, ܳଵଶ, ܳଶଵ. Figure 4.20(b) depicts the change of shear-tension coupling factor over the applied 4% longitudinal strain range. The coupling starts with a sharp decrease followed with a nearly constant trend. It could be hypothesized that in the straightening phase of yarns, the shear deformation increases the crimping of yarns while the tensioning decreases the crimping (i.e., a high interaction between the two influencing factors), whereas in the stretching phase there is no appreciable further variation in the crimping, resulting in a constant coupling factor.  4.5 Summary of findings This chapter included a comprehensive experimental study and an analytical framework to characterize different variations of inherent coupling within woven fabrics. To be more precise, regarding the technical challenges reported in the literature to conducting shear frame tests, the advantages of using needles over fully clamped boundary condition were evidenced. Moreover, a characterization plan was suggested to capture the significance of coupling in woven fabrics. Upon determination of the trends and significance of different coupling modes on the mechanical behavior of a fibreglass woven fabric, existing conflicts in the literature could also be resolved and explanations were provided as to their cause. The effect of yarn tensioning on the shear rigidity of woven fabrics should be viewed as the greatest importance when developing forming simulations of woven fabrics, in that such coupling factor could reach up to 50  (i.e. 50 times magnification of the shear rigidity) under a practical yarn elongation of 2%. The effect will be magnified further at large strains. In practice, during fabric forming e.g. the yarns tension may be induced to blank-holder forces. Tension-tension coupling is deemed as the second important type to be taken into account, as it showed a cumulative effect on both the effective normal moduli (up to 3 times change) and 92  the effective Poisson’s ratio of the fabric (up to 10 times change). The shear-tension coupling would be viewed as the least critical coupling. Nevertheless, in terms of design of the final cured composite part, accounting for the latter coupling is still deemed critical in order to predict reliable performance levels of the manufactured parts. Namely, under the simultaneous tension-shear mode (e.g. similar to forming of double-curvature shapes under blank holder pressure [7, 17]), it can be seen that the non-orthogonal angle between yarns due to shearing can reduce yarns’ effective tensile modulus.    93  Chapter 5: Proposing a Coupled Non-Orthogonal Constitutive Model for Woven Fabrics 5.1 Overview Inherent coupling modes and their notable magnitudes (Chapter 4) have not been implemented to date in the forming simulations of fabric reinforcements. Coupling should be incorporated in numerical optimization routines in order to accurately predict the deformation of the material under complex forming set-ups, and more importantly to predict realistic yarn tension levels that can suppress wrinkles. Towards this goal, the present chapter proposes and implements a new coupled non-orthogonal model which predicts not only the stress-strain path, but also the critical point (shear wrinkling) of a plain woven fabric, under combined loading conditions similar to the draping processes. Furthermore, the chapter will reveal that the concept of inherent coupling raises a new issue in fabric forming simulations; the load history dependency of the fabric. Accordingly, the constitutive model is enhanced to a hypoelastic form to capture the load path dependency of the forming material. Finally, the constitutive model is integrated with the presented analytical criterion in Chapter 3 to account for the occurrence of wrinkling, based on the fabric properties and the level of tension applied during forming. To show its application, the model is implemented in ABAQUS via a UMAT code to predict the stress and strain fields as well as the onset of wrinkling under large shear deformation. 5.2 The meaning of coupling in the course of constitutive modeling         As stated in the previous chapters, coupling affects the effective mechanical properties of woven fabrics, namely the stiffness and strength. Recalling the stress-strain relationship based on the Hook’s law presented in Eq. 2.2, in order to take the inherent coupling into account, Eq. 2.2 must be developed in a way that each component of stiffness matrix is a function of the strains the strains that are not multiplied by the aforementioned stiffness component. To exemplify, ܦଵଵ which is multiplied by ߝଵ should depend on ߝଶ or ߝଵଶ to take into account an inherent coupling. Notice that if ܦଵଵ is a constant number, the tensile behavior of the material (ߪଵversus ߝଵ) would be linear, while a non-linear response is captured when ܦଵଵis function of ߝଵ (see also Figure 2.5). 94  5.3 The proposed coupled non-orthogonal constitutive model       Based on multi-scale mechanisms discussed in the previous chapters, an ideal fabric constitutive model should take the following list of features into consideration:  Yarns relative rotation using a local non-orthogonal coordinate system;  Non-linear tensile behavior of the fabric: ܦ௜௜ ൌ ݂ሺߝ௜௜ሻ, 	݅ ൌ 1,2;  Non-linear shear behavior of the fabric: ܦ଺଺ ൌ ݂ሺߛଵଶሻ;  Interaction between warp and weft yarns (tension-tension coupling):      ܦ௜௜ ൌ ݂൫ߝ௝௝൯, 	݅	ܽ݊݀	݆ ൌ 1,2; and  Effect of the in-plane tension on the shear rigidity of fabric (tension-shear coupling):  	ܦ଺଺ ൌ ݂൫ߝଵଵ, 		ߝଶଶ൯.   Note that the above-mentioned issues come into effect when normal strains are non-negative. The stiffness for compression behavior of fabrics is often considered to be a small constant value, given the weakness of dry yarns against compression. Updating the direction of yarns during shear deformation can be achieved by employing covariant and contravariant coordinate systems, as depicted in Figure 5.1. The bases of the local covariant and contravariant systems can be related to the global Cartesian coordinate system as shown in Eqs. 5.1 (a) and (b), using the transformation matrix presented in Eq. 5.1(c) [39-42]. ߜ௝௜ is the Kronecker delta while ݃௜, ݃௜, and ௝݁ are the base vectors of the local covariant system, local contravarient system, and global orthogonal coordinate system, respectively.   (a) (b) Figure 5.1  Using the non-orthogonal coordinate system to trace the fibers rotation under large shear. ࣂ is the angle between warp/weft yarns in the deformed configuration, and ࢻ is half of the engineering strain (ࢽ૚૛) calculated for each element during finite element implementations using a regular shell element  95  ݃௜ ൌ ௜ܲ௝ ௝݁       5.1(a) ݃௜. ݃௝ ൌ ߜ௝௜       5.1(b) ܲ ൌ ൣ ௜ܲ௝൧ ൌ ൤ cos α sin αcosሺα ൅ θሻ sinሺα ൅ θሻ൨ 5.1(c)   Eqs. 5.2 (a, b, c) [39-42] express how the stress and strain components in the global orthogonal coordinate system can be found based on the stresses and strains in the local non-orthogonal coordinate system.  ߪ ൌ ்ܴܶ. ߪ෤ 5.2(a)ߝ̃ ൌ ܴܶ. ߝ 5.2(b)ܴܶ ൌ ቎ሺ ଵܲଵሻଶ ሺ ଶܲଵሻଶ 2 ଵܲଵ ଶܲଵሺ ଵܲଶሻଶ ሺ ଶܲଶሻଶ 2 ଶܲଶ ଵܲଶଵܲଵ ଵܲଶ ଶܲଶ ଶܲଵ ଵܲଶ ଶܲଵ ൅ ଵܲଵ ଶܲଶ቏ൌ ቎ܿ݋ݏଶα ܿ݋ݏଶሺα ൅ θሻ 2 cos α cosሺα ൅ θሻݏ݅݊ଶα ݏ݅݊ଶሺα ൅ θሻ 2 sin α sinሺα ൅ θሻsin α cos α sinሺα ൅ θሻ cosሺα ൅ θሻ sinሺ2α ൅ θሻ቏ 5.2(c)  Where ߪ and ߝ represent the stress and strain tensors in the global orthogonal coordinate system, while the local stress and strain tensors are denoted by ߪ෤ and ߝ̃, respectively. ߪ෤௜௝ are the contravariant components of the stress tensor in the covariant coordinate system. ߝ௜̃௝ are the covariant components of strain tensor. Moreover, TR is the transformation matrix whose components are expressed in detail in Eq. 5.2(c).     The stress-strain relationship in the local non-orthogonal system should hold the following form to consider the above addressed coupled behavior of woven fabrics. Namely, each stiffness function is decomposed into two components; the pure modulus (ܦ෩଴ሻ and the coupling induced modulus (ܦ෩௖ሻ. ൥ߪ෤ଵଵߪ෤ଶଶߪ෤ଵଶ൩ ൌ ቎ܦ෩଴ଵଵሺߝଵ̃ଵሻܦ෩௖ଵଵሺߝଵ̃ଵ, ߝଶ̃ଶሻ ܦ෩଴ଵଶሺߝଶ̃ଶሻܦ෩௖ଵଶሺߝଵ̃ଵ, ߝଶ̃ଶሻ 0ܦ෩଴ଶଵሺߝଵ̃ଵሻܦ෩௖ଶଵሺߝଵ̃ଵ, ߝଶ̃ଶሻ ܦ෩଴ଶଶሺߝଶ̃ଶሻܦ෩௖ଶଶሺߝଵ̃ଵ, ߝଶ̃ଶሻ 00 0 ܦ෩଴଺଺ሺߛ෤ଵଶሻܦ෩௖଺଺ሺߝଵ̃ଵ, ߝଶ̃ଶሻ቏ ൥ߝଵ̃ଵߝଶ̃ଶߛ෤ଵଶ൩ (5.3)96  5.4 Model identification   To identify the model functions, test data acquired in Chapter 4 was utilized. However, for validation purposes, some additional tests were  carried out per Table 5.1. Owing to the nearly balanced behavior of the tested fabric, other stiffness functions not shown in Table 5.1 were determined from symmetry. For instance, the obtained relation between ܦ෩଴ଵଵ and ߝଵ̃ଵ can be employed for finding the relationship between ܦ෩଴ଶଶ and ߝଶ̃ଶ. An analytical framework proposed in Chapter 4 was used to transform the global force-displacement measurements to the local non-orthogonal stress-strain values. In the sub-sections to follow, details of coupling effects on constituting the component of the stiffness functions will be discussed. Table 5.1. The identification strategy proposed to determine the stiffness functions in the constitutive model and validate the model # Experiment Goal 1 Tensile test in direction 1 with no transverse pre-tension To identify ܦ෩଴ଵଵand ܦ෩଴ଶଵ 2 Tensile test in direction 1 with constant transverse pre-tension (1, 2, 3, and 4%) To determine ܦ෩௖ଵଵand ܦ෩௖ଶଵ 3 Picture frame test without pre-tension To identify ܦ෩଴଺଺ 4 Picture frame test with pre-tension in one and two directions To determine ܦ෩௖଺଺ 5 Biaxial tensile loading To validate the model 6 Simultaneous biaxial tension-picture frame test To validate the model  5.5 Considering the load history dependency of woven fabrics   The loading path dependency of the stress-strain state of the dry fabric is a critical issue linked to the coupling, and not addressed in the earlier literature. Namely, the underlying hypothesis is that the instantaneous magnitude of stiffness modulus at each loading instant depends on the instantaneous values of all corresponding strain components. Hence, the loading path would affect the final stress state. If there was no such dependency, according to the presumed model in Eq. (5.3), for any given strain state, there should be only one distinct stress state, no matter how the final strain values has been reached. To assess the hypothesis, 97  Table 5.2 and Figure 5.2(a) compare the stress state for an identical strain state (2%, 2%) between two loading scenarios: a longitudinal tensile test up to 2% strain with a 2% transverse pre-tension, and a biaxial tensile test up to 2% strain in both longitudinal and transverse directions at the same time. The clear difference between these two tests in Table 5.2 and Figure 5.2(a) proves the load history dependency of the woven fabric. In spite of the initial jump in the curve with 2% transverse pre-tension, which is due to the Poisson’s ratio effect, the stress at the ending point (2%, 2%) is higher in the biaxial tensile test. That is, for a given deformation level, the growth of the force – stiffness – is higher in the biaxial loading. In fact, it is observed that more compliant behavior in the sequential loading condition due to the considerable increase in the crimping of yarns in the longitudinal direction from the first step of loading. In other words, as discussed in Chapter 4, the transverse elongation raises the crimping of the longitudinal yarns, causing more compliant response. As a result, in the sequential biaxial tensile test, because a considerable transverse tension is applied firstly and then tension in the longitudinal direction is imposed, more compliant behavior should be expected.   In addition to the above load-history dependency in the material tensile behavior, it is found that the tension-shear coupling is also affected by the loading path. To demonstrate, a comparison was drawn between a simultaneous biaxial tensile-picture frame test and a picture frame test with 4% pre-tension on both warp and weft directions, as presented in Table 5.3 and Figure 5.2(b). While the tension was kept constant over the shear loading in the picture frame test with 4% pre-tension, in the simultaneous experiment the tension was concurrently increased from 0 to 4% as the shear angle grew from 0 to 12 degree (0 to 0.2 rad). In the PF test with constant 4% biaxial pre-tension, because the fabric specimen is subjected to high level of tension from the first steps of loading, the dissipated energy is significantly higher, causing higher shear rigidity, and eventually the higher shear stress from the initial steps of shear per Table 5.3 and Figure 5.2(b). A practical meaning of this effect is. that e.g. the required tension to prevent wrinkling in a given 3D draping process is dependent on whether the tension has been applied before starting the forming process or in the middle of forming process.   98  Table 5.2. Different biaxial stress states at an identical strain state, showing the presence of loading path dependency in the fabric behavior under individual deformation modes. Strain ሺߝ௫, 	ߝ௬ሻ Stress in biaxial tensile test ሺߪ௫, ߪ௬ሻ ሺܯܲܽሻ Stress in sequential tensile test ሺߪ௫, 	ߪ௬ሻ ሺܯܲܽሻ (0.02, 0.02) (3.46, 3.46) (2.08, 2.95) Table 5.3. Different shear stress states at an identical biaxial strain state, showing the presence of loading path dependency in the fabric behavior under combined loading modes. Strain ሺߝ௫, 	ߝ௬, 	ߛሻ Shear stressሺܯܲܽሻ in simultaneous test up to 4% pre-tension Shear stressሺܯܲܽሻ in shear test with 4% pre-tension (0.04,0.04,0.2) 0.65 4.08  (a) (b) Figure 5.2 Different response of the woven fabric to different loading paths with the same final deformation condition, proving the presence of load history dependency in woven fabric; (a) Comparison between biaxial tensile test and uniaxial tensile test with 2% transverse pre-tension, (b) Comparison between simultaneous shear-tension test and picture frame test with 4% biaxial pre-tension   In view of the proven loading path dependency above, the fabric stress-strain relationship in Eq. 5.3 should be re-written in an incremental form as shown in Eq. 5.4. It should be noted that the stiffness functions in the hypoelastic model in Eq. 5.4 are now the tangential moduli, i.e. the instantaneous slope of the stress-strain curve. 99  ൥݀ߪ෤ଵଵ݀ߪ෤ଶଶ݀ߪ෤ଵଶ൩ൌ ቎ܦ෩଴ଵଵሺߝଵ̃ଵሻܦ෩௖ଵଵሺߝଵ̃ଵ, ߝଶ̃ଶሻ ܦ෩଴ଵଶሺߝଶ̃ଶሻܦ෩௖ଵଶሺߝଵ̃ଵ, ߝଶ̃ଶሻ 0ܦ෩଴ଶଵሺߝଵ̃ଵሻܦ෩௖ଶଵሺߝଵ̃ଵ, ߝଶ̃ଶሻ ܦ෩଴ଶଶሺߝଶ̃ଶሻܦ෩௖ଶଶሺߝଵ̃ଵ, ߝଶ̃ଶሻ 00 0 ܦ෩଴଺଺ሺ2ߝଵ̃ଶሻܦ෩௖଺଺ሺߝଵ̃ଵ, ߝଶ̃ଶሻ቏ ൥݀ߝଵ̃ଵ݀ߝଶ̃ଶߛ෤ଵଶ൩(5.4)  5.6 Determination of the stiffness functions   A phenomenological approach is used to define the shape of the stiffness functions in Eq. 5.4 based on the multi-scale behavior of woven fabrics and the underlying sources of couplings shown in Figure 5.3. In the proposed functions for each stiffness component, each term represents a specific characteristic of the meso-scale behavior of woven fabrics. Regarding the general behavior of fabrics, the two phases of the tensile behavior and four phases of the shear of fabrics are considered. It is to note that the sources of coupling affecting the stiffness components of the fabric cab be interconnected; for instance, per Figure 5.3 the tensile behavior of the fabric in a given direction may be required to be dependent on the bending stiffness of transverse yarns, which is a function of the applied strain on the transverse yarns. Thus, one specific term should be considered for this meso-level mechanism in the corresponding macro-level stiffness function. The general form of stiffness functions identified are presented in Eqs. (5-5) to (5-10), where H is the Heaviside function. The unknown coefficients of functions were curve-fitted (Appendix A) for the tested fabric, based on the results of experiments presented in Table 5.1, except the biaxial tensile test simultaneous biaxial tensile-shear tests. The latter two experiments, which impose the most complex loading conditions on the specimens, were kept for validation purposes; Table 5.1 describes precise role of each experiment in determination of corresponding stiffness functions.   100   Figure 5.3 Concept map for different types of coupling and their effects on the mechanical behavior of woven fabrics ܦ෩଴௜௜ ൌ ቊܦ෩଴ି௦௧௥௔௜௚௛௧௘௡௜௡௚௜௜ ߝ௜௜ ൑ ߝ௧௥௔௡௦௜௧௜௢௡	௣௢௜௡௧ܦ෩଴௜௜൫ߝ௧௥௔௡௦௜௧௜௢௡ ௣௢௜௡௧൯ ߝ௜௜ ൐ ߝ௧௥௔௡௦௜௧௜௢௡ ௣௢௜௡௧    (5.5) ܦ෩௖௜௜൫ߝ௜௜, ߝ௝௝൯ ൌ 1 ൅	ܦ෩௖ି௖௥௜௠௣௜௡௚௜௜ ൫ߝ௜௜, ߝ௝௝൯൅ ܪሺߝ௝௝ െ ߝ௧௥௔௡௦௜௧௜௢௡ ௣௢௜௡௧ሻܦ෩௖ି௕௘௡ௗ௜௡௚௜௜ ൫ߝ௜௜, ߝ௝௝൯ (5.6) ܦ෩଴௜௝ ൌ ൝ܦ෩଴ି௦௧௥௔௜௚௛௧௘௡௜௡௚௜௝ ߝ௝௝ ൑ ߝ௧௥௔௡௦௜௧௜௢௡	௣௢௜௡௧ܦ෩଴௜௝൫ߝ௧௥௔௡௦௜௧௜௢௡ ௣௢௜௡௧൯ ߝ௝௝ ൐ ߝ௧௥௔௡௦௜௧௜௢௡ ௣௢௜௡௧ (5.7) ܦ෩௖௜௝ ൌ 1 ൅ ܦ෩଴ି௜௡௧௘௥௔௖௧௜௢௡௜௝  (5.8) ܦ෩଴଺଺ ൌ ܦ෩଴ି௦௧௔௧௜௖	௙௥௜௖௧௜௢௡଺଺ ൅ ܦ෩଴ି௞௜௡௘௧௜௖	௙௥௜௖௧௜௢௡଺଺ ܪ൫ߛ െ ߛ௧௥௔௡௦௜௧௜௢௡	௣௢௜௡௧൯൅ ܦ෩଴ି௖௢௠௣௔௖௧௜௢௡ ௢௙ ௬௔௥௡௦଺଺ ܪ൫ߛ െ ߛ௟௢௖௞௜௡௚൯ ߛ ൏ ߛ௪௥௜௡௞௟௜௡௚ (5.9) ܦ෩௖଺଺ ൌ 1 ൅	ܦ෩௖ି௦௧௔௧௜௖	௙௥௜௖௧௜௢௡଺଺൅ ܪ൫ߛ െ ߛ௡௘௪	௧௥௔௡௦௜௧௜௢௡	௣௢௜௡௧൯ܦ෩௖ି௦௛௢௥௧௘௡௜௡௚	௘௙௙௘௖௧଺଺൅ ܪ൫ߛ െ ߛ௧௥௔௡௦௜௧௜௢௡ ௣௢௜௡௧൯ܦ෩௖ି௞௜௡௘௧௜௖ ௙௥௜௖௧௜௢௡଺଺  (5.10) 101    The above coupling-induced stiffness functions may in fact be viewed as correction (normalized) factors for the stiffness components, implying that they do not have a unit. Additionally, there are specific boundary conditions that have to be met by the model, specifically due to the coupling. To explain, let’s momentarily assume that there is no coupling in an arbitrary material such as UDs. The value of the coupling induced functions must then be equal to 1 in all deformation modes, such that the model can be simplified to the generalized Hook’s law. In a fabric material with inherent coupling, only under pure (individual) loading cases such as uniaxial tensile test and picture frame test without pre-tension,  the coupling factors must be equal to 1 (but not under the combined loading modes). Mathematically, these boundary conditions are written as follows: ܦ෩௖௜௜൫ߝ௜௜, ߝ௝௝൯ ൌ 1 ݂݅ ߝ௝௝ ൌ 0 5.11(a) ܦ෩௖௜௝൫ߝ௜௜, ߝ௝௝൯ ൌ 1 ݂݅ ߝ௜௜ ൌ 0 5.11(b) ܦ෩௖଺଺ሺߝଵଵ, ߝଶଶ, ߛሻ ൌ 1 ݂݅ ߝଵଵ ൌ ߝଶଶ ൌ 0 5.11(c)  5.7 Validity of the model As discussed in Chapters 2-4, there are a number of multi-scale mechanisms contributing to the deformation of fabrics, causing distinct meso-scale architectural configurations, and in turn different stresses at identical strain states. According to Table 5.1, comparing a specific pair of experiments could identify exclusively the mathematical form of the stiffness coefficients for each particular deformation mode. To examine the general effectiveness and precision of the idendtified model in Section 5.6, a biaxial tensile test (in which tensile deformation is applied on both warp as well as weft yarns at the same time) and a combined loading test (where the fabric undergoes both shear and tensile deformation at the same time), were selected for the validation purposes (Table 5.1). These two validation tests are deemed among the most complex loading conditions, while the simpler ones were utilized in the latter section to identify the stiffness functions. The incremental model of Eq. 5.4 was implemented in Matlab and its results are presented in Figures 5.4 and 5.5. First, Figure 5.4 compares the model prediction and the experimental data for the same loading cases whose results had been used to identify the unknown stiffness factors. Subsequently, Figure 5.5 demonstrates the capabality of the model 102  for simultaneous loading cases which would be more similar to real forming processes where both biaxial tension and shear deformations are present. According to this figure, next to visual inspection, the validity of the model can be justified from two mechanistic point of views as follows.  (a)  (b)  (c) Figure 5.4 Examining the accuracy of the model for datasets employed in the model identification stage; a) under sequential tensile test with 2% transverse pre-tension, , b) under sequential tensile test with 2% transverse pre-tension, c) under picture frame test with 3% biaxial pre-tension 103    (a) (b) Figure 5.5 The validity of the proposed model via comparison between the accuracy of the coupled and uncoupled models using independent datasets, a) for biaxial tensile test, b) for simultaneous biaxial tensile-shear test 5.7.1 The capability of the model to predict the effect of tension-tension coupling  At first glance over Figure 5.5(a), it seems that the predictions of the coupled and uncoupled models for this loading case are not quite different. According to Eq. 5.4, the stress applied to the material under biaxial loading is the summation of two stress components, ܦ෩ଵଵ݀ߝଵ̃ଵ and ܦ෩ଵଶ݀ߝଶ̃ଶ. Recalling the consequences of tension-tension coupling (Section 4.4.1), this coupling mode reduces and increases the overall magnitude of ܦ෩ଵଵ and ܦ෩ଵଶ, respectively.  Such reduction in the longitudinal stiffness and growth in the extension-extension coupling compensate each other; hence, no significant difference between the coupled an uncoupled model predictions is observed.  There is, however, an important distinction between them in Figure 5.5(a): the slope of curves in the stretching phase. Despite the highly accurate prediction of both models up to 2%, there exists considerable deviation of both modeling curves from the experimental one. While the coupled model has slightly better prediction after 2% up to 3.5% elongation, the uncoupled model predicts the endpoint (the stress at (4%, 4%) strain) with a higher precision. This fact is because the uncoupled model overestimates the stiffness of fabric, which compensates for the deviation and finally causes more comparable prediction with the experimental data for the ending points.  It can be stated that the coupled model provides slightly more precise results under the biaxial mode, specially because the slope of the straight line in the stretching phase is similar to the experimental one. Notice that the observed constant deviation of the coupled model stems from the nature of the incremental form in Eq. (5.4); that is, ignorable errors in the 104  estimation of tangent modulus at each of the steps are accumulated. More closely, the reason of this difference is in fact the earlier occurrence of transition from the first phase to the second phase in the biaxial loading compared to the uniaxial loading. The introduced coupled model focuses only on the effect of couplings on the stiffness moduli in Eq. 5.4, while the transition point is assumed to be independent of the coupling mode. This transition point was earlier observed to be nearly unchanged in the sequential biaxial tensile experiments (Figure 4.11(a)). However, in the biaxial tensile loading, because both groups of yarns tend to unbend (decrimp) and lose their waviness, the yarns of each yarn group resist the unbending of the other one, causing earlier initiation of the second phase. To capture this effect, the coupled model was further modified by adjusting the transition point from 0.025 to 0.02. The prediction of the updated coupled model closely followed the experimental behavior (Figure 5.5(a)). 5.7.2 The effectiveness of the model to predict tension-shear coupling Figure 5.5(b) compares the coupled and uncoupled models with the experimental results for the simultaneous loading mode. According to this figure, the coupled model has considerably better accuracy in comparison to the uncoupled one, which suggests the significance of taking tension-shear coupling into account. 5.8 A coupled numerical model for woven fabrics  The presented coupled constitutive model in Section 5.5, can now be integrated with the criterion for shear wrinkling of fabrics developed in Chapter 3 in order to arrive at a more comprehensive material model for simulation purposes. Recalling Figure 2.4, a comprehensive material model should predict not only the in-plane stress-strain response, but also its critical end point: wrinkling of woven fabric in terms of coupling. Employing such a material model in numerical simulations could help to predict the deformed shape of fabrics (stress and strain fields) in general, and the wrinkles location and shape in particular. Figure 5.6 demonstrates the simulation procedure implemented via a user material subroutine (UMAT) in ABAQUS. According to this figure, the coupled non-orthogonal hypoelastic model (Eq 5.4) is used in each time increment to measure the stress and strain values in each element. Then, the shear wrinkling criterion (Eqs. 3.18 or 3.19) is checked for each single element. If there exists low and ignorable pressure on the blank holder, the deformation would be more similar to BE test and Eq. 3.18 should be used. On the other hand, if high pressure is applied on the blank holders, 105  Eq. 3.19 would be more appropriate. The fulfillment of the criterion condition (i.e., Eq. 3.18 or Eq. 3.19) means the occurrence of wrinkling, implying that the forming process parameters such as applied pressure on blank holders should be optimized to avoid such wrinkle initiations.                         Figure 5.6 The suggested simulation framework for woven fabrics 5.8.1. Implementation of the suggested simulation framework in ABAQUS A customized material subroutine (Appendix B) was written in ABAQUS/Standard, where S4R elements were employed for forming simulation purposes, as suggested in [86]. At each increment, the strain components as well as incremental strain components in the global orthogonal coordinate system were obtained and converted to the strain components in the local non-orthogonal coordinate system using Eqs. 5.2(b&c). Thereafter, the pure moduli and coupling induced moduli terms were calculated in terms of the new strain values, based on which the stiffness matrix could be updated. Subsequently, the local non-orthogonal and the global orthogonal incremental stress components were determined. Given the stress state in each previous increment and the obtained increase in the stress value, the stress in the current step was calculated.  Moreover, an additional state variable was defined based on the introduced criterion for wrinkling onset (Eq. 3.25) to identify the vulnerable locations of the preform to wrinkling. Namely, if in an element the proposed criterion was fulfilled, the state variable value was changed from zero to one. Accordingly, the manufacturing process should be modified and simulation is re-conducted.  5.8.2. Validity of the numerical model The written UMAT code (Appendix B) was implemented in ABAQUS, and the stress-strain curves were obtained for the experiments presented in Table 5.1. Thereafter, the Determine stress and strain fields using a coupled non‐orthogonal hypoelasticconstitutive model.Define a state variable for the wrinkling criterion and then check the fulfilment of the criterion for each single element in each time increment. If the criterion is fulfilled, the manufacturing process parameters should be altered and the forming is resimulated.106  numerical and analytical results were compared; Figure 5.7(a & b) indicate a fair agreement between the UMAT implemented in ABAQUS and the coupled model results in terms of stress-strain behavior. In addition, the functionality of the state variable to predict wrinkling onset was assessed using simulation of a bias extension test. Figure 5.7(C) demonstrates the correct prediction of wrinkling initiation in the bias extension test, using the developed procedure per Fig 5.6. Bias extension test was selected for the validation of correct prediction of wrinkling location because the current model is capable of predicting the in-plane coupled behavior of fabrics; there are two in-plane tests to capture wrinkling, namely bias extension and picture frame tests.  The shear angle of all the elements in a simulated picture frame test is ideally the same, implying no chance of the assessment of the effectiveness of the model given that all the elements reach the wrinkling angle at the same time. However, by modelling bias extension test in which there is heterogeneous deformation field, the validity of the model in prediction of wrinkling location can be examined.                                              (a)                       (b)  (c) (d) Figure 5.7 Validation of the UMAT code implemented in ABAQUS in terms of stress-strain behavior (a, b & c) and wrinkling prediction (d); a) Accurate prediction of longitudinal stress under sequential tensile test with 2% transverse pre-tension, b) The transverse stress under the sequential tensile test with 2% transverse pre-tension c) Schematic of stress in direction 1 of the specimen under sequential tensile test with 2% transverse pre-tension, d) Correct prediction of wrinkling onset in the woven fabric under bias extension test using the developed material model 107  5.8.3. A remark on low bending rigidity of fabrics As discussed in Section 3.3.2.4, one of the characteristics of woven fabrics is their significantly lower bending stiffness compared to their tensile modulus [87]. In other words, the bending rigidity obtained using classical beam theory based on the tensile modulus and the cross section of the materials, as currently used in several finite element software programs, is much higher than the real flexural stiffness value of the fabric at dry level. Low bending rigidity of dry fabrics is due to their discrete nature –fabrics are made of a number of wavy yarns which can slip over each other. More importantly, the discontinuity of yarns which constitute thousands of not perfectly tightened filaments to each other is another source of low bending rigidity. Recently, new test methods were developed to accurately characterize low bending rigidity of woven fabrics [30, 88-91]. Also, as implemented in the material model in this chapter, the compressive stiffness of fabrics should be presumed to be a low constant value (33MPa which was the slope of the initial instants of the decrimping phase of fabric tensile behavior), while the tensile stiffness should be considered to be much higher and variable due to the coupling. Such asymmetric (tensile versus compression) behavior of fabrics can be another source of their low bending rigidity, specially for thicker or consolidated fabrics. Although this asymmetry was implemented in the presented UMAT model (Appendix B), it may not accurately predict the bending behavior of fabrics since other reason of low bending rigidity, the discrete nature, has been overlooked. Conducting a number of bending experiments, the validity of the presented material model against bending modes of fabrics should be assessed in the future studies. Moreover, the influence of tension on the overall flexural rigidity of fabrics, i.e. tension-bending coupling, should be considered in the future studies.    Finally, it is worth adding that the bending stiffness of fabrics is distinct from that of single yarns which was addressed in wrinkling analyses in Chapter 3. Namely, the bending of woven fabric is dependent on bending of individual yarns as well as other parameters such as the extent of crimp. Another difference between these two bending rigidities is that the bending stiffness of yarns affects the ‘shear wrinkling’ which is a meso-scale, local deformation distortion mode upon fiber locking, whereas the bending of woven fabric would show its 108  influence on wrinkles induced due to the macro-scale global compression forces (i.e., ‘compression wrinkling’), as shown in Figure 5.8.                      Figure 5.8 Comparison between two modes of wrinkles in a woven fabric under a 3D forming test. Compression wrinkle: A global phenomenon; Shear wrinkle: A local phenomenon. This trial illustrative draping test was done a TWINTEX® TPP60N22P-060 fabric with hemispherical punch with diameter of 120mm 5.9 Summary of findings This chapter presented a new coupled hypoelastic constitutive model—stress-strain relationship—to mimic the in-plane mechanical behavior of woven fabrics under coupling, based on the understandings and data in Chapters 3 and 4. The underlying sources of coupling were incorporated in the stress-strain relationship and the model was identified and validated via a step-by-step procedure. Rooted in the coupling effect,  the dependency of the response of woven fabrics on the loading path was proven. Accordingly, the model was enhanced to a hypoelastic formulation to capture this effect. Although the model was trained with the datasets from simple loading cases such as uniaxial extension and sequential loading tests, it could predict the behavior of woven fabrics under more complex combined loading conditions including bixial tensile test and simultaneous biaxial tensile-shear test. Finally, the material model was completed by integration of the hypoelastic constitutive relations with the wrinkling criterion presented in Chapter 3. The final model was successfully implemented in an ABAQUS simulation.   109  Chapter 6: Conclusion 6.1 Summary Textile fabric composites enjoy advantages such as better out-of-plane impact resistance and superior formability compared to their unidirectional counterparts. Consequently, they are often used in applications where high impact resistance is required or a complex shape needs to be formed. The analyses of this class of composite materials, however, is cumbersome due to the multi-scale architecture of fabrics and the ensuing couplings between different families of yarns under different deformation modes. This thesis aimed to bring a new insight into understanding and modeling the mechanical behavior of woven fabric materials under their inherent couplings.  In Chapter 2, the concept of inherent coupling was introduced in detail and distinguished from the commonly used coupling definition in mechanics of materials. Chapter 3 presented an enhanced understanding of approaches used to characterize the shear behavior of woven fabrics, the most dominant deformation of woven fabrics subjected to forming processes. In fact, although the bias extension test and picture frame tests have been used over the past decades for this purpose, proposing a new theoretical ‘pure’ shear’ theory through a multi-scale study explained why woven fabrics experience distinct deformation fields under these two characterization modes. Moreover, distinct equations to predict the locking and wrinkling angles under BE and PF tests were proposed for plain weaves, based on the established shear theory. A fair agreement between the analytical and experimental results were observed, suggesting the accuracy of the theory. The results on a plain carbon fabric showed that the locking and wrinkling arise 30 degrees earlier under the BE test as compared to the PF test. Finally, a practical industrial guideline to select appropriate characterization methods for characterization purposes was outlined.    Subsequently, in Chapter 4, a comprehensive experimental plan along with an analytical framework to characterize the mechanical behavior of fabrics in general, and different types of couplings in particular, was introduced. The analytical procedure transformed the global forces and displacements to local non-orthogonal stresses and strains. Thereafter, the test results on a typical plain PP/glass fabric were analyzed and interpreted in terms of various multi-scale deformation sources. Also, this chapter could resolve the existing 110  conflicts in the literature related to tension-tension and tension-shear coupling modes. Results showed that tension-shear coupling and tension-tension coupling are the most important modes which should be taken into account. To be more specific, shear modulus can be magnified by 50 times based on a practical applied tension (up to 2%). Also, tension-tension coupling could increase the longitudinal modulus and the Poisson’s ratio by 3 and 10 times, respectively.  Eventually, Chapter 5 consolidated the enhanced knowledge and experimental results achieved from the previous chapters into a mathematical form:  a new form of the fabric stress-strain relationship. In comparison with the past uncoupled modeling approach, the new coupled non-orthogonal hypoelastic constitutive model was agreeably able to predict the response of fabric under complex in-plane loading conditions, such as simultaneous tensile-shear which is similar to real forming processes. Finally, it was shown how this model can be combined with wrinkle onset criterion in Chapter 3 for realistic numerical simulation purposes.  Specific contributions of the work may be summarized as follows.  6.2 Contribution to knowledge  Distinguishing between the inherent coupling and the general coupling presented in Hook’s law through defining the inherent coupling in the course of stress-strain relationship;  Introducing a new theory for the shear behavior of woven fabrics, called multi-scale pure shear deformation;  Revealing the distinction between fabric multi-scale deformation under bias extension and picture frame characterization tests;  Proposing new equations for the prediction of locking and wrinkling angles of plain woven fabrics under the bias extension and picture frame test;  Suggesting a preliminary industrial guideline for suitable characterization and optimization of boundary conditions during forming; 111   Determination of the role of each coupling type (tension-tension, tension-shear, and shear-tension) in the mechanical behavior of woven fabrics and understanding the underlying multi-scale sources;  Proposing a new analytical procedure and experimental strategy to correctly characterize the mechanical behavior of woven fabrics under combined loading modes;  Resolving the existing conflicts observed in the literature regarding the influence of coupling on the mechanical behavior of woven fabrics, using the developed analytical framework;  Developing a new coupled non-orthogonal hypoelastic model to more predict the behavior of woven fabrics under individual and combined tension and shear modes; and  Implementing a preliminary numerical model for capturing both in-plane response as well as the wrinkling onset in woven fabrics under coupling. 6.3 Limitations Although this thesis attempted to fill the gap between the academic and technical/industrial levels of the knowledge in woven fabrics area, there are a number of limitations which should be addressed in the future studies.  Only the plain architecture of woven fabrics was investigated in this study. It is expected that the significance of coupling would be different for other weave patterns such as twill and satin; however, the general forms of introduced characterization procedure and the constitutive model would be still applicable to new fiber architectures. Moreover, the derived equations for locking and wrinkling should be adjusted for new weave patterns, and also to take into account the non-linear tensile behavior of yarns.  The unloading of woven fabrics has not been considered in this study. However, due to the existence of friction in woven fabrics, in particular during shear behavior, the unloading of woven fabrics may cause some hysteresis effects. 112   The constitutive model is capable of prediction of in-plane behavior of woven fabrics, while the bending behavior of woven fabrics may not be accurately predicted by this model. It was attempted to predict the low bending rigidity of woven fabrics by developing an asymmetric model, as suggested as an effective way in [30]; however, it is expected to exist another mode of coupling, tension-bending coupling, which was proposed and introduced in this study, but not taken into account. 6.4 Future work recommendations  The following recommendations are suggested as possible future research directions in the field:  Studying other types of weave architectures;  Proposing a new boundary condition for picture frame test to make the condition of the experiment more similar to real forming processes;  Assessing the effect of manufacturing process parameters such as temperature, tool-fabric friction, and boundary conditions on the meso-scale deformation fields – including the contribution of intra-yarn shear – and in turn optimization of the forming process;  Conducting an integrated micro-meso numerical study to assess the presented hypotheses about the sources of each coupling mode;  Enhancing the proposed second-order beam formulation for analyzing bending rigidity of dry fabrics, e.g. by adding non-linear behavior of yarns and considering the effect of tension on increasing the filament-filament interaction/friction within the yarns.   Extension of the presented numerical model to include an accurate bending behavior of dry woven fabrics;  Integration of the implemented material model with an appropriate numerical model which includes the manufacturing process parameters such as tool-part interactions; 113   Taking the unloading effect into account to predict the influence of pre-conditioning on better forming of woven fabrics;  Examination of the presented material model to predict different modes of wrinkling in woven fabrics under actual 3D forming set-ups.    114  Bibliography [1] Dixit A, Mali HS. 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Compos Part A. 2017;97:128-40.    122  Appendices Appendix A: Details of the obtained stiffness functions The coefficients of the stiffness functions were found employing non-linear regression analysis. The identified functions are as follows: ܦ෩଴௜௜ ൌ ቊെ0.4518e12ߝ௜̃௜ହ 	൅ 3.2748e10ߝ௜̃௜ସ െ 7.3238e8ߝ௜̃௜ଷ ൅ 6.6648e6ߝ௜̃௜ଶ െ 1.3571e4ߝ௜̃௜ଵ 	൅ 	0.3344e2 ߝଵଵ ൑ 0.03ܦ෩଴௜௜ሺ0.03ሻ ߝଵଵ ൐ 0.03  A.1   ܦ෩௖௜௜൫ߝ௜௜, ߝ௝௝൯ ൌ 1 െ 21.13 ∗ ߝ௝௝ଵ.ଷ െ ܪሺߝ௜௜ െ 0.025ሻ ∗ ሺ0.64 ∗ ߝଶଶ଴.ଶ଼ሻ ൅ ܪ൫ߝ௝௝ െ 0.025൯ሺ2 െ 54∗ ߝ௜௜ሻ   A.2  ܦ෩଴௜௝ ൌ 2.6 ൅ 1200 ∗ ߝ݆݆ െ ܪ൫ߝ݆݆ െ 0.02൯ ∗ ൫ߝ݆݆ െ 0.02൯ ∗ 1200   A.3  ܦ෩௖௜௝ ൌ 1 ൅ 211ߝ௜௜ A.4  ܦ෩଴଺଺ ൌ 0.2063 െ 0.1089ܪሺߛ െ 0.07ሻ ൅ 0.18ܪሺߛ െ 0.67ሻ  A.5  ܦ෩௖଺଺ ൌ 1 ൅ 1700ሺߝଵଵ ൅ ߝଶଶሻ ൅ ܪሺߝଵଵ െ 2.5ሻ14597ሺߝଵଵ െ 0.025ሻଵ.ଶଶ൅ ܪሺߝଶଶ െ 2.5ሻ14597ሺߝଶଶ െ 0.025ሻଵ.ଶଶ ൅ ܪሺߝଵଵ െ 2.5ሻܪሺߝଶଶ െ 2.5ሻ12000ሾሺߝଵଵ െ 0.025ሻሺߝଶଶ െ 0.025ሻሿ଴.ହ+ ܪሺߛ െ 0.017ሻሾെ1570ሺߝଵଵ ൅ ߝଵଵሻ െܪሺߝଵଵ െ 2.5ሻ15685ሺߝଵଵ െ 0.025ሻଵ.ଷ െ ܪሺߝଶଶ െ 2.5ሻ15685ሺߝଶଶ െ 2.5ሻଵ.ଷ െ ܪሺߝଵଵ െ 2.5ሻܪሺߝଶଶ െ 2.5ሻ11353ሾሺߝଵଵ െ 2.5ሻሺߝଶଶ െ 2.5ሻሿ଴.ହA.6     123  Appendix B: UMAT code for the developed material model for woven fabrics This section provides the developed UMAT code implemented in ABAQUS for the simulation of woven fabrics under the new constitutive model and wrinkle onset criterion. The description of the code, including the theory and the algorithm, were presented in Chapter 5. C **************************************************************** C **************************************************************** C                        UMAT FOR Woven Fabircs C                          UMAT_WOVEN FABRICS C            User Material Subroutine for Coupled Non-Orthogonal  C                 Constitutive Modeling of Woven Fabrics  C    C    January, 2017 C    Version  1.1  C **************************************************************** C    Applications: C                2-D Planer Stress/Strain C                Conventional Shell Element S4 C **************************************************************** C    Authors: C                Masoud Hejazi & Masoud Haghi Kashani C                Abbas Hosseini C                Farrokh Sassani C               Frank ko C                Abbas Milani C C    Department of Mechanical Engineering, C    University Of British Columbia, Canada  C            C  C ---------------------------------------------------------------- C       SUBROUTINE UMAT(STRESS,STATEV,DDSDDE,SSE,SPD,SCD,      1 RPL,DDSDDT,DRPLDE,DRPLDT,      2 STRAN,DSTRAN,TIME,DTIME,TEMP,DTEMP,PREDEF,DPRED,CMNAME,      3 NDI,NSHR,NTENS,NSTATV,PROPS,NPROPS,COORDS,DROT,PNEWDT,      4 CELENT,DFGRD0,DFGRD1,NOEL,NPT,LAYER,KSPT,JSTEP,KINC) C       INCLUDE 'ABA_PARAM.INC' C       CHARACTER*80 CMNAME       DIMENSION STRESS(NTENS),STATEV(NSTATV),      1 DDSDDE(NTENS,NTENS),DDSDDT(NTENS),DRPLDE(NTENS),      2 STRAN(NTENS),DSTRAN(NTENS),TIME(2),PREDEF(1),DPRED(1),      3 PROPS(NPROPS),COORDS(3),DROT(3,3),DFGRD0(3,3),DFGRD1(3,3),      4 JSTEP(4) 124  c C LOCAL ARRAYS C ---------------------------------------------------------------- C DOBAR -  PURE TENSILE STIFNESS MODULES  C DCBAR -  COUPLING TENSILE STIFNESS INDUCED MODULUS C DOSBAR-  PURE SHEAR STIFNESS MODULES  C DCSBAR-  COUPLING SHEAR STIFNESS INDUCED MODULUS C TRANSF-  TRANSFORMATION MATRIX C DBAR-    JACOBIAN NON-ORTHOGONAL MATRIX C STRANL-  NON-ORTHOGONAL STRAIN ARRAY C STREESL- NON-ORTHOGONAL STRESS ARRAY C       DIMENSION DOBAR(4), DCBAR(4), DOSBAR(1), DCSBAR(1),      1 STRANL(NTENS), STRESSL(NTENS), TRANSF(NTENS,NTENS),      2 TRANSFT(NTENS,NTENS),      3 DBAR(NTENS,NTENS), DSTRANL(NTENS), DENTAL(NTENS,NTENS),      4 DDSDDET(NTENS,NTENS), DELTA(1) C C       PARAMETER(ONE=1.D0, TWO=2.D0, THREE=3.D0, FOUR=4.D0, FIVE=5.D0,      1 AII=0.4518D12, BII=3.2748D10, CII=7.3238D8, DII=6.6648D6,      2 EII=1.3571D4, FII=0.3344D2, GII=27.47D0,      3 EPSILONII=0.03D0, OPTH=1.3D0, ZPSF=0.83D0, ZPTE=0.28D0,      4 ZPTF=0.025D0, FIFFOUR=54.D0, TPS=2.6D0, THUND=1200.D0,      5 ZPZT=0.02D0, THOO=211.D0, ZPTZS=0.1587D0, ZPOZE=0.1071D0,      6 ZPZS=0.07D0, SVHU=1700.D0, TAH=2.5D0, OFFN=14597.D0,      7 OPTT=1.22D0, TWTH=12.D3, HALF=0.5D0, OFSZ=1570.D0,      8 OFSE=15685.D0, OOTF=11353.D0, ZPZOS=0.017D0, ZERO=0.D0,       9 TOL=1.D-10, ANGS=0.921D0) C C TRANSFORMATION MATRIX, ORTHOGONAL TO NONORTHOGONAL C-----------------------------------------------------------------  C       TRANSF(1,1)=COS(STRAN(3)/TWO)**TWO       TRANSF(1,2)=SIN(STRAN(3)/TWO)**TWO       TRANSF(1,3)=SIN(STRAN(3))       TRANSF(2,1)=SIN(STRAN(3)/TWO)**TWO       TRANSF(2,2)=COS(STRAN(3)/TWO)**TWO       TRANSF(2,3)=SIN(STRAN(3))       TRANSF(3,1)=SIN(STRAN(3)/TWO)*COS(STRAN(3)/TWO)       TRANSF(3,2)=SIN(STRAN(3)/TWO)*COS(STRAN(3)/TWO)       TRANSF(3,3)=ONE                        DO K1=1, NTENS         DO K2=1, NTENS         TRANSFT(K1,K2)=TRANSF(K2,K1)         END DO       END DO 125  C C TRANSFORMING TO NONORTHOGONAL STRANL AND DSTRANL C --------------------------------------------------------------- C       DO K1=1, NTENS         STRANL(K1)=ZERO         DSTRANL(K1)=ZERO       END DO              DO K1=1, NTENS         DO K2=1, NTENS           STRANL(K1)=STRANL(K1)+TRANSFT(K1,K2)*(STRAN(K2)+DSTRAN(K2))         END DO       END DO              DO K1=1, NTENS         DO K2=1, NTENS           DSTRANL(K1)=DSTRANL(K1)+TRANSFT(K1,K2)*DSTRAN(K2)         END DO       END DO C C C---------------------------------------------------------------- C --------------------------------------------------------------- C --------------------------------------------------------------- C FUNCTIONS DOBAR DCBAR DOSBAR DC C      -------------------------------------------------- C DOBAR.. C -----------------------------------------------------------------        DOBAR(1)=-AII*STATEV(1)**FIVE+BII*STATEV(1)**FOUR      1 -CII*STATEV(1)**THREE+DII*STATEV(1)**TWO-EII*STATEV(1)+FII       IF(STATEV(1).GT.EPSILONII) THEN         DOBAR(1)=-AII*EPSILONII**FIVE+BII*EPSILONII**FOUR      1 -CII*EPSILONII**THREE+DII*EPSILONII**TWO-EII*EPSILONII+FII       END IF       IF(STATEV(1).LT.ZERO) THEN         DOBAR(1)=FII       END IF       C            DOBAR(2)=-AII*STATEV(2)**FIVE+BII*STATEV(2)**FOUR      1 -CII*STATEV(2)**THREE+DII*STATEV(2)**TWO-EII*STATEV(2)+FII       IF(STATEV(2).GT.EPSILONII) THEN         DOBAR(2)=-AII*EPSILONII**FIVE+BII*EPSILONII**FOUR      1 -CII*EPSILONII**THREE+DII*EPSILONII**TWO-EII*EPSILONII+FII       END IF       IF(STATEV(2).LT.ZERO) THEN         DOBAR(2)=FII       END IF 126   C            DOBAR(3)=(TPS+THUND*STATEV(2)-THUND*(STATEV(2)-ZPZT))/OPTH       IF(STATEV(2).LT.ZPZT) THEN         DOBAR(3)=(TPS+THUND*STATEV(2))/OPTH       END IF       IF(STATEV(2).LT.ZERO) THEN         DOBAR(3)=TPS       END IF C             DOBAR(4)=(TPS+THUND*STATEV(1)-THUND*(STATEV(1)-ZPZT))/OPTH       IF(STATEV(1).LT.ZPZT) THEN         DOBAR(4)=(TPS+THUND*STATEV(1))/OPTH       END IF       IF(STATEV(1).LT.ZERO) THEN         DOBAR(4)=TPS       END IF C ----------------------------------------------------------------- C C DCBAR C -----------------------------------------------------------------       DCBAR(1)=ONE-GII*(STATEV(2)+TOL)**OPTH       IF(STATEV(1).GT.ZPTF) THEN         DCBAR(1)=DCBAR(1)-ZPSF*(STATEV(2)+TOL)**ZPTE       END IF       IF(STATEV(2).GT.ZPTF) THEN         IF(STATEV(1).LT.EPSILONII.OR.STATEV(1).EQ.EPSILONII) THEN           DCBAR(1)=DCBAR(1)+(TWO-FIFFOUR*STATEV(1))         END IF         IF(STATEV(1).GT.EPSILONII) THEN         DCBAR(1)=DCBAR(1)+(TWO-FIFFOUR*EPSILONII)         END IF       END IF                     DCBAR(2)=ONE-GII*(STATEV(1)+TOL)**OPTH       IF(STATEV(2).GT.ZPTF) THEN         DCBAR(2)=DCBAR(2)-ZPSF*(STATEV(1)+TOL)**ZPTE       END IF       IF(STATEV(1).GT.ZPTF) THEN         IF(STATEV(2).LT.EPSILONII.OR.STATEV(2).EQ.EPSILONII) THEN           DCBAR(2)=DCBAR(2)+(TWO-FIFFOUR*STATEV(2))         END IF         IF(STATEV(2).GT.EPSILONII) THEN         DCBAR(2)=DCBAR(2)+(TWO-FIFFOUR*EPSILONII)         END IF       END IF            DCBAR(3)=ONE+THOO*STATEV(1) 127  C            DCBAR(4)=ONE+THOO*STATEV(2)              IF(STATEV(2).LT.ZERO.OR.STATEV(1).LT.ZERO) THEN        DO K1=1, 4          DCBAR(K1)=ONE        END DO       END IF C ----------------------------------------------------------------- C ----------------------------------------------------------------- C C DOSBAR AND DCSBAR C -----------------------------------------------------------------       DOSBAR(1)=ZPTZS C             DCSBAR(1)=ONE+SVHU*(STATEV(1)+STATEV(2))       If(STATEV(1).GT.ZPTF) THEN         DCSBAR(1)=DCSBAR(1)+OFFN*(STATEV(1)-ZPTF)**OPTT       END IF C             IF(STATEV(2).GT.ZPTF) THEN         DCSBAR(1)=DCSBAR(1)+OFFN*(STATEV(2)-ZPTF)**OPTT C                 IF(STATEV(1).GT.ZPTF) THEN           DCSBAR(1)=DCSBAR(1)      1    +TWTH*((STATEV(1)-ZPTF)*(STATEV(2)-ZPTF))**HALF         END IF C                 END IF C       C             IF(STATEV(3).GT.ZPZOS) THEN         DCSBAR(1)=DCSBAR(1)-OFSZ*(STATEV(1)+STATEV(2)) C                 If(STATEV(1).GT.ZPTF) THEN           DCSBAR(1)=DCSBAR(1)-OFSE*(STATEV(1)-ZPTF)**OPTH         END IF C               IF(STATEV(2).GT.ZPTF) THEN           DCSBAR(1)=DCSBAR(1)-OFSE*(STATEV(2)-ZPTF)**OPTH C                   IF(STATEV(1).GT.ZPTF) THEN             DCSBAR(1)=DCSBAR(1)      1      -OOTF*((STATEV(1)-ZPTF)*(STATEV(2)-ZPTF))**HALF C                     END IF         END IF       END IF C      128        IF(STATEV(1).LT.ZERO.OR.STATEV(1).EQ.ZERO) THEN         IF(STATEV(2).LT.ZERO.OR.STATEV(2).EQ.ZERO) THEN           DOSBAR(1)=ZPTZS           IF(STATEV(3).GT.ZPZS)Then             DOSBAR(1)=DOSBAR(1)-ZPOZE           END IF           DCSBAR(1)=ONE         END IF       END IF C         C -------------------------------------------------------------------------- C -------------------------------------------------------------------------- C C C ASSEMBELING NONORTHOGONAL JACOBIAN MATRIX  C --------------------------------------------------------------------------       DBAR(1,1)=DOBAR(1)*DCBAR(1)       DBAR(1,2)=DOBAR(3)*DCBAR(3)       DBAR(1,3)=ZERO       DBAR(2,1)=DOBAR(4)*DCBAR(4)       DBAR(2,2)=DOBAR(2)*DCBAR(2)       DBAR(2,3)=ZERO       DBAR(3,1)=ZERO       DBAR(3,2)=ZERO       DBAR(3,3)=DCSBAR(1)*DOSBAR(1)  C--------------------------------------------------------------------------- C -------------------------------------------------------------------------- C --------------------------------------------------------------------------  C CALCULATING ORTHOGONAL JACOBIAN MATRIX AND STRESS  C ------------------------------------------------------------------------       C       DO K1=1, NTENS c        STRESS(K1)=ZERO         DO K2=1, NTENS          DDSDDET(K1,K2)=ZERO          DENTAL(K1,K2)=ZERO         END DO       END DO              DO K1=1, NTENS         DO K2=1, NTENS           DO K3=1, NTENS            DENTAL(K1,K2)=DENTAL(K1,K2)+TRANSF(K1,K3)*DBAR(K3,K2)           END DO         END DO       END DO  129                 DO K1=1, NTENS         DO K2=1, NTENS           DO K3=1, NTENS             DDSDDET(K1,K2)=DDSDDET(K1,K2)+DENTAL(K1,K3)*TRANSFT(K3,K2)           END DO         END DO       END DO               DO K1=1, NTENS         DO K2=1, NTENS           DDSDDE(K1,K2)=DDSDDET(K1,K2)         END DO       END DO              DO K1=1, NTENS         DO K2=1, NTENS           STRESS(K1)=STRESS(K1)+DDSDDE(K1,K2)*DSTRAN(K2)         END DO       END DO C CALCULATING NONORTHOGONAL STRESSES C---------------------------------------------------------------------------       DO K1=1, NTENS         STRESSL(K1)=ZERO       END DO       DO K1=1, NTENS         DO K2=1, NTENS           STRESSL(K1)=STRESSL(K1)+TRANSFT(K1,K2)*STRESS(K2)         END DO       END DO C-------------------------------------------------------------------------- C-------------------------------------------------------------------------- C        DO K1=1, NTENS         STATEV(K1)=STRANL(K1)         STATEV(NTENS+K1)=STRESSL(K1)       END DO   C WRINKLE PREDECTION C--------------------------------------------------------------------------       DELTA(1)=ABS(STRESSL(1))**(ONE+HALF)/(ABS(STRANL(1))**(ONE+HALF)+      1 STRANL(1)**HALF-TANH(ABS(STRANL(1))**HALF))       IF(STATEV(3).LT.-ACOS(ANGS).OR.STATEV(3).GT.DELTA(1)) THEN         STATEV(3*NTENS)=ONE       END IF       STATEV(NSTATV)=DELTA(1)    C ------------------------------------------------------------------------- C--------------------------------------------------------------------------       C             RETURN 130        END 

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