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An efficient unit cell based numerical model for continuum representation of fabric systems Shahkarami Noori, Ali 2006

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A N EFFICIENT UNIT CELL BASED N U M E R I C A L M O D E L FOR C O N T I N U U M REPRESENTATION OF FABRIC SYSTEMS  by A l i Shahkarami N o o r i B . S c , Sharif U n i v e r s i t y of T e c h n o l o g y , 1995 M . A . S c The U n i v e r s i t y of B r i t i s h C o l u m b i a , 1999 v  A THESIS SUBMITTED I N P A R T I A L F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF  PHILOSOPHY in  T H E F A C U L T Y OF G R A D U A T E STUDIES (Civil Engineering)  T H E UNIVERSITY OF BRITISH C O L U M B I A J u l y 2006  © A l i S h a h k a r a m i N o o r i , 2006  Abstract Flexible p e r s o n a l a r m o u r systems are u s e d e v e r y d a y b y p o l i c e forces, m i l i t a r y personnel, p a r a m e d i c s , a n d others i n their l i n e of d u t y . T h e i n t r o d u c t i o n of h i g h - p e r f o r m a n c e p o l y m e r - b a s e d fibres a n d y a r n s has p a v e d the w a y for the d e s i g n of l i g h t w e i g h t , yet strong b o d y a r m o u r s that c a n defeat a w i d e range of threats. H o w e v e r , despite the r e l a t i v e l y h i g h cost of materials u s e d i n b u l l e t resistant vests, their d e s i g n is c o m p l e t e l y based o n e x p e r i m e n t s . A d d i t i o n of s i m u l a t i o n c a p a b i l i t y to the d e s i g n process of b u l l e t resistant vests s h o u l d i n t r o d u c e significant s a v i n g s i n the time a n d cost associated w i t h this a c t i v i t y a n d l e a d to m o r e o p t i m i z e d a r m o u r systems. A n o v e l n u m e r i c a l a p p r o a c h is i n t r o d u c e d i n this thesis that is c o m p r i s e d of t w o d i s t i n c t n u m e r i c a l m o d e l s d e v e l o p e d i n p a r a l l e l a n d v e r i f i e d against each other. E x p e r i m e n t a l data are u s e d to characterize the b e h a v i o u r of fabrics a n d to v a l i d a t e the p r e d i c t i o n s of the n u m e r i c a l m o d e l s . T h i s c o m b i n a t i o n of the numerical  and  e x p e r i m e n t a l efforts  has  resulted  i n a unique modelling  technique to capture the d e f o r m a t i o n a l response of fabric panels. A  detailed 3 D m o d e l of the fabric u n i t cell is created  ,  to investigate  the  interaction of the c r o s s i n g y a r n s u n d e r extensional i n - p l a n e d i s p l a c e m e n t s . T h i s c o n t i n u u m - b a s e d m o d e l e x p l i c i t l y considers the g e o m e t r y of the y a r n s as a n assembly of s o l i d a n d b a r elements a n d u p o n successful c a l i b r a t i o n , predicts the response of the fabric u n i t cell u n d e r the a p p l i e d d i s p l a c e m e n t s . An  efficient 2 D s h e l l crossover  m o d e l is also d e v e l o p e d  that  implicitly  incorporates the e x t e n s i o n a l a n d shearing response of the c r o s s i n g yarns. T h e g o v e r n i n g c o n s t i t u t i v e equations for this s h e l l element are d e r i v e d c o n s i d e r i n g s i n u s o i d a l c o m p r e s s i b l e y a r n s subjected to s y m m e t r i c d i s p l a c e m e n t s i n the fabric's p l a n e . T h e efficient s h e l l elements are u s e d to create finite element m o d e l s of fabric u n d e r static a n d d y n a m i c l o a d i n g scenarios. S i m u l a t i o n s of single a n d m u l t i - p l y fabric targets subjected  to i m p a c t are  carried o u t a n d the m o d e l p r e d i c t i o n s are c o m p a r e d against the data f r o m i n s t r u m e n t e d b a l l i s t i c experiments. T h e m o d e l is f o u n d to be successful i n c a p t u r i n g the d e f o r m a t i o n a l b e h a v i o u r of fabric targets. P a r a m e t r i c analysis is p e r f o r m e d o n v a r i o u s geometric a n d m e c h a n i c a l properties of the fabric to investigate the s e n s i t i v i t y of its response to s u c h values. R e c o m m e n d a t i o n s are m a d e for setting u p m o d e l s of fabric to be u s e d i n i n d u s t r i a l d e s i g n process, i n o r d e r to m a k e the a p p r o a c h available to a r m o u r designers.  Table of Contents Abstract  ii  Table of C o n t e n t s  iii  L i s t of Tables  vi  L i s t of F i g u r e s  viii  L i s t of S y m b o l s  xxiv  Acknowledgements  xxxi  Dedication  xxxii  C h a p t e r 1 - D y n a m i c A n a l y s i s of F a b r i c Systems 1.1  M o d e l l i n g B a l l i s t i c Fabrics - A H i s t o r i c B a c k g r o u n d  1 1  1.1.1  Discrete m o d e l s  3  1.1.2  Unit-cell continuum models  6  1.2  C u r r e n t State-of-the-Art  8  1.3  Research Objectives  8  1.4  Methodology  9  1.5  O u t l i n e of the Thesis  11  Chapter 2 - Experimental Data 2.1  15  Fabric M a t e r i a l  16  2.2 Transverse C o m p r e s s i o n of Fabrics  16  2.2.1  Transverse C o m p r e s s i o n Experiments  17  2.2.2  Transverse C o m p r e s s i o n M o d e l  18  2.3  Ballistic I m p a c t E x p e r i m e n t s  23  2.4  Summary  24  Chapter 3 - 3D Fabric Crossover M o d e l  33  3.1  Introduction  33  3.2  3D Fabric Unit-cell M o d e l s  34  3.3  Explicit Finite Element Analysis  37  3.4  M o d e l Basics  40  3.4.1  Discrete B a r E l e m e n t s  41  3.4.2  C o n t i n u u m Solid Elements  44  -iii-  3.4.3  D e t e r m i n a t i o n of M a t e r i a l Properties  49  3.4.3.1  S i n g l e y a r n tensile test s i m u l a t i o n  50  3.4.3.2  Transverse c o m p r e s s i o n test s i m u l a t i o n  51  3.4.3.3  B i a x i a l Response of the 3 D C r o s s o v e r M o d e l  52  3.5  Advantages and Limitations  53  3.6  Summary  55  Chapter 4 - 2D Shell Crossover M o d e l  63  4.1  Introduction  63  4.2  Shell-Based Fabric Unit-cell M o d e l s  64  4.3  2 D S h e l l F a b r i c U n i t - c e l l - M o d e l Basics  68  4.3.1  In-plane E x t e n s i o n a l Response  68  4.3.2  Shear R e s p o n s e  77  4.3.3  D e v e l o p m e n t of U s e r M a t e r i a l M o d e l for S h e l l E l e m e n t s  82  4.3.4  D e t e r m i n a t i o n of the i n p u t parameters  89  4.4  4.3.4.1  E x t e n s i o n a l Response of the F a b r i c U n i t - c e l l  89  4.3.4.2  Transverse C o m p r e s s i o n of the Y a r n s  89  4.3.4.3  Shear Response of the Fabric U n i t - c e l l  90  Simulations and Verifications  91  4.4.1  C o m p a r i s o n w i t h 3 D crossover m o d e l  91  4.4.2  C o m p a r i s o n w i t h pin-jointed cable m o d e l  94  4.4.3  B i a s E x t e n s i o n Test S i m u l a t i o n s  96  4.4.4  N o t e s o n the N u m e r i c a l Efficiency of S h e l l U C  98  4.5  Summary  100  C h a p t e r 5 - V a l i d a t i o n of the Static a n d D y n a m i c R e s p o n s e P r e d i c t i o n s of Fabrics  119  5.1  Introduction  119  5.2  Ballistic I m p a c t R e s p o n s e of Fabric Panels  120  5.2.1  Ballistic I m p a c t S i m u l a t i o n : 1-ply Targets  120  5.2.2  B a l l i s t i c I m p a c t S i m u l a t i o n : M u l t i - p l y Targets  124  5.2.3  D i s c u s s i o n of the Ballistic Impact S i m u l a t i o n Results  128  5.2.4  O v e r a l l C o m p a r i s o n of V a r i o u s Kevlar® 129 F a b r i c s  129  5.2.4.1  U n i t - c e l l m e c h a n i c a l response  130  5.2.4.2  Iso-velocity fabric i m p a c t s i m u l a t i o n s  131  5.3  B i a x i a l fabric e x t e n s i o n s i m u l a t i o n s  137  5.4  Summary  141  -iv-  C h a p t e r 6 - S e n s i t i v i t y A n a l y s i s a n d N u m e r i c a l Case-Studies  178  6.1  Introduction  178  6.2  Fabric Energy A b s o r p t i o n M e c h a n i s m  179  6.3  Parametric Study  181  6.3.1  Effect of Y a r n Stiffness  182  6.3.2  Effect of Y a r n C r i m p  183  6.3.3  Effect of F a b r i c A r e a l D e n s i t y  185  6.3.4  Discussion  186  6.4  S e n s i t i v i t y to O t h e r Parameters  188  6.4.1  Transverse C o m p r e s s i o n of Y a r n s  188  6.4.2  Shear Properties  192  6.4.3  F a b r i c S l i p p a g e at the B o u n d a r y  194  6.4.4  Inter-Layer G a p  197  6.5  Pencilling Phenomenon  198  6.6  R e c o m m e n d a t i o n s for I n d u s t r i a l A p p l i c a t i o n  201  6.7  Summary  203  Chapter 7 - Summary, Conclusions and Future W o r k  231  7.1  Summary  231  7.2  Conclusions  233  7.3  C o n t r i b u t i o n s to the State-of-the-Art  235  7.4  R e c o m m e n d a t i o n s for F u t u r e W o r k  238  • Bibliography Appendix A -  243 3 D crossover M e s h Pre-processor  250  A . l Fabric Cross-sectional Micro-images  250  A . 2 F i n i t e E l e m e n t U n i t - c e l l M e s h Pre-processor  253  Appendix B B. l  Pseudo-Codes  265  Beam Element  265  B.2 S o l i d E l e m e n t  266  B.3 S h e l l E l e m e n t  267  -V-  List of Tables Table 2 - 1 : P r o p e r t i e s of the Kevlar® 129 p l a i n w e a v e fabric panels investigated i n this s t u d y  26  Table 2 - 2: T h e f i t t i n g parameters a a n d b for the transverse c o m p r e s s i o n of the fabric m o d e l ( E q u a t i o n 2 - 7 ) for single a n d m u l t i - p l y panels of Kevlar® 129  26  Table 3 - 1 : B e a m e l e m e n t i n p u t parameters u s e d for the 3 D crossover m o d e l of S-720 fabric u n i t - c e l l  56  Table 3 - 2: S o l i d e l e m e n t i n p u t parameters u s e d for the 3 D crossover m o d e l of S-720 fabric u n i t - c e l l  56  Table 4 - 1 : S h e l l i n p u t parameters for the i n - p l a n e shear m o d e l of a l l six Kevlar® 129 fabric types  102  Table 4 - 2: I n p u t m a t e r i a l properties of the steel R C C projectile u s e d i n the impact simulations  102  Table 4 - 3: C o m p a r i s o n of e x e c u t i o n times for s i m u l a t i o n of i m p a c t of a n R C C projectile o n square panels of Kevlar® 129 fabric u s i n g cable ( p i n jointed), 2 D s h e l l crossover a n d 3 D s o l i d / b a r crossover fabric models  103  Table 5 - 1 : I n p u t v a l u e s u s e d to set u p shell-based fabric m o d e l s to s i m u l a t e instrumented ballistic impact experiments  143  Table 5 - 2: Interlayer gap d e t e r m i n e d f r o m the transverse c o m p r e s s i o n test data p e r f o r m e d o n 4 - p l y fabric s a m p l e s  144  Table 5 - 3: G e o m e t r i c a n d m e c h a n i c a l properties of the p l a i n w e a v e glass fabric panels s t u d i e d b y B u e t - G a u t i e r a n d Boisse [37]  144  Table 5 - 4: S h e l l element i n p u t parameters u s e d to c a r r y o n s i m u l a t i o n of b i a x i a l l o a d i n g of fabric panels s t u d i e d b y B u e t - G a u t i e r a n d Boisse [37]  145  -vi-  Table 6 - 1 : M e c h a n i c a l p r o p e r t i e s of the d u m m y b a c k i n g m a t e r i a l u s e d to investigate p e n c i l l i n g p h e n o m e n o n  206  Table 6 - 2 : Elastic m o d u l u s a n d d e n s i t y of the y a r n s c o n s i d e r e d i n the s i m u l a t i o n s p e r f o r m e d to investigate the p e n c i l l i n g p h e n o m e n o n . 206  Table 7 - 1 : C o m p a r i n g the 3 D crossover m o d e l d e v e l o p e d i n this s t u d y to that of the current literature  242  Table 7 - 2 : C o m p a r i n g the 2 D shell m o d e l d e v e l o p e d i n this s t u d y to that of the current literature  242  Table A - 1 : F a b r i c a n d w e a v e parameters o b t a i n e d f r o m the m i c r o - i m a g e s (data p r o v i d e d b y K o n g s h a v n [42])  258  Table A - 2: Y a r n c r i m p v a l u e s a n d v o i d ratios c a l c u l a t e d f r o m the m i c r o i m a g e s (data p r o v i d e d b y K o n g s h a v n [42])  259  Table A - 3: M a t e r i a l a n d g e o m e t r i c a l properties i n p u t to the pre-processor for S-720G  259  -vii-  List of Figures F i g u r e 1 - 1 : D i s c r e t e m o d e l l i n g a p p r o a c h a d o p t e d b y m a n y researchers to s t u d y the d y n a m i c b e h a v i o u r of fabrics  13  F i g u r e 1 - 2 : C o n t i n u u m m o d e l l i n g a p p r o a c h , further s u b d i v i d e d i n t o 3 D s o l i d m o d e l s a n d u n i t - c e l l s h e l l m o d e l s , u s e d b y m a n y to capture the d y n a m i c response of fabrics a n d fabric c o m p o s i t e s  13  F i g u r e 1 - 3 : A s u m m a r y of the n u m e r i c a l a p p r o a c h presented i n this s t u d y . . 14  F i g u r e 2 - 1 : E x p e r i m e n t a l set-up for the transverse c o m p r e s s i o n tests c o n d u c t e d b y K o n g s h a v n [42]  27  F i g u r e 2 - 2: T r a n s v e r s e c o m p r e s s i v e force-displacement response of S-720 p a n e l s c o n d u c t e d b y K o n g s h a v n for fabric p a c k s of 1 to 16 plies [42]  27  F i g u r e 2 - 3 : T r a n s v e r s e c o m p r e s s i v e response of a s i n g l e - p l y of S-720 fabric before a n d after extracting the c o m p l i a n c e of the T e f l o n ™ layers ( e x p e r i m e n t a l data o b t a i n e d f r o m K o n g s h a v n [42])  28  F i g u r e 2 - 4 : L i n e a r f u n c t i o n representing the r e l a t i o n s h i p b e t w e e n the m i n i m u m c o m p r e s s i o n v o l u m e of a u n i t area of the fabrics (obtained f r o m c o m p r e s s i o n m o d e l ) a n d their areal d e n s i t y  28  F i g u r e 2 - 5 : F i t t i n g p a r a m e t e r K" as a f u n c t i o n of the t h i r d p o w e r of areal d e n s i t y (A/)  for the six different Kevlar® 129 fabrics  29  F i g u r e 2 - 6 : S c h e m a t i c of the test fixture u s e d i n the b a l l i s t i c e x p e r i m e n t s [51]. 29 F i g u r e 2 - 7: D u a l E L V S system setup w i t h t r a c k i n g the d i s p l a c e m e n t of the projectile s t r i k i n g the front-side of the target (front E L V S ) a n d the tip of the d e f o r m a t i o n cone o n the b a c k - s i d e of the target (back E L V S ) [51]  30  F i g u r e 2 - 8 : D i s p l a c e m e n t - t i m e response of 1-layer S-726 p a n e l s i m p a c t e d at 96 m / s a n d 110 m/s o b t a i n e d f r o m a n E L V S - i n s t r u m e n t e d b a l l i s t i c e x p e r i m e n t (performed b y C e p u s [42])  -viii-  30  Figure 2  9: V e l o c i t y - t i m e response of 1-layer S-726 panels i m p a c t e d at 96 m/s a n d 110 m / s o b t a i n e d f r o m a n E L V S - i n s t r u m e n t e d b a l l i s t i c e x p e r i m e n t ( p e r f o r m e d b y C e p u s [42])  Figure 2  31  10: E n e r g y absorbed-time response of 1-layer S-726 panels i m p a c t e d at 96 m / s a n d 110 m/s o b t a i n e d f r o m a n E L V S - i n s t r u m e n t e d ballistic e x p e r i m e n t (performed b y C e p u s [42])  Figure 2  31  11: Projectile strike v e l o c i t y versus its r e s i d u a l v e l o c i t y i n the ballistic i m p a c t tests for the 1-ply S-720 panels ( E x p e r i m e n t s p e r f o r m e d b y C e p u s [42], data s u m m a r i z e d b y the author)  Figure 2  32  12: Projectile strike energy versus the absorbed e n e r g y b y the target i n the b a l l i s t i c i m p a c t tests for the 1-ply S-720 panels (Experiments p e r f o r m e d b y C e p u s [42], data s u m m a r i z e d b y the author)  Figure 3  32  1: schematic representation of L S - D Y N A e x p l i c i t finite element analysis [56]  57  F i g u r e 3 2: S o l i d (top left) a n d bar (top right) elements c o m b i n e d to m a k e u p a 3 D m o d e l of a y a r n (bottom)  57  F i g u r e 3 3: H o u r g l a s s - t y p e d e f o r m a t i o n shapes (left) e l i m i n a t e d b y the a d d i t i o n of b a r elements (right)  58  F i g u r e 3 4: F o r c e - d i s p l a c e m e n t response of a single c r i m p e d y a r n s h o w i n g the i n s t a b i l i t y of the y a r n m o d e l w i t h o u t the b a r elements  58  F i g u r e 3 5: S c h e m a t i c representation of fibre r e a l i g n m e n t a n d c o m p r e s s i o n i n a y a r n subjected to transverse c o m p r e s s i o n  59  F i g u r e 3 6: R e l a t i o n s h i p b e t w e e n transverse elastic m o d u l u s a n d the fibre v o l u m e fraction for the S-720 w a r p y a r n Figure 3  59  7: S i n g l e - y a r n tensile test p r e d i c t i o n s for S-720 w a r p a n d weft yarns w i t h 2.18% a n d 1.39% c r i m p , respectively  60  F i g u r e 3 8: S i m u l a t i o n of the transverse c o m p r e s s i o n test of a fabric u n i t cell (note that o n l y a quarter of the u n i t cell is s h o w n d u e to symmetry)  60  F i g u r e 3 9: T r a n s v e r s e c o m p r e s s i o n response of S-720 fabric u n i t cell, e x p e r i m e n t s c o m p a r e d to p r e d i c t i o n of the 3 D crossover m o d e l . 61  -ix-  F i g u r e 3 - 1 0 : T e n s i l e force d e v e l o p e d i n the w a r p a n d weft y a r n of a single crossover m o d e l extracted f r o m S-720 fabric  61  F i g u r e 3 - 1 1 : W a r p a n d weft y a r n tensile response surfaces as a f u n c t i o n of y a r n displacements, calculated for S-720 fabric u n i t - c e l l  62  F i g u r e 4 - 1 : I l l u s t r a t i o n of the b i a x i a l crossover m o d e l of a single p l a i n w e a v e fabric crossover w i t h linear yarns o r i g i n a l l y d e v e l o p e d b y K a w a b a t a et a l . [57]  104  F i g u r e 4 - 2 : 3 D finite element m e s h of a single fabric crossover (left) a n d the e q u i v a l e n t m a t h e m a t i c a l representation of the y a r n s b y their centrelines (right)  104  F i g u r e 4 - 3 : A schematic m o d e l of a y a r n i n the fabric (left) i n contact w i t h the c r o s s i n g y a r n a n d a free b o d y d i a g r a m s h o w i n g the forces a c t i n g o n it (right)  105  F i g u r e 4 - 4 : C o s i n e - b a s e d shape of a n arbitrary y a r n i n a p l a i n w e a v e fabric. 105 F i g u r e 4 - 5 : W a r p a n d weft centreline geometry before (green) a n d after (red) deformation  106  F i g u r e 4 - 6 : Shear m o d e l d e v e l o p e d for the fabric u n i t - c e l l b a s e d o n a picture frame shear test  106  F i g u r e 4 - 7 : C o m p a c t i o n of the y a r n s as a result of the crossover s h e a r i n g (before (left) a n d after (right) shearing). N o t e that the fabric thickness a n d c o n f i g u r a t i o n of y a r n centreline are a s s u m e d to r e m a i n u n c h a n g e d d u r i n g shear d e f o r m a t i o n  107  F i g u r e 4 - 8 : Secant shear m o d u l u s , G, as a f u n c t i o n of the shear s t r a i n y, r e p r e s e n t i n g the trellis m e c h a n i s m  107  F i g u r e 4 - 9 : D e f i n i t i o n of the l o c a l element a n d m a t e r i a l axes c o n v e n t i o n a s s u m e d for s h e l l elements i n L S - D Y N A  108  F i g u r e 4 - 1 0 : D e f i n i t i o n of element stresses a n d their r e l a t i o n s h i p w i t h those i n d i v i d u a l y a r n s i n a n e q u i v a l e n t s m e a r e d s h e l l a p p r o a c h (w a n d /  i n the figure represent the w a r p a n d weft yarns, respectively). 108  -X-  F i g u r e 4 - 1 1 : F l o w c h a r t of the U M A T r o u t i n e s p e c i f y i n g the constitutive r e l a t i o n s h i p of the smeared fabric m a t e r i a l of the representative s h e l l element (see A p p e n d i x B for m o r e details)  109  F i g u r e 4 - 1 2 : T e n s i o n - d i s p l a c e m e n t response of S-726 single c r i m p e d w a r p a n d weft y a r n s p r e d i c t e d b y the representative s h e l l element  109  F i g u r e 4 - 1 3 : D e f o r m e d shape a n d contours of element d e f o r m a t i o n ( v o i d ratios) u n d e r transverse c o m p r e s s i o n (left) a n d b i a x i a l extension (right) l o a d i n g s i m u l a t i o n s u s i n g the 3 D crossover m o d e l (note that o n l y one quarter of the y a r n is s h o w n for i l l u s t r a t i o n purposes)  110  F i g u r e 4 - 1 4 : T r a n s v e r s e force-displacement response p r e d i c t i o n s (along w i t h the best fit) of the 3 D crossover m o d e l of S-720 u n i t - c e l l o b t a i n e d f r o m s i m u l a t i o n s of transverse c o m p r e s s i o n tests a n d b i a x i a l t e n s i o n tests, s h o w i n g the difference i n the c o m p r e s s i v e response of the y a r n s  110  F i g u r e 4 - 15: B i a x i a l response of S-726 u n i t - c e l l as a f u n c t i o n of w a r p d i s p l a c e m e n t (weft y a r n fixed, d = 0 ) , p r e d i c t e d b y the 3 D 2  crossover m o d e l a n d the 2 D s h e l l m o d e l  Ill  F i g u r e 4 - 16: V a r i a t i o n s of h a n d h w i t h w a r p d i s p l a c e m e n t , p r e d i c t e d b y the x  2  s h e l l m o d e l d u r i n g the b i a x i a l extension of a n S-726 u n i t - c e l l illustrated i n Figure 4 - 1 5  Ill  F i g u r e 4 - 1 7 : E n e r g y stored i n the u n i t - c e l l as a f u n c t i o n of w a r p d i s p l a c e m e n t , c o m p a r i n g the p r e d i c t i o n s of the s h e l l crossover a n d 3 D s o l i d m o d e l s d u r i n g the b i a x i a l extension of a n S-726 u n i t - c e l l illustrated i n Figure 4 - 1 5  112  F i g u r e 4 - 1 8 : Y a r n cross-sections before (left) a n d after (right) d e f o r m a t i o n d u e to the i n t e r a c t i o n of the t w o y a r n s i n b i a x i a l extension, for the b i a x i a l extension of a n S-726 u n i t - c e l l i l l u s t r a t e d i n F i g u r e 4 - 1 5 . 112 F i g u r e 4 - 19: D e f o r m e d p r o f i l e of the w a r p y a r n p r e d i c t e d b y the 3 D s o l i d m o d e l c o m p a r e d w i t h a sine c u r v e (note that o n l y half of the y a r n is s h o w n ) , for the b i a x i a l extension of a n S-726 u n i t - c e l l i l l u s t r a t e d in Figure 4 - 1 5  113  -xi-  Figure 4-20: Deformed profile of the weft yarn predicted by the 3D solid model compared with a sine curve (note that only half of the yarn is shown), for the biaxial extension of an S-726 unit-cell illustrated in Figure 4 - 15  113  Figure 4-21: Velocity-time response of the projectile impacting a single-layer S-720 target, predicted using shell-based and cable-based model (yarn crimp not considered)  114  Figure 4 - 22: Comparison of a yarn reaction force in an impact simulation on a single-layer S-720 target, predicted using shell-based and cablebased model (yarn crimp not considered)  114  Figure 4 - 23: Velocity-time response of the projectile impacting a single-layer S-720 target, predicted using shell-based and cable-based model (yarn crimp considered)  115  Figure 4 - 24: Schematics of the picture frame shear testing methodology  115  Figure 4-25: Schematic representation of undeformed fabric (top) in the bias extension test and the creation of shear zones after deformation (bottom) [67]  116  Figure 4 - 26: Contours of warp/weft angle in the deformed balanced S-728 fabric in the bias extension simulation  116  Figure 4 - 27: Unstable deformation mode of the S-728 fabric without considering the shear locking in bias extension simulation  117  Figure 4-28: Contours of contact force between the yarns, computed for biaxial extension of an S-720 fabric crossover (-0.05 < d < 0.05 i  where z = l,2)  117  Figure 4-29: Contours of the number of iterations needed to gain convergence, when calculating the yarn tensile forces as a function of the symmetric displacements applied to the S-720 fabric unit-cell (-0.05<d,<0.05 where i = 1,2)  118  Figure 4 - 30: Contours of the number of iterations needed to gain convergence after bypassing iterations for negative displacement pairs developed for S-720 fabric unit-cell (-0.05 < d, < 0.05 where z = l,2)  118  -xii-  Figure 5 - 1 : Areal density comparison of the six Kevlar® 129 panels investigated. Further information on these panels is provided in Chapter 3  146  Figure 5 - 2: Crimp comparison of the six Kevlar® 129 panels investigated. Further information on these panels is provided in Chapter 3.. 146 Figure 5-3: Crimp ratio (warp/weft) comparison of the six Kevlar® 129 panels investigated. Further information on these panels is provided in Chapter 3  147  Figure 5-4: Velocity-time response of an R C C projectile in a non-perforating impact simulation of a 1-layer S-720 target struck at 92.5 m/s... 147 Figure 5-5: Velocity-time response of an R C C projectile in a perforating impact simulation of a 1-layer S-720 target struck at 155 m/s.... 148 Figure 5-6: Velocity-time response of an R C C projectile in a non-perforating impact simulation of a 1-layer S-728 target struck at 76 m/s  148  Figure 5-7: Velocity-time response of an R C C projectile in a perforating impact simulation of a 1-layer S-728 target struck at 170 m/s.... 149 Figure 5-8: Transverse deformation profile (left) and corresponding fringe plot (right) for a 1-ply S-720 panel impacted at 100 m/s after 225 us  149  Figure 5-9: Comparison of the normalized reaction force in warp and weft of S-720 fabric  150  Figure 5 -10: Velocity-time response of an R C C projectile in a non-perforating impact simulation of a 1-layer S-726 target struck at 96 m/s  150  Figure 5-11: Velocity-time response of an R C C projectile in a perforating impact simulation of a 1-layer S-726 target struck at 174 m/s.... 151 Figure 5 -12: Velocity-time response of an R C C projectile in a non-perforating impact simulation of a 1-layer S-727 target struck at 40 m/s  151  Figure 5 -13: Velocity-time response of an R C C projectile in a perforating impact simulation of a 1-layer S-727 target struck at 158 m/s.... 152 Figure 5 -14: Velocity-time response of an R C C projectile in a non-perforating impact simulation of a 1-layer S-731 target struck at 116 m/s.... 152  -xiii-  F i g u r e 5 - 1 5 : V e l o c i t y - t i m e response of a n R C C projectile i n a perforating i m p a c t s i m u l a t i o n of a 1-layer S-731 target struck at 132 m / s . . . . 153 F i g u r e 5 - 1 6 : V e l o c i t y - t i m e response of a n R C C projectile i n a n o n - p e r f o r a t i n g i m p a c t s i m u l a t i o n of a 1-layer S-745 target struck at 110 m / s . . . . 153 F i g u r e 5 - 1 7 : V e l o c i t y - t i m e response of a n R C C projectile i n a perforating i m p a c t s i m u l a t i o n of a 1-layer S-745 target s t r u c k at 215 m / s . . . . 154 F i g u r e 5 - 1 8 : Transverse d e f o r m a t i o n profile (left) a n d c o r r e s p o n d i n g fringe p l o t (right) for a 1-ply S-731 p a n e l i m p a c t e d at 100 m / s after 225 |as  154  F i g u r e 5 - 1 9 : C o m p a r i s o n of the n o r m a l i z e d reaction force i n w a r p a n d weft of S-726 fabric  155  F i g u r e 5 - 20: P o s t - m o r t e m e x a m i n a t i o n of the 1-layer S-726 panels i m p a c t e d at 175 m/s  155  F i g u r e 5 - 2 1 : Schematics of the m e c h a n i c a l m o d e l c o n s i d e r e d for the transverse c o m p r e s s i o n of single a n d t w o - p l y fabric specimens. 156 F i g u r e 5 - 22: C a l c u l a t i o n of i n t e r - p l y gap f r o m transverse c o m p r e s s i o n experiments, o b t a i n e d for 4 - p l y S-728 fabric  156  F i g u r e 5 - 2 3 : A schematic a r r a n g e m e n t of m u l t i p l e layers m a d e f r o m s h e l l elements to construct a 4 - p l y S-726 4-layer m o d e l (quarter of the m o d e l is s h o w n d u e to s y m m e t r y )  157  F i g u r e 5 - 24: V e l o c i t y - t i m e response of a n R C C projectile i n a n o n - p e r f o r a t i n g i m p a c t s i m u l a t i o n of a 4-layer S-726 target struck at 158 m / s . . . . 157 F i g u r e 5 - 25: V e l o c i t y - t i m e response of a n R C C projectile i n a n o n - p e r f o r a t i n g i m p a c t s i m u l a t i o n of a 4-layer S-728 target struck at 91 m/s  158  F i g u r e 5 - 26: V e l o c i t y - t i m e response of a n R C C projectile i n a n o n - p e r f o r a t i n g i m p a c t s i m u l a t i o n of a 4-layer S-731 target s t r u c k at 136 m / s . . . . 158 F i g u r e 5 - 27: V e l o c i t y - t i m e response of a n R C C projectile i n a n o n - p e r f o r a t i n g i m p a c t s i m u l a t i o n of a n 8-layer S-726 target struck at 204 m / s . . 159 F i g u r e 5 - 28: V e l o c i t y - t i m e response of a n R C C projectile i n a p e r f o r a t i n g i m p a c t s i m u l a t i o n of a n 8-layer S-728 target struck at 162 m / s . . 159  -xiv-  F i g u r e 5 - 29: V e l o c i t y - t i m e response of a n R C C projectile i n a p e r f o r a t i n g i m p a c t s i m u l a t i o n of a n 8-layer S-731 target struck at 142 m / s . . 160 F i g u r e 5 - 3 0 : V e l o c i t y - t i m e response of a n R C C projectile i n a perforating i m p a c t s i m u l a t i o n of a 16-layer S-726 target struck at 243 m / s . . 160 F i g u r e 5 - 3 1 : V e l o c i t y - t i m e response of a n R C C projectile i n a perforating i m p a c t s i m u l a t i o n of a 16-layer S-728 target struck at 170 m / s . . 161 F i g u r e 5 - 32: V e l o c i t y - t i m e response of a n R C C projectile i n a perforating i m p a c t s i m u l a t i o n of a 16-layer S-731 target struck at 120 m / s . . 161 F i g u r e 5 - 33: W a r p a n d weft tension surfaces as a f u n c t i o n of the i n - p l a n e e x t e n s i o n a l displacements, d e v e l o p e d for S-720 fabric  162  F i g u r e 5 - 34: E n e r g y absorbed per u n i t area (J/m ) b y S-720 fabric u n d e r 2  s y m m e t r i c a l l y a p p l i e d extension, expressed i n terms of strain i n the w a r p a n d weft directions; 3 D surface (left) a n d 2 D contour p l o t (right)  162  F i g u r e 5 - 35: C o n t o u r s of strain energy stored i n a u n i t area of S-720 a n d S-728 fabrics as a f u n c t i o n of the w a r p a n d weft strains  163  F i g u r e 5 - 36: E n e r g y - s t r a i n c o n t o u r plots of a u n i t area of S-726, S-727, S-731 a n d S-745 fabrics  163  F i g u r e 5 - 37: V e l o c i t y - t i m e p r e d i c t i o n of the shell-based fabric m o d e l s h o w i n g the response of a l l six Kevlar® 129 panels i m p a c t e d b y a n R C C at 100 m / s  164  F i g u r e 5 - 38: Target's energy a b s o r p t i o n p l o t t e d against projectile d i s p l a c e m e n t o b t a i n e d f r o m n u m e r i c a l s i m u l a t i o n of the R C C projectile i m p a c t i n g the six Kevlar® 129 p a n e l s at 100 m / s  164  F i g u r e 5 - 3 9 : K i n e t i c to i n t e r n a l energy ratio of the target's a b s o r b e d energy p l o t t e d against projectile d i s p l a c e m e n t o b t a i n e d f r o m n u m e r i c a l s i m u l a t i o n of the R C C projectile i m p a c t i n g the six Kevlar® 129 panels at 100 m/s  165  F i g u r e 5 - 40: P r o p a g a t i o n of the strain w a v e e v i d e n t f r o m the e v o l u t i o n of the y a r n tension a l o n g the y a r n at different time instances i n a n i m p a c t s i m u l a t i o n of R C C projectile o n a s i n g l e y a r n of S-731 fabric m o d e l l e d b y the s h e l l elements  -XV-  165  F i g u r e 5 - 4 1 : Transverse d e f o r m a t i o n of a n S-731 single y a r n i m p a c t e d b y an R C C , s h o w i n g the g r o w t h of the d e f o r m a t i o n cone  166  F i g u r e 5 - 42: P r o p a g a t i o n of the strain w a v e e v i d e n t f r o m the e v o l u t i o n of the y a r n tension a l o n g the y a r n at different t i m e instances i n a n i m p a c t s i m u l a t i o n of R C C projectile o n a n S-731 single y a r n ( w i t h the a d d i t i o n of the mass of the c r o s s i n g yarns) m o d e l l e d b y the s h e l l elements  166  F i g u r e 5 - 4 3 : P r o p a g a t i o n of the strain w a v e e v i d e n t f r o m the e v o l u t i o n of the y a r n t e n s i o n a l o n g the y a r n at different time instances i n a n i m p a c t s i m u l a t i o n of R C C projectile o n a c r i m p e d ( c r i m p = 8.16%) S-731 single y a r n ( w i t h the a d d i t i o n of the m a s s of the crossing yarns) m o d e l l e d b y the shell elements  167  F i g u r e 5 - 44: T r a n s v e r s e d e f o r m a t i o n of a c r i m p e d ( c r i m p = 8.16%) S-731 single y a r n i m p a c t e d b y an R C C , s h o w i n g the g r o w t h of the d e f o r m a t i o n cone  167  F i g u r e 5 - 45: P r o p a g a t i o n of the strain w a v e e v i d e n t f r o m the e v o l u t i o n of the tension a l o n g the w a r p y a r n i n the i m p a c t s i m u l a t i o n of a n R C C projectile o n a 1-ply S-720 fabric m o d e l l e d b y the s h e l l elements. 168 F i g u r e 5 - 46: P r o p a g a t i o n of the strain w a v e e v i d e n t f r o m the e v o l u t i o n of the tension a l o n g the weft y a r n i n the i m p a c t s i m u l a t i o n of a n R C C projectile o n a 1-ply S-720 fabric m o d e l l e d b y the s h e l l elements. 168 F i g u r e 5 - 47: Transverse d e f o r m a t i o n of a 1-ply S-731 p a n e l p l o t t e d for a w a r p a n d weft y a r n i n the i m p a c t s i m u l a t i o n of a n R C C at strike v e l o c i t y of 100 m/s  1.69  F i g u r e 5 - 48: L o n g i t u d i n a l w a v e speed m e a s u r e d f r o m the n u m e r i c a l m o d e l of the fabric p l o t t e d against the y a r n c r i m p  169  F i g u r e 5 - 49: Transverse w a v e speed m e a s u r e d f r o m the n u m e r i c a l m o d e l of the fabric p l o t t e d against the y a r n c r i m p  170  F i g u r e 5 - 5 0 : A v e r a g e l o n g i t u d i n a l w a v e speed m e a s u r e d f r o m the n u m e r i c a l m o d e l of the fabric for the 100 m/s i s o - v e l o c i t y i m p a c t simulations  170  -xvi-  F i g u r e 5 - 5 1 : C r i t i c a l v e l o c i t y of the 1-ply Kevlar® 129 p a n e l s o b t a i n e d f r o m e x p e r i m e n t a l data a n d estimated b y the shell-based n u m e r i c a l model  171  F i g u r e 5 - 52: P o s t - m o r t e m e x a m i n a t i o n of the Kevlar® panels; p e r f o r a t i o n b y y a r n p u l l o u t (left) a n d b y y a r n r u p t u r e (right)  171  F i g u r e 5 - 53: E r r o r i n the p r e d i c t i o n of critical v e l o c i t y of the 1-ply Kevlar® 129 panels c o m p a r e d to e x p e r i m e n t a l data, p l o t t e d against their average c o u n t n u m b e r  172  F i g u r e 5 - 54: E n e r g y - s t r a i n c o n t o u r plots of a u n i t area of the six Kevlar® 129 fabric panels as a f u n c t i o n of fabric strain, also s h o w i n g the strain p a i r s c o r r e s p o n d i n g to the m o s t critical elements a r o u n d the projectile (shells w i t h highest energy) d u r i n g a n R C C i m p a c t s i m u l a t i o n at 100 m/s  172  F i g u r e 5 - 55: T r a n s v e r s e d e f o r m a t i o n of the a n S-726 4-layer target i m p a c t e d at 100 m/s s h o w i n g the increased interaction of adjacent layers i n the alternating p a c k i n g m e t h o d  173  F i g u r e 5 - 56: C o n t o u r s of transverse d e f o r m a t i o n of the t h i r d l a y e r a w a y f r o m the strike face of a n S-726 4-layer target i m p a c t e d at 100 m/s s h o w i n g the benefit of alternating p a c k i n g m e t h o d  173  F i g u r e 5 - 57: C o m p a r i s o n of the projectile response for different p a n e l o r i e n t a t i o n i n S-726 4-layer target  174  F i g u r e 5 - 58: C r o s s - s h a p e d fabric s p e c i m e n u s e d i n the b i a x i a l e x p e r i m e n t s c o n d u c t e d b y B u e t - G a u t i e r a n d Boisse [37]  174  F i g u r e 5 - 59: P r e d i c t i o n s of the n u m e r i c a l m o d e l o n the b i a x i a l l o a d i n g of the b a l a n c e d p l a i n w e a v e glass fabric, after c a l i b r a t i n g the i n p u t p a r a m e t e r s to m a t c h the i s o - d i s p l a c e m e n t l o a d i n g case (k = 1). 175 F i g u r e 5 - 6 0 : P r e d i c t i o n s of the n u m e r i c a l m o d e l o n the b i a x i a l l o a d i n g of the b a l a n c e d p l a i n w e a v e glass fabric, e x t e n d e d to other t w o l o a d i n g scenarios ( k = 0.5 & 2)  175  F i g u r e 5 - 6 1 : P r e d i c t i o n s of the n u m e r i c a l m o d e l o n the w a r p extension of the non-balance p l a i n w e a v e glass fabric, after c a l i b r a t i n g the i n p u t p a r a m e t e r s to m a t c h the i s o - d i s p l a c e m e n t l o a d i n g case ( k = 1). 176  -xvii-  F i g u r e 5 - 62: P r e d i c t i o n s of the n u m e r i c a l m o d e l o n the weft extension of the n o n - b a l a n c e p l a i n w e a v e glass fabric, after c a l i b r a t i n g the i n p u t p a r a m e t e r s to m a t c h the i s o - d i s p l a c e m e n t l o a d i n g case (k = 1). 176 F i g u r e 5 - 63: P r e d i c t i o n s of the n u m e r i c a l m o d e l o n the w a r p extension of the non-balance p l a i n w e a v e glass fabric, e x t e n d e d to other t w o l o a d i n g scenarios ( k = 0.5 & 2)  177  F i g u r e 5 - 64: P r e d i c t i o n s of the n u m e r i c a l m o d e l o n the weft e x t e n s i o n of the non-balance p l a i n w e a v e glass fabric, e x t e n d e d to other t w o l o a d i n g scenarios ( k = 0.5 & 2)  177  F i g u r e 6 - 1 : V e l o c i t y - t i m e response of the R C C projectile i m p a c t i n g a 1-ply S728 fabric at 100 m/s. T w o zones of b e h a v i o u r are i d e n t i f i e d , d o m i n a t e d b y m a s s (zone I) a n d stiffness (zone II) of the target. 207 F i g u r e 6 - 2 : C o n t o u r s of i n - p l a n e fabric v e l o c i t y i n w a r p (right) a n d weft (left) d i r e c t i o n s r e a c h i n g the p a n e l b o u n d a r i e s , p r e d i c t e d for the i m p a c t of a n R C C projectile o n a 1-ply S-728 fabric at 100 m / s  207  F i g u r e 6 - 3 : C o n t o u r s of t e n s i o n i n w a r p (right) a n d weft (left) y a r n s r e a c h i n g the p a n e l b o u n d a r i e s , p r e d i c t e d for the i m p a c t of a n R C C projectile o n a 1-ply S-728 fabric at 100 m / s  208  F i g u r e 6 - 4 : K i n e t i c / s t r a i n energy ratio of the total e n e r g y a b s o r b e d b y a 1-ply S-728 fabric i m p a c t e d at 100 m/s b y a n R C C projectile  208  F i g u r e 6 - 5 : T e n s i o n - t i m e response of the weft y a r n of a 1-ply S-728 fabric i m p a c t e d at 100 m / s b y a n R C C projectile  209  F i g u r e 6 - 6 : T e n s i o n - d i s p l a c e m e n t response of a n S-728 u n i t - c e l l l o a d e d i n the w a r p d i r e c t i o n w i t h fixed weft y a r n , s h o w i n g the effects of ± 2 0 % change i n the elastic m o d u l u s of the y a r n s  209  F i g u r e 6 - 7 : V e l o c i t y - t i m e response of the R C C projectile i m p a c t i n g a 1-ply S728 fabric at 100 m/s, s h o w i n g the effects of ± 2 0 % change i n the elastic m o d u l u s of the yarns  -xviii-  210  F i g u r e 6 - 8 : Strain-time response of the central weft y a r n of a n S-728 i n a n i m p a c t s i m u l a t i o n of a n R C C s t r i k i n g a 1-ply target at 100 m/s, s h o w i n g the effects of m o d u l u s change  210  F i g u r e 6 - 9 : C r i m p m e a s u r e m e n t p e r f o r m e d i n d e p e n d e n t l y o n the six K e v l a r ® 129 fabric p a n e l s b y t w o different i n d i v i d u a l s , s h o w i n g a m a x i m u m d e v i a t i o n of 20% f r o m each other  211  F i g u r e 6 - 1 0 : T e n s i o n - d i s p l a c e m e n t response of a n S-728 u n i t - c e l l l o a d e d i n the w a r p d i r e c t i o n w i t h fixed weft y a r n , s h o w i n g the effects of ± 2 0 % change i n the y a r n c r i m p  211  F i g u r e 6 - 1 1 : V e l o c i t y - t i m e response of the R C C projectile i m p a c t i n g a 1-ply S728 fabric at 100 m/s, s h o w i n g the effects of ± 2 0 % change i n the yarn crimp  212  F i g u r e 6 - 1 2 : Strain-time response of an S-728 central weft y a r n i n a n i m p a c t s i m u l a t i o n of a n R C C s t r i k i n g a 1-ply target at 100 m/s, s h o w i n g the s e n s i t i v i t y of the fabric response to c r i m p  212  F i g u r e 6 - 1 3 : V e l o c i t y - t i m e response of the R C C projectile i m p a c t i n g a 1-ply S728 fabric at 100 m/s, s h o w i n g the effects of ± 2 0 % change i n the fabric areal d e n s i t y  213  F i g u r e 6 - 1 4 : Strain-time response of an S-728 central weft y a r n i n a n i m p a c t s i m u l a t i o n of a n R C C s t r i k i n g a 1-ply target at 100 m/s, s h o w i n g the sensitivity of the fabric response to areal d e n s i t y  213  F i g u r e 6 - 1 5 : I n i t i a l 100 us of the v e l o c i t y - t i m e response of the R C C projectile i m p a c t i n g a 1-ply S-728 fabric at 100 m/s, s h o w i n g the effects of ± 2 0 % change i n the fabric elastic m o d u l u s  214  F i g u r e 6 - 1 6 : I n i t i a l 100 us of the v e l o c i t y - t i m e response of the R C C projectile i m p a c t i n g a 1-ply S-728 fabric at 100 m/s, s h o w i n g the effects of ± 2 0 % change i n the fabric c r i m p v a l u e  214  F i g u r e 6 - 1 7 : I n i t i a l 100 p.s of the v e l o c i t y - t i m e response of the R C C projectile i m p a c t i n g a 1-ply S-728 fabric at 100 m/s, s h o w i n g the effects of ± 2 0 % change i n the fabric areal density  215  F i g u r e 6 - 18: T e n s i l e response of the w a r p y a r n of a n S-728 fabric u n i t - c e l l subjected to d i s p l a c e m e n t i n the w a r p d i r e c t i o n (weft y a r n fixed), a s s u m i n g compressible, n o n - c o m p r e s s i b l e a n d d e c o u p l e d crossover y a r n s  215  -xix-  F i g u r e 6 - 1 9 : T e n s i l e response of the weft y a r n of a n S-728 fabric u n i t - c e l l subjected to d i s p l a c e m e n t i n the w a r p d i r e c t i o n (weft y a r n fixed), assuming compressible, non-compressible a n d decoupled crossover y a r n s  216  F i g u r e 6 - 20: Surfaces of e n e r g y stored i n a n S-728 fabric u n i t - c e l l subjected to w a r p a n d weft displacement, a s s u m i n g c o m p r e s s i b l e , n o n c o m p r e s s i b l e a n d d e c o u p l e d crossover y a r n s  216  F i g u r e 6 - 2 1 : E n e r g y stored i n a n S-728 fabric u n i t - c e l l subjected to d i s p l a c e m e n t i n the w a r p d i r e c t i o n (weft y a r n fixed), a s s u m i n g c o m p r e s s i b l e , n o n - c o m p r e s s i b l e a n d d e c o u p l e d crossover yarns. 217 F i g u r e 6 - 22: V e l o c i t y - t i m e response of a n R C C projectile i m p a c t i n g a 1-ply S728 fabric at 100 m/s, p r e d i c t e d a s s u m i n g c o m p r e s s i b l e , n o n c o m p r e s s i b l e a n d d e c o u p l e d crossover y a r n s  217  F i g u r e 6 - 23: E n e r g y - t i m e response of a n R C C projectile i m p a c t i n g a 1-ply S728 fabric at 100 m/s, p r e d i c t e d a s s u m i n g c o m p r e s s i b l e , n o n c o m p r e s s i b l e a n d d e c o u p l e d crossover y a r n s  218  F i g u r e 6 - 24: R a t i o of the k i n e t i c to i n t e r n a l energy c o m p o n e n t s stored i n the fabric i n the i m p a c t s i m u l a t i o n of a n R C C projectile s t r i k i n g a 1p l y S-728 fabric at 100 m/s, p r e d i c t e d a s s u m i n g c o m p r e s s i b l e , n o n - c o m p r e s s i b l e a n d d e c o u p l e d crossover y a r n s  218  F i g u r e 6 - 25: T e n s i l e response of the w a r p y a r n of a n S-731 fabric u n i t - c e l l subjected to d i s p l a c e m e n t i n the w a r p d i r e c t i o n (weft y a r n fixed), a s s u m i n g compressible, n o n - c o m p r e s s i b l e a n d d e c o u p l e d crossover y a r n s  ;  219  F i g u r e 6 - 26: T e n s i l e response of the weft y a r n of a n S-728 fabric u n i t - c e l l subjected to d i s p l a c e m e n t i n the weft d i r e c t i o n ( w a r p y a r n fixed), a s s u m i n g compressible, n o n - c o m p r e s s i b l e a n d d e c o u p l e d crossover y a r n s  219  F i g u r e 6 - 27: Surfaces of energy stored i n a n S-731 fabric u n i t r c e l l subjected to w a r p a n d weft displacement, a s s u m i n g c o m p r e s s i b l e , n o n c o m p r e s s i b l e a n d d e c o u p l e d crossover y a r n s  220  F i g u r e 6 - 28: E n e r g y stored i n a n S-731 fabric u n i t - c e l l subjected to d i s p l a c e m e n t i n the w a r p d i r e c t i o n (weft y a r n fixed), a s s u m i n g  -XX-  c o m p r e s s i b l e , n o n - c o m p r e s s i b l e a n d d e c o u p l e d crossover yarns. 220 F i g u r e 6 - 29: E n e r g y stored i n a n S-731 fabric u n i t - c e l l subjected to d i s p l a c e m e n t i n the weft d i r e c t i o n ( w a r p y a r n fixed), a s s u m i n g c o m p r e s s i b l e , n o n - c o m p r e s s i b l e a n d d e c o u p l e d crossover yarns. 221 F i g u r e 6 - 30: V e l o c i t y - t i m e response of a n R C C projectile i m p a c t i n g a 1-ply S731 fabric at 100 m/s, p r e d i c t e d a s s u m i n g c o m p r e s s i b l e , n o n c o m p r e s s i b l e a n d d e c o u p l e d crossover y a r n s  221  F i g u r e 6 - 3 1 : R a t i o of the k i n e t i c to i n t e r n a l energy c o m p o n e n t s stored i n the fabric i n the i m p a c t s i m u l a t i o n of a n R C C projectile s t r i k i n g a 1p l y S-731 fabric at 100 m/s, p r e d i c t e d a s s u m i n g compressible, n o n - c o m p r e s s i b l e a n d d e c o u p l e d crossover y a r n s  222  F i g u r e 6 - 32: M a x i m u m shear strain o b s e r v e d i n the six Kevlar® 129 panels i m p a c t e d b y a n R C C f l y i n g at 100 m/s, c o m p a r e d to the stiffening a n d l o c k i n g shear strains, y a n d y x  2  222  F i g u r e 6 - 33: V e l o c i t y - t i m e response of a n R C C projectile i m p a c t i n g a 1-ply S728 fabric at 100 m/s, s h o w i n g t w o extreme cases of l o w a n d h i g h fabric shear m o d u l u s values  223  F i g u r e 6 - 34: D e f o r m a t i o n patterns s h o w n b y contours of z - d i s p l a c e m e n t c o n s i d e r i n g l o w (left) a n d h i g h (right) fabric shear m o d u l u s . . . . 223 F i g u r e 6 - 35: V e l o c i t y - t i m e response of a n R C C projectile i m p a c t i n g a 1-ply S728 fabric at 100 m/s, s h o w i n g the effect of stiffening a n d l o c k i n g angles o n the response of the projectile  224  F i g u r e 6 - 36: B o u n d a r y s l i p p a g e m a r k e d d u r i n g a b a l l i s t i c i m p a c t e x p e r i m e n t o n a 1-ply S-728 panels at a strike v e l o c i t y of 36.9 m / s  224  F i g u r e 6 - 37: B o u n d a r y reaction i n the weft d i r e c t i o n of a n S-728 1-ply fabric i m p a c t e d b y a n R C C projectile at 100 m/s  225  F i g u r e 6 - 38: S i m u l a t i o n of s l i p at the b o u n d a r y of a n S-728 1-ply fabric t h r o u g h p r e s c r i b i n g the i n - p l a n e m o t i o n of the n o d e s at the b o u n d a r i e s (weft s l i p s h o w n here)  -xxi-  225  F i g u r e 6 - 39: V e l o c i t y - t i m e response of a n R C C projectile i m p a c t i n g a 1-ply S728 fabric at 100 m/s, s h o w i n g the effect of weft b o u n d a r y slip o n the response of the projectile  226  F i g u r e 6 - 40: R e a c t i o n force-time response of the central weft y a r n i n a n i m p a c t s i m u l a t i o n of a n R C C projectile s t r i k i n g a 1-ply S-728 fabric at 100 m/s  226  F i g u r e 6 - 4 1 : V e l o c i t y - t i m e response of a n R C C projectile i m p a c t i n g a 1-ply S728 fabric at 100 m/s, s h o w i n g the effect of s l i p p i n g time o n the deceleration of the projectile  227  F i g u r e 6 - 42: V e l o c i t y - t i m e response of a n R C C projectile i m p a c t i n g a 1-ply S728 fabric at 36 m/s w i t h a n d w i t h o u t b o u n d a r y slip, c o m p a r e d to the b a l l i s t i c e x p e r i m e n t a l data (5 m m weft s l i p b e t w e e n 200 a n d 650 us)  227  F i g u r e 6 - 43: V e l o c i t y - t i m e response of a n R C C projectile i m p a c t i n g a 16-ply S728 fabric at 170 m/s, s h o w i n g the effect of i n t e r l a y e r gap o n the projectile deceleration  228  F i g u r e 6 - 44: V e l o c i t y - t i m e response of a n R C C projectile i m p a c t i n g a 16-ply S728 fabric at 170 m/s, s h o w i n g responses of its tip, tail, a n d centre of m a s s  228  F i g u r e 6 - 4 5 : D i s p l a c e m e n t - t i m e response of a n 8 g R C C projectile i m p a c t i n g 4p l y a n d m a s s - e q u i v a l e n t 1-ply targets at 300 m / s  229  F i g u r e 6 - 4 6 : D i s p l a c e m e n t - t i m e response of a n 8g R C C projectile i m p a c t i n g 4p l y a n d 1-ply targets at 300 m/s  229  F i g u r e 6 - 47: P o s t - m o r t e m d e f o r m a t i o n o b s e r v e d i n the b a c k i n g m a t e r i a l i n the i m p a c t s i m u l a t i o n of a n 8g R C C projectile s t r i k i n g a 4 - p l y (top) a n d 1-ply (bottom) S-728 targets at 300 m/s  230  F i g u r e A - 1: C r o s s - s e c t i o n a l m i c r o - i m a g e s of S-728 p a n e l (top: w a r p , b o t t o m : weft) [42]  260  F i g u r e A - 2: M e a s u r e m e n t of v a r i o u s d i m e n s i o n s o n the m i c r o - i m a g e s of S-728 (warp) panels [42]  260  -xxii-  F i g u r e A - 3: Y a r n o v e r l a y i n fabric S-731; cross-section (top) a n d p l a n v i e w (bottom) [42]  261  F i g u r e A - 4: M i c r o - i m a g e of the S-731 fabric cross-section, cut at a n angle [42]. 261 F i g u r e A - 5: Process of creating a single y a r n crossover i n the pre-processor. 262 F i g u r e A - 6: S i n u s o i d a l y a r n profile u s e d to create the y a r n centreline i n the m e s h pre-processor  262  F i g u r e A - 7: Sine-based (left) or e l l i p t i c a l (right) cross-section d e f i n i t i o n of yarns  263  F i g u r e A - 8: Process of generating the F . E . m e s h of a fabric u n i t - c e l l  263  F i g u r e A - 9: C r o s s - s e c t i o n a l m i c r o - i m a g e of S-720 weft fabric c o m p a r e d to the generated finite element m e s h  -xxiii-  264  List of Symbols Chapter 2 - Experimental Data a  F i t t i n g constant i n the p r o p o s e d c o m p r e s s i o n m o d e l  b  F i t t i n g constant i n the p r o p o s e d c o m p r e s s i o n m o d e l  d  Transverse c o m p r e s s i v e d e f o r m a t i o n of the fabric  m  M a s s of fibres  n  T h e e x p o n e n t of the p r o p o s e d c o m p r e s s i o n m o d e l  p  Transverse pressure a p p l i e d to the fabric  p  D e f i n e d as "latent p r e s s u r e " at z e r o p r e s s u r e  t  I n i t i a l fabric thickness  v  C u r r e n t v o l u m e of the fabric  c  0  Q  v  I n i t i a l v o l u m e of the fabric  0  A  L o a d i n g area i n the transverse c o m p r e s s i o n tests  A  A r e a l d e n s i t y of the fabric  E  Elastic m o d u l u s of the fibres  d  F  Transverse c o m p r e s s i v e force A p p l i e d to the fabric  K  A d i m e n s i o n l e s s constant (see S e c t i o n  K'  C o n s t a n t u s e d i n the transverse c o m p r e s s i o n m o d e l (see  c  Section K"  2.2.2)  2.2.2)  A f i t t i n g constant i n the transverse c o m p r e s s i o n m o d e l d e v e l o p e d i n this s t u d y  P  Pressure a p p l i e d to the fabric i n the p r o p o s e d transverse compression model  V  M a x i m u m available fibre v o l u m e fraction  V  F i b r e v o l u m e fraction o f the fabric  a  f  V  mia  M i n i m u m c o m p r e s s e d v o l u m e o f the fabric i n compression  -xxiv-  V  I n i t i a l fibre v o l u m e fraction  V  C u r r e n t v o l u m e of the fabric i n c o m p r e s s i o n  B  A constant d e t e r m i n e d e m p i r i c a l l y  Y  E x p o n e n t i n the general p r e s s u r e - v o l u m e r e l a t i o n s h i p for  0  the fabrics p  D e n s i t y of fibres  a  A p p l i e d b u l k stress of the fabric  b  Chapter 3 - 3 D Fabric Crossover M o d e l a  M o d e l parameter for the transverse m o d u l u s of the s o l i d  E  elements b  M o d e l p a r a m e t e r for the transverse m o d u l u s of the s o l i d  E  elements d  N o r m a l i n - p l a n e d i s p l a c e m e n t a p p l i e d to the fabric u n i t  x  cell i n the l o c a l 1-direction (warp) d  N o r m a l i n - p l a n e d i s p l a c e m e n t a p p l i e d to the fabric u n i t  2  c e l l i n the l o c a l 2 - d i r e c t i o n (weft) x  M a t e r i a l coordinate s y s t e m i n the d e f o r m e d state  A"  C r o s s - s e c t i o n a l area of a s o l i d element p e r p e n d i c u l a r to the y a r n a x i a l d i r e c t i o n at time-step n  A  0  I n i t i a l cross-sectional area of a s o l i d element p e r p e n d i c u l a r to the y a r n a x i a l d i r e c t i o n  C  S p e e d of s o u n d w a v e i n the m a t e r i a l  E  I n i t i a l m o d u l u s of a s o l i d element i n the a x i a l y a r n  a  direction E  c  U p p e r l i m i t of the transverse m o d u l u s of the s o l i d elements  E"  M o d u l u s i n the y a r n a x i a l d i r e c t i o n at time-step n  -XXV-  Secant modulus of a solid element i n the transverse direction (local 2-dir.) at time-step n Secant modulus of a solid element i n the transverse direction (local 3-dir.) at time-step n Deformation gradient matrix Shear modulus of the solid element material i n the plane perpendicular to / local axis Jacobian, defined as the determinant of F Reduction factor for the shear modulus of the solid elements Reduction factor for the transverse yarn modulus when subjected to tension N o m i n a l length of an element i n the finite element mesh Length of a solid element i n the yarn axial direction at time-step n Initial length of a solid element i n the yarn axial direction Volume of a solid element at time-step n Initial Volume of a solid element Solid element volume corresponding to the m a x i m u m transverse modulus Yarn's fibre volume fraction at time-step n Initial fibre volume fraction of the yarn Material coordinate system i n the undeformed state Logarithmic normal strain i n the yarn axial direction at time-step n Instantaneous strain to failure of the yarn Component / of the solid element stress vector at timestep n  -xxvi-  At  Time step size i n explicit finite element analysis  Chapter 4 - 2D Shell Crossover M o d e l ( ),  Quantity referring to warp (/ = 1) and weft (/' = 2)  a  Constant used i n the definition of the transverse compression model  b  Constant used i n the definition of the transverse compression model  cr  C r i m p value of yarn i of the unit cell  d  Transverse compression of yarn /' of the unit cell  d  Extensional in-plane displacement applied to the fabric  t  Ci  t  unit cell i n / direction /,  Objective function obtained from the equilibrium of yarn contact forces  f  Objective function obtained from the compatibility of the  2  distance between the yarn centrelines g  Transverse gap between the warp and weft yarns of a unit cell  hj  Centreline height of yarn / at the centre of the unit cell  h  Initial centreline height of yarn / at the centre of the unit  0i  cell m  Yarn count i n / direction (threads per unit length)  p  In-plane gap between neighbouring parallel yarns  q"  Vector defining the / material direction at time step n  q" and q,"  Components of the material vector / i n the local x and y  i  directions at time step n u  In-plane normal distance between parallel yarns  -xxvii-  u  In-plane normal distance between parallel yarns at the  x  onset of parallel yarn contact u  In-plane normal distance between parallel yarns at shear  2  locking w,  Unit cell length the i direction  x, y, z  Local coordinate system of the representative shell element  x  t  Unit cell coordinate axis along yarn i  z  t  Unit cell coordinate axis perpendicular to the fabric plane  A  Equivalent compacted cross-sectional area of a yarn  p  A  Solid cross-sectional area of a yarn  A  Cross-sectional area of the shell element representing the  s  sh  fabric unit cell A  Solid cross-sectional area of yarn / of the unit cell  sj  E  Elastic modulus of yarn i of the unit cell  t  F  Contact force of yarn / of the unit cell  F"  Incremental deformation gradient at time step n  G,  Initial shear modulus of the unit cell  G  Unit cell shear modulus for strains beyond shear locking  Ci  2  /_  2x2 unit matrix  J  Jacobian matrix, with components J  kl  - —  dh,  (where  k = 1,2 and / = 1,2)  L  Side-length of the square fabric sample used i n the picture-frame shear test  T  {  Tension i n yarn / of the unit cell  S  t  Length of yarn / of the unit cell  -xxviii-  Initial crimped length of yarn /' of the unit cell Slope between the fabric's plane and the centreline of yarn / at the edge of the unit cell Instantaneous in-plane angle between the warp and weft yarns Initial in-plane angle between the warp and weft yarns (90° for plain weave fabric) Warp-to-weft angle corresponding to the onset of contact between parallel yarns i n a sheared fabric Warp-to-weft angle corresponding to the onset of shear locking (full compaction of parallel yarns) i n a sheared fabric Shear strain corresponding to the onset of contact between parallel yarns Shear strain corresponding to the onset of shear locking Stretch increment in the / material direction at time step n Angle between the material vector / and the local x axes at time step n Stress matrix obtained from the yarn i n the /' material direction at time step n Stress matrix obtained from the yarn i n the / material direction at time step n, transformed into the local coordinate system Total average stress matrix i n at time step n Element on the row 1 and column 1 of the /' material direction strain increment matrix at time step n N o r m a l strain increment i n the local x -direction at time step n  -xxix-  As"  N o r m a l strain increment i n the local y -direction at time step n  As  n xy  Shear strain increment i n the local xy -direction at time step n  As"  Strain increment matrix at time step n  Af,"  Strain increment matrix i n the i material direction at time step n  A"  Total stretch i n the / material direction at time step n  0,"  Transformation matrix between the local and material coordinate systems at time step n  Chapter 5 - Ballistic Response of Fabric Panels "d  Total transverse compression of an n -ply fabric specimen  K  Spring stiffness representing the inter-ply fabric  c  ff  deformation K  fm  Spring stiffness representing the deformation resulting from the interaction of the fabric and the loading cell  d  Inter-ply fabric transverse deformation  d  Transverse deformation resulting from the interaction of  ff  fm  the fabric and the loading cell ./V  Number of layer i n a multi-ply fabric pack  C  Speed of sound i n material  E  Yarn elastic modulus  p  Yarn material density  k  Ratio of warp to weft strain applied i n the biaxial extension experiments (see Section 5.3)  -XXX-  Acknowledgements This thesis was made possible with the unlimited support and guidance of my supervisors, especially D r . Reza Vaziri, to w h o m I am deeply grateful. Thank you Reza, for the opportunity and for trusting me throughout this journey. I would also like to thank D r . A n o u s h Poursartip w h o supported me over the years and helped me along when I needed it the most. M a n y thanks go to the other members of m y supervisory committee, D r . Goran Fernlund and Dr. Terje Haukaas, for their role throughout the course of this thesis. I w o u l d like to thank the Natural Sciences and Engineering Research Council of Canada, Defence Research and Development Canada, DuPont Advanced Fiber Systems Inc., Pacific Safety Products Inc., and The University of British Columbia for the financial support of this project. I shared this experience w i t h many other people, to w h o m I am thankful. I would like to thank D r . Elvis Cepus, M r . Roger Bennett and M s . Ingrid Kongshavn w h o have been instrumental i n exposing me to the w o r l d of ballistic experiments. I w o u l d like to thank all the past and current members of the Composites Group for their support and friendship, especially D r . Anthony Floyd and M r . M e h d i Haghshenas for their friendship and for the endless technical discussions. A n d finally I w o u l d like to thank m y family, specially m y mother, w h o has always believed i n me and given me her unconditional love.  -xxxi-  Dedication  To my mother, who instilled in me the determination to follow my dreams, and devoted her life to ensure I get a chance to do so. And to my late father, who always believed in me and is greatly missed.  -xxxii-  Chapter 1 - Dynamic Analysis of Fabric Systems  CHAPTER 1 -  D Y N A M I C ANALYSIS O FFABRIC SYSTEMS  Modelling the ballistic response of fabrics has been the focus of many studies over the years. In this chapter, a review of the various modelling approaches undertaken to capture the dynamic behaviour of fabrics is provided and an overall picture of the evolution of body armour analysis and design is sketched. Following this introduction, the goals of the current study is discussed and the overall modelling methodology and its several components are introduced.  1.1  M O D E L L I N G BALLISTIC FABRICS - A HISTORIC B A C K G R O U N D  Fabric-based body armour systems are used by police officers, military personnel and many others to reduce the risk of threats they are exposed to everyday i n their line of duty. Historically, body armours were made from metals and mainly used by soldiers to provide protection i n the battles of ancient wars. W i t h the advancement i n weapons and the rising need to defeat these harmful threats, the metallic body armours grew bigger and heavier, to the point that they were almost impractical, as they critically limited the mobility and flexibility of the soldiers (for a full review of the armour history,  Chapter 1 - Dynamic Analysis of Fabric Systems  see [1]). Introduction of soft personal body armours i n the form used today was made possible b y the introduction of high-performance fibres and fabrics i n the early 1970's. Since then, many studies have been focused on the understanding of the complex mechanical behaviour of woven fabrics and on quantifying their performance as the building block of fabric-based armour systems. However, it is only fair to say that the knowledge of such behaviour is still underdeveloped and there are numerous avenues to be explored on the dynamic behaviour of w o v e n fabrics. The designers of bullet-proof vests follow strict guidelines to demonstrate the capability of the armour to defeat the threats they are designed for. The design process is highly experiment-based despite the high price of the material used in the bullet-proof vests. This is perhaps due to the absence of a robust numerical tool to analyse the ballistic performance of fabrics. Availability of a reliable fabric model can help the designers to save the cost of material by redirecting the design and manufacturing i n a desired direction, as well as by reducing the time required to demonstrate the capability of their design i n defeating a set of preconceived threats. Prediction of the complex behaviour of fabrics under  dynamic loading  scenarios has been the centre of many studies over the years. Fabrics used i n a variety of military and civil applications pose a certain challenge for the designers due to their specific non-homogeneity and multifaceted interaction of their components. The ever-increasing demand for high-performance fabrics over the past few decades has revealed the immaturity of the design tools and inability to fully optimize the armour systems. This has created much interest  Chapter 1 - Dynamic Analysis of Fabric Systems  over the past decades i n exploring various aspects of the fabric's ballistic performance. The studies geared towards predicting the dynamic behaviour of fabrics can be divided into analytical and numerical approaches. Since this thesis is focused on the development of a new computational approach, only a review of the relevant numerical approaches i n the literature is presented i n this chapter. The numerical models developed to capture the response of fabric systems has evolved significantly over the past decades. However, predicting the behaviour of these constantly-evolving lightweight structures under high-speed impact is a formidable task. M a t u r i n g of finite element analysis methodology along with the massive improvement of the available computational resources i n the form of faster computers and large storage devices have enabled the modellers to incorporate more details of the structure and materials into their consideration. This has resulted i n the development of different generations of fabric models, categorized here as Discrete models and Unit-cell C o n t i n u u m models, discussed separately below.  1.1.1 Discrete models The Discrete modelling technique is one of the earliest approaches considering the fabric as an assembly of discrete masses connected to each other by cable or spring elements (see Figure 1 - 1). This net-like mass-spring approximation of the fabrics provided a simple and efficient platform for studying the dynamic behaviour of fabric systems. Although many different versions of discrete models have been developed over the years, they all share a very similar mathematical basis. While spatial characteristics of the fabric are more or less at  Chapter 1 - Dynamic Analysis of Fabric Systems  the discretion of the user, the temporal discretization is controlled by the convergence of the mathematical solution of the physical system. Stable results are obtained only if the time step size is smaller than the one needed for the strain wave to travel the smallest element nominal dimension i n the finite element mesh (see Section 3.3). U p o n impact, nodes directly i n contact w i t h the projectile start to move w i t h it, and the equilibrium of the nodes are sought based on the impulse-momentum equations. A t each time-step, the coordinates of the nodes are updated from their velocities and the tensile forces i n the cables are determined. The calculations are completed by determining the new nodal velocities and the analysis moves to the next time-step. The same procedure is followed for the duration of the impact event. The numerical model allows the user to obtain various target and projectile information during the event and gain a better understanding of the mechanics of the fabric deformation. Work of Roylance et al. [2] was fundamental i n introducing the discretized mass-spring model of fabrics and more or less transformed the ballistic textile industry to focus more efforts on this modelling approach. For this purpose, effects of the projectile geometry, yarn bending, slippage at the crossovers, etc. were initially ignored. Roylance et al. [2, 3] used this model to investigate the energy absorption mechanisms of fabric panels and concluded that despite the limitations of the approach, it is a useful tool to study the behaviour of fabric systems and can be used i n the preliminary design of armour systems. After Roylance et al. [2], many other researchers used the discrete modelling approach to study various characteristics of fabric panels. Some examples of  Chapter 1 - Dynamic Analysis of Fabric Systems  such studies can be found i n the works of Billon and Robinson [4], and Johnson et al. [5]. Tan, Shim and co-workers [6-11] extensively used the discrete masscable model to study the effects of fabric crimp, boundary conditions, projectile obliquity and a variety of other factors on the behaviour of fabric targets. Shahkarami et al. [1, 12] developed a similar pin-jointed mass-string model for impact simulation of fabric, called T E X I M , and studied the effect of fabric boundary conditions on the overall energy absorption of the system and perforation of the projectile. T E X I M was later used by N o v o t n y et al. [13, 14] to study the behaviour of hybrid panels, comprised of two different materials, and their possible advantages over ordinary monolithic fabric packs i n absorbing the impact energy. The major shortcoming of the discrete models is their failure to capture the biaxial nature of the fabric response. To overcome this deficiency, some researchers have introduced crimped yarns w i t h a zigzag profile where the crossing yarns interact during the deformation. Cunniff, Ting and co-workers [15, 16] developed a numerical model where the warp and weft yarns were non-coplanar to capture the crimp i n the yarns. The warp and weft yarn coupling was then incorporated via spring elements. Tan and Ching [17] presented a discrete model where the fabric was created from a network of linear bar elements, w i t h the warp and weft yarns initially i n contact with, but not jointed to, one another. This model allowed for the consideration of yarn crimp, as well as the sliding contact between the crossing yarns. A l l the modelling attempts using the discrete approach, some of which discussed above, have tremendously contributed to the general knowledge on  Chapter 1 - Dynamic Analysis of Fabric Systems  the ballistic behaviour of fabrics. However, the discrete modelling techniques are oversimplified for the current needs of the high-performance  fabric  industry and are abandoned by many scientists i n favour of more complex models, discussed i n the next section.  1.1.2 Unit-cell continuum models The unit cell models are generally based on a continuum representation of the fabrics. Based on the scale of such representation,  different modelling  techniques are formulized (see Figure 1 - 2 ) . Some 3D models of fabric and fabric unit cells usually incorporate the weave details at the yarn level i n their structure definition. In these models, individual yarns are meshed based on their respective geometry inside the weave, and proper material properties are assigned to them. The complexity of these models and the level of structural details  they  deformational  consider  make  behaviour  of  them fabric  an excellent tool for analysing the unit  cells.  However,  the  explicit  representation of the yarns and the amount of details they consider eventually leads to an extreme inefficiency, u p to a point that they become impractical for real-size problems. Shockey et al. [18, 19] presented the results of their study on the fabric barriers to contain turbine engine fragments of the aircrafts. To complement the extensive experimental effort carried out i n that study, a detailed 3D model and a simple shell model were also developed. The detailed 3D model was perhaps one of the first attempts to capture the deformational response of the fabrics by explicitly modelling the yarns i n the weave. In a separate study, D u a n et al. [2023] recently used the same modelling approach to investigate the influence of  Chapter 1 - Dynamic Analysis of Fabric Systems  intra-ply friction (i.e. friction between the crossing yarns) on the energy absorption characteristics of fabric panels. Similar modelling technique was used by Boisse and co-workers [24-27], Peng and Cao [28], and others. In all these cases, the 3D unit cell model of the fabric or fabric composites appeared to be very inefficient and limited to studying only the fabric unit cell or fabric specimen w i t h very small dimensions. To overcome the prohibiting inefficiency of the 3D model, shell-based unit cell models were developed for simulation of fabrics i n ballistic impact and for the analysis of w o v e n composites forming process. In this approach, the fabric unit cell is replaced by a representative shell element, whose material constitutive relationship is a representation of the orthotropic fabric behaviour i n a smeared approach. The material model developed for the shell element can be as simple as a basic orthotropic model (equivalent of the discrete pin-jointed model) to a fully inclusive mechanistic model of the fabric crossover. A s a result, a varying level of complexify can be achieved by these models w h i c h w o u l d inversely affect their efficiency. Examples of such models include the work by Tabiei and Ivanov [29, 30], K i n g et al. [31-33], L i m et al. [34], Tanov and Brueggert [35, 36], Buet-Gautier and Boisse [37], Xue et al. [38], L u o [39], Srubbs [40], Simons and co-workers [18, 41], etc. Overall, the unit cell fabric modelling approaches are used more widely i n a variety of applications. The efficient shell-based models appear to be extremely versatile i n terms of their applicability and efficiency. The shell elements can be formulated  to be as simple as the discrete bar models, or almost as  comprehensive as the 3D solid models, thus offering a wide spectrum of  Chapter 1 - Dynamic Analysis of Fabric Systems  options to the user. A s a result, the shell-based continuum models currently seem to be a perfect modelling approach for a wide range of applications.  1.2  CURRENT STATE-OF-THE-ART  The need for a robust numerical model that can truly capture the realistic response of the fabric structure without sacrificing efficiency still exists. Despite the development of various modelling techniques to date, a robust numerical approach that is efficient and can incorporate various deformational and failure mechanisms of fabrics is still lacking i n the engineering community. The pool of available ballistic experimental data on various fabric types and target configurations is extremely large, leading to a vast discontinuity between the experimental fabric research and the advancement of the numerical modelling techniques. However, the significant evolution of the fabric models over the past decade has the promise of the numerical models that are an inseparable part of the armour analysis and design process i n the near future.  1.3  RESEARCH OBJECTIVES  The main objective of this study is to develop an efficient and robust numerical analysis tool to aid w i t h the design of fabric-based personal protective systems. The availability of such predictive tools w o u l d be useful for the engineers and designers of body armours to reduce the time and material costs involved i n arriving at an optimum solution.  Chapter 1 - Dynamic Analysis of Fabric Systems  To achieve the above goals, this work intends to link the valuable experimental data on the mechanical behaviour of fabrics under a variety of loading scenarios to the numerical analysis, both i n terms of providing input values and for validation purposes. The computational study established is aimed to be fully exercised, i n order to better understand the characteristics of the fabric targets and to contribute to the current body of knowledge on their ballistic behaviour and the complex mechanics of bullet-proof vests.  1.4  METHODOLOGY  The work presented i n this thesis is an effort to establish a new technique to evaluate the performance of fabrics under quasi-static and dynamic loadings. The methodology proposed is a mix of numerical and experimental effort, shown i n Figure 1 - 3 . The main component of the current approach is the highly efficient shell element that represents a fabric unit-cell, capturing the response of a single yarn crossover i n the case of plain weave fabric. In parallel to the 2D shell model, a more detailed finite element representation of the fabric unit cell is provided through a 3D solid-based model of the yarns i n a single fabric crossover. This 3D crossover model is used to provide some of the input variables to the 2D shell crossover model. Moreover, it serves to verify the 2D shell's predictions of the fabric biaxial response. In doing so, detailed experimental measurements are used to provide the information needed by these two models and for validation of their predictions. The verification of the 2D shell crossover response using the 3D crossover model predictions and its validation using the experimental data w o u l d provide confidence i n the  Chapter 1 - Dynamic Analysis of Fabric Systems  accuracy of the numerical model proposed and makes it a useful tool to create detailed multi-ply models of fabric subjected to different loading scenarios i n a variety of applications. The main tasks that are envisaged to achieve the objectives of this study are as follows: •  Introduce a new constitutive model for the yarn material represented by solid elements i n the 3D model of fabric crossover. This constitutive model w o u l d be capable  of capturing the transverse mechanical  response of the yarns woven into a fabric as they interact w i t h each other under the applied loading. •  Formulate the mechanical response of a single yarn crossover under biaxial and shear loading conditions and implement this mathematical representation into a single shell element that represents an individual fabric unit cell.  •  Use the developed shell elements to create real-size models of fabrics and study their behaviour under quasi-static and dynamic loadings. This task w o u l d not only serve as a validation for the 2D shell crossover model, but it w o u l d also provide more insight into the complex behaviour of fabrics.  •  Utilize the shell elements to perform parametric studies, i n order to identify the sensitivity of the fabric behaviour to a variety of geometrical and mechanical properties. This w o u l d help identify the influencing factors affecting the energy absorption and deformational characteristics  -10-  Chapter 1 - Dynamic Analysis of Fabric Systems  of fabric structures and facilitate the design of more efficient fabric systems.  1.5  OUTLINE OF T H E THESIS  This thesis consists of seven chapters that i n addition to the current first chapter, provides the experimental data (Chapter 2), the details of the model development, verification and validation (Chapters 3 to 5), the sensitivity analysis and case-studies (Chapter 6), and the conclusions and future work (Chapter 7). Chapter 2 provides the information regarding the experimental data used throughout the thesis. This chapter covers some general information on the material that is the focus of this study, along w i t h the methodology used to measure the transverse  compression of the fabrics (tests performed  by  Kongshavn [42]). A phenomenological model is developed to capture  the  fabric's transverse compression behaviour. The ballistic impact experiments (performed by Cepus [42]) are introduced and the basics of the data acquisition system, E L V S (Enhanced Laser Velocity System) are briefly discussed. Chapter 3 presents the development of the 3D solid crossover model. The components of the 3D crossover model are presented and the simplifying assumptions made to formulate the axial and transverse mechanical behaviour of the yarns i n a single fabric crossover are provided. The biaxial nature of the fabric crossover extensional response is discussed and the interaction of the yarns during such loading is studied.  -11-  Chapter 1 - Dynamic Analysis of Fabric Systems  Chapter 4 discusses the details of the 2D shell crossover model development, along w i t h its formulation and implementation into a shell element's material constitutive model. The predictions of the shell elements are then verified against those of the 3D solid model and the pin-jointed cable model on a unit cell and fabric level under biaxial and ballistic impact loadings. Chapter 5 presents the validation of the static and dynamic response predictions of the 2D shell crossover model through comparisons w i t h the relevant experimental data. The efficiencies of the model are highlighted and its limitations are identified by comparing the numerical simulations to fabric biaxial extension and ballistic impact data. Chapter 6 presents the sensitivity analysis performed on a variety of fabric's geometrical and mechanical properties, i n an effort to determine the influence of different fabric parameters on its static and dynamic responses. The 2D shell crossover is then used to study pencilling phenomenon, a problem that is the side-effect of the recent advancements i n the design of the personal body armours. Chapter 7 provides a summary of the findings throughout the work presented in this thesis, along w i t h the conclusions that can be drawn from those findings. Recommendations are made for the refinement of the modelling approach proposed and to find solutions to the unanswered questions.  -12-  Chapter 1 - Dynamic Analysis of Fabric Systems  Discrete Models Leveli  Level i+l  Level  Level _/+l  Roylance etal. (1973) Tan, Shim, and coworkers (1995-2005) Shahkarami (1999) Johnson etal. (1999) Billon and Robinson (2001) Cunniff, Ting and coworkers (1998-1999) F i g u r e 1-1:  Discrete modelling approach adopted b y m a n y researchers to study the d y n a m i c behaviour o f fabrics.  C o n t i n u u m Models 3 D Solid Models  Unit-Cell Shell Models  Tabiei & Ivanov (2002-2004) King et al. (2003-2005) Lim et al. (2003) Tanov & Brueggert (2001-2003) Luo (2000) Stubbs(1989) Shockey & coworkers (2000-2001) F i g u r e 1-2:  Shockey etal. (2000-2001) Duan et al. (2005-2006) Boisse & coworkers (2000-2005) Peng and Cao (2002)  C o n t i n u u m m o d e l l i n g a p p r o a c h , f u r t h e r s u b d i v i d e d i n t o 3D s o l i d a n d unit-cell shell models, used b y m a n y to capture the d y n a m i c of fabrics a n d fabric  composites.  -13-  models response  Chapter 1 - Dynamic Analysis of Fabric Systems Verification Stage  Fabric Micro-images (Appendix A )  F.E. Mesh Pre-processor (Appendix A )  3D Fabric Crossover Model (Chapter 3)  Compaction Tests (Chapter 2) 2D Shell Crossover Model (Chapter 4) Ballistic Experiments (Chapter 2)  Fabric Simulation (Chapter 5)  Simulation Stage  Figure 1 - 3: A summary of the numerical approach presented in this study.  -14-  Chapter 2 - Experimental Data  CHAPTER 2 -  EXPERIMENTAL DATA  This chapter highlights the experimental data that was extensively used i n the development  and  validation of the  fabric numerical model  presented  throughout this thesis. Only the results obtained from special experiments, such as transverse compression tests and ballistic tests, are covered here. Other data generated from standard tests, such as fibre density, elastic modulus, strain-to-failure, etc., have been taken from various sources i n the literature. The original data of the transverse compression test analysed i n Section 2.2 were obtained by Kongshavn [42]. The ballistic impact experiments on single and multi-ply panels discussed i n Section 2.3 were performed by Cepus [42]. These  experimental  data  were  generated  as  part  of  a  combined  experimental/numerical research project sponsored by the Natural Sciences and Engineering Research Council of Canada (NSERC) and DuPont Advanced Fiber Systems Inc. It should be noted once again that the experimental data presented i n this chapter are obtained by Kongshavn and Cepus [42] i n the case of transverse compression and ballistic tests, respectively. The author used these data to  -15-  Chapter 2 - Experimental Data  perform model development and experimental data fitting, as presented i n the following sections.  2.1  FABRIC MATERIAL  Kevlar® is perhaps one of the most common high-strength fibres w i d e l y used i n ballistic applications. Kevlar® para-aramid fibres were first manufactured by DuPont i n 1965. Kevlar® fibres are made of highly oriented chains of polyparaphenylene  terephthalamide  molecules w i t h strong inter-chain bonding  [43]. The specific arrangement of polymeric chains has given lightweight Kevlar® fibre such superior physical properties that make it the material of choice i n a variety of applications. In addition to applications i n protection and defence industry, Kevlar® is widely used i n tires, fibre optics and aircraft construction. The study presented here focuses mainly on fabrics w o v e n from Kevlar® 129 yarns for ballistic applications. The yarn densities of the fabrics tested i n ballistic experiments range from 840 to 3000 denier, and the fabrics are all woven into a plain weave structure. Basic properties of the panels as reported by the manufacturer or measured i n the laboratory are presented i n Table 2 - 1 .  2.2  TRANSVERSE COMPRESSION OF FABRICS  Transverse yarn deformation is one of the influential parameters that affect the overall response of fabrics during ballistic impact events. Compression of the crossing yarns as they interact under in-plane extensional displacements that result from the dynamic loading of single and multi-layer fabric targets  -16-  Chapter 2 - Experimental  Data  determines the degree of their compliance and hence their energy absorption capabilities. This interaction, which affects the overall response of fabric systems, is commonly neglected by many researchers. This section discusses the development of a suitable mathematical model to capture the transverse compression of the yarns i n the Kevlar® fabrics studied using the experimental data obtained by Kongshavn [42]. 2.2.1  Transverse Compression Experiments  The experimental set-up for the transverse compression tests is depicted i n Figure 2 - 1 . Squares of 55 m m by 55mm were cut from the fabrics and stacked on top of each other w i t h the warp yarns running i n the same direction. Samples of 1, 4, 8, and 16-ply packs were prepared and placed between two steel platens of size 19mm by 19mm i n an MTS™ Servohydraulic machine. Teflon™ layers were placed at the interface of the loading head and the top and bottom layers, to minimize the friction between them and allow in-plane movement of fabrics. The displacement of the loading head was measured using an Instron™ extensometer, and the load applied to the target was measured from the load-cell mounted on the loading head. Figure 2 - 2 shows the raw data for transverse compressive force-displacement response of 1 to 16-ply panels of S-720 fabric. A s seen i n the figure, the loading of the panels at a constant displacement-controlled rate of 1 mm/minute was stopped at 7 k N . The loading was maintained at 7 k N for 0.2 minutes, followed by unloading phase back to zero force at the same rate. The loading and unloading  branches  showed  different  characteristics,  which  is mainly  attributed to the compaction of the panels during the loading process. A s  -17-  Chapter 2 - Experimental Data  expected, the pack shows an increasingly more compliant response w i t h the addition of more plies. Since Teflon™ was added to the pack, the influence of the Teflon™ panels should be extracted from the experimental data. Compressive tests were performed on samples of only two layers of Teflon™ to obtain the transverse compliance of the Teflon™ along w i t h that of the test set up. Pure compressive response of the fabrics can thus be obtained by subtracting the Teflon™ response from that of the fabric pack with added layers of Teflon™, as shown in Figure 2 - 3 .  2.2.2 Transverse Compression Model The experimental data from the transverse compression tests were studied by only considering the loading part of the tests, since no significant transverse unloading takes place during a ballistic event. In order to obtain a suitable realistic model for the transverse behaviour of fabrics, similar work previously done by other researchers i n the literature needs to be reviewed. Numerous studies have been conducted on the transverse compression of the yarns and fabrics. These can mostly be found i n the textile science literature pertaining to various applications i n fabric industry (Van W y k [44], Hoffman and Beste [45]) or to manufacturing of textile composites (Gutowski et al. [46], Saunders et al. [47, 48], Pearce and Summerscales [49]). These models generally adopt either an empirical or micromechanical approach. In 1946, van W y k [44] presented a compression model to capture the pressure-volume response of wool, where special attention was given to its "handle" as a part of w o o l  -18-  Chapter 2 - Experimental Data  quality assessment. V a n W y k ' s model is a relationship based on a power law, where the pressure is related to the cube of the inverse of the volume i n the high pressure range as follows:  KEm  I  3  1_  (2-1)  where p is the pressure, E is the modulus of elasticity of the fibres, m and p are respectively the mass and density of wool, v and v are the current and 0  initial volumes, and K is a dimensionless constant. V a n W y k concluded that the above equation is similar to the general relation employed by others w i t h an exponent of y = 3 i n the general form given below: ( v V v oy  (p + Po) = Po  (2-2)  v  where p  0  was termed "latent pressure" of w o o l at zero pressure. For the low  pressure range, a different compressive behaviour dominated the response that was best represented by a simple inverse relation: K'Em f 1 3  P  3  2 V  1  (2-3)  0  with K' being a constant. Hoffman and Beste [45] also observed that the response of fabrics to pressure can be expressed by a power function, w i t h the exponent changing suddenly from 5/4 at low pressures to 3 at high pressures. The compressibility of the reinforcing fabrics i n the processing of composite materials has been the focus of many studies, since the compression of the fibre-bed/fabric reinforcement w o u l d greatly affect the permeability and other  -19-  Chapter 2 - Experimental Data  properties of the composite structure during both its processing and load bearing life. One of the pioneering works i n this field was performed by Gutowski [46] w h o provided a micro-mechanical model to capture the compression of fibre beds based on bending of single fibres. For instance, Gutowski derived the following relation between the applied bulk stress, cr  br  and the fibre volume fraction, V , i n the absence of any axial stress i n the fibre f  bundle:  (2-4)  where E is the elastic modulus of the fibres, V and V are respectively the 0  a  initial and the m a x i m u m available fibre volume fractions, and B- is a constant determined empirically. Pearce and Summerscales [49] studied the compression of glass-fibre woven fabrics used i n R T M (Resin Transfer Molding) processes. In their study, they used a power-law expression to fit the measured pressure to the fibre volume fraction of the fabric. The exponent of the power function used i n data fitting was found to be i n the range of 7 to 11. From these experiments, they reported that the transverse thickness change due to the applied pressure w o u l d not result i n a lateral spreading. They also observed noticeable relaxation i n the fabrics under pressure, and achieved higher fibre volume fraction after each loading cycle. Saunders et al. [47] performed compression tests on dry and wet glass woven fabrics w i t h different weave structure (plain, twill, five-harness  -20-  Chapter 2 - Experimental Data  satin), and concluded that the response is best represented by a power function. The exponent of the function obtained from the fitting to the experimental data was found to be between 9 and 10.5 i n this study. Saunders et al. commented on the effect of additional layers on the response of the fabrics, and concluded that the higher compression i n multi-layer packs can be related to nesting of the layers, specifically i n plain woven fabrics. One major common finding of all studies cited above is that the relationship between the pressure applied to the fabric and fibre volume fraction can be best expressed b y a power function. For the Kevlar® 129 fabrics studied i n this work, the power function adopted has the following mathematical form:  where P is the applied pressure, V is the fabric current volume, V  min  is a  m i n i m u m compression volume, K" is a fitting constant, and n is an exponent. The m i n i m u m compressive volume, V , is chosen such that a best fit w i t h an min  exponent of 3 is obtained. The m i n i m u m volume and the exponent are merely fitting parameters; however, they reflect the properties of the fabric yarns and the specific weave characteristics. This is evident i n Figure 2 - 4 , where a linear relationship seems to exist between the m i n i m u m volume, v , and the areal min  density of the six different Kevlar® 129 fabrics investigated. The parameter K" obtained from the fitting of the above function to the experimental data also shows dependence to the cube of the fabric areal density,  A. d  This dependence,  as shown i n Figure 2 - 5, is critical i n  -21-  Chapter 2 - Experimental  Data  extrapolating the transverse compression model to any other plain weave Kevlar® 129 fabric i n the absence of transverse compression test data. The best-fit line shown i n Figure 2 - 5 is used to determine the transverse compression response of the fabrics. The best-fit line for the one-ply panels is presented below: K" = c A x  d  +c = 12259• A 2  3 d  -4X10  1 0  (2-6)  with c, and c being the fitting constants. H a v i n g established the relationship 2  between the pressure and the volume of the fabric panels, their transverse force-displacement relation can be obtained k n o w i n g the initial or final thicknesses of the fabric packs, as:  (2-7)  where the parameter a is  a  =  'o-% A  (2-8)  with t being the initial fabric thickness and A the loading area. The parameter 0  b can also be calculated as follows:  (2-9)  Final values of a and b are provided i n Table 2 - 2 for 1, 4, 8, and 16 plies of fabric. The transverse compression model is used to calibrate the transverse properties of the yarns i n the 3D solid crossover model (Section 3.4.3.2), and  -22-  Chapter 2 - Experimental Data  eventually to obtain true compression response of yarns subjected to biaxial i n plane loading during ballistic events.  2.3  BALLISTIC IMPACT EXPERIMENTS  Ballistic impact experiments were performed on all six Kevlar® 129 fabric types with properties presented i n Table 2 - 1 . The experiments, conducted by Cepus [42], were instrumented w i t h two Enhanced Laser Velocity System (ELVS), placed on the strike and exit faces of the target [50]. The projectiles used i n these experiments were flat-nosed cylindrical steel projectiles (Right Circular Cylinders, RCC), w i t h a base diameter of 5.5 m m and length of 42 m m . The projectiles were partially cored to reduce their weight to 3 grams. The panels were held i n a fixture depicted i n Figure 2 - 6 w i t h an opening of 203mm by 203 mm (8"x8"), w h i c h is capable of providing various levels of end fixity for each side of the panel. The E L V S  systems  allow for continuous measurement  of the  projectile  displacement during an impact event [50]. Figure 2 - 7 shows a schematic of the dual E L V S system w i t h laser sheet placed on both sides of the target. The diode laser produces a sheet of light that is collimated i n the horizontal and vertical directions passing through two piano-cylindrical lenses. This provides a sheet of light w i t h a uniform intensity and constant w i d t h and thickness. The light intensity of the sheet is then measured by a photo detector and recorded as voltage by an oscilloscope. A s the projectile enters the sheet before impacting the target, the recorded voltage drops d o w n since the intensity of the light entering the photo detector reduces. The location of the projectile before and  -23-  Chapter 2 - Experimental Data  during the impact is then traced from these voltage recordings and its displacement time-history is obtained. Figure 2 - 8 to Figure 2 - 1 0 show typical displacement, velocity, and energy-time curves for R C C projectiles impacting 1-ply S-726 targets at 96 m/s and 110 m/s resulting i n a non-perforating and a perforating  impact respectively. The displacement-time  response  of  the  projectile is directly calculated from the voltage-time data, w h i l e the velocitytime  information is obtained  through  numerical differentiation  of  the  displacement data. The absorbed energy by the target is determined from the reduced velocity of the projectile. Further differentiation of the velocity-time data provides the acceleration-time response of the projectile, and i n parallel, the resultant decelerating force imposed on the projectile by the target. Traditional plots such as residual vs. strike velocity {V$-Vr) and absorbed vs. strike energy (£ -E«) can also be plotted as a testament of the inclusiveness of S  the E L V S data acquisition system, as presented i n Figure 2 - 1 1 and Figure 2 12 for S-720 panels. The E L V S data obtained from the instrumented ballistic impact experiments are used later i n Chapter 6 to validate the predictions of the shell-based fabric model developed i n this study.  2.4  SUMMARY  The details of the laboratory test data used throughout this thesis were introduced i n this chapter. The experiments reported  are the  transverse  compression of fabrics and instrumented ballistic impact tests, performed by Kongshavn and Cepus [42], respectively. The fabric compression tests were conducted to characterize the transverse behaviour of yarns w i t h i n the fabric. These tests were done on fabric packs of 1 to 16 plies using a standard M T S  -24-  Chapter 2 - Experimental Data  machine, and the load carrying capacity of the fabric versus its transverse deformation  was  recorded.  A mathematical  function was fitted to  the  experimental data w h i c h w i l l be used later i n this thesis to obtain certain input parameters to the numerical models of fabric unit-cell. Representative ballistic impact experiments on single fabric targets and the basics of E L V S data acquisition system were introduced. The data from the ballistic impact tests w i l l be used to measure the success of the numerical model developed i n this study capturing the true dynamic behaviour of the fabrics.  -25-  Chapter 2 - Experimental Data Table 2-1: Properties of the Kevlar 129 plain weave fabric panels investigated in this study. S-720  S-726  S-727  S-728  S-731  S-745  Yarn (denier )  1420  840  1000  1500  1000  3000  Warp count (tpi*)  20  26  26  17  31  17  Weft count (tpi*)  20  27  26  17  31  17  266.5  203.7  238.9  232.4  293.0  474.0  Warp crimp (%)  2.18  3.44  3.38  1.76  8.16  5.27  Weft crimp (%)  1.39  0.50  0.74  0.94  0.56  1.38  Panel 1  Areal density (g/m ) 2  denier: mass of 9000 meters of the yarn in grams. *tpi: threads per inch. Table 2-2: The fitting parameters a and b for the transverse compression of the fabric model (Equation 2-7) for single and multi-ply panels of Kevlar 129.  (  Layers  cl (mm)  b  (mm.NW)  1  1.21E+04  -3.92E+04  4  4.43E+05  -1.03E+06  8  1.96E+06  5.5E+06  16  2.21E+07  0  -26-  Chapter 2 - Experimental Data  Figure 2-1: Experimental set-up for the transverse compression tests conducted by Kongshavn [42].  10  i  1-ply  4-ply  8-ply  12-ply  16-ply  Gauge Displacement (mm) Figure 2-2: Transverse compressive force-displacement response of S-720 panels conducted by Kongshavn for fabric packs of 1 to 16 plies [42].  -27-  Chapter 2 - Experimental Data  8000 T  1.35  Transverse Displacement (mm)  Figure 2-3: Transverse compressive response of a single-ply of S-720 fabric before and after extracting the compliance of the Teflon™ layers (experimental data obtained from Kongshavn [42]).  0.4 Vol Panel S-720 S-726 S-727 S-728 S-731 S-745  0.3  E E, c _E  0.2  m i n  (9'm )  (mm)  266 203 239 232 293 474  0.176 0.140 0.170 0.139 0.191 0.355  2  2  y  y  my® y  O >  yi  y  y  y  y  y  0.1  0.0  A  0  i 100  1  1  200  300  400  :  1  500  Areal Density (g/m ) 2  Figure 2-4: Linear function representing the relationship between the minimum compression volume of a unit area of the fabrics (obtained from compression model) and their areal density.  -28-  Chapter 2 - Experimental Data  1.6E+12  Figure 2-5: Fitting parameter K" as a function of the third power of areal density (A )for the six different Kevlar® 129 fabrics. d  7.9 mm (=/„") square rods (x4) 12.7 mm (V, ) N/C coarse thread, counter bored alien screws (x 16) -  Front Plate  inside surface  Back Plate  Figure 2-6: Schematic of the test fixture used in the ballistic experiments [51].  -29-  Chapter 2 - Experimental Data  Figure 2-7: Dual ELVS system setup with tracking the displacement of the projectile striking the front-side of the target (front ELVS) and the tip of the deformation cone on the back-side of the target (back ELVS) [51].  Figure 2-8: Displacement-time response of 1-layer S-726 panels impacted at 96 m/s and 110 m/s obtained from an ELVS-instrumented ballistic experiment (performed by Cepus [42]).  -30-  Chapter 2 - Experimental Data  Figure 2-9: Velocity-time response of 1-layer S-726 panels impacted at 96 m/s and 110 m/s obtained from an ELVS-instrumented ballistic experiment (performed by Cepus [42]).  Figure 2 - 10: Energy absorbed-time response of 1-layer S-726 panels impacted at 96 m/s and 110 m/s obtained from an ELVS-instrumented ballistic experiment (performed by Cepus [42]).  -31-  Chapter 2 - Experimental Data  300 -i  100  200  300  Strike Velocity, V (m/s) 8  Figure 2-11: Projectile strike velocity versus its residual velocity in the ballistic impact tests for the 1-ply S-720 panels (Experiments performed by Cepus [42], data summarized by the author).  20  T  Figure 2 -12: Projectile strike energy versus the absorbed energy by the target in the ballistic impact tests for the 1-ply S-720 panels (Experiments performed by Cepus [42], data summarized by the author).  -32-  Chapter 3- 3D Fabric Crossover Model  CHAPTER 3 -  3.1  3D F A B R I C C R O S S O V E R M O D E L  INTRODUCTION  One of the most detailed modelling techniques  to capture the  dynamic  behaviour of fabrics is to model the individual yarns as continuum bodies, woven into the fabrics i n a specific architecture. This approach takes advantage of the realistic yarn geometry definition by explicitly including the individual yarns i n a w o v e n fabric i n the finite element model. A s a result,  the  complicated interactions of the yarns are inherently captured i n such analysis. This chapter reviews the basics of a 3D model specifically developed to capture the mechanical response of a plain-weave fabric crossover. The specific constitutive  relationship  implemented  i n the  beam  and  solid  element  components of the model is discussed. The calibration process including input data determination for the 3D model is discussed and the advantages and limitations of the approach are highlighted.  -33-  Chapter 3- 3D Fabric Crossover Model  3.2  3D FABRIC UNIT-CELL M O D E L S  Finite element analysis methods have proven to be invaluable tools to simulate a wide range of physical phenomena by means of mathematical formulations. In order to achieve that, a series of simplifying assumptions are to be made and a number of initial parameters need to be defined. It is rather intuitive that the outcome of any simulation is only as good as suitability of the initial assumptions and the accuracy of the input parameters. Since the finite element models are only mathematical approximations of the physical reality, there are always some degree of deviation from the true response of a system (other than the inherent error of the F.E. formulations) that stems from user's knowledge of the physical event and the wealth of details considered. Several researchers have attempted to incorporate the many details of the complicated structure of a fabric through continuum representation of the yarns i n the weave. In these unit-cell based 3D models, solid (brick) elements replace the fibres i n a yarn i n a smeared approach to capture the behaviour of dry fabrics or fabric composites. In simulating fabric composite manufacturing, many of the models developed focus on determination of fabric deformation during the processing of textile composites. These fabrics might be i n dry state (in the case of Resin Transfer Molding), or i n a fluid/semi-fluid resin base (in case of prepreg draping). Regardless of the application, the fabric may be subjected to extreme shearing and some stretching, leading to yarn rotation and fabric compaction. In the case of ballistic impact, dry fabrics are used to arrest the flying objects through membrane stretching. In this application, the fabric undergoes  a large amount of stretch and a moderate  -34-  amount of shear  Chapter  3- 3D Fabric  Crossover  Model  depending on the boundary conditions. Considering the resemblances of the above mentioned applications, similar 3D solid models of fabrics have been developed i n both fields. Shockey et al. [18, 19] presented a combined experimental/numerical study to evaluate  the  performance  of high strength  polymeric fabric panels  in  containing the turbine engine fragments on commercial aircrafts. A l o n g with impact and quasi-static tests, a detailed 3D solid model of fabrics was developed to study the response of fabric panels. The model was implemented in L S - D Y N A and the material model employed for the solid elements was chosen to be elastic orthotropic, with reduced shear and transverse normal moduli. The profile and shape of the yarns i n the fabric were obtained from cross-sectional imaging studies of the fabrics, while their properties were calibrated from the mechanical properties tests performed on single yarns. The failure model implemented was based on a cumulative damage model to capture the gradual rupture of the fibres i n the yarn as it is stretched to failure. The model developed by Shockey et al. [18, 19] was successful i n providing insight into the physics of fabric energy absorption; however, it lacked a robust constitutive model for the yarns, specifically due to their simplified transverse compression response. Shockey et al. further reported that the computational requirements for their detailed 3D model are considerable due to the large number of nodes and elements, significantly limiting the applicability of the model to larger fabric targets.  Duan et al. [20-23] developed a model very similar to that of Shockey et al. [18, 19] for a square patch of fabric of size 32.7 m m x 32.7 m m . The aim of their  -35-  Chapter  3- 3D Fabric  Crossover  Model  study was to consider the yarn-to-yarn and fabric-projectile friction  to  investigate the energy absorption and failure of the fabric. The finite element fabric mesh used for their study was made up of 46,800 eight-noded solid elements. The findings of their studies indicated that the addition of friction would enhance the energy absorption of the fabric and reduce the projectile residual velocity. The role of boundary conditions on the performance of the target was also investigated. The study by Duan et al. further highlights the inefficiency of the 3D solid modelling, since a small panel size and very high impact velocity (800 m/s) was needed to decrease the execution time to within a practical range. In  fabric composite manufacturing, it is highly advantageous  to have a  numerical model to predict the deformed fabric configuration; to know the final angle between the yarns which w o u l d affect the permeability of the reinforcement and determine the final mechanical properties of the composite. Boisse and co-workers. [24-27] presented a meso/macro-mechanical approach to capture the deformation of plain and twill weave panels. In their approach, they take advantage of the available biaxial and shear response of the fabrics to calibrate a detailed 3D solid model of fabric unit-cell. A parallel mesomechanical model powered by an analytical model of plain weave yarn crossover is developed to simulate the forming process of fabric composites. In effect, Boisse et al. use the 3D solid model validated b y the experimental data to either feed the biaxial response to an equivalent representative element, or to verify the response of their biaxial analytical model.  -36-  Chapter 3- 3D Fabric Crossover Model  Peng and Cao [28] adopted a unit-cell approach to predict the nonlinear elastic properties of fabric composites. In this study, they used the homogenization method to estimate the moduli of the yarns i n the composite based on the properties of its constituents. Numerical tests were then performed on the 3D unit-cell to obtain the extensional and trellis behaviour of the fabric. These responses were then correlated with an equivalent shell element to evaluate its effective properties. A l l the models discussed above suffer from the extreme inefficiency problem acknowledged by many researchers. The 3 D unit-cell proves to be very powerful due to the wealth of geometry and material details it considers, while for modeling practical engineering systems, the resulting inefficiency proves to be its Achilles heal. The following sections highlight the basics and results of the 3D solid crossover model developed i n this study, as a part of the overall modelling approach discussed earlier (Figure 1-3).  3.3  EXPLICIT FINITE E L E M E N T ANALYSIS  Any  structure, i n theory, can be represented by a series of mathematical  equations along w i t h specific boundary conditions, solution of which can characterize the state of the body under the specific physical conditions. Solving these mathematical equations, however, is not an easy task. Finite element analysis method is an efficient approach for analysing the state of a physical body under an applied set of loading and boundary conditions. The analysis procedure is to discretize a continuum body into a finite number of components (elements) connected to each other at certain points along their  -37-  Chapter 3- 3D Fabric Crossover Model  edges, usually referred to as nodes. Temporal and spatial field parameters are considered and their relationship with the mechanical properties of the body is formulated into mathematical equations. Temporal and spatial discretization of the problem w o u l d lead to a solution of the mathematical equations resulting in the values of the field parameters at the common nodes of the elements. Within  the  domain of an element,  the  values  of all parameters  are  approximated through the adoption of appropriate shape functions. In a structure, since the true state of stress and strain i n the material w o u l d correspond to the m i n i m u m energy of the system, any other approximation of this state w o u l d lead to higher energy than the real case. A s a result, the finite element solution w o u l d always exhibit a stiffer response compared to that of the real physical system. To analyse the dynamic response of any structure, temporal discretization should be performed on the equations of motion to deal w i t h the time dependency of the mathematical equations. Details on these well-established methods, briefly discussed i n this section, can be found i n any finite element references [52, 53]. The choice of temporal discretization scheme for the finite element analysis is usually based on the nature of the dynamic event. In general, two main methods are usually adopted, commonly referred to as Implicit  and  Explicit  time-integration  schemes.  Implicit  schemes  take  advantage of the trapezoidal rule to approximate the time derivatives i n the equation of motion. The result is a discretized scheme that is unconditionally stable, and the time-step-size is only defined based on accuracy considerations. Explicit schemes, on the other hand, utilize the central difference method for temporal discretization. The analysis w i t h this method is conditionally stable,  -38-  Chapter  3- 3D Fabric  Crossover  Model  meaning that the accuracy and convergence of the solution is time-step-size dependent. The criterion to achieve stability i n the explicit time-integration scheme is based on the speed of sound wave i n the material, C , as follows: L  At<—  C  where  (3-1)  is the time step size, L is the smallest element length. The time-step-  size of a finite element problem is chosen to be equal to the smallest value calculated amongst all the elements i n the model. The explicit scheme is suitable for wave propagation problems where the response is dominated by high-frequency modes. In these types of events, the time scale is usually very small and i n the same order as the wave propagation across an element. For these problems, where the behaviour of the wave front is important, the explicit time-integration  method is more suitable since the stability requirement leads  to a time-step-size during which the wave in the system can travel less than the smallest dimension of any element i n the model. In structural dynamics problems, the response of the system is usually dominated by lower frequency components. For these problems, implicit methods are more suitable since the analysis time is m u c h longer than i n the wave propagation problems and the time-step size obtained considering accuracy criteria is usually much larger than the largest time-step size from the explicit methods. This study is based on developing constitutive relationships for elements that are aimed to represent the behaviour of yarns and fabrics. For this purpose, LSD Y N A explicit finite element code is used as the main analysis tool, since the impact problems investigated here are more of a wave propagation nature. The main reason i n choosing L S - D Y N A as the analysis platform is its versatility and  -39-  Chapter  3- 3D Fabric  Crossover  Model  excellent capability i n simulating impact events, combined w i t h its wellestablished  contact  algorithms  that w o u l d  allow  for  a  more  realistic  representation of the colliding bodies. Figure 3 - 1 shows a schematic of the analysis process i n L S - D Y N A , as an explicit wave propagation code. L S - D Y N A allows the users to define the material constitutive relationship for beam, shell and solid elements within a user-defined material model ( U M A T ) . This option is fully exploited i n this chapter to modify the behaviour of beam and solid elements to match specific characteristics of the fibres of a yarn. The following sections provide the basic components of a yarn model and highlight the details of U M A T development for beams and solid elements.  3.4  M O D E L BASICS  A 3D micro-mechanical model of a fabric unit-cell is developed to capture the biaxial response of plain weave fabrics. A s seen i n Figure 3 - 2 , the model is based on representation of a single fabric crossover and is comprised of solid (brick) and bar elements. This approach, developed for the plain weave fabrics, is not weave specific and can be readily applied to any other fabric structure. The finite element mesh of the unit-cell is created b y a specialized preprocessor presented i n A p p e n d i x A . The basic assumptions considered i n the creation of the yarn cross-sections and longitudinal profiles are obtained from the study of the micro-images of the fabric cross-sections. A s a result, it is assumed that the yarn profiles follow a sine function, while their cross-section can be elliptical or sinusoidal. It appears from the fabric micro-images that the cross-sectional areas of the yarns are considerably larger than the solid area of  -40-  Chapter  3-3D  Fabric  Crossover  Model  the fibres, i m p l y i n g that there is a significant amount of voids between the fibres. Thus the cross-sectional area of the yarns i n the finite element mesh is inflated to account for the voids between fibres. The pre-processor is also capable of generating crossover meshes of unbalanced fabrics w i t h varying dimensions and crimp values i n warp and weft yarns. However, one of the basic limitations of the pre-processor is its inability to consider varying yarn cross-sections along its profile. A s a result, yarn overlays cannot be created in the finite element mesh of the crossovers. The key to the success of the fabric crossover model is capturing the true response of the individual yarns that form a crossover. The finite element model of each yarn is made from a combination of bar and solid elements. The bar elements represent the axial stiffness of the fibres i n the yarn and make up the tensile stiffness of the yarns. The solid elements, on the other hand, provide the transverse compressive resistance of the yarns, as they interact w i t h each other i n a fabric crossover. The following sections highlight the basics of the bar and solid elements developed for the finite element representation of the yarns. 3.4.1 Due  Discrete Bar Elements to the complex nature of the yarns as an assembly of fibres, many of their  mechanical properties are not uniquely available. In finite element analysis, the absence of a meaningful mechanical property w o u l d cause instabilities i n the calculations and lead to spurious modes of deformation without significant energy  absorption.  This  phenomenon  occurs  commonly  for  strongly  orthotropic materials and specifically i n 3D solid elements of yarns as reported by  some researchers (e.g. Gasser et al. [26]). This phenomenon is shown i n  -41-  Chapter  3- 3D Fabric  Crossover  Model  Figure 3 - 3 (right) for a single crimped yarn made of orthotropic solid elements with low shear modulus under tensile loading. The unstable load-displacement response of such a yarn is also shown i n Figure 3 - 4 . Boisse et al. [25] state that "because some rigidities are nearly equal to zero, some numerical instabilities often appear. This is avoided by hourglass control." Lack of sufficient shear stiffness i n the yarns w o u l d lead to unstable element shapes that closely resemble hourglass deformation modes that occur i n under-integrated reduced-integration)  (or  elements. A s a result, one possible solution to this  problem could be the addition of hourglass-control stiffness component to an element's stiffness w h i c h w o u l d resist the initiation of such deformations. Due to the non-physical nature of the added hourglass stiffness, it is preferred that the energy consumed by these modes of deformation be limited to a small amount. This fact is usually overlooked by many users, leading to unreliable structural responses. Reinforcing the solid elements by addition of rebars is an efficient method of controlling  unrealistic hourglass-like deformation  modes.  In  doing so,  reinforcing bar elements are added to the yarns i n the axial direction to play the role of the reinforcement. Figure 3 - 3 (left) shows the stable deformed shape of the same crimped yarn under axial displacement after the addition of the reinforcing bars. The axial tensile force i n the yarn also shows a monotonic rise with increasing displacements, as depicted i n Figure 3 - 4 . The reinforcing bar elements are formulated to provide the axial tensile stiffness of the yarns. A s these bars should closely represent the axial response of a bundle of fibres, they w o u l d have the same tensile modulus and area of the  -42-  Chapter  3- 3D Fabric  Crossover  Model  fibre bundle they represent (see Table 3 - 1 ) , while exhibit no resistance (due to fibre buckling) i n compression. A user material routine ( U M A T ) is developed to modify the response of beam elements i n L S - D Y N A (see flowchart i n Appendix B). The goal of this transformation is to take advantage of the many capabilities of beam elements such as element erosion, while implementing the non-conventional tension, compression and shear behaviour of the fibre bundles i n the reinforcing bars. In general, erosion of elements from the mesh in L S - D Y N A is performed based on a user-defined criterion that prompts the code to physically remove an element from the finite element mesh. In this process, the code eliminates the element from the calculation while retains its mass properties on the nodes attached to it. For the case of beams, the erosion is activated based on an instantaneous failure strain criterion, where the beam is eliminated from the mesh when its axial strain exceeds a predefined failure strain. The U M A T developed takes advantage of a visco-elastic material model along with an instantaneous strain-to-failure criterion. The U M A T is considered i n conjunction w i t h a beam element based on the Hughes-Liu beam formulation available i n L S - D Y N A . The specific type of beam formulation is based on a degenerated 8-noded iso-parametric solid element formulation, although the choice of beam formulation is not expected to affect the results i n the current analysis. In order to convert the behaviour of the beam elements to that of a bar, the bending and shear resistance of the beam should be eliminated. This is achieved through considering a single integration point i n the thickness of the beam, since lack of stress gradient through the thickness of the beam eliminates the bending moments i n the beam. The shear stresses and axial stiffness of the  -43-  Chapter  3- 3D Fabric  Crossover  Model  beam i n compression are also set to very small values manually. The final result is a beam element that along w i t h the U M A T developed behaves similar to a cable that is ideal for representing the axial response of the fibres i n the yarn. 3.4.2  C o n t i n u u m S o l i d Elements  The key part i n developing a successful 3D model of the fabric is the yarn's constitutive model. The structure of a non-twisted yarn is based on an assembly of long parallel fibres bundled (entangled) together. A s a result, the individual fibres are more or less free to move and interact w i t h each other within a yarn, only exhibiting a high axial stiffness i n tension. In axial compression, o w i n g to the absence of any lateral confinement, the fibres of a single yarn easily buckle and show little resistance. This lack of bonding between the fibres also translates into near zero shearing or bending stiffness i n the yarn. In the transverse direction, the presence of voids between the fibres along w i t h their extensive relative motion leads to an extremely complex nonlinear response. The challenge i n setting up a realistic model of a fabric crossover is to provide a material model that can capture this complex behaviour of yarns, i n all possible loading modes and directions. The solid elements used i n this study are the reduced integration constant stress solid elements. In this type of elements, the volume integration is carried out using Gaussian approximation considering only a single integration point. This leads to an element that provides a high efficiency compared to other elements, e.g. fully integrated (2x2 integration point) element. However, the one-point  quadrature  algorithm is  accompanied  -44-  by  some  zero-energy  Chapter  3- 3D Fabric  Crossover  Model  deformation (hourglass) modes i n the elements that lead to unstable results. For more information on the constant stress solid elements and other available choices the reader is referred to the L S - D Y N A Theoretical M a n u a l [54]. While the yarns  exhibit strong overall orthotropic  characteristics,  their  behaviour can be considered isotropic i n the transverse direction. In this study, it is assumed that the axial and transverse yarn responses are completely decoupled, and the Poisson's ratios are assumed to be zero i n all directions. A s mentioned previously, the axial stiffness of the yarns is provided by the bar elements, thus the modulus of solid elements along the axis of the yarn is set to a small value. However, due to the presence of voids i n the yarn and occurrence of large strains i n the transverse direction, the rule of mixtures is used to adjust the modulus of the solid elements i n the yarn axial direction according to the yarn cross-sectional area. A t any instant of time t (time-step n), the axial modulus (E") is obtained from current fibre volume fraction (V ) f  as follows: E" =  (3-2)  where (E ) is the initial modulus of the solid elements along the yarn axis. The a  fibre volume fraction is monitored as the yarns cross-section deforms due to contact and the axial modulus is updated at each time-step. U s i n g the basics of continuum mechanics, the compression of the yarns are tracked using the deformation gradient, F, defined as (using indicial notation):  (3-3)  -45-  Chapter 3- 3D Fabric Crossover Model  where X  and x correspond to the material coordinates before and after  deformation, respectively. The volume of an element can be calculated as follows: V" = JV  (3-4)  where V" and V° are the calculated current and initial volumes respectively, and J is the Jacobian defined as the determinant of F J = \E\  (3 - 5)  Considering the current and old axial dimensions of the yarn L" and L°, one can estimate the current yarn cross-section, A" from its original value, A° as rQ\  f  A"  J-  (3-6)  L"  The updated fibre volume fraction can then be calculated as:  v; =  (3-7)  JL%  K  Adopting an incremental strain algorithm, the total logarithmic strain is calculated by the accumulation of the strain increments as follows:  (3-8)  J where e" is the logarithmic strain at time-step n along the yarn axis. Using the two equations above, we can calculate the updated nominal fibre volume fraction as:  v; =  expi  J  -46-  (3-9)  Chapter 3- 3D Fabric Crossover Model  In the transverse direction, the material representing the fibres i n a yarn is assumed to be isotropic. There is not much information available on the transverse modulus of Kevlar® 129 fibres. The transverse elastic modulus is assumed to be a function of the nominal fibre volume fraction of the yarns i n a non-linear fashion. The behaviour of the yarn subjected to transverse loading is schematically shown i n Figure 3 - 5 . Based on the loading conditions (tension versus compression) and its amplitude, an effective transverse secant modulus is determined. The conditional function defining the instantaneous transverse modulus is presented below. 1. If V" <V°, the yarn is subjected to transverse tension. In reality, this type of loading w o u l d separate the fibres of a non-twisted yarn (such as the Kevlar® 129 yarns i n the six fabrics studied here) and not much resistance is displayed by the yarn. A s a result, the yarn is assumed to have a very soft response defined as: E" = El = K E 2  r  (3 -10)  c  where E is an upper limit for the modulus and K is a reduction factor. c  r  2. If V° < V" < V , where V is referred to as the m i n i m u m nominal fibre c  c  volume fraction and is merely a fitting parameter, then the yarn easily deforms under the applied compressive loading. The deformation of the yarn is combined of fibre displacement and realignment, along w i t h some transverse fibre compression. This zone is similar to the initial soft response of foams. The transverse modulus i n this zone is assumed to follow a nonlinear function of the form:  -47-  Chapter 3- 3D Fabric Crossover Model  (3-H) f  where a and b are calculated below: E  E  aE = E (b -V ) c  E  (3 - 12)  c  (3 - 13)  \-K  r  It should be noted that V can be higher than the m a x i m u m packing ratios c  for square or hexagonal packing, since it is assumed that a certain level of fibre compression does also occur at V and beyond. c  3. If V" >V , then the yarn is subjected to high compression. The resistance C  in this case is high, as the fibres of the yarn have reached m a x i m u m packing ratio and have been  compressed  transversely.  The transverse fabric  modulus is at its maximum value, E  c  (3 - 14)  Figure 3 - 6 shows the transverse modulus of the yarns as a function of the nominal fibre volume fraction for the input values presented on the figure inset. It should be noted that the nominal fibre volume fraction is merely a fitting parameter and can exceed the value of 1.0 i n the presence of fibre compression, since it is calculated using the compressed volume of the yarn compared to the uncompressed net area of the fibres i n the yarn. A d o p t i n g the above function for the transverse modulus, the fitting parameters V , E , and c  c  K are determined by simulating the 1-layer transverse compression tests and r  the predicted force-displacement response to that of the experiments. The  -48-  Chapter  3- 3D Fabric  Crossover  Model  transverse elastic modulus estimated is then used to calculate the state of stress as follows: a" = E" e" 2  where E  2  where i = 2,3  (3-15)  is the secant modulus defining the transverse stress i n the yarn as a  function of applied strains. As mentioned earlier, the shear stiffness of a yarn is generally extremely low due to the lack of any bonding between the fibres. The shear resistance of the yarn should have some dependency to the axial and transverse state of stress as they are affected by the fibre-to-fibre interaction and friction. This is ignored i n this study due to absence of any experimental or analytical evidence, and the shear m o d u l i i n all directions are set to a low value. These moduli are determined from the compacted transverse elastic modulus E as follows: c  where K  G  is a reduction factor, and i = 1,2,3. In this study, this reduction factor  is assumed to be K  G  = 0 . 1 % to simulate the lack cohesion between the fibres of  a yarn i n shear. The shear modulus of this magnitude w o u l d not be able to provide any significant resistance to the yarn deformation, similar to the behaviour of a real yarn.  3.4.3  Determination of Material Properties  Perhaps the biggest challenge i n setting up the 3D solid model of fabrics is determination of the materials properties. The behaviour of the yarns i n a fabric is so complicated that the modellers are forced to estimate some values or  -49-  Chapter  3 - 3D Fabric  Crossover  Model  resort to model calibration using experimental data. The following sections discuss the procedure to obtain the input parameters for solid and beam elements.  3.4.3.1 Single yarn tensile test simulation Tensile tests are performed on single yarns to validate the axial properties of the yarns. In solids, the modulus i n the direction of the yarn axis is set to a very low  value since the bar elements are taking all the tension. The axial tensile  modulus of the bars is set equal to the modulus of the fibres and their crosssectional area estimated from the solid area of the fibres. This ensures that the virtual yarns exhibit the same stiffness as the actual ones i n the axial direction. Figure 3 - 7 shows the single yarn response of the crimped warp and weft yarns of S-720 panel. It can be seen that the yarns show very little resistance while they are crimped and stiffen u p when taut. The tensile responses also reflect the difference i n the crimp values of the warp and weft yarns, as the S-720 panel has non-balanced crimp values. The corresponding responses predicted by an analytical model are superimposed  on the graph for comparison. This  analytical model assumes zero stiffness for strains less than the crimp value and linear Kevlar® yarn stiffness for strains higher than the crimp strain (see bar properties provided i n Table 3 - 1 for the S-720 fabric). G o o d agreement is observed between the analytical model and the predictions of the 3D finite element model of the crimped yarn. Erosion of the bar elements are also incorporated i n this model which results i n the failure of the yarns at a predefined failure strain value (taken to be 3% i n this study, as seen i n Table 3 1).  -50-  Chapter 3- 3D Fabric Crossover Model  3.4.3.2 Transverse compression test simulation Determination of the transverse properties of yarns i n a w o v e n fabric is one of the most challenging aspects of yarn modelling. Overlooked b y many of the existing numerical models, the transverse  response  of yarns affects the  interaction of the crossing yarns and defines the shape of the nonlinear biaxial response of a fabric. This fact can perhaps be further explained by comparing the yarn interaction for two extreme cases: transversely incompressible versus over-compressible  yarns.  In fabrics woven from  incompressible  yarns,  stretching of the yarns i n one direction leads only to pure crimp exchange between them while the transverse distance between the yarn centrelines remains constant. For over-compressible yarns, interaction of the crossing yarns due to in-plane extension disappears i n their transverse deformation. A s a result, the fabrics woven from over-compressible yarns exhibit decoupled warp and weft yarn responses, while the ones w i t h incompressible yarns show the highest level of coupling between the two directions. Regardless of the level of yarn transverse compliancy, capturing the true transverse yarn response is a crucial step i n developing a realistic fabric model. This fact is also reiterated by other researchers, e.g. Boisse et al. [25] w h o quotes "The transverse law appears to be an important point for a good modelling of the unit w o v e n shell."  Some researchers have tried to design experiments that w o u l d realistically capture the transverse compressive behaviour of yarns. Perhaps one of the earliest investigations into this matter is that of Kawabata [55]. The proposed method, referred to as the wire method, consists of hanging the yarn on a horizontal steel wire at a certain angle. Weights are added to the yarn successively and the change i n the yarn thickness is measured as a function of  -51-  Chapter 3- 3D Fabric Crossover Model  the contact force. This method, however, fails to represent the true state of yarn in the fabric and capture the interaction of the yarns as they get compressed. Despite the efforts to establish a standard experiment for measurement of transverse compressive response of yarns i n a w o v e n fabric, little can be said about their success since none of them can produce the true compressive behaviour of yarns during the biaxial loading of a fabric (as seen i n Figure 3 8). Several simulations of the compression experiments  were conducted  to  calibrate the transverse m o d u l i of the solid elements. In these simulations, the modelled yarn crossover is transversely compacted between two rigid plates and the contact force exerted on the plates by the yarns is plotted against the applied transverse displacement. In effect, the value of the transverse modulus is back calculated from the transverse compression data (discussed i n Section 2.2.1), by matching the forcedisplacement response of the yarn crossover predicted from the numerical simulation to that of the experiments. The comparison of the two responses is presented i n Figure 3 - 9 for a single crossover of S-720 fabric.  3.4.3.3 Biaxial Response of the 3D Crossover Model The 3D crossover model assembled from yarns w i t h solid and bar elements is used to study the biaxial response of fabrics at a unit-cell level. The contact between the yarns is activated through surface-to-surface option i n L S - D Y N A which is based on a penalty method, i.e. placing normal interface springs between the contacting bodies. The friction between the yarns is ignored and  -52-  Chapter 3- 3D Fabric Crossover Model  symmetric displacements are applied to the yarns i n the biaxial deformation mode. Figure 3 - 1 0 shows the development of tensile force i n the warp yarn as it is displaced while the weft yarn is kept fixed at both ends. Comparing the tensile response of the warp yarn inside the weave (crossover response) and outside (single crimped yarn response), it appears that the yarn exhibits a stiffer response inside the fabric due to the interaction w i t h the crossing (weft) yarn. A t the same time, tension also builds up i n the weft yarn as a result of this interaction. Due to the biaxial state of the in-plane extensional response i n the fabrics, the tension i n each yarn is a function of its o w n displacement (d ) as x  well as that of the crossing yarn (d ). This results i n response surfaces for each 2  yarn, as shown i n Figure 3 - 1 1 . These surfaces are a product of various geometrical and mechanical properties of the yarns and fabric, and are the basis of the specific characteristics of a fabric. The response surfaces obtained here are used i n the following chapter to verify the predictions of the mesomechanical model developed to capture the interaction of the crossing yarns and the biaxial behaviour of fabrics.  3.5  A D V A N T A G E S A N D LIMITATIONS  The 3D crossover model incorporates many details of the yarns i n the weave and provides a large amount of information on the mechanics of their deformation under various applied loads. Due to the generality of the assumption made i n setting up the 3D crossover model, there is almost no limitation on the loading scenarios that could be considered.  -53-  Chapter  The  3-3D  Fabric  Crossover  Model  3D solid-bar modeling approach brings along several challenges and  limitations. Some of these issues are listed below: • Generating the shape of the yarns as they are i n the fabric is not an easy task. Plain fabric cross-section micro-imaging (see A p p e n d i x A ) reveals that the yarns i n the warp and weft directions interact even before any loading is applied. This is evident from the non-uniform profiles, yarn overlays, etc. • The properties of the continuum representation of the yarns cannot be easily determined. This is mainly due to the complex, highly nonhomogeneous nature of the yarns and lack of understanding of the physics of fibre interactions i n a yarn. • Due to the small scale of the fabric unit-cell and the number of solid and bar elements used to capture the shape of the yarns, the finite element mesh of fabrics, even w i t h small dimensions, w o u l d contain a large number of elements. This results i n unrealistic computational requirements leading to extremely  time-consuming  analysis  even  for  models  with  modest  dimensions. Despite all the limitations discussed above, the 3D solid-bar model is perhaps one of the best tools to get insight into the behaviour of yarns i n fabrics of different weave structure. W i t h the everyday advancement of computing machines, this issue w i l l become less and less limiting and 3D models w i l l perhaps replace the other simplified approaches i n the near future.  -54-  Chapter 3-3D  3.6  Fabric Crossover Model  SUMMARY  This chapter introduced the basics of a detailed 3D model of the fabric unit-cell (a single crossover i n the case of plain weave fabric) that is comprised of bar and solid elements. Representing bundles of fibres i n the yarn, the bar elements are aligned w i t h the yarn axial direction and w o u l d provide its tensile stiffness. The solid elements are placed i n the space between these bar elements and provide the soft transverse compressible response of the yarns. The modeling technique established can be applied to any type of fabric, since it is not dependent on the geometry of the yarns or the fabric structure. The 3D crossover model was successful i n capturing the biaxial response of fabrics due to the interaction of the crossing yarns. The 3D crossover model w i l l be used i n the following chapter to verify the predictions of the computationally more efficient 2D shell crossover model.  -55-  Chapter 3- 3D Fabric Crossover Model Table 3-1: Beam element input parameters used for the 3D crossover model of S-720 fabric unit-cell. Parameter  Warp  Weft  Elastic Modulus (GPa)  96.0E+09  96.0E+09  Number of bars/yarn  44  38  Yarn Density (denier)  1420  1420  Fibre Area (mm )  1.096E-01  1.096E-01  Bar Area (mm )  2.49E-03  2.88E-03  3.0%  3.0%  2  2  Failure strain, e  f  Table 3-2: Solid element input parameters used for the 3D crossover model of S-720 fabric unit-cell. Parameter  Warp  Weft  E (Pa)  1.0E+06  1.0E+06  E (Pa)  2.0E+09  2.0E09  0.02  0.02  0.70  0.62  1.00  1.00  a  c  K  -56-  Chapter 3-3D  Fabric Crossover Model  Update velocities  Update displacement  T Start  Apply force boundary conditions  User material models  w  Process elements  Figure 3-1: schematic representation of LS-DYNA explicit finite element analysis [56].  Figure 3-2: Solid (top left) and bar (top right) elements combined to make up a 3D model of a yarn (bottom).  -57-  Chapter 3- 3D Fabric Crossover Model  Figure 3-3: Hourglass-type deformation shapes (left) eliminated by the addition of bar elements (right).  0.00  0.01  0.02  0.03  0.04  Yarn Displacement (mm)  Figure 3-4: Force-displacement response of a single crimped yarn showing the instability of the yarn model without the bar elements.  -58-  Chapter 3-3D Fabric Crossover Model _L VJ < V  ®©©©©®©®©®® ®®©®©®®©®®©®© ®®®®©©®©®®@®@® ®®@®®®©®®®©®® ©®©©©®@©©o  Tension  f  (very low resistance)  V° < V" < V Compression c  A  (increasing resistance)  1  0OOOOO8 OGOOO O G OOOOOOOO  _ ob o o o o o o  ,  _0_Q_0_0_0„0_G_0_®_®_®_®_®„ g> O.O O O O Q O O ® ® ®  /  1  a& G © © 0 0g"°  o o o o o o©  Q OOOOOOO Original Yarn  Compression V" > V (highest resistance)  0  0  0  0  Q  c  t  Figure 3-5: Schematic representation offibrerealignment and compression in a yarn subjected to transverse compression.  2.5 n  0.0 I 0.7  0.8  1  .  1  0.9  1  1.1  Nominal Fibre Volume Fraction  Figure 3-6: Relationship between transverse elastic modulus and thefibrevolume fraction for the S-720 warp yarn.  -59-  Chapter 3 - 3D Fabric Crossover Model  Figure 3-7: Single-yarn tensile test predictions for S-720 warp and weft yarns with 2.18% and 1.39% crimp, respectively.  Figure 3-8: Simulation of the transverse compression test of a fabric unit cell (note that only a quarter of the unit cell is shown due to symmetry).  -60-  Chapter 3-3D  Fabric Crossover Model  50 -.  0.00  0.04  0.08  0.12  0.16  0.20  Transverse Displacement (mm)  Figure 3-9: Transverse compression response of S-720 fabric unit cell, experiments compared to prediction of the 3D crossover model.  Figure 3 - 10: Tensile force developed in the warp and weft yarn of a single crossover model extracted from S-720 fabric.  -61-  Chapter 3-3D  Fabric Crossover Model  Figure 3-11: Warp and weft yarn tensile response surfaces as a function of yarn displacements, calculated for S-720 fabric unit-cell.  -62-  Chapter  4 - 2 D Shell  CHAPTER 4-  4.1  Crossover  Model  2DSHELL CROSSOVER  MODEL  INTRODUCTION  In this chapter, a meso-mechanical model of fabric crossover is developed that is implemented as a constitutive material model for an efficient shell element i n L S - D Y N A to capture the biaxial response of a fabric unit-cell. A comprehensive review of the relevant literature is first presented, to highlight the strengths and shortcomings of the unit-cell models currently studied by other scientists. The details of the shell constitutive model are then presented, followed by the verification of its response through comparison w i t h the 3D unit-cell discussed in the previous chapter. A critical review of the capabilities and limitations of the model is presented along with some future recommendation for further improvements of the model.  -63-  Chapter 4-2D  4.2  Shell Crossover Model  SHELL-BASED FABRIC UNIT-CELL M O D E L S  Most fabrics possess a periodic geometry due to the specific pattern that the yarns are w o v e n into, making them an excellent candidate for unit-cell analysis approach. M a n y researchers have tried to duplicate the mechanical response of fabrics i n efficient membrane/shell elements. The elements developed are generally a continuum representation of the fabric through smearing the effects of warp and weft yarns into a two dimensional homogeneous material. While pin-jointed bar models of fabrics generally consider the response of warp and weft yarns to be decoupled, shell/membrane elements can accommodate the biaxial behaviour of the fabrics through implementation of  appropriate  constitutive models. This section includes a brief review of some of the unit-cell models that have employed shell/membrane elements as the basis to predict the fabric response. Fabrics exhibit a non-linear structural response when stretched i n the in-plane direction. Perhaps some of the earliest models of fabric crossover to capture the biaxial behaviour  of the  fabrics response  under  extensional and  shear  deformation modes are those of Kawabata et al. [57-59]. In a series of papers published i n 1973, Kawabata et al. presented simple analytical models to capture the biaxial and uniaxial behaviour of symmetrically loaded yarn crossovers, as w e l l as their shear response. Considering a linear representation of the yarn centrelines (Figure 4 - 1), the equilibrium of the crossing yarns under biaxial extension was satisfied through balancing the yarns tensile forces with the  contact force developed between  them. Kawabata et al. [57]  accommodated the transverse deformation of the yarns due to the inter-yarn  -64-  Chapter  4-2D  Shell  Crossover  Model  contact force i n determining the transverse position of the yarns satisfying the equilibrium of forces and compatibility of the deformations. This model was expanded i n a following paper [58] to incorporate the bending resistance of yarns, most dominant i n the uniaxial extension of the fabrics. The model developed to capture the shear response of the fabric [59] is discussed i n Section 4.3.2. Since the early work of Kawabata et al. [57], many other studies have adopted a similar, if not identical, approach to capture the biaxial response of fabrics using finite element analysis approach. Ivanov and Tabiei [30] developed a model that i n their words was "a computational micro-mechanical material model for loosely w o v e n fabric", with a unit-cell consisting of a crossover with the yarns interacting under the applied displacements. The yarn material was considered to be visco-elastic and its response modelled by a three-element spring/dashpot model. The model was implemented in the shell element and used to simulate ballistic impact experiments on Kevlar® 129 fabric targets. K i n g et al. [31, 32] published the details of a continuum-based model of a fabric unit-cell used to capture the deformational response of w o v e n fabrics. The model relies on the selection of a representative model of weave geometry along w i t h a constitutive relationship to capture the biaxial response of the fabric crossover. In this multi-scale approach, K i n g et al. adopted a similar geometry to that of Kawabata et al. [57] to identify the in-plane extensional response of the yarns. The yarns were represented by an assembly of trusses pin-jointed together to make up the geometry of a certain weave. Other aspects of fabric response such as lateral compression and bending of yarns were  -65-  Chapter  4-2D  Shell  Crossover  Model  considered through individual springs i n the unit-cell. W i t h the knowledge of macroscopic deformations and displacements of the fabric at any time, the internal forces can be calculated from the unique configuration of the yarn network by m i n i m i z i n g the energy stored i n the system. K i n g et al. [57] compared and obtained good agreement between the predictions of their numerical model and the laboratory data for a variety of different experiments. A detailed mesoscopic model for biaxial analysis of fabric deformation during the forming process was developed by Boisse et al. [27]. The initial geometry of yarns was represented by a combination of circular arcs and straight lines. The tensile response of the yarn was measured from simple tension tests. The model considered for transverse crushing law of the yarns was dependent on both the compressive state of the yarn and its axial tension. The friction between the yarns were considered implicitly to prevent relative sliding of the yarns, however, it was assumed that the frictional dissipative energy was too small compared to other energy components. Boisse et al. used this model to determine the forming of a fabric w i t h a square punch and die. They reported that their biaxial unit-cell model was i n good agreement w i t h the experimental data and the predictions of a 3D micromechanical unit-cell model they developed for the same purpose (discussed i n Chapter 4). Xue  et al. [38] proposed a combined micro/macro-mechanical non-orthogonal  constitutive model to capture the large deformation behaviour of the fabrics i n the thermo-forming process. The constitutive model developed considered details of the weave such as the dimensions of the fibres, yarns and the unitcell, their material properties and the orientation of the yarns. The shear  -66-  Chapter  4-2D  Shell  Crossover  Model  properties of the fabric were then obtained from a mechanistic analysis of the unit-cell and the tensile properties of the fabric was obtained from a simplified analytical model. The tensile model is based on incorporating the interaction of the crossing yarns through correlation parameters determined by performing fabric uniaxial tensile tests. Stubbs [40] presented the details of a model developed to capture the response of  coated fabrics under multi-axial loading. This model represented  the  interlacing yarns and the added stiffness of the coating material through an assembly of truss elements. Stubbs' approach is very similar to that of Kawabata et al. [57] i n principle, with only the addition of two horizontal bars (springs) connecting the yarn ends that w o u l d characterize the effect of the resin coating of the fabric. Luo  [39] developed a constitutive model for the unit-cell of fabric-reinforced  flexible composites under biaxial loads. The response of the unit-cell was obtained using a strain energy approach, accounting for the yarn deformation through rigid body displacement resulting from the crimp exchange between the interacting yarns, and their cross-sectional changes under the contact force. Tanov and Brueggert [36] proposed a model based on a representative cell approach to capture the extreme shearing of loosely w o v e n fabrics during the expansion of side airbags i n cars. The representative cell was embedded into a User Material model for membrane elements i n L S - D Y N A . The axial response of the yarns was represented by four bar members along the sides of the element and the shear response including the locking of fabric was captured by diagonal springs. Tanov and Brueggert considered the rotation of the fibres  -67-  Chapter  4 - 2 D Shell  Crossover  Model  during the shearing of the fabric, but seem to have overlooked the interaction of the crossing yarns during extension. The model developed b y Tanov and Brueggert seems to be successful i n capturing the response of airbags and appear to be i n good agreement with the experimental measurements. The continuum-based fabric unit-cell models discussed above are a few of many useful approaches  to simulate the biaxial response of fabrics. The  invaluable research and scientific development i n the field of computational modeling of fabrics, some of which briefly discussed above, has  been  instrumental i n the evolution of the approach presented i n this thesis.  4.3  2 D SHELL FABRIC UNIT-CELL - M O D E L BASICS  The  2D shell crossover  model  developed  is based  on  a  continuum  representation of the fabric unit-cells using shell elements. The model captures the micromechanics of the fibres and yarns i n the fabric through a smeared continuum approach. In establishing the basis of the model, it is assumed that the in-plane extensional mode of deformation is decoupled from the shearing mode. A s a result, the constitutive relations developed for these two modes of deformation are derived separately. The following sections review the material models developed to capture the extensional and shearing response of a plain weave fabric unit-cell comprised of a single yarn crossover.  4.3.1 In-plane Extensional Response Plain weave fabrics exhibit strong biaxial behaviour w h e n extended i n their plane. A n analytical model is developed here to capture this biaxial response  -68-  Chapter 4-2D  Shell Crossover Model  considering the interaction of warp and weft yarns at fabric crossovers. The model presented here is inspired by Kawabata's linear crossover model [57], with further expansion to accommodate the non-linear geometry of the yarns. The model is based on tracking the location of the yarn centrelines (with geometries illustrated i n Figure 4 - 2), as the two yarns interact under the applied displacements. The initial geometry determined  by  its  initial  crimp value,  of each yarn centreline is  obtained  from  the  laboratory  measurements. The analytical model developed here is aimed to capture the interaction of the warp and weft yarns and to determine their in-plane extensional response. In establishing the mathematical formulation of the model, several simplifying assumptions are considered, as listed below: • It is assumed that the yarn profiles of the w o v e n fabric are initially sinusoidal, based on the observations made through studying the fabric cross-sectional micro-images (see Appendix A ) . It is further assumed that the yarn profiles stay sinusoidal at all times, independent from the details of micro and macroscopic unit-cell displacements. • The in-plane warp and weft displacements are assumed to be symmetrically applied at the two yarn ends, preventing any sliding between the two crossing yarns. This assumption results i n an effectively pin-jointed model, since no relative in-plane yarn displacement occurs at the point of contact between the two yarns. • A s the warp and weft yarns interact, a contact force develops along their entire contact length while stretching under the applied displacements. It is assumed that there is no friction between the contacting yarns so that the  -69-  Chapter 4-2D  Shell Crossover Model  contact force at any point is perpendicular to the surface of the yarns. For the analysis of the crossover mechanics, only the resultant contact force acting at the centre-point of the crossover is considered. • Figure 4 - 3 shows the free-body diagram of a yarn, w i t h a continuous support system representing the effects of the crossing yarn. Since the contact between the two yarns is assumed to be frictionless, the supports act as rollers. Transverse compression of the yarns is illustrated by the springs between the supports and the yarn centreline. Due to symmetry, the resultant contact force acting on the yarn only has a component i n the z direction (normal to the fabric plane). The absence of friction between the yarns results i n a constant tensile force along the yarn. Global equilibrium of the forces acting on the yarn is satisfied when the z -component of the yarn tension equals the resultant contact force, summed over the entire length of the yarn. Taking the origin of the coordinates to be at the centre of the unit-cell, the shape of the yarn can be expressed with a cosine function, as depicted i n Figure 4 - 4 . For a unit-cell of w i d t h 2w i n the i direction (/' = 1,2 for warp and weft, i  respectively), the profile of the yarn centreline can be expressed as: n  f  /(x,.)=Z, =/2,.COS  x  2w,  K  (4-1)  i  j  where h is the height of the yarn centreline at the centre of the unit-cell t  (x, =0). The parameter w, (in mm) can be calculated from the thread count of the crossing yarns, w (in threads/mm), as ;  -70-  Chapter 4-2D  Shell Crossover Model  (4-2)  =—  2w,  nij  where j - 2,1 defines the perpendicular direction to yarn /'. The initial value of h can be calculated mathematically from the crimped length of the yarn. t  Considering the yarn profile function presented i n Equation 4 - 1 , the length, S , of the yarn at any moment can be calculated from the equation below. t  n  sin  2w  2w.  dx  (4-3)  The initial length of the yarn can also be determined from yarn crimp, cr : {  S =2w (l 0i  t  + cr ) t  (4-4)  From Equations 4 - 3 and 4 - 4, the nonlinear equation needed to calculate the value of h is obtained: 0l  (4-5)  It is basically assumed that the yarns maintain their sinusoidal profile during deformation. Application of a set of symmetric displacements to the warp and weft yarns w o u l d lead to the development of contact force between them as a result of their interaction w i t h each other. The consequent motion of the central point of the crossover i n the out-of-plane direction is an outcome of this mesomechanical interaction i n a fabric crossover. Figure 4 - 5 depicts the geometry of the yarns i n the warp and weft directions before and after the application of symmetric displacements. Note that the two perpendicular yarns are shown in the same plane for illustration purposes, t h e geometry of the yarns after the  -71-  Chapter 4-2D  Shell Crossover Model  application of symmetric displacements d and d can be expressed i n terms of x  2  their new heights h and h . The challenge at this point is to determine the x  2  values of the two unknowns, h and h for a given set of displacements d and x  2  x  d. 2  For any displacement d applied to the yarn /' (/ = 1,2) of the crossover, there is {  an  equilibrium configuration with  a yarn height  of  /z,. The applied  displacement leads to the development of a tensile force, T  if  force, F,  and a contact  i n the yarn. This contact between the two yarns leads to their  deformation i n the transverse direction denoted b y d . B y equilibrium, the c  resultant contact forces acting on the two yarns should be equal; i n other words: K =K t  2  (4-6)  The tensile forces developed i n the yarns due to their in-plane deformations are functions of the yarn material and geometrical properties. The bending/shear mechanical properties of the yarns are usually very small compared to their axial tensile stiffness and thus considered to be negligible i n this study. However, due to the potential numerical instabilities resulting from this assumption, the user is provided with the option of including some bending resistance to the yarns as a percentage of the yarn tensile stiffness. This has proven to be useful i n significantly reducing the numerical oscillations, especially where the fabric is subjected to l o w levels of load or when the yarns lose contact due to buckling i n a compressed unit-cell. For the case of Kevlar®  -72-  Chapter 4-2D  Shell Crossover Model  129 material, the tensile behaviour of the yarns is assumed to simply follow a linear elastic model, defined by the equation below:  (4-7)  T,=-^-(S,-S ) OI  where £, is the elastic modulus of the fibres i n yarn / , A is the cross-sectional si  area of the fibres i n the yarn, and S  and S are the initial and the current  0i  t  length of the yarn. The equivalent contact force that develops between the two yarns, F , can be estimated by enforcing the equilibrium of yarn /', as follows: c  F =27,. sin a,  (4-8)  Ci  where a is the yarn-end slope (see Figure 4 - 5 ) . This slope is calculated using t  the first derivative of the yarn shape function calculated at the yarn end, i.e. dz tana, =  7th:  —  2w,  dw  Therefore, tana, sina, =-==  =  Vl + tan a,.  .  7th:  '  =  (4-10)  ^(2w, f + fa, )  2  2  The relationship between the transverse contact force acting o n the yarns and their compression d can be generally expressed as: ci  d =Q{F ) Cl  Ci  (4-11)  The compression model developed for the transverse mechanical response of the yarns was discussed i n detail i n Chapter 3. This relationship can be expressed i n the following form:  -73-  Chapter 4-2D  Shell Crossover Model  (4 - 1 2 )  where a and b are constants. The method to determine a and b is discussed later i n this chapter. In addition to the equilibrium condition expressed i n Equation 4 - 6 ,  the  geometric compatibility of the yarn centerlines should also be satisfied. For this purpose, the original and current shape of the yarns (h  x  and h ), 2  transverse  deformation of the two yarns and the initial gap, g, between them are considered i n the compatibility expression given below: h +h =h +h -d -d -g x  2  0x  02  cX  (4-13)  c2  Equations 4 - 6 and 4 - 1 3 are used to determine the two unknowns h and h . x  2  Due to the complex nature of these two equations, an explicit closed-form solution is not readily obtainable. Therefore, h and h are calculated using a x  2  numerical approach that is capable of solving such multi-variable non-linear system of equations. The Newton-Raphson technique is used to estimate the unknowns h and h by solving a system of nonlinear equations applied to the x  2  objective functions fx and f2, defined as: fx=F -F Ci  fl  = \  +  h  2+ c d  +  l  (4-14)  Ci  d  c  2  +g~K  ~Kl  (4-  1 5  )  Details of the Newton-Raphson iterative technique can be found i n the literature (e.g. see [60]). The first step i n setting up the iterative scheme is to determine the Jacobian matrix, J, defined as:  -74-  Chapter 4-2D  Shell Crossover Model  J=  3/,  3/,  d\  dh  2  df  df  d\  dh  2  (4-16)  2  2  The components of the Jacobian matrix can be calculated from the partial derivatives of equations developed earlier. The terms o n the first row of the Jacobian matrix J  u  and J  can be determined as:  l2  H  J,  dh  x  d („ dh  „  \  SF  C2  dh,  dh  2  (4-17)  dk  dh  x  2  (4-18)  since the contact force acting o n a yarn is only a function of the geometry of that yarn, i n other words: 5F  5 F  dh  =0  dh  2  (4-19)  x  The derivative of the contact force w i t h respect to the height is needed for each yarn to be substituted i n Equations 4 - 1 7 and 4 - 18. Differentiating Equation 4 - 8 w i t h respect to h using the chain rule results i n the following relation: ;  ^  =2sina ^ (  dh  dk  s  2 7 : * ^  +  dk  (4-20)  Differentiating Equation 4 - 1 0 results i n d(sina ) _  4TTW,  i  dh  t  "[(2  and using Equation 4 - 7 ,  -75-  (4-21)  ) +(^,) J 2  W /  2  5  Chapter 4-2D Shell Crossover Model dT  t  =  EA t  dh  dS  si  t  S  t  (4-22)  dh  0i  t  Using Leibniz Integral Rule [61] 6(z)  b(z)  a(z)  a{z)\ oz  da | - ")'f{x, z)dx== "]^dx + f{b(z), z)^- - f{a(z), z) f(x,z)dx UZ  the term  dh,  t  \  oz  i  (4-23)  oz  can be evaluated as  dS,  '-} M-  dh7  2w,  0  1+  ' 7th,  \  \2w,)  \ >  \ i  J  2w  71 X  . 2 sin  (4-24)  \dx  X  r s m  r  2  )  2w  Thus, Equation 4 - 2 2 can be evaluated as  dT  EA  dh,  S  t  t  0i  n^  f si  \  2  w  i J  M-  X dx  rsm  1+ \ 2  w  \  71  sin  2 w  i  (4-25)  J  X  U  \ >  J  2W  Equations 4 - 2 1 and 4 - 2 5 are inserted into 4 - 20 to evaluate J  u  and J . The n  other components of the Jacobian matrix can now be determined:  J  2 i  =^r = d\ ^r( ^ 2 dh h  +h  +  c  d  +  c  d  l  (4-26)  +g-Kx-K)  2  x  using the chain rule,  _ 4 J d  J 21  u  =  dh,  d(dJdF  u  Ci  (4-27)  dF„ dh  dF  d(d )  d\  dF„  c  The value of — ^ is obtained from Equation 4 - 20, and from differentiation of Equation 4 - 1 2 :  -76-  c  is evaluated  Chapter  4-2D  Shell  Crossover  Model d(d ) _b  1  Ci  Similarly for J  22  (4 - 28)  we have 4dJ  df  = 1+ J 22 dh dh 2  d{dJdF  =  u  dF  2  2  Ci  c  (4-29)  dh  2  where d{dj_b dF  C2  1 ~ 3 (  F  (4 - 30)  J i  With all the components of the Jacobian matrix determined, one can perform an iterative numerical scheme to find the values of h and h for a given set of d x  2  x  and d according to the algorithm outlines i n A p p e n d i x B. Other physical and 2  geometrical characteristics of the crossover such as the contact force and yarn tension can then be calculated, resulting i n a biaxial response that incorporates the interaction of the yarns i n a single crossover.  4.3.2  Shear Response  One of the distinctive attributes of plain weave fabrics is their unconventional shear behaviour. Due to the specific arrangement of the yarns i n the weave structure, the fabric can shear easily u p to a certain locking angle where the compacted yarns resist further shearing. M a n y scientists i n the textile industry and the field of textile composites manufacturing have studied this behaviour, some of w h i c h are reviewed here. Kawabata et al. [59] introduced a semi-empirical model to capture the shear deformation response of fabrics. The overall shear behaviour of fabrics under  -77-  Chapter 4-2D  Shell Crossover Model  biaxial tensile and shear deformations was approximated based on utilizing a linear empirical relationship to estimate the torque required to change the angle between the crossing yarns. The u n k n o w n parameters i n this empirical relationship, comprised of a frictional  and an elastic component,  were  calibrated through experimental measurements. Prodromou and Chen [62] studied the relationship between shear angle and wrinkling of textile composite performs. They performed frame shear tests on glass fabrics w i t h plain weave, four- and eight-harness satin weaves. Their experimental results showed that two distinct regions can be distinguished on the load versus shear angle response of the fabrics: prior to locking where the trellis mechanism dominates the response and post-locking where the fabric starts to wrinkle out-of-plane due to high shear modulus i n the fabric. They found that more conformable weaves such as satin w o u l d reach the shear locking at larger deformations, as is also the case for smaller yarn size at a given yarn count. The shear tests performed by Prodromou and Chen [62] indicated that the relative motion of the yarns at the centre of a fabric crossover does not occur during the trellising, meaning that the crossing yarns can effectively be considered pinned. Based on these observations, Prodromou and Chen [62] proposed a method for calculating the locking angle. In their proposed pin-jointed model, they assumed that shear locking occurs when the space between the parallel yarns closes, and moved on to calculate the locking angle by setting the distance between the yarns of the deformed fabric equal to their width. Based on the results obtained from this mechanistic pin-jointed model, they concluded that other factors not considered such as friction and  -78-  Chapter  4-2D  Shell  Crossover  Model  changes i n the yarn w i d t h and their spacing can result i n a different measured locking angle compared to the predicted value. McBride and Chen [63] proposed a pin-jointed shear model that w o u l d consider the geometry of the fabric unit-cell w i t h sinusoidal yarn shapes during the shearing deformation. In this model, transverse yarn compaction was included as a deformation mechanism i n addition to the trellising. The model predictions of yarns w i d t h and fibre volume fraction as a function of shear angle  showed  good  agreement w i t h the  observed  values,  despite  the  inaccuracies stemming from other factors not accounted for, such as friction and fibre waviness within a yarn. The model presented i n this study is motivated by the two models of Prodromou and C h e n [62] and McBride and Chen [63]. A trellis mechanism is considered where the shear response of a crossover is subdivided into three distinct regions. Initially, due to the presence of gap between parallel yarns, the lone resisting mechanism against shearing is considered to be friction, leading to a very small initial shear modulus (referred to here as zone I). Further shearing of the unit-cell w o u l d give rise to the shear modulus, as the yarns start to interact and compact each other transversely (zone II). Eventually, the compaction of the yarns reaches a maximum value, after w h i c h the shear modulus is at its m a x i m u m and further shearing deformation of the fabric w o u l d lead to w r i n k l i n g (zone III). The information obtained from the picture-frame tests are used to set up a shear model for the plain weave fabric. The fabric geometry changes from its initial state to deformed state are schematically shown i n Figure 4 - 6. A s the  -79-  Chapter  4-2D  Shell  Crossover  Model  fabric shears, the warp-to-weft angle (3 decreases from its original value, J3 ., 0  thus reducing the spacing between the adjacent parallel yarns. This distance, u, and  its reduction are shown i n Figure 4 - 6 . The relationship between the shear  angle and yarn spacing can be derived from the fabric deformed geometry, as:  sin/? = Y  (4-31)  where L is the side-length of the fabric i n picture frame shear test. The presence of in-plane gap, p, between adjacent yarns i n the undeformed configuration allows the fabric to shear freely to a certain angle, /?,. A t this angle, the inter-yarn gap diminishes and the parallel yarns start to get i n contact w i t h each other. Angle /?, can be calculated as  siny0,=-2—^  (4-32)  Li  where u  0  is the initial distance between the yarns. Further shearing of the  fabric leads to gradual compaction of the yarns, up to an angle j3 , where the 2  yarns reach their m a x i m u m compaction. This angle can be calculated from the following equation:  sin/? =^2  (4-33)  where u corresponds to the yarns spacing at m a x i m u m compaction. In order 2  to calculate u , certain assumptions need to be made. It is assumed that the 2  fabric thickness, as reported i n many other studies, remains more or less constant and is decoupled from  the shear deformation. Based on this  assumption, it is concluded that the amplitude of the sine function representing  -80-  Chapter 4-2D Shell Crossover Model  the yarn profiles remains constant during shear. It is further assumed that the yarn cross-sectional shape becomes elliptical at the point of full compaction. In reality, the yarn compaction w o u l d not be uniform along its length due to the localized interaction of the crossing yarns and their specific configuration i n a highly sheared fabric. To account for this effect, an area reduction factor, K , is s  considered w h i c h results i n a yarn packed area of . U s i n g the schematic of the yarns i n the sheared unit cell shown i n Figure 4 - 7 , the compacted area can be calculated from the formula below:  (4-34)  A =K A =^2 l  p  The values of u  2  s  s  calculated from the above equation can be substituted i n  Equation 4 - 33 to determine the locking angle j5 . 2  Using the locking angles /?, and /? , the corresponding shear strains y and y 2  x  2  can be determined. The shear stress-strain relationship is formulated such that the instantaneous shear strain y is related to the secant shear modulus G b y the function below (also schematically shown i n Figure 4 - 8 ) :  G, G  =  0<y<y  x  G +^^{y-y ) x  x  G  (4-35)  y,<y<y  2  y>y  2  2  A similar relation exists between the shear strain / and shear modulus for values smaller than zero, i n other words, the shear model is the same for positive and negative values of shear strain.  -81-  Chapter  4.3.3  4 - 2 D Shell  Crossover  Model  Development of User Material Model for Shell Elements  The main goal of this work is to develop an efficient shell element that performs similar to that of a yarn crossover. This shell element takes advantage of a meso-mechanical model of the fabric crossover and can be categorized between the simple pin-jointed cable model and the detailed 3D fabric crossover model of the fabric unit-cell both i n terms of complexity and efficiency. In other words, the shell crossover model w o u l d be able to capture many details of the weave and yarns that the pin-jointed bar model w o u l d overlook, while it is significantly more efficient numerically compared to the 3D fabric crossover model. This section discusses the details of the User Material M o d e l ( U M A T ) development to provide the constitutive relationship for the smeared shell element that represents a fabric unit-cell. In  developing  the  UMAT  for  the  shell element,  several  simplifying  assumptions are made based on the physical behaviour of a fabric. Since fabrics generally have very l o w bending and shear stiffness, it is assumed that their out-of-plane stiffness is insignificant compared to their in-plane stiffness. The in-plane extensional and shear responses of the fabric are decoupled and treated separately based on the mathematical models developed i n the preceding sections. The interface between the main engine of the finite element code, L S - D Y N A and the user developed material routine, U M A T , is simple and efficient. A t each time step, L S - D Y N A passes the element strain increments evaluated at the shell element integration point (centre of the element for the constant stress shell elements used i n this study) i n the local coordinate system to the user  -82-  Chapter 4-2D  Shell Crossover Model  material routine. In return, the components of the element stress tensor are computed by the U M A T and returned to the code for continuation of the analysis. The goal i n setting up the U M A T is to use the strain increments to determine  the  unit-cell  displacements  and  warp-to-weft  angle  needed,  respectively, by the extensional and the shear models discussed. The basic knowledge of stretches i n the crossing yarns and their relative angles are essential for determination of the forces developed i n the crossover. To capture the extension and rotation of warp and weft yarns at each time step, two vectors are attached to the crossing virtual yarns, whose configuration is constantly updated throughout the analysis. The deformation gradient, F", pertaining to the incremental changes i n the displacement field between the two consecutive time steps n and n — \ can be assembled from the strain increments as follows:  F =(l n  + 2A£ J *L N  5  + A_" =  l + As "  Asxy  0  \ + As "  where As" is the tensor of the strain increments w i t h components As",  (4 - 36)  As"  and As " i n the local coordinate system, xyz, and I is the unit matrix. In the xy  convention used i n this study, the local x -axis is defined parallel to the  - n  2  edge, as shown i n Figure 4-9. The z -axis is taken to be normal to the element mid-plane and the y -axis is determined by the cross product y = zxx  (4-37)  This local axes system i n L S - D Y N A is based on the shell node numbering, as defined i n the User's M a n u a l [54]. The material axes, on the other hand, are  -83-  Chapter 4-2D  Shell Crossover Model  assumed to be parallel to the warp and weft yarns. The strain increments obtained from the deformation of the element need to be converted into the unit-cell displacement i n the material directions (along warp and weft yarns) to determine  the  biaxial  response  of the  crossover.  In  the  un-deformed  configuration, the material and local axes coincide i n the case of uniform rectangular elements. A s the element undergoes large displacements and rotations, the angle between the local and the material axes (defined by the warp and the weft) changes. The angle between the material and local axes of the element is thus updated at each time step i n order to determine the displacements applied to the warp and weft yarns. To track the configuration of the material axes i n the local coordinate system during the analysis, two vectors q  m  and q  02  are considered parallel to the warp  and weft yarns, respectively. The coordinates of these two vectors are updated at each time step i n the local coordinate system (Figure 4 - 9 )  using the  deformation gradient as follows: n  ,-,M  <h_ =£  n-l  <h_  (4-38)  where q " and q " are the / material vector (/ = 1,2 for warp and weft) at time _l  t  steps n and n-l,  t  respectively. The angle between each material vector and the  local x -axis can be determined as follows: 6," =taiT  where  -84-  (4 - 39)  Chapter  4-2D  Shell  Crossover  Model I  n  n  (4 - 40)  The strain increment tensor needs to be transformed into the direction of the two material vectors. The appropriate transformation matrix is calculated for each material vector based on its current angle: cos0,"  sin0,"  (4-41)  -sm.9" cosO"  The strain increments are subsequently rotated to the material directions using the transformation matrix as follows:  As/  1  (4-42)  =^"__"@ "  T  ±  The displacements along the yarns can then be calculated using the basics of continuum mechanics. The incremental stretch of a unit vector i n the material directions from time step n — 1 to n can be calculated from the equation below:  K  = Vl + 2A<,  * 1 + As"  u  (4 - 43)  The total stretch i n the material directions is updated at each time step using the incremental stretch values of Equation 4 - 43, as shown below:  A "=A "-V I  I  (4-44)  Displacement along yarn / at time step n w o u l d then be equal to  J;=(A;-I)W,  (4-45)  where w, is the initial unit cell w i d t h obtained from Equation 4 -2. K n o w i n g the displacements d and d , the meso-mechanical crossover model can be used to x  2  calculate the tensile forces developed i n the warp and weft yarns. In order to determine equivalent stress components that w o u l d result from these yarn  -85-  Chapter 4 - 2D Shell Crossover Model  tensions, it is assumed that each yarn occupies only half of the shell thickness. A s depicted i n Figure 4 - 1 0 , independent stress tensors i n the top and bottom halves of the smeared shell are assembled that represent the contribution of the warp and weft (fill) yarns to the overall stress tensor. The components of the yarn tensors are calculated from the forces exerted on the yarn, i n this case, the axial yarn tension: T 0  A, 0  (4 - 46)  0  where A, is the cross-sectional area of the equivalent top or bottom halfthickness. The above tensor can also accommodate future model developments in a case where definition of other yarn stress components (e.g. resulting from contact between the yarns due to shearing of the crossover, etc.) is required. The individual yarn stress tensors are then transformed back into the local coordinate system using the transformation matrix, 0 , :  ^ , " = ^ " 0 ,  (4-47)  The total element stress tensor is the volume average of the two yarn stress tensors, as calculated below:  ^S=^Y^  (4-48)  However, an extra term should be added to the in-plane shear stress that w o u l d represent the interaction of the crossing yarns i n shear, as discussed i n Section 4.3.2. This additional shear term can easily be determined from the  -86-  Chapter  4 - 2 D Shell  Crossover  Model  current angle between the two yarns, calculated from the updated coordinates of the two material vectors. Failure of individual yarns i n warp and weft directions is activated based on an instantaneous failure criterion, i.e. comparing the tensile force of every yarn to their ultimate tensile load reported by the yarn manufacturer. Once the yarn tension has surpassed its ultimate tensile load, it is flagged and permanently eliminated from the shell element. A s a result, the shell element is only capable of carrying load i n the direction where the yarn has not failed. Once both the warp and weft yarns tensile forces exceed their failure threshold, they are eliminated from the shell element, which is no longer capable of carrying any extensional load. Such a shell element is then eroded from the finite element mesh. In general finite element terms, erosion of an element implies that the code removes the element from the calculation while retains its mass properties on the nodes attached to it. Every node that becomes unconstrained as a result of eroding all the elements attached to it is also eliminated from the analysis. Erosion of shell elements i n L S - D Y N A can be activated from within U M A T based on a user-defined criterion that prompts the code to physically eliminate an element from the finite element mesh. Erosion of the shell elements with extensive unrealistic deformation (usually occurring due to the failure of one yarn only) is also triggered via a user-controlled parameter, to avoid numerical instabilities. The representative shell element, consisting of the analytical model described above, provides the biaxial extensional and shear responses of the fabric unitcell under the applied deformation. Figure 4 - 1 1 shows a simple flowchart  -87-  Chapter  4-2D  Shell  Crossover  Model  generally highlighting the process of calculating element stresses at each time step from the applied strains (more details can be found i n A p p e n d i x B). L S - D Y N A provides an extensive library of shell elements w i t h different formulations. The main two element types considered for this study are Hughes-Liu (HL)  and Belytschko-Tsay (BT) four-noded shell elements:  BT shell elements are one of the most efficient elements implemented i n L S D Y N A . Taken from the work by Belytschko et al. [64], these elements take advantage of a combined co-rotational and velocity-strain formulation to provide an efficient algorithm for small element strains. The formulation is based on a co-planar geometry, so erroneous results may result from noncoplanar deformations. H L elements, taken from the work b y Hughes and Liu [65, 66], are based on an incrementally  objective formulation, incorporating transverse  thickness thinning. Since no geometric assumptions  are  shear  and  considered i n  developing the formulation of the H L elements, these shells can handle warpage when the nodes are not coplanar. However, the under-integrated H L elements w i t h 4066 operations are more demanding computationally than the BT  elements w i t h 725 mathematical operations [54]. In this study, the  selectively-reduced H L formulation is used for the shell element, although the response of the model does not seem to be specifically affected by the choice of the shell element formulation (HL or BT).  Chapter 4- 2D Shell Crossover Model  4.3.4 Determination of the input parameters The input parameters for the representative shell model are determined either directly from the results of mechanical tests o n yarns or indirectly from the analysis of the 3D unit-cell. The process of obtaining the input data is highlighted i n this section.  4.3.4.1 Extensional Response of the Fabric Unit-cell The extensional response of the individual yarns is determined b y simulating single-yarn extensional tests i n the warp and weft directions. The values of elastic modulus and failure strain of the yarns are supplied to the shell elements and the single yarn tensile response of the warp and weft yarns are simulated and compared to the simple analytical prediction of the tensile response of crimped single yarns (previously discussed i n Section 3.4.3.1). Figure 4 - 1 2 shows the tension-displacement response of the S-726 warp and weft yarns w i t h crimp values reported i n Table 2 - 1 . G o o d agreement was found between the crimped yarn tensile responses obtained from the above two methods.  4.3.4.2 Transverse Compression of the Yarns The transverse compression behaviour of yarns during biaxial extension inside the fabric is extremely complicated. Examination of the yarns responses i n the 3D crossover model revealed that the relationship between the contact force between the yarns and their deformation changes based o n the type of loading the fabric is subjected to. This is especially true for the two loading cases of transverse compression and biaxial extension of the fabric since the different contact areas and yarn geometries (as seen i n Figure 4 - 13) lead to different  -89-  Chapter  4-2D  Shell  Crossover  Model  compressive responses. The implication of this discrepancy is that the test results from the fabric compression tests cannot be used directly for the fabric biaxial extension simulations. In order to obtain a yarn compression response that truly represents the transverse behaviour of yarns i n unit-cell biaxial extension, the 3D crossover model is used. After calibrating the transverse mechanical properties of the yarns (see Section 3.4.3.2), biaxial tests are performed and the contact force developed between the two yarns is monitored against the compression of the yarns. The transverse compressive responses of the yarns are plotted i n Figure 4 - 1 4 for the two loading cases. The best fit to the compression data obtained from biaxial simulation of the unit-cell using the 3D crossover model is fed into the shell element as a refined transverse forcedisplacement response of the yarns. 4.3.4.3 Shear Response of the Fabric Unit-cell The response of the unit-cell to pure shear deformation follows the model discussed i n Section 4.3.2. The shear modulus i n the locked position, G , is 2  calculated based on the assumption that the fabric crossover deformed beyond the locking angle behaves as an isotropic media upon further deformation. G  2  is thus chosen to be equal to:  E A.  ^ G, = where A and A s  s h  2A  -  (4 - 49)  s h  are the yarn and shell cross-sectional areas, respectively, and  E is the yarn elastic modulus. The initial shear modulus G is chosen to be a x  fraction of G (e.g. G = 0 . 1 % x G i n most cases), since the yarns easily rotate 2  x  2  with respect to each other prior to shear locking. L i m i t strains y and y x  -90-  2  are  Chapter 4-2D  Shell Crossover Model  evaluated from the weave geometric parameters, specifically the initial "gap between the yarns and the yarn width. For this purpose, the area reduction factor, K  s  needed to be estimated. In this study, K  s  was evaluated by simply  shearing a single fabric layer by hand on a flat frictionless surface until the occurrence of wrinkles. The shear angles corresponding to the onset of wrinkling were then recorded for all six fabrics and u p o n further analysis, it was found that an area reduction factor of 0.65 w o u l d best capture  the  measured locking angles. This compaction ratio is smaller than the packing ratio of 0.907 for hexagonal and 0.785 for square packing of cylindrical fibres i n a yarn. This discrepancy was attributed to the fact that the specific weave geometry and the interlacing of warp and weft yarns w o u l d introduce an extremely non-uniform compaction along the length of each yarn, leading to a lower equivalent compaction ratio. The input values to the shear model for all six Kevlar® 129 fabric types are presented i n Table 4 - 1 .  4.4  SIMULATIONS A N D VERIFICATIONS  The representative shell element developed i n the previous sections is checked against various other models to verify the accuracy of its predictions. This section reviews the important details of such verifications and highlights the capabilities and limitations of the shell crossover model.  4.4.1  Comparison w i t h 3D crossover model  One of the main fabric modes of deformation i n ballistic impact is its in-plane biaxial extension. A n y continuum model of woven fabrics should be able to  -91-  Chapter  4-2D  Shell  Crossover  Model  realistically capture the coupled warp and weft behaviour and their interaction as they are loaded. This section presents the verification studies of the 2D crossover model embedded i n the representative  shell element with the  predictions of the 3D crossover model presented i n Chapter 3. The tensile response of the warp and weft yarns of a fabric unit-cell predicted using the shell element is compared to that of the 3D crossover model. Figure 4 - 15 shows the yarn tensions i n the warp and weft directions of S-726 fabric unit-cell as a function of the warp displacement applied to the crossover while the weft yarn is held fixed. Other mechanical and/or geometrical parameters governing the interaction of the two yarns can also be investigated. Figure 4 16 depicts the changes i n the heights of the yarn mid-plane profile (h and h ) x  2  showing the crimp exchange between the two crossing yarns w i t h the warp displacement applied. It can be seen that w i t h increasing warp displacement,  h decreases and the interaction of the two yarns results i n increasing h . The x  2  work required to stretch the fabric unit-cell is also plotted as a function of the warp displacement i n Figure 4 -17. It can be seen i n these figures that the two models predict very similar responses for the fabric unit-cell despite their different approaches. However, the subtle differences between the two models stem from the many assumptions embedded i n their specific approach. Some of these differences are listed below: • Yarn cross-section can change i n the 3D solid model due to the interaction of the two yarns and the presence of the contact force. This is shown in Figure 4 - 18, for a single S-726 fabric unit-cell during biaxial extension simulation using the 3D crossover model. The change i n the cross-sectional  -92-  Chapter  4-2D  Shell  Crossover  Model  shape of the yarn can further result i n a non-uniform contact force distribution along the yarns and lead to non-prismatic yarn cross-sections. This phenomenon is considered to have a minimal effect on the response of the unit-cell and thus ignored i n the shell crossover model. • One of the major assumptions i n developing the extensional response of the fabric crossover is that the yarn profiles remain sinusoidal throughout the deformation. This theory can easily breakdown, especially if the crosssectional shape of the yarns change and lead to an irregular contact between the yarns, e.g. contact concentrated only i n the middle area of the yarn. Figure 4 - 1 9 and Figure 4 - 20, respectively, show the profiles of warp and weft yarns extracted from S-726 fabric unit-cell predicted by the 3D crossover model and shell element. A s evident from these figures, while the predictions of the two models match well for the warp profile, this cannot be said i n the case of the weft yarn. • The transverse  compression model implemented  i n the  extensional  response of the shell elements assumes identical compression responses for the two yarns. This is clearly not the case i n the 3D solid model due to the differences i n the warp and weft yarn shapes and their initial void ratio, which i n turn can affect the response of the unit-cell. • The transverse  compression model implemented i n the shell element  expresses the transverse deformation of the yarn only as a function of the contact force. The dependence of the compressive model on the ratio of warp to weft displacement is ignored i n the shell model, while this is intrinsically included i n the 3D crossover model.  -93-  Chapter  4.4.2  4-2D  Shell  Crossover  Model  Comparison w i t h pin-jointed cable m o d e l  The ballistic response of a single layer of fabric is predicted using shell elements and compared w i t h that of the cable-based pin-jointed model. The discrete pin-jointed bar models are one of the earliest numerical models developed to capture the response of fabrics i n a variety of applications. A s discussed i n Chapter 2, the cable models of fabric are based on representing the fabric layer as a series of masses and cable elements initially arranged i n orthogonal directions. In this study, the elements used to construct the p i n jointed cable model of fabric are identical to the bar elements used to represent the axial stiffness of the fibres i n the 3D crossover model. The major difference between the shell-based and cable-based fabric models is the biaxial nature of the response,  i.e. the biaxial extensional  response  embedded i n the shell elements is ignored i n the cable model due to the decoupled nature of the warp and weft yarns i n that approach. This decoupled warp/weft behaviour of the cable model can be reproduced i n the shell-based fabric model by over-softening the transverse compression response of the yarns and eliminating the interaction of the warp and weft yarns. In a way, the cable-based pin-jointed fabric model can be recovered from the shell-based model as a special case. In order to show that the discrete cable model is truly a special case of the biaxial shell model, the dynamic response of a fabric panel i n ballistic impact predicted by the shell-based and cable-based models are compared. The impact events modelled i n this thesis are simulations of the instrumented ballistic impact experiments performed at The University of British Columbia [42],  -94-  Chapter  4-2D  Shell  Crossover  Model  details of w h i c h are presented i n Chapter 2. Targets considered are squareshaped w i t h dimensions of 203 m m by 203 m m (8" b y 8"), made of a single or multiple layers of fabric. The projectile is meshed to the exact geometry of the actual R C C projectiles used i n the experiments (dimensions given i n Section 2.3), using a kinematic plasticity model of L S - D Y N A (material type 3, i.e. M A T _ P L A S T I C _ K I N E M A T I C ) to capture the mechanical behaviour of steel (see Table 4 - 2 ) . Due to Symmetry, only one quarter of the projectile and the target is simulated to reduce  the computational cost of the analyses.  Proper  constraints were applied to the target and the projectile to simulate fixed boundary and symmetry conditions. The projectile was assigned an initial velocity and contact was activated between the target and projectile, as well as the several layers of the multi-ply fabric target. Figure 4 - 2 1 shows the velocity-time history of the projectile impacting a single-ply S-720 panel at 100 m/s for a case where the crimp i n the panel is ignored. The predictions from the two approaches are clearly very similar, with some subtle differences due to the disparities between the two modeling approaches. These differences are better depicted i n Figure 4 - 2 2 where the reaction force at the boundary of a warp yarn passing through the impact point is shown. The shell based model seems to be more stable and less noisy compared to the cable-based model. Similar comparisons can be made after inclusion of the crimp i n the shell-based and cable-based models. Figure 4 - 2 3 shows such a comparison, where the prediction of the decoupled shell model seems to be very close to that of the  -95-  Chapter  4-2D  Shell  Crossover  Model  cable-based model, as expected. The overall stiffness and energy absorption of the fabric increases u p o n the inclusion of the coupling between warp and weft unit-cell yarns i n the shell model. This implies that the cable-based model is ignoring an important feature of the fabric behaviour since it can only provide a decoupled warp/weft response, which can highly influence the performance of plain weave fabric panels.  4.4.3 The  Bias Extension Test Simulations complex shear deformation characteristics of fabrics have been the focus of  many studies. Fabrics are subjected to very large shear strains i n a variety of applications such as thermoforming/stamping as a part of the processing of fabric composites. Deformation of the fabric under shear leads to realignment of fibres/yarns w h i c h i n turn w o u l d alter the mechanical properties of the finished product. For this reason, and to avoid the occurrence of defects such as fabric wrinkling, etc., a thorough understanding of the fabric shear behaviour is essential i n these applications. Several experimental methods  are established to characterize the  shear  behaviour of fabrics. Amongst these methods, two have been widely studied and commonly used to determine the pre- and post-locking moduli and locking angle of fabrics: picture frame shear and bias extension tests. In picture frame shear testing methodology, a fabric is subjected to shearing deformation inside a square frame w i t h hinges at its corners (see Figure 4 - 24). A s the fabric is loaded along on of its diagonals, the frame allows for rotation of the sides while keeping them parallel to each other. The load applied to the fabric is then measured and shearing properties of the fabric is extracted from its picture  -96-  Chapter  4-2D  Shell  Crossover  Model  frame load-displacement data. In the bias extension tests, a rectangular sample is cut w i t h the warp and weft yarns aligned to make a 45° angle with the sides of the sample. The fabric sample is then loaded i n tension and its deformational behaviour as w e l l as the force-displacement response are studied. Based on the kinematics of the bias extension test [67], the fabric can be divided into three distinct shearing regions (zones A , B and C i n Figure 4 - 25) during the test. Assuming no yarn slippage occurs during the test, the fabric  remains  undeformed i n zone A due to the constraint applied by the clamping plate at the top and bottom of the sample, while it is i n a state of pure shear i n zone C. The deformation angle i n zone B is the average of the other two zones, A and C. This section investigates the success of the shear model implemented i n the 2D shell crossover model i n predicting the response of fabrics i n bias extension test simulations. Simulation of the bias extension test was performed  on a  rectangular S-728 panel w i t h a length of 143.5 m m and a w i d t h of 74 m m . i n order to obtain symmetric deformation profiles, the fabric was balanced i n these simulations by using identical warp and weft yarn properties. The failure of yarns was also deactivated to study the elastic shearing response of the shellbased fabric model i n a highly deformed case. Figure 4 - 2 6 shows the fringe plot of the warp/weft angle after stretching the fabric by 5 m m on each side. A s seen i n the figure, the three zones (A, B, and C) are distinctly visible and the warp/weft angle i n region B is equal to the average of regions A and C . As highlighted i n Section 4.3, the overall shearing resistance of the fabric stems from the rotation of the yarns, as well as the locking of the fabric i n a trellis  -97-  Chapter  4-2D  Shell  Crossover  Model  mechanism. Eliminating the trellis mechanism from the shell element shear response w o u l d transform them to behave similar to the cable-based pinjointed models where only the rotation of the yarns is considered. A s seen i n Figure 4 - 27, lack of shear locking i n bias extension test simulation results i n an unstable mode of deformation, where bands of elements at 45° have undergone extreme shearing and the rest of the fabric has stayed undeformed. This observation further emphasises the inability of the cable-based pin-jointed models i n capturing the true shear response of the fabrics i n simulating quasistatic and dynamic events.  4.4.4  Notes on the Numerical Efficiency of Shell UC  One of the most important factors directly affecting the functionality of a model is its numerical efficiency. In general, as a model gets more complex and more detailed, the computational resources needed to perform a certain set of required analysis amplify. This is clearly the case for all the modeling approaches presented i n this study. A comparison of the execution time taken to analyse several fabric models utilizing the three modeling approaches (cablebased fabric, 2D crossover shell and 3D solid and bar fabric models) is presented i n Table 4 - 3 . Intuitively, the simple cable-based model is the most efficient, while the 3D solid-bar model is extremely inefficient, to the point that it is almost impractical for panels with realistic dimensions. Shell-based fabric models are significantly less time-consuming i n comparison to the 3D solid-bar models of fabrics, but are more numerically intensive compared to the cablebased models. However, the inefficiency of the shell model compared to the  -98-  Chapter 4-2D  Shell Crossover Model  cable-based model is justifiable since they incorporate several details of the weave and yarns i n capturing the mechanical response of the fabrics. The efficiency of the shell model is highly dependent on the fabric properties and the user-specified input parameters. Since the calculation of yarn tensile forces i n the U M A T is an iterative process, the number of cycles needed to achieve convergence is usually dependent on the fabric geometry  and  mechanical properties, as well as on the applied displacements. Figure 4 - 2 8 shows the contours of contact force between the yarns as a function of the symmetric displacements applied to a single yarn crossover i n the S-720 fabric unit-cell, considering the failure of the yarns based on the instantaneous failure criterion. A s seen i n the figure, the yarns interaction is limited to the applied displacements that fall within the area identified as region 1. For displacement pairs outside the boundaries of this region no interaction between the yarns occurs, due to either separation of the crossing yarns i n region 2 or failure of one or both yarns i n region 3. Figure 4 - 2 9 shows the number of iterations needed to achieve convergence for the range of displacements considered. Overlaying the boundaries of region 1 on this plot reveals that many displacement pairs outside this region require a higher number of iterations to satisfy the convergence criteria, while the displacements that result i n significant resistance from the unit-cell (in region 1) usually achieve convergence after only a few iterations. A n efficiency model is thus developed i n the U M A T where the code is instructed to perform the Newton-Raphson iteration scheme only for displacements that fall within region 1. However, using the efficiency model requires some pre-processing to  -99-  Chapter 4-2D  Shell Crossover Model  identify the parameters that identify the boundaries of region 1. A simpler efficiency model can also be activated that w o u l d bypass the iterations for negative displacement pairs only (Figure 4 - 30). In the impact analysis of shell-based fabric models, the displacement pairs that elements experience are mostly located i n region 1 and its close vicinity. The most time-consuming elements i n terms of iteration convergence are the ones where the applied displacement pairs cause subtle contact between the two yarns, located around the curved edge of region 1. In these cases, the number of iterations needed to reach convergence is quite high, mainly due to the fact that the numerical round-off errors become significant compared to the numeric value of the  target functions. The tensile forces  associated  w i t h these  displacements are, however, very small and inconsequential i n the overall performance of the panel. A s a result, a large portion of the execution time is spent performing iterations to calculate forces that w o u l d not contribute much to the overall energy absorption. This emphasizes the importance of the development of more intelligent efficiency model that w o u l d bypass these time-consuming elements, and save a considerable amount of time and computing power throughout the analysis process.  4.5  SUMMARY  A shell element based on a continuum representation of a fabric unit-cell, comprised of a single crossover has been developed i n this chapter. The proposed model takes advantage of several simplifying  assumptions  to  formulate the biaxial response of a fabric crossover. The interaction of the  -100-  Chapter  4-2D  Shell  Crossover  Model  yarns, whose shapes are based on sine functions (identical to the observed fabric cross-sections), is considered under displacements that are symmetrically applied to the yarns ends. The biaxial response of the pinned yarn crossover was implemented i n the constitutive model developed i n the commercial finite element software, L S - D Y N A . The developed shell-based model has been verified against the 3-D model and proven to be significantly more efficient in terms of the required numerical resources. This efficiency was reflected i n the comparison of the execution times reported for a benchmark impact problem. The shell model developed here, aside from its simplifying assumptions, is very successful i n capturing the in-plane response of a yarn crossover. The model framework is such that it can be easily expanded to consider the bending of the yarns, or take advantage of a more realistic shear model. This expansion should be performed along w i t h relevant experimental effort that w o u l d calibrate the new properties needed by the model.  -101-  Chapter  4-2D  Shell  Crossover  Model  Table 4-1: Shell input parameters for the in-plane shear model of all six Kevlar 129 fabric types. Panel  S-720  S-726  S-727  S-728  S-731  S-745  G, (Pa)  9.67E6  1.07E7  1.15E7  8.66E6  1.17E7  1.14E7  G  9.67E9  1.07E10  1.15E10  8.66E9  1.17E10  1.14E10  Y\  0.24  0.15  0.19  0.12  0.35  0.26  Yi  0.49  0.40  0.40  0.70  0.40  0.50  2  (Pa)  Table 4-2: Input material properties of the steel RCC simulations.  projectile used in the impact  Property (units)  Value  Density (g/cm )  7.922  2  Elastic modulus  193.0  (GPa)  Poisson's ratio Yield stress  0.30  (MPa)  220.0  -102-  Chapter  4 - 2D Shell  Table 4-3:  Crossover  Model  C o m p a r i s o n o f execution times for simulation o f impact o f a n R C C p r o j e c t i l e o n s q u a r e p a n e l s o f K e v l a r ® 129 f a b r i c u s i n g c a b l e 2D s h e l l c r o s s o v e r a n d 3D s o l i d / b a r c r o s s o v e r f a b r i c  Panel  Dimension (mm)  Elements  Duration of impact  number  type  (us)  (pin-jointed),  models.  No. of  Execution  CPUs  time (sec)  S-720,1-ply  100.33  18,723  cable  250  1  1,243  S-720,1-ply  100.33  6,241  shell  250  1  1,932  S-720,1-ply  25.4  160,000  solid  42  3  157,424  S-7265, 4-ply  100.66  132,251  cable  400  4  5,297  S-7265, 4-ply  100.66  44,084  sheU  400  4  15,258  -103-  Chapter 4-2D  Shell Crossover Model  y X  Weft yarn Figure 4-1: Illustration of the biaxial crossover model of a single plain weave fabric crossover with linear yarns originally developed by Kawabata et al. [57].  t  z y  X  Figure 4-2: 3D finite element mesh of a single fabric crossover (left) and the equivalent mathematical representation of the yarns by their centrelines (right).  -104-  Chapter 4-2D Shell Crossover Model  yarns  Figure 4 - 3: A schematic model of a yarn in the fabric (left) in contact with the crossing yarn and a free body diagram showing the forces acting on it (right).  -105-  Chapter 4-2D  Shell Crossover Model  -106-  Chapter 4-2D  Shell Crossover Model  Figure 4-7: Compaction of the yarns as a result of the crossover shearing (before (left) and after (right) shearing). Note that the fabric thickness and configuration of yarn centreline are assumed to remain unchanged during shear deformation.  . G  G  \ -Y  2  "Yi  Yi  2  Y2  Figure 4-8: Secant shear modulus, G , as a function of the shear strain y, representing the trellis mechanism.  -107-  Chapter 4-2D  Shell Crossover Model  Initial configuration  Deformed configuration  Figure 4-9: Definition of the local element and material axes convention assumed for shell elements in LS-DYNA.  Figure 4 - 10: Definition of element stresses and their relationship with those individual yarns in an equivalent smeared shell approach ( i r and / in the figure represent the warp and weft yarns, respectively).  -108-  Chapter 4-2D  Shell Crossover Model  E l e m e n t strain i n c r e m e n t s  from L S - D Y N A  1  User Material M o d e l  R e a d yarn/fabric i n p u t data  W a r p and weft orientation  Determine d, & d  B i a x i a l extensional  Trellis M o d e l D e t e r m i n e shear r e s p o n s e  2  model  Newton-Raphson scheme  Stress c a l c u l a t i o n  E l e m e n t stresses R e t u r n to  LS-DYNA  Figure 4-11: Flowchart of the UMAT routine specifying the constitutive relationship of the smeared fabric material of the representative shell element (see Appendix B for more details).  -109-  Chapter 4 - 2D Shell Crossover Model  Figure 4 - 13: Deformed shape and contours of element deformation (void ratios) under transverse compression (left) and biaxial extension (right) loading simulations using the 3D crossover model (note that only one quarter of the yarn is shown for illustration purposes).  50 i  0.00  0.04  0.08  0.12  0.16  Transverse Displacement (mm) Figure 4 -14: Transverse force-displacement response predictions (along with the best fit) of the 3D crossover model of S-720 unit-cell obtained from simulations of transverse compression tests and biaxial tension tests, showing the difference in the compressive response of the yarns.  -110-  Chapter 4-2D  Shell Crossover Model  Figure 4 - 15: Biaxial response of S-726 unit-cell as a function of warp displacement (weft yarn fixed, d = 0), predicted by the 3D crossover model and the 2D shell model. 2  Figure 4 - 16: Variations of h and h with warp displacement, predicted by the shell model during the biaxial extension of an S-726 unit-cell illustrated in Figure 4 - 15. x  2  -Ill-  Chapter 4-2D  Shell Crossover Model  Figure 4 - 17: Energy stored in the unit-cell as a function of warp displacement, comparing the predictions of the shell crossover and 3D solid models during the biaxial extension of an S-726 unit-cell illustrated in Figure 4  15.  Figure 4 -18: Yarn cross-sections before (left) and after (right) deformation due to the interaction of the two yarns in biaxial extension, for the biaxial extension of an S-726 unit-cell illustrated in Figure 4 -15.  -112-  Chapter 4-2D  Shell Crossover Model  0.04  0.00  0.10  0.20  0.30  x  0.40  0.50  (mm)  Figure 4 - 19: Deformed profile of the warp yarn predicted by the 3D solid model compared with a sine curve (note that only half of the yarn is shown), for the biaxial extension of an S-726 unit-cell illustrated in Figure 4 - 15.  -0.08  1  y (mm) Figure 4 - 20: Deformed profile of the weft yarn predicted by the 3D solid model compared with a sine curve (note that only half of the yarn is shown), for the biaxial extension of an S-726 unit-cell illustrated in Figure 4 -15.  -113-  Chapter 4-2D  120  Shell Crossover Model  i  0  50  100  150  200  250  Time (us)  Figure 4-21: Velocity-time response of the projectile impacting a single-layer S-720 target, predicted using shell-based and cable-based model (yarn crimp not considered).  Figure 4 - 22: Comparison of a yarn reaction force in an impact simulation on a singlelayer S-720 target, predicted using shell-based and cable-based model (yarn crimp not considered).  -114-  Chapter 4-2D  Shell Crossover Model  120  |  90  o o  0> 60 Shell-based decoupled  .a,o  2  Q.  30  model response  Shell-based biaxial  model  response  200  100  300  Time (LIS) Figure 4 - 23: Velocity-time response of the projectile impacting a single-layer S-720 target, predicted using shell-based and cable-based model (yarn crimp considered).  Before Shearing  After Shearing  Figure 4-24: Schematics of the picture frame shear testing methodology.  -115-  Chapter 4-2D  Shell Crossover Model  shearing regions Figure 4 - 25: Schematic representation of undeformed fabric (top) in the bias extension test and the creation of shear zones after deformation (bottom) [67].  -116-  Chapter 4-2D  Shell Crossover Model  Figure 4 - 27: Unstable deformation mode of the S-728 fabric without considering the shear locking in bias extension simulation.  -0.04  -0.02  0 d1 ( m m )  0.02  0.04  Figure 4 - 28: Contours of contact force between the yarns, computed for biaxial extension of an S-720 fabric crossover (- 0.05 <d < 0.05 where i = 1,2). t  -117-  Chapter 4-2D  Shell Crossover Model  -0.04  -0.02  0  d1 (mm)  0.02  0.04  Figure 4 - 29: Contours of the number of iterations needed to gain convergence, when calculating the yarn tensile forces as a function of the symmetric displacements applied to the S-720 fabric unit-cell ( - 0.05 < d < 0.05 where / = 1,2 ). t  d1  (mm)  Figure 4-30: Contours of the number of iterations needed to gain convergence after bypassing iterations for negative displacement pairs developed for S-720 fabric unit-cell (- 0.05 <d < 0.05 where i = 1,2). t  -118-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  C H A P T E R 5-  VALIDATION DYNAMIC  OF  T H E  RESPONSE  STATIC  PREDICTIONS  A N D OF  FABRICS  5.1  INTRODUCTION  The meso-mechanical shell fabric model developed i n the previous chapter is utilized to study the behaviour of fabrics under two different loading scenarios. In the first part of this chapter, the performance of various Kevlar® 129 panels subjected to ballistic impact of R C C projectiles is investigated. For this purpose, data from instrumented ballistic tests on single and multi-ply fabric targets are used to validate the predictions of the numerical model. The second part of this chapter deals w i t h simulation of the in-plane biaxial extension tests obtained from the literature on balanced and non-balanced fabric panels woven from yarns of glass filaments. The aim of these simulations is to highlight the capabilities of the shell crossover model as w e l l as to probe its limitations i n capturing the behaviour of fabric panels w i t h various structures. The findings  -119-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  of each of these two sections are concluded and summarized at the end of the chapter.  5.2  BALLISTIC IMPACT RESPONSE OF FABRIC PANELS  The meso-mechanical model of fabrics developed and presented  i n the  previous chapter is used here to study the ballistic behaviour of Kevlar® 129 panels. Simulations of ballistic impact experiments on six fabrics panels presented i n Section 2.3 (performed by Cepus [42]) are carried out and the energy absorption capacities of the targets are studied. The predictions of the numerical model are presented separately for single and multi-ply target systems, followed b y the discussion of the numerical observations.  5.2.1  Ballistic Impact Simulation: 1-ply Targets  The first step i n examining the ability of the model to capture the ballistic response of fabric panels is to simulate ballistic impact events on 1-ply target systems. Due to the lack of inter-ply interaction i n single-ply targets, direct conclusions can be d r a w n on the success of the representative shell elements on mimicking the biaxial extensional response of the fabric during impact-induced deformations. This section covers the study of ballistic impact on six different Kevlar® 129 fabric weaves presented earlier i n Chapter 2. The input values chosen for these test simulations are presented i n Table 5 - 1 . The geometrical symmetry of the target and projectile is used to only consider a quarter of the panel and the projectile i n the finite element analysis through application of proper boundary conditions. The geometrical dimensions of each  -120-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  shell element are set equal to that of a single yarn crossover of the fabric it is meant to represent. The density of the shells is also adjusted to match the total weight of the crossing yarns within its boundaries, resulting i n an areal density closely matching the laboratory measurements (see Figure 5 - 1 ) . The contact between the projectile and the target is based on a frictionless surface-to-surface contact model i n L S - D Y N A [54] activated between the shell elements of the target and solid elements of the projectile. R C C (Right Circular Cylinder) projectiles were meshed into a geometry that closely resembled the shape of the actual steel projectiles used i n the experiments [51]. Most input values for the shell elements are obtained directly from the laboratory measurements, except for the constants of the transverse compression model of the yarns, which are obtained from the 3D unit-cell model. Initial examination of the six Kevlar® 129 fabrics reveals critical differences among them. Figure 5 - 2 shows the value of the total crimp (obtained by the addition of the warp and weft crimp) for the fabrics studied, i n order to gain some relative understanding of the level of yarn undulation amongst them. Since almost all panels studied here have identical yarns and mostly equal count numbers i n the warp and weft directions, the level of imbalance of each panel w o u l d be determined by the difference of yarn undulation between the two directions. Plotting the ratio of warp and weft crimp as shown i n Figure 5 3, it can be seen that S-720 and S-728 panels have the lowest warp to weft crimp ratios, while S-731 and S-726 panels possess a large ratio showing a significant level of imbalance i n the warp and weft directions. The investigation of the impact response of 1-ply panels is thus further divided into two main  -121-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  categories: relatively balanced fabrics (S-720 & S-728) and highly imbalanced panels (S-726, 727, 731, & 745.) The S-720 and S-728 panels have a combination of l o w total crimp and warp/weft ratio amongst the fabrics studied. This implies that the yarns i n the warp and weft directions have more or less the same geometries and given that they have equal linear density (denier), intuitively, they are expected to almost equally provide the load-bearing capacity of the fabric. The velocity-time response prediction of the R C C projectile impacting a 1-ply S-720 fabric panel at a non-perforating speed of 92.5 m/s and a perforating velocity of 155 m/s are shown in Figure 5 - 4 and Figure 5 - 5 , respectively. While the numerical model seems to be successful i n capturing the impact response of S-720 fabric i n the initial stages of impact, it seems to over-estimate the energy absorption capacity of the fabric i n the latter portion of the event (see Figure 5 - 4). It was also observed that at higher strike velocity leading to perforation (e.g. Figure 5 - 5), the failure prediction of the numerical model is somehow delayed compared to the experimental data. The sources of such discrepancies are discussed later i n this chapter (see Section 5.2.3). Similar conclusions can be drawn for S-728 fabric panel. Figure 5 - 6 and Figure 5 - 7 , respectively, show a non-perforating and a perforating event at impact velocities of 76 m/s and 170 m/s. The deformation profiles (seen i n Figure 5 - 8) of the fabrics appear to be symmetric, translating into a balanced distribution of all energy components between the warp and weft yarns. Further examination of the S-720 and S-728 panels during impact reveals that there are slight differences between the warp  -122-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  and weft responses, evident from the propagation of the tensile wave along the warp and weft yarns and the transverse deformation of the panels. Figure 5 - 9 shows the time-history of the reaction force acting at the boundary of a yarn that goes through the impact point. A s seen i n this figure, there are minor differences between the reaction forces and a delay i n the arrival of the strain wave (indicated by the sudden increase i n reaction force) travelling along the warp yarn compared to that of the weft yarn. However, this difference is not significant enough to cause major disparities between the two main material directions and the S-720 and S-728 fabrics can be considered to be relatively balanced. The other four fabric types have significantly higher crimp values i n the warp direction compared to that of weft. Sample model predictions of the projectile velocity-time response for these panels are shown i n Figure 5 - 1 0 to Figure 5 17. The agreement between the numerical predictions and experimental data seems to be best i n the initial part of the response, w i t h the numerical model slightly over-estimating the energy absorption capacity of the panels. The existing warp to weft crimp disparity i n these fabrics leads to a nonsymmetric dynamic behaviour that is readily evident from the  transverse  deformation profile of the panels (shown for S-731 i n Figure 5 - 18). This can also be seen i n the reaction force time-histories plotted i n Figure 5 - 1 9 for an S726 warp and a weft yarn going through the impact point. While the reaction in the weft yarn (with 0.50% undulation) rises quickly at about 30 us, the warp yarn (at 3.44% crimp) does not see any significant increase until after the 100 us mark. This clearly indicates the dissimilar tension development i n the warp  -123-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  and weft directions, and as a result, the asymmetric energy distribution i n the target. Post-mortem examination of the S-726 panel (Figure 5 - 20) revealed significant yarn slippage at the boundaries holding the weft (low crimp direction) yarns, w h i c h is supported by the numerical observation that higher reaction force is needed to contain the weft yarns. A t higher velocities, the weft yarns w o u l d fail sooner than the warp (high crimp) yarns due to their higher tension, after w h i c h point the warp yarns w o u l d resist the motion of the projectile, slipping and sliding i n the fabric as they are pulled by the projectile (hence a wider de-crimped zone seen i n Figure 5 - 20). The asymmetry inherent i n the non-balanced fabrics translates into an efficiency reduction of the fabric system if only the energy absorption is concerned i n the absence of any yarn failure. This w i l l be further discussed i n Section 5.2.4.2.  5.2.2  Ballistic Impact Simulation: M u l t i - p l y Targets  Personal armour systems, i n practice, are usually made from multiple layers of high performance fabrics to defeat the specific types of threats they are exposed to. A s a result, a major part of the fabric target energy absorption stems from the interaction of individual layers with each other i n the pack. To study such interaction existing i n a real vest, the shell crossover model is used to simulate the ballistic impact experiments on multi-ply Kevlar® 129 targets. This section reviews the comparison of the numerical predictions versus the instrumented test data.  -124-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  One  of the  challenges  of modelling multi-layer fabric  targets is  the  determination of the gap between the adjacent layers. In order to obtain an approximation of the distance between the mid-surfaces of successive layers, transverse compression test data on multi-ply fabrics are used. Figure 5 - 2 1 shows a schematic of the mechanical model considered i n this study to estimate the initial gap between the layers. For a 1-ply model, the interaction of the material between the fabric mid-plane and the surface of the steel loading platen is represented by a spring w i t h a non-linear stiffness of K . fm  A d d i t i o n of  one more layer of fabric to the system introduces a supplementary inter-fabric spring w i t h a stiffness of K ,  which represents the interaction of adjacent  ff  layers during compression. A s a result, one can obtain the pure inter-fabric response of the panels under transverse compression b y extracting a 1-ply force-displacement response from that of the multi-ply system. Let us assume that under an arbitrary compressive force F applied to an N -ply fabric pack, c  the total compression turns out to be "d . Considering the system of springs i n c  Figure 5 - 21, the total compression can be divided into two main components: "d =2d +(N-\)d c  where d  fm  fm  ff  (5-1)  is the fabric-metal interaction component of deformation and d  ff  is  the fabric-fabric component. For the case of a 1-ply system, N = \ and we have: d =2d  (5-2)  1  c  fm  To obtain the fabric-fabric component of the compression, we subtract the 1-ply response from the N -ply, as follows: "d - d =(N-\)d x  c  c  ff  -125-  (5-3)  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  A n d thus we can conclude that  "d -d d =—  (5-4)  ff  This process is illustrated i n Figure 5 - 22, where the compression data for a 4ply S-728 panel is used to determine the inter-fabric compression response. The value of gap considered between adjacent layers for all multi-ply targets of each fabric is chosen to be equal to its inter-ply deformation corresponding to the maximum load applied i n the 4-ply transverse compression experiments. Table 5 - 2 provides the calculated inter-ply gap values for the three panels used  i n the  multi-ply ballistic impact experiments.  Subsequent  to  the  implementation of the inter-ply gap, the contact between the adjacent layers i n L S - D Y N A is activated through the  AUTOMATIC_SURFACE_TO_SURFACE  option. The contact algorithm used for this purpose is based on the penalty method. This method is based on placing virtual springs normal to the interface of the contacting surfaces, and proved to be successful i n maintaining the contact between the layers of the fabric pack, as well as that of the projectile and the target. The configuration of the layers and the projectile is shown i n Figure 5 - 2 3 for a quarter of the problem due to symmetry. The multi-ply experimental data available to this investigation were limited to only three fabrics types: S-726, S-728, and S-731. In the analysis performed on the multi-layer targets, the apparent stiffness of the fabric panels are correlated to their rate of energy absorption and to projectile deceleration, i.e. a more rapid projectile velocity drop is associated w i t h higher target stiffness. Figure 5 - 24 to Figure 5 - 2 6 show the velocity-time response of the projectile impacting  -126-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  a 4-ply target of S-726, S-728, and S-731 fabrics, respectively. It can thus be seen that the  predictions seem  to be  slightly stiffer  than  the  experimental  measurements, as was the case for 1-ply targets. Similarly, velocity-time plots of the R C C projectile impacting 8-ply targets are shown i n Figure 5 - 27 to Figure 5 - 29, and 16-ply targets i n Figure 5 - 30 to Figure 5 - 32, respectively for S-726, S-728, and S-731 fabrics. It seems that the over-estimation of the target stiffness becomes more pronounced as the number of layers increase. This can be due to under-prediction of the interlayer gap obtained from the method above. Further investigation of the deformed shapes and energy absorption characteristics of the multi-layer panels seem to result i n the same conclusions discussed for the case of 1-ply fabric targets. A m o n g the three panels simulated i n multi-ply configuration, the S-728 panel is the most balanced. In the case of non-balanced fabrics (S-726 and S-731), the deformation cone appeared to be highly non-symmetric, as was the case i n the 1-ply targets. The failure of the yarns i n multi-ply fabrics, however, led to  premature  perforation of the targets i n many cases. Examination of the strains and stresses of the shell elements of the multi-layer models revealed the presence of a significant amount of numerical noise i n the tensile response of yarns. The numerical oscillations observed appeared more pronounced i n comparison to the 1-ply models, w h i c h can stem from the interaction of the fabric layers i n the simulations. The excessive noise resulting from the contact algorithms caused the premature failure of the yarns and hence the early perforation of the targets. This problem should be further  -127-  looked into and remedied by  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  implementation of more advanced failure criteria, or through averaging or damping out the oscillations.  5.2.3  D i s c u s s i o n o f the B a l l i s t i c I m p a c t S i m u l a t i o n R e s u l t s  Some of the fabric panels simulated using shell elements appeared to be stiffer than what  was  measured  i n the  experiments.  Furthermore,  while the  perforation of the multi-ply targets occurred prematurely due to the presence of numerical noise, the failure prediction of the 1-ply models seemed to occur consistently at a later instance compared to the experiments. The sources of such discrepancies can be linked to several major disparities between the numerical model and the actual fabrics, as listed below: • The input parameters corresponding to the mechanical properties of the yarns are taken from the ones reported by the manufacturer for virgin yarns before the weaving process. It is k n o w n that the weaving of fabrics causes some damage to the yarn i n the form of fibre breakage and fibrillation, leading to deteriorated yarn properties [68, 69] i n the form of reduced stiffness and tensile strength. The reduction i n the yarn stiffness can lead to a softer response of the panel and along w i t h the diminished yarn strength, lead to earlier perforation by the projectile i n the case of 1-ply targets. To be able to consider the effects of weaving on the properties of the fabrics, laboratory experiments need to be performed on the yarns extracted from the woven panels. • The compliance of the experimental fixture and the frequent slippage of the fabric at the boundaries are not considered i n the impact simulations  -128-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  presented i n this study. It is k n o w n that the slippage of the fabric at the boundary plays a major role i n the energy absorption of the targets [42]. Even small slippages at the fixed edges of the target can soften the response of the target and should be considered i n a separate study. • Due to the assumption of symmetry i n the extensional responses of the fabric unit-cells, the slippage of the yarns at the crossover level is not allowed. Post-mortem examination of the fabric targets suggests that yarn slippage occurs w i t h i n a zone including the yarns that go through the impact point.  Ignoring the  yarn slippage  can  introduce  additional  constraints on the motion of the yarns and increase the apparent stiffness of the fabric model. • The numerical model, however, proved to be successful i n capturing the biaxial response of the fabrics and approximating their energy absorption characteristics. The availability of this meso-mechanical fabric model allows for a close internal examination of the targets during the impact event and identification of a fabric's critical material and geometrical parameters.  5.2.4  Overall C o m p a r i s o n of Various Kevlar® 129 Fabrics  Aside from the success of the sine-based yarn crossover meso-mechanical model developed i n this study i n capturing the ballistic response of fabric panels, one needs to be able to compare the performance of fabrics w i t h various weave parameters. In order to do so, the unit-cell responses and ballistic performances of all six Kevlar® 129 fabrics that are the focus of this study are investigated.  -129-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  5.2.4.1 Unit-cell mechanical response Studying the mechanical behaviour of Kevlar® 129 panels w i t h  different  geometrical and weave properties under biaxial deformation can underscore some of the strengths and weaknesses of each fabric system. The basic characteristics of the panels are thus identified by comparing the warp and weft  tension  surfaces  as  their  unit-cell is  subjected  to  extensional  displacements. These surfaces, shown in Figure 5 - 3 3 for S-720 fabric system, implicitly incorporate many aspects of the fabrics such as yarn crimp i n the two directions, and transverse interaction of the crossing yarns. A more comprehensive term w o u l d be the strain energy stored i n the fabric as a function of the applied displacements. This energy is calculated by measuring the external work done to stretch the fabric unit-cell to a k n o w n pair of warp and weft displacements. Such energy surface can be plotted against the warp and weft strains as a three-dimensional surface or as a contour plot (see Figure 5 - 34). It should be noted that the energy calculation is only shown for the nofailure zone, where the strains i n the warp and weft yarns fall below the failure strain (of 3%, considered i n these figures). The value of absorbed energy is set to zero outside this zone for illustration purposes. Comparison of these energy contour plots for all six Kevlar® 129 fabric panels is shown i n Figure 5 - 3 5 and Figure 5 - 36. A s can be seen, the size of the no-failure zone is different amongst the fabrics, indicating the varying extension capacity before yarn failure occurs. Since the yarn failure strain is the same i n all panels, it is only intuitive that the size of the no-failure zone be directly correlated to the level of crimp i n the fabric. Furthermore, the level of energy stored i n the system also varies amongst the six fabric systems, appearing to be a direct function of the yarn  -130-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  linear density. Based on these contour plots, one can conclude that a panel with higher energy absorption capacity w o u l d be superior to others. This w i l l be further discussed after the performance of these fabrics is assessed i n ballistic impact simulations.  5.2.4.2 Iso-velocity fabric impact simulations In order to further investigate the performances of the fabrics studied, nonperforating impact simulations were performed on identical panels of various fabric systems at a strike velocity of 100 m/s. Investigating the impact behaviour of fabric panels under iso-velocity conditions w o u l d reveal the differences between their energy absorption characteristics, differentiating the superior fabric systems from the others. Figure 5 - 3 7 shows the projectile's velocity time-history impacting the six different Kevlar® 129 panels at a strike velocity of 100 m/s. From the reduction i n the projectile velocity, the energy absorbed by the target can be determined. However, one measure of the efficiency of the target is the energy that it can absorb at a certain projectile displacement. Figure 5 - 3 8 shows the plot of total energy absorbed by the fabric against the projectile displacement. It is evident from this figure that S-728 is capable of stopping the projectile w i t h the smallest back-face deformation, while S-731 is the most compliant fabric type i n defeating the projectile. Looking further into the energy absorbed by the target and its components, it appears (Figure 5 - 39) that i n the panels w i t h high total crimp value (S-745 and S-731) the kinetic to internal energy component ratio is much larger than other panels. Panels with lower crimp value more or less have equal distribution of internal and kinetic  -131-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  energy. A more in-depth study needs to be performed, however, to highlight more differences amongst the six fabric types. Before performing an in-depth study on woven fabrics, it is essential to verify some of the predictions of the numerical code. For this matter, single-yarn impact simulations were carried out and predictions of the numerical code were compared w i t h the well-established analytical solutions. It is k n o w n that when a yarn is transversely impacted, a longitudinal strain wave is generated that travels at a speed calculated as:  (5-5)  where C is the speed of sound i n the material, E is the yarn elastic modulus, and p is the yarn material density. Material swept by this strain wave moves towards the impact point at a constant speed. In the wake of this strain wave, a transverse wave front develops after which the motion of the material changes from in-plane to transverse at a speed equal to that of the projectile. This behaviour can clearly be seen i n the single yarn impact simulation performed using the shell elements. The longitudinal wave speed calculated for an impact on an S-731 yarn is i n perfect agreement w i t h the closed form solution. The propagation of longitudinal wave and the development of the transverse cone are also shown i n Figure 5 - 4 0 and Figure 5 - 4 1 and their wave front speed is calculated. It can be seen that the transverse deformation cone has linear sides w i t h a slope that show slight reduction i n time. For the case of woven fabrics, the longitudinal wave speed decreases due to the presence of  -132-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  the crossing yarns. Roylance et al. [70] suggests that the strain wave speed should be reduced by a factor of V 2 , as shown below:  (5-6)  After adjusting the mass of the shell elements to include the crossing yarns, a reduction i n the longitudinal wave speed was observed w h i c h matched the value obtained from Roylance's suggested equation, as shown i n Figure 5 - 4 2 . Yarns are rarely straight i n a woven fabric. For the next step, transverse impact on a single crimped yarn is studied. In the absence of any analytical solution for this case, the effect of yarn undulation is investigated using the numerical model. Presence of slack i n the yarn due to the undulation causes a reduction i n the longitudinal strain wave speed, as every segment of the yarn needs to straighten out before it is able to carry the tension on to the next segment (Figure 5 - 43). The transverse wave speed (U) also gets inversely affected i n the presence of yarn undulation. Furthermore, it is observed that the transverse deformation cone has a steeper side-slope i n the crimped single-yarn model compared to the no-crimp case (Figure 5 - 44). The fabric deformational response shows many similarities to the single-yarn impact cases discussed above. The longitudinal and transverse waves are generated as the projectile strikes the target, with a propagation speed that can be calculated from the motion of the nodes along the yarns. The longitudinal wave front is observed to have a distinct sharp front that flattens out as it propagates along a yarn due to wave reflection, refraction and transmission at the crossover points. This is shown for the S-720 panel i n the plot of warp and  -133-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  weft tension along the central yarn going through the impact point i n Figure 5 45 and Figure 5 - 46. The transverse wave front speed is also observed to be affected by the crimp i n the system. This is the main reason behind the nonsymmetric shape of the deformation cone in the unbalanced panels such as S731. A s seen i n Figure 5 - 47, there is a significant difference i n the transverse wave speed between the warp and the weft direction of S-731. The warp direction w i t h higher crimp value clearly has a sharp side angle (similar to the case of single crimped yarn impact discussed earlier) and a lower transverse wave speed, while the less undulated weft direction behaves similar to the nocrimp single yarn impact response. The reduction of the wave speeds seems to be an inverse function of the yarn crimp. A s a matter of fact, a power relation seems to exist between the longitudinal wave speed and the yarns crimp, as shown i n Figure 5 - 4 8 for the C values measured for the warp and weft yarns of all panels impacted at 100 m/s. A similar trend is found for the transverse wave speed, U, as shown in Figure 5 - 4 9 . Some studies suggest that the energy absorption characteristics of a fabric panel seem to be influenced by the longitudinal wave speed. Comparing the ratio of kinetic to internal energy components i n the six Kevlar® 129 fabrics presented i n Figure 5 - 3 9 and the average (warp and weft) longitudinal wave speed plotted i n Figure 5 - 50 it can be seen that as the average longitudinal wave speed i n a fabric increases, there is a more tendency for the energy components (kinetic and strain energies) to equally contribute to the overall energy absorption of the panels.  -134-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  Perforation of the fabric panels as a result of yarn failure is of a significant importance i n the study of their ballistic performance. A s discussed earlier i n Section 5.2.3, the  numerical model generally  overestimates  the  energy  absorption capacity of the panels, resulting i n a higher critical velocity than the experimental results, as seen i n Figure 5 - 5 1 . B y definition, critical velocity is the projectile speed above w h i c h the target is perforated and below which the projectile is arrested by the fabric. Due to the deterministic nature of the numerical model, this critical velocity is always a constant value for a specific fabric system. O n the other hand, due to the influence of many factors on the perforation of the projectile such as the material flaws i n the fabric, the experimental critical velocity is not easy to determine. In this study, the experimental critical velocity is calculated by taking the average of the highest non-perforating and lowest perforating strike velocities recorded for each fabric. The discrepancy between the predicted critical velocity and the measured value seems to be influenced by the tightness of the fabric. A s shown i n Figure 5 - 53, the difference between the predicted and measured critical velocity diminishes as the fabric average count number increases. This is mainly attributed to the fact that tighter weaves reduce the occurrence possibility of deformation mechanisms ignored i n the shell model, mainly yarn slippage and nonsymmetric loading of the unit cells. Post-mortem examination of different perforated panels reveal the difference in the projectile penetration mechanism, as illustrated i n Figure 5 - 52, i.e. projectile impacting the panels w i t h looser weave w o u l d p u l l the yarns and wedge through them, while the yarns w o u l d rupture i n the tighter weaves as the projectile perforates the target.  -135-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  Although the failure prediction of the model is not i n complete agreement w i t h the experimental observation (Figure 5 - 51), they can still be used to investigate the trends amongst the panels. Using the energy surfaces previously shown in Figure 5 - 3 5 and Figure 5 - 36, one can determine the remaining energy absorption capacity of a panel before any yarn failure occurs. For the case of 100 m/s iso-velocity impact simulations, the energy absorbed by the elements surrounding the projectile at the most critical time (closest to failure) are overlaid on the energy contour plots i n Figure 5 - 54. It can be concluded from these plots that while a panel like S-745 still has a significant amount of energy absorption capability, S-731 is very close to experience failure of its weft yarn (maximum d ). Such analysis also shows the importance of having balanced 2  fabrics, since the capacity of both warp and weft directions are utilized optimally before yarn failure occurs. A s mentioned before, the lack of symmetry i n the imbalance panels w o u l d result i n a less than optimum ballistic performance. However, i n the multilayer packs, this asymmetry can be reduced by increasing the interaction of the consecutive layers. Since the asymmetric deformation cone shapes of every layer tends to nest i n the similar profile of its successive layers, one method of increasing the inter-ply interaction is to prevent this nesting phenomenon. Alternating the warp and weft alignments of neighbouring fabric layers within the multi-ply pack w o u l d significantly increase the contact between adjacent layers, and w o u l d result into a deformation profile that is more symmetric. This hypothesis is examined for the case of a 4-layer S-726 panel, with the deformed shapes illustrated i n Figure 5 - 5 5 for the similar and alternating  -136-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  packing systems. The deformed shape of the individual layers, specifically those located i n the middle of the pack, take a more symmetric shape, as shown in Figure 5 - 5 6 for the deformation of the third layer of fabric away from the strike face of the S-726 4-ply target. The target w i t h alternating packing sequence also exhibits a higher ballistic efficiency, evident from its improved energy absorption capacity (see the velocity-time response of the projectile i n Figure 5 - 57).  5.3  BIAXIAL FABRIC EXTENSION SIMULATIONS  In order to examine the capabilities and limitations of the shell crossover model, an attempt was made to simulate biaxial loading experiments on plain weave fabrics. The experimental data used for this validation exercise is taken from the article of Buet-Gautier and Boisse [37], w h o presented the results of several biaxial extension tests on woven composite reinforcements. These results included tests on balanced and non-balanced plain weave glass fabric and twill weave carbon fabric. For the purposes of this study, only the test results on plain weave fabrics are investigated. The biaxial tensile testing device used by Buet-Gautier and Boisse [37] for these experiments is capable of subjecting the fabric to simultaneous prescribed displacements i n the warp and weft directions. Square cross-shaped fabric patches w i t h dimensions shown i n Figure 5 - 5 8 were tested and the results were reported i n terms of load measured per yarn versus fabric strain. The geometric and mechanical properties of a balanced plain weave glass fabric studied i n the biaxial experiments for different strain ratios (k) is provided i n  -137-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  Table 5 - 3 . Since this specific fabric is balanced i n all aspects, the warp and weft responses are identical and only one set of the results are reported. Table 5 - 3 also provides the properties of a non-balance fabric made from significantly different glass yarns i n the warp and weft directions subjected to biaxial loading. Buet-Gautier and Boisse [37] proposed a meso-mechanical model based on the linear model by Kawabata [55] and were successful i n predicting the response of the fabrics investigated compared to the experimental data. However, they adopted a transverse compression model that is a function of the yarns  axial strain, whose constants were back-calculated from  the  experimental data for equal warp and weft loading (k = 1). The sine-based meso-mechanical model developed here is used to capture the biaxial response of the fabrics tested by Buet-Gautier and Boisse [37]. To obtain the parameters of the transverse compression model, an inverse calculation approach was used, similar to that of Buet-Gautier and Boisse [37]. The tensile response of the glass yarns were considered to be linear, despite their apparent initial nonlinearities. The bending of yarns was also ignored i n the calculations. The input values calculated from the panel properties (such as yarn heights, h ) i  and fitted (in the case of transverse compression parameters,  a and b) are  presented i n Table 5 - 4 . Figure 5 - 5 9 shows the single yarn tensile response prediction, as w e l l as that of the fabric where the perpendicular direction is free. Also, the fabric response prediction for iso-displacement loading case (k = 1) fitted to the experimental data is depicted i n the same figure. W i t h all the material and mechanical parameters determined, the fabric's biaxial  -138-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  extensional response is obtained using the shell element. A s seen i n Figure 5 60, the predictions are i n good agreement w i t h the experimental data. Similar analysis is performed on a non-balanced glass fabric, where the properties of the warp and weft yarns are significantly different. Figure 5 - 6 1 and Figure 5 - 6 2 show the tensile response of the warp and weft extensional responses of single yarns and that of the non-balance glass fabric w i t h free boundary conditions i n the perpendicular direction. Similar to other cases, the transverse compression model is calibrated w i t h the biaxial data for k = 1. The predictions for other warp to weft strain ratios (& = 0 . 5 & 2 ) show some discrepancies. A s seen i n Figure 5 - 63, while the response prediction of the model for the warp direction for the ratio k = 2 matches that of the experiment, the prediction for k = 0.5 loading case is not consistent w i t h the test data. A similar trend is also observed for the fabric tension i n the weft direction (Figure 5 - 64), showing some discrepancies between the model predictions and the experimental data. Probable sources of such disagreement are listed below: • The sine-based formulation of the current meso-mechanical model is based on the assumption that the crossing yarns are i n contact w i t h each other over a significant portion of their lengths, i.e. little gap exists between parallel yarns. A s a result, the sine-based model is usually w e l l suited for tight fabrics w i t h yarns of high linear density and count numbers (as is the case for the Kevlar® 129 fabrics investigated i n the ballistic experiments). The non-balanced glass fabric studied by Buet-Gautier and Boisse [37] do not match the above description, pushing the sine-based model to the limits of its validity. For these fabrics, the Kawabata [55] straight-yarn model is  -139-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  better suited, since the contact is concentrated i n the middle area of the unitcell and the yarns more or less maintain a linear profile. • The transverse compression force-displacement model implemented by Buet-Gautier and Boisse [37] et al. incorporates the axial strain of the yarns in determining the value of their transverse stiffness. This dependency is related to the change i n the contact area of the crossing yarns and the ability of the fibres/filaments to move with respect to each other as they are transversely compressed. The transverse compression model developed i n this thesis for the crossing yarns ignores any such dependency. • The force-strain response of the yarns considered is linear i n the current model, while the extension tests on single glass yarns show nonlinearities at low strain values. U s i n g the linear tensile response for the yarns leads to unrealistic values for transverse model parameters after fitting the model prediction to the experimental data for  k = 1. Poor correlations with  experiments thus result when the predictions for other loading ratios are carried out. • The determination of the beginning of the curve i n the experimental data is not an easy task. For these experiments, Buet-Gautier and Boisse [37] chose to use a kinematic criterion where the beginning of the curve is marked at the instant where all the yarns (sagging initially under self-weight) arrive at the mid-plane of the fabric. Slight shifting of the curve to the left and right can alter the error i n the numerical predictions, specifically for the k = 0.5 case of Figure 5 - 6 3 and Figure 5 - 64.  -140-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  5.4  SUMMARY  The efficient shell element developed i n the previous chapter was used to perform ballistic impact simulations on single and multi-ply fabric targets. The results were compared w i t h the data from instrumented ballistic impact experiments and found to be i n good agreement i n capturing the general ballistic response of the fabrics. It was also found that the numerical predictions are always slightly stiffer that the tested panel. This discrepancy was attributed to such factors as using mechanical properties of the virgin yarns, the choice of transverse  compression  model,  and  ignoring  the  compliance  of  the  experimental fixture and sliding at the boundaries. The six Kevlar® 129 fabric types were also compared w i t h each other at the unit-cell level, as w e l l as the full fabric scale. It was found that the energy absorption of the panels are highly affected by the yarn undulation and that panels w i t h balanced warp and weft properties exhibit a more optimized response. It was also indicated that alternating the warp and weft alignment i n a multi-layer pack made from unbalanced fabric w o u l d improve its energy absorption  characteristics.  However, the  shell elements  proved  to  be  unsuccessful i n capturing the failure of the yarns and projectile perforation i n multi-layer targets, as the excessive oscillation caused by the contact algorithms lead to premature failure of the yarns. The possibility of incorporating a failure model other than instantaneous erosion strain ought to be investigated. The biaxial response of fabrics under quasi-static extensional loading was studied using the sine-based shell crossover model. The experimental data for  -141-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  this analysis was taken from the article by Buet-Gautier and Boisse [37], w h o applied biaxial displacements i n the warp and weft directions (at a variety of different ratios) and measured the load required to do so. While the shell crossover model was successful i n capturing the response of a balanced glass fabric, it d i d not appear to be as successful i n the case of an unbalanced fabric. Further investigation of the results and the panels tested showed that the specific properties of the weave and the difference between the warp and weft yarns used  i n the unbalanced  glass fabric w o u l d  invalidate the basic  assumptions of the shell crossover model and hence lead to erroneous results.  -142-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  T a b l e 5-1:  I n p u t v a l u e s u s e d to set u p s h e l l - b a s e d f a b r i c m o d e l s to s i m u l a t e instrumented ballistic impact experiments.  S-720  Properties  S-726  S-727  S-728  S-731  S-745  96.0  Elastic modulus (GPa) Shell thickness (mm)  0.43  0.31  0.34  0.44  0.38  0.67  F-d parameter a (mm)  0.045  0.05  0.0425  0.0044  0.046  0.082  F-d parameter b (mm.uN )  5.8  5.0  5.5  7.8  2.5  9.0  Yarn (denier)  1420  840  1000  1500  1000  3000  Warp count (tpi*)  20  26  26  17  31  17  Weft count (tpi*)  20  27  26  17  31  17  Warp crimp (%)  2.18  3.44  3.38  1.76  8.16  5.27  Weft crimp (%)  1.39  0.50  0.74  0.94  0.56  1.38  HOI (mm)  0.12  0.11  0.12  0.13  0.15  0.22  H02 (mm)  0.10  0.04  0.05  0.09  0.04  0.11  Failure strain  0.03  0.03  0.03  0.03  0.03  0.03  255.2  201.4  238.1  231.1  282.3  473.7  1/3  Areal density (g/m ) 2  (*tpi: t h r e a d s p e r i n c h )  -143-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics Table 5-2: Interlayer gap determined from the transverse compression test data performed on 4-ply fabric samples.  Fabric Panel  Interlayer gap (mm)  S-726  0.10  S-728  0.12  S-731  0.14  Table 5-3: Geometric and mechanical properties of the plain weave glass fabric panels studied by Buet-Gautier and Boisse [37]. Balanced glass weave  Non-balance glass weave  warp  0.22  0.22  weft  0.22  0.16  warp  0.40  0.42  weft  0.40  0.65  warp  350  850  weft  350  80  warp  38000  75000  weft  38000  8900  Panel  Count (tlireads/rrirn)  C r i m p (%)  Tenacity (N)  Stiffness* (N)  (^Stiffness is defined as the slope of the tension-strain curve of the yarns)  -144-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  Table 5-4: Shell element input parameters used to carry on simulation of biaxial loading of fabric panels studied by Buet-Gautier and Boisse [37]. Balanced glass weave  Non-balance glass weave  warp  0.18  0.26  weft  0.18  0.23  warp  4.55  6.25  weft  4.55  4.55  warp  0.075  0.21  weft  3.50  16.0  Panel  h (mm)  Shell dimension (mm) a (mm) b (rnm.uN" ) 3  -145-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  • Measured Ad • Calculated Ad  720  726  727  731  728  745  Fabric Type  Figure 5-1: Areal density comparison of the six Kevlar 129 panels investigated. Further information on these panels is provided in Chapter 3.  10  —  720  726  [  727  _____  728  731  745  Fabric Type  Figure 5-2: Crimp comparison of the six Kevlar® 129 panels investigated. Further information on these panels is provided in Chapter 3.  -146-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  720  726  727  728  731  745  Fabric Type  Figure 5-3: Crimp ratio (warp/weft) comparison of the six Kevlar 129 panels investigated. Further information on these panels is provided in Chapter 3.  -50  50  150  250  350  450  T i m e (us)  Figure 5-4: Velocity-time response of an RCC projectile in a non-perforating impact simulation of a 1-layer S-720 target struck at 92.5 m/s.  -147-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  200 ELVS data S-720-01p-09 (0  E u o o > O CL  180  Figure 5-5: Velocity-time response of an RCC projectile in a perforating impact simulation of a 1-layer S-720 target struck at 155 m/s.  80 i  -20  J  Time (us)  Figure 5-6: Velocity-time response of an RCC projectile in a non-perforating impact simulation of a 1-layer S-728 target struck at 76 m/s.  -148-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  200  w  E o o >  o  '•5 O  0)  'o  v.  Q.  -50  50  100  150  200  250  300  T i m e (|as)  Figure 5-7: Velocity-time response of an RCC projectile in a perforating impact simulation of a 1-layer S-728 target struck at 170 m/s.  -149-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  Figure 5 - 9 : Comparison of the normalized reaction force in warp and weft of S-720 fabric.  -20  J  T i m e (us)  Figure 5 - 10: Velocity-time response of an R C C projectile in a non-perforating impact simulation of a 1-layer S-726 target struck at 96 m/s.  -150-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  -80-  -20  20  60  100  140  Time (us)  Figure 5-11: Velocity-time response of an RCC projectile in a perforating impact simulation of a 1-layer S-726 target struck at 174 m/s.  60  -20  i  J  Time (us)  Figure 5 - 12: Velocity-time response of an RCC projectile in a non-perforating impact simulation of a 1-layer S-727 target struck at 40 m/s.  -151-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  200  (0  E o o  > 0)  o  "5* Q.  100  -50  150  300  T i m e ((is)  Figure 5 - 13: Velocity-time response of an RCC projectile in a perforating impact simulation of a 1-layer S-727 target struck at 158 m/s.  140 n  Time (us)  Figure 5 - 14: Velocity-time response of an RCC projectile in a non-perforating impact simulation of a 1-layer S-731 target struck at 116 m/s.  -152-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  160 -,  Figure 5 - 15: Velocity-time response of an RCC projectile in a perforating impact simulation of a 1-layer S-731 target struck at 132 m/s.  140 -i  -20  J  T i m e (us)  Figure 5 - 16: Velocity-time response of an RCC projectile in a non-perforating impact simulation of a 1-layer S-745 target struck at 110 m/s.  -153-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  240 ELVS Data S-745-01p-01  E o  o  a> >  o  +3 O (D  O i_  Q.  Figure 5 - 17: Velocity-time response of an RCC projectile in a perforating impact simulation of a 1-layer S-745 target struck at 215 m/s.  Figure 5 - 18: Transverse deformation profile (left) and corresponding fringe plot (right) for a 1-ply S-731 panel impacted at 100 m/s after 225 ps.  -154-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  Time ( u s )  Figure  5 - 19:  Comparison  o fthe normalized  reaction force i n w a r p a n d weft o f  S-726  fabric.  Y a r n s l i p p a g e at the boundary  |  Narrow slippage  zone wide  weft (low  Figure  5 - 20:  zone  crimp)  w a r p (high  slippage  crimp)  P o s t - m o r t e m e x a m i n a t i o n o f the 1-layer m/s.  -155-  S-726  panels i m p a c t e d at  175  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  Figure 5-21: Schematics of the mechanical model considered for the transverse compression of single and two-ply fabric specimens.  Figure 5 - 22: Calculation of inter-ply gap from transverse compression experiments, obtained for 4-ply S-728 fabric.  -156-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  Figure 5-23: A schematic arrangement of multiple layers made from shell elements to construct a 4-ply S-726 4-layer model (quarter of the model is shown due to symmetry).  200  -50  i  J  Time (us) Figure 5 - 24: Velocity-time response of an RCC projectile in a non-perforating impact simulation of a 4-layer S-726 target struck at 158 m/s.  -157-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  100 n  T i m e (us)  Figure 5 - 25: Velocity-time response of an RCC projectile in a non-perforating impact simulation of a 4-layer S-728 target struck at 91 m/s.  Figure 5 - 26: Velocity-time response of an RCC projectile in a non-perforating impact simulation of a 4-layer S-731 target struck at 136 m/s.  -158-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  260  n  Time (us)  Figure 5 - 27: Velocity-time response of an RCC projectile in a non-perforating impact simulation of an 8-layer S-726 target struck at 204 m/s.  200  -50  i  J  T i m e (us)  Figure 5 - 28: Velocity-time response of an RCC projectile in a perforating impact simulation of an 8-layer S-728 target struck at 162 m/s.  -159-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  180  (0  E  o o >  o  u  "o" 0.  350  Figure 5-29: Velocity-time response of an RCC projectile in a perforating impact simulation of an 8-layer S-731 target struck at 142 m/s.  320  £  o o > u a> Q.  200  T i m e (us)  Figure 5 - 30: Velocity-time response of an RCC projectile in a perforating impact simulation of a 16-layer S-726 target struck at 243 m/s.  -160-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  200 -|  160  Time (us)  Figure 5-31: Velocity-time response of an RCC projectile in a perforating impact simulation of a 16-layer S-728 target struck at 170 m/s.  140 -i  Time (us)  Figure 5 - 32: Velocity-time response of an RCC projectile in a perforating impact simulation of a 16-layer S-731 target struck at 120 m/s.  -161-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  Figure 5-33: Warp and weft tension surfaces as a function of the in-plane extensional displacements, developed for S-720 fabric.  Figure 5-34: Energy absorbed per unit area (J/m ) by S-720 fabric under symmetrically applied extension, expressed in terms of strain in the warp and weft directions; 3D surface (left) and 2D contour plot (right). 2  -162-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  Warp fabric Strain  Warp fabric Strain  Figure 5 - 3 5 : Contours of strain energy stored in a unit area of S-720 and S-728 fabrics as a function of the warp and weft strains.  S-731  0.04  S-745  0.06  0.08  0.02  Warp fabric Strain  0.04  0.021 0.018 0.016 0.013 0.010 0.008 0.005 0.003  0.06  0.08  0.1  Warp fabric Strain  Figure 5 - 36: Energy-strain contour plots of a unit area of S-726, S-727, S-731 and S745 fabrics.  -163-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics 140 i  T i m e (us)  Figure 5 - 37: Velocity-time prediction of the shell-based fabric model showing the response of all six Kevlar® 129 panels impacted by an R C C at 100 m/s.  16 -  0  5  10  15  20  25  Projectile Displacement (mm) Figure 5 - 38: Target's energy absorption plotted against projectile displacement obtained from numerical simulation of the R C C projectile impacting the six Kevlar® 129 panels at 100 m/s.  -164-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  Figure 5 - 39: Kinetic to internal energy ratio of the target's absorbed energy plotted against projectile displacement obtained from numerical simulation of the RCC projectile impacting the six Kevlar® 129 panels at 100 m/s. 30  i  S-731  25  time = 20 us  \  L  single-yam impact numerical " 8050 tTl/S  C  Canalytical = 7900 m/S  20 c o c 15 d>  \  £ 10  V f  time = 15 us  \  time = 10 us  20 40 60 80 Distance f r o m t h e Impact Point (mm)  100  Figure 5 - 40: Propagation of the strain wave evident from the evolution of the yarn tension along the yarn at different time instances in an impact simulation of R C C projectile on a single yarn of S-731 fabric modelled by the shell elements.  -165-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  S-731 single5-yarn g  20  40  60  impact  = 685  m/s  100  80  Distance f r o m t h e Impact Point (mm)  Figure 5 - 41: Transverse deformation of an S-731 single yarn impacted by an RCC, showing the growth of the deformation cone.  50  S-731 single-yarn  impact  (crossing yarns  added)  Canalytical = 5650 m / S  OT  c  0)  re >-  t i m e = 20  20  40  60  80  (is  100  Distance f r o m t h e Impact Point (mm)  Figure 5-42: Propagation of the strain wave evident from the evolution of the yarn tension along the yarn at different time instances in an impact simulation of RCC projectile on an S-731 single yarn (with the addition of the mass of the crossing yarns) modelled by the shell elements.  -166-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  S-731 single-yarn impact (crimp 8.16%) Cnumerical =  1 2 7  0  m /  S  time = 30 us time = 25 us time = 20 us ; « « o—» « » « » • o « o a 20  40  60  80  100  Distance f r o m the Impact Point (mm)  Figure 5-43: Propagation of the strain wave evident from the evolution of the yarn tension along the yarn at different time instances in an impact simulation of RCC projectile on a crimped (crimp = 8.16%) S-731 single yarn (with the addition of the mass of the crossing yarns) modelled by the shell elements.  S-731 single-yarn impact (crimp = 8.16%)  20 40 60 80 Distance f r o m the Impact Point (mm)  100  Figure 5 - 44: Transverse deformation of a crimped (crimp = 8.16%) S-731 single yarn impacted by an RCC, showing the growth of the deformation cone.  -167-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  Figure 5 - 45: Propagation of the strain wave evident from the evolution of the tension along the warp yarn in the impact simulation of an RCC projectile on a 1ply S-720 fabric modelled by the shell elements. 30  T  S-720 fabric impact (weft yarn) crimp = 1.39% C =2267 m/s a  c o '35 c  re >it 0)  v  g  20  1 0  time =40 u s  time =3 0 us\  5  time =2 0 u s time =1 0 u s  20  4 0  6 0  8 0  100  Distance f r o m t h e I m p a c t P o i n t ( m m )  Figure 5 - 46: Propagation of the strain wave evident from the evolution of the tension along the weft yarn in the impact simulation of an RCC projectile on a 1ply S-720 fabric modelled by the shell elements.  -168-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  Figure 5 - 47: Transverse deformation of a 1-ply S-731 panel plotted for a warp and weft yarn in the impact simulation of an RCC at strike velocity of 100 m/s.  (0  1 8000 o  CL  Warp Direction  0  Weft direction  —  •a Q) Q>  0  Best fit  6000  CO « >  CO  ~ro 4000 H CO  "co c T3  3 2000  "c5> o  4  Yarn Crimp (%)  6  Figure 5 - 48: Longitudinal wave speed measured from the numerical model of the fabric plotted against the yarn crimp.  -169-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  800  O Warp Direction •  1 —  600  Weft Direction Best fit  —  I \  \ 400  V  •% V  o  {2 200 D  1  !  4  1  1 10  6  Y a r n C r i m p (%)  Figure 5 - 49: Transverse wave speed measured from the numerical model of the fabric plotted against the yarn crimp.  3500 XI 0) 0)  3000  a  CO a> > n  2500  «  _  3  E  " 5> c »  '  2000  1500  o  1000 D) re  >  500  < 720  726  727  728  731  745  Panel  Figure 5 - 50: Average longitudinal wave speed measured from the numerical model of the fabric for the 100 m/s iso-velocity impact simulations.  -170-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  250  "3"  n  i  c o  !P  O  200  y  £ D_  4  150  y  a> TJ O  -  n  Vent  Panel  100  o  S-720 S-726 S-727 S-728 S-731 S-745  a>  E  50  > 50  100  V  crit  150  (m/s)  Test Model 99 108 116 84 128 144 200  154 128 142 134 124 192 250  , Experimental Data (m/s)  Figure 5 - 51: Critical velocity of the 1-ply Kevlar 129 panels obtained from experimental data and estimated by the shell-based numerical model.  Figure 5 - 52: Post-mortem examination of the Kevlar panels; perforation by yarn pullout (left) and by yarn rupture (right).  -171-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  70 Panel  1  t  50  c o  +3  C o u n t  AV„  S-720  20  55  S-726  26.5  20  S-727  26  26  S-728  17  50  S-731  31  - 4  S-745  17  49  O  2  Q.  3 0  > c O  1 0  c 111 20  15  30  2 5  35  -10  Count (threads/inch) Figure 5 - 53: Error in the prediction of critical velocity of the 1-ply Kevlar 129 panels compared to experimental data, plotted against their average count number.  S-726  0.02  0.04  0.06  0.01  to  0.02  Warp fabric Strain  0.04  S-727  0.06 0.08  0  0.02  Warp fabric Strain  0.04  0.00 0.08  Warp fabric Strain  S-745  S-728  C  i  0.021 0.018 0.016 0.013 0.010 0.008 0.005 0.003  0 04  3 0  0.02  0.04  0.06  0.08  Warp fabric Strain  0.02  0.04  0.06  0.08  Warp fabric Strain  0.1  0.02 "0  0.02  0.04  0.06  0.08  Warp fabric Strain  Figure 5 - 54: Energy-strain contour plots of a unit area of the six Kevlar 129 fabric panels as a function of fabric strain, also showing the strain pairs corresponding to the most critical elements around the projectile (shells with highest energy) during an RCC impact simulation at 100 m/s.  -172-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  Parallel System  Alternating System  Figure 5 - 55: Transverse deformation of the an S-726 4-layer target impacted at 100 m/s showing the increased interaction of adjacent layers in the alternating packing method.  ft*tj)* Ltmli  -ijt»..»»  <««»»*»  Alternating System  Parallel System  Figure 5 - 56: Contours of transverse deformation of the third layer away from the strike face of an S-726 4-layer target impacted at 100 m/s showing the benefit of alternating packing method.  -173-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  120  n  0J 0  j  ;  50  100  1  150  ;  200  Time (us)  Figure 5 - 57: Comparison of the projectile response for different panel orientation in S-726 4-layer target.  I  230 m m  I  70 m m  50 m m  40 m m  Figure 5 - 58: Cross-shaped fabric specimen used in the biaxial experiments conducted by Buet-Gautier and Boisse [37].  -174-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  Figure 5 - 59: Predictions of the numerical model on the biaxial loading of the balanced plain weave glass fabric, after calibrating the input parameters to match the iso-displacement loading case (k = 1).  Figure 5 - 60: Predictions of the numerical model on the biaxial loading of the balanced plain weave glass fabric, extended to other two loading scenarios (jfc = 0 . 5 & 2 ) .  -175-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics  Figure 5-61: Predictions of the numerical model on the warp extension of the nonbalance plain weave glass fabric, after calibrating the input parameters to match the iso-displacement loading case (k = 1).  Figure 5-62: Predictions of the numerical model on the weft extension of the nonbalance plain weave glass fabric, after calibrating the input parameters to match the iso-displacement loading case ( k = 1).  -176-  Chapter 5 - Validation of the Static and Dynamic Response Predictions of Fabrics 200 -]  0  0.2  0.4  0.6  0.8  Strain (%) Figure 5-63: Predictions of the numerical model on the warp extension of the nonbalance plain weave glass fabric, extended to other two loading scenarios  (Jfc = 0 . 5 & 2 ) .  Figure 5 - 64: Predictions of the numerical model on the weft extension of the nonbalance plain weave glass fabric, extended to other two loading scenarios  (* = 0.5&2).  -177-  Chapter 6 - Sensitivity Analysis and Numerical  CHAPTER 6-  SENSITIVITY  Case-Studies  ANALYSIS  A N D NUMERICAL  CASE-STUDIES  6.1  INTRODUCTION  Determination of the material and geometric properties of the fabric panel is not a trivial task. Almost all mechanical and geometrical properties of a fabric have some level of uncertainty associated w i t h them. Since the analysis procedure established i n this thesis is based on a deterministic approach, one needs to incorporate this lack of knowledge on the accuracy of the input values in the interpretation of the model predictions. The basic step i n assisting the user to better assess the level of confidence i n the numerical results is to understand the sensitivity of the fabric response to various input parameters. A parametric study is thus performed to better understand the dependency of the fabric response on input parameters. This chapter aims to provide some understanding into the physics of the ballistic response of the woven fabrics through examination of the influence of various parameters on a fabric's  dynamic behaviour.  -178-  Furthermore,  the  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  numerical model is used later i n the chapter to qualitatively explore the roots of pencilling phenomenon, a by-product of the weight reduction i n the personal armours due to the use of new yarns and fibres w i t h superior material properties.  6.2  FABRIC ENERGY ABSORPTION M E C H A N I S M  The overall energy transferred into a fabric target during an impact event is stored  i n the  forms  of internal (strain)  and  kinetic energies. Detailed  examination of the history of the absorbed energy by the targets reveals two distinct characteristic behaviours. A s illustrated i n Figure 6 - 1 , the velocity time-history response of the projectile impacting a single-ply S-728 target at 100 m/s can be divided into two distinct zones of behaviour. In Zone I, the projectile velocity shows a gradual decrease w i t h time w i t h a shallow slope, while the deceleration i n Zone II seems to be more rapid. Further examination of the fabric response shows that throughout Zone I, the fabric deformation eliminates the inherent "slack" i n the system as a result of the yarn crimp and initiates the interaction of the warp and weft yarns. This can be seen i n snapshots of the fabric's in-plane motion (see contours of in-plane velocities i n Figure 6 - 2 ) and similarly warp and weft tensile force (depicted i n Figure 6 - 3 ) at the instance when the longitudinal strain wave reaches the warp and weft boundaries. Due to the difference between the values of warp and weft crimp, this wave w o u l d travel at different speeds i n the two main fabric directions. Dependency of the longitudinal and transverse wave speeds on the yarn crimp was previously discussed i n Chapter 6 (see Figure 6 - 4 8 and Figure 6 - 49).  -179-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  Considering the kinetic and internal energy distribution i n the fabric, the overlap of the two zones of behaviour can be better illustrated, shown i n Figure 6 - 4 as the ratio of the kinetic to internal energy components. It appears that the fabric absorbs energy primarily i n the form of kinetic energy i n the initial stages of impact. Overlaying the zones identified on the velocity-time response plot over the energy ratio-time curve further reinforces the hypothesis that the kinetic energy is the dominant component of energy i n the fabric during the initial stages of impact as the crimped yarns are stretching taut and the projectile's velocity loss is mainly due to the momentum exchange with the fabric. The stiffness of the stretched fabric yarns w o u l d come into play i n the later stages of impact when the warp and weft yarns are fully taut and engaged in the energy absorption. A similar trend can also be seen i n the development of tension i n the warp and weft yarns. A s shown i n Figure 6 - 5 , the tension (plotted for the weft yarn of S728 i n this case) suddenly builds up i n the yarns u p o n impact due to momentum exchange between the projectile and the fabric. The yarn tension then more or less remains constant (or increasing w i t h at a l o w rate) within zone I as the yarn is being stretched and the crimp being taken out, after which the tension starts increasing monotonically i n zone II. Similar observations are reported i n other studies [1] and the nature of behaviour is attributed w i t h the strain wave generation, propagation and reflection. The plot shown i n Figure 6 - 5 only covers the initial 100 LIS of the impact to focus on the initial response of the yarns; the tensile force w i l l later reach a peak value and start decreasing, given that it remains below the failure limit. Depending on the projectile's striking velocity, failure of the yarns i n a penetrating impact can occur at the  -180-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  initial peak (for high impact velocities) before any significant strain wave propagation i n the target occurs (zone I), or later i n zone II (moderate strike velocities) w i t h i n the increasing segment of the yarn tensile response. A s a result, the failure of yarns and eventually the perforation of the fabric i n each of these two scenarios are affected by different sets of fabric properties. Thus, the influencing parameters need to be identified, as discussed i n the following sections.  6.3  As  PARAMETRIC STUDY  mentioned  earlier, all the  measured  and  calculated geometric  and  mechanical properties of the fabric fed into the shell-based fabric model have various levels of uncertainties associated w i t h them. The analysis methodology established here is based on a deterministic approach; however, i n order to increase the user's confidence i n the predictions of the numerical model, the effects of these inherent uncertainties on the outcome of the analysis need to be investigated. In addition, considering the two zones of behaviour discussed above, one can optimize the performance of a fabric system k n o w i n g the influence of various mechanical and geometric parameters on the overall energy absorption of the system. This section highlights the influence of some of these parameters on the performance prediction of the fabric systems, and reviews the variability level of each parameter i n an effort to identify their overall importance.  -181-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  6.3.1  Effect of Y a r n Stiffness  A s mentioned previously, the input material properties employed to perform impact simulations on single and multi-ply fabric targets are based on the properties of the virgin yarns. It is well k n o w n that the process of weaving the yarns into fabric deteriorates these properties due to the wear and abrasion caused by the contact between the yarns and different components of the manufacturing machinery. One of the main properties affected b y this process is the yarn stiffness. Both the elastic modulus and cross-sectional area of the yarns w o u n d be affected where fibres of the yarn are damaged during weaving [68, 69]. The yarn stiffness reduction can be directly measured i n the laboratory by comparing the tension-displacement response of the virgin yarns w i t h that of yarns extracted from the woven fabric. However, such data were not available for the study performed here and therefore properties of virgin yarns were used throughout the analyses. This section investigates the influence of the yarn stiffness on the overall response of the fabric. This sensitivity analysis is performed by varying the elastic modulus of the yarns only, as a representative of the yarn stiffness. In the case of Kevlar® 129, the fibres are reported to have an elastic modulus of 96.0 G P a i n the virgin state. The effect of a nominal ±20% change i n the elastic modulus of the yarns on the biaxial warp tension-displacement response (fixed weft) of an S-728 fabric unit-cell is shown i n Figure 6 - 6. A s expected, a higher elastic modulus results i n a stiffer force-displacement response as seen i n this figure. The same trend is also reflected on the ballistic response of the fabric, as depicted i n Figure 6 - 7 , showing the projectile velocity-time response predicted  -182-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  for impact of an R C C projectile on a 1-ply S-728 fabric at 100 m/s. It can be seen that a rather significant reduction of 20% i n the yarn stiffness w o u l d only slightly affect the deceleration of the projectile. Investigating the evolution of strain i n the yarns passing through the impact point, one can predict the penetration characteristics of a fabric target. A s seen i n the plot of the yarn strain-time response of Figure 6 - 8 , the deformation of the yarns is affected by the modulus reduction throughout the impact, w i t h the lower modulus resulting i n a higher strain level. This w o u l d imply that stiffness degradation of the yarns causes the fabric to fail sooner for almost all the strike velocities, causing failure both i n zone I and zone II. A s a result, it can be concluded that the stiffness reduction w o u l d affect both the energy absorption characteristics and critical velocity of a fabric.  6.3.2  Effect of Y a r n C r i m p  Undulation of the yarns i n a woven fabric is perhaps one of the most critical parameters affecting its highly non-linear biaxial behaviour. The presence of yarn crimp is mainly responsible for the deviation of the fabric response from that of a flat net-like structure and is the fundamental reason behind the inherent "slack" i n an un-stretched fabric. Realistic representation of the yarn undulation i n a w o v e n fabric is thus the essential part of any crimp measurement methodology. Determination of yarn crimp i n the warp and weft directions is not a trivial task. Several factors such as over-handling of the fabric, choice of the measurement technique, and human factors can highly affect the measured crimp values of identical fabrics. Figure 6 - 9  -183-  shows an example of slight  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  human-affected  measurement  disparities  resulting  from  two  crimp  measurements performed by the author and those reported by Kongshavn [42] for the same set of Kevlar® 129 fabrics. Despite the good agreements between the results, a difference of up to 20% exists between the two measured crimp values. In order to probe the sensitivity of the unit-cell mechanical behaviour and the fabric's impact response to the variability i n the yarn crimp values, changes of ±20% are considered i n the warp and weft crimp values of the S-728 fabric. A s seen i n Figure 6 - 1 0 , an increase in the yarn crimp w o u l d result i n a softer i n plane extensional response. Similar conclusions can be d r a w n from the nonperforating ballistic response of the S-728 subjected to impact of an R C C projectile at a strike velocity of 100 m/s (Figure 6 - 11). Studying the strain timehistory of the central weft yarn going through the impact point (Figure 6 -12) reveals that although increasing the crimp reduces the energy  absorption  capacity of the fabric at a given time or projectile displacement, it results i n a lower strain level i n the yarns throughout the impact event. This suggests that the critical velocity of a fabric w o u l d increase w i t h the introduction of more crimp into it. However, this improvement w o u l d come at a cost of increasing the back-face deformation of the fabric target. In a way, addition of crimp to the fabric is equivalent to adding more "ductility" to the system, leading to more overall  energy  absorption  w i t h an  increase  i n the  fabric's  transverse  deformation. It should be noted that this analysis is based on a pin-jointed crossover approach and other deformation mechanisms such as yarn slippage, which is specifically amplified by higher yarn crimp values, are not considered.  -184-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  6.3.3  Effect of Fabric A r e a l Density  One of the fundamental components of the dynamic response of any structure is its inertial (mass) properties. In the case of fabrics, the mass properties of the targets are usually represented by their areal density, defined as the mass of the system per unit area. In most cases, the laboratory measurements and the theoretically calculated areal densities of Kevlar® 129 panels are i n good agreement (Figure 6 - 1 for the case of six Kevlar® 129 panels studied here), showing small and often negligible disparities (a m a x i m u m of 4% i n the case of S-720). This small margin of error can be translated into a high level of confidence i n the theoretical values used i n the numerical model. In order to further investigate the influence of a fabric's areal density on its overall energy  absorption  characteristics,  an approach  similar to those  employed for elastic modulus and yarn crimp sensitivity is employed. Ballistic response of a 1-ply S-728 fabric target under the impact of an R C C projectile flying at 100 m/s was studied while perturbing its areal density by ±20%. Figure 6 - 1 3 shows the sensitivity of the projectile deceleration on the changes of areal density. A s expected, the heavier fabric is capable of absorbing more energy at a given time. The strain time-history of the central yarn i n the weft direction shown in Figure 6 - 1 4 indicates that despite differences i n the response for various values of areal density, no immediate improvement can be seen in the yarn strain evolution and the variability i n the areal density is not expected to dramatically change the critical velocity of the fabric target. Considering the above sensitivity analysis, areal density is perhaps one of the least concerning input parameters, and a reasonable level of confidence is  -185-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  associated w i t h its measured and calculated values. The possible sources of error i n areal density stems from the inaccuracies i n panel size and weight measurements, crimp measurement (in the case of calculated areal density), and the uncertainty i n the yarn linear density reported by the manufacturer.  6.3.4  Discussion  Elastic modulus and crimp of the yarns and areal density of the fabric are three basic input parameters that determine the mechanical behaviour of fabrics under static and dynamic loadings. The analysis above shows that the influence of these three parameters on the energy absorption of the fabric (evident from the velocity time-history of the projectile) is more or less similar. However, there are different levels of accuracy associated w i t h each of these parameters. Elastic modulus of the fabric yarns is the parameter that gets affected by several factors i n the manufacturing process, from constructing individual yarns out of fibres to weaving yarns into fabrics. Every step of  the  manufacturing process deteriorates the properties of the yarns to a certain degree, the amount of w h i c h should be determined from the examination of the yarns extracted from the final product. Crimp, on the other hand, is created during the weaving process and even i n the final product can be affected by factors such as handling of the fabric. It is suggested that these two parameters be measured for every fabric individually to acquire a high level confidence in their values. Further investigation of the effects of other disturbing factors, such as stitching during the production of personal armours, on their overall performance is also highly advantageous to the accuracy of the numerical predictions. Areal density, on the other hand, seems to be more easily  -186-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  determinable and i m p r o v i n g accuracies on geometry and crimp values of yarns can lead to a better agreement between measured  and calculated areal  densities. Two zones of behaviour were identified i n this section, during which the response of the fabric is dominated by various parameters. Examining the sensitivity of projectile velocity deceleration to any of the  above-studied  parameters (elastic modulus, yarn crimp and areal density), little difference can be noticed within zone II (after 60 \is i n the case of S-728). To better understand the differences i n zone I, the projectile velocity-time response i n initial 100 us of the impact event is plotted for varying elastic modulus, yarn crimp and areal density i n Figure 6 - 1 5 , Figure 6 - 1 6 and Figure 6 -17, respectively. It appears that the only parameter that most affects the energy absorption of the fabric in zone I is the areal density, where the projectile decelerates more rapidly as the fabric's mass increases. The critical velocity of a fabric is inversely affected by the level of tension i n the yarns. From the sensitivity analysis performed here, one can conclude that for an equal value of strain-to-failure, the fabric w i t h higher elastic modulus w o u l d have higher critical velocity (see lower tension for higher modulus i n Figure 6 8). A similar conclusion can be drawn for fabrics w i t h higher crimp (see Figure 6 - 12). The same can not be said for changes i n the areal density, as no clear advantage can be gained i n yarn tension history before its peak (Figure 6 -14) with the change i n areal density. A s mentioned previously, the failure of the yarns i n zone I occurs at strike velocities significantly higher than the critical velocity of a 1-ply fabric. In  -187-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  reality, personal armours are made of many layers of fabric to withstand the impact of threats they are exposed to. The critical velocity for a multi-ply fabric pack is significantly higher than that of a single-ply target. A s a result, the range of projectile strike velocities that a multi-ply pack is able to defeat can cause a single-ply target to instantaneously fail upon impact (zone I). Since the response of the fabric i n zone I seems to be more efficiently improved by the areal density (Figure 6 - 17), it is more economically advantageous to place fabric materials w i t h inferior mechanical properties but equal areal density on the strike face of a multi-ply fabric armour, as suggested by Cunniff [71]. This is a basic example of the advantages provided by the numerical model i n the design and manufacturing of body armours.  6.4  SENSITIVITY T O O T H E R PARAMETERS  6.4.1 Transverse Compression of Yarns Perhaps the most unpredictable aspect of yarn mechanical behaviour inside a woven fabric is determining its transverse mechanical response. Individual Kevlar® 129 yarns (or any other yarn w i t h untwisted fibre structure for that matter) are loosely made from juxtaposition of parallel fibres without any cohesion and are not capable of sustaining any transverse loading. Inside the fabric weave, however, the constraint provided by the weave structure changes the transverse behaviour of the yarns. A s a result, the transverse mechanical properties of the yarns are highly dependent on many factors, such as weave tightness, voids between the fibres i n the yarns, and transverse properties of the fibres themselves. Unfortunately, there is still no robust testing method  -188-  Chapter 6 - Sensitivity  Analysis  and Numerical  Case-Studies  available to measure the transverse compressive behaviour of the yarns directly inside the fabric. The method employed i n this thesis to predict this transverse yarn behaviour from the 3D crossover model is highly dependent on the success of the finite element model and susceptible to the possible inaccuracies in the numerical model. K n o w i n g the influence of this aspect of modeling on the overall response of a fabric can help the user determine how much this uncertainty w o u l d affect the final conclusions of his/her analysis. This section aims to address this issue b y exploring the effects of transverse yarn properties on the in-plane extensional and ballistic response of fabric panels. The biaxial response of a fabric crossover w i t h compressible yarns w o u l d fall between two extreme cases: crossover model w i t h non-compressible yarns and one with decoupled warp and weft yarns. The true fabric response w i t h the inclusion of yarn compressibility (compressible model) is thus compared to these two cases on unit-cell and fabric levels to investigate the sensitivity of the response to the transverse compression properties of the yarns. For this purpose, S-728 and S-731 fabrics are considered and the in-plane extensional responses of their unit-cells are investigated. S-728 represents a highly balanced low-crimp fabric while S-731 is an example of an unbalanced high-crimp one amongst the six Kevlar® 129 fabrics studied i n this thesis. Figure 6 - 1 8 and Figure 6 - 1 9 show the warp and weft tensile response as a function of warp extension i n a unit-cell loading case where the weft yarn is held fixed. A s can be seen, the decoupled response of the warp yarn i n this loading scenario is comprised of an initial no-resistance zone up to the point where the yarns are stretched tight, after w h i c h the response is identical to that  -189-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  of a single straight (taut) yarn. The weft yarn does not experience any tension in the decoupled model due to the absence of warp-weft yarn interaction, when the unit-cell is being loaded i n the warp direction only. The incompressible model, on the other hand, exhibits pure crimp exchange and w i t h no change i n the transverse distance between the yarn centerlines, and intuitively shows the stiffest response amongst the three modeling approaches. A s expected, the response of the unit-cell w i t h compressible yarns falls i n between the two other cases. This can be seen i n the plot of energy surfaces predicted by the three models i n Figure 6 - 20. A slice from these surfaces (parallel to the warp displacement axis at zero weft displacement) w i l l result i n the graph of Figure 6 - 2 1 where the energy stored in the unit-cell is plotted versus the warp displacement for fixed weft yarn. These figures show that the energy surface corresponding to the model with compressible yarns is consistently i n the middle of the two other models. Performing impact simulations on single-ply S-728 panels also indicate similar results. While the panels w i t h compressible and decoupled yarns show the stiffest and most compliant responses, respectively, the one w i t h compressible yarns falls between these two cases (see Figure 6 - 2 2 and Figure 6 - 23). However, investigating the components of the overall energy absorbed by the target (Figure 6 - 24) reveals that the panel w i t h decoupled warp and weft yarns has a significantly larger kinetic energy component compared to the other two models, mainly due to the freedom of the fabric material to move without considerable resistance, a direct result of the absence of warp-weft yarn interaction.  -190-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  A similar analysis is performed on the highly imbalanced S-731 fabric. Figure 6 - 25 and Figure 6 - 2 6 show the tensile response of the warp and weft yarns under extension, respectively, w i t h the perpendicular yarn held fixed i n each case. The response of the S-731 unit-cell model w i t h compressible yarns appears to be very close to that of the model w i t h decoupled yarns, especially for the case of low-crimp weft yarn loading. The overall absorbed energy surfaces of the unit-cell w i t h compressible, non-compressible and decoupled warp and weft assumption shown i n Figure 6 - 2 7 also indicate the close proximity of the fabric behaviour w i t h compressible yarns to that of decoupled warp and weft. The energy absorbed by the unit-cell subjected to extension only i n the warp direction (Figure 6 - 28) is closer to the decoupled yarns model than  the non-compressible  yarns, while the  same energy  plot of  the  compressible model and decoupled yarns model overlap for weft extension (Figure 6 - 29). A similar behaviour is observed for a 1-ply panel impacted by an R C C projectile at 100 m/s, where the projectile velocity-time response of the fabric with compressible yarns appear much closer to that of the decoupled warp and weft model than the model w i t h non-compressible yarns (Figure 6 - 30). It appears that the high level of crimp i n the fabric, along w i t h significant crimp imbalance between the warp and weft directions leads to the low yarn interaction, causing them to respond more or less i n a decoupled fashion. The balance of kinetic to internal energy components i n the impact simulation shows a larger kinetic component, similar to that of a fabric w i t h decoupled warp and weft yarns (Figure 6 - 31).  -191-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  6.4.2  Shear Properties  Shear properties of the fabric are amongst the major factors influencing the overall behaviour of fabrics. The fabric shear model adopted i n this study is based on the trellis mechanism (see Chapter 5, Figure 4 - 8 ) .  The four  characteristic parameters that construct fabric shear response are the stiffening and locking shear strains (y  x  and y ) and initial and locked shear moduli (G, 2  and G ) and are determined using a simplified mechanistic approach that 2  considers the geometry and physical interaction of parallel and crossing yarns (see Section 4.3.2). In the absence of any experimental data on the shear behaviour of the six Kevlar® 129 fabrics investigated here, all input values w o u l d merely provide an approximation of the real fabrics' shear response. In this section, results of a sensitivity analysis on the shear model input values are presented for the ballistic impact of R C C projectiles on 1-ply fabric targets. A compilation of m a x i m u m shear strains i n each of the six Kevlar® 129 fabric types reached during the impact simulation of an R C C projectile striking the target at 100 m/s is provided i n Figure 6 - 3 2  alongside their respective  stiffening and locking shear strains (y and y ). A s is evident from this picture, x  2  the maximum shear strain i n all fabrics (except S-731) is slightly higher than the stiffening shear strain value. In order to further investigate the influence of the shear response of shell elements on the overall response of the target, two extreme cases w i t h low and high shear resistance of the trellis mechanism is considered. The panel chosen for this study was S-728 fabric impacted by an R C C at 100 m/s. For the low shear resistance case, the locked shear modulus is set equal to the lower initial modulus (i.e. G -G x  -192-  2  8.66 M P a for S-728),  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  leading to a very soft linear response of the shells i n shear. Conversely i n the high resistance scenario, the initial modulus is set equal to that of the locked configuration (i.e. G, = G = 8.66 M P a for S-728) to obtain high shear resistance 2  in all the shell elements throughout the fabric deformation. Figure 6 - 3 3 shows a plot of the projectile's velocity prediction i n the impact simulation of a 1-ply S-728 target comparing the trellis fabric model w i t h the low and high resistance models discussed above. A s seen, the response of the fabric w i t h trellis model is more or less identical to that of the low shear resistance case, since the maximum shear deformation of the fabric observed i n the simulations is significantly smaller than the locking strain. O n the other hand, the high modulus case results i n a much stiffer response and decelerates the projectile relatively more rapidly. The deformation profile of the fabrics also changes noticeably w i t h the increase in the shear stiffness. Figure 6 - 3 4 diamond-shape  transverse  shows such a difference, where the  cone i n the fabric w i t h trellis shear model is  transformed into an almost circular cone i n the high-shear fabric. While the rhomboidal shape of the cone is a signature of the l o w shear orthotropic fabrics in ballistic impact, a transverse cone w i t h a circular base is the fundamental characteristic of an isotropic membrane structure. Figure 6 - 3 5 shows the l o w dependence of the S-728 fabric's dynamic response on the stiffening and locking strain values. A s seen i n this figure, reducing these limiting strains to 30% of their original value does not significantly affect the deceleration of the projectile. It can be concluded that i n the case of the six Kevlar® 129 fabrics studied here i n the ballistic impact w i t h fixed boundary  -193-  Chapter 6 - Sensitivity  Analysis and Numerical Case-Studies  conditions, the shear mode of deformation is not a dominant energy absorbing mode. This, however, might not be true for other fabrics w i t h lower shear locking angles. In addition, other boundary conditions can also result i n higher shear deformation of the fabrics and lead to a higher dependency of a fabric's ballistic performance to its shear properties. The user should pay particular attention to the fact that the shear model i n this study is simplified for practical reasons. It is suggested that the mechanical behaviour of fabrics under shearing modes of deformation be studied using a more elaborate approach if the model proposed i n this study is being utilized i n a different application where significant shearing of fabrics is likely to occur.  6.4.3  Fabric Slippage at the Boundary  Perfect clamping of the fabric i n the fixed boundaries is difficult to achieve for single and multi-ply targets. Effect of clamping pressure on the ballistic performance of various fabric targets was investigated [42] and it was observed that although higher pressures can significantly reduce the slipping of the fabric at the fixed boundaries, there is usually a certain level of slippage observed i n most panels. Clearly, the slippage of the fabric at the boundaries would directly affect its ballistic performance and any analysis/simulation of such impact event should incorporate this phenomenon. In order to investigate this boundary slippage, the fixed edges of the target were traced over the fabric before and after the impact. Figure 6 - 3 6 shows a typical boundary slippage that occurred i n a 1-ply S-728 panel at a strike velocity of 36.9 m/s. Post-mortem examination of the tested panels showed a highly variable amount of slipping, both i n terms of slip length and its width.  -194-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  However, it was observed that the w i d t h of the slip zone is commonly proportional to the amount of fabric pull-in (slip length). The boundary force profile of a 1-ply S-728 fabric impacted at 100 m/s is shown in Figure 6 - 3 7 for the weft yarns. A s seen i n this figure, the tension builds up at the boundary as the impact progresses. The central yarns seem to carry significantly higher reaction forces, which w o u l d later stabilize at a lower tension level as the distance from the central yarn increases. The w i d t h of the high tension zone w o u l d be affected by the projectile size and other fabric and impact properties. The boundary slip during the impact was incorporated i n the numerical simulations by allowing the boundary nodes to move i n during the event, w i t h a triangular profile depicted i n Figure 6 - 38. The w i d t h of the slipping zone was chosen to be 60 mm, i n order to cover all the yarns with relatively high reaction forces. Two  characteristic  variables  defining this boundary  slip  during these  simulations were assumed to be the pull-in amount and the time interval in which the slip occurs. Intuitively, the slipping w o u l d soften the ballistic response of the panels, as shown i n Figure 6 - 3 9 for the weft yarn slippage during the impact of an R C C projectile on a 1-ply S-728 fabric at a strike velocity of 100 m/s. To obtain these results, the time interval during which the weft slippage occurs was chosen to be the initial 0-200 [is. Despite this fact, it can be observed that there is little effect on the deceleration of the projectile in zone I, and that the slippage w o u l d mostly influence the response of the fabric in zone II. However, it is clear that the slip time-interval of 0-200 us is not  -195-  Chapter  6 - Sensitivity  Analysis  and Numerical  Case-Studies  realistic since the tension i n the central yarns is too l o w (if not zero) i n the initial stages of impact. Figure 6 - 4 0 shows the reaction force applied to the central weft yarn at the fixed boundary of the S-728 target. It can be seen that the peak force applied to the yarn is within the 150-250 LIS timeframe. The influence of the slip timeinterval on the velocity-time response of the projectile is depicted i n Figure 6 41. It can be seen clearly that the onset of softening of the fabric response is directly linked to the beginning of the slip time-interval. It was previously discussed i n Chapter 6 that the model prediction of the fabric response was consistently stiffer than the E L V S measurement, specifically i n the latter stages of impact where a clear slope-change was evident i n the projectile velocity-time response (e.g. see Figure 5 - 6 and Figure 5 - 10). This sudden softening of the target can be attributed to the boundary slippage. Figure 6 - 4 2  shows the numerical predictions for the impact of an R C C  projectile at a strike velocity of 36 m/s on a 1-ply S-728 fabric, w i t h and without boundary slip consideration, compared to the E L V S data. The value of boundary slip considered i n this simulation was 5 m m , w h i c h is the average of the slip values measured during the post-mortem examination of this specific panel. The slipping was assumed to occur linearly between the times of 200 to 650 us. It can be seen that inclusion of the boundary slip improves the predictions of the model. However, the mechanics of such slip is not k n o w n at this point and an in-depth study of the slipping phenomenon is required both on the experimental and numerical fronts, before more conclusions can be drawn.  -196-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  6.4.4  Inter-Layer G a p  Interlayer gap is an important parameter affecting the impact simulation of multi-ply fabric targets. A method was previously discussed i n Chapter 6 to determine the gap between the layers using the data obtained from the transverse  compression experiments  of multi-ply fabric specimens.  The  numeric value of interlayer gap obtained from that approach for the six fabrics studied ranged between 0.10 m m and 0.14 m m . The model predictions of the projectile deceleration during the impact indicated a fabric response that appeared to be reasonably stiffer than the E L V S measurements. One of the sources of such discrepancy was associated w i t h the small value of initial interlayer gap between the layers of the fabric pack. In order to further investigate the effects of interlayer gap on the behaviour of the fabric pack, sensitivity of the velocity-time response of the projectile impacting a 16-ply S-728 fabric at 170 m/s to different values of interlayer gap is determined. The simulations are performed for the initial gap of 0.12 m m (as determined from the transverse compression data, see Table 5 - 2), as well as for a gap of 0.5 m m and 1.0 m m added to the initial interlayer gap. A s seen i n Figure 6 - 43, the addition of 0.5 m m gap significantly reduces the abrupt oscillations i n the velocity-time plot of the projectile, eliminating the sudden jumps i n the response. The source of such sudden jumps i n the response is linked to the amplitude of the stress wave generated i n the projectile upon impact that travels from its tip to the tail. Comparing the deceleration of the projectile tip and tail to that of its centre of mass (see Figure 6 - 44) for the impact simulation discussed above, it is evident  -197-  Chapter  6 - Sensitivity  Analysis  and Numerical  Case-Studies  that the two ends of the projectile oscillate around its centre of mass. Since the front E L V S measurements provide the deceleration of the projectile's tail and it appears to be smoother than the model response predictions w i t h the gap value calculated from the transverse compression data, it is concluded that i n reality there is more spacing between the adjacent plies of a multi-ply fabric pack. The reason behind the inaccuracies i n the interlayer gap calculated from the transverse compression data is due to the lack of information on the starting point of the test data. Since the initial thickness of the fabric specimen is unknown, it is highly probable that a major portion of the initial part of the compression data is discarded, leading to interlayer gap values that are significantly smaller than the real values. This topic should be considered i n more detail i n the future, i n order to determine the realistic interlayer gap that exists i n multi-ply fabric targets.  6.5  PENCILLING P H E N O M E N O N  Restricting the Behind A r m o u r Blunt Trauma (BABT) to an acceptable level is one of the important goals of any bullet-proof vest design. Lewis et al. [72, 73] reported on the observation of a pencilling-type injury while investigating the response of personal armour systems at l o w velocities. The characteristics of the  back-face  deformations  observed  deviated  significantly  from  the  conventional shallow hemispherical crater and showed a sharp and highly localized deformation mode. Further studies were carried out and similar results were produced i n ballistic impact tests firing 9 m m and 0.357" M a g n u m ammunitions at armour systems made from 20 plies of Kevlar® fabrics on a  -198-  Chapter  6 - Sensitivity  Analysis  and Numerical  Case-Studies  variety of different tissue simulants (Plastilina®, ballistic gelatine, ballistic soap), as w e l l as tests using p i g tissue and cadaver as the backing. Lewis et al. concluded from their studies that this type of injuries should be considered i n the design of bullet-proof vests. They argued that while the new designs of bullet-proof vests pass the acceptance criteria on the depth of back-face penetration, this deformation is highly localized compared to the shallow hemispherical deformation observed i n the traditional vests. They concluded that this sharp localized deformation can cause puncture wounds i n the body and can cause significant underlying injuries specifically over the ribcage. They linked the occurrence of the pencilling phenomenon to the light weight and flexibility of the new bullet-proof vests. In order to investigate the mechanisms that trigger such behaviour, ballistic impact response of an R C C  projectile striking a 4-ply target was compared to  that of a 1-ply fabric, both supported by a block of backing material to incorporate the interaction of the armour w i t h the body. The backing block was meshed as a cube w i t h dimensions of 300 m m x 300 m m x 180 m m . The material model used for the backing block was the elasto-plastic material model i n L S - D Y N A w i t h material properties provided i n Table 6 - 1 . The  RCC  projectile was considered to be rigid w i t h a mass of 8.1 grams (equal to that of a 0.357" M a g n u m bullet). Since the improvements i n the yarn mechanical properties i n recent years have resulted i n bullet-proof vests that are lighter than the traditional ones ([72, 73]), the elastic modulus and areal density of the 1-ply fabric was varied to probe the influence of each of the two parameters on the depth and shape of the back-face deformation.  -199-  Chapter  6 - Sensitivity  Table 6 - 2  Analysis  and Numerical  Case-Studies  provides the specifics of the numerical cases that were run to  investigate the effects of elastic modulus and areal density on the back-face deformation. The failure of the yarns i n the target was not considered i n this study to prevent the projectile from perforating the target. The projectile's strike velocity was set to 300 m/s to match that of the 0.357" M a g n u m bullet. Figure 6 - 4 5 shows the displacement of the projectile for simulations of impact on  4-ply target and backing only cases, as well as a mass-equivalent 1-ply  target (areal density multiplied by four) w i t h and without the increase i n elastic modulus. The high-modulus equivalent 1-ply case was considered to account for  the higher stiffness of the 4-ply target to that of a 1-ply target. A s seen i n  this figure, depth of penetration increases by moving from 4 to 1-ply target, indicating the effect of target thickness. Figure 6 - 4 6  shows the projectile  displacement obtained from the simulation of impact on 4-ply target, as well as 1-ply targets w i t h and without modulus increase. It should be noted that the areal density of the 4-ply target is four times those of the 1-ply targets. It is evident from this plot that reduction of the target mass significantly affects the projectile penetration depth. This mass reduction also affects the shape of the deformation i n the backing material. A s seen i n Figure 6 - 47, the shape of the deformation crater i n the backing material is steeper i n the lighter target (1ply)  than the heavier one (4-ply). From the above comparisons, one can  conclude that the occurrence of the pencilling phenomenon w i l l be highly dependent on the mass of the layers resisting the motion of the projectile (nonperforated layers i n the case of partial penetration), as well as the in-plane mobility (i.e. flexibility) of those layers, since they tend to bunch up around the projectile and conform to the steep and narrow shape of the deformation cone.  -200-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  In order to reduce the pencilling effect i n the body armours, the in-plane displacement of the individual layers should be constrained. This can be done through various methods, such as more efficient stitching of the layers together. Inclusion of layers w i t h high shear resistance (low shear locking angle) at the armour-body interface should also decrease the dominance of this mode of deformation, since high in-plane shear w o u l d reduce the flexibility of the target and resist the deformation of the bottom layers that have the tendency to shear significantly to wrap around the projectile and amplify the pencilling deformation.  6.6  RECOMMENDATIONS FOR INDUSTRIAL APPLICATION  The goal of this study, as mentioned i n Section 1.3, is to provide an efficient numerical tool for the analysis of fabric targets that is of use to personal armour designers. The verification, validation and sensitivity studies performed i n this thesis prior to this point indicate the ability of the 2D shell crossover model to successfully predict the deformational response of fabric panels. The overall modelling approach established i n this thesis is capable of p r o v i d i n g solutions to industrial design problems. This section summarizes the requirements for setting up a fabric model and the analysis of an impact event i n an industrial setting. Design of a fabric starts w i t h the choice of warp and weft yarns, their linear density, the weave type (plain, twill, etc.), and fabric thread count i n warp and weft directions. Once the fabric is woven, several measurements can be made to determine the fabric crimp and in-plane gap between the yarns i n the two  -201-  Chapter  6 - Sensitivity  Analysis  and Numerical  Case-Studies  directions. Knowledge of these parameters is sufficient for setting up the shellbased model of the fabric. The steps required to set up a model are presented below: • Calculate the fabric unit cell dimensions. A fabric unit cell is made of a single yarn crossover and its length i n each direction is calculated from the thread count i n the opposite direction. Each shell element w o u l d represent a single fabric unit cell, thus the fabric can be meshed from these elements w i t h k n o w n dimensions. • Determine the material properties (stiffness and strength) of the yarns. If mechanical properties of the yarns after weaving are not available, the properties of virgin yarns can be used, considering the sensitivity of the response to the yarn stiffness (Figure 6-7). • Calculate the initial yarn heights ( h ) using their length and crimp values t  (Equation 4 - 5 ) . The shell thickness is then determined as the summation of the yarn heights multiplied by two. • Determine the shear locking angle. This is done using the gap between parallel yarns, according to the process illustrated i n Section 4.3.2. • In the case of multi-ply targets, determine the interlayer gap. This can be done either through performing fabric transverse  compression tests,  manually measuring the thickness of the multi-ply fabric pack, or assuming  a  pre-determined  interlayer  -202-  gap.  Regardless  of  the  Chapter  6 - Sensitivity  methodology,  Analysis  and Numerical  however,  the  user  Case-Studies  should always consider  the  gap  sensitivity analysis performed earlier (Figure 6-43). • Calculate the parameters required for the transverse compressive model of the yarns. These parameters are perhaps the most difficult to determine. For  panels of Kevlar® 129, the transverse compression model derived i n  this thesis by fitting the predictions to the compression test data of the six panels studied can be used. If the fabric transverse compression data is not available, analysis of the two extreme cases of fabric w i t h decoupled and w i t h non-compressible yarns can be carried out to obtain the range of the response (see Figure 6 - 2 2 and Figure 6 - 30) and use engineering judgment to determine a reasonable approximation for the behaviour of the target. Following the above steps, the user w o u l d have all the input data needed to set up a finite element model of the target and carry out the simulation of the ballistic event. The designer can use this model to help identify the weaknesses of a specific design and possibly improve/optimize the armour performance, though more w o r k still needs to be done  6.7  Due  SUMMARY  to the uncertainties associated w i t h the determination of many mechanical  and geometric fabric properties, sensitivity of the fabric response to these parameters  was  investigated  in  this  chapter.  The  energy  absorption  mechanisms dominating the dynamic response of the fabric were investigated  -203-  Chapter  6 - Sensitivity  Analysis  and Numerical  Case-Studies  and two distinct zones of behaviour were identified. The effects of various input parameters on the  overall response  were  examined within  this  framework. The influence of yarn elastic modulus and crimp, and fabric areal density were investigated. It appeared that while the areal density is the only parameter that w o u l d noticeably affect the zone I response, all three parameters w o u l d equally affect the zone II response. While determination of the fabric areal density and yarn elastic modulus are more trivial through laboratory measurement and mechanical testing of the yarns extracted from the fabric, yarn crimp is more difficult to measure as it can be affected by many factors such as handling of the fabric, etc. It was found that for a constant strain-to-failure value, increasing the elastic modulus and yarn crimp can lead to a higher critical velocity. Transverse compressive behaviour of the yarns appeared to have significant influence on the interaction of the yarns. Yarns that are transversely compliant transform a fabric's response through reducing the interaction of the crossing yarns. This is more dominant i n unbalanced high crimp fabrics than the balanced low crimp ones. Determination of the transverse behaviour of yarns inside the fabric is a difficult process and the user should determine the bounds of  the response by performing simulations w i t h decoupled and/or non-  compressive yarns. Sensitivity of the ballistic response of the fabrics to the components of the shear model was investigated. A s discussed, the shearing of the fabric i n the specific ballistic experiments simulated i n this study seemed to be w i t h i n the low-shear portion of the fabrics' response. A s a result, increasing the shear resistance up  -204-  Chapter  6 - Sensitivity  Analysis  and Numerical  Case-Studies  to a certain limit d i d not affect the ballistic response of the fabric. It was also suggested that a more sophisticated shear model be implemented i n the shell elements, i n the cases where the proposed model is being used i n applications where the fabric w o u l d undergo high shear deformations. The influence of the almost inevitable fabric slippage at the fixed boundaries during the ballistic impact experiments on the overall response of the fabric was investigated i n Section 6.4.3. It was observed that the slippage of the fabric at the boundaries can significantly affect the response of the fabric, and was suggested that the experimental information on the boundary slippage be precisely recorded and analysed. This phenomenon not only w o u l d affect the energy absorption of the fabrics but can also affect the critical velocity determination for a specific fabric target. Pencilling phenomenon, a common characteristic back-face deformation of the new lightweight body armours was also investigated i n this chapter. The numerical simulations confirmed that the  occurrence  of the  pencilling  behaviour is linked to the light weight and flexibility of the target, and found that addition of high-shear-resistant  layers can reduce the probability of  occurrence of such phenomenon.  -205-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  Table 6 - 1 : Mechanical properties of the dummy backing material used to investigate pencilling phenomenon.  Property  Value  E (MPa)  100.0  Density (g/mm )  3.0E-03  Y i e l d strain  0.20  Erosion strain  0.95  2  Table 6-2: Elastic modulus and density of the yarns considered in the simulations performed to investigate the pencilling phenomenon.  Simulation  Target  Elastic modulus (GPa)  Total A r e a l density  Case 1  1-ply  480.0  1155  Case 2  1-ply  96.0  1155  Case 3  1-ply  480.0  231  Case 4  1-ply  96.0  231  Case 5  4-ply  96.0  1155  -206-  (g/m ) 2  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  0  50  100  150  200  250  300  Time (us)  Figure 6-1: Velocity-time response of the RCC projectile impacting a 1-ply S-728 fabric at 100 m/s. Two zones of behaviour are identified, dominated by mass (zone I) and stiffness (zone II) of the target.  Figure 6-2: Contours of in-plane fabric velocity in warp (right) and weft (left) directions reaching the panel boundaries, predicted for the impact of an RCC projectile on a 1-ply S-728 fabric at 100 m/s.  -207-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  Figure 6-4: Kinetic/strain energy ratio of the total energy absorbed by a 1-ply S-728 fabric impacted at 100 m/s by an R C C projectile.  -208-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  120  i  0  20  40  60  80  100  Time (us) Figure 6-5: Tension-time response of the weft yarn of a 1-ply S-728 fabric impacted at 100 m/s by an RCC projectile.  Figure 6-6: Tension-displacement response of an S-728 unit-cell loaded in the warp direction with fixed weft yarn, showing the effects of ±20% change in the elastic modulus of the yarns.  -209-  Chapter  6 - Sensitivity  120  Analysis  and Numerical  Case-Studies  !  0  50  100  150  200  250  300  T i m e (us)  Figure 6-7: Velocity-time response of the RCC projectile impacting a 1-ply S-728 fabric at 100 m/s, showing the effects of ±20% change in the elastic modulus of the yarns.  Figure 6-8: Strain-time response of the central weft yarn of an S-728 in an impact simulation of an RCC striking a 1-ply target at 100 m/s, showing the effects of modulus change.  -210-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  5.0 I Measurement 1 I Measurement 2  720  726  Kevlar® 129 Panel  Figure 6-9: Crimp measurement performed independently on the six Kevlar® 129 fabric panels by two different individuals, showing a maximum deviation of 20% from each other.  300 Cr = 0.8 Cr„  Cr  avg  = 1.35%  O 200  '35 c CD  Decreasing crimp Cr'  re >a 100  5 0.00  0.01  0.02  0.03  0.04  Warp Displacement, d i (mm)  Figure 6 -10: Tension-displacement response of an S-728 unit-cell loaded in the warp direction with fixed weft yarn, showing the effects of ±20% change in the yarn crimp.  -211-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  120  i  Time (us) Figure 6 - 11: Velocity-time response of the RCC projectile impacting a 1-ply S-728 fabric at 100 m/s, showing the effects of ±20% change in the yarn crimp.  2.0% i  Time (us) Figure 6 - 12: Strain-time response of an S-728 central weft yarn in an impact simulation of an R C C striking a 1-ply target at 100 m/s, showing the sensitivity of the fabric response to crimp.  -212-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  120  i  Time (us)  Figure 6 - 13: Velocity-time response of the RCC projectile impacting a 1-ply S-728 fabric at 100 m/s, showing the effects of ±20% change in the fabric areal density.  Figure 6 - 14: Strain-time response of an S-728 central weft yarn in an impact simulation of an R C C striking a 1-ply target at 100 m/s, showing the sensitivity of the fabric response to areal density.  -213-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  105  i  80 J 0  ,  ,  ,  1  1  20  40  60  80  100  T i m e (us)  Figure 6 - 15: Initial 100 us of the velocity-time response of the R C C projectile impacting a 1-ply S-728 fabric at 100 m/s, showing the effects of ±20% change in the fabric elastic modulus.  105 -,  T i m e (us)  Figure 6 -16: Initial 100 us of the velocity-time response of the R C C projectile impacting a 1-ply S-728 fabric at 100 m/s, showing the effects of ±20% change in the fabric crimp value.  -214-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  105  -i  80  -I  ,  ,  ,  ,  ,  0  20  40  60  80  100  T i m e (us)  F i g u r e 6 - 17:  I n i t i a l 100 u s o f t h e v e l o c i t y - t i m e r e s p o n s e o f t h e R C C p r o j e c t i l e i m p a c t i n g a 1-ply  S-728 f a b r i c  at  100 m / s ,  s h o w i n g t h e effects o f  ±20%  change i n the fabric areal density.  Figure  6 -18:  Tensile response o f the w a r p yarn o f a n  S-728  fabric unit-cell subjected to  d i s p l a c e m e n t i n t h e w a r p d i r e c t i o n (weft y a r n fixed), a s s u m i n g compressible, non-compressible and decoupled crossover  -215-  yarns.  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  120  i  1-ply S-728 Fabric 100  Yarn: 1500 denier Count: 17 tpi Crimp: 1.35%(avg) Modulus: 96.0 GPa  Non-Compressible Yarns  Compressible Yarn 40 H  0.00  0.01  0.02  W a r p Displacement,  0.03  0.04  (mm)  Figure 6 -19: Tensile response of the weft yarn of an S-728 fabric unit-cell subjected to displacement in the warp direction (weft yarn fixed), assuming compressible, non-compressible and decoupled crossover yarns.  Figure 6 - 20: Surfaces of energy stored in an S-728 fabric unit-cell subjected to warp and weft displacement, assuming compressible, non-compressible and decoupled crossover yarns.  -216-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  1-plv S-728 Fabric  0.00  0.01  0.02  Warp Displacement,  0.03  0.04  (mm)  Figure 6 - 21: Energy stored in an S-728 fabric unit-cell subjected to displacement in the warp direction (weft yarn fixed), assuming compressible, noncompressible and decoupled crossover yarns.  Figure 6 - 22: Velocity-time response of an R C C projectile impacting a 1-ply S-728 fabric at 100 m/s, predicted assuming compressible, non-compressible and decoupled crossover yarns.  -217-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies 16  n  0  5  10  15  20  Projectile Displacement (mm)  Figure 6 - 23: Energy-time response of an RCC projectile impacting a 1-ply S-728 fabric at 100 m/s, predicted assuming compressible, non-compressible and decoupled crossover yarns.  Figure 6 - 24: Ratio of the kinetic to internal energy components stored in the fabric in the impact simulation of an RCC projectile striking a 1-ply S-728 fabric at 100 m/s, predicted assuming compressible, non-compressible and decoupled crossover yarns.  -218-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  Figure 6 - 25: Tensile response of the warp yarn of an S-731 fabric unit-cell subjected to displacement in the warp direction (weft yarn fixed), assuming compressible, non-compressible and decoupled crossover yarns.  Figure 6 - 26: Tensile response of the weft yarn of an S-728 fabric unit-cell subjected to displacement in the weft direction (warp yarn fixed), assuming compressible, non-compressible and decoupled crossover yarns.  -219-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  Figure 6 - 27: Surfaces of energy stored in an S-731 fabric unit-cell subjected to warp and weft displacement, assuming compressible, non-compressible and decoupled crossover yarns.  0.00  0.01  0.02  0.03  0.04  0.05  Warp Displacement, d i (mm)  Figure 6 - 28: Energy stored in an S-731 fabric unit-cell subjected to displacement in the warp direction (weft yarn fixed), assuming compressible, noncompressible and decoupled crossover yarns.  -220-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  0.004 -i  0.000  0.005  0.010  0.015  W e f t Displacement, d ( m m ) 2  Figure 6 - 29: Energy stored in an S-731 fabric unit-cell subjected to displacement in the weft direction (warp yarn fixed), assuming compressible, noncompressible and decoupled crossover yarns.  Figure 6 - 30: Velocity-time response of an R C C projectile impacting a 1-ply S-731 fabric at 100 m/s, predicted assuming compressible, non-compressible and decoupled crossover yarns.  -221-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  Figure 6-31: Ratio of the kinetic to internal energy components stored in the fabric in the impact simulation of an RCC projectile striking a 1-ply S-731 fabric at 100 m/s, predicted assuming compressible, non-compressible and decoupled crossover yarns.  Figure 6 - 32: Maximum shear strain observed in the six Kevlar® 129 panels impacted by an R C C flying at 100 m/s, compared to the stiffening and locking shear strains, y and y . x  2  -222-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  Figure 6 - 33: Velocity-time response of an RCC projectile impacting a 1-ply S-728 fabric at 100 m/s, showing two extreme cases of low and high fabric shear modulus values.  Figure 6 - 34: Deformation patterns shown by contours of z-displacement considering low (left) and high (right) fabric shear modulus.  -223-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  Figure 6 - 35: Velocity-time response of an RCC projectile impacting a 1-ply S-728 fabric at 100 m/s, showing the effect of stiffening and locking angles on the response of the projectile.  Figure 6 - 36: Boundary slippage marked during a ballistic impact experiment on a 1ply S-728 panels at a strike velocity of 36.9 m/s.  -224-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies 120  i  0  5  10  15  20  25  30  Distance from the Central Yarn (mm) Figure 6 - 37: Boundary reaction in the weft direction of an S-728 1-ply fabric impacted by an R C C projectile at 100 m/s.  Figure 6 - 38: Simulation of slip at the boundary of an S-728 1-ply fabric through prescribing the in-plane motion of the nodes at the boundaries (weft slip shown here).  -225-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  Figure 6 - 39: Velocity-time response of an RCC projectile impacting a 1-ply S-728 fabric at 100 m/s, showing the effect of weft boundary slip on the response of the projectile.  120 -|  £  a>  90  u  c o o ra O 0.  60  Peak Rection at the Boundary  S-728 Fabric  4-  Yarn: 1500 denier Count = 17 tpi Cr'avg , = 4.36% V =100 m/s  Q) 30  a  100  150  200  250  300  350  400  T i m e (us)  Figure 6 - 40: Reaction force-time response of the central weft yarn in an impact simulation of an RCC projectile striking a 1-ply S-728 fabric at 100 m/s.  -226-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  Figure 6 - 43: Velocity-time response of an RCC projectile impacting a 16-ply S-728 fabric at 170 m/s, showing the effect of interlayer gap on the projectile deceleration.  200 16-plv  S-728  Fabric  Yarn: 1500 denier Count: 17 tpi Crimp: 1.35%(avg) Modulus: 96.0 GPa  1 io  Projectile's Centre of Mass  o  CD  > CD  2 0_  100  Figure 6 - 44: Velocity-time response of an RCC projectile impacting a 16-ply S-728 fabric at 170 m/s, showing responses of its tip, tail, and centre of mass.  -228-  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  70  n  0  SO  100  150  200  250  300  350  Time (us)  Figure 6 - 45: Displacement-time response of an 8g RCC projectile impacting 4-ply and mass-equivalent 1-ply targets at 300 m/s.  70 -,  0  50  100  150  200  250  300  350  Time (us)  Figure 6 - 46: Displacement-time response of an 8g RCC projectile impacting 4-ply and 1-ply targets at 300 m/s.  -229-  •  Chapter 6 - Sensitivity Analysis and Numerical Case-Studies  1-ply target 5E.A,  1 if! 1  1-ply target  '1  ft  ^* ^ d  ^tv  •  Figure 6-47: Post-mortem deformation observed in the backing material in the impact simulation of an 8g RCC projectile striking a 4-ply (top) and 1-ply (bottom) S-728 targets at 300 m/s.  -230-  Chapter 7 - Summary, Conclusions and Future Work  C H A P T E R 7-  SUMMARY,  CONCLUSIONS  A N D FUTURE  WORK  7.1  SUMMARY  The work presented i n this thesis introduces a new approach to the numerical modelling of textile materials that are specifically designed for ballistic applications. This approach takes advantage of the available experimental information  and  the  development  of  novel  modelling techniques  to  characterize the mechanical behaviour of a single fabric crossover embedded i n a fabric unit cell i n order to simulate ballistic impact experiments on multi-ply packs of fabric. To study the behaviour of the fabric unit cell, a detailed 3D finite element model is developed that explicitly considers the warp and weft yarns i n a single crossover. This 3D crossover model consists of bar and solid elements with specialized constitutive models that capture the axial and  transverse  properties of the yarns. The 3D crossover model proved successful i n capturing many details of fabric geometry and yarn interaction during the fabric  -231-  Chapter 7 - Summary, Conclusions and Future Work  deformation, but appeared extremely inefficient due to the large amount of computational resources that it needed to carry out the analysis, to a point that it is impractical for simulations of real size (full size) fabric. A n efficient shell element is also developed to represent the mechanical response of a single yarn crossover i n a more efficient manner. The material model developed for this shell element is based on a sine-based representation of the yarn profiles and considers their interaction as they deform under the applied displacements. While the response predictions of the shell elements agrees well w i t h those obtained from the 3D fabric crossover model, they are significantly more efficient to run thus making them suitable for the simulation of real-size multi-ply fabric panels. The efficient shell elements are used to perform simulations of ballistic impact on single and multi-ply fabric targets. The predictions of the model are compared w i t h the experimental data obtained from instrumented ballistic experiments. Overall, the model has proved to be successful i n capturing the essence of the dynamic response of fabric targets and where discrepancies have been detected their source have been probed. A detailed sensitivity study has been performed on several geometric and mechanical properties of the fabric. This analysis revealed many characteristic features of the fabrics that are influential i n their specific energy absorption properties during a ballistic event. The model has also been used to study the pencilling phenomenon and the factors that affect its occurrence i n lightweight armour systems.  -232-  Chapter 7 - Summary, Conclusions and Future Work  7.2  CONCLUSIONS  The study presented i n this thesis has resulted i n a robust numerical tool for simulating the complex deformational behaviour of fabrics under impact. The numerical model enables the user to perform an in-depth investigation of the physics of fabric deformation and gain knowledge on the importance of various design parameters. In general, the 3D crossover model proved to be invaluable i n determining the mechanical behaviour of the fabric particularly under loading scenarios that do not lend themselves to impact experiments. Development of the constitutive model for solid elements that provides the transverse mechanical response of the yarns paves the way for studying the unit cell response of any fabric weave, as long as the shape of the yarns i n the weave is k n o w n . In other words, there is no limit on applicability of the 3D crossover model, as long as the finite element mesh of the fabric unit cell can be generated. Development of the 2D shell crossover model has been instrumental i n carrying out the simulations of ballistic impact experiments and the related sensitivity studies efficiently. The efficiency of the shell elements makes them a practical choice for creating models of real-life dimensions and analyzing fullscale impact events. The shell elements are highly versatile and can be easily extended  to  incorporate  other  weave  types  for  future  research  development. Some of the specific conclusions drawn from this thesis are listed below:  -233-  and  Chapter 7 - Summary, Conclusions and Future Work  • The transverse mechanical response of the yarns inside the fabric has proved to be very difficult to characterize. The data from transverse compression tests used i n this study provide a useful guide on how to calibrate such a yarn response. Despite the usefulness of the transverse compression tests, more study needs to be performed to better understand the transverse behaviour of yarns, i n particular, and that of the fabric as an assembly of yarns. • The method developed i n this study for inter-layer gap determination i n multi-ply fabric systems takes advantage of the data from the transverse fabric compression tests. However, it appeared  that the gap values  calculated result i n multi-ply panel response predictions that are much stiffer than those of the real panels. It is speculated that i n reality, the gap between the layers is larger than the ones estimated, and this discrepancy is associated w i t h the truncation of the initial segment of the transverse force-displacement data available to this study. • The success of a numerical simulation is highly dependent on the validity of its underlying assumptions. Therefore, it is extremely important that these assumptions be based on reality and remain valid throughout the analysis. The assumption of sine-based yarn profile i n developing the material model for the shell elements proved not to be valid for loose fabrics (low count and small yarn linear density) and resulted i n a deviation of the predictions from the experimental measurements. • Perhaps one of the most important fabric parameters that could influence the appearance of different deformational and failure mechanisms is yarn  -234-  Chapter 7 - Summary, Conclusions and Future Work  crimp. This factor, overlooked by the majority of previous studies, directly controls a fabric's energy absorption characteristics by affecting its longitudinal  and  transverse  wave  speeds. Furthermore,  the  crimp  imbalance i n warp and weft yarns causes the overloading of one direction compared to the other one and leads to a non-optimal fabric system. • T w o distinct zones of behaviour have been identified for the energy absorption of the fabrics. Specifically, it is found that the amount of energy absorbed  i n the initial stages of impact (zone I) is mostly  influenced by the mass of the fabric. This reinforces the suggestion by Cunniff [71] to place material w i t h inferior mechanical properties on the strike face of the multi-ply fabric targets, since the projectile perforation i n these top layers occur i n zone I of their energy absorption response. Overall, it was concluded that i n order to truly optimize the energy absorption characteristics of a fabric panel, one needs to consider many fabric and yarn properties  and their relative influence on the overall fabric mechanical  response. The numerical models developed i n this thesis provide suitable tools for investigating the fabric behaviour and can reduce the reliance on the experimental efforts i n the design and optimization process of the personal armour systems.  7.3  CONTRIBUTIONS TO T H E STATE-OF-THE-ART  The strength of this study is its ability to successfully merge the experimental data and their interpretation w i t h the numerical models to obtain a realistic  -235-  Chapter 7 - Summary, Conclusions and Future Work  representation of the fabric's static and dynamic deformational behaviour. Availability of the experimental measurements during the impact event [51] provides a unique opportunity to validate the detailed time history of the predictions of various quantities of interest. Typically, the success of ballistic response predictions are based on h o w well they predict the residual velocity of the projectile as it exits the target or essentially the total energy absorbed by the target. The high resolution validation process afforded b y the E L V S system can expose the smallest flaws i n the numerical model, ones that w o u l d not be discovered i n the absence of such detailed experimental data. The result of this study is a computational model that has met such high standards for validation and the limitations of its applicability are well established. Furthermore, all the variables used i n this approach are based on the physical reality of the fabric, whose values have been calibrated by appropriate experimental measurements. The main advantage of the work conducted throughout this thesis is the way i n which the two unit cell models (2D shell crossover and 3D fabric crossover models) complement each other. The detailed 3D fabric crossover model not only provides information on the transverse compression of the yarns as input to the 2D shell crossover model, it also serves to verify the predictions of the shells on the mechanical response of a single fabric crossover. The 3D crossover model provides an invaluable tool to gauge the suitability of the assumptions made i n deriving the fundamentals of the shell model, as w e l l as identifying the limits of their validity. The formulation of the shell elements based on the sine-based yarn profiles developed i n this study is unique, and is i n accordance w i t h the yarn shapes  -236-  Chapter 7 - Summary, Conclusions and Future Work  observed i n the cross-sectional micro-images of fabrics [42], as well as the predictions of the detailed 3D crossover model. Traditionally, yarns with trusslike bilinear configuration have been used i n the representative shell elements [57]. These models lead to theoretical yarn geometries that are different from what is observed i n the real fabrics and given the high dependency of the response on the yarns configuration, result i n inconsistent crossover response. Furthermore, the comparison between the 3D fabric crossover model and the 2D shell elements w o u l d not be meaningful due to their significantly different geometries. The fabric models assembled from the shell elements w i t h sinebased formulation have been shown to capture the ballistic response of fabric targets successfully, making them a good candidate for future  studies.  Compared to the traditional discrete models, the shell prediction of the fabric response under ballistic impact shows a slight improvement for the fabrics considered i n this study. However, this is due to the fact that the coupled biaxial behaviour of the crossing yarns considered along w i t h the inclusion of yarn transverse compression affect the fabric response i n opposite ways. Therefore, little difference is observed i n the projectile deceleration prediction compared to that of the discrete pin-jointed model that fails to capture these two features of fabric behaviour. A s a result, the shell crossover model developed here provides a more realistic mechanics of the fabric compared to that obtained from the discrete model.  The contributions made to the 3D crossover and the 2D shell crossover models are summarized i n Table 7 - 1 and Table 7 - 2 . For a comparison of the model efficiencies, the reader is referred to Table 4 - 3 for execution times required to  -237-  Chapter 7 - Summary, Conclusions and Future Work  complete the analysis of impact on fabric models constructed from the 2D and 3D crossover models, as w e l l as the discrete pin-jointed model. Overall, the numerical model provided i n this study is a robust state-of-the-art tool for analysing the mechanical response of fabric structures, i n general, and their ballistic behaviour, i n particular.  RECOMMENDATIONS FOR FUTURE W O R K  7.4  The approach introduced i n this thesis is only a stepping stone towards a fully inclusive and comprehensive model of fabrics. Despite the success of the proposed models i n fulfilling the initial goals, improvements are needed to be made  in  order  to  expand  the  applicability of  the  approach.  The  recommendations for future w o r k are: • One of the influential factors i n the mechanical response of fabrics was found to be the transverse compression of the yarns. Further studies should be conducted on designing more suitable experiments to capture the yarns true transverse behaviour inside a fabric. Measurement of the fabric thickness before and after the experiments was found to be extremely valuable, specifically i n determining the inter-layer gap i n multi-ply fabric packs. • The transverse compression model needs to be revisited, and a mechanicsbased model be developed to capture the interaction of the yarns i n a specific weave structure during compression. The model implemented i n the solid elements to capture the transverse mechanical response of the  -238-  Chapter 7 - Summary, Conclusions and Future Work  yarn also needs to be improved and based on the physics of interaction between the individual fibres of a yarn. • Degradation of the material and geometrical properties of yarns during the weaving process should be further investigated. A s mentioned before, it is k n o w n that the weaving process damages the yarns and causes degradation of their mechanical properties  [68, 69], w h i c h i n turn  introduce an undetermined amount of error i n the predictions of the numerical model. The true properties of the yarns need to be determined by performing experiments on the yarns extracted from the fabric. • One of the assumptions made i n establishing the crossover models of fabric i n this study is the assumption of frictionless contact between the crossing yarns. Effects of this inter-yarn friction on the overall behaviour of a yarn crossover should be further  investigated  and the  error  introduced by such assumption be evaluated. • The shear component of the fabric unit cell was found to have little effect on the energy absorption of the fabrics for the loading cases studied i n this thesis. However, this might not be true for other applications where the fabric is subjected to extreme shear deformations, or fabrics with low shear locking angles. The shear model implemented i n the shell elements is based on a simple trellis mechanism. A more mechanistic shear model that considers  the  interaction of the  yarns  deformation mechanism needs to be developed.  -239-  during the  scissoring  Chapter 7 - Summary, Conclusions and Future Work  • One of the shortcomings of the model developed is its oversimplified yarn failure criterion. The instantaneous strain-to-failure model incorporated i n the shell element is highly sensitive to any oscillations of numerical noise that is introduced i n the calculations from a variety of sources such as contact. A s a result, premature failure of yarns was observed, specifically for  targets w i t h higher  number  of layers.  Implementation  of  an  appropriate failure model that is not affected by the oscillations i n the response is highly beneficial to the accuracy of the predictions, especially in the penetrating impact simulations. • The efficiency of the model can be further increased by implementing a more optimized mathematical solution to the differential equations. The biaxial behaviour of a yarn crossover can ideally be represented by the response surfaces that provide the tension history of the yarns as a function of the applied displacements. Instead of calculating the response at each time-step during the analysis, this can be done as a pre-processing step and implemented into the shell U M A T as a direct mathematical function or look up table, to further reduce the execution time. • Some of the simplifying assumptions used i n developing the material constitutive model of the shell elements should be further investigated. For instance, the assumptions of symmetric unit cell displacements and the sine shape of the yarn profile should be relaxed to determine their effects on the unit cell and ultimately fabric response. • The approach employed i n this study is a purely deterministic one. In reality, the material and geometric properties of the fabrics and yarns  -240-  Chapter 7 - Summary, Conclusions and Future Work  have some uncertainty associated w i t h them, due to the presence of flaws, material handling, measurement techniques, etc. Incorporation of the uncertainties i n numerical analysis is a subject of great interest i n the field of computational mechanics. Prediction of the ballistic response of fabrics w o u l d greatly benefit from the current research i n this area.  -241-  Chapter 7 - Summary, Conclusions and Future Work Table 7-1: Comparing the 3D crossover model developed in this study to that of the current literature.  Property  Current 3D M o d e l  3D M o d e l s i n Literature  Elements  Solid and bar  Solid  Spurious modes of deformation  Eliminated  Present, reduced by hourglass control  Yarn material model  Orthotropic  Orthotropic  Yarn transverse material model  Coupled non-linear model (dependent on fibre volume fraction)  Decoupled linear elastic (Poisson's ratio =0)  Transverse mechanical properties  Calibrated w i t h compression experiments  Generally not calibrated  Table 7-2: Comparing the 2D shell model developed in this study to that of the current literature.  Property  Current Shell M o d e l  Shell Models i n Literature  Warp and weft yarns  Coupled  C o u p l e d and decoupled  Y a r n shape  Sinusoidal  M a i n l y linear  Transverse compression calibration  3D crossover model, or experiments  Generally not calibrated  Verification  3D crossover model  none  -242-  Bibliography  Bibliography 1. Shahkarami, A . 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(1998), L S - D Y N A Theoretical Manual, Livermore Software Technology Corporation.  -247-  Bibliography  57. Kawabata, S., M . N i w a , and H . K a w a i (1973), "The finite-deformation theory of plain-weave fabrics Part I: The biaxial-deformation theory," Journal of the Textile Institute, 64(2), 21-46.  58. Kawabata, S., M . N i w a , and H . K a w a i (1973), "The finite-deformation theory of plain-weave fabrics Part II: The uniaxial-deformation theory," Journal of the Textile Institute, 64(2), 47-61.  59. Kawabata, S., M . N i w a , and H . K a w a i (1973), "The finite deformation theory of plain-weave fabrics Part III: The shear-deformation theory," Journal of the Textile Institute, 64(2), 62-85.  60. Press, W . H . , S. A . Teukolsky, W . T. Wetterling, and B. P. Flannery (1992), Numerical recipes i n F O R T R A N 77: The art of scientific computing, Cambridge University Press. 61. Wylie, C . R. and L . C . Barrett (1995), Advanced engineering mathematics, M c G r a w - H i l l , Inc. 62. Prodromou, A . G . and J. Chen (1997), "On the relationship between shear angle and wrinkling of textile composite preforms," Composites Part A: Applied Science and Manufacturing,  28(5), 491-503.  63. McBride, T. M . and J. Chen (1997), "Unit-cell geometry i n plain-weave fabrics during shear deformations," Composites Science and Technology, 57(3), 345-351. 64. Belytschko, T. B., J. L i n , and C . S. Tsay (1984), "Explicit algorithms for nonlinear dynamics of shells," Computational Methods in Applied Mechanical Engineering,  42, 225-251.  65. Hughes, T. J. R. and W . K . L i u (1981), "Nonlinear finite element analysis of shells: Part I. Three-dimensional shells," Computer Methods in Applied Mechanics and Engineering,  26, 331-362.  66. Hughes, T. J. R. and W . K . L i u (1981), "Nonlinear finite element analysis of shells- Part II. Two-dimensional shells," Computer Methods in Applied Mechanics and Engineering,  27,167-181.  67. Lebrun, G., M . N . Bureau, and J. Denault (2003), "Evaluation of biasextension and picture-frame test methods for the measurement of intraply shear properties of PP/glass commingled fabrics," Composite Structures, 61(4), 341-352.  -248-  Bibliography  68. Lee, L., S. Rudov-Clark, A . P. Mouritz, M . K . Bannister, and I. Herszberg (2002), "Effect of weaving damage on the tensile properties of threedimensional w o v e n composites," Composite Structures, 57, 405-413. 69. Rudov-Clark, S., A . P. Mouritz, L . Lee, and M . K . Bannister (2003), "Fibre damage i n the manufacture of advanced three-dimensional woven composites," Composites: Part A, 34, 963-970. 70. Roylance, D., P. Hammas, J. Ting, H . C h i , and B. Scott (1995), "Numerical M o d e l i n g of Fabric Impact," Proceedings of High Strain Rate Effects on Polymer, Metal and Ceramic Matrix Composites and Other Advanced  Materials, 155-160. 71. Cunniff, P. M . (1999), "Decoupled response of textile body armor," Proceedings of 18th International Symposium on Ballistics, 814-821.  72. Lewis, E. A . , I. Horsfall, and C. Watson (2006), "Pencilling: A novel behind armour blunt trauma injury," Proceedings of 22nd International Symposium on Ballistics, 1326-1332.  73. Lewis, E. A . (2005), "Pencilling: A novel behind armour blunt trauma injury," P h D Thesis, Cranfield University. 74. Chapman, S. J. (2000), M A T L A B Programming for Engineering, Brooks/Cole Publishing Company. 75. Lomov, S. V., A . V . Gusakov, G . Huysmans, A . Prodromou, and I. Verpoest (2000), "Textile geometry preprocessor for meso-mechanical models of w o v e n composites," Composites Science and Technology, 60, 2083-2095. 76. Brown, D., M . Morgan, and R. Mcllhagger (2003), " A system for the automatic generation of solid models of w o v e n structures," Composites: Part A, 34,511-515.  -249-  Appendix A-3D  Crossover Mesh Pre-processor  APPENDIX A - 3D CROSSOVER MESH PRE-PROCESSOR Creation of the finite element mesh of a physical model is one of the major steps i n the mechanical analysis of any system. This task gets even more cumbersome w h e n the geometries of the bodies studied are complicated. To create the unit-cell of a woven fabric, multiple trials of mesh generation might be needed due to the existing uncertainties on the accuracy of some input parameters. Availability of a specialized pre-processor to create the mesh of a fabric unit-cell w o u l d greatly facilitate the creation of a consistent F.E. mesh of a yarn crossover. A finite element mesh pre-processor was developed that w o u l d create unitcells of plain-weave fabrics. In developing the pre-processor, basic assumptions are derived from investigating cross-sectional micro-images of various fabric panels. The following sections highlight the findings from the micro-images and the basics of the 3D unit-cell pre-processor.  A . l FABRIC CROSS-SECTIONAL MICRO-IMAGES Geometrical properties of yarn i n the woven fabric are one of the basic information needed to develop 3D models of fabric unit-cells. A yarn's  -250-  Appendix A-3D  Crossover Mesh Pre-processor  geometry and profile i n a woven fabric is mainly influenced by its material properties and the weave architecture, as well as many other processing parameters. A s a result, the final shape of yarns i n the weave is a product of the above-mentioned factor and the interaction of the orthogonally woven yarns. The woven profile and  configuration of the yarns i n the weave  was  investigated by optical microscopy of fabric samples. This section discusses the highlights of the procedure and the results obtained from imaging the crosssections of various plain-weave fabrics of Kevlar® 129. In order to obtain images of the fabric cross-section, 4cm by 4cm squares of fabric were cut from the fabrics and submerged i n a translucent araldite epoxy resin [42]. After the resin cured, the samples were sectioned parallel to the warp and weft directions, exposing the cross-section of the w o v e n fabric. Images of the weave architecture and yarn profiles were then obtained by observing the samples i n a Wilde™ dissecting microscope. A sample of the warp and weft images obtained for the S-728 fabric is presented i n Figure A - 1 . The digital images obtained from the optical microscope were analysed and various parameters such as crimp and relative position of the yarns were measured, as seen i n Figure A - 2. The dimensions measured on the microimages are summarized i n Table A - 1 . From the geometrical values measured, many fabric and weave properties such as yarn crimp, thread count, and void content of the yarns can be estimated using the fabric cross-sectional microimages. The yarn crimp and void ratios determined from the micro-images are provided i n Table A - 2. The crimp values obtained from the micro-images  -251-  Appendix A - 3D Crossover Mesh Pre-processor  appeared generally to have a larger value than the hand measured ones. This discrepancy was related to the facts that the yarn crimp i n the images might have altered during cutting and handling of fabric samples, or may reflect local effects. The yarns may have also absorbed some liquid during the potting procedure, further changing the yarn crimp observed i n the images. The void content of the yarn was calculated using the cross-sectional area of the yarn (A ) measured from image analysis compared the total theoretical solid T  area of the fibres i n the yarn (A ) obtained from the yarn linear density. The f  yarn's initial void is thus calculated from the following equation:  Void =  (A-l)  The void ratios calculated show a surprisingly high level of v o i d i n the yarns (57% to 71.5%) leading to the suspicion that the yarns might have absorbed some liquid during the potting process. A n important observation i n the micro-images was the presence of yarn overlay. A s seen i n Figure A - 3, the fibres i n the yarns tend to spread apart at the centre of each crossover point, thus overlaying their adjacent yarns. This leads i n to a non-uniform yarn cross-section and is mostly observed i n the panel direction w i t h l o w crimp value. However, despite all the discrepancies observed, the micro-images of fabric cross-section provide invaluable qualitative information on the yarn shapes and profiles i n the woven fabric. This information was used to establish the  -252-  Appendix A-3D  Crossover Mesh Pre-processor  basis for the generation  of the F.E. mesh of the yarn crossovers  and  development of the pre-processor, as presented i n the following section. The location of the cut is usually aligned w i t h the centerline of a warp or weft yarn. However, there is usually some error involved w i t h the sectioning process. Depending on the normal distance of the cutting line from the yarn centreline, the thickness yarn profile visible i n the image, as well as the crosssectional dimensions and their alignment gets affected. The change i n the profile thickness and yarn alignment is clearly evident i n Figure A - 4, where the cutting plane makes an angle with the warp yarn.  A.2 FINITE ELEMENT UNIT-CELL MESH PRE-PROCESSOR A pre-processor was developed i n M A T L A B [74] environment to create the geometry and F.E. meshes of plain-weave fabric unit-cells, later used to analyse the biaxial response of fabrics. A s mentioned earlier, the pre-processor takes advantage  of  the  findings  from  the  cross-sectional  micro-images  to  approximate the shape of the yarns i n the fabric. There have been several studies on developing pre-processors to create the profile of yarns i n a certain fabric weave structure. L o m o v et al. [75] presented a pre-processor named C E T K A (meaning net i n Russian) that is capable of predicting the yarn profile i n complex 3-dimensional weave patterns. The algorithm is later  further  developed  to become  the  foundation  for a  commercially available pre-processor called WISETEX. The fabric topography is described and the equilibrium of yarns i n the weave is satisfied through mathematical representation of warp/weft interaction i n a fabric. Mechanical model are thus developed for plain fabric and textile composite unit-cell. This  -253-  Appendix A-3D  Crossover Mesh  Pre-processor  pre-processor is capable of creating models of highly complex weave structure, requiring a wealth of yarn and fabric information, some of w h i c h might not be easily obtainable. Brown et al. [76] also developed a pre-processor that generates solid models of fabrics. The aim of Brown's study was to provide a system to generate solid models of the w o v e n fabric and extract geometric data from it to optimize the architecture of the fabric composite reinforcement. The options considered for the cross-sectional shape and yarn profile are discussed, leading to the utilization of sinusoidal shape functions. The cross-sectional micro-images of the fabric cross-sections were used to set the framework for the pre-processor development. Qualitatively, it can be concluded that the yarn profiles can be realistically represented by sinusoidal functions. The yarn cross-sections appeared to be either elliptical or sine-based. It was also observed that the yarn cross-section may change along the yarn, depending on the interaction between the crossing yarns. This change can be so sever that lead to the yarn overlay, as discussed earlier. The input data used i n the pro-processor are the fabric and yarn geometrical parameters obtained from the laboratory measurements. Figure A - 5 shows a general overview of the process of creating a single yarn crossover. The first action undertaken by the pre-processor is to calculate the cross-section and profile of the yarns from the input values provided. The profile created is then discretized and intermediate cross-sections are generated  along the yarn,  paving the way to the addition of solid elements and generation of the fabric unit-cell.  -254-  Appendix A-3D  Crossover Mesh Pre-processor  Several simplifying assumptions are made i n order to create the finite element mesh of the yarns that closely match the observations from the cross-sectional micro-images. The basic assumptions are listed below: • Yarn profiles are considered to be best represented by a sine function. • Yarn cross-sections are considered to be elliptical, or sine-shaped. • Yarn cross-sections (perpendicular to its centerline) stay constant along the yarn (i.e. no yarn overlay). The yarn profile (see Figure A - 6) is defined by the following function: z = a sin(b x) 2  2  Parameter b is calculated from setting the period of the sine function equal to 2  the axis-to-axis distance of two adjacent yarns i n the perpendicular direction (unit-cell length), w , as follows: 2  w  2  is directly calculated from the yarn count number. The amplitude of the  sine function, a , is a function of the crimped length of the yarn, S . The 2  cr  length of the sine curve is set equal to the yarn crimped length, as follows:  o  Two options are considered for the yarn cross-section: sinusoidal and elliptical (Figure A - 7).  -255-  Appendix A-3D  Crossover Mesh Pre-processor  In both cases, the w i d t h of the yarn, a , is determined from the unit-cell width, x  w , and the gap between parallel yarns, g : x  x  b is determined from the yarn area considerations, such that the voided area x  of the yarn does not interpenetrate the geometry of the crossing yarn. Figure A - 8 depicts the process of producing the F.E. mesh of a fabric unit-cell and  all its components. Basically, the 3D model of the fabric crossover is  comprised of 3-D solid (brick) elements. A set of input referred to as "control input data" specifies the additional components to be included i n the unit-cell model. The user-controlled options are the addition of linear bar elements to represent the axial stiffness of the yarns, and shell elements to the yarn ends. The user can also choose to create a unit-cell model that includes one or four crossovers of plain weave fabrics. The pre-processor provides node and element data i n formats recognizable by either L S - D Y N A  or  ABAQUS  commercial finite element codes. The first step i n creating the unit-cell mesh is to create the geometry of the yarns through calling the subroutine that locates the nodes defining the volume of the yarns. The pre-processor then proceeds with filling the volume of the yarns w i t h solid elements. If prompted by the user, the linear beams and yarn-end shell elements are added to the yarn i n the next step. In the final step, the yarns are plotted for visual inspection by the user and the input files are created. One important characteristic of every finite element mesh is the element size. The pre-processor w o u l d automatically calculate the element size if prompted,  -256-  Appendix A-3D  Crossover Mesh Pre-processor  or w o u l d use some user-defined values to create the F.E. mesh of the fabric unit-cell. The node numbering is designed such that the elements' local 2direction is aligned w i t h the yarn's axial direction. Nodes located at the yarn ends are grouped i n node-sets and are utilized to define the yarn motion i n the finite element analysis. The geometrical information obtained from the study of cross-sectional microimages of all fabric panels are used to create finite element meshes of their unitcells. One of such unit-cells is presented i n Figure A - 9, where the micro-image of the warp cross-section of S-720G fabric is compared w i t h the finite element mesh created using information obtained from the micro-images. It can be seen that the finite element mesh generated match the yarn profile and cross-section seen i n the actual micro-image.  -257-  Appendix A - 3 D Crossover Mesh Pre-processor  Table A - 1: Fabric and weave parameters obtained from the micro-images (data provided by Kongshavn [42]).  Panel  Total Crossing Yarn's Thickness Thickness (mm) (mm)  Direction  Yarn Width (mm)  Yarn Thickness (mm)  warp  1.37  0.32  0.19  0.51  0.78  weft  1.23  0.33  0.14  0.47  0.78  warp  1.08  0.19  0.14  0.33  1.05  weft  0.95  0.30  0.10  0.40  1.03  warp  1.18  0.28  0.12  0.40  0.99  weft  0.92  0.29  0.13  0.42  1.02  warp  1.49  0.33  0.13  0.46  0.64  weft  1.64  0.32  0.10  0.42  0.69  warp  0.98  0.27  0.15  0.42  1.22  weft  0.74  0.33  0.14  0.47  1.50  warp  1.80  0.55  0.11  0.66  0.68  weft  1.43  0.59  0.17  0.76  0.60  Threads (per mm)  720  726  727  728  731  745  -258-  Appendix A-3D  Crossover Mesh Pre-processor  Table A - 2: Yarn crimp values and void ratios calculated from the micro-images (data provided by Kongshavn [42]).  Panels 720  7265  7272  728  731  745  Direction  Crimp (%)  Void Ratio (%)  warp  3.21  63.9  weft  2.68  60.4  warp  4.21  57.5  weft  1.48  64.9  warp  2.04  70.3  weft  2.13  61.0  warp  3.05  65.3  weft  2.70  69.3  warp  9.90  63.5  weft  3.04  54.1  warp  10.16  70.0  weft  2.68  63.2  Table A - 3: Material and geometrical properties input to the pre-processor for S-720G. Property  Warp  Weft  Yarn (denier)  1420  1420  Count (tpi)  20  20  Crimp (%)  3.21  2.68  Cross-section  Sine  Sine  -259-  Appendix A-3D  Crossover Mesh Pre-processor  Figure A - 1: Cross-sectional micro-images of S-728 panel (top: warp, bottom: weft) [42].  Figure A - 2: Measurement of various dimensions on the micro-images of S-728 (warp) panels [42].  -260-  Appendix A-3D  Crossover Mesh Pre-processor  Appendix A-3D  Crossover Mesh Pre-processor  create y a r n profile  V  a n d cross-section  create nodes, define y a r n v o l u m e  create 3D solid elements defining the y a r n V  V s i m i l a r l y create the crossing y a r n F . E . mesh is complete!  F i g u r e A - 5: P r o c e s s o f c r e a t i n g a s i n g l e y a r n c r o s s o v e r i n t h e p r e - p r o c e s s o r .  Unit-cell width, w  F i g u r e A - 6:  S i n u s o i d a l y a r n profile used to create the y a r n c e n t r e l i n e i n the m e s h processor.  -262-  pre-  Appendix A-3D  Crossover Mesh Pre-processor  2 •• a cos(&, y) x  +  (Z \  A)  =1  Figure A - 7: Sine-based (left) or elliptical (right) cross-section definition of yarns.  Figure A - 8: Process of generating the F.E. mesh of a fabric unit-cell.  -263-  Appendix A-3D  Crossover Mesh Pre-processor  Figure A - 9: Cross-sectional micro-image of S-720 weft fabric compared to the generated finite element mesh.  -264-  Appendix  B-  Pseudo-Codes  APPENDIX B -  PSEUDO-CODES  The following sections present the algorithms that are implemented i n the User Material Models of L S - D Y N A for beams and solids of the 3D fabric crossover model and the shells of the 2D shell crossover model.  B.l  B E A M ELEMENT 1. R e a d £  =s  the +As  current  strain  rates,  As",  calculate  the  total  strains  .  2.  Check the failure of the beams: i f s" > s  3.  C a l c u l a t e the axial stress component using a viscoelastic constitutive  f  model:  cr," = cr,"" + EAe + ju-fEpl^-,  where  1  x  t h e n erode the element.  At  //  is  the  viscosity  coefficient. 4. Eliminate axial compression: i f a" <0 t h e n a" =R xcr " where R is c  x  the compression reduction factor. 5. Include yarn crimp: i f s" < s  cr  t h e n a" = 0.01 x a".  6. C a l c u l a t e shear components: a" = cr"' + R x As" where / = 4 , 6 . x  s  -265-  c  Appendix B - Pseudo-Codes  7. The updated stress tensor is r e t u r n e d to L S - D Y N A to continue the analysis.  B.2 SOLID ELEMENT 1. R e a d s  2.  n  the  current  n-1  =s  Call  *  strain  rates,  As",  the  calculate  total  strains  n  + As  .  the deformation gradient subroutine, r e a d F" and c a l c u l a t e the  Jacobian J" = F" 3. C a l c u l a t e average cross-sectional change i n the 1-3 plane (2-dir aligned J" w i t h the yarn axis) by AA = •  exp'  v;  4. C a l c u l a t e the current fibre volume fraction: V " = - —. J  f  f  AA  5. C a l c u l a t e the elastic modulus i n the yarn axis direction (2-dir): if s " > 0 2  then E = R E V " 2  c  f  else E =  f  2  E V ". f  f  6. C a l c u l a t e the transverse elastic modulus: if V" < V ° (transverse yarn f  expansion)  then  E =RE x  T  C  where  R  T  is a reduction factor.  Else  E, = — - — . Set E, = E,. b-V 1  n  3  1  f  7. C a l c u l a t e the stress tensor: a" = E^"  where / = 1...6 (shear m o d u l i E , 4  E , and E inputted by the user). 5  6  8. The updated stress tensor is r e t u r n e d to L S - D Y N A to continue the analysis.  -266-  Appendix B - Pseudo-Codes  B.3 SHELL ELEMENT 1. Read the current strain rates (increments), As" =  2. Read q" {  _1  the  = [g, "  material  coordinates  As "  Ay"  x  (warp  o  A;  and  weft  £  vectors,  q " J ) from the history variables.  _1  _1  t  3. Calculate  the  F"=I + As" =  incremental  deformation  1+A * /  Ay"  0  1 + Aff,"  gradient  4. Update the location of the warp and weft vectors b y q " = F" q " t  5. Calculate the transformation matrix 0 , " =  cos6>" -sintf,"  6"  It.  =tan"  from  1  t  smQ" , cosf9,"  where  and / = 1,2 represents the warp and weft directions.  \ 'y J q  6. Calculate the strain tensor transformed to the warp or weft direction Af," = 0 , " A e f O " 7. Calculate  the  . incremental  stretch  along  each  yarn  from  A" =- j\ + 2As " ml + ASj", where Ae" is the transformed strain tensor s  i  x  component i n the yarn direction. 8. Calculate the total stretch of each yarn b y  A,"=  A"~ A,", L  where  A,""  1  is  obtained from the history variables. 9. Calculate  the  unit-cell  displacement  along  each  yarn  from  d? =(A,"-l)w , where w . is the initial half-length of the unit-cell. 0i  0(  10. Read the range of d" for which the iterations should be performed. I f d" out of the specified range then F " =0 (indicating the absence of c  contact between the yarns) and go to line 25 (bypass the iterations).  -267-  Appendix B - Pseudo-Codes  11. I f any of the two yarns have previously failed then determine the tension i n the other yarn from the single yarn response (7) = ^ ( w , . -S )  if w,>S ,  L  0l  else 7) =0) and go to line 25 (skip the  0l  iterations). 12. Set up the Newton-Raphson iterative scheme. Set initial values of /z,. 13.  Start loop  over  +  sin  2w  K  Calculate  iter,  u  2w,  K  the  current yarns lengths  from  dS,  dx and its derivative  dh  j  14.  Calculate  E A the yarns tensile force from T, = ' ' (S  15.  Calculate  the yarn angle with the fabric plane at the edges of the unit  -S ).  i  0i  rdi.  cell from sin a, =  V(2",) +(A) 2  2  16.  Calculate the  yarn contact force, F = 2T s i n « .  17.  Calculate the  transverse yarn compression, d  18.  Calculate the  equilibrium objective function, / , = F  c  Ci  the  f =h +h +d x  2  Ci  20. Calculate x  t0  +d  Ci  tolerance  Tol = K , x|F  =a•  -F  Ci  19. Calculate 2  (  i  C)  compatibility  objective  function,  +g-h -h . 0l  02  value  for  + F | and Tol = K Ci  C2  2  lol  each  objective  x\h + h \, where K x  2  tol  function:  = 0.001 i n this  study. 21. Check the solution convergence criteria. I f f < Tol and f x  (convergence is achieved) go to line 25.  -268-  x  2  < Tol then 2  Appendix B - Pseudo-Codes  22.  Else  calculate  (convergence not satisfied)  dh = Jx f where J  dfx  3/,  dh  dh  fa  fa  dh  dh  x  x  23.  Calculate new  24.  Go to 13.  and / =  2  the h, correction values,  V." fi  2  /z, using h = h-dh.  25. Check for contact between the yarns. If contact force between the yarns are negative (i.e. F <0) c  (7; = ^ L ( W , - S  O  I  if w,  )  26. Check for yarn failure.  then  >S ,  If  27. If both yarns have failed  0I  else  calculate the single yarn =0).  T,  7) > Tw,. then 7] = 0 and flag the yarn failed.  then erode  the shell element from the mesh. Z  28.  Calculate the  response  stress tensor for each yarn using a "  0  A 0  t  0  29. Transform the stress tensors back to the shell local coordinate system: CT, = 0 ,  30.  31.  <J 0 , T  Calculate the  average stress tensor: a " =  CT, +CT  2  avg  Calculate the shear  0  avg 2  component from the trellis mechanism and add it to  the in-plane shear stress. The trellis shear component (Equation 4-35) as a function of the shear strain,  If \y"\<y then r x  else if  12  y <y" <y x  2  (where y" = y"~ + Ay") is: l  " = T " - + G y" x  1 2  then r  x2  f  =T " 1 2  _ 1  +G y" + ° ~ 2  Yi-Yx  -269-  -ft)  G x  x  2  Appendix  B-  else  Pseudo-Codes  T ;=T r +G X  x  x  l Y  » {G -G y-^ +  x  2  32. The updated stress tensor cr" -  is r e t u r n e d to LSavg  2  D Y N A to continue the analysis.  -270-  

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