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On constitutive modelling of fibre-reinforced composite materials Vaziri, Reza 1989

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ON CONSTITUTIVE MODELLING OF FIBRE-REINFORCED COMPOSITE MATERIALS By REZA VAZIRI B .Sc . (Eng . ) , U n i v e r s i t y of London, U . K . , 1982 M . A . S c , U n i v e r s i t y of B r i t i s h Columbia, 1985 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept t h i s thes i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1989 © Reza V a z i r i , 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this .thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada Date ^ DE-6 (2/88) ABSTRACT A r e l a t i v e l y s i m p l e b u t c o m p r e h e n s i v e c o n s t i t u t i v e mode l i s p r e s e n t e d h e r e i n f o r p r e d i c t i n g t h e n o n l i n e a r b e h a v i o u r o f l a m i n a t e d c o m p o s i t e s t r u c t u r e s c o m p r i s i n g l a y e r s o f u n i d i r e c t i o n a l a n d / o r b i d i r e c t i o n a l ( e . g . woven) f i b r e - r e i n f o r c e d m a t e r i a l s (FRMs) . The FRM l a y e r i s t r e a t e d as an o r t h o t r o p i c b u t homogeneous c o n t i n u u m u n d e r g o i n g i s o t h e r m a l i n f i n i t e s i m a l d e f o r m a t i o n . The p r o p o s e d c o n s t i t u t i v e mode l f o r s i n g l e l a y e r s o f FRM i s b u i l t w i t h i n t h e f ramework o f r a t e - i n d e p e n d e n t t h e o r y o f o r t h o t r o p i c e l a s t o p l a s t i c i t y . The c o n s t i t u t i v e e q u a t i o n s so d e v e l o p e d , a r e t h e n s u p e r i m p o s e d u s i n g t h e c l a s s i c a l l a m i n a t i o n t h e o r y , t o a r r i v e a t t h e g o v e r n i n g r e s p o n s e r e l a t i o n s f o r m u l t i l a y e r l a m i n a t e s . The mode l i n v o k e s a 3 - p a r a m e t e r q u a d r a t i c y i e l d s u r f a c e and t h e a s s o c i a t e d f l o w r u l e o f p l a s t i c i t y . D u r i n g p l a s t i c f l o w t h e e v o l u t i o n o f t h e y i e l d s u r f a c e i n t h e s t r e s s s p a c e i s d e s c r i b e d b y a n o n -p r o p o r t i o n a l change i n t h e p a r a m e t e r s o f t h e i n i t i a l y i e l d f u n c t i o n . A 3 - p a r a m e t e r q u a d r a t i c f a i l u r e s u r f a c e s i m i l a r i n f o rm t o t h a t o f t h e i n i t i a l y i e l d s u r f a c e i s d e f i n e d t o mark t h e u p p e r l i m i t o f p l a s t i c f l o w . Once f a i l u r e i s r e a c h e d , i t i s i d e n t i f i e d as f i b r e o r m a t r i x mode o f f a i l u r e d e p e n d i n g on t h e r e l a t i v e m a g n i t u d e o f v a r i o u s s t r e s s r a t i o t e r m s a p p e a r i n g i n t h e f a i l u r e c r i t e r i o n . I n t h e p o s t - f a i l u r e m o d e l l i n g , b o t h b r i t t l e and d u c t i l e t y p e o f b e h a v i o u r a r e c o n s i d e r e d i n t h e d i r e c t i o n o f t h e o f f e n d i n g s t r e s s . U n i d i r e c t i o n a l and b i d i r e c t i o n a l FRM l a y e r s a r e t r e a t e d w i t h i n t h e same g e n e r a l f ramework w i t h t h e e x c e p t i o n t h a t y i e l d i n g (and f a i l u r e ) i n t h e s e l a y e r s a r e assumed t o be g o v e r n e d b y d i f f e r e n t c r i t e r i a , n a m e l y , H i l l ' s and P u p p o - E v e n s e n ' s y i e l d (and f a i l u r e ) c r i t e r i a , r e s p e c t i v e l y . To c o m p l e t e l y q u a n t i f y t h e p r o p o s e d e l a s t i c - p l a s t i c - f a i l u r e mode l t h r e e p i e c e s o f e x p e r i m e n t a l s t r e s s - s t r a i n c u r v e s a r e r e q u i r e d , n a m e l y , t h e u n i a x i a l s t r e s s - s t r a i n c u r v e s a l o n g t h e two p r i n c i p a l a x e s o f o r t h o t r o p y , and t h e i n - p l a n e s h e a r s t r e s s - s t r a i n c u r v e . Once e s t a b l i s h e d , t h e s e s t r e s s -s t r a i n c u r v e s a r e r e p r e s e n t e d b y b i l i n e a r a p p r o x i m a t i o n s t h u s c l e a r l y d e f i n i n g t h e k e y p a r a m e t e r s u n d e r t h e v a r i o u s l o a d i n g p r o g r a m s . No p r o v i s i o n s a r e made f o r t h e d i f f e r e n c e b e t w e e n t e n s i l e and c o m p r e s s i v e r e s p o n s e s . B a s e d on t h e p r o p o s e d m o d e l , c o n s t i t u t i v e e q u a t i o n s a r e p r o p e r l y f o r m u l a t e d . A n o n l i n e a r f i n i t e e l e m e n t code i s d e v e l o p e d t o i n c o r p o r a t e t h e d e r i v e d c o n s t i t u t i v e e q u a t i o n s . The p r o g r a m i s b a s e d on t h e c o n v e n t i o n a l d i s p l a c e m e n t method f i n i t e e l e m e n t p r o c e d u r e u s i n g two d i m e n s i o n a l 8 - n o d e i s o p a r a m e t r i c e l e m e n t s . The n o n l i n e a r i t i e s i n t h e e q u i l i b r i u m e q u a t i o n s a r e h a n d l e d b y a m i x e d i n c r e m e n t a l and N e w t o n - R a p h s o n i t e r a t i v e p r o c e d u r e . A n a l y s i s r e s t a r t and c y c l i c l o a d i n g c a p a b i l i t i e s a r e a l s o i n c l u d e d t o e x p a n d t h e p r o g r a m ' s u s e f u l n e s s . The p e r f o r m a n c e o f t h e p r o g r a m and t h e e f f e c t i v e n e s s o f t h e mode l a r e v e r i f i e d f o r a number o f i n - p l a n e l o a d i n g p a t h s a p p l i e d t o a w i d e v a r i e t y o f l a m i n a t e d FRMs w i t h and w i t h o u t g e o m e t r i c d i s c o n t i n u i t i e s . The f a v o u r a b l e c o m p a r i s o n s o f t h e mode l t o e x p e r i m e n t a l r e s u l t s a v a i l a b l e i n t h e l i t e r a t u r e s u p p o r t t h e v a l i d i t y o f t h e m o d e l . - i i i -TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF FIGURES v i i i LIST OF TABLES xv LIST OF SYMBOLS x v i ACKNOWLEDGEMENTS xx CHAPTER 1 - INTRODUCTION 1 1.1 Introduct ion 1 1.2 Types of Composites and Basic Terminology 2 1.3 General Remarks on the Mechanical Propert ies of F i b r e Composites 5 1.4 Purpose and Scope of the Present Study 9 CHAPTER 2 - REVIEW OF THE LITERATURE 10 2.1 Introduct ion 10 2.2 Background 10 2.3 C o n s t i t u t i v e Model l ing of Undamaged Composites 12 2.3.1 Micromechanics Approach 12 2.3.2 Minimechanics Approach 13 2.3.3 Macromechanics Approach 15 2.4 I n i t i a l F a i l u r e 24 2.5 C o n s t i t u t i v e Model l ing of Damaged Composites 28 2.6 Ult imate F a i l u r e 30 - i v -TABLE OF CONTENTS (Continued) CHAPTER 3 - THEORETICAL FOUNDATIONS OF THE PROPOSED CONSTITUTIVE MODEL 31 3.1 Introduct ion 31 3.2 D e s c r i p t i v e Out l ine of the Model 31 3.3 General Formulation of the S ingle Layer C o n s t i t u t i v e Equations 33 3.3.1 E l a s t i c Regime 33 3.3.2 P l a s t i c Regime 36 3.3.3 P o s t - F a i l u r e Regime , 49 3.4 Plane Stress Formulation of the Single Layer C o n s t i t u t i v e Equations 51 3.4.1 E l a s t i c Regime 51 3.4.2 P l a s t i c Regime 53 3.4.3 P o s t - F a i l u r e Regime 61 3.5 M u l t i l a y e r Laminates 62 CHAPTER 4 - FINITE ELEMENT FORMULATION 67 4.1 Introduct ion 67 4.2 Governing Equations of F i n i t e Element Analys i s 67 4.2.1 Isoparametric Element Representation 68 4.2.2 Element S t i f f n e s s Formulation 69 4.2.3 S t r u c t u r a l S t i f f n e s s Formulation 73 4.3 Numerical So lu t ion of Nonlinear E q u i l i b r i u m Equations 74 4.4 Numerical Implementation of the A n i s o t r o p i c E l a s t i c -P l a s t i c - F a i l u r e Model 78 4.4.1 E l a s t i c - P l a s t i c Formulation 79 4.4.2 P o s t - F a i l u r e Formulation 87 - v -TABLE OF CONTENTS (Continued) CHAPTER 5 - NUMERICAL RESULTS AND DISCUSSIONS 90 5.1 Introduct ion 90 5.2 V e r i f i c a t i o n of the F i n i t e Element Program 90 5.2.1 Thick-Wal led I so trop ic Cy l inder Under Interna l Pressure 92 5.2.2 Combined Tension and Tors ion of an I so tro ipc T h i n -Walled Tube 93 5.2.3 Perforated I so trop ic Sheet Subjected to Remote Uniform Tension 95 5.2.4 Conclusions 98 5.3 Response P r e d i c t i o n of Laminated Composite Coupons 98 5.3.1 U n i a x i a l Loading 99 5.3.2 B i a x i a l Loading 105 5.3.3 C y c l i c Loading 107 5.3.4 Conclusions 109 5.4 Perforated Orthotropic Plates Subjected to Remote Uniform Tension 109 5.4.1 E l a s t i c Ana lys i s 110 5.4.2 E l a s t i c - P l a s t i c Ana lys i s 112 5.4.3 E l a s t i c - P l a s t i c - F a i l u r e Analys i s 119 5.4.4 Conclusions 123 CHAPTER 6 - SUMMARY AND CONCLUSIONS 124 6.1 Summary 124 6.2 Concluding Remarks 126 6.3 Further Areas of Research 127 - v i -TABLE OF CONTENTS ( C o n t i n u e d ) REFERENCES 129 APPENDICES: A - DETERMINATION OF THE ANISOTROPIC PARAMETERS OF THE YIELD FUNCTION 138 B - VARIATION OF ANISOTROPIC PARAMETERS WITH STRAIN-HARDENING . . 143 C - DERIVATION OF THE EFFECTIVE PLASTIC STRAIN INCREMENT de P . . 149 FIGURES 152 - v i i -LIST OF FIGURES Page Figure 3.1 Idea l i zed s t r e s s - s t r a i n curve showing d i f f e r e n t stages of the proposed e l a s t i c - p l a s t i c - f a i l u r e model 152 3.2 Transverse matrix cracking i n a s ing le layer of U/D FRM 152 3.3 Nomenclature for s ing le layers of FRM: a - B i d i r e c t i o n a l , b - U n i d i r e c t i o n a l 153 3.4 Puppo-Evensen y i e l d surfaces i n the o 6=0 plane for b i -d i r e c t i o n a l layers with X=Y and various values of the parameter A 154 3.5 O r i e n t a t i o n of layer coordinate axes with respect to laminate coordinates 155 3.6 I l l u s t r a t i o n of an n- layered laminate along with a t y p i c a l in -p lane deformed geometry 156 4.1 Quadratic i soparametric element 157 4.2 Tangent s t i f f n e s s method for a s ing l e v a r i a b l e problem 158 4.3 Incremental e l a s t o p l a s t i c s tress computation for an i n i t i a l l y e l a s t i c point 159 5.1 F i n i t e element mesh for the ana lys i s of an i s o t r o p i c e l a s t o p l a s t i c c y l i n d e r under i n t e r n a l pressure 160 5.2 Pressure P versus inner and outer wa l l displacements u a and u^ for the problem of F i g . 5.1 160 5.3 Progress ion of y i e l d i n g for the problem of F i g . 5.1 161 5.4 Radia l d i s t r i b u t i o n of hoop s tress at various pressures for . the problem of F i g . 5.1 162 5.5 Numerical model and the s tress path for the ana lys i s of an i s o t r o p i c t h i n - w a l l e d tube subjected to combined tension and t o r s i o n 163 5.6 S t r a i n path for an i s o t r o p i c t h i n tube subjected to combined tens ion and t o r s i o n 164 - v i i i -LIST OF FIGURES (Continued) Page 5.7 Nomenclature for the perforated p la te problem 165 5.8 F i n i t e element mesh used to analyze a quadrant of an i s o t r o p i c per forated p la te 165 5.9 The s t r e s s - s t r a i n curve i n pure tens ion for Aluminum a l l o y 57S [Theocaris and Marketos, 1964] 166 5.10 Nondimensional graph of mean s tress against maximum s t r a i n for the i s o t r o p i c perforated p la te shown i n F i g . 5.8 167 5.11 Comparison of the computed and experimental ly determined p l a s t i c zone growth for the i s o t r o p i c perforated p l a t e subjected to e l a s t o p l a s t i c loading 168-169 5.12 E f f e c t i v e s tress contours at var ious load l eve l s for the i s o t r o p i c per forated p l a t e subjected to e l a s t o p l a s t i c loading 170-171 5.13a S t r a i n p r o f i l e at the net s ec t ion for the i s o t r o p i c perforated p la te subjected to am = 0.47 o 0 172 5.13b Stress p r o f i l e at the net s ec t ion for the i s o t r o p i c perforated p la te subjected to am = 0.47 o 0 172 5.14 Long i tud ina l t e n s i l e s t r e s s - s t r a i n curve for a s ing le layer of U/D Boron/Epoxy 173 5.15 Transverse t e n s i l e s t r e s s - s t r a i n curve for a s ing le layer of U/D Boron/Epoxy 173 5.16 Shear s t r e s s - s t r a i n curve for a s ing le layer of U/D Boron/Epoxy 174 5.17 T e n s i l e s t r e s s - s t r a i n curve for [0/90] Boron/Epoxy laminate . . 174 5.18 T e n s i l e s t r e s s - s t r a i n curve for [+45/-45] Boron/Epoxy laminate 175 5.19a T e n s i l e s t r e s s - s t r a i n curve for [+30/-30] Boron/Epoxy laminate 175 5.19b Stress paths i n the +30 deg layer during u n i a x i a l loading of [+30/-30] B/Ep laminate: H i l l ' s f a i l u r e c r i t e r i o n 176 5.19c Stress paths i n the +30 deg layer during u n i a x i a l loading of [+30/-30] B/Ep laminate: Maximum stress f a i l u r e c r i t e r i o n 177 5.20 T e n s i l e s t r e s s - s t r a i n curve for [+60/-60] Boron/Epoxy laminate 178 5.21a T e n s i l e s t r e s s - s t r a i n curve for [+20/-20] Boron/Epoxy laminate 178 - i x -L I S T OF FIGURES (Continued) Page 5.21b Stress paths i n the +20 deg layer during u n i a x i a l loading of [+20/-20] B/Ep laminate: H i l l ' s f a i l u r e c r i t e r i o n 179 5,21c Stress paths i n the +20 deg layer during u n i a x i a l loading of [+20/-20] B/Ep laminate: Maximum s tress f a i l u r e c r i t e r i o n . . . . 180 5.22 T e n s i l e s t r e s s - s t r a i n curve for [0/+45/-45/90] Boron/Epoxy laminate 181 5.23 T e n s i l e s t r e s s - s t r a i n curve for [0/+60/-60] Boron/Epoxy laminate 181 5.24 T e n s i l e s t r e s s - s t r a i n curve for [0 3 /45/-45] Boron/Epoxy laminate 182 5.25 T e n s i l e s t r e s s - s t r a i n curve for [65 3 /20/-70] Boron/Epoxy laminate 182 5.26a Basic s t r e s s - s t r a i n curves for a B/D layer made of 181 glass f a b r i c and po lyes ter r e s i n 183 5.26b T e n s i l e s t r e s s - s t r a i n curve at 45 deg to the f i b r e d i r e c t i o n s for the mater ia l of F i g . 5.26a 183 5.27a Basic s t r e s s - s t r a i n curves for a B/D layer made of 162 glass f a b r i c and polyes ter r e s i n 184 5.27b T e n s i l e s t r e s s - s t r a i n curve at 45 deg to the f i b r e d i r e c t i o n s for the mater ia l of F i g . 5.27a 184 5.28a Basic s t r e s s - s t r a i n curves for a B/D layer made of 143 glass f a b r i c and po lyes ter r e s i n 185 5.28b T e n s i l e s t r e s s - s t r a i n curve at 45 deg to the f i b r e d i r e c t i o n s for the mater ia l of F i g . 5.28a 185 5.29 P r e s s u r e - s t r a i n curve for [0/60/-60] Gr/Ep tube under i n t e r n a l pressure 186 5.30 P r e s s u r e - s t r a i n curve for [0/60/-60] Gr/Ep tube under combined i n t e r n a l pressure and pre-torque 186 5.31 Torque-shear s t r a i n curve for [0/60/-60] Gr/Ep tube with p r e - l o a d of i n t e r n a l pressure 187 5.32 T e n s i l e s t r e s s - s t r a i n curve for [0/45/-45] B / A l laminate under three load cycles .' 187 5.33 Geometry and the f i n i t e element model for a quadrant of an or tho trop ic per forated sheet 188 - x -LIST OF FIGURES (Continued) Page 5.34 E l a s t i c c i r c u m f e r e n t i a l s tress d i s t r i b u t i o n around the hole for a U/D B / A l l a y e r : a - F ibres perpendicular to the load d i r e c t i o n , b - F ibres along the load d i r e c t i o n 189 5.35 E l a s t i c c i r c u m f e r e n t i a l s tress d i s t r i b u t i o n around the hole for a U/D B/Ep layer : a - F ibres perpendicular to the load d i r e c t i o n , b - F ibres along the load d i r e c t i o n 190 5.36 Geometry of the t e s t specimen used by R i z z i et a l . (1987) for experimental determination of the s t r e s s - s t r a i n behaviour of a perforated 90-deg layer of U/D B / A l 191 5.37 Transverse t e n s i l e s t r e s s - s t r a i n curve for a s i n g l e layer of U/D B / A l [Kenaga et a l . , 1987] 192 5.38 Long i tud ina l s t r a i n d i s t r i b u t i o n along the net s ec t ion for various remote load l eve l s imposed on a perforated 90-deg layer of U/D B / A l 193-194 5.39 Development of p l a s t i c zones for a perforated 90-deg layer of U/D B / A l subjected to e l a s t o p l a s t i c loading 195 5.40 Nondimensional e f f e c t i v e s tress contours for a per forated 90-deg layer of U/D B / A l subjected to e l a s t o p l a s t i c loading . . 196 5.41 Res idual l o n g i t u d i n a l s t r a i n d i s t r i b u t i o n along the net s ec t ion of a per forated 90-deg layer of U/D B / A l a f t er unloading from various load l e v e l s 197 5.42 D i s t r i b u t i o n of r e s i d u a l s tress components along the net s ec t ion of a perforated 90-deg layer of U/D B / A l a f ter unloading from various load l eve l s 198 5.43 Development of p l a s t i c zones i n the 90-deg layer of a perforated [0/90] B / A l laminate subjected to e l a s t o p l a s t i c loading 199-200 5.44 Development of p l s t i c zones i n the 0-deg layer of a per forated [0/90] B / A l laminate subjected to e l a s t o p l a s t i c i loading 201-202 5.45 Nondimensional e f f e c t i v e s tress contours for the 90-deg layer of a per forated [0/90] B / A l laminate subjected to e l a s t o p l a s t i c loading 203-204 5.46 Nondimensional e f f e c t i v e s tress contours for the 0-deg layer of a per forated [0/90] B / A l laminate subjected to e l a s t o p l a s t i c loading 205-206 - x i -LIST OF FIGURES (Continued) Page 5.47 Development of p l a s t i c zones i n a [0/90] F p / A l laminate with a hole [Bahae i -E l -Din and Dvorak, 1980] 207 5.48 Long i tud ina l s tress d i s t r i b u t i o n along the net sec t ion for each layer of a perforated [0/90] B / A l laminate 208 5.49 Long i tud ina l s tress at point A versus the appl i ed load for the 0 deg layer of a perforated [0/90] B / A l laminate 209 5.50 Load versus d e f l e c t i o n at point B for a perforated [0/90] B / A l laminate 209 5.51 Long i tud ina l r e s i d u a l s tress d i s t r i b u t i o n along the net sec t ion for each layer of a perforated [0/90] B / A l laminate due to unloading from am = 20 k s i 210 5.52 Predic ted damage progress ion for a per forated 90-deg U/D B/Ep layer : D u c t i l e matrix 211 5.53 Predic ted damage progress ion for a perforated 0-deg U/D B/Ep l a y e r : D u c t i l e F ibre 212 5.54 Stress path at point A for the 0-deg layer of a perforated [0/90] B/Ep laminate: B r i t t l e f i b r e 213 5.55 Load versus d e f l e c t i o n at point B for a perforated [0/90] B/Ep laminate 213 5.56a Predic ted damage progress ion for the 90-deg layer of a [0/90] B/Ep laminate: D u c t i l e f i b r e and d u c t i l e matrix 214 5.56b Pred ic ted damage progress ion for. the 0-deg layer of a [0/90] B/Ep laminate: D u c t i l e f i b r e and d u c t i l e matrix 215 5.57a Pred ic ted damage progress ion for the 90-deg layer of a [0/90] B/Ep laminate: D u c t i l e f i b r e and b r i t t l e matrix 216 5.57b Pred ic ted damage progress ion for the 0-deg layer of a [0/90] B/Ep laminate: D u c t i l e f i b r e and b r i t t l e matrix 217 5.58a Pred ic ted damage progress ion for the 90-deg layer of a [0/90] B/Ep laminate: B r i t t l e f i b r e and d u c t i l e matrix 218 5.58b Pred ic ted damage progress ion for the 0-deg layer of a [0/90] B/Ep laminate: B r i t t l e f i b r e and d u c t i l e matrix 219 5.59 Change i n s tress d i s t r i b u t i o n along the net s ec t ion during the process of b r i t t l e f i b r e f a i l u r e i n the 0-deg layer of a perforated [0/90] B/Ep laminate 220 - x i i -LIST OF FIGURES (Continued) Page 5.60 Load versus d e f l e c t i o n at po int B for a perforated [45/-45] B/Ep laminate 220 5.61a Pred ic ted damage progress ion for the +45-deg layer of a [45/-45] B/Ep laminate: D u c t i l e matrix 221 5.61b Predic ted damage progress ion for the -45-deg layer of a [45/-45] B/Ep laminate: D u c t i l e matrix 222 5.62a Predic ted damage progress ion for the +45-deg layer of a [45/-45] B/Ep laminate: B r i t t l e matrix 223 5.62b Predic ted damage progress ion for the -45-deg layer of a [45/-45] B/Ep laminate: B r i t t l e matrix 224 5.63 Load versus d e f l e c t i o n at point B for a perforated [0/45/-45/90] B/Ep laminate 225 5.64a Predic ted damage progress ion for the 90-deg layer of a [0/45/-45/90] B/Ep laminate: D u c t i l e f i b r e and d u c t i l e matrix 226 5.64b Predic ted damage progress ion for the 0-deg layer of a [0/45/-45/90] B/Ep laminate: D u c t i l e f i b r e and d u c t i l e matrix 227-5.64c Predic ted damage progress ion for the +45-deg layer of a [0/45/-45/90] B/Ep laminate: D u c t i l e f i b r e and d u c t i l e matrix 228 5.64d Pred ic ted damage progress ion for the -45-deg layer of a [0/45/-45/90] B/Ep laminate: D u c t i l e f i b r e and d u c t i l e matrix 229 5.65a Pred ic ted damage progress ion for the 90-deg layer of a [0/45/-45/90] B/Ep laminate: D u c t i l e f i b r e and b r i t t l e matrix 230 5.65b Pred ic ted damage progress ion for the 0-deg layer of a [0/45/-45/90] B/Ep laminate: D u c t i l e f i b r e and b r i t t l e matrix 231 5.65c Pred ic ted damage progress ion for the +45-deg layer of a [0/45/-45/90] B/Ep laminate: D u c t i l e f i b r e and b r i t t l e matrix 232 5.65d Pred ic ted damage progress ion for the -45-deg layer of a [0/45/-45/90] B/Ep laminate: D u c t i l e f i b r e and b r i t t l e matrix 233 5.66a Pred ic ted damage progress ion for the 90-deg layer of a [0/45/-45/90] B/Ep laminate: B r i t t l e f i b r e and d u c t i l e matrix 234 5.66b Pred ic ted damage progress ion for the 0-deg layer of a [0/45/-45/90] B/Ep laminate: B r i t t l e f i b r e and d u c t i l e matrix 235 5.66c Pred ic ted damage progress ion for the +45-deg layer of a [0/45/-45/90] B/Ep laminate: B r i t t l e f i b r e and d u c t i l e matrix 236 - x i i i -LIST OF FIGURES (Continued) Page 5.66d Predic ted damage progress ion for the -45-deg layer of a [0/45/-45/90] B/Ep laminate: B r i t t l e f i b r e and d u c t i l e matrix 237 B . l A c t u a l s t r e s s - s t r a i n curve and i t s b i l i n e a r approximation . . . . 238 B.2 B i l i n e a r s t r e s s - p l a s t i c s t r a i n curve 238 B.3 V a r i a t i o n of the p r i n c i p a l a n i s o t r o p i c s trength parameters with the e f f e c t i v e s tress 239 - x i v -LIST OF TABLES Page Table 3.1 Comparison between tensor , engineering and contracted 35 nota t ion for s tresses and s t r a i n s 3.2 F a i l u r e i d e n t i f i c a t i o n procedure 60 3.3 P o s t - f a i l u r e incremental c o n s t i t u t i v e matrix [Q^] for both b r i t t l e and d u c t i l e f a i l u r e mode 61 3.A Released s tress vector {o^ } during b r i t t l e type of f a i l u r e . . . . 62 4.1 Sampling coordinates and weighting factors for one-dimensional Gaussian quadrature 73 4.2 Sequence of the i t e r a t i v e s o l u t i o n technique 77 5.1 Load path data for t e n s i o n - t o r s i o n t e s t on an i s o t r o i p c tube . . 94 5.2 Exact and computed s t r a i n s for combined tens ion and t o r s i o n of an i s o t r o p i c tube 95 5.3 Input mater ia l proper t i e s for a s ing l e layer of U/D B/Ep 99 5.4 Input mater ia l proper t i e s for a s ing l e layer of B/D Glass f a b r i c / p o l y e s t e r r e s i n 104 5.5 Input mater ia l proper t i e s for a s ing le layer of U/D Gr/Ep 106 5.6 Input m a t e r i a l proper t i e s for a s ing l e layer of U/D B / A l 108 - xv -LIST OF SYMBOLS A l i s t of important symbols i s compiled here. A l l symbols are defined i n the text when they f i r s t appear. A . . ; [ A ] a n i s o t r o p i c s trength parameters w r i t t e n i n t e n s o r i a l nota t ion and matrix format, r e s p e c t i v e l y A 6 elemental area a inner radius of the i s o t r o p i c t h i c k - w a l l e d c y l i n d e r a^;{a} p l a s t i c flow vector def ined by Eq . (3.26) [B] s tra in-displacement matrix b outer radius of the i s o t r o p i c t h i c k - w a l l e d c y l i n d e r b. def ined by Eq . (B.12) e p ep C . . ; C . . ; C . general ized e l a s t i c ; p l a s t i c ; and e l a s t o - p l a s t i c mater ia l s t i f f n e s s tensor c i def ined by E q . (B.12) E^;E^, ;E e l a s t i c , tangent and p l a s t i c moduli r e f e r r i n g to a generic 1 * 1 o^-e^ curve e . . s t r a i n tensor {F} ex ternal force vector f 0 ; f ; f i n i t i a l y i e l d ; subsequent y i e l d ; and f a i l u r e funct ion G;G^,;G e l a s t i c ; tangent; and p l a s t i c moduli r e f e r r i n g to an in -p lane ^ shear s t r e s s - s t r a i n curve g p l a s t i c p o t e n t i a l [H]^ matrix def ined by E q . (4.14) H;H' def ined by Eqs. (3.22) and (3.33) [J] Jacobian matrix [Kp] tangent s t i f f n e s s matrix k 0 ; k ; k u e f f e c t i v e i n i t i a l y i e l d ; subsequent y i e l d ; and f a i l u r e s tress L h a l f - l e n g t h of a sheet with a c e n t r a l hole L^ tensor defined by Eq . (A.3) M t o t a l number of subincrements in to which the s t r a i n increment i s d iv ided {N} vector of in -p lane s tress resu l tants n t o t a l number of layers through the laminate thickness - x v i -L I S T OF SYMBOLS ( C o n t ' d ) E "D Q"0 f Q. . ; Q . . ;Q. . ;Q. . plane s tress e l a s t i c ; p l a s t i c ; e l a s t o p l a s t i c ; and post -^ 1*' 1 ^ f a i l u r e mater ia l s t i f f n e s s tensors q order of Gaussian i n t e g r a t i o n {R} re leased force vector defined by E q . (4.51) R radius of the hole defined i n F i g s . (5.8) and (5.33) r i def ined by Eq . ( B . l l ) S 0 ; S ; S i n i t i a l y i e l d ; subsequent y i e l d ; f a i l u r e s tress i n pure shear s i def ined by Eq . ( B . l l ) [T] transformation matrix defined by Eq . (3.63) t t o t a l laminate thickness t^ thickness of the k ^ layer U 0 ; U i n i t i a l and subsequent y i e l d s tress of a 45° o f f - a x i s specimen u displacement along the x -ax i s u r a d i a l displacement Q V elemental volume v displacement along the y -ax i s v „ l o n g i t u d i n a l displacement at point B i n F i g . (5.33) weighting factor for Gauss i n t e g r a t i o n p l a s t i c work per u n i t volume w width defined i n F i g s . (5.8) and (5.33) X 0 ; X ; X u i n i t i a l y i e l d ; subsequent y i e l d ; and f a i l u r e s tress along the p r i n c i p a l mater ia l d i r e c t i o n , x1 x1,x2,x3 p r i n c i p a l axes of orthotrpoy for a s ing le layer x , y , z laminate coordinate system Y0;Y;Y^ i n i t i a l y i e l d ; subsequent y i e l d and f a i l u r e s tress along the p r i n c i p a l mater ia l d i r e c t i o n , x 2 u tensor descr ib ing the o r i g i n of the y i e l d surface B def ined by Eq . (4.30) r o . ; r . ; r i n i t i a l y i e l d ; subsequent y i e l d ; and f a i l u r e s tress tensor i r shear s t r a i n i n plane of the laminate 'xy r A incremental quant i t i e s {5} displacement vector - x v i i -LIST OF SYMBOLS (Cont'd) P e f f e c t i v e p l a s t i c s t r a i n e. s t r a i n tensor defined i n Table 3.1 1 ex 5 e v x and y components of o v e r a l l laminate s t r a i n s e p de^;de^ e l a s t i c and p l a s t i c s t r a i n increment tensors r.l,lJ def ined by Eq . (5.5) i-| n a t u r a l coordinate of an isoparametric element 9 Angle between the layer and laminate coordinate axes defined i n F i g . (3.5) K hardening parameter def ined by Eqs. (3.21a,b) A i n t e r a c t i o n factor i n the Puppo-Evensen c r i t e r i o n def ined by Eq . (3.54c) •- • dX hardening parameter def in ing the length of the p l a s t i c s t r a i n increment vector de? I u defined by Eq . (3.55) \ ) 1 2 ; \ ) J 1 major and minor Poisson's r a t i o E n a t u r a l coordinate of an isoparametric element p 1 ; p 2 def ined by Eq . (5.6) a e f f e c t i v e s tress defined by Eq . (3.17) am remote s tress app l i ed to perforated p la tes o Q c i r c u m f e r e n t i a l s tress [a } s tress vector at contact with the i n i t i a l y i e l d l sur face def ined i n F i g . (4.3) a. s tress tensor defined i n Table 3.1 I ax'>ay x and y components of o v e r a l l laminate s tresses {AO13} p l a s t i c s tress increment vector define by Eq . (4.31) x shear s tress i n the o v e r a l l laminate coordinate system xy J $ s tra in -energy dens i ty funct ion defined by Eq . (3.4) <JK shape functions for an 8-node isoparametric element X def ined by E q . (A.3) vector of the unblanaced forces defined by E q . (4.17) w angle defined by Eq . (4.47) - x v i i i -L I S T OF SYMBOLS ( C o n t ' d ) S u f f i x 0 i n i t i a l y i e l d value u f a i l u r e value e e l a s t i c or elemental quant i t i e s depending on the context used p p l a s t i c ep e l a s t o p l a s t i c r quant i t i e s r e l a t e d to the r ^ i t e r a t i o n k quant i t i e s r e l a t e d to the k ^ layer of a laminate ra quant i t i e s r e l a t e d to the m*"*1 s t r a i n subincrement ( ) ' quant i t i e s transformed from p l y to laminate coordinates {} vector quant i t i e s [] matrix quant i t i e s - x i x -ACKNOWLEDGEMENT I w i s h t o e x p r e s s my d e e p e s t g r a t i t u d e t o my r e s e a r c h a d v i s o r s , D r . M.D. O l s o n and D r . D . L . A n d e r s o n , f o r s u g g e s t i n g t h i s t h e s i s t o p i c and f o r t h e i r g u i d a n c e and c o n t i n u e d e n c o u r a g e m e n t , w i t h o u t w h i c h t h i s work w o u l d n o t have b e e n r e a l i z e d . A p p r e c i a t i o n i s a l s o e x p r e s s e d t o t h e S u p e r v i s o r y Commi t tee members, D r . A . P o u r s a r t i p and D r . H. Ramsey, f o r t h e i r t i m e , h e l p f u l s u g g e s t i o n s and comments. The f i n a n c i a l s u p p o r t p r o v i d e d b y t h e C a n a d i a n Depa r tment o f N a t i o n a l D e f e n c e t h r o u g h a c o n t r a c t f r o m t h e D e f e n c e R e s e a r c h E s t a b l i s h m e n t S u f f i e l d i s g r a t e f u l l y a c k n o w l e d g e d . I am, o f c o u r s e , g r e a t l y i n d e b t e d t o my f e l l o w g r a d u a t e s t u d e n t s b o t h p a s t and p r e s e n t who have n o t o n l y c o n t r i b u t e d t o my e d u c a t i o n b u t have made my t i m e a t t h e U n i v e r s i t y v e r y e n j o y a b l e . I n p a r t i c u l a r I w o u l d l i k e t o r e c o r d my t h a n k s t o B r y a n F o l z , P a r e s h P a t t a n i , Bob S c h u b a k , S a r a t h A b a y a k o o n , K e v i n M c T a g g a r t , G e r a r d C a n i s i u s , and K w o k - F a i Cheung . S p e c i a l t h a n k s go t o my f r i e n d s , S i a v o c h e M o u s s a v i and A r d e s h i r R i a h i f o r b e i n g a v a i l a b l e a t t h e t i m e when t h e i r m o r a l s u p p o r t was most n e e d e d . The s k i l l f u l and prompt t y p i n g o f M r s . K e l l y Lamb comb ined w i t h h e r p a t i e n c e and good humour a r e d e e p l y a p p r e c i a t e d . I am t r u l y i n d e b t e d t o h e r ( i n c l u d i n g f i n a n c i a l l y ! ) f o r h e r e f f o r t s i n m a k i n g t h i s t h e s i s p r e s e n t a b l e . F i n a l l y , I w o u l d l i k e t o t a k e t h i s o p p o r t u n i t y t o e x p r e s s my d e e p e s t g r a t i t u d e t o my l o v i n g p a r e n t s and k i n d b r o t h e r , H a n s , f o r t h e c a r e , e n c o u r -agement and c o n t i n u e d s u p p o r t t h a t I have r e c e i v e d f r o m them t h r o u g h o u t my l i f e . - x x -1 CHAPTER 1 INTRODUCTION 1.1 Introduct ion The p o s s i b i l i t y of manufacturing mater ia l s with des i rab le mechanical proper t i e s by r e i n f o r c i n g a matrix mater ia l with strong f i b r e s having high e l a s t i c extensional modulus has r e c e n t l y rece ived a great deal of a t t e n t i o n , both experimental and t h e o r e t i c a l . Recent a c t i v i t y has l a r g e l y been st imulated by the development of new types of high s trength f i b r e s , but the idea i s a very o l d one. A r t i f i c i a l mater ia l s such as f i b r e g l a s s and r e i n f o r c e d concrete which are of t h i s type have been a v a i l a b l e for some t ime, and many n a t u r a l m a t e r i a l s , for example wood and bone, are e s s e n t i a l l y of t h i s character . Composite mater ia l s i n v o l v i n g f i b r e reinforcement are cont inuing to replace t r a d i t i o n a l mater ia l s at a r a p i d r a t e . The d r i v i n g force for t h i s replacement i s due to t h e i r outstanding s p e c i f i c propert ie s ( i . e . h igh s trength and s t i f fnes s - to -we ight r a t i o s ) . These superior proper t i e s of composite mater ia l s over the monol i th ic metals makes these mater ia l s very a t t r a c t i v e for weight and s t i f f n e s s s e n s i t i v e s t r u c t u r e s . The a p p l i c a t i o n s range from sports equipment, automotive p a r t s , a i r c r a f t and aerospace s tructures to h igh performance m i l i t a r y s tructures (e .g . ground, underwater and space v e h i c l e s ) . The r a p i d growth rate of t h i s f i e l d e n t a i l s a good understanding of the mechanics of composites so that they may be e f f i c i e n t l y u t i l i z e d i n engineering a p p l i c a t i o n s . With the present s tate of development of f i n i t e -element computer programs, the problem of modell ing the mechanical behaviour of composites remains one of the most d i f f i c u l t challenges i n the f i e l d of 2 composite s t r u c t u r a l engineering. Mater ia l s behaviour re fers to m u l t i -d i m e n s i o n a l s t r e s s - s t r a i n ^ " r e l a t i o n s which adequate ly describe the bas ic c h a r a c t e r i s t i c s of the mater ia l subjected to monotonic and c y c l i c l oad ing . The emphasis i n t h i s thes i s i s p laced upon the c o n s t i t u t i v e modell ing i n a n a l y t i c a l and numerical ana lys i s of composite s t r u c t u r e s . We s h a l l be concerned p r i m a r i l y with continuum theories and "macroscopic" models of mater ia l behaviour. However, the proper t i e s of the composite mater ia l s we consider derive u l t i m a t e l y from the proper t i e s and geometrical arrangements of t h e i r cons t i tuents . Thus i n formulating the macroscopic mechanical proper t i e s of the composite i t i s impossible to d i sregard e n t i r e l y the proper t i e s of i t s cons t i tuents . Therefore for background and mot ivat ion we begin i n t h i s in troductory chapter with a b r i e f d i s c u s s i o n of some of the proper t i e s of composite m a t e r i a l s , and the way i n which these depend on the proper t i e s of t h e i r cons t i tuents . We do not , however, attempt to give i n any way a comprehensive account of the great volume of work which has been done i n studying the i n t e r a c t i o n s between const i tuents of composites, or determin-ing the proper t i e s of composites i n terms of t h e i r cons t i tuents . Although these problems are of great importance, they-are outs ide the scope of t h i s t h e s i s . 1.2 Types of Composites and Basic Terminology A composite mater ia l i s defined to be any mater ia l cons i s t ing of two or more d i s t i n c t const i tuents (or phases) . For the sake of convenience, one of the phases w i l l be r e f e r r e d to as the "matrix", while the others as the The s t r e s s - s t r a i n r e l a t i o n s are a l so r e f e r r e d to as c o n s t i t u t i v e r e l a t i o n s as they describe the mechanical c o n s t i t u t i o n of the m a t e r i a l . 3 "reinforcement". Even though reinforcement impl ies strengthening of the m a t e r i a l , the term i s used to denote any phase that i s imbedded i n the matr ix . Thus cracks and voids are inc luded w i th in t h i s term. The main types of reinforcement are p a r t i c l e s , chopped (or d i scont inuous) , continuous f ibres and f l akes . Although f lakes and p a r t i c l e s have become important const i tuents i n many composite systems, f i b r e reinforcement dominates the f i e l d and are by far the most extens ive ly analysed. For the remainder of t h i s t h e s i s , emphasis w i l l be p laced on f i b r e - r e i n f o r c e d mater ia l s , (FRM). It i s use fu l at t h i s stage to examine the const i tuents of these m a t e r i a l s . F i b r e - r e i n f o r c e d mater ia l s are u s u a l l y d i v i d e d in to three broad groups according to the matrix mater ia l s : p l a s t i c (e .g . epoxies) ; metal (e .g . aluminum and magnesium); and ceramic. The r o l e of the matrix mater ia l i s to b ind the r e i n f o r c i n g f i b r e s together in to a s o l i d mass and therefore enable the t rans fer of load to the f i b r e s . The matrix a lso permits ease of f a b r i c a t i o n in to a des i red c o n f i g u r a t i o n . In many s t r u c t u r a l a p p l i c a t i o n s the f i b r e proper t i e s are the most important and the matrix may be chosen based on cost and minimum weight. There are , however, a s i g n i f i c a n t number of a p p l i c a t i o n s i n a i r c r a f t , spacecraf t , e t c . where the matrix must possess p a r t i c u l a r p r o p e r t i e s . In these cases the mater ia l i s subjected to h igh temperatures, where p l a s t i c matrix composites are unusable, so that e i t h e r metal or i n the case of extreme temperatures, ceramic matrix composites must be cons idered. Commonly used f ibres are g la s s , boron, k e v l a r , and carbon (graphi te ) . The f i b r e s may be continuous, i n which case each f i b r e extends through a body from one boundary to another, or discontinuous i n the form of chopped f i b r e s . Continuous FRM can be made e i ther by a l i g n i n g a l l the f ibres i n one d i r e c t i o n ( u n i d i r e c t i o n a l ) , or weaving a c l o t h ( b i d i r e c t i o n a l ) . The chopped f i b r e composites are s t a t i s t i c a l l y i s o t r o p i c and t h e i r a n a l y t i c a l 4 t r e a t m e n t does n o t p o s e a g r e a t d i f f i c u l t y . W i t h t h a t i n m i n d much o f what w i l l be d i s c u s s e d i n t h i s t h e s i s i m p l i e s c o m p o s i t e s made f r o m c o n t i n u o u s f i b r e s . U n i d i r e c t i o n a l FRMs have e x c e p t i o n a l s t r e n g t h and s t i f f n e s s p r o p e r t i e s i n t h e d i r e c t i o n o f f i b r e s , h o w e v e r , t h e i r p r o p e r t i e s i n any o t h e r d i r e c t i o n a r e r a t h e r p o o r . The o v e r a l l p r o p e r t i e s a r e n o r m a l l y i m p r o v e d b y l a m i n a t i n g s i n g l e p l i e s " ' ' w i t h d i f f e r e n t r e i n f o r c i n g d i r e c t i o n s . T h i s l e a d s t o what i s c a l l e d " l a m i n a t e d c o m p o s i t e s " o r " l a m i n a t e s " . The l a m i n a t e i s t a i l o r e d t o j u s t meet s p e c i f i c r e q u i r e m e n t s . By a p p r o p r i a t e c o n s i d e r a t i o n o f t h e l o a d s and t h e i r d i r e c t i o n s , a l a m i n a t e c a n be c o n s t r u c t e d o f i n d i v i d u a l p l i e s i n s u c h a manner as t o j u s t r e s i s t t h o s e l o a d s and no m o r e . I n t h i s r e s p e c t i s o t r o p i c m a t e r i a l s a r e u s u a l l y i n e f f i c i e n t b e c a u s e e x c e s s s t r e n g t h and s t i f f n e s s i s i n e v i t a b l y a v a i l a b l e i n some d i r e c t i o n . B i d i r e c t i o n a l woven f a b r i c s have i n f e r i o r m e c h a n i c a l p r o p e r t i e s ( i n t h e f i b r e d i r e c t i o n ) t o t h e i r u n i d i r e c t i o n a l c o u n t e r p a r t s . T h i s i s due i n p a r t t o t h e w e a v i n g p r o c e s s w h i c h may c a u s e f i b r e damage. L a y e r s o f woven f a b r i c a r e t h e r e f o r e o f t e n u s e d as f i l l e r l a y e r s where s t r e n g t h and s t i f f n e s s a r e n o t c r i t i c a l . The m a i n r e a s o n s f o r t h e u s e o f t h e s e m a t e r i a l s a r e e a s e o f h a n d l i n g ( w i t h c o n s e q u e n t r e d u c t i o n i n l a b o u r c o s t s ) , and t h e a b i l i t y o f f a b r i c t o c o n f o r m t o c o m p l e x s h a p e s . They a r e e x t e n s i v e l y u s e d i n b o a t c o n s t r u c t i o n and s h i p s u p e r s t r u c t u r e s . Woven c o m p o s i t e s a r e a l s o known t o r e s u l t i n b e t t e r c o n t a i n m e n t o f i m p a c t damage and i m p r o v e d r e s i d u a l p r o p e r -t i e s a f t e r i m p a c t compared w i t h nonwoven m a t e r i a l s ( S m i t h , C . S . , 1 9 8 6 ) . Some a u t h o r s u s e t h e t e r m " l a m i n a " t o d e n o t e a s i n g l e l a y e r o r p l y . However , t h i s t e r m i n o l o g y i s n o t u s e d h e r e , s i n c e i t i s e a s i l y c o n f u s e d w i t h t h e t e r m " l a m i n a t e " , w h i c h means a l l o f t h e l a y e r s bonded t o g e t h e r . 1.3 General Remarks on the Mechanical Properties of Fibre Composites The aim of t h i s s ec t ion i s to convey i n a q u a l i t a t i v e manner a general understanding of the mechanical behaviour of composites. The behaviour of composite mater ia l s can be s tudied t h e o r e t i c a l l y from three l e v e l s of magni f i ca t ion . They are i ) Micromechanics - which considers the problems of l o c a l i n t e r a c t i o n s at the in ter faces between the f i b r e and the matrix phase. For the study of i n t e r f a c e problems a t t en t ion i s given to a s ing l e f i b r e and i t s surrounding matrix m a t e r i a l . For purposes of ana lys i s the usual procedure i s to regard the composite as an assemblage of c i r c u l a r c y l i n d r i c a l f ibres surrounded by concentr ic hollow c y l i n d e r s of matrix m a t e r i a l . Examples of the use of t h i s approach can be found i n the survey a r t i c l e s by Hashin (1983), Franc i s and Bert (1975), and Chamis and Sendeckyj (1968). Apart from using them for comparison purposes, the micromechanical analyses w i l l not be pursued i n t h i s t h e s i s . i i ) Minimechanics - which re la t e s the proper t i e s of the composite to the i n d i v i d u a l proper t i e s of the f i b r e and the matr ix . A mathematical model of a composite i s constructed by applying any of the r h e o l o g i c a l proper-t i e s (e .g . e l a s t i c , v i s c o e l a s t i c , p l a s t i c , e t c . ) to the f i b r e and any one to the matr ix . Geometry of the phases are not taken in to account here . In t h i s approach the f i b r e and matrix are u s u a l l y separated and rearranged i n ser i e s or i n p a r a l l e l as may be appropr ia te . i i i ) Macromechanics - which describes the behaviour of the composite by continuum models without d i r e c t reference to the proper t i e s of the i n d i v i d u a l cons t i tuents . In other words the FRM i s t reated as a homo-geneous a n i s o t r o p i c continuum with some average proper t i e s known from experiments. Such an' approach which i s fol lowed i n t h i s thes i s i s 6 a p p e a l i n g t o t h e e n g i n e e r o r d e s i g n e r who r e q u i r e s r e a s o n a b l y s i m p l e ( y e t r e a l i s t i c ) methods o f s t r e s s and s t r a i n a n a l y s i s o f c o m p o s i t e m a t e r i a l s . T h i s a p p e a r s t o be a t v a r i a n c e w i t h t h e a p p r o a c h o f t h e m e t a l l u r g i s t who w i s h e s t o g i v e an a c c u r a t e d e s c r i p t i o n o f t h e mechan isms w h i c h t a k e p l a c e . I t c a n be s e e n t h a t t h e m a c r o s c o p i c b e h a v i o u r i s a c o n s e q u e n c e o f t h e b e h a v i o u r on t h e m i n i s c a l e . I n t u r n t h e b e h a v i o u r on t h e m i n i s c a l e depends upon t h e b e h a v i o u r on t h e m i c r o s c a l e and so on down ( D r u c k e r , 1 9 7 5 ) . Under s u f f i c i e n t l y s i m p l e c o n d i t i o n s , we c a n p r o c e e d w i t h c o n f i d e n c e one s t e p up o r down i n s c a l e . Most o f t e n i t i s n o t p o s s i b l e t o g i v e more t h a n a q u a l i t a t i v e p r e d i c t i o n o f t h e i n f l u e n c e o f one l e v e l on t h e n e x t . F o r example i t i s l i k e l y t o be d i f f i c u l t t o p r e d i c t q u a n t i t a t i v e l y t h e m e c h a n i c a l p r o p e r t i e s o f a FRM i n t e r m s o f t h e p r o p e r t i e s o f t h e c o n s t i t u e n t s . However , i t i s h e l p f u l t o have a t l e a s t a q u a l i t a t i v e a p p r e c i a t i o n o f m a t e r i a l b e h a v i o u r a t t h e m i c r o and m i n i s c a l e s i n o r d e r t o f o r m u l a t e t h e m a c r o s c o p i c t h e o r y w h i c h w i l l be d e v e l o p e d i n l a t e r c h a p t e r s . W i t h t h i s i n m i n d we g i v e a b r i e f d i s c u s s i o n on t h e g e n e r a l p r o p e r t i e s o f FRMs. The i n t r o d u c t i o n o f a f a m i l y o f f i b r e s i n a d e f i n i t e o r i e n t a t i o n i n t h e m a t r i x i m m e d i a t e l y i n t r o d u c e s a p r e f e r r e d d i r e c t i o n i n t h e m a t e r i a l . Thus e v e n i f t h e c o n s t i t u e n t s o f a f i b r e r e i n f o r c e d c o m p o s i t e a r e i s o t r o p i c , t h e c o m p o s i t e i t s e l f w i l l be m a c r o s c o p i c a l l y a n i s o t r o p i c . I n d e e d i f t h e r e a r e l a r g e d i f f e r e n c e s b e t w e e n t h e m e c h a n i c a l p r o p e r t i e s o f t h e f i b r e and m a t r i x t h e n t h e p r o p e r t i e s a l o n g and p e r p e n d i c u l a r t o f i b r e w o u l d be q u i t e d i f f e r -e n t . T h i s g i v e s r i s e t o a " s t r o n g l y a n i s o t r o p i c " m a t e r i a l ( s u c h a s F i b r e - r e i n f o r c e d c o m p o s i t e l a y e r s , w h e t h e r u n i d i r e c t i o n a l o r b i d i r e c t i o n a l , a r e a l m o s t i n v a r i a b l y o r t h o t r o p i c p o s s e s s i n g t h r e e p l a n e s o f m a t e r i a l symmetry . M o r e o v e r , s i n c e t h e f i b r e s a r e g e n e r a l l y a t random l o c a t i o n s t h e u n i d i r e c t i o n a l FRM i s m a c r o s c o p i c a l l y t r a n s v e r s e l y i s o t r o p i c . 7 u n i - d i r e c t i o n a l c o m p o s i t e s ) . By t h e same t o k e n a m a t r i x r e i n f o r c e d b y two f a m i l i e s o f f i b r e s ( s u c h as w o v e n - c l o t h r e i n f o r c e d c o m p o s i t e s ) a r e c o n s i d e r e d as " w e a k l y a n i s o t r o p i c " on t h e m a c r o s c o p i c l e v e l . I n a r e a l c o m p o s i t e , t h e m a t r i x m a t e r i a l has a l o w s t i f f n e s s and s t r e n g t h compared t o t h e f i b r e . F i b r e s g e n e r a l l y e x h i b i t l i n e a r e l a s t i c b e h a v i o u r . M e t a l m a t r i x . m a t e r i a l s e x h i b i t e l a s t i c - p l a s t i c b e h a v i o u r and p o l y m e r i c m a t r i c e s u s u a l l y a r e v i s c o e l a s t i c i f n o t v i s c o p l a s t i c . C o n s i d e r a u n i d i r e c t i o n a l FRM. A l o n g t h e f i b r e d i r e c t i o n t h e p r o p e r t i e s o f t h e c o m p o s i t e i s p r e d o m i n a n t e l y t h a t o f t h e f i b r e r e s u l t i n g i n f i b r e f r a c t u r e ( o r m u l t i p l e m a t r i x c r a c k i n g i f t h e u l t i m a t e s t r a i n o f t h e m a t r i x i s l o w e r t h a n t h a t o f t h e f i b r e ) . I n t h e d i r e c t i o n t r a n s v e r s e t o t h e f i b r e s , t h e i n h e r e n t m i s m a t c h o f s t i f f n e s s e s b e t w e e n f i b r e and t h e m a t r i x r e s u l t s i n t h e d e v e l o p m e n t o f h i g h l o c a l s t r a i n c o n c e n t r a t i o n s i n t h e m a t r i x . To accommodate t h e s e h i g h l o c a l s t r a i n s w i t h o u t i n d u c i n g l o c a l f a i l u r e s , m a t r i x m a t e r i a l s a r e n o r m a l l y s e l e c t e d w h i c h have h i g h s t r a i n s t o f a i l u r e . T h i s p e r m i t s l o c a l p l a s t i c f l o w t o o c c u r i n r e g i o n s o f h i g h s t r a i n c o n c e n t r a t i o n c a u s i n g a r e d i s t r i b u t i o n o f s t r e s s e s ( s t r e s s r e l i e f ) . T h u s , e v e n i f t h e a v e r a g e a p p l i e d t r a n s v e r s e s t r e s s on t h e c o m p o s i t e i s r e l a t i v e l y l o w , l o c a l s t r e s s e s and s t r a i n s w i t h i n t h e c o m p o s i t e m a t e r i a l may have e x c e e d e d t h e e l a s t i c l i m i t . A t t h i s p o i n t t h e m a t r i x f l o w s o r f r a c t u r e s ( a c c o r d i n g t o t h e d e g r e e o f d u c t i l i t y o f t h e m a t r i x m a t e r i a l ) . The s t r a i n t o f a i l u r e o f t h e c o m p o s i t e i n s u c h c a s e s i s v e r y s m a l l compared t o t h a t o f t h e m a t r i x . S h e a r r e s p o n s e i n t h e m a t e r i a l p r i n c i p a l d i r e c t i o n s e x h i b i t s c o n s i d e r -a b l e n o n l i n e a r i t y i n d i c a t i n g t h e dominance o f t h e ( s o f t ) m a t r i x m a t e r i a l H e r e t h e t e r m " t r a n s v e r s e " i s r e s e r v e d f o r t h e i n - p l a n e d i r e c t i o n w h i c h i s p e r p e n d i c u l a r t o t h e l o n g i t u d i n a l d i r e c t i o n . Out o f p l a n e d i r e c t i o n w i l l be r e f e r r e d t o as " t h i c k n e s s " (as o p p o s e d t o t r a n s v e r s e ) d i r e c t i o n . 8 u n d e r s u c h l o a d i n g s . I t i s r e a d i l y a p p a r e n t t h a t t h e s e s h e a r s t r e s s components a r e p r e s e n t e v e n when t h e c o m p o s i t e l a y e r i s s u b j e c t e d t o n o r m a l s t r e s s e s a t an a n g l e t o t h e p r i n c i p a l m a t e r i a l a x e s . The n o n l i n e a r s h e a r r e s p o n s e i s t h e r e f o r e a m a j o r , p e r h a p s t h e m a j o r , s o u r c e o f n o n l i n e a r i t i e s i n t h e r e s p o n s e o f c o m p o s i t e l a m i n a t e s . Any a n a l y s i s w h i c h hopes t o p r o v i d e r e a l i s t i c a s s e s s m e n t s o f t h e s t r e s s e s and s t r a i n s i n v a r i o u s p l i e s o f t h e l a m i n a t e d c o m p o s i t e s must a c c o u n t f o r t h e n o n l i n e a r s h e a r r e s p o n s e . T h e r e a r e many ways i n w h i c h a FRM l a y e r may f a i l . These may be i n t h e f o r m o f m a t r i x c r a c k s , f i b r e f r a c t u r e , i n t e r f a c e s e p a r a t i o n ( i . e . f i b r e p u l l o u t ) and l o c a l p l a s t i f i c a t i o n i n t h e m a t r i x . Some o f t h e s e f a i l u r e mechanisms r e s e m b l e b r i t t l e f r a c t u r e o f t h e c o m p o s i t e , w i t h l o w e n e r g y a b s o r p t i o n , w h i l e o t h e r s p r o d u c e a d u c t i l e t y p e o f f r a c t u r e w i t h t h e a b s o r p t i o n o f a l a r g e q u a n t i t y o f e n e r g y . E x p e r i m e n t a l e v i d e n c e shows t h a t f o r l a m i n a t e s c o n s i s t i n g o f p o l y m e r i c m a t r i x f i b r e c o m p o s i t e s , u n d e r s t a t i c o r c y c l i c l o a d , t h e r e a r e two m a j o r t y p e s o f c r a c k s : (a) i n t r a l a m i n a r ( o r i n t r a p l y ) c r a c k s w i t h i n c e r t a i n p l i e s ; and (b) i n t e r l a m i n a r ( o r i n t e r p l y ) c r a c k s w h i c h d e v e l o p on p l a n e s be tween p l i e s . M e t a l m a t r i x c o m p o s i t e s u s u a l l y d e f o r m p l a s t i c a l l y and do n o t e x h i b i t e x t e n s i v e m a t r i x c r a c k i n g u n d e r m o n o t o n i c l o a d s , b u t a r e q u i t e s u s c e p t i b l e t o m a t r i x f a t i g u e c r a c k i n g when s u b j e c t e d t o c y c l i c l o a d i n g (Dvorak and J o h n s o n , 1 9 8 0 ) . I n t r a l a m i n a r c r a c k s a r e e i t h e r s h o r t c r a c k s ( p e r p e n d i c u l a r t o t h e f i b r e s ) t h a t r u p t u r e t h e f i b r e s and debond f i b r e m a t r i x i n t e r f a c e s , o r , l o n g c r a c k s ( p a r a l l e l t o t h e f i b r e s ) t h a t t r a v e r s e f r o m edge t o edge and a r e e s s e n t i a l l y n o r m a l t o t h e p l a n e o f t h e p l y . I n t e r l a m i n a r c r a c k s w h i c h debond t h e p l y i n t e r f a c e s e i t h e r o r i g i n a t e a t l a m i n a t e edges due t o t h e p r e s e n c e o f h i g h v a l u e s o f i n t e r l a m i n a r n o r m a l and s h e a r s t r e s s , o r , a t r e g i o n s o f h i g h t r a n s v e r s e s h e a r i n l a m i n a t e d p l a t e s u n d e r b e n d i n g . 9 From t h e above b r i e f d i s c u s s i o n i t i s c l e a r t h a t t h e c o n s i d e r a t i o n o f n o n l i n e a r b e h a v i o u r i s i m p o r t a n t e v e n f o r c o m p o s i t e m a t e r i a l s s u b j e c t e d t o a p p l i e d s t r e s s e s w h i c h a r e i n t e n d e d t o r e m a i n b e l o w t h e a p p a r e n t e l a s t i c l i m i t o f t h e c o m p o s i t e . F o r l a m i n a t e d s t r u c t u r e s d e s i g n e d t o an u l t i m a t e s t r e n g t h c r i t e r i o n , t h e n e e d f o r c o n s i d e r a t i o n o f n o n l i n e a r b e h a v i o u r , i n p a r t i c u l a r i n e l a s t i c e f f e c t s i n c l u d i n g f a i l u r e , i s e v e n more a p p a r e n t . 1.4 P u r p o s e and Scope o f t h e P r e s e n t S t u d y The b a s i c o b j e c t i v e i n t h i s t h e s i s i s t o d e v e l o p a r e l a t i v e l y c o m p r e -h e n s i v e p l a s t i c i t y - b a s e d m a c r o s c o p i c c o n s t i t u t i v e mode l f o r t h e i n d i v i d u a l l a y e r o f u n i d i r e c t i o n a l and b i d i r e c t i o n a l FRMs. T h i s mode l i s t o be u s e d i n t h e n o n l i n e a r a n a l y s i s o f l a m i n a t e s h a v i n g an a r b i t r a r y number o f s u c h l a y e r s w i t h v a r i o u s f i b r e o r i e n t a t i o n s . F i n i t e e l e m e n t a n a l y s e s w i l l be d e v e l o p e d f o r t h e c a s e o f s y m m e t r i c l a m i n a t e s s u b j e c t e d t o membrane l o a d i n g . I t w i l l be shown t h a t p l a s t i c i t y t h e o r y , when n o t i n t e r p r e t e d t o o n a r r o w l y , i s a v e r y f l e x i b l e mode l - one t h a t c a n be u s e d t o d e s c r i b e a w i d e v a r i e t y o f b e h a v i o u r i n c l u d i n g c r a c k i n g o f p o l y m e r i c c o m p o s i t e s . C h a p t e r 2 a ims a t r e v i e w i n g some o f t h e immense e x i s t i n g body o f t h e l i t e r a t u r e on c o n s t i t u t i v e m o d e l l i n g o f f i b r e - r e i n f o r c e d c o m p o s i t e s . C h a p t e r 3 o u t l i n e s t h e t h e o r e t i c a l f o r m u l a t i o n o f t h e p r o p o s e d c o n s t i t u t i v e m o d e l . C h a p t e r 4 d e s c r i b e s t h e i m p l e m e n t a t i o n o f t h e c o n s t i t u t i v e mode l i n a two d i m e n s i o n a l f i n i t e e l e m e n t p r o g r a m u s e d t o p e r f o r m p r o g r e s s i v e f a i l u r e a n a l y s i s o f c o m p o s i t e l a m i n a t e s . The n u m e r i c a l a n a l y s i s d e v e l o p e d i s t h e n a p p l i e d t o a s e r i e s o f p r o b l e m s and t h e r e s u l t s a r e compared w i t h a w i d e r a n g e o f e x p e r i m e n t a l and o t h e r n u m e r i c a l r e s u l t s i n C h a p t e r 5 . C h a p t e r 6 o u t l i n e s t h e c o n c l u s i o n s t h a t c a n be drawn f r o m t h e r e s u l t s o f t h e p r o p o s e d t h e o r y . The a p p l i c a b i l i t y o f t h e c o n s t i t u t i v e mode l i s d i s c u s s e d and f u r t h e r a r e a s o f r e s e a r c h a r e s u g g e s t e d . 10 CHAPTER 2 REVIEW OF THE LITERATURE 2.1 Introduct ion Although l a r g e - s c a l e f i n i t e element software packages now have a wide range of a p p l i c a t i o n i n s tress ana lys i s of composites ( G r i f f i n , 1982), inade-quate mater ia l models are often one of the major obstacles for a r igorous a n a l y s i s . E r r o r s assoc iated with mater ia l proper t i e s are u s u a l l y far greater than errors inherent i n the numerical methods of so lv ing the f i e l d equations. A large v a r i e t y of models have been proposed to charac ter i ze the s t r e s s -s t r a i n and f a i l u r e behaviour of f i b r e re in forced mater ia l s (FRMs) under mult idimensional s tress s ta tes . A l l these models have c e r t a i n inherent advantages and disadvantages which depend to a large degree on t h e i r p a r t i c u -l a r a p p l i c a t i o n . The objec t ive of t h i s sec t ion i s to present a summary of various proposed mater ia l models of FRM and to determine the range of t h e i r a p p l i c a b i l i t y , r e l a t i v e merits and l i m i t a t i o n s . No survey or l i s t of references can even approach complete coverage of such a wide f i e l d and the references c i t e d here a f f o r d only a glimpse of the extensive l i t e r a t u r e a v a i l a b l e and are by no means exhaustive. 2.2 Background In construct ing a c o n s t i t u t i v e model for laminated composites the under-l y i n g po int of view i s that the laminate response must be understood i n terms of the behaviour at the p l y l e v e l . In other words the mechanical behaviour of a s ing le layer forms the bas ic b u i l d i n g block i n the ana lys i s of laminated s tructures (cons i s t ing of several i n d i v i d u a l l a y e r s ) . The gross behaviour of p e r f e c t l y e l a s t i c laminated p la tes under bending and s t re tch ing deformations can be exac t ly analyzed i n terms of the p l y proper t i e s and t h e i r s tacking 11 a r r a n g e m e n t . These a n a l y s e s a r e b a s e d on t h e c l a s s i c a l l a m i n a t i o n t h e o r y (CLT) w h i c h i s w e l l known i n t h e c o m p o s i t e s l i t e r a t u r e . The b a s i c a s s u m p t i o n i n CLT i s t h a t t h e r e i s a p e r f e c t b o n d i n g a t t h e p l y i n t e r f a c e s and t h a t e a c h p l y , w h i c h i s c o n s i d e r e d t o be homogeneous and a n i s o t r o p i c , i s u n d e r a s t a t e o f p l a n e s t r e s s . Once t h e s t r a i n f i e l d i s p r e s c r i b e d ( s u c h as u n i f o r m s t r a i n i n c a s e o f membrane l o a d i n g , and l i n e a r d i s t r i b u t i o n a c c o r d i n g t o t h e K i r c h h o f f - L o v e h y p o t h e s i s i n t h e c a s e o f b e n d i n g ) , t h e s t r e s s f i e l d i s d i r e c t l y d e t e r m i n e d b y t h e s t r e s s - s t r a i n r e l a t i o n s o f e a c h p l y . However , i n i n e l a s t i c l a m i n a t e s s u c h a d i r e c t d e t e r m i n a t i o n o f t h e s t r e s s f i e l d f r o m t h e s t r a i n d i s t r i b u t i o n i s n o t p o s s i b l e due t o t h e p a t h dependent n a t u r e o f t h e s t r e s s - s t r a i n r e l a t i o n s . T h i s does n o t p o s e an i n s u r m o u n t a b l e p r o b l e m i n v i e w o f t h e power o f modern n u m e r i c a l c o m p u t a t i o n . The c o n c l u s i o n i s t h a t a n a l y s i s o f "undamaged" l a m i n a t e s , e v e n f o r r a t h e r c o m p l i c a t e d c a s e s o f m a t e r i a l b e h a v i o u r ( o f i n d i v i d u a l p l i e s ) c a n be c a r r i e d o u t . The p r o b l e m s emerge when one c o n s i d e r s "damage" and " f a i l u r e " . C o m p o s i t e l a m i n a t e s u n d e r s t a t i c o r c y c l i c l o a d i n g e x p e r i e n c e f a i l u r e i n one o r more l a y e r s e a r l y d u r i n g t h e l o a d i n g p r o c e s s , w h i c h i n most c a s e s does n o t l e a d t o t h e f a i l u r e o f t h e e n t i r e l a m i n a t e . Such a f a i l u r e i m p l i e s damage i n t e r m s o f c r a c k d i s t r i b u t i o n w i t h i n t h e f a i l e d p l y ( o r p l i e s ) . To d e t e r m i n e s u b s e q u e n t f a i l u r e s i t i s n e c e s s a r y t o p e r f o r m s t r e s s a n a l y s i s o f t h e "damaged" l a m i n a t e . Thus t h e l a m i n a t e must be s u b j e c t e d t o a p r o g r e s s i v e f a i l u r e a n a l y s i s u n t i l i t s l o a d c a r r y i n g c a p a c i t y i s e x h a u s t e d . Such a n a l y t i c a l d e t e r m i n a t i o n o f t h e f a i l u r e l o a d s o f a l a m i n a t e i s a h i g h l y c o n t r o v e r s i a l i s s u e and r e m a i n s one o f t h e most c h a l l e n g i n g a r e a s o f c u r r e n t r e s e a r c h . I t i s a p p a r e n t f r o m t h e above d i s c u s s i o n t h a t t h e s c o p e o f r e s e a r c h on t h e c o n s t i t u t i v e b e h a v i o u r o f c o m p o s i t e l a m i n a t e s may be d i v i d e d i n t o f o u r m a i n s e q u e n t i a l s u b j e c t s : 12 • Behaviour of the undamaged laminate, i . e . p r e - f a i l u r e . • Onset of damage, i . e . f i r s t p l y or i n i t i a l f a i l u r e . • Behaviour of the damaged laminate, i . e . post f i r s t p l y f a i l u r e . • Ult imate f a i l u r e . The remainder of t h i s sec t ion gives a b r i e f review of the a v a i l a b l e l i t e r a t u r e on the a n a l y t i c a l treatment of the top ic s i temized above. 2 . 3 C o n s t i t u t i v e M o d e l l i n g o f Undamaged C o m p o s i t e s A n a l y t i c a l s tudies of the problem of mechanical behaviour of composite mater ia l s have been approached from any one of three bas ic l e v e l s , namely, the micromechanics l e v e l , the minimechanics l e v e l and the macromechanics l e v e l . The l inear -range proper t i e s of composites, approached from a l l three l e v e l s , are quite we l l understood and w e l l documented i n a l l the standard texts on mechanics of composites (see for example the survey a r t i c l e by Hashin (1983)). On the other hand, the nonl inear behaviour of f i b r e compo-s i t e s i s much more complicated and i t s r igorous treatment d i d not s t a r t u n t i l the e a r l y 1970s. The complexity of nonl inear response exh ib i t ed by compo-s i t e s helps to exp la in the d i v e r s i f i e d methodologies employed by numerous inves t i ga tors i n formulating c o n s t i t u t i v e theories for these m a t e r i a l s . Unless otherwise s ta t ed , the d i s cuss ion i n t h i s s ec t ion i s concerned with the behaviour of a s i n g l e p l y . 2 . 3 . 1 M i c r o m e c h a n i c s A p p r o a c h Rigorous micromechanical models based on the mathematical theory of p l a s t i c i t y have appeared i n the l i t e r a t u r e . Notable among these are the f i n i t e element approaches of Adams (1970, 1974), Foye and Baker (1971), Foye (1973), L i n et a l . (1972) and Dvorak et a l . (1973, 1974) to determine i n i t i a l 13 y i e l d surfaces and subsequent s t r e s s - s t r a i n curves of u n i d i r e c t i o n a l l y -r e i n f o r c e d composites. Aboudi (1984) considered a model of a square array of f i b r e s with square sect ions i n terms of l i n e a r approximating f i e l d s i n c l u d i n g v i s c o p l a s t i c e f f e c t s . In a recent paper Aboudi (1986) summarized h i s c o n s t i -t u t i v e theory and gave a l i s t of references of previous works. A l l the above papers take in to account the complicated geometry of the composite on the microsca le . Although such d e t a i l e d inves t iga t ions are e s s e n t i a l for an understanding of the i n e l a s t i c behaviour of FRMs they are unfortunate ly very complicated and require a r e l a t i v e l y large computer f a c i l i t y and s i g n i f i c a n t amounts of computer time. Thus, they can be p r o h i b i t i v e l y expensive to u t i l i z e ex tens ive ly i n the s tress ana lys i s of large sca le s t r u c t u r e s . 2.3.2 Minimechanics Approach Approaches i n t h i s category t y p i c a l l y r e l a t e s tresses to s t r a i n s i n terms of p h y s i c a l parameters such as the f i b r e volume content and the mater ia l proper t i e s of the f i b r e and the matr ix . These s tresses and s t r a i n s are most often average (or composite) values over representat ive volume elements which are large compared to t y p i c a l phase region dimensions (e .g . f i b r e diameters and spac ings) . Analyses based on t h i s approach are therefore not t r u l y micromechanical , s ince they do not provide a d e s c r i p t i o n of the l o c a l s tress and s t r a i n gradients w i th in the composite. Hence the term minimechanics i s used here to encompass a l l such approaches. The general groundwork for determination of o v e r a l l mechanical proper-t i e s of FRMs from t h e i r const i tuents has been l a i d out by H i l l (1964) for a c lass of t ransverse ly i s o t r o p i c mater ia l s with e l a s t i c f ibres and e l a s t o -14 p l a s t i c matr ices . Using a s e l f - c o n s i s t e n t scheme (SCS)*, bounds were obtained for the o v e r a l l moduli and flow s tress at any stage of deformation. Based on the deformation theory of p l a s t i c i t y and SCS, Huang (1971) pred ic ted the o v e r a l l transverse e l a s t i c - p l a s t i c u n i a x i a l s t r e s s - s t r a i n curve for a u n i d i r e c t i o n a l FRM comprised of r i g i d f ibres and an e l a s t i c - p l a s t i c matr ix . Dvorak and B a h e i - e l - d i n (1979) modified the SCS of H i l l i n c a l c u l a t i n g i n t e r n a l s tress f i e l d s , o v e r a l l and l o c a l y i e l d surfaces , instantaneous modul i , thermal c o e f f i c i e n t s , p l a s t i c s t r a i n s and thermal microstresses for t ransverse ly i s o t r o p i c m a t e r i a l s . Dvorak and B a h e i - e l - d i n (1982) l a t e r used a simple model to a r r i v e at three dimensional c o n s t i t u t i v e r e l a t i o n s for e l a s t i c - p l a s t i c deformation of u n i d i r e c t i o n a l f ibrous composites. In t h e i r approximate treatment, which s i m p l i f i e s the geometry of the micros t ruc ture , each:of the f i b r e s i s assumed to be of very small diameter, so that although the f ibres occupy a f i n i t e volume f r a c t i o n of the composite, they do not i n t e r f e r e with matrix deformation i n the transverse and l o n g i t u d i n a l d i r e c t i o n s . The f i b r e s were regarded as e l a s t i c embedded i n an i s o t r o p i c e l a s t o p l a s t i c matrix of Mises-type with kinematic hardening. As a r e s u l t , a n a l y t i c a l expressions were obtained for the y i e l d cond i t i ons , hardening r u l e s , and flow ru les for the composite aggregate i n terms of l o c a l proper t i e s and volume f rac t ions of the phases. In a subsequent paper B a h e i - e l - d i n and Dvorak (1982) used t h e i r model of e l a s t i c - p l a s t i c behaviour of u n i d i r e c t i o n a l FRM to der ive c o n s t i t u t i v e equations of laminate p la tes under in -p lane mechanical load ing . A n a l y t i c a l c a l c u l a t i o n s based on t h e i r model were compared with se lec ted experimental r e s u l t s on Boron/Aluminum *For a d e s c r i p t i o n of t h i s scheme and further references see p . 59 of Chris tensen (1979). 15 (B/Al) laminates. In order to obta in good agreement they had to use p l a s t i c proper t i e s of the matrix that departed s i g n i f i c a n t l y from those of the true unre inforced aluminum. The reasons for t h i s discrepancy were a t t r i b u t e d to the inherent d e f i c i e n c i e s i n t h e i r mater ia l model. Min (1981) used a s i m i l a r l y simple model to a r r i v e at a plane s tress d e s c r i p t i o n of the e l a s t o p l a s t i c response of u n i d i r e c t i o n a l l y re in forced metal matrix composites. In t h i s study, based on Hoffman's model (1979), the f i b r e was assumed to have s t i f f n e s s only i n the a x i a l d i r e c t i o n , whereas the matrix was considered to be an e l a s t i c - p e r f e c t l y p l a s t i c mater ia l obeying the von Mises y i e l d c r i t e r i o n and i t s assoc iated flow r u l e . This l ed to a work-hardening type response for the o v e r a l l behaviour of the composite s i m i l a r to that reported by B a h e i - e l - d i n and Dvorak. A few numerical examples were presented and shown to compare favourably with the r e s u l t s of experiments performed on Graphite/Aluminum (Gr/Al ) composites under simple in -p lane u n i a x i a l and b i a x i a l loading cond i t ions . 2 . 3 . 3 Macromechanics Approach In t h i s sec t ion we s h a l l be p r i m a r i l y concerned with "continuum" theories which describe the behaviour of the FRM on the macroscopic s c a l e . The models to be discussed t rea t the composite as a mater ia l i n i t s own r i g h t , without d i r e c t reference to the proper t i e s of the i n d i v i d u a l c o n s t i -tuents . In these models the FRM i s regarded as an an i so trop ic continuum, with appropriate o v e r a l l (average) proper t i e s known from experiments or micromechanical analyses . The several macroscopic approaches for def in ing the s t r e s s - s t r a i n behaviour of FRMs under var ious s tress states can be convenient ly c l a s s i f i e d as belonging to f i ve main groups: 16 i ) L i n e a r E l a s t i c i t y i i ) N o n l i n e a r E l a s t i c i t y i i i ) V i s c o e l a s t i c i t y i v ) I n c r e m e n t a l P l a s t i c i t y v ) E n d o c h r o n i c P l a s t i c i t y . The f o l l o w i n g o u t l i n e s some o f t h e e x i s t i n g c o n s t i t u t i v e m o d e l s u n d e r t h e above h e a d i n g s . i ) L i n e a r E l a s t i c i t y I n s p i t e o f i t s s h o r t c o m i n g s , t h e l i n e a r e l a s t i c i t y t h e o r y i s b y f a r t h e most commonly u s e d m a t e r i a l mode l f o r c o m p o s i t e s i n t h e p r e f a i l u r e r a n g e . The e m p h a s i s on l i n e a r e l a s t i c i t y r e f l e c t s i t s u s e f u l n e s s and i m p o r t a n c e , n o t o n l y as a b a s i c t h e o r y b u t a l s o f o r p r o v i d i n g t h e means t o d e v e l o p p r a c t i c a l d e s i g n m e t h o d s . The b a s i c c o n c e p t o f l i n e a r e l a s t i c i t y as a p p l i e d t o FRMs i s w e l l e s t a b l i s h e d (see f o r example J o n e s ( 1 9 7 5 ) ) . The c o r r e s p o n d i n g d e s i g n a s p e c t s a r e a l s o w e l l a d v a n c e d . By no means , h o w e v e r , c a n a l l p r a c t i c a l FRMs be i d e a l i z e d as b e h a v i n g a c c o r d i n g t o l i n e a r e l a s t i c i t y t h e o r y . Many t y p e s o f c o m p o s i t e m a t e r i a l s i n v o l v e c o n s t i t u t i v e b e h a v i o u r t h a t i s d i s t i n c t l y n o n l i n e a r ( e l a s t i c o r i n e l a s t i c ) i n a t l e a s t one o f t h e p r i n c i p a l m a t e r i a l d i r e c t i o n s . The f o l l o w i n g d e a l s w i t h t h e o r i e s t h a t c o v e r s u c h n o n -l i n e a r i t i e s . i i ) N o n l i n e a r E l a s t i c i t y C o n t e m p o r a r y FRMs g e n e r a l l y c o n s i s t o f e l a s t i c b r i t t l e f i b r e s s u c h as g l a s s , b o r o n o r g r a p h i t e i n r e l a t i v e l y s o f t m a t r i x m a t e r i a l s s u c h as e p o x y o r a l u m i n u m . F o r t h e s e m a t r i x m a t e r i a l s i t i s r e a s o n a b l e t o a n t i c i p a t e t h a t a t a c e r t a i n l o a d i n g s t a t e t h e m a t r i x w i l l b e g i n t o e x h i b i t n o n l i n e a r e f f e c t s . 17 The degree of n o n l i n e a r i t y var i e s from composite to composite depending on the type of matrix m a t e r i a l . Polymer matrix composites r e i n f o r c e d by u n i d i r e c t i o n a l f i b r e s u s u a l l y e x h i b i t an appreciable amount of n o n l i n e a r i t y i n shear and only a s l i g h t n o n l i n e a r i t y i n tension transverse to the f i b r e s . On the other hand, metal matrix composites such as B / A l have strong t r a n s -verse and shear n o n l i n e a r i t i e s . The n o n l i n e a r i t i e s for a l l of these mater ia l s are more pronounced with increas ing temperature. Various i n v e s t i g a t o r s have attempted to inc lude mechanical property n o n l i n e a r i t i e s i n ana lys i s of composite m a t e r i a l s . P e t i t and Waddoups (1969) devised an incremental method (using a piecewise l i n e a r approximation) for nonl inear ana lys i s of laminates. According to t h i s method, an increment of average laminate s tress (or stresses) i s p laced on the laminate , and by using the i n i t i a l laminate compliance matr ix , the f i r s t increment i n the laminate s t r a i n s i s c a l c u l a t e d with the assumption that the laminate behaves l i n e a r l y over the appl i ed s tress increment. The increment i n the laminate s t r a i n s i s added to any previous s t r a i n s to determine the current t o t a l laminate s t r a i n . As the incremental loading proceeds, the i n d i v i d u a l p l y s t r a i n s are monitored, and, by r e f e r r i n g to the bas ic p l y s t r e s s - s t r a i n curves , the corresponding p l y tangent moduli and s t i f fnes se s for the s t r a i n l eve l s present are c a l c u l a t e d . The Petit-Waddoups method requires far too many input data and t h e i r incremental scheme i s unduly complicated. S t a r t i n g with a complementary energy dens i ty funct ion for a l i n e a r e l a s t i c m a t e r i a l , Hahn and T s a i (1973) added a fourth order terra i n shear to model the nonl inear shear behaviour and regarded a l l other s t r e s s - s t r a i n curves as l i n e a r . The method which was a p p l i c a b l e to u n i d i r e c t i o n a l l a y e r s , was subsequently extended to laminated composites by Hahn (1973). Hashin, 18 Bagchi and Rosen (1974) proposed a deformation type theory* i n conjunction with the Raraberg-Osgood (1943) representat ion of s t r e s s - s t r a i n r e l a t i o n s to approximate the n o n l i n e a r i t i e s . In t h e i r ana lys i s the s t r a i n s i n each p l y were s p l i t in to e l a s t i c and i n e l a s t i c components. Furthermore, the transverse and shear s tresses were allowed to i n t e r a c t i n the i n e l a s t i c range while i n e l a s t i c s t r a i n s i n the f i b r e d i r e c t i o n were neglected. The deforma-t i o n theory used i n t h e i r ana lys i s has the obvious de f i c i ency of f a i l i n g to account for l o a d - h i s t o r y e f fec t s and of p o s s i b l y causing c o n t i n u i t y and uniqueness problems i n the case of nonproport ional loading (see for example Kachanov, 1971, p . 54). Sandhu (1976) introduced an incremental method that used piecewise cubic sp l ine i n t e r p o l a t i o n functions to represent the bas i c nonl inear s t r e s s - s t r a i n data . L ike the P e t i t and Waddoups' a n a l y s i s , Sandhu 1s method required the complete p l y t e n s i l e and compressive s t r e s s -s t r a i n data under l o n g i t u d i n a l , transverse and shear loading as input . Sandhu's model attempted to compensate for the t r i a x i a l s tress e f f ec t (which was absent i n Petit-Waddoups' model) by de f in ing equivalent s t r a i n s . A s l i g h t inconvenience of the Sandhu's ana lys i s i s that i t requires b i a x i a l loading to determine normal and transverse tangent moduli for plane s tress load ing . The model a l so lacks prov i s ions for s tress i n t e r a c t i o n i n shear. Jones and Morgan (1977) developed a mater ia l model i n which the nonl inear mechanical proper t i e s were expressed as functions of the s t r a i n energy *The deformation theory (also c a l l e d J 2 deformation theory, octahedral shear deformation, t o t a l theory) developed i n 1924 by Hencky assumes that there i s a one-to-one correspondence between the s tress and s t r a i n . It i s known that i n the case of p r o p o r t i o n a l l oad ing , that i s , a l l s tresses at a point grow simultaneously i n a f i xed r a t i o to one another, deformation theory i s simply an i n t e g r a t i o n of the incremental p l a s t i c i t y theory. I t has a lso been shown by Kachanov (1971) that the governing equation of deformation theory c o r r e s -ponds to a nonl inear e l a s t i c c o n s t i t u t i v e representat ion . 19 dens i ty . They argued that under raultiaxial loading the s t r a i n energy capac i ty of the mater ia l can exceed the u n i a x i a l s t r a i n energy c a p a c i t i e s . Using some ad-hoc modi f icat ions (to prevent the mater ia l from v i o l a t i n g thermodynamic c o n s t r a i n t s ) , they extrapolated the s t r e s s - s t r a i n and mechanical p r o p e r t y - s t r a i n energy curves i n a rather complicated manner. The complexity of t h i s method d i d not prove to produce be t ter r e s u l t s when compared with the Hahn-Tsai method. Nahas (1984) employed a technique s i m i l a r to that of Sandhu i n a d d i t i o n to the secant modulus concept to p r e d i c t the nonl inear behaviour of laminates . Recent ly , Takahashi and Chou (1987) adopted the piecewise l i n e a r approximation of P e t i t and Waddoups and used Four ier s er i e s expansion of the experimental r e s u l t s to model the nonl inear shear s t r e s s - s t r a i n r e l a t i o n s of i n d i v i d u a l p l i e s . i i i ) V i s c o e l a s t i c i t y Composite mater ia l s which have one or more polymeric const i tuents (such as resinous matrix mater ia l s ) e x h i b i t a considerable amount of time-dependent mechanical behaviour. Th i s behaviour, termed v i s c o e l a s t i c i t y , increases i n s i g n i f i c a n c e with e levated temperature. In the case of m e t a l l i c matrix m a t e r i a l s , such as aluminum, time-dependent e f fec t s are genera l ly n e g l i g i b l e unless e levated temperature or h igh s t r a i n rate condi t ions are considered. A summary and review of the l i t e r a t u r e on v i s c o e l a s t i c behaviour and ana lys i s of composites are given by Schapery (1974). More recent developments i n t h i s f i e l d can be found i n the survey a r t i c l e by Hashin (1983). 20 iv) Incremental P l a s t i c i t y A l l the macroscopic c o n s t i t u t i v e models mentioned so far suffer from c e r t a i n inherent l i m i t a t i o n s . In p a r t i c u l a r they cannot p r e d i c t d i s s i p a t i v e ( i r r e v e r s i b l e ) e f f ec t s charac ter i zed by permanent s t r a i n accumulation, a shortcoming that becomes apparent when the mater ia l experiences unloading at large s tresses . The incremental theory of p l a s t i c i t y i s a we l l developed c o n s t i t u t i v e representat ion that accounts, i n p r i n c i p l e , for the s tress h i s t o r y dependent behaviour and r e s i d u a l s t r a i n s due to unloading. A more extensive d i s cuss ion of the incremental theory of p l a s t i c i t y i s given i n Chapter 3. While considerable work apparently has been done i n the area of composite e l a s t i c i t y , the study of i t s e l a s t o p l a s t i c behaviour i s s t i l l very l i m i t e d . Studies of p l a s t i c a l l y a n i s o t r o p i c mater ia l s i n the context of the incremental theory of p l a s t i c i t y were begun by H i l l (1950), who f i r s t postu lated the form of a y i e l d cond i t ion based on the von-Mises c r i t e r i o n for i s o t r o p i c p l a s t i c m a t e r i a l s . H i l l ' s y i e l d condi t ion was devised to account for the d i f ferences of y i e l d s tress i n r o l l e d s t e e l sheet i n the r o l l i n g and transverse d i r e c t i o n s . In h i s formulat ion , H i l l introduced s i x parameters to account for o r t h o t r o p i c symmetry of the m a t e r i a l . However, he considered only i s o t r o p i c hardening which r e s u l t s i n a p r o p o r t i o n a l change of the s i x or tho trop ic parameters during hardening. Hu (1956) extended H i l l ' s theory to the ana lys i s of p l a s t i c flow of a n i s o t r o p i c bodies with s t ra in -harden ing . In t h i s work the a n i s o t r o p i c parameters of the y i e l d c r i t e r i o n were considered as constant during p l a s t i c deformation. Hu (1958) general ized the Tresca maximum shear s tress c r i t e r i o n to study the p l a s t i c flow of an i so trop ic bodies . Whang (1969) general ized H i l l ' s c r i t e r i o n by suggesting a 21 non-proport ional r u l e for changing the a n i s o t r o p i c parameters during the hardening process . This was based on the assumption that for equal amounts of p l a s t i c work produced during s t r e s s - s t r a i n tes ts i n each of the p r i n c i p a l d i r e c t i o n s , the e f f e c t i v e s tress l e v e l reached would be the same. Whang's approach was used by V a l l i a p p a n (1971) i n the f i n i t e element so lut ions of severa l a n i s o t r o p i c e l a s t o - p l a s t i c s t r u c t u r e s . Shih and Lee (1978) proposed an extension of H i l l ' s formulat ion to account for the d i s t o r t i o n of the y i e l d surface for d i f f e r i n g strengths i n tens ion and compression and the e f f e c t i v e s i ze of the loading surface . The a n i s o t r o p i c parameters of the y i e l d funct ion were determined from monotonic loading tests on Z i r c a l o y m a t e r i a l s . It was observed that these parameters, which were respons ib le for the d i s t o r t i o n of the y i e l d surface , tended to reach constant values with increas ing p l a s t i c s t r a i n . G r i f f i n , Kamat and Herakovich (1981) employed H i l l ' s c r i t e r i o n and i t s assoc iated flow r u l e i n a three dimensional f i n i t e element program to analyze the i n e l a s t i c t e n s i l e response of u n i d i r e c t i o n a l o f f - a x i s FRMs. The Ramberg-Osgood representat ion was used to approximate the bas ic s t r e s s - s t r a i n r e l a t i o n s . For the purpose of computing the hardening modulus ( i . e . the slope of the e f f e c t i v e s tress versus e f f e c t i v e p l a s t i c s t r a i n diagram) p r o p o r t i o n a l loading was assumed. Such assumptions lack p h y s i c a l arguments. The method suggested i s rather complicated and requires much experimental data for the eva luat ion of var ious parameters. Kenaga, Doyle and Sun (1987) used a plane s tress or tho trop ic e l a s t i c - p l a s t i c formulat ion based on a four parameter quadrat ic y i e l d funct ion to charac ter i ze the nonl inear behaviour of u n i d i r e c t i o n a l B / A l FRMs. A number of o f f - a x i s t e n s i l e t es t s were performed and a t r i a l and error procedure was employed to determine the a n i s o t r o p i c parameters that best f i t t e d the data . Leewood, Doyle and Sun (1987) implemented the above 22 formulat ion i n a two-dimensional f i n i t e element program to analyze the e l a s t i c - p l a s t i c behaviour of m u l t i l a y e r laminates . Higher order (than quadratic) y i e l d funct ions for a n i s o t r o p i c mater ia l s have a l so been suggested by some authors (e .g . Dubey and H i l l i e r (1972); Gotoh (1977); and Rees (1984)'). The theories descr ibed above may be i n t e r p r e t e d as genera l i za t ions of p l a s t i c flow theories for i s o t r o p i c m a t e r i a l s , with enough a r b i t r a r y mater ia l parameters b u i l t i n to account for as many c lasses of mater ia l symmetry as de s i red . These theories thus contain no elements which can account p h y s i c -a l l y for the presence of f i b r e s i n a d u c t i l e matr ix . A survey of the l i t e r a t u r e , however, reveals that t h e o r e t i c a l analyses have appeared which recognize the presence of s t i f f f ibres by cons tra in ing the deformation of an e s s e n t i a l l y i s o t r o p i c p l a s t i c m a t e r i a l . In t h i s manner the d i f f i c u l t i e s assoc iated with a n i s o t r o p i c p l a s t i c i t y are avoided. The bas ic idea of represent ing a FRM by a continuum model of t h i s nature o r i g i n a t e d with the paper by Adkins and R i v l i n (1955) who trea ted the problem of a r u b b e r - l i k e incompressible mater ia l r e i n f o r c e d with inextens ib le f i b r e s . Mulhern, Rogers and Spencer (1967) adopted a somewhat s i m i l a r procedure as Adkins and R i v l i n , but app l i ed i t to p l a s t i c rather than e l a s t i c s o l i d s . They proposed a continuum model for descr ib ing the mechanical behaviour of a r i g i d - p l a s t i c mater ia l r e i n f o r c e d by a s ing l e family of inextens ib le f i b r e s . Whereas Mulhern et a l . (1967) treated the composite as a t ransverse ly i s o t r o p i c r i g i d p l a s t i c s o l i d , Prager (1969) viewed the composite as cons i s t ing of an i s o t r o p i c r i g i d - p l a s t i c matrix constrained by inextens ib le f i b r e s . Mulhern, Rogers and Spencer (1969) l a t e r re laxed the r i g i d i t y assumption and permitted e l a s t i c composite s t r a i n s i n the f i b r e d i r e c t i o n . While the aforementioned papers are based on Mises i s o t r o p i c y i e l d cond i t ion or on i t s an i so trop ic 23 modi f i ca t ion by H i l l (1950), Lance and Robinson (1971) developed a model of an incompressible r i g i d - p l a s t i c mater ia l of the type assumed by Mulhern et a l . , but obeying an a n i s o t r o p i c modi f i ca t ion of Tresca ' s y i e l d c o n d i t i o n . Dvorak and Rao (1976) proposed a continuum theory for axisymmetric p l a s t i c deformation of u n i d i r e c t i o n a l f ibrous composites, cons i s t ing of e l a s t i c f ibres and an e l a s t i c - p e r f e c t l y p l a s t i c matr ix . The ir theory accounted for both the p l a s t i c e x t e n s i b i l i t y of the composite i n the f i b r e d i r e c t i o n , and for the p l a s t i c d i l a t a t i o n ( in the presence of e l a s t i c deformation of the f i b r e s ) . It should be emphasized tha t , i n a l l the above papers, the f ibres are assumed to have a cons tra in ing e f f ec t on the y i e l d i n g of the matr ix . Such d e t a i l s as f i b r e s tresses ( i . e . the part of the o v e r a l l s tress c a r r i e d by f ibres themselves) have been ignored. The theor ies suggested are there -fore continuum theor ies and must not be confused with the minimechanics approaches o u t l i n e d i n sec t ion 2 .3 .2 . v) Endochronic P l a s t i c i t y In the preceding s e c t i o n , the c l a s s i c a l incremental theory of p l a s t i c i t y was used as the bas i s for developing c o n s t i t u t i v e models for FRMs. Funda-mental ly , the incremental theory assumes the existence of a y i e l d c r i t e r i o n coupled with a hardening r u l e to define the subsequent y i e l d surfaces . However, i t i s often d i f f i c u l t to determine the prec i se values of the y i e l d s tresses and define appropriate hardening r u l e s . A theory that does not require the existence of a y i e l d condi t ion and i s therefore free from harden-ing r u l e s , i s the endochronic theory of p l a s t i c i t y developed o r i g i n a l l y by V a l a n i s (1971) for the d e s c r i p t i o n of mechanical behaviour of metals . Using V a l a n i s ' concept, Pindera and Herakovich (1983) extended the theory to t ransverse ly i s o t r o p i c media i n order to describe the response of graphi te -24 polyimide o f f - a x i s t e n s i l e coupons under monotonic and c y c l i c load ing . They demonstrated the a p p l i c a b i l i t y of the endochronic theory by obtaining good c o r r e l a t i o n with the observed experimental data . I t should be noted, however, that the endochronic theory i s not without i t s l i m i t a t i o n s and some serious c r i t i c i s m s of i t has been r a i s e d by R i v l i n (1981). 2.4 I n i t i a l F a i l u r e F a i l u r e c r i t e r i a for composite mater ia l s are more d i f f i c u l t to postulate than for i s o t r o p i c m a t e r i a l s . The a n a l y t i c a l determination of the strength of composites on the bas is of micromechanics methods i s extremely complex, perhaps to the point of being regarded as an i n t r a c t a b l e problem. On the other hand, i t i s a lso i m p r a c t i c a l to resolve the problem by experimentation alone s ince the number of tes ts required to develop the f u l l f a i l u r e surface would be extremely l arge . The remaining a l t e r n a t i v e i s to construct a n a l y t i c a l f a i l u r e c r i t e r i a i n terms of macrovariables , such as average s tresses or s t r a i n s . Over the l a s t twenty years , a s i g n i f i c a n t number of f a i l u r e c r i t e r i a for many a n i s o t r o p i c mater ia l s have been proposed. Extensive surveys of the c r i t e r i a as app l i ed to composite mater ia l s are presented by T s a i and Hahn (1975), Wu (1974), Rowlands (1985), Nahas (1986), Craddock and Champagne (1985), Fan (1987), and Labossiere and Neale (1987a). A l l the e x i s t i n g f a i l u r e c r i t e r i a tend to be phenomenological and empir i ca l i n nature , not mechanist ic . The intended use of most of these c r i t e r i a was mainly the p r e d i c t i o n of the strength i n "single layers" of FRMs under complex loading cond i t i ons . None of the a v a i l a b l e a n i s o t r o p i c s trength c r i t e r i a represents observed r e s u l t s s u f f i c i e n t l y accurate ly to be employed conf ident ly by them-selves i n p r a c t i c e . Several of the theor ies suf fer from the inconvenience of 25 r e q u i r i n g b i a x i a l information as bas ic input data . Some of the most popular f a i l u r e c r i t e r i a w i l l be discussed i n the fo l lowing . The s implest f a i l u r e c r i t e r i a are the maximum stress and maximum s t r a i n c r i t e r i a . According to these theories f a i l u r e of a layer occurs when any s ing l e s tress or s t r a i n component i n the p r i n c i p a l mater ia l axes d i r e c t i o n s reaches i t s corresponding ul t imate value regardless of the values of the other components. The maximum s tress ( s tra in) c r i t e r i a are not r e a l i s t i c s ince they d i sregard the combined e f fec t s of s tresses ( s tra ins ) on f a i l u r e , * and therefore overestimate the s trength of the mater ia l under combined s tress ( s t r a i n ) . Both c r i t e r i a , however, are simple to u t i l i z e i n p r a c t i c e and are capable of determining the mode of f a i l u r e of the f a i l e d p l y . The l a t t e r f a c i l i t a t e s the study of the behaviour of the laminate a f ter the f i r s t p l y f a i l u r e . A convenient mathematical representat ion of f a i l u r e c r i t e r i a that accounts for the i n t e r a c t i o n of s tresses (or s t ra ins ) i s i n terms of p o l y -nomials i n s tress (or s t ra ins ) . I t i s then necessary to determine the c o e f f i c i e n t s of the polynomial i n terms of t es t r e s u l t s which can be convenient ly obtained i n the laboratory , such as u n i a x i a l tension or compression, pure shear, e t c . In one of the f i r s t contr ibut ions to the subject T s a i (1965) assumed that H i l l ' s (1950) quadrat ic y i e l d c r i t e r i o n for o r t h o t r o p i c p l a s t i c mater ia l s could be used as a f a i l u r e c r i t e r i o n . Hoffman (1967) added l i n e a r terms to account for d i f f e r e n t t e n s i l e and compressive u l t imate s t res ses . The disadvantage of these c r i t e r i a i s that they are based on the assumption that hydros ta t i c pressure has no e f fec t on the f a i l u r e . *It should be noted that although the maximum s t r a i n c r i t e r i o n i s an independent mode c r i t e r i o n i n s t r a i n space i t accounts for the i n t e r a c t i o n of s tresses (due to the Poisson r a t i o e f fect ) i n s tress space. 26 While such an assumption may be a good approximation for i n i t i a l y i e l d i n g of a meta l , i t i s c e r t a i n l y not v a l i d for FRMs. R e a l i z i n g the shortcomings of the previous f a i l u r e c r i t e r i a T s a i and Wu (1971) proposed a general tensor polynominal c r i t e r i o n i n terms of s tresses . In t h e i r notat ion f a i l u r e of the mater ia l w i l l occur when the fo l lowing condi t ion i s met, F . o . + F . . 0 . 0 . + F . . . 0 . 0 . 0 . + . . . = 1 ( i . j . k = 1,2 6) (2.1) i i I J l j l j k l j k J . Here o. are the components of the s tress tensor and the c o e f f i c i e n t s F . , F . . , I R I i j If' ijk' e t c . are the components of the s t r e n g t h t e n s o r s , c a l c u l a t e d from experimental data . A l l components are r e f e r r e d to the mater ia l p r i n c i p a l axes and the fo l lowing contracted tensor nota t ion ( in the sense of Green) i s used, (2.2) Thus F . and o. are second o r d e r tensors . S i m i l a r l y , F . . i s a fourth order i i J i j tensor with 21 independent components. A l l h igher-order tensors appearing i n Eq . (2.1) fo l low the same general character . T e n s o r i a l c r i t e r i a s i m i l a r to that of Tsai-Wu had been proposed e a r l i e r i n the Russian l i t e r a t u r e (see for example Rowlands (1985)). Although the polynomial (2.1) can be expanded to any degree, the number of s trength parameters r i s e s cons iderably for each a d d i t i o n a l degree i n c l u d e d . To reduce the number of experiments required to obta in the s trength parameters, u s u a l l y only l i n e a r and quadrat ic terms are r e t a i n e d . * In t h i s case a l l of the c o e f f i c i e n t s i n E q . (2 .1) , except the *Some authors have advocated the i n c l u s i o n of higher order terms (see for example Tennyson et a l . (1978) and Ashkenazi (1965)), but the e f f o r t i n determining the corresponding strength parameters hard ly j u s t i f i e s the gain i n accuracy. 27 cross terra c o e f f i c i e n t s , F l a , F l s and F 1 3 , can be found from simple, s ing l e s t ress component t e s t s . T s a i and Wu propose to determine the cross term c o e f f i c i e n t s by running b i a x i a l f a i l u r e t e s t s . Unfortunate ly , such tests are complicated and expensive. Indeed i t should be expected that the i n t e r a c t i o n parameters, w i l l be dependent upon the signs of the s tresses and thus w i l l not be unique (see the d i scuss ion of quadrat ic f a i l u r e c r i t e r i a by Wu (1974)). Moreover, the al lowable values of these parameters are l i m i t e d by bounding condit ions to ensure that the f a i l u r e envelope i s c losed (see Sect ion 3 for d e t a i l s ) . Recent ly , Labossiere and Neale (1987b) have proposed a l t e r n a t i v e methods of c a l c u l a t i n g the strength parameters. Reddy and Pandey (1987) have examined the accuracy of the var ious f a i l -ure c r i t e r i a discussed above i n p r e d i c t i n g the i n i t i a l f a i l u r e (or f i r s t p l y f a i l u r e ) of laminated composite p la tes under in -p lane or bending loads . They concluded that a l l these f a i l u r e c r i t e r i a were equivalent i n t h e i r p r e d i c t i o n of f a i l u r e when laminates were subjected to in -p lane load . For laminates subjected to bending, the maximum s t r a i n and H i l l ' s c r i t e r i a were found to p r e d i c t d i f f e r e n t f a i l u r e l o c a t i o n and f a i l u r e loads to the other c r i t e r i a . The Tsai-Wu t e n s o r i a l f a i l u r e c r i t e r i o n has the advantage of being i n v a r i a n t under coordinate transformat ion. Furthermore, Wu (1974) has shown that a l l other s tress based f a i l u r e theories ( inc lud ing the maximum s tress c r i t e r i o n ) are the degenerate cases of the tensor polynomial c r i t e r i o n given by Eq . (2 .1) . A major shortcoming of these polynomial-based f a i l u r e c r i t e r i a , however, i s that they p r i m a r i l y p r e d i c t the onset, but not the mode, of f a i l u r e . In view of the d iverse f a i l u r e mechanisms that are opera-t i v e i n a composite m a t e r i a l , t h i s shortcoming i s p a r t i c u l a r l y severe. I d e n t i f i c a t i o n of the mode of damage i s a required feature i f the f a i l u r e c r i t e r i o n i s to be use fu l for progress ive f a i l u r e ana lys i s of f i b r e composite 28 laminates by computational procedures. Motivated by the foregoing need, Hashin (1980) proposed separate quadrat ic f a i l u r e c r i t e r i o n to d i s t i n g u i s h between the f i b r e dominated f a i l u r e mode and the matrix dominated f a i l u r e mode i n u n i d i r e c t i o n a l FRMs. These c r i t e r i a had d i f f e r e n t forms for t e n s i l e and compressive s tresses . One of the advantages of Hashin's approach was that troublesome b i a x i a l t es t data were not needed for the evaluat ion of various strength parameters. It should a l so be noted that Hashin's f a i l u r e c r i t e r i a were intended to i d e n t i f y only the f a i l u r e modes w i t h i n a s ing l e p l y ( i . e . in tra laminar modes of f a i l u r e ) . Subsequently, Lee (1982) used a s i m i l a r c r i t e r i o n to d i s t i n g u i s h delamination a n a l y t i c a l l y from other modes. Another way of d i f f e r e n t i a t i n g between the modes of f a i l u r e has been to ascr ibe f a i l u r e to e i t h e r matrix or f i b r e depending on the r e l a t i v e magnitude of the various terms appearing i n Eq . (2.1) (see for example, Chiu (1969)). 2.5 C o n s t i t u t i v e Model l ing of Damaged Composites The major damage which develops i n laminates under s t a t i c or c y c l i c loading i s i n the form of in ter laminar and in tra laminar cracks . The former develop gradua l ly and s lowly between the p l i e s . The l a t t e r appear suddenly and i n large numbers i n p l i e s i n which the s tresses reach c r i t i c a l values def ined by the i n i t i a l f a i l u r e c r i t e r i a . The main macroscopic e f fec t of such cracks on laminate proper t i e s i s reduct ion of s t i f f n e s s . Many approaches have been used i n the past to model the behaviour of damaged (or cracked) laminates . A t t e n t i o n has mostly been d i r e c t e d towards the e f f e c t of in tra laminar cracks on the load carry ing capac i ty of laminates under in -p lane loading cond i t ions . Depending upon the mode of f a i l u r e p r e d i c t e d by the f a i l u r e c r i t e r i a , many authors have suggested a softening of response i n the d i r e c t i o n i n which the f a i l u r e has occurred. P e t i t and 29 Waddoups (1969) modelled the softening by g iv ing the tangent moduli r e l a -t i v e l y high negative va lues . Chiu (1969) considered instantaneous s tress r e l a x a t i o n i n the f a i l e d l a y e r s . Such p o s t - f a i l u r e models have been used by many authors (see for example the survey a r t i c l e by Nahas (1986)). Swanson and C h r i s t o f o r o u (1987) proposed an empir i ca l express ion, i n terms of an e f f e c t i v e s t r a i n , for softening due to matrix crack ing . Chang and Chang (1987) assumed that upon matrix cracking i n a u n i d i r e c t i o n a l l a y e r , only the transverse modulus and the Poisson's r a t i o reduce to zero while the proper-t i e s i n the other d i r e c t i o n s remain unchanged. For f i b r e f a i l u r e they postu lated that both the l o n g i t u d i n a l and shear moduli reduce according to the Weibul l d i s t r i b u t i o n , whereas the transverse modulus and Poisson's r a t i o van i sh . A t t e n t i o n i n recent years has focused on the r igorous determination of s t i f f n e s s reduc t ion , p r i m a r i l y for the case of transverse tens ion cracking i n the matr ix . The analyses have been concerned mainly with c r o s s - p l y laminates i n which only the 9 0 ° p l i e s are cracked. A prec i se estimate of the s t i f f n e s s reductions for laminates with general layups i s present ly not a v a i l a b l e . Perhaps the s implest model i s that of Highsmith and Rei f sne ider (1982) who devised a simple shear lag method to evaluate s t i f f n e s s reduct ion due to cracks . Another method of ana lys i s i s due to Laws, Dvorak and Hejaz i (1983) who employed the s e l f - c o n s i s t e n t scheme for the p r e d i c t i o n of the e f f e c t i v e s t i f f n e s s of a cracked p l y . This method i s based on the s o l u t i o n of the problem of a s ing l e crack embedded i n an i n f i n i t e medium. A d i f f e r e n t approach which was proposed by T a l r e j a (1985, 1986) i s based on a continuum damage theory i n which the mater ia l i s character ized by a set of vector f i e l d s each represent ing a damage mode. The r e s u l t i n g c o n s t i t u t i v e equations contain numerous parameters which must be determined experimental ly . Hashin 30 ( 1 9 8 5 , 1 9 8 6 , 1 9 8 7 ) u s e d t h e v a r i a t i o n a l method on t h e b a s i s o f t h e p r i n c i p l e o f minimum c o m p l e m e n t a r y e n e r g y f o r t h e a n a l y s i s o f c r a c k e d l a m i n a t e s . The m a j o r s h o r t c o m i n g o f t h e s e s o - c a l l e d r i g o r o u s m o d e l s i s t h a t t h e y l a c k g e n e r a l i t y o f t h e l o a d i n g and c o n f i g u r a t i o n , and t h e i r a c t u a l u s e i s o f t e n c o m p l i c a t e d b y t h e r e q u i r e m e n t f o r numerous e x p e r i m e n t a l l y d e t e r m i n e d q u a n t i t i e s . 2.6 Ult imate F a i l u r e The f i n a l p o i n t t o be a d d r e s s e d i s t h e u l t i m a t e f a i l u r e o f l a m i n a t e s . The p r o b l e m o f u l t i m a t e f a i l u r e i n l a m i n a t e s c a n be a p p r o a c h e d b y m o n i t o r i n g t h e g r o w t h o f damage z o n e s u n t i l f a i l u r e o c c u r s e i t h e r b y e x c e s s i v e d e b o n d i n g o r f i b r e f r a c t u r e o f p r i m a r y l o a d - c a r r y i n g p l i e s . A number o f p r o g r e s s i v e f a i l u r e a n a l y s e s have b e e n p r e s e n t e d i n t h e l i t e r a t u r e . These r e q u i r e as i n p u t i n f o r m a t i o n t h e c o m p l e t e c o n s t i t u t i v e p r o p e r t i e s o f t h e i n d i v i d u a l p l i e s , and u s e t h e c l a s s i c a l l a m i n a t i o n t h e o r y (CLT) t o t r a c e t h e o v e r a l l l o a d - d i s p l a c e m e n t r e s p o n s e up t o t h e u l t i m a t e s t a t e . Among t h e most r e c e n t s t u d y i n t h i s a r e a a r e t h e p a p e r s b y Lee' ( 1 9 8 2 ) , Sandhu e t a l . ( 1 9 8 3 ) , Swanson and C h r i s t o f o r o u ( 1 9 8 7 ) , Ochoa and Engb lom ( 1 9 8 7 ) , T a k a h a s h i and Chou ( 1 9 8 7 ) , and Chang and Chang ( 1 9 8 7 ) . One o f t h e v e r y few f a i l u r e c r i t e r i a w h i c h h a s s u c c e s s f u l l y been a p p l i e d t o t h e l a m i n a t e as a " w h o l e " i s t h e P u p p o - E v e n s o n (1972) q u a d r a t i c f a i l u r e c r i t e r i o n . T h i s t h e o r y i s a d i r e c t l a m i n a t e a n a l y s i s w h i c h makes no c o n s t i -t u t i v e a s s u m p t i o n and does n o t i n v o l v e l a m i n a t i o n t h e o r y . I t has b e e n shown b y H i i t t e r , S c h e l l i n g and K r a u s s (1974) t h a t f o r g l a s s / e p o x y l a m i n a t e s l o a d e d b i a x i a l l y , t h e Puppo and E v e n s e n a n a l y s i s p r e d i c t s t h e o b s e r v e d f a i l u r e r e s u l t s q u i t e w e l l . 31 CHAPTER 3 THEORETICAL FOUNDATIONS OF THE PROPOSED CONSTITUTIVE MODEL 3 . 1 I n t r o d u c t i o n The p r i m a r y o b j e c t i v e o f t h e p r e s e n t c h a p t e r i s t o d e v e l o p a r e l a t i v e l y c o m p r e h e n s i v e c o n s t i t u t i v e mode l f o r p r o g r e s s i v e f a i l u r e a n a l y s i s o f l a m i n a t e d c o m p o s i t e s t r u c t u r e s c o m p r i s i n g l a y e r s o f u n i d i r e c t i o n a l a n d / o r b i d i r e c t i o n a l ( i . e . woven f a b r i c ) FRMs. The m o d e l , w h i c h a t t e m p t s t o c o v e r t h e e n t i r e s t r e s s h i s t o r y , i s e s s e n -t i a l l y r e p r e s e n t a t i v e o f t h e m e c h a n i c a l b e h a v i o u r o f one l a y e r . To t h i s e n d , t h e f i r s t p a r t o f t h i s c h a p t e r i s d e v o t e d t o t h e d e r i v a t i o n o f t h e c o n s t i t u -t i v e e q u a t i o n s f o r s i n g l e l a y e r s o f u n i d i r e c t i o n a l and b i d i r e c t i o n a l FRM. I n t h e f i n a l p a r t t h e s e e q u a t i o n s a r e comb ined w i t h c l a s s i c a l l a m i n a t i o n t h e o r y t o f o r m t h e c o m p l e t e c o n s t i t u t i v e r e l a t i o n s f o r m u l t i l a y e r l a m i n a t e s . F o r c o n c i s e n e s s and b r e v i t y , t h e f o r m u l a t i o n s a r e e x p r e s s e d i n t e n s o r i a l ( i n d i c i a l ) n o t a t i o n , w h e r e v e r p o s s i b l e . M a t r i x c o n s t i t u t i v e e q u a t i o n s a r e a l s o p r e s e n t e d f o r d i r e c t f i n i t e e l e m e n t i m p l e m e n t a t i o n , d e t a i l s o f w h i c h a r e p r o v i d e d i n C h a p t e r 4 . 3 . 2 D e s c r i p t i v e O u t l i n e o f t h e M o d e l P h y s i c a l l y , t h e n o n l i n e a r a n d / o r i r r e v e r s i b l e d e f o r m a t i o n o f FRMs c a n be c a u s e d b y i n h e r e n t m a t e r i a l n o n l i n e a r i t i e s o f t h e i n d i v i d u a l c o n s t i t u e n t s , damage a c c u m u l a t i o n due t o f i b r e o r m a t r i x c r a c k i n g , i n t e r f a c i a l d e b o n d i n g , o r any c o m b i n a t i o n o f t h e a b o v e . These phenomena may be d e s c r i b e d m a c r o -s c o p i c a l l y w i t h i n t h e f ramework o f p l a s t i c i t y t h e o r y , t h u s p r o v i d i n g t h e i m p e t u s f o r t h e e l a s t i c - p l a s t i c - f a i l u r e mode l p r o p o s e d h e r e . 32 I t i s i n s t r u c t i v e t o p r e s e n t f i r s t a d e s c r i p t i v e o u t l i n e o f t h e f u n c -t i o n i n g o f t h e mode l b e f o r e i t s a n a l y t i c a l f o r m u l a t i o n . We a t t e m p t t o d e s c r i b e t h e b e h a v i o u r o f FRMs i n t e r m s o f t h e " c l a s s i c a l i n c r e m e n t a l t h e o r y o f p l a s t i c i t y " . I n o r d e r t o do t h i s , we i g n o r e t h e d e t a i l e d s t r u c t u r e o f t h e m a t e r i a l and assume t h a t i t i s p e r m i s s i b l e t o c o n s i d e r s t r e s s and s t r a i n as a v e r a g e s t a k e n t h r o u g h o u t a r e p r e s e n t a t i v e vo lume w h i c h i s i t s e l f t a k e n t o c o r r e s p o n d t o a p o i n t i n a c o n t i n u u m . We s t a y e n t i r e l y w i t h i n t h e most f a m i -l i a r f ramework o f i n c r e m e n t a l p l a s t i c i t y , w i t h s m a l l d i s p l a c e m e n t s and no v i s c o s i t y , c r e e p o r t h e r m o e l a s t i c i t y . M o r e o v e r , o n l y t h e m e c h a n i c a l b e h a -v i o u r o f t h e m a t e r i a l u n d e r s t a t i c o r q u a s i - s t a t i c l o a d i n g w i l l be c o n s i d e r e d ( i . e . s t r a i n r a t e e f f e c t s on y i e l d i n g a r e i g n o r e d ) . I t w i l l be assumed t h a t any d e g r a d a t i o n w h i c h o c c u r s due t o p l y y i e l d i n g o r p l y f a i l u r e i s r e s t r i c t e d t o t h a t p l y and i s n o t t r a n s m i t t e d t o a d j a c e n t p l i e s . A l s o no a t t e m p t w i l l be made t o mode l t h e i n t e r l a m i n a r e f f e c t s . The mode l p r o p o s e d i n t h i s s t u d y may b e s t be d i v i d e d i n t o t h r e e r e g i m e s : t h e e l a s t i c r e g i m e , t h e p l a s t i c r e g i m e and t h e p o s t - f a i l u r e r e g i m e . I n t h i s mode l t h e l i n e a r e l a s t i c s t r e s s - s t r a i n r e l a t i o n i s u s e d f i r s t u n t i l t h e comb ined s t a t e o f s t r e s s r e a c h e s an i n i t i a l y i e l d s u r f a c e w h i c h marks t h e b e g i n n i n g o f p l a s t i c f l o w . F u r t h e r l o a d i n g p r o d u c e s p l a s t i c r e s p o n s e u n t i l f a i l u r e i s r e a c h e d . F o r s i m p l i c i t y t h e i n i t i a l y i e l d c r i t e r i o n and t h e f a i l -u r e c r i t e r i o n a r e assumed t o have s i m i l a r f u n c t i o n a l fo rms i n s t r e s s s p a c e . Between t h e i n i t i a l y i e l d i n g s t a t e and t h e f a i l u r e s t a t e , t h e c o n s t i t u t i v e r e l a t i o n s a r e e x p r e s s e d i n i n c r e m e n t a l f o r m b a s e d on t h e a s s o c i a t e d f l o w r u l e o f p l a s t i c i t y t h e o r y . When f a i l u r e i s r e a c h e d i t i s a s c r i b e d t o e i t h e r m a t r i x o r f i b r e d e p e n d i n g on t h e r e l a t i v e m a g n i t u d e o f t h e v a r i o u s s t r e s s r a t i o s a p p e a r i n g i n t h e c r i t e r i o n . To s i m u l a t e p o s t - f a i l u r e b e h a v i o u r , two t y p e s o f f a i l u r e modes a r e d e f i n e d , n a m e l y , b r i t t l e and d u c t i l e . F o r t h e 33 b r i t t l e f rac ture mode, the layer i s assumed to lose i t s e n t i r e r i g i d i t y and s trength i n the dominant s tress d i r e c t i o n . For the d u c t i l e f rac ture mode, the layer re ta ins i t s s trength but loses a l l of i t s s t i f f n e s s i n the f a i l u r e d i r e c t i o n . D i f f e r e n t stages of the proposed e l a s t i c - p l a s t i c - f r a c t u r e model mentioned above can be i l l u s t r a t e d schemat ica l ly on an i d e a l i z e d u n i a x i a l s t r e s s - s t r a i n curve shown i n F i g . 3 .1 . In what follows a complete set of e l a s t o p l a s t i c c o n s t i t u t i v e r e l a t i o n s w i l l f i r s t be developed i n a general form, appropriate for any or tho trop ic mater ia l whose behaviour f a l l s w i th in the framework of time and temperature independent incremental p l a s t i c i t y . These formulations w i l l subsequently be s p e c i a l i z e d to the s ing le layers of u n i d i r e c t i o n a l and b i d i r e c t i o n a l FRMs under plane s t r e s s , s ince the l a t t e r i s the condi t ion that normally p r e v a i l s i n p l i e s of a laminate s u f f i c i e n t l y d i s tant from s i n g u l a r i t i e s . 3.3 General Formulation of the Single Layer C o n s t i t u t i v e Equations 3.3.1 E l a s t i c Regime In the i n i t i a l loading s ta te , the FRM layer i s t reated as a homogeneous and o r t h o t r o i p c l i n e a r e l a s t i c continuum. Let x 1 ( x 3 and x 3 denote a l o c a l orthogonal Cartes ian axes, the axes of x a and x 3 being i n the mid-plane of the layer and x 3 i n the thickness d i r e c t i o n . These axes co inc ide with the p r i n c i p a l axes of orthotropy. A l l subsequent d iscuss ions and der iva t ions w i l l be r e f e r r e d to t h i s l o c a l coordinate system. In the l i n e a r e l a s t i c range there i s a one-to-one a n a l y t i c a l r e l a t i o n b e t w e e n t h e s t r e s s t e n s o r a., and the s t r a i n t e n s o r e . . . T h i s can be expressed as (Sokolnikof f , 1956) 34 ° i j " C ? j k S e k £ ( i ' J ' k ' f i - 1 ' 2 ' 3 ) <3.1) where C . . , „ i s a f o u r t h o r d e r s t i f f n e s s t e n s o r whose components are the e l a s t i c constants , or moduli of the m a t e r i a l , and repeated ind ices imply summation. Equation (3.1) i s a n a t u r a l g e n e r a l i z a t i o n of Hooke's law, and i t i s used i n a l l developments of the l i n e a r theory of e l a s t i c i t y . Inasmuch as the components . and e „ are symmetr ic , the tensor of e l a s t i c c o n s t a n t s C ^ . ^ i s symmetric with respect to the f i r s t two and the l a s t two i n d i c e s , i . e . C i j k 2 C j i k 2 C i j i k C j i 2 k ( 3 , 2 ) Such symmetry considerat ions reduce the maximum number of independent e l a s t i c constants from 81 to 36. Green (1839) asserted that for an e l a s t i c body there ex i s t s a p o t e n t i a l funct ion $ with the property o. . = 3$/3e. . (3.3) For a l i n e a r e l a s t i c body, <J> co inc ides with the s tra in-energy dens i ty funct ion so that * = 1/2 C ? . , „ e. . e. = 1/2 o. . e. . (3.4) where E q . (3.1) i s used i n the l a s t step. I t can be deduced from Eqs. (3.3) and (3.4) that the order of the p a i r s of subscr ipts i j and k i are interchangeable , so that 35 Thus, under the above r e s t r i c t i o n , the 36 independent e l a s t i c constants can now be reduced to 21 such constants for the most general case of an an i so -t r o p i c e l a s t i c body. I f there are e l a s t i c symmetries i n c e r t a i n d i r e c t i o n s Q o f the m a t e r i a l , then the number of independent constants C . . , 0 i n Eq . (3.1) 1 J k x can be further reduced. To avoid deal ing with double sums, i t i s convenient to wr i te Hooke's law i n contracted notat ion as ° i = C i j e j ( i , J " = 1 , 2 6 ) ( 3 ' 6 ) The r e l a t i o n s h i p s between the contracted and tensor notat ions are given i n Table 3.1 below. Table 3.1 Comparison between tensor , engineering and contracted nota t ion for stresses and s t r a i n s . Stresses Stra ins Tensor Contracted Tensor Engineering Contracted ° n ° i e n e n e i ° 2 2 ° 2 e 22 G2 2 £ 2 ° 3 3 ° 3 e 33 e 33 £ 3 ° 2 3 = T23 °* T1 3 e « ° 3 1 = T31 2 e 3 1 r , i ° 1 2 = T12 °6 2 e 1 2 £ 6 36 - I f t h e m a t e r i a l has two m u t u a l l y p e r p e n d i c u l a r p l a n e s o f e l a s t i c symmet ry , i t i s c o n s i d e r e d o r t h o t r o p i c . F o r s u c h m a t e r i a l s , t h e r e a r e o n l y n i n e i n d e p e n d e n t e l a s t i c c o n s t a n t s C ^ . , and t h e s t r e s s - s t r a i n r e l a t i o n (Eq . 3 . 6 ) i n m a t r i x f o r m becomes 1 2 e 2 2 1 3 e 2 3 e S y m m e t r i c 0 0 0 0 0 0 0 ' 5 S 0 0 0 0 0 ( 3 . 7 ) A s s u m i n g t h a t t h e m a t e r i a l c o e f f i c i e n t s r e m a i n c o n s t a n t d u r i n g t h e d e f o r m a t i o n p r o c e s s , t h e i n c r e m e n t a l e l a s t i c c o n s t i t u t i v e r e l a t i o n s h i p t a k e s t h e f o l l o w i n g f o r m do . = C ? . d e . ( i , j = 1 ,2 6) ( 3 . 8 ) where do . and d e . a r e t h e s t r e s s - and s t r a i n - i n c r e m e n t t e n s o r s , l l 3.3.2 P l a s t i c Regime A t h e o r y o f p l a s t i c i t y i s a p r o c e d u r e b y w h i c h a s e t o f c o n s t i t u t i v e e q u a t i o n s f o r m u l t i a x i a l s t r e s s - s t a t e s c a n be d e r i v e d f r o m u n i a x i a l s t r e s s - s t r a i n t e s t d a t a . Such a t h e o r y a c c o u n t s , i n p r i n c i p l e , f o r t h e s t r e s s h i s t o r y dependent b e h a v i o u r . D i s s i p a t i v e r e s p o n s e c h a r a c t e r i z e d b y permanent s t r a i n a c c u m u l a t i o n c a n be e v a l u a t e d . The t h e o r y a c c o u n t s f o r 37 unloading and r e l o a d i n g , and for the i n t e r a c t i o n of s t res ses . Some references on the subject of p l a s t i c i t y are the books by H i l l (1950), Kachanov (1971), Mendelson (1968), and the review a r t i c l e by Naghdi (1960). The fo l lowing four bas i c ingredients of p l a s t i c i t y theory are proposed for use with FRMs. In a d d i t i o n , item (v) i s added to describe the onset of f a i l u r e : i ) An i n i t i a l y i e l d surface , bounding the part of the s tress space w i t h i n which deformation i s pure ly e l a s t i c . i i ) A hardening r u l e , spec i fy ing the modi f i ca t ion of the y i e l d surface i n the course of p l a s t i c deformation. i i i ) A flow law, i n d i c a t i n g the d i r e c t i o n of the incremental p l a s t i c s t r a i n vec tor . iv ) A p l a s t i c modulus or hardening modulus, i . e . the r a t i o between increments of "ef fect ive stress" and "ef fect ive p l a s t i c s t r a i n " . This r a t i o i s assumed to be independent of loading d i r e c t i o n . v) A f a i l u r e surface , de f in ing an upper bound of the p l a s t i c regime i n s tress space. In what fo l lows , the a n a l y t i c a l formulat ion of the foregoing items w i l l be presented. i ) I n i t i a l y i e l d The s implest g e n e r a l i z a t i o n of the y i e l d c o n d i t i o n for p l a s t i c a l l y a n i s o t r o p i c mater ia l s i s the general quadrat ic funct ion given by (cf . Baltov and Sawczuk, 1965; Shih and Lee, 1978) f (a. . , a . . , A . „ ,k) = 0 (3.9a) 38 where f E A i j k * ( o i j - a i j ) ( o u " k I ( 3 ' 9 b ) In Eq . (3 .9b) , A ^ . ^ g d , j ,k ,2 = 1,2,3) denotes the fourth order tensor of a n i s o t r o p i c s trength parameters which describes the shape of the y i e l d s u r f a c e , and the t ensor . describes the o r i g i n of the y i e l d surface . The e f f e c t i v e s i ze of the y i e l d surface i s given by the s ca lar parameter k which stands for a reference y i e l d s t r e s s . In contracted nota t ion Eq . (3.9a,b) can be rewr i t t en as f ( o i , a i , A i j ,k) E A ^ (o^-a^) (o^-a..) - k 2 = 0 ( i , j = 1,2 6) (3.9c) G e n e r a l l y , y i e l d s tresses are d i f f e r e n t i n tens ion and compression. It should be noted that a „ i n Eq . (3.9b) (or a., i n Eq . (3.9c)) accounts for the strength d i f f e r e n t i a l between t e n s i l e and compressive y i e l d s tresses by s h i f t i n g the o r i g i n of the y i e l d surface . Since the mater ia l cannot d i s t i n g u i s h between p o s i t i v e and negative shear s tresses (provided that the reference frame coincides with the p r i n c i p a l axes of orthotropy) , a l l odd powers (one, i n t h i s case) of shear s tress terms i n E q . (3.9c) must van i sh . From the above arguments and symmetry cons iderat ions , one can wri te A „ and i n the fo l lowing matrix form [A] = 0 0 0 A j 2 ^ J 3 0 0 0 A3 3 0 0 0 A , , 0 0 A 5 5 0 Symmetric A (3.10) 39 (a) = ( a 1 a2 a 3 0 0 0} The f u n c t i o n w h i c h e x p r e s s e s t h e i n i t i a l y i e l d c o n d i t i o n s f o r an o r t h o t r o p i c m a t e r i a l c a n be d e f i n e d as where t h e s u f f i x 0 d e n o t e s t h e i n i t i a l v a l u e o f t h e q u a n t i t y t o w h i c h i t i s a t t a c h e d . T h e p a r a m e t e r s A ? . , a? a n d k n m u s t b e o b t a i n e d f r o m e x p e r i m e n t s . C I J I 0 C A p p e n d i x A p r o v i d e s t h e d e t a i l s o f e v a l u a t i n g t h e s e p a r a m e t e r s . i i ) Subsequent y i e l d and h a r d e n i n g r u l e A f t e r i n i t i a l y i e l d i n g , i t i s n e c e s s a r y t o d e f i n e c o n d i t i o n s f o r s u b s e -quent y i e l d i n g o f t h e m a t e r i a l u n d e r c h a n g i n g l o a d . B e c a u s e s t r e s s s t a t e s o u t s i d e t h e y i e l d s u r f a c e a r e n o t a d m i s s i b l e , an i n c r e a s e i n l o a d must be a c c o u n t e d f o r i n one o f two w a y s . I n an i d e a l p l a s t i c m a t e r i a l t h e p o i n t o f i n t e r e s t m e r e l y t r a n s l a t e s on t h e unchanged i n i t i a l y i e l d s u r f a c e , w h i l e i f some f o r m o f h a r d e n i n g i s a l l o w e d , t h e y i e l d s u r f a c e w i l l change shape a n d / o r p o s i t i o n i n a manner d i c t a t e d b y t h e h a r d e n i n g r u l e and t h e s t r e s s p o i n t w i l l be somewhere on t h e s u b s e q u e n t y i e l d s u r f a c e . The p r o g r e s s i v e d e f o r m a t i o n o f t h e m a t e r i a l i s d e s c r i b e d i n t e r m s o f e v o l u t i o n o f t h e f a m i l y o f y i e l d ( o r l o a d i n g ) s u r f a c e s g i v e n b y a ? , A ? . , k 0 ) = 0 (3.11) f ( o i , O i ( K ) , A . . ( K ) , k ( K ) ) = 0 ( 3 . 12 ) 40 where K termed the hardening parameter i s a s u i t a b l y defined p l a s t i c i n t e r n a l v a r i a b l e which can be r e l a t e d to some measure of p l a s t i c deformation or p l a s t i c work. Re la t ive to an a r b i t r a r y po int i n s tress space, a s tate of "loading" i s s p e c i f i e d by f = 0; d< f 0 ; and O f / a o ^ ) d o i > 0 During "neutral loading" f = 0; dK = 0; and (af/Sc^) d o i = 0 F i n a l l y during "unloading" we have f = 0; dK = 0; and O f / a o ^ ) d o i < 0 By a l lowing ou, A „ and k i n Eq . (3.12) to vary as some funct ion of the hard-ening parameter K var ious kinds of hardening models can be s imulated. The two p r i n c i p a l models of s t r a i n hardening behaviour that have been developed are the "kinematic" and " i so trop ic" hardening models. I so trop ic hardening occurs when the i n i t i a l y i e l d surface expands uniformly during p l a s t i c flow. For kinematic hardening the y i e l d surface does not change i t s i n i t i a l shape and o r i e n t a t i o n but t rans la te s i n the s tress space l i k e a r i g i d body. This concept was introduced by Prager (1955) and l a t e r modified by Z i e g l e r (1959). The kinematic hardening model accounts for the Bauschinger 41 e f f e c t * , w h i c h i s an e x p e r i m e n t a l l y o b s e r v e d phenomenon u n d e r c y c l i c l o a d i n g s i t u a t i o n s . When K ( K ) = c o n s t a n t ( i . e . , 3k /3 i< = 0 ) a n d A „ ( K ) = c o n s t a n t ( i . e . , 3 A ^ . / 3 K ; = 0) i n E q . ( 3 . 1 2 ) , we o b t a i n t h e c a s e o f k i n e m a t i c w o r k - h a r d e n i n g . I f o n t h e o t h e r h a n d 3 a ^ / 3 < = 0 ; 3 A ^ ^ / 3 K = 0 a n d k( ic) i s a m o n o t o n i c a l l y i n c r e a s i n g f u n c t i o n , i s o t r o p i c w o r k h a r d e n i n g o c c u r s . The l a t t e r c a n a l t e r n a t i v e l y b e a t t a i n e d b y a l l o w i n g t h e A ^ p a r a m e t e r s t o v a r y i n a p r o p o r t i o n a l manner w h i l e S O U / S K = 0 and 3 k / 3 K = 0 . More r e c e n t l y , a t t e m p t s have b e e n made a t c o m b i n i n g t h e two p r e c e d i n g m o d e l s i n a m i x e d h a r d e n i n g f o r m u l a t i o n ( A x e l s s o n and S a m u e l s s o n , 1 9 7 9 ) , i n o r d e r t o o b t a i n a more r e a l i s t i c r e p r e s e n t a t i o n o f e l a s t o p l a s t i c b e h a v i o u r . I n t h i s c a s e , t h e y i e l d s u r f a c e e x p e r i e n c e s t r a n s l a t i o n and u n i f o r m e x p a n s i o n i n a l l d i r e c t i o n s , i . e . i t r e t a i n s i t s o r i g i n a l s h a p e . G e n e r a l " a n i s o t r o p i c " h a r d e n i n g c a n be i n t r o d u c e d b y l e t t i n g a l l t h e p a r a m e t e r s , A ^ a n d k v a r y w i t h K . Such a t h e o r y a c c o u n t s f o r changes i n t h e shape o f t h e y i e l d l o c u s d u r i n g p l a s t i c d e f o r m a t i o n , a f e a t u r e t h a t i s a b s e n t i n t h e h a r d e n i n g m o d e l s d i s c u s s e d so f a r . I t s h o u l d be n o t e d t h a t t h e k i n e m a t i c h a r d e n i n g mode l i s a s p e c i a l c a s e o f a n i s o t r o p i c h a r d e n i n g whereby t h e a n i s o t r o p y i s i n d u c e d b y B a u s c h i n g e r e f f e c t . I n t h e p r e s e n t s t u d y , s p e c i a l t y p e s o f i s o t r o p i c and a n i s o t r o p i c h a r d e n -i n g m o d e l s a r e u s e d . The l a t t e r a l l o w s f o r a n o n p r o p o r t i o n a l change o f t h e y i e l d v a l u e s and t h u s l e a d s t o a d i s t o r t e d shape o f t h e y i e l d s u r f a c e ( i . e . , S A ^ / S K f1 c o n s t a n t ) w h i l e 3 O U / 3 K = 0 and 3 k / 3 < = c o n s t a n t . The m a t h e m a t i c a l d e t a i l s o f t h e s e m o d e l s a r e p r e s e n t e d i n A p p e n d i x B. *The B a u s c h i n g e r e f f e c t r e f e r s t o a p a r t i c u l a r t y p e o f d i r e c t i o n a l a n i s o t r o p y i n d u c e d b y p l a s t i c d e f o r m a t i o n , n a m e l y , t h a t an i n i t i a l p l a s t i c d e f o r m a t i o n o f one s i g n r e d u c e s t h e r e s i s t a n c e o f t h e m a t e r i a l w i t h r e s p e c t t o a s u b s e q u e n t p l a s t i c d e f o r m a t i o n o f t h e o p p o s i t e s i g n . 42 i i i ) Flow r u l e A f t e r i n i t i a l y i e l d i n g the mater ia l behaviour w i l l be p a r t l y e l a s t i c (recoverable) and p a r t l y p l a s t i c ( i r r e c o v e r a b l e ) . I t i s a fundamental assumption i n the i n f i n i t e s i m a l theory of p l a s t i c i t y that the increment of the t o t a l s t r a i n tensor i n the p l a s t i c range may be decomposed in to e l a s t i c e D and p l a s t i c components de^ and de? by simple superpos i t i on , so that de. = de? + de? ( i = 1,2 6) (3.13) i i i The e l a s t i c s t r a i n increment i s r e l a t e d to the s tress increment by the general ized Hooke's law (Eq. (3 .8 ) ) , i . e . do. = C ? . de? ( i , j = 1,2 6) (3.14) l I J j In order to der ive the r e l a t i o n s h i p between the p l a s t i c s t r a i n increment and the s tress increment a further assumption about the mater ia l must be made. In p a r t i c u l a r i t w i l l be assumed that the p l a s t i c s t r a i n increment i s p r o p o r t i o n a l to the s tress gradient of a func t ion , g, termed the p l a s t i c p o t e n t i a l * , so that de? = dX I s - (3.15) 1 oo. 1 * P l a s t i c p o t e n t i a l funct ion i s analogous to the p o t e n t i a l funct ion i n i d e a l f l u i d flow and the s t r a i n - or complementary-energy dens i ty funct ion i n l i n e a r e l a s t i c i t y . A t h e o r e t i c a l bas i s for i t s existence i s developed by H i l l (1950). 43 where dX i s a p o s i t i v e s ca lar parameter which can vary throughout the deformation process . Eq . (3.15) i s termed the "flow ru le" s ince i t governs the p l a s t i c flow a f t er y i e l d i n g . The gradient of the p o t e n t i a l surface 3 g / 8 o . d e f i n e s the d i r e c t i o n of the p l a s t i c s t r a i n increment vector de?, 6 1 r • l while the length i s determined by dX. The flow r u l e i s termed "associated" i f the p l a s t i c p o t e n t i a l surface has the same shape as the current y i e l d or loading surface , i . e . f E g (cf . Bland, 1957). The l a t t e r has a s p e c i a l s i g n i f i c a n c e i n the mathematical theory of p l a s t i c i t y , s ince for t h i s case c e r t a i n v a r i a t i o n a l p r i n c i p l e s and uniqueness theorems can be formulated. Since there i s very l i t t l e experimental evidence on subsequent y i e l d surfaces , e s p e c i a l l y for FRMs, the assoc iated flow r u l e w i l l be app l i ed i n the fo l lowing work. In t h i s case E q . (3.15) becomes de? = dX (3.16) 1 oo. l E q . (3 .16) i s a l s o termed the n o r m a l i t y cond i t ion s ince 9f/3o^ i s a vector d i r e c t e d outward and normal to the y i e l d surface at the s tress point under cons idera t ion . iv ) E f f e c t i v e s tress and e f f e c t i v e s t r a i n For the work-hardening theory of p l a s t i c i t y to be of any p r a c t i c a l use, we must r e l a t e the hardening parameter K i n the loading funct ion (Eq. (3.12)) to experimental u n i a x i a l s t r e s s - s t r a i n curves . To t h i s end we are looking for some s tress v a r i a b l e , c a l l e d "ef fect ive s tress" , which i s a funct ion of the s t res ses , and some s t r a i n v a r i a b l e , c a l l e d "ef fect ive s t r a i n " , which i s a funct ion of the p l a s t i c s t r a i n s , so that they can be used to corre la t e the t e s t r e s u l t s obtained by d i f f e r e n t loading programs. 44 I f from here on the strength d i f f e r e n t i a l i n tens ion and compression i s i g n o r e d ( i . e . , = 0) p a r t l y for the sake of s i m p l i c i t y and also due to a lack of adequate knowledge of the p o s t - y i e l d behaviour of FRMs, i t i s then convenient to define the e f f e c t i v e s tress o as fo l lows: - 2 o = A . . o . o . ( i , j = 1 , 2 , . . . , 6 ) (3.17) i j l ] Therefore the loading funct ion can be rewr i t t en as f ( o . , A . . ( K ) , k(K)) = a2(a., A . . ( K ) ) - k'dc) = 0 (3.18) X X J X X J The d e f i n i t i o n of e f f e c t i v e p l a s t i c s t r a i n e p i s not quite as s imple . Two methods are genera l ly used. One defines the e f f e c t i v e p l a s t i c s t r a i n increment i n terms of the s p e c i f i c p l a s t i c work increment, dW P, i n the form dWP = o. de? (3.19) 1 x = k d i p The second method defines the e f f e c t i v e p l a s t i c s t r a i n increment as a metric i n the space of p l a s t i c s t r a i n s e?. In t h i s case we can i n t u i t i v e l y wri te an expression for de P as ( d e p ) 1 = A * , de? de? (3.20) where A * , i s a matrix of c o e f f i c i e n t s . It i s shown i n Appendix C that A * , i s i n fac t the inverse of the matrix A . . provided | A . . | t 0. The hardening parameter K appearing i n the loading funct ion (Eq. (3.12)) i s commonly def ined e i ther as o r 45 K = i p = JdP ( 3 . 21a ) K = W P = fo± de? (3 .21b ) where t h e i n t e g r a t i o n s a r e p e r f o r m e d o v e r a p p r o p r i a t e s t r a i n p a t h s . The f i r s t d e f i n i t i o n c o m p l i e s w i t h t h e s t r a i n h a r d e n i n g h y p o t h e s i s w h i l e t h e s e c o n d i s i n a c c o r d w i t h t h e work h a r d e n i n g h y p o t h e s i s . I n e i t h e r c a s e K i s a h i s t o r y dependent p a r a m e t e r . F o r a l g e b r a i c c o n v e n i e n c e t h e s t r a i n -h a r d e n i n g h y p o t h e s i s i s a d o p t e d i n t h e p r e s e n t s t u d y . A c c o r d i n g l y , t h e e f f e c t i v e y i e l d s t r e s s - e f f e c t i v e p l a s t i c s t r a i n r e l a t i o n has t h e g e n e r a l f o r m k = H ( e p ) ( 3 . 2 2 ) Then upon d i f f e r e n t i a t i o n we o b t a i n t h e s l o p e dk H' = — ( 3 . 2 3 ) d i p e v a l u a t e d a t t h e c u r r e n t v a l u e o f t h e e f f e c t i v e y i e l d s t r e s s k . The f u n c t i o n H , and i t s d e r i v a t i v e H' ( t e r m e d t h e p l a s t i c o r h a r d e n i n g modu lus a s s o c i a t e d w i t h t h e r a t e o f e x p a n s i o n o f t h e y i e l d s u r f a c e ) a r e d e r i v a b l e f r o m a g e n e r -a l i z e d e f f e c t i v e s t r e s s - e f f e c t i v e s t r a i n d i a g r a m w h i c h i s u s u a l l y i d e n t i f i e d w i t h one o f t h e s t r e s s - s t r a i n c u r v e s a l o n g t h e p r i n c i p a l m a t e r i a l d i r e c t i o n s . F o r a b i l i n e a r s t r e s s - s t r a i n d i a g r a m a d o p t e d i n t h i s s t u d y , H ' i s a c o n s t a n t and we have k = k 0 + H ' e P W i t h a v i e w t o d e t e r m i n i n g d X , we u s e t h e f l o w r u l e ( E q . ( 3 . 1 6 ) ) i n E q . ( 3 . 2 0 ) t o o b t a i n 46 ( d e V = A * (dX)' | f - | f (3.24) 11 oo. oo. 1 J Now from Eqs. (3.17) and (3.18) we have If - 2 k a. (3.25) oo. 1 I where a. = f 5 - = f A . . o. (3.26) l 3o^ k i j j are the components o f a p l a s t i c f low v e c t o r i n the s t r e s s s p a c e , o 1 ( O j °" 6. I n t r o d u c i n g E q s . (3.25) and (3.26) i n t o E q . (3 .24) and n o t i n g that A ^ A£j = (see Appendix C) , y i e l d s de p dX = g - ( 3 . 2 7 ) With the a i d of Eqs . (3.25) and (3.27) , the flow r u l e given by E q . (3.16) may be re-expressed as de? = de P a. (3.28) We can now proceed with the d e r i v a t i o n of the incremental e l a s t o p l a s t i c s t r e s s - s t r a i n r e l a t i o n s h i p s . 6 . . i s the Kronnecker-del ta def ined as , 6 . . = 0 1 for i f j for i = j 47 During p l a s t i c l oad ing , both i n i t i a l y i e l d and subsequent s tress s tates must s a t i s f y the y i e l d c o n d i t i o n , i . e . df = 0. Therefore , p l a s t i c flow i s governed by the fo l lowing consistency condi t ion* df = § £ - do. + f f — dA. . + | f dk = 0 (3.29) 3o. l 3A. i i 3k I t i s i n t e r e s t i n g to note that i n a l l the previous work on a n i s o t r o p i c p l a s t i c i t y ( e . g . Whang, 1969; V a l l i a p p a n , 1971) the terra 3 f / 3 A „ was not i n c l u d e d i n the f o r m u l a t i o n even though the assumption of vary ing A ^ . ' s was made. Consistent with the s t r a i n hardening hypothesis given by E q . (3.21a), the parameters A „ and k can i n general be functions only of the accumulated e f f e c t i v e p l a s t i c s t r a i n e P . Thus, we may wri te dk = H' d e p 3 A . . 3 A . . d A . . = - r v 1 dk = -TV1 H' de p (3.30) » - ^ - L U i ^"P i j 3k 3k A l s o from E q s . (3.13) and (3.14) we can wri te the s tress increment do^ as do. = C ? . (de. - de P ) (3.31) R e c a l l i n g the flow r u l e given by E q . (3.28) , the above equation can be recast in to the fo l lowing form *This i s the term used by Prager (1949) which demands that loading from a p l a s t i c s tate must lead to another p l a s t i c s ta te . 48 da± = (de. - a. dee) (3.32) S u b s t i t u t i n g Eqs. (3.30) and (3.32) in to Eq . (3.29) and not ing that | J - = 2k a. ; = o . o . and | | = -2k (3.33) oO. 1 o A . . 1 1 oK 1 I J J one can rewrite the consistency condi t ion (Eq. (3.29)) as a . C e , d e P = 1 1-1 de. (3.34) uH' + a C e a J ^ m mn n where a l l the ind ices take on the values 1,2 6 and the parameter u i s given by 3A. . * • 1 " k -# ( 3 - 3 5 ) F i n a l l y , s u b s t i t u t i o n of Eq . (3.34) i n Eq . (3.32) leads to the fo l lowing e l a s t o p l a s t i c c o n s t i t u t i v e equations do. = C? p. de. = (C?. - C?. ) de. (3.36) where C? P . i s the e l a s t o p l a s t i c m a t e r i a l s t i f f n e s s t e n s o r and C? . i s the p l a s t i c mater ia l s t i f f n e s s tensor defined as c p _ = C i k a k a g C 2 j 1 J ' uH' + a C e a ^ m mn n (3.37) 49 The n e g a t i v e s i g n a p p e a r i n g i n E q . ( 3 .36 ) c l e a r l y r e p r e s e n t s t h e d e g r a d a t i o n o f t h e m a t e r i a l s t i f f n e s s due t o p l a s t i c f l o w . v ) F a i l u r e A f a i l u r e c r i t e r i o n d e f i n e s t h e maximum s t r e n g t h o f t h e FRM l a y e r u n d e r any p o s s i b l e c o m b i n a t i o n o f s t r e s s e s . I t i s assumed t h a t t h i s c r i t e r i o n i s n o t i n f l u e n c e d b y t h e d e f o r m a t i o n h i s t o r y ( i . e . i t i s p a t h - i n d e p e n d e n t ) and c a n be p o s t u l a t e d a p r i o r i . U n l e s s o t h e r w i s e s t a t e d , i n t h e p r e s e n t s t u d y t h e f a i l u r e s u r f a c e i s assumed t o have a s i m i l a r f u n c t i o n a l f o r m t o t h a t o f t h e y i e l d l o c u s , i . e . f ( o . . A ? . , k ) E A ? , a.a. - k* = 0 ( 3 . 3 8 ) U 1 I J ' U I J 1 J u where t h e s u f f i x u s t a n d s f o r t h e u l t i m a t e v a l u e s o f t h e q u a n t i t i e s t o w h i c h i t i s a t t a c h e d . The p a r a m e t e r s A ? , and k a r e m a t e r i a l c o n s t a n t s w h i c h c a n i j u be r e l a t e d t o t h e u l t i m a t e s t r e s s e s o b t a i n e d f r o m t h e same b a s i c t e s t s u s e d t o o b t a i n t h e y i e l d v a l u e s . C r i t e r i a o f t h e t y p e g i v e n above p r e d i c t t h e o n s e t o f f a i l u r e b u t p r o v i d e no i n f o r m a t i o n r e g a r d i n g t h e modes o f f a i l u r e . The l a t t e r a r e i m p o r t a n t i n s t u d y i n g t h e b e h a v i o u r o f t h e m a t e r i a l beyond i n i t i a l f a i l u r e . The f a i l u r e mode i d e n t i f i c a t i o n p r o c e d u r e u s e d i n t h e p r e s e n t s t u d y w i l l be f u l l y d i s c u s s e d i n s e c t i o n 3 . 4 f o r p l a n e s t r e s s s i t u a t i o n s . 3.3.3 P o s t - F a i l u r e Regime To c o m p l e t e t h e c o n s t i t u t i v e m o d e l , we a l s o n e e d t o d e f i n e t h e p o s t -f a i l u r e b e h a v i o u r f o r d i f f e r e n t f a i l u r e modes. F o r FRMs, f a i l u r e may t a k e one o f two f o r m s , n a m e l y , f i b r e f r a c t u r e o r m a t r i x c r a c k i n g . S e v e r a l a p p r o a c h e s c a n be e m p l o y e d f o r c r a c k m o d e l l i n g i n t h e p o s t - f a i l u r e r e g i o n . 50 T h e s e may be c l a s s i f i e d as t h e smeared c r a c k i n g m o d e l , and t h e d i s c r e t e c r a c k i n g m o d e l . The p a r t i c u l a r c h o i c e o f t h e mode l depends upon t h e p u r p o s e o f t h e a n a l y s i s . I n g e n e r a l , i f o v e r a l l l o a d - d e f l e c t i o n b e h a v i o u r i s d e s i r e d , w i t h o u t r e g a r d t o c o m p l e t e l y r e a l i s t i c c r a c k p a t t e r n s and l o c a l s t r e s s e s , t h e s m e a r e d - c r a c k mode l i s p r o b a b l y t h e b e s t c h o i c e . I f d e t a i l e d l o c a l b e h a v i o u r i s o f i n t e r e s t t h e u s e o f t h e d i s c r e t e mode l may p r o v e n e c e s s a r y . However , f o r most s t r u c t u r a l e n g i n e e r i n g a p p l i c a t i o n s , t h e smeared c r a c k i n g mode l i s g e n e r a l l y a d o p t e d . Such r e p r e s e n t a t i o n , w h i c h i s f a v o u r e d i n t h e p r e s e n t w o r k , assumes t h a t t h e c r a c k e d c o m p o s i t e r e m a i n s a c o n t i n u u m , i . e . t h e c r a c k s a r e smeared o u t i n a c o n t i n u o u s f a s h i o n . The e f f e c t s o f s u c h c r a c k s on t h e b e h a v i o u r o f t h e f a i l e d l a y e r i s t h e r e d u c t i o n o f s t i f f n e s s a n d / o r s t r e n g t h i n c e r t a i n d i r e c t i o n s . The f i b r e f r a c t u r e i s assumed t o c a u s e a l o s s i n l o a d c a r r y i n g c a p a c i t y ( i . e . s t r e n g t h and s t i f f n e s s ) i n t h e f i b r e d i r e c t i o n . T h i s c o r r e s p o n d s w i t h t h e b r i t t l e f r a c t u r e mode shown i n F i g . 3 . 1 . M a t r i x c r a c k i n g ; on t h e o t h e r h a n d , i s assumed t o be c a u s e d b y s h e a r a n d / o r t r a n s v e r s e t e n s i o n . These c r a c k s a r e fo rmed p a r a l l e l t o t h e f i b r e d i r e c t i o n s as shown s c h e m a t i c a l l y i n F i g . 3 . 2 . When t h e c r a c k e d l a y e r i s w i t h i n a m u l t i l a y e r l a m i n a t e t h e c r a c k o p e n i n g d i s p l a c e m e n t s w i l l be c o n s t r a i n e d b y t h e a d j a c e n t l a y e r s . T h i s l e a d s t o a g r a d u a l l o s s o f l o a d c a r r y i n g c a p a c i t y o r s o f t e n i n g o f t h e f a i l e d l a y e r i n t h e o v e r a l l s e n s e . The e x t e n t o f s o f t e n i n g , marked b y t h e s l o p e o f t h e d e s c e n d i n g p a r t o f t h e s t r e s s - s t r a i n c u r v e , i s g e n e r a l l y dependent on t h e o r i e n t a t i o n o f t h e a d j a c e n t p l i e s i n a l a m i n a t e , i . e . i t i s l a m i n a t e -d e p e n d e n t . T h i s l e a d s t o c o n s i d e r a b l e c o m p l i c a t i o n s i n t h e d e v e l o p m e n t o f c o n t i n u u m c o n s t i t u t i v e t h e o r i e s f o r FRM l a y e r s beyond i n i t i a l f a i l u r e . I n t h e p r e s e n t work t h e two e x t r e m e c a s e s o f b r i t t l e and d u c t i l e f r a c t u r e mode ls shown i n F i g . 3 . 1 a r e c o n s i d e r e d s u f f i c i e n t t o s e r v e as bounds t o t h e a c t u a l b e h a v i o u r a f t e r m a t r i x c r a c k i n g . The d e t a i l s o f t h e p o s t - f a i l u r e m o d e l l i n g u n d e r p l a n e s t r e s s c o n d i t i o n s a r e o u t l i n e d i n S e c t i o n 3 . 4 f o r u n i d i r e c t i o n a l and b i d i r e c t i o n a l l a y e r s . 3 . 4 P l a n e S t r e s s F o r m u l a t i o n o f t h e S i n g l e L a y e r C o n s t i t u t i v e E q u a t i o n s The p l i e s o f a l a m i n a t e a r e g e n e r a l l y u n d e r t h e s t a t e o f p l a n e s t r e s s a t s u f f i c i e n t d i s t a n c e f r o m t h e f r e e e d g e s . T h e r e f o r e t h e d i s c u s s i o n i n t h i s s e c t i o n w i l l be d e v o t e d t o s p e c i a l i z a t i o n o f t h e p r e c e d i n g c o n s t i t u t i v e e q u a -t i o n s t o p l a n e s t r e s s s i t u a t i o n s . Under s u c h c o n d i t i o n s , t h e t h r o u g h -t h i c k n e s s s t r e s s e s c a n be n e g l e c t e d ( i . e . , 0 3 = 0 ^ = 0 5 = 0 ) . 3 . 4 . 1 E l a s t i c Regime The e l a s t i c c o n s t i t u t i v e r e l a t i o n g i v e n b y E q . ( 3 . 7 ) may now be w r i t t e n a s : Q11 0 e i ° 2 = 0 ( 3 . 39 ) - ° 6 - 0 0 Q ? 6 - - £ 6 -6 w h e r e t h e c o m p o n e n t s o f t h e r e d u c e d e l a s t i c s t i f f n e s s m a t r i x . c a n be r e l a t e d t o t h e . c o n s t a n t s b y u s i n g o 3 = 0 i n E q . ( 3 . 7 ) . Thus Q?2 = Q j i • c f , - ce12ce23/ce33 ( 3 . 4 0 ) 52 The Q^j c o e f f i c i e n t s can a l so be represented i n the more common engineering form as (see e .g . Jones, 1975) (3.41) Q ! 6 - o where E t and E , are the e l a s t i c moduli i n the x t and x , d i r e c t i o n s , r e s p e c t i v e l y ; \ > 1 2 » \ J 2 i a r e Poisson's r a t i o s ; and G i s the in -p lane shear modulus. In the nota t ion for Poisson's r a t i o s the f i r s t index re f er s to the d i r e c t i o n of imposed s t r a i n and the second index r e f e r s to the response d i r e c t i o n . The Poisson's r a t i o s \> 1 2 and \ j J X are r e l a t e d through the r e c i p r o c a l r e l a t i o n s h i p \ J n E , = v J 1 E 1 (3.42) leaving a t o t a l of four independent e l a s t i c constants to describe the planar orthotropy . In symbolic matrix n o t a t i o n , the incremental e l a s t i c c o n s t i t u t i v e r e l a t i o n s h i p for the plane o r t h o t r o p i c case i s {do} = [Q e] {de} (3.43) 53 where T [a] = {a1 o a o6} {e} = { £ j e 2 e6} (3.44) T 6 are the s t r e s s and s t r a i n v e c t o r s and [Q ] i s the 3 x 3 e l a s t i c mater ia l s t i f f n e s s matr ix . 3 .4.2 P l a s t i c Regime The y i e l d c r i t e r i o n given by E q . (3.18) can now be w r i t t e n i n matrix nota t ion as f ( {a} , [A] ,k) E o 2 ( {a} , [A]) - k 2 = 0 (3.45) where a 2 ( {o} , [A]) = {o}T [A] {o} (3.46) and [A] = A n 0 A 1 2 A „ 0 0 0 A (3.47) 6 6 —I Although any polynomial i n the form of E q . (3.45) represents a surface i n the s tress space, not a l l the surfaces are admiss ible y i e l d surfaces . A geometric i n t e r p r e t a t i o n i s use fu l for examining the cons tra int s on the form of Eq . (3.45). S p e c i f i c a l l y , i n the plane s tress s ta te , open-ended y i e l d surfaces are p h y s i c a l l y impossible s ince they imply i n f i n i t e s trength for 54 some state of s t r e s s . * To ensure the boundedness of the y i e l d surface i n the alfa2,a6 s tress space we must have A ^ O j O j > 0 ( i . j = 1,2 6) (3.48) f o r an a r b i t r a r y s t r e s s t e n s o r o^. Mathemat i ca l ly , the i n e q u a l i t y (3.48) impl ies that the matrix of a n i s o t r o p i c s trength parameters [A] has to be p o s i t i v e d e f i n i t e . Since [A] i s symmetric, the necessary and s u f f i c i e n t condi t ions for i t to be p o s i t i v e d e f i n i t e are that a l l i t s p r i n c i p a l minors be p o s i t i v e . Thus, the components of [A] must s a t i s f y the fo l lowing i n e q u a l i t i e s : A i x A , , > 0 (3.49) A , , A 6 6 > 0 Geometr ica l ly , the p o s i t i v e def in i teness of [A] ensures that the y i e l d surface i s an e l l i p s o i d . The i n t e r a c t i o n parameter A l a defines the angular o r i e n t a t i o n of the e l l i p s o i d (with respect to the o a and o 3 axes) , and determines the lengths of i t s major and minor axes. The range of al lowable values for A 1 3 , however, i s l i m i t e d by the bounding condit ions (3.49) and t h i s i n turn can often r e s u l t i n u n s a t i s f a c t o r y y i e l d (or f a i l u r e ) surfaces . * In three-dimensional s tates of s tress a loading state ex i s t s for which most mater ia l s e x h i b i t e s s e n t i a l l y i n f i n i t e s trength . In f a c t , any c r i t e r i o n neg lec t ing the e f fec t of hydros ta t i c s t r e s s , such as the von Mises c r i t e r i o n (cf . Mendelson, 1968), i s such a case. 55 The incremental e l a s t o p l a s t i c c o n s t i t u t i v e r e l a t i o n s given by Eqs. (3.36) and (3.37) can be expressed i n matrix notat ion as: {do} = [Q e p]{de} = ([Q e ] - [QP]){de} - ([Q 6 ] - ^](a)(af[f] ) {de} (3.50) uH' + {a} i[Q e]{a} where [Q P ] and [ Q e p ] are the plane s tress p l a s t i c and e l a s t o p l a s t i c mater ia l s t i f f n e s s matr ices ; and {a} = 1/k [A]{o} (3.51) u = 1 - l / 2 k {o}T [3A/3k] {o} The treatment of u n i d i r e c t i o n a l (U/D) and b i d i r e c t i o n a l (B/D) layers d i f f e r s i n the way t h e i r y i e l d (and f a i l u r e ) surfaces are def ined . Otherwise the above formulations are common to both these cases. We s h a l l now define separate y i e l d (and f a i l u r e ) functions for U/D and B/D l a y e r s . (a) B i d i r e c t i o n a l layers Let us consider the plane s tress form of H i l l ' s y i e l d c r i t e r i o n (1950) which reads (see Appendix A ) : ° i a ° 2 a 1 1 1 ° 6 j (jj-) + " % + yT " t r ) ° i a 2 + <s"> = 1 ( 3 ' 5 2 ) where X , Y , Z are the y i e l d s tresses under u n i a x i a l loading along each of the three p r i n c i p a l axes of anisotropy ( F i g . 3 .3a); and S i s the corresponding 56 y i e l d value of in -p lane shear s tress o 6 . In the absence of information about Z , the y i e l d s tress U for u n i a x i a l tens ion at 45° to the f ibres can be used to obta in an estimate for Z (see Appendix A ) . Under these circumstances, comparing Eqs. (3.45) and (3.52) for the four tes t condit ions gives k J k J k* A I I = Y T » A 2 2 = yi ' A 6 6 = c T » a n c * (3.53) A = r- - (— + — + — )1 k* " 1 2 L U2 V X J Y 2 S J One of the disadvantages of t h i s c r i t e r i o n i s that i t requires four mater ia l constants to define the y i e l d f u n c t i o n , and quite often four separate tes ts are not a v a i l a b l e . In a d d i t i o n for many mater ia l s i t proved d i f f i c u l t to s a t i s f y Eq . (3.49) i . e . the boundedness c r i t e r i a . This motivated a search for a c r i t e r i o n that required fewer input data for i t s eva luat ion and d i d not have as r e s t r i c t i v e a condi t ion as that given by Eq . (3.49) . The Puppo-Evensen c r i t e r i o n (1972) (which was o r i g i n a l l y suggested for the p r e d i c t i o n of f a i l u r e i n m u l t i l a y e r laminates) s a t i s f i e s the above requirements and a lso possesses features that are i d e a l for modell ing a wide range of b i d i r e c t i o n a l FRMs. This c r i t e r i o n , i n the plane s tress case, can be w r i t t e n as: ° i 2 y °i °> °* * °e a ( X _ ) " A ( Y ° X~ T + A ( Y ~ ) + ( F ) = 1 (3.54a) ° 1 2 V °i °« °* 2 ° 6 2 A ( X " ) " A ( r X~ Y ~ + lT] + ( S " } = 1 (3.54b) 57 where A = 3SVXY (3.54c) Thus, i n t h i s case the matrix of a n i s o t r o p i c s trength parameters takes one of the fo l lowing two forms depending on which one of the c r i t e r i a (3.54a) or (3.54b) i s dominant ( i . e . corresponds with the greater e f f e c t i v e s tress a ) : 1_ 3 S_l_ 2 XY 3 [A] = k Symm. XY 3 1_ (3.55a) [A] = k X 3 Y Symm. 3 2 1 _S_1 X 3 Y 1_ S 1 (3.55b) F i g u r a t i v e l y , Eqs . (3.54 a,b) descr ibe a p a i r of e l l i p s o i d s i n the three dimensional s tress space (alta3,a6). The y i e l d surface i s taken to be the inner surface r e s u l t i n g from the i n t e r s e c t i o n of these e l l i p s o i d s . The p a r a -meter A i s numer ica l ly equal to u n i t y for an i s o t r o p i c mater ia l obeying the von Mises y i e l d c r i t e r i o n ( i . e . X=Y=v'3 S) and tends to zero for the case of a f a b r i c - l i k e mater ia l made up of strong f ibres i n the x t and x 2 d i r e c t i o n s but with very weak matrix m a t e r i a l . In the former case, the y i e l d surface reduces to the von Mises e l l i p s e and i n the l a t t e r i t approaches the square shape ( i . e . , n o n - i n t e r a c t i n g stresses) i n the o 6 = 0 p lane . A t y p i c a l b i -d i r e c t i o n a l l y - r e i n f o r c e d mater ia l f a l l s somewhere i n between these two extreme cases , i . e . 0 < A < 1. F i g . 3.4 i l l u s t r a t e s the progress ive change 58 of the y i e l d surface i n the o g = 0 p lane , for some representat ive values of the parameter A. The surfaces are p l o t t e d for the p a r t i c u l a r case of equal s trength f i b r e s , i . e . X = Y. The condi t ion for the boundedness of the y i e l d surface (Eq. (3.49)) now becomes < i X I 3 Y and (3.56) 4 Y 3  b < 3 X For s trong ly a n i s o t r o p i c mater ia l s (e .g . u n i d i r e c t i o n a l composites i n which X >> Y) at l eas t one of the i n e q u a l i t i e s i n (3.56) may be v i o l a t e d . This makes the 3-parameter Puppo-Evensen 1s c r i t e r i o n unsui table for p r e d i c t i o n of y i e l d i n g i n U/D m a t e r i a l s . The fo l lowing i s devoted to treatment of such m a t e r i a l s . b) U n i d i r e c t i o n a l layers Here we propose the 3-parameter H i l l ' s c r i t e r i o n for t ransverse ly i s o t r o p i c medium. This extension of H i l l ' s c r i t e r i o n , which was f i r s t suggested by A z z i and T s a i (1965) , i s based on the concept that the u n i d i r e c t i o n a l f i b r e s (a l igned i n the x1 d i r e c t i o n as shown i n F i g . 3.3b) are randomly d i s t r i b u t e d and hence the proper t i e s i n the x 2 and x 3 d i r e c t i o n s are the same. Thus Y = Z and the 4-parameter H i l l ' s c r i t e r i o n given by Eq . (3.52) reduces to 59 The matrix [A] now reads: [A] = k 1 X 1 1 1_ 2 X J T T T - o T7T 0 _ Symm. yT 0 1 (3.58) I t ' can be shown that the surface represented by Eq . (3.57) s a t i s f i e s the c losure condit ions of Eq . (3.49) , provided that Y < 2X. The l a t t e r cond i t ion i s r e a d i l y met by the u n i d i r e c t i o n a l FRMs for which X > Y. I t should be noted that when Eqs. (3.54 a,b) and (3.57) are used as i n i t i a l y i e l d c r i t e r i a the quant i t i e s X , Y , S and k are replaced by t h e i r i n i t i a l values X 0 , Y 0 , S 0 and k 0 . During subsequent y i e l d i n g , i f the evo lut ion of the y i e l d surface fol lows the a n i s o t r o p i c hardening model described i n Appendix B, then the i n d i v i d u a l y i e l d values i n terms of the current e f f e c t i v e y i e l d s tress k become X (k) Y 3 ( k ) 2 w -n 2 2 2 S (k) = £ (k - k 0 ) + s 0 where E , E and G a r e t h e p l a s t i c m o d u l i i n t h e x , - d i r e c t i o n , P i P , P 1 x 2 - d i r e c t i o n , and in -p lane shear r e s p e c t i v e l y . These moduli remain constant P i 2 2 2 JT- (k " k „ ) + X„ (3.59) P2 2 2 2 — (k - k 0 ) + Y 0 60 for the b i l i n e a r s t r e s s - s t r a i n curves considered here and can be expressed i n terms of the e l a s t i c and tangent moduli as (see Appendix B) E . E A 1 5P l = E 1 - E A 1 E 2 E T j (3.60) P 2 E 2 ~ E T G P G - G T In order to use Eqs. (3.54a,b) and (3.57) as f a i l u r e c r i t e r i a , X , Y , S and k must be r e p l a c e d by t h e i r u l t i m a t e v a l u e s X ,Y ,S and k . F a i l u r e i s r J u u u u asc irbed to e i t h e r the matrix or f i b r e depending on the r e l a t i v e magnitude of the various s tress r a t i o terms appearing i n the f a i l u r e c r i t e r i o n . The f a i l u r e mode i d e n t i f i c a t i o n procedure i s shown i n Table 3 .2 , for both U/D and B/D l a y e r s . Table 3.2 F a i l u r e I d e n t i f i c a t i o n Procedure Layer Type Condi t ion F a i l u r e Mode U n i d i r e c t i o n a l ng > ng and lig > ng Otherwise l^ -l > L^-l rg \\ f i b r e matrix ( tens i le ) matrix (shear) B i d i r e c t i o n a l | o 1 / X u l i s greatest Io 3 /Y^| i s greatest | a 6 / S u l i s greatest f i b r e i n xr f i b r e i n x 3 matrix 61 3 . 4 . 3 Post-Failure Regime A f t e r f a i l u r e , new i n c r e m e n t a l c o n s t i t u t i v e r e l a t i o n s h i p s must be d e r i v e d . T h i s i s a c c o m p l i s h e d b y m o d i f y i n g t h e e l a s t i c i t y m a t r i x [Q e ] s u c h t h a t a t t h e f a i l u r e l o c a t i o n t h e m a t e r i a l c a n n o t c a r r y any a d d i t i o n a l t e n s i l e s t r e s s n o r m a l t o t h e c r a c k p l a n e o r s h e a r s t r e s s p a r a l l e l t o t h e c r a c k p l a n e . T a b l e 3 .3 summar i zes t h e f o r m o f t h e p o s t - f a i l u r e e l a s t i c i t y m a t r i x ( d e n o t e d by [ Q f ] ) f o r b o t h U/D and B/D l a y e r s . T a b l e 3 . 3 . P o s t - F a i l u r e I n c r e m e n t a l C o n s t i t u t i v e M a t r i x [Q ] f o r B o t h B r i t t l e and D u c t i l e F a i l u r e M o d e l s L a y e r Type F a i l u r e Mode [ Q f ] U n i - d i r e c t i o n a l F i b r e ( x 1 - d i r e c t i o n ) M a t r i x Q 2 J = E 2 ; a l l o t h e r Q[. = 0 = E : ; a l l o t h e r Q5 . = 0 B i - d i r e c t i o n a l F i b r e ( X j - d i r e c t i o n ) F i b r e ( x 2 - d i r e c t i o n ) M a t r i x Q f a = E 3 ; a l l o t h e r Q * = 0 = E 1 ; a l l o t h e r = 0 Q i i = E X 1 f and a l l o t h e r Q = 0 f i j Q a a = E a J These m a t r i c e s a p p l y w h e t h e r t h e f a i l u r e p r o c e s s i s assumed t o be b r i t t l e o r d u c t i l e . However , i n t h e f o r m e r c a s e t h e r e l e v a n t s t r e s s components on t h e c r a c k e d p l a n e s j u s t b e f o r e f a i l u r e must be removed a b r u p t l y and r e d i s t r i b u t e d t o a d j a c e n t u n c r a c k e d m a t e r i a l o f t h e e n t i r e s t r u c t u r e . T h e v e c t o r o f r e l e a s e d s t r e s s e s ( d e n o t e d b y (o^)) i s t a b u l a t e d i n T a b l e 3 .4 62 f o r v a r i o u s c a s e s . I t s h o u l d be n o t e d t h a t f o r t h e d u c t i l e mode o f f r a c t u r e {o^} v a n i s h e s . T a b l e 3 . 4 R e l e a s e d S t r e s s V e c t o r {o^} D u r i n g B r i t t l e Type o f F a i l u r e L a y e r Type F a i l u r e Mode [ o f ] U n i - d i r e c t i o n a l F i b r e ( x 1 - d i r e c t i o n ) M a t r i x { o l t 0 , o 6 } T {0, o a , o 6 } T B i - d i r e c t i o n a l F i b r e ( x 1 - d i r e c t i o n ) F i b r e ( x 2 ~ d i r e c t i o n ) M a t r i x {alt 0 , o 6 } T {0, o a , o 6 } T {0, 0 , o 6 } T 3 . 5 M u l t i l a y e r L a m i n a t e s The c o n s t i t u t i v e e q u a t i o n s f o r s i n g l e l a y e r s o f FRM d e v e l o p e d i n t h e p r e v i o u s s e c t i o n s a r e now a p p l i e d t o l a m i n a t e s c o m p r i s i n g a number o f l a y e r s t o d e t e r m i n e l a m i n a t e r e s p o n s e . I n t h e f o l l o w i n g a l a m i n a t e s u b j e c t e d t o a g e n e r a l s t a t e o f membrane s t r e s s w i l l be t r e a t e d . The e q u a t i o n s w h i c h a r e d e v e l o p e d d e f i n e t h e o v e r a l l r e s p o n s e o f a l a m i n a t e u n d e r a g i v e n s e t o f membrane l o a d s . B e c a u s e o f t h e p o t e n t i a l f o r p l a s t i c f l o w i n e a c h l a y e r t h e g o v e r n i n g e q u a t i o n s a r e p r e s e n t e d i n d i f f e r e n t i a l f o r m . L e t t h e l a m i n a t e c o n s i s t o f homogeneous o r t h o t r o p i c t h i n l a y e r s , e a c h e x h i b i t i n g p l a n e s t r e s s b e h a v i o u r . The i n c r e m e n t a l e l a s t i c and e l a s t o p l a s t i c c o n s t i t u t i v e r e l a t i o n s i n p r i n c i p a l m a t e r i a l c o o r d i n a t e s ( i . e . x l f x 2 ) f o r a s i n g l e l a y e r a r e g i v e n b y E q s . ( 3 . 4 3 ) and ( 3 . 5 0 ) , r e s p e c t i v e l y . I n any o t h e r c o o r d i n a t e s y s t e m i n t h e p l a n e o f t h e l a y e r ( s a y x , y i n F i g . 3 . 5 ) , t h e i n c r e m e n t a l s t r e s s - s t r a i n r e l a t i o n s c a n be w r i t t e n as 63 where do x do y dx L xy j de x de y dr x y ( 3 .61 ) do x do y dx L x y J [T] dOj do , do . de V d e j de = [ T ] T d e . y J d r x y . _ 2 d £ 6 _ [Q ' ] = [T] [Q][T] ( 3 .62 ) and [T] i s t h e t r a n s f o r m a t i o n m a t r i x g i v e n b y [T] = 2 3 1 c o s 8 s i n 0 ^ s i n 2 6 2 2 I s i n 9 c o s 0 - ^ s i n 2 6 - s i n 2 9 s i n 2 9 c o s 2 9 ( 3 . 6 3 ) w i t h 9 b e i n g t h e a n g l e " f r o m " t h e x - a x i s " t o " t h e X j - a x i s ( see F i g . 3 . 5 ) . I n E q . ( 3 . 6 2 ) [Q] s t a n d s f o r e l a s t i c , e l a s t o p l a s t i c and p o s t - f a i l u r e c o n s t i t u t i v e m a t r i c e s , w h i c h e v e r i s a p p r o p r i a t e . The i n c r e m e n t a l s t r e s s - s t r a i n r e l a t i o n s i n a r b i t r a r y c o o r d i n a t e s g i v e n b y E q . ( 3 . 6 1 ) , a r e u s e f u l i n t h e d e f i n i t i o n o f t h e l a m i n a t e s t i f f n e s s e s b e c a u s e o f t h e a r b i t r a r y o r i e n t a t i o n o f t h e c o n s t i t u e n t l a y e r s . E q . ( 3 .61 ) c a n b e t h o u g h t o f a s a s e t o f c o n s t i t u t i v e r e l a t i o n s f o r t h e k*"*1 l a y e r o f a m u l t i l a y e r e d l a m i n a t e , and t h u s c a n be w r i t t e n as 64 {do ' ) k = [ Q ' ] k {de ' ) k (3.64) where (3.65) x y 'xy The laminate i s assumed to cons i s t of p e r f e c t l y bonded layers and the displacements are continuous across the layer boundaries ( interfaces) so that no layer can s l i p r e l a t i v e to another. I t i s further assumed that the s tresses i n any layer are constant, but d i f f e r e n t i n the d i f f e r e n t l a y e r s . These are the usual type of assumptions made i n the c l a s s i c a l laminat ion theory (see e .g . Jones, 1975). A t y p i c a l laminate i s p i c t u r e d i n F i g . 3.6 along with i t s deformed shape. Let the laminate be r e f e r r e d to a f i xed system of coordinates x , y and z as shown i n F i g . 3 .6 . This w i l l henceforth be r e f e r r e d to as the laminate or s t r u c t u r a l coordinate system. The incremental s t r e s s - s t r a i n r e l a t i o n s for each layer can be expressed as where { d e ' ° } i s the laminate s t r a i n increment vec tor , which i s the same for each l a y e r . The r e s u l t a n t laminate forces per u n i t width are obtained by in tegra t ing the s tress components of each layer through the t o t a l thickness of the laminate. In the incremental sense, they w i l l take the fo l lowing form: {do ' ) k = [ Q ' ] f c {de 1 0} (3.66) 65 dN x dN dN t /2 J - t / 2 do do dz (3.67) xy dx xy J k Since the s tresses are constant through the thickness of each layer then t h i s becomes dN do X X n dN = I do y k=l y dN dx xy xy (3.68) -"k where t^ i s the t h i c k n e s s o f the k t ^ l a y e r , and n i s the t o t a l number of l a y e r s . When the layer incremental s t r e s s - s t r a i n r e l a t i o n s , Eq . (3.66) , are subs t i tu ted i n Eq . (3.68), the r e s u l t i n g incremental laminate f o r c e - s t r a i n r e l a t i o n s take the fo l lowing form: dN dN dN xy J n I k=l [ Q ' ] k \ de £ de' dr° xy (3.69a) or n {dN} = I [Q'] k=l k ^ { d e ' ° } (3.69b) The o v e r a l l response of the laminate w i l l be a f fec ted by each l a y e r ' s s t i f f n e s s c o n t r i b u t i o n [ Q 1 ] ^ . During e l a s t i c deformation, [ Q ' l ^ = [Q' 66 and d u r i n g e l a s t o p l a s t i c d e f o r m a t i o n L Q ' ] ^ = [ Q ' 6 ^ ] ^ ' I t s h o u l d be n o t e d t h a t t h e a n a l y s i s g i v e n above i s b a s e d on p l a n e s t r e s s c o n d i t i o n s i n i n d i v i d u a l l a y e r s . T h i s i s a v a l i d a s s u m p t i o n i f : a) The l o a d s on t h e l a m i n a t e a r e s t a t i c a l l y e q u i v a l e n t t o i n - p l a n e f o r c e s (membrane f o r c e s ) and p r o d u c e n e i t h e r b e n d i n g n o r t w i s t i n g moments. b) The l a m i n a t e has a c e r t a i n " s t a c k i n g s e q u e n c e " o f l a y e r s w h i c h d e f i n e s a s o - c a l l e d s y m m e t r i c l a m i n a t e . c) The " f r e e - e d g e e f f e c t s " a r e n e g l i g i b l e . The " s t a c k i n g s e q u e n c e " r e f e r r e d t o i n (b) i s an a r r a n g e m e n t i n w h i c h t h e l a m i n a t e has a m i d d l e p l a n e o f g e o m e t r i c a l and m a t e r i a l symmetry . The l a y e r s a r e a r r a n g e d i n p a i r s w i t h r e s p e c t t o t h e p l a n e o f symmetry . The l a y e r s o f s u c h p a i r have e q u a l t h i c k n e s s e s , same d i s t a n c e s f r o m m i d d l e p l a n e , and a r e o f t h e same m a t e r i a l w i t h i d e n t i c a l o r i e n t a t i o n o f m a t e r i a l a x e s , 8 . I n a n o n - s y m m e t r i c l a m i n a t e a p p l i c a t i o n o f membrane f o r c e s w i l l i n g e n e r a l p r o d u c e b e n d i n g and t w i s t i n g o f t h e l a y e r s and t h u s a p l a n e s t a t e o f s t r e s s w i l l n o t be r e a l i z e d . The " f r e e edge e f f e c t s " r e f e r r e d t o i n (c) a r e c o n f i n e d t o a " b o u n d a r y l a y e r " w h i c h e x t e n d s i n w a r d f rom t h e f r e e edge t o a d i s t a n c e a p p r o x i m a t e l y e q u a l t o t h e l a m i n a t e t h i c k n e s s . W i t h i n t h i s r e g i o n , t h e i n t e r l a m i n a r s t r e s s e s become p r o n o u n c e d and t h e l a m i n a t e i s i n a t h r e e - d i m e n s i o n a l s t a t e o f s t r e s s ( c f . P i p e s and P a g a n o , 1 9 7 0 ) . 67 CHAPTER 4 FINITE ELEMENT FORMULATION 4.1 Introduct ion I n r e c e n t y e a r s , t h e f i n i t e e l e m e n t method h a s emerged as t h e most p o w e r f u l g e n e r a l method o f s t r u c t u r a l a n a l y s i s and h a s p r o v i d e d e n g i n e e r s w i t h a t o o l o f v e r y w i d e a p p l i c a b i l i t y . I n d e e d , w i t h o u t t h i s c o m p u t a t i o n a l t o o l , i t i s v e r y d i f f i c u l t and e x p e n s i v e t o g a i n i n s i g h t i n t o t h e n o n l i n e a r b e h a v i o u r o f m a t e r i a l s i n g e n e r a l and l a m i n a t e d FRMs i n p a r t i c u l a r . The f o l l o w i n g d i s c u s s i o n i s m a i n l y c o n c e r n e d w i t h t h e n u m e r i c a l i m p l e -m e n t a t i o n o f t h e e l a s t i c - p l a s t i c - f a i l u r e mode l ( d e v e l o p e d i n t h e p r e v i o u s c h a p t e r ) i n t o a d i s p l a c e m e n t b a s e d f i n i t e e l e m e n t p r o g r a m COMPLY (COMpos i te P L a s t i c Y i e l d i n g ) . The c h a p t e r i s d i v i d e d i n t o t h r e e p a r t s . The f i r s t p a r t b r i e f l y d e s c r i b e s t h e b a s i c s t e p s o f t h e f i n i t e e l e m e n t a n a l y s i s . I n t h e s e c o n d p a r t t h e t e c h n i q u e a d o p t e d f o r s o l u t i o n o f n o n l i n e a r e q u i l i b r i u m e q u a -t i o n s i s o u t l i n e d . The t h i r d and f i n a l p a r t p r e s e n t s t h e n u m e r i c a l t e c h n i q u e i m p l e m e n t e d t o p e r f o r m t h e r e q u i r e d c o m p u t a t i o n s f o r t h e p r o p o s e d c o n s t i t u t i v e m o d e l . 4.2 Governing Equations of F i n i t e Element Analys i s The p r i n c i p l e s o f t h e f i n i t e e l e m e n t method a r e now w e l l e s t a b l i s h e d ( Z i e n k i e w i c z , 1971) and w i l l n o t be d i s c u s s e d h e r e i n g r e a t d e t a i l . I n t h i s s e c t i o n we summar i ze b r i e f l y t h e g e n e r a l t e c h n i q u e u s e d i n t h e f i n i t e e l e m e n t method f o r s o l v i n g p r o b l e m s w i t h m a t e r i a l n o n l i n e a r i t y . I n d e r i v i n g t h e b a s i c e q u a t i o n s o f t h e c u r r e n t f i n i t e e l e m e n t a n a l y s i s a two d i m e n s i o n a l i s o p a r a m e t r i c f o r m u l a t i o n i s u s e d t h r o u g h o u t . 68 Relat ions descr ib ing the approximate e q u i l i b r i u m equations w i l l be presented here. The o u t l i n e of the numerical scheme used to solve these genera l l y nonl inear equations w i l l be deferred to Sect ion 4.3 . 4 . 2 . 1 I s o p a r a m e t r i c e l e m e n t R e p r e s e n t a t i o n It i s genera l ly thought that the i soparametric numerica l ly integrated element i s more e f f i c i e n t and accurate i n e l a s t o - p l a s t i c ana lys i s than simple low order elements (Nayak and Zienkiewicz , 1972; Nagtegaal et a l . , 1974). For t h i s reason, the conventional 8-node (quadratic) i soparametric element i s implemented i n the f i n i t e element code COMPLY. Isoparametric elements are those for which the func t iona l representat ion of deformation i s employed i n representat ion of the element geometry. A t y p i c a l curved element with the nodal po ints being numbered for the purposes of d i scuss ions i s shown i n F i g . 4 .1 . Also shown i n the same f igure i s the parent element of square shape with coordiates E, r\, ranging from -1 to 1 on t h e i r boundaries . I f u and v denote the in -p lane displacements i n the x and y d i r e c t i o n s r e s p e c t i v e l y , then the v a r i a t i o n of displacement w i th in an element can be w r i t t e n as u = 8 8 v = Z d>.v. i - 1 1 1 (4.1) where u . , v . are the n o d a l displacement v a r i a b l e s for the i node, and <j>. i i i are the assoc iated shape functions given by • For corner nodes: = \ (1 + S E . K l + nn .MK. + n n . " D ( i = 1,3,5,7) 69 • F o r m i d s i d e n o d e s : = (1 + £ £ . ) ( ! - n») + (1 + nn.)(l - &) (i = 2,4,6,8) (4.2) i n w h i c h E^, a r e t n e n o d a l c o o r d i n a t e s i n t h e £-n p l a n e . By d e f i n i t i o n o f i s o p a r a m e t r i c e l e m e n t s t h e above shape f u n c t i o n s a r e a l s o u s e d t o map t h e e l e m e n t g e o m e t r y i n t h e x - y p l a n e , i . e . 8 8 x y = i $±y± • 1 = 1 (4.3) where x ^ , y ^ a r e t h e c o o r d i n a t e s o f t h e nodes i n t h e x - y p l a n e . 4.2.2 Element S t i f f n e s s Formulation The b a s i c s t e p i n any f i n i t e e l e m e n t a n a l y s i s i s t h e d e r i v a t i o n o f t h e n o d a l f o r c e - d i s p l a c e m e n t r e l a t i o n s h i p . To d e r i v e t h i s r e l a t i o n we must s a t i s f y t h r e e c o n d i t i o n s no m a t t e r what t y p e o r shape o f e l e m e n t i s i n v o l v e d . These a r e : i ) The c o m p a t i b i l i t y c o n d i t i o n ( i . e . s t r a i n - d i s p l a c e m e n t r e l a t i o n s ) i i ) The e q u i l i b r i u m c o n d i t i o n i i i ) The s t r e s s - s t r a i n r e l a t i o n s i ) S t r a i n - D i s p l a c e m e n t R e l a t i o n s e T L e t {6^} = {u^, v^} be t h e v e c t o r o f n o d a l d i s p l a c e m e n t v a r i a b l e s f o r t h e node o f a t y p i c a l e l e m e n t e i n t h e f i n i t e e l e m e n t mesh . The i n - p l a n e e T s t r a i n f i e l d w i t h i n t h i s e l e m e n t , {e'} = {e , e , r } , i s t h e n r e l a t e d t o ' x y ' x y t h e n o d a l p o i n t d i s p l a c e m e n t v e c t o r as 70 8 { e } e = I [ B . ] { f i . } 6 = [B] {o } e i = l 1 1 ( 4 . 4 ) where [B^] i s t h e s t r a i n - d i s p l a c e m e n t m a t r i x * f o r t h e i ^ node g i v e n b y [ B . ] S d ^ / S x S d ^ / S y a ^ / a x ( i = 1 , 2 8) ( 4 . 5 ) and [B] = [ [ B ^ . E B , ] [ B 8 ] ] {&} = { u 1 ( V J , u , , . . . , v 8 } The C a r t e s i a n shape f u n c t i o n d e r i v a t i v e s u s e d i n E q . ( 4 . 5 ) may be e x p r e s s e d i n t e r m s o f t h e E.n. d e r i v a t i v e s o f du b y t h e t r a n s f o r m a t i o n a d ^ / a x = [ J ] " 1 a^/ae a /^ay a ^ / a n ( 4 . 6 ) where [J ] i s t h e i n v e r s e o f t h e J a c o b i a n m a t r i x [J ] = ax/as ay/as' _ 3 x / 3 n a y / a n , . 8 3(j). ill ^ X i 8 34 . I ^ x . i - 1 3 n 1 8 3d>. 8 3d>. i i i " y * ( 4 . 7 ) * C o n s i d e r a t i o n h e r e i s l i m i t e d t o s m a l l d e f o r m a t i o n s i t u a t i o n s where t h e s t r a i n s c a n be assumed t o be i n f i n i t e s i m a l ( i . e . L a g r a n g i a n and E u l e r i a n g e o m e t r i c d e s c r i p t i o n s c o i n c i d e ) and t h e s t r a i n - d i s p l a c e m e n t m a t r i x r e m a i n s c o n s t a n t d u r i n g t h e d e f o r m a t i o n p r o c e s s . i i ) E q u i l i b r i u m C o n d i t i o n By a p p l y i n g t h e p r i n c i p l e o f v i r t u a l work o r any o t h e r s i m i l a r e n e r g y p r i n c i p l e , t h e e q u i l i b r i u m e q u a t i o n f o r a s i n g l e e l e m e n t c a n be w r i t t e n as J [B] {o'} e dV = {F} 6 , ( 4 . 8 ) e T e w h e r e f a ' } = {a , a , T } i s t h e e l e m e n t a l s t r e s s v e c t o r , {F} i s t h e x y x y c o n s i s t e n t ( e x t e r n a l ) l o a d v e c t o r and V \ i s t h e e l e m e n t a l v o l u m e . i i i ) C o n s t i t u t i v e R e l a t i o n s B e f o r e a s o l u t i o n c a n p r o c e e d f u r t h e r , a c o n s t i t u t i v e r e l a t i o n s h i p b e t w e e n {o 1 } o f E q . ( 4 . 8 ) and {e 1 } o f E q . ( 4 . 4 ) h a s t o be e s t a b l i s h e d . F o r t h e p r e s e n t p l a n e s t r e s s a n a l y s i s o f l a m i n a t e d F R M s , t h e m a t e r i a l n o n l i n e a r i t y i s i n c o r p o r a t e d i n t o t h e s t r e s s - s t r a i n r e l a t i o n i n t h e f o l l o w i n g i n c r e m e n t a l f o r m { d o ' } * = [ Q ' ] J l d e ' } J = [ Q ' ] J { d e " } " ( 4 . 9 ) w h e r e [ Q 1 ] ^ i s t h e c o n s t i t u t i v e m a t r i x ( f o r t h e c u r r e n t s t r e s s l e v e l ) a s s o c i a t e d w i t h t h e k ^ l a y e r o f t h e l a m i n a t e ( see E q . ( 3 . 6 6 ) ) . S u b s t i t u t i n g E q . ( 4 . 9 ) i n t h e i n c r e m e n t a l f o r m o f E q . ( 4 . 8 ) and u s i n g E q s . ( 4 . 4 ) and ( 3 . 6 9 b ) g i v e s f o r e a c h e l e m e n t {dF} 6 = J [ B ] T {dN}edA = [K^l {do} 6 ( 4 . 10 ) A 8 72 where n [ l y 6 = I t k / [ B ] T [ Q ' ] 6 [B] dx dy (4.11) k=l ~ .e A i s the element t a n g e n t i a l s t i f f n e s s m a t r i x . In Eq . (4.11) A e denotes the surface area of the element under cons iderat ion and the summation i s taken over the t o t a l number of l a y e r s , n , through the th ickness . Due to the nonl inear nature of the integrand i n Eq . (4.11), the i n t e g r a t i o n must be c a r r i e d out numer ica l ly . This w i l l be attempted i n the fo l lowing . Using the transformation r e l a t i o n dxdy = det[J] d£dn (4.12) where det[J] i s the determinant of the Jacobian matrix [J] , we can rewrite Eq . (4.11) as n 1 1 n u 6 = i t, ; ; im? dsdn (4 .13) 1 k=l K -1 -1 K where [H]^ i s defined as [ H ] 6 = [HtE.n)]6. = [ B ] T [ Q ' ] k [B] det [J] (4.14) Now, us ing Gaussian quadrature with q x q sampling points the element tangent s t i f f n e s s matrix can be evaluated as n q q [ K p ] 8 = I X I [ H C ^ . n . n J t k W i W (4.15) k=l j= l i = l -1 J 73 w h e r e ( E - . n . ) i s a s a m p l i n g p o s i t i o n w i t h i n t h e e l e m e n t and W.,W. a r e t h e a s s o c i a t e d w e i g h t i n g f a c t o r s . W i t h r e g a r d t o one d i m e n s i o n a l i n t e g r a l s , q Gauss e v a l u a t i o n p o i n t s a r e s u f f i c i e n t t o i n t e g r a t e e x a c t l y a p o l y n o m i a l o f d e g r e e ( 2 q - l ) . T a b l e 4 . 1 o f f e r s a b r i e f summary o f some o f t h e d e t a i l s o f G a u s s i a n q u a d r a t u r e . T a b l e 4 . 1 S a m p l i n g C o o r d i n a t e s and W e i g h t i n g F a c t o r s f o r O n e - D i m e n s i o n a l G a u s s i a n Q u a d r a t u r e r + l q r H(E) di = X W. H (1.) -1 1=1 1 1 q i W i 1 ( l i n e a r ) 1 0 2 2 ( c u b i c ) 1 + i / n 1 2 - 1 /V3 1 3 ( q u i n t i c ) 1 0 8 / 9 2 + V 3 / 5 5 / 9 3 - V 3 / 5 5 / 9 4.2.3 S t r u c t u r a l S t i f f n e s s Formulation O n c e t h e e l e m e n t s t i f f n e s s m a t r i c e s [Kj,] have been c a l c u l a t e d i n t h e g l o b a l ( s t r u c t u r a l o r l a m i n a t e ) c o o r d i n a t e s , t h e s t r u c t u r a l t a n g e n t s t i f f -n e s s m a t r i x [ K ^ ] , w h i c h r e l a t e s t h e l o a d i n c r e m e n t {dF} t o t h e n o d a l d i s p l a c e m e n t i n c r e m e n t {d&} o f t h e c o m p l e t e s t r u c t u r e , c a n be fo rmed b y t h e s y s t e m a t i c a d d i t i o n o f e l e m e n t s t i f f n e s s e s . Thus {dF} = [K j ] {do} (4 .16 ) 74 4 . 3 Numerical So lu t ion of Nonlinear E q u i l i b r i u m Equations A large number of numerical schemes have been proposed i n the l i t e r a t u r e for the i n t e g r a t i o n of equations (4.16) , a l l of which are based on piecewise l i n e a r i z a t i o n of the nonl inear equations over a f i n i t e number of steps. The two main categories of methods are pure ly i t e r a t i v e or pure ly incremental . In the incremental (step by step) method the s t ruc ture i s loaded i n small increments and for every step of loading a new s t r u c t u r a l s t i f f n e s s matrix i s ca l cu la ted from the updated mater ia l ( cons t i tu t ive ) matr ices . However, s ince the flow theory of p l a s t i c i t y i s based on " d i f f e r e n t i a l " steps and the load must be added i n " f i n i t e " steps , the s t i f f n e s s of the s t ruc ture i s e a s i l y overestimated and the e q u i l i b r i u m condit ions are v i o l a t e d . In t h i s case the incremental procedure must be combined with a s u i t a b l e i t e r a t i v e process i n order to s a t i s f y e q u i l i b r i u m at each load step. During the general stage of the i n c r e m e n t a l / i t e r a t i v e s o l u t i o n process , the e q u i l i b r i u m equations (4.16) w i l l not be exac t ly s a t i s f i e d and a system of r e s i d u a l (unbalanced) forces (\|)} w i l l e x i s t such that i n which the subscr ip t r s i g n i f i e s the i t e r a t i o n cyc le number w i th in a p a r t i c u l a r load increment and {P} = (P(o)} i s the i n t e r n a l equivalent force vector given by = {F} - {P(fl)> r r (4.17) (4.18) where the in tegra t ions are c a r r i e d out over the whole s t ruc ture using the usual element-by-element procedure and standard assembly r u l e s . Note that i n Eqs. (4.17) and (4.18) the t o t a l (accumulated) l e v e l as opposed to 75 incremental l e v e l of various force and s tress quant i t i e s are considered. A l so due to the n o n l i n e a r i t y of the problem the in tegrat ions i n E q . (4.18) must be c a r r i e d out numer ica l ly by the Gaussian quadrature scheme described e a r l i e r . L e t us now c o n s i d e r the T a y l o r s e r i e s expansion of {\|>}r about W r _ i ( i . e . the known s o l u t i o n from previous i t e r a t i o n ) so that » > r " » ' r - l + US} {6} , r -1 UA} - {5} ,) + (4.19) r r-1 Putt ing {A6} = {6} r - {6} r_ 1 (4.20) and then truncat ing the expansion to the l i n e a r term y i e l d s W r = WT_i - [ K T ] r . 1 {A6} (4.21) where r K _ i = M i l T r - l 3(6} 1 6 } 3(6} r -1 {6} r -1 (4.22) i s the t a n g e n t i a l s t i f f n e s s matrix of the e n t i r e s t ruc ture evaluated at the b e g i n n i n g o f the r^"*1 i t e r a t i o n . T h i s i s c a l c u l a t e d using the element by element assembly of s t i f fnes se s given by Eqs. (4.11) or (4 .15) . I t i s des i red that = 0, thus [ K T ] r _ 1 {A6} = {^}r_1 (4.23) 76 T h i s i s t h e l i n e a r i n c r e m e n t a l e q u i l i b r i u m e q u a t i o n w h e r e i n t h e r e s i d u a l s {\|>} c a n be v i s u a l i z e d as c o r r e c t i v e n o d a l f o r c e s r e q u i r e d t o b r i n g t h e assumed d i s p l a c e m e n t p a t t e r n i n t o n o d a l e q u i l i b r i u m . The s e t o f s i m u l t a n e o u s e q u a t i o n s ( 4 .23 ) c a n be s o l v e d * f o r {AS} and t h e i m p r o v e d s o l u t i o n w i l l be W r = ( 6 } r _ 1 + {Afl} ( 4 . 24 ) The u p d a t e d d i s p l a c e m e n t s {6}^ a r e u s e d t o e v a l u a t e t h e c u r r e n t s t r e s s e s {o}^ a n d h e n c e t h e r e s i d u a l f o r c e s {\j)) r f r o m E q s . ( 4 . 1 7 ) . The i t e r a t i o n p r o c e s s i s r e p e a t e d u n t i l t h e s e r e s i d u a l f o r c e s p r a c t i c a l l y v a n i s h , o r e q u i v a l e n t l y t h a t { 6 } r _ p a n < 3 ^ r a r e e f f i c i e n t l y c l o s e . F o r t h e p u r p o s e s o f t h i s s t u d y we w i l l assume t h a t t h e n u m e r i c a l p r o c e s s has c o n v e r g e d i f (Owen and H i n t o n , 1980) \ \ W r ii wl W„ V- = [ 1 - ] <: TOLER ( 4 . 2 5 ) I U F J I I L { F } T ( F } where TOLER i s a s m a l l p r e s e t t o l e r a n c e l i m i t . The c r i t e r i o n ( 4 .25 ) s t a t e s t h a t c o n v e r g e n c e o c c u r s i f t h e E u c l i d i a n norm o f t h e r e s i d u a l f o r c e s becomes l e s s t h a n TOLER t i m e s t h e norm o f t h e t o t a l a p p l i e d f o r c e s . The above i t e r a t i v e p r o c e d u r e , o f t e n t e r m e d t h e N e w t o n - R a p h s o n m e t h o d , i s i l l u s t r a t e d s c h e m a t i c a l l y i n F i g . 4 . 2 f o r a s i n g l e v a r i a b l e s i t u a t i o n . I t s h o u l d be n o t e d t h a t i n t h i s s c h e m e t h e t a n g e n t s t i f f n e s s m a t r i x [K^,] i s u p d a t e d and s o l u t i o n o f t h e f u l l e q u a t i o n s y s t e m i s o b t a i n e d f o r e a c h i t e r a -t i o n . A v a r i a n t on t h e above a l g o r i t h m i s o f f e r e d b y t h e " i n i t i a l s t i f f n e s s " scheme i n w h i c h t h e o r i g i n a l s t r u c t u r a l s t i f f n e s s m a t r i x i s emp loyed a t e a c h * I n t h e c u r r e n t compute r code COMPLY t h e s o l u t i o n method i s b a s e d on t h e G a u s s i a n e l i m i n a t i o n p r o c e d u r e . 77 stage of the i t e r a t i o n process . This reduces the computational costs per i t e r a t i o n but unfortunate ly a l so reduces the rate of convergence of the process . In p r a c t i c e the optimum algori thm i s genera l ly provided by updating the s t i f fnes se s at se lec ted i t e r a t i v e i n t e r v a l s only (Owen and Hinton, 1980). In the present work the pure Newton-Raphson method i s employed whereby the tangent s t i f f n e s s matrix i s re-evaluated at each i t e r a t i o n of each load increment. The complete sequence of the i t e r a t i v e process described above i s summarized i n Table 4 .2 . Table 4.2 Sequence of the I t e r a t i v e So lu t ion Technique 1. Begin new load increments, {F} = {F} + {AF}. Set i t e r a t i o n counter r = 1 and {\|))r_^ = {AF} + {\J>} , where {\|>} i s the e q u i l i b r i u m c o r r e c t i o n from previous increment. 2. Evaluate the new tangent ia l s t i f f n e s s matrix [Kj.] r_-^. 3. Solve tKj ] r _2 {A6} = W r _ i by Gauss e l i m i n a t i o n . 4. Set {6} r = {6} r_ x + {A6}. 5. For each Gauss point c a l c u l a t e the increment i n s t r a i n { A e ' ° } = [BHAfi}. 6. Estimate the increments i n s tress at each Gauss point i n each layer {Ao1}^ = [Q']k{Ae' 0 } and hence evaluate the t o t a l s tress value {o'}k = {o'jk.-L + {Ao'} k . The stresses { o ' } £ must be adjusted to account for any p l a s t i c behaviour or f a i l u r e of the layer (see Sect ion 4 .4) . 7. n k Evaluate the s tress resu l tants {N}_ = X [a'}t: t v . at each Gauss r k=l r k p o i n t . 8. Determine the r e s i d u a l force vector W r = {F} - J" [B]T{N} dA. A 9. I f convergence i s achieved according to c r i t e r i o n (4.25), set {\J)} = {\|)}rand go to step (1) to perform the next load increment. Otherwise set r = r+1 and go to step (2) to perform the next i t e r a t i o n . 78 4 . 4 Numerical Implementation of the An i so trop ic E l a s t i c - P l a s t i c - F a i l u r e Model As was observed i n the previous s e c t i o n , the f i n i t e element scheme solves the displacement equations of e q u i l i b r i u m i n an incremental fashion . Hence, the c o n s t i t u t i v e laws presented e a r l i e r (Chapter 3) that dealt with d i f f e r e n t i a l ( i n f i n i t e s i m a l ) s tress and s t r a i n quant i t i e s must now be used approximately to r e l a t e small but f i n i t e increments i n s tresses and s t r a i n s . A f i n i t e element model may cons is t of several sampling points ( layers and Gauss points) at which the p l a s t i c i t y computations are performed for every load step and c o r r e c t i v e i t e r a t i o n . Analyses of such models must permit r e l a t i v e l y large load steps to maintain e f f i c i e n c y of the s o l u t i o n . These large increments place severe demands on the p l a s t i c i t y rout ines to maintain accuracy, numerical s t a b i l i t y and e f f i c i e n c y . The importance of the p r e c i s i o n with which c o n s t i t u t i v e e l a s t o p l a s t i c r e l a t i o n s are in tegrated has motivated a number of recent s tud ies , e .g . (Krieg and K r i e g , 1977; Schreyer et a l . , 1979; O r t i z and Popov, 1985; Franchi and Genna, 1987; Dodds, 1987). The genera l ized t r a p e z o i d a l r u l e as described by O r t i z and Popov (1985) encompasses the three most popular methods for i n t e g r a t i o n of the e l a s t o -p l a s t i c c o n s t i t u t i v e equations. These methods are known as: (1) tangent p r e d i c t o r - r a d i a l c o r r e c t o r ; (2) mean normal or secant s t i f f n e s s ; and (3) e l a s t i c p r e d i c t o r - r a d i a l r e t u r n . In the present work the f i r s t method i s employed to in tegrate the incremental s t r e s s - s t r a i n law. As shown by Schreyer et a l . (1979) t h i s method, i f used with subincrementation of the s t r a i n increment vector (as i n the present a n a l y s i s ) , i s very accurate for plane s tress cond i t ions . I t i s the aim of t h i s sec t ion to ou t l ine the s tress computation technique used i n the current f i n i t e element program COMPLY. The 79 n u m e r i c a l p r o c e d u r e d e s c r i b i n g t h e f a i l u r e and p o s t - f a i l u r e s t r e s s a n a l y s e s w i l l a l s o be c o v e r e d . A . A . I E l a s t i c - P l a s t i c Formulation D u r i n g e a c h i t e r a t i o n o f e a c h l o a d i n c r e m e n t an e l e m e n t , o r p a r t o f an e l e m e n t , may y i e l d . A l l s t r e s s and s t r a i n q u a n t i t i e s a r e m o n i t o r e d a t e a c h G a u s s i a n i n t e g r a t i o n p o i n t i n e v e r y l a y e r o f e v e r y e l e m e n t . C o n s e q u e n t l y an e l e m e n t c a n behave e l a s t i c a l l y a t some p o i n t s and e l a s t i c - p l a s t i c a l l y a t o t h e r s . F o r e v e r y i t e r a t i o n o f a g i v e n l o a d i n c r e m e n t i t i s n e c e s s a r y t o a d j u s t t h e s t r e s s and s t r a i n t e r m s u n t i l t h e y i e l d c r i t e r i o n and t h e c o n s t i t u t i v e l a w s a r e s a t i s f i e d . The p r o c e d u r e a d o p t e d i s d e s c r i b e d b e l o w . C o n s i d e r t h e s i t u a t i o n e x i s t i n g ( a t a p o i n t * ) f o r t h e r ^ i t e r a t i o n of a n y p a r t i c u l a r l o a d i n c r e m e n t . The s t r e s s components { ° } r _ ^ a n < i t n e p a r a -m e t e r s d e s c r i b i n g t h e y i e l d s u r f a c e a r e a l l known f r o m t h e s o l u t i o n a t t h e e n d o f t h e ( r - 1 ) 1 " * 1 i t e r a t i o n . A l s o k n o w n a r e t h e c o m p o n e n t s o f t h e new s t r a i n i n c r e m e n t s {Ae ' ° } . The l a t t e r , w h i c h i s g i v e n i n t h e o v e r a l l ( l a m i n a t e ) c o o r d i n a t e s y s t e m , must a p p r o p r i a t e l y be t r a n s f o r m e d ( u s i n g E q s . ( 3 .62 ) and ( 3 . 6 3 ) ) t o g i v e t h e i n c r e m e n t a l s t r a i n components i n t h e p r i n c i p a l m a t e r i a l d i r e c t i o n s . Such t r a n s f o r m a t i o n i s e s s e n t i a l s i n c e t h e y i e l d and f a i l u r e c r i t e r i a d e s c r i b e d e a r l i e r ( C h a p t e r 3) i n v o l v e t h e s t r e s s and s t r a i n c o m p o n e n t s , w h i c h a r e r e f e r r e d t o t h e m a t e r i a l c o o r d i n a t e s o f a p a r t i c u l a r l a y e r . F o r t h e p u r p o s e s o f t h e f o l l o w i n g d i s c u s s i o n we assume t h a t a l l t h e s t r e s s and s t r a i n q u a n t i t i e s have a l r e a d y b e e n t r a n s f o r m e d t o t h e m a t e r i a l * I t i s u n d e r s t o o d t h a t t h e s t r e s s c o m p u t a t i o n i s p e r f o r m e d f o r e v e r y p o i n t ( i . e . f o r e a c h Gauss s t a t i o n i n e v e r y l a y e r ) i n v o l v e d i n n u m e r i c a l i n t e g r a -t i o n o v e r t h e v o l u m e . Thus s u f f i x e s e and k p r e v i o u s l y u s e d t o d e n o t e g e n e r i c e l e m e n t s and l a y e r s , r e s p e c t i v e l y , w i l l h e n c e f o r t h be s u p p r e s s e d f o r c l a r i t y . 80 c o o r d i n a t e s o f t h e p o i n t u n d e r c o n s i d e r a t i o n . T h u s , i n what f o l l o w s , T T {Ae} = { A e ^ A e 2 , A e 6 } ; {Ao} = {Lo1, A o 2 , A o 6 } . I n t h e f i r s t s t e p o f t h e n u m e r i c a l a l g o r i t h m , an e l a s t i c e s t i m a t e {Ao } f o r t h e s t r e s s i n c r e m e n t i s computed as {Ao 6 } = [ Q 6 ] { A e ] ( 4 .26 ) Q A s e t o f e l a s t i c t r i a l s t r e s s e s {o }^ i s t h e n c a l c u l a t e d b y a c c u m u l a t i n g t h e t o t a l s t r e s s . The r e s u l t i s { o 6 ) r = { o ) r _ 1 + {Ao 6 } ( 4 . 2 7 ) These t r i a l s t r e s s e s a r e t h e n t e s t e d w i t h r e s p e c t t o t h e i n i t i a l y i e l d s u r f a c e f 0 ( { o } , [ A ° ] , k 0 ) = a' ({a},[A"]) - k* = 0 ( 4 . 2 8 ) where t h e m a t r i x [ A 0 ] i s g i v e n b y E q s . ( 3 . 5 5 a , b ) o r ( 3 . 5 8 ) ( i n w h i c h t h e y i e l d s t r e s s e s , X , Y , S and k a r e r e p l a c e d b y t h e i r i n i t i a l v a l u e s X 0 , Y 0 , S 0 and k 0 ) d e p e n d i n g on w h e t h e r t h e l a y e r i s b i d i r e c t i o n a l o r u n i d i r e c t i o n a l , r e s p e c t i v e l y . I f t h e t r i a l s t r e s s e s do n o t v i o l a t e t h e y i e l d c r i t e r i o n ( 4 . 2 8 ) , i . e . f 0 ( { o } , [ A 0 ] , k 0 ) £ 0 , t h e n t h e e l a s t i c b e h a v i o u r a s s u m p t i o n h o l d s and t h e t h e f i n a l s t r e s s e s {a} a t t h e end o f t h e r i t e r a t i o n a r e i n d e e d {a } . O t h e r -r r w i s e , t h e i n i t i a l y i e l d s u r f a c e has b e e n c r o s s e d d u r i n g t h e t r i a l s t r e s s i n c r e m e n t a t i o n . T h i s i s s h o w n s c h e m a t i c a l l y i n F i g . 4 . 3 . I f {o } d e n o t e s 81 the s tress s tate at the point where the assumed s tress path comes in to contact with the i n i t i a l y i e l d surface , then we can wri te {oC} = {o ) r _ 1 + B{Ao®} ; 0 £ B < 1 (A.29) where B{Ao } i s the p o r t i o n o f the s t r e s s increment at which the p l a s t i c b e h a v i o u r i s f i r s t encountered, i . e . f 0 ({a } , [ A ° ] , k 0 ) = 0. This condi t ion leads to a quadrat ic equation for the determination of B. However, a simple approx imate v a l u e o f B can be obtained by a l i n e a r i n t e r p o l a t i o n i n o (Owen and Hinton , 1980), that i s , (A.30) a - a , r r -1 where o® = (oe}^  [ A 0 ] { o e ) r , and a ^ = {a}J_ 1[A 0]{o} r_ 1. I t should be o b s e r v e d t h a t t h e p a t h from { ° } r _ ^ t o 1 ° } c o n s t i t u t e s f u l l y e l a s t i c r e s p o n s e . The r e m a i n i n g p o r t i o n of s t r e s s , (1-B) {Ao } r e s u l t s i n a s tress s tate that l i e s outs ide the i n i t i a l y i e l d surface and consequently must be adjusted by al lowing p l a s t i c deformation to occur. Once B has been deter -mined from E q . (A.30) , the p l a s t i c s tress increment can be c a l c u l a t e d as {Ae} {AoP} = J [QP]{de} (A.31) B{Ae} This p l a s t i c s tress increment i s required to res tore the assumed e l a s t i c s tress increment 82 {Ae} {Aoe} = J [Q e]{de} = [Qe]{Ae} (A.32) 0 to the correc t e l a s t o p l a s t i c values as requ ired by the c o n s t i t u t i v e equation (3.50). Therefore , s u b s t i t u t i n g incremental changes for i n f i n i t e s i m a l s i n E q . (3.50), we have {Ae} B{Ae} {Ae} {Ao} = J ([Q e ] - [QP]){de} = J [Q e]{de} + J ( [Q e ] - [QP]){de} 0 0 B{Ae} {Ae} {Ae} = J [ Q e ] { d e } - J [QP]{de} = {Ao6} - {Ac?} (A.33) 0 B{Ae} The cons truc t ion of the e l a s t o p l a s t i c s tress increment vector {Ao} i s i l l u s t r a t e d s c h e m a t i c a l l y i n F i g . A . 3 . I f {AeP} denotes the v e c t o r o f p l a s t i c s t r a i n increments, which i s unknown at t h i s stage, then i n view of the r e l a t i o n {Ao} = [Q e]({Ae} - {Aep}) (A.3A) and Eq . (A.33) , one can wr i te {Ao15} = [Q e]{Ae p} (A.35) S i n c e the p l a s t i c mater ia l s t i f f n e s s [QP] v a r i e s with the current state of s t r e s s , the computation of the p l a s t i c s tress increment given by Eq. (A.31) requires a numerical i n t e g r a t i o n . Various algorithms have been 83 designed for t h i s purpose. The simplest approximation of Eq . (4.31) i s obtained by using a one-step, forward Euler i n t e g r a t i o n method: {AoP} = (1-B)[Q p]{Ae} (4.36) The i n t e g r a t i o n process used above i s admissible i f small load (and s tra in ) increments are a p p l i e d . The fact that the d i r e c t i o n of the p l a s t i c flow i s only correc t i n the beginning of the increment can lead to a s i g n i f i c a n t error i n the f i n a l o r i e n t a t i o n of the s tress vector i n the s tress space. Therefore to al low for r e l a t i v e l y large load (and s t r a i n ) increments a more accurate i n t e g r a t i o n procedure i s d e s i r a b l e . This can be achieved by d i v i d i n g the e l a s t o - p l a s t i c p o r t i o n of the s t r a i n increment v e c t o r , (1-8){Ae} i n t o M e q u a l s u b - i n c r e m e n t s and r e f o r m i n g the p l a s t i c matrix [C^3] at the beginning of each subincrement. Accord ing ly , for each sub in terva l we have {Ae} = {Ae}/M (4.37) m and {Ao e } m = ( l -B) [Q e ] {Ae} m = {Ao e}/M m m {Ao15} = (1-B) [Q p] {Ae} = {Ao^/M m m (4.38) where {Ae} , {Ao6} and {Ao^ 1} are the m*"*1 s t r a i n , e l a s t i c s tress and p l a s t i c m m m * s tress subincrements, r e s p e c t i v e l y . The input quant i t i e s used i n the c a l c u -l a t i o n of [Q P ] (see Eq . 3.50) are the accumulated ( tota l ) s tress components {a} , , the e f f e c t i v e y i e l d s tress k , ; the a n i s o t r o p i c parameters [A(k)] , m-1 m-1 m-1 and u , , a l l of which are evaluated at the end of the ( m - l ) ^ 1 subincrement. m-1 84 It should be noted that for ra = 1 ( i . e . at the onset of workhardening) , we have {o}0 = {oC} [ A ( k ) ] 0 = [A"] (4.39) , 1 , c , T r 3 A , | f c, " 1 - - 2 * 7 { o 1 [ 3 k ] | k = k 0 { ° } The y i e l d s tress k Is updated according to the amount of p l a s t i c work p r o d u c e d d u r i n g the m t ^ subincreraent . For the b i l i n e a r s t r e s s - s t r a i n representat ion considered i n t h i s study one can wri te k = k , + H* AW p /k , (4.40) m m-1 m-1 where AWP = {o}T , {AeP} (4.41) m-i m i s the increment of p l a s t i c work done per u n i t volume. In Eq . (4.41) the v e c t o r o f p l a s t i c s t r a i n increments {Aep} can be determined from Eq . (4.35) m as fol lows {Ae p} r a = [Q e] {Ao p ) m (4.42) In order to update the a n i s o t r o p i c parameters [A(k)] we adopt the approach of Jensen et a l . (1966) or Whang (1969), the d e t a i l s of which are 85 o u t l i n e d i n A p p e n d i x B. A c c o r d i n g t o t h i s method t h e new l e v e l s o f y i e l d s t r e s s X , Y and S r e a c h e d d u r i n g p l a s t i c f l o w a r e (see E q . 3 . 5 9 ) : E 2 P i 2 2 2 m = [ k m " ^ + X o E 2 P2 2 2 2 Y m = [ k m " ^ + Y o n 2 n 2 2 2 Sm = IT [ k m " ^ + S 0 I t s h o u l d be e m p h a s i z e d t h a t any one o f t h e s t r e s s - s t r a i n c u r v e s , O j - e l t O j - e , a n d o 6 - e 6 c a n b e p r e s c r i b e d a s t h e e f f e c t i v e s t r e s s (a) - e f f e c t i v e s t r a i n (e) d i a g r a m , i . e . t h e a n a l y s i s i s i n d e p e n d e n t o f t h e c h o i c e made. H a v i n g e s t a b l i s h e d t h e u p d a t e d v a l u e s o f t h e y i e l d s t r e s s e s , t h e m a t r i x o f a n i s o t r o p i c p a r a m e t e r s [ A ] ^ a t t h e e n d o f t h e m ^ s u b i n c r e m e n t c a n be f o u n d f r o m E q s . ( 3 . 5 5 a , b ) o r ( 3 . 5 8 ) , as t h e c a s e may b e . * The u p d a t e d y i e l d f u n c t i o n now becomes f ( { o L .CAl .k ) = {o}^ [A] {0} - k ' ( 4 . 44 ) m m m m m m m m I n g e n e r a l , t h e s t r e s s s t a t e ( o } m w i l l be o u t s i d e t h e u p d a t e d y i e l d s u r f a c e a n d we e x p e c t f ^ 0 . T h i s s m a l l d e p a r t u r e f rom t h e y i e l d s u r f a c e w i l l be c u m u l a t i v e . To p r e v e n t a r t i f i c i a l h a r d e n i n g , a c o r r e c t i o n must be made t o r e s t o r e t h e s t r e s s e s t o t h e c o r r e c t y i e l d s u r f a c e . Such a c o r r e c t i o n i s *The u p d a t e d a n i s o t r o p i c p a r a m e t e r s must a l w a y s be t e s t e d w i t h r e s p e c t t o t h e i n e q u a l i t i e s ( 3 . 4 9 ) t o c h e c k f o r t h e boundedness o f t h e new y i e l d s u r f a c e . 86 a c h i e v e d b y s i m p l y s c a l i n g t h e s t r e s s e s [a] t o t h e y i e l d s u r f a c e . The m a p p r o p r i a t e s c a l i n g f a c t o r i s r e a d i l y s e e n t o be s = k / V { o r [A] {o} m m m m (A .45 ) The above p r o c e s s i s r e p e a t e d f o r a l l s u b i n c r e m e n t s l e a d i n g t o t h e f o l l o w i n g e x p r e s s i o n f o r t h e s t r e s s e s a t t h e end o f t h e r*"*1 i t e r a t i o n M (°)r = MT.X + P ^ o - 6 } + I d A o e } m - {AoP} ) m=l ( 4 . 4 6 ) w h e r e {Ao } , { A o * j a r e g i v e n b y E q . ( 4 . 3 8 ) i n t e r m s o f t h e known s t r a i n m m e J i n c r e m e n t v e c t o r {Ae}. O b v i o u s l y t h e g r e a t e r t h e number o f s t e p s M i n t o w h i c h t h e p l a s t i c p o r t i o n o f s t r a i n i n c r e m e n t ( 1 -B ) {Ae } i s d i v i d e d , t h e g r e a t e r t h e a c c u r a c y . H o w e v e r , t h e e x p e n s e o f r e f o r m i n g [Q P ] and u p d a t i n g t h e s t r e s s e s f o r many s t e p s , may l e a d t o e x c e s s i v e c o m p u t a t i o n t i m e s . C l e a r l y a b a l a n c e must be s o u g h t and t o t h i s end s e v e r a l c r i t e r i a have b e e n p r o p o s e d t o s e l e c t an opt imum number o f s u b i n c r e m e n t s M. S c h r e y e r e t a l . (1979) s e l e c t M b y l i m i t i n g t h e a n g u l a r d i f f e r e n c e b e t w e e n a n o r m a l v e c t o r { 3 f / 3 o } 0 a t t h e c u r r e n t c o n t a c t p o i n t {o } and a n o r m a l v e c t o r { 3 f / 3 o } 1 a t t h e c o n t a c t p o i n t computed w i t h a s i n g l e s t e p ( i . e . , M=l) e s t i m a t e . T h i s a n g l e w h i c h m e a s u r e s t h e change i n p l a s t i c f l o w d i r e c t i o n (due t o c u r v a t u r e o f t h e y i e l d s u r f a c e ) w i t h i n an i n c r e m e n t , i s c a l c u l a t e d as uj = cos l 3 o 0 1 ' 3 o J ( 4 . 4 7 ) where I I i n d i c a t e s t h e m a g n i t u d e o f a v e c t o r . 87 Schreyer et a l . (1979) of fered the fo l lowing simple formula for the number of subincrements: M = 1 + u / i (4.48) where u i s given i n degrees and 2 i s a post ive number chosen on the bas is of numerical experience. A l t e r n a t i v e c r i t e r i a for determination of M are given by Bushnel l (1976), Nyssen (1981), and Owen and Hinton (1980). However, that of Schreyer et a l i s pre ferred i n the present study. I t should be remarked i n passing that for i n t e g r a t i o n points that have a l r e a d y y i e l d e d i n the previous i t e r a t i o n { ° } r _ ^ = (o }, i . e . 8 = 0 and the s tress computation procedure described above appl ies i d e n t i c a l l y . Also during the i t e r a t i v e process , i f the e f f e c t i v e s tress at a Gauss point f a l l s below the y i e l d value at the end of the previous load increment, that point i s assumed to be e l a s t i c a l l y unloading, i . e . {AoP} = 0. This concludes the numerical implementation of the e l a s t i c - p l a s t i c region of the c o n s t i t u t i v e model. The next subsect ion i s devoted to the numerical treatment of f a i l u r e and p o s t - f a i l u r e behaviour. 4.4.2 P o s t - F a i l u r e Formulation In each constant s t r a i n increment {Ae}, the updated s tate of s tress ( ° } r at an i n t e g r a t i o n po int of a f i n i t e element needs to be examined with respect to the f a i l u r e c r i t e r i o n f u ( { o } , [ A u ] , k u ) = {o}T [AU] {o} " k^ = 0 (4.49) 88 w h e r e t h e m a t r i x [ A U ] i s g i v e n b y E q s . ( 3 . 5 5 a , b ) and ( 3 . 5 8 ) f o r B/D and U/D l a y e r s , r e s p e c t i v e l y , w i t h X ^ , Y , S u and k^ r e p l a c i n g X . Y . S and k . I f t h e s t r e s s c o m b i n a t i o n v i o l a t e s t h e f a i l u r e c r i t e r i o n , t h e n c r a c k s a r e d e f i n e d i n t h e r e g i o n s o f t h e i n t e g r a t i o n p o i n t u n d e r c o n s i d e r a t i o n . To a c c o u n t f o r t h e p r e s e n c e o f c r a c k s i n s u c c e e d i n g i t e r a t i o n s ( o r i n c r e m e n t s o f l o a d i n g ) , t h e p o s t - f a i l u r e e l a s t i c i t y m a t r i x [Q^] t a b u l a t e d i n T a b l e 3 . 3 i s u s e d so t h a t a t t h a t i n t e g r a t i o n p o i n t t h e e l e m e n t c a n n o t c a r r y any more i n c r e m e n t s o f s t r e s s i n c e r t a i n d i r e c t i o n s . T h i s i s a common f e a t u r e f o r b o t h b r i t t l e and d u c t i l e f r a c t u r e m o d e l s as d e s c r i b e d i n C h a p t e r 3 . T h e r e f o r e , once f a i l u r e o c c u r s , t h e s u b s e q u e n t i n c r e m e n t a l s t r e s s - s t r a i n r e l a t i o n s h i p c a n be w r i t t e n as {Ao} = [ Q f ] {Ae} ( 4 . 5 0 a ) i f t h e b e h a v i o u r i s e l a s t i c , o r {Ao} = ( [ Q f ] - [ Q f ] C a } { a } ^ [ Q f ] ) { A e } ( A > 5 Q b ) ufl' + { a r [ Q r ] { a } i f t h e b e h a v i o u r i s p l a s t i c . I f t h e f a i l u r e i s o f b r i t t l e t y p e , h o w e v e r , t h e a p p r o p r i a t e s t r e s s e s ( n o r m a l o r p a r a l l e l t o t h e c r a c k , as t h e c a s e may be) a t t h e i n t e g r a t i o n p o i n t j u s t b e f o r e f a i l u r e a r e r e l e a s e d c o m p l e t e l y * and t h e r e a f t e r t h e p o i n t i s assumed t o l o s e i t s r e s i s t a n c e a g a i n s t any f u r t h e r d e f o r m a t i o n i n t h e I n most s i t u a t i o n s where f a i l u r e o c c u r s i n t h e p r i m a r y l o a d - c a r r y i n g p a r t o f t h e s t r u c t u r e , t h e s t r e s s e s must be r e l e a s e d g r a d u a l l y t o f a c i l i t a t e c o n v e r g e n c e o f t h e i t e r a t i v e p r o c e s s . 89 f a i l u r e d i r e c t i o n . I n t h i s c a s e , t h e n o d a l p o i n t r e l e a s e d - f o r c e v e c t o r i n t h e s t r u c t u r e ( o r l a m i n a t e ) c o o r d i n a t e s y s t e m c a n be w r i t t e n as {R} = J [ B ] T [ T ] T {o f } dV (A .51 ) V w h e r e t h e v e c t o r o f r e l e a s e d s t r e s s e s { o f } i s t a b u l a t e d i n T a b l e 3 . A f o r v a r i o u s b r i t t l e modes o f f a i l u r e . The r e s i d u a l f o r c e v e c t o r now t a k e s on t h e f o l l o w i n g f o r m W = {F} + {R} - {P} (A .52 ) The r e l e a s e d f o r c e s {R} c a n t h e r e f o r e be i n t e r p r e t e d as known n o d a l l o a d s , w h i c h i n a d d i t i o n t o t h e a c t u a l e x t e r n a l l o a d s {F} , must be e q u i l i b r i a t e d b y t h e i n t e r n a l f o r c e s {P}. T h i s c o r r e s p o n d s t o t h e r e d i s t r i b u t i o n o f t h e r e l e a s e d s t r e s s e s f r o m f a i l e d i n t e g r a t i o n p o i n t s t o u n f a i l e d p o i n t s o f t h e e n t i r e s t r u c t u r e . D u r i n g t h e p r o c e s s o f s t r e s s r e d i s t r i b u t i o n , a n o t h e r p o i n t may f a i l e v e n t h o u g h t h e a p p l i e d l o a d {F} r e m a i n s c o n s t a n t . I f t h e f a i l u r e s p r e a d s t h r o u g h o u t t h e s t r u c t u r e , a s i n g u l a r o r n e g a t i v e s t r u c t u r a l s t i f f n e s s m a t r i x may a p p e a r . O r , t h e N e w t o n - R a p h s o n i t e r a t i o n o f u n b a l a n c e d e x t e r n a l f o r c e s may become d i v e r g e n t and an e q u i l i b r i u m s t a t e c a n n o t be r e a c h e d . F o r t h e s e c a s e s , t h e s t r u c t u r e i s c o n s i d e r e d t o have c o l l a p s e d and t h e c o m p u t a t i o n i s t e r m i n a t e d . To o b t a i n t h e c o l l a p s e l o a d w i t h i n n a r r o w l i m i t s t h e s i z e o f l o a d i n c r e m e n t s must be r e f i n e d when t h e s t r u c t u r e i s a b o u t t o c o l l a p s e . Use o f t h e r e s t a r t f a c i l i t y i n c l u d e d i n t h e p r e s e n t code i s p a r t i c u l a r l y u s e f u l i n t h i s r e s p e c t . 90 CHAPTER 5 NUMERICAL RESULTS AND DISCUSSIONS 5.1 Introduct ion A g e n e r a l t h e o r y o f t h e i n e l a s t i c i t y o f FRMs h a s b e e n o u t l i n e d i n t h e p r e c e d i n g c h a p t e r s w h i c h w o u l d a p p e a r t o e x h i b i t c o n s i d e r a b l e p o t e n t i a l as a b a s i s f o r a s y s t e m a t i c mode l o f t h e o b s e r v e d m a t e r i a l b e h a v i o u r . To c o m p l e t e l y f u l f i l l t h e o b j e c t i v e s o f t h i s t h e s i s , i t r e m a i n s t o c h e c k t h e a b i l i t y o f t h e p r o p o s e d t h e o r y t o r e p r o d u c e ( w i t h i n t h e bounds o f t h e t h e o r e t i c a l a s s u m p t i o n s ) t h e a v a i l a b l e d a t a and t h e s u i t a b i l i t y o f t h e mode l f o r u s e i n f i n i t e e l e m e n t c o m p u t a t i o n s . I n t h i s c h a p t e r , t h e s e i s s u e s a r e a d d r e s s e d b y c o m p a r i n g some t e s t s o f t h e mode l t o e x p e r i m e n t a l d a t a and v a r i o u s o t h e r n u m e r i c a l r e s u l t s . T h e r e a r e t h r e e m a j o r p o r t i o n s o f t h i s c h a p t e r . I n t h e f i r s t p o r t i o n t h e g e n e r a l p e r f o r m a n c e and f u n c t i o n i n g o f t h e f i n i t e e l e m e n t code COMPLY i s v a l i d a t e d b y c o n d u c t i n g n u m e r i c a l a n a l y s e s o f a few w e l l known example p r o b -lems i n v o l v i n g i s o t r o p i c m a t e r i a l s . These p r o b l e m s m e r e l y s e r v e t o v e r i f y t h e i s o t r o p i c e l a s t o p l a s t i c a n a l y s i s c a p a b i l i t y o f t h e COMPLY p r o g r a m b e f o r e i t c a n be a p p l i e d w i t h c o n f i d e n c e t o p r o b l e m s i n v o l v i n g a n i s o t r o p i c l a y e r e d m e d i a . The s e c o n d p o r t i o n o f t h e c h a p t e r i s c o n c e r n e d w i t h t h e a p p l i c a t i o n o f t h e computer p r o g r a m t o s e l e c t e d l a m i n a t e coupon s p e c i m e n s s u b j e c t e d t o v a r i o u s l o a d i n g c o n d i t i o n s . The o b j e c t i v e o f t h i s n u m e r i c a l s t u d y i s t o d e t e r m i n e t h e a d e q u a c y o f t h e p r o p o s e d c o n s t i t u t i v e mode l i n d e s c r i b i n g t h e b a s i c r e s p o n s e c h a r a c t e r i s t i c s o f l a m i n a t e d FRMs. The t h i r d and f i n a l p o r t i o n o f t h e c h a p t e r c o v e r s t h e f i n i t e e l e m e n t ( i n - p l a n e ) a n a l y s e s o f l a m i n a t e d c o m p o s i t e p l a t e s w i t h c e n t e r e d c i r c u l a r h o l e . T h i s g e o m e t r y , w h i c h p r o v i d e s n o n - u n i f o r m y i e l d i n g as w e l l as m u l t i - a x i a l s t r e s s s t a t e s , i s 91 an i d e a l a p p l i c a t i o n f o r t r a c i n g t h e o v e r a l l l o a d - d e f l e c t i o n r e s p o n s e i n t h e p o s t - f a i l u r e r e a l m ( i . e . beyond t h e f a i l u r e o f one e l e m e n t ) and c o n s e q u e n t l y t e s t i n g t h e c a p a b i l i t y o f t h e COMPLY p r o g r a m i n p e r f o r m i n g p r o g r e s s i v e f a i l u r e a n a l y s i s o f l a m i n a t e d s t r u c t u r e s . To f a c i l i t a t e c o m p a r i s o n t o e x p e r i m e n t , t h e o r y and o t h e r n u m e r i c a l t e c h -n i q u e s , a l a r g e number o f p r o b l e m s have b e e n c h o s e n f r o m w o r k s i n t h e open l i t e r a t u r e . S i n c e t h e s e p r o b l e m s do n o t come f r o m any s i n g l e s o u r c e t h e y have n o t b e e n o r i g i n a l l y s p e c i f i e d i n one u n i f i e d u n i t s y s t e m . I n o r d e r t o make an e a s y c o m p a r i s o n w i t h t h e s e r e s u l t s , t h e u n i t s as s p e c i f i e d i n t h e o r i g i n a l s o u r c e have b e e n a d o p t e d . 5 . 2 V e r i f i c a t i o n o f t h e F i n i t e E l e m e n t P r o g r a m I n o r d e r t o a s s e s s t h e a c c u r a c y o f e l a s t o p l a s t i c s o l u t i o n s o b t a i n e d b y t h e f i n i t e e l e m e n t code COMPLY, i t i s n e c e s s a r y t o c o n s i d e r p r o b l e m s t h a t have s o l u t i o n s o f known v a l i d i t y . I n t h i s s e c t i o n we i n v e s t i g a t e t h r e e s u c h p r o b l e m s a l l o f w h i c h i n v o l v e i s o t r o p i c m a t e r i a l s o b e y i n g t h e v o n M i s e s ' y i e l d c r i t e r i o n . The f i r s t example c o n s i d e r s an i n f i n i t e l y l o n g t h i c k - w a l l e d c y l i n d e r l o a d e d b y an i n t e r n a l p r e s s u r e c a u s i n g e l a s t o - p l a s t i c d e f o r m a t i o n s . I n t h e s e c o n d example t h e c a p a b i l i t y o f t h e p r o g r a m t o h a n d l e more c o m p l e x ( n o n - p r o p o r t i o n a l ) l o a d i n g p a t h s i s t e s t e d b y c o n d u c t i n g t h e a n a l y s i s o f a t h i n - w a l l e d t u b e s u b j e c t e d t o comb ined t o r s i o n and t e n s i o n . The l a s t example i n v e s t i g a t e s t h e e l a s t o - p l a s t i c b e h a v i o u r o f a t h i n s h e e t c o n t a i n i n g a c i r c u l a r h o l e u n d e r remote u n i f o r m t e n s i o n . The p u r p o s e o f t h e p r o b l e m s c o n s i d e r e d i n t h i s s e c t i o n i s t o examine t h e d e v e l o p m e n t o f p l a s t i c d e f o r m a -t i o n , t h u s no l i m i t i s s p e c i f i e d f o r f a i l u r e s t r a i n s and s t r e s s e s . 92 5 .2 .1 Thick-Wal led I so trop ic Cy l inder Under Internal Pressure The c y l i n d e r i s assumed t o be i n f i n i t e l y l o n g so t h a t t h e c o n d i t i o n o f p l a n e s t r a i n p r e v a i l s i n t h e a x i a l d i r e c t i o n . The m a t e r i a l i s e l a s t i c -p e r f e c t l y p l a s t i c o b e y i n g t h e v o n M i s e s 1 y i e l d c r i t e r i o n . E a r l i e r s o l u t i o n s t o t h i s p r o b l e m has b e e n o b t a i n e d b y Hodge and W h i t e (1950) u s i n g f i n i t e d i f f e r e n c e s , and b y H i l l (1950) who emp loyed t h e T r e s c a y i e l d c r i t e r i o n i n o r d e r t o s i m p l i f y t h e s o l u t i o n o f t h e g o v e r n i n g e q u a t i o n s . G r a p h i c a l r e p r e s e n t a t i o n s o f t h e s o l u t i o n have a l s o been g i v e n b y P r a g e r and Hodge (1951) . I n t h e f o l l o w i n g , t h e s e s o l u t i o n s , r e g a r d e d as e x a c t , f o r m t h e b a s i s o f c o m p a r i s o n w i t h t h e p r e s e n t n u m e r i c a l r e s u l t s . Due t o a x i a l symmet ry , o n l y a q u a r t e r o f t h e c y l i n d e r needs t o be c o n s i d e r e d . The c y l i n d e r shown i n F i g . 5 .1 was a n a l y z e d w i t h t h e p r o g r a m COMPLY u s i n g t h e same t y p e o f f i n i t e e l e m e n t ( i . e . , 8 - n o d e d i s o p a r a m e t r i c e l e m e n t ) and t h e same e l e m e n t s u b d i v i s i o n as t h a t o f Owen and H i n t o n (1980) . The f o l l o w i n g n u m e r i c a l v a l u e s were a s s i g n e d t o t h e g e o m e t r i c and m a t e r i a l p a r a m e t e r s i n v o l v e d : I n n e r r a d i u s , a = 100 mm O u t e r r a d i u s , b = 200 mm 5 E l a s t i c m o d u l u s , E = 2 .1 x 10 MPa P o i s s o n ' s r a t i o , \J = 0.3 U n i a x i a l y i e l d s t r e s s , o 0 = 240 MPa P l a s t i c m o d u l u s , H' = 0 The s o l u t i o n was o b t a i n e d u s i n g a 2 x 2 Gauss i n t e g r a t i o n r u l e . The i n t e r n a l p r e s s u r e , P , was g r a d u a l l y i n c r e a s e d u n t i l t h e p l a s t i c c o l l a p s e o f t h e c y l i n d e r was a t t a i n e d a t P = 185 MPa. The l a t t e r i s marked b y t h e d i v e r g e n c e o f t h e i t e r a t i v e p r o c e s s f o r an i n c r e m e n t a l l o a d i n c r e a s e . F i g u r e 5 .2 shows 93 that the c a l c u l a t e d p r e s s u r e - r a d i a l displacement curve f a l l s s l i g h t l y below that obtained by Hodge arid White (1950). For comparison, i t i s i n t e r e s t i n g to note that the exact values of the i n i t i a l and ul t imate (col lapse) pressure l e v e l s are 104 and 192 MPa, r e s p e c t i v e l y . The spreading of the p l a s t i c zone, corresponding to se lec ted load l eve l s P = 80, 120, 140 and 180 MPa, i s shown i n F i g . 5 .3 , where the y i e l d e d Gauss po ints are i n d i c a t e d by f u l l squares. Figure 5.4 depicts the c i r c u m f e r e n t i a l (hoop) s t r e s s , Og d i s t r i b u t i o n for the above s p e c i f i e d pressures va lues . The peaks on the t h e o r e t i c a l curves mark the p o s i t i o n of the e l a s t o - p l a s t i c boundaries . A reasonably good agreement between the numercial and exact s o l u t i o n i s ev ident . 5 . 2 . 2 Combined T e n s i o n and T o r s i o n o f an I s o t r o p i c T h i n - W a l l e d Tube The p h y s i c a l problem being solved i s that of a th in -wa l l ed c y l i n d r i c a l tube which i s subjected to a x i a l tens ion and t o r s i o n . We consider a tube which i s f i r s t s tressed i n tens ion from a s tress free s tate to i n c i p i e n t y i e l d and i s subsequently twisted under constant a x i a l s t r e s s . At a given value of the shear s t r e s s , x, the extens ional s t r a i n e and the shear s t r a i n y are given by the r e l a t i o n s h i p s ° 0 3 X 2 ° o e = — 2 n t l + l l - ] + _ (5.1) 3 ° o - T T = „ T [T " — tan"* ( W 3 / o 0 ) ] + } (5.2) V 3 b where G i s the e l a s t i c shear modulus for \) = 0 .3 . The above exact s o l u t i o n for i s o t r o p i c hardening mater ia l s has been der ived by H i l l (1950). We can use i t here to quant i fy our f i n i t e element s o l u t i o n e r r o r s . Since the s tress d i s t r i b u t i o n i n the tube, remote from the ends, i s constant everywhere ( i . e . 94 no s p a t i a l v a r i a t i o n ) , a s ing le mater ia l point i s s u f f i c i e n t to model . the problem. This can be accomplished by a s ing l e plane s tress i soparametric element. The numerical model together with the loading path i s shown i n F i g . 5.5 while the mater ia l constants are l i s t e d below: E = 28300 k s i E T = 280 k s i M = 0.3 o 0 = 26.25 k s i The coordinates of the points 0, A and B along the load path are tabulated i n Table 5 .1 . A l l the data used i n the present ana lys i s i s taken from Dodds (1987) who adopted the e l a s t i c p r e d i c t o r - r a d i a l r e t u r n a lgor i thm for the s tress computations. Table 5.1 Load Path Data for Tens ion-Tors ion Test on an I so trop ic Tube Points i n F i g . 5.5 o (ksi) (ksi) 0 0 0 A 26.25 0 B 26.25 24 Since the e l a s t o p l a s t i c n o n l i n e a r i t i e s are very severe i n the case considered, t h i s problem can provide a benchmarking example for t e s t i n g the accuracy of our nonl inear f i n i t e element code. To compute the s o l u t i o n , the a x i a l s tress was increased to o 0 i n the f i r s t load step. The shear s tress was then increased to 24 k s i using a constant s i z e increment Ax. Two analyses were performed corresponding to 95 Ax = 2 k s i and AT = A k s i . C o n s i d e r i n g t h e n o n l i n e a r i t i e s i n v o l v e d , t h e l a t t e r l o a d i n c r e m e n t s a r e r a t h e r l a r g e l e a d i n g t o a s e v e r e t e s t o f t h e p r e s e n t n u m e r i c a l m e t h o d . A t o l e r a n c e o f 0 . 0 1 % was u s e d t o e n s u r e t h e c o n v e r g e n c e o f t h e i t e r a t i v e s o l u t i o n a t e a c h l o a d s t e p . T a b l e 5 .2 compares t h e computed and e x a c t s t r a i n s . A g r a p h i c a l r e p r e s e n t a t i o n o f t h e e x a c t s t r a i n p a t h a l o n g w i t h t h e p r e s e n t r e s u l t s i s a l s o p r o v i d e d i n F i g . 5 . 6 . T a b l e 5 .2 E x a c t and Computed S t r a i n s f o r Combined T e n s i o n and T o r s i o n o f an I s o t r o p i c Tube T E x a c t A t = 2 . 0 k s i AT = A . O k s i t k s i ) e x 1 0 * r x 10» e x 1 0 J r x 1 0 * e x 1 0 A r x 1 0 * 0 0 . 0 9 3 0 . 0 0 0 0 . 0 9 3 0 . 0 0 0 0 . 0 9 3 0 . 0 0 0 A 0 . A 0 5 0 . 1 3 1 0 . A 0 8 0 . 1 2 6 0 . A 0 A 0 . 1 7 9 8 1 . 2 3 A 0 . 7 5 2 1 .251 0 . 6 8 A 1 . 2 5 0 0 . 7 1 2 12 2 . 3 5 2 2 . 0 7 3 2 . A 0 0 1 . 9 2 6 2 . A 2 0 1 .891 16 3 . 5 6 8 A . 0 5 7 3 . 6 5 A 3 . 8 3 8 3 . 7 0 7 3 . 7 3 7 20 A . 7 7 3 6 . 5 7 2 A . 8 9 7 6 . 2 9 6 A . 9 8 7 6 . 1 3 9 2A 5 . 9 1 7 9 . A 8 2 6 . 0 7 6 9 . 1 6 A 6 . 2 0 0 8 . 9 6 A 5 . 2 . 3 P e r f o r a t e d I s o t r o p i c Shee t S u b j e c t e d t o Remote U n i f o r m T e n s i o n A t h i n p e r f o r a t e d s h e e t w i t h i s o t r o p i c e l a s t o p l a s t i c m a t e r i a l p r o p e r t i e s i s m o d e l l e d and a n a l y z e d n u m e r i c a l l y f o r t h e c a s e o f a u n i f o r m l y d i s t r i b u t e d l o a d a p p l i e d a t t h e edges remote f r o m t h e h o l e . T h i s p r o b l e m i s o f i n t e r e s t f o r two i m p o r t a n t r e a s o n s . F i r s t , i t i s one o f t h e few n o n - t r i v i a l p r o b l e m s f o r w h i c h a d e q u a t e t h e o r e t i c a l and e x p e r i m e n t a l s o l u t i o n s e x i s t . S e c o n d , i t p r o v i d e s a benchmark f r o m w h i c h o r t h o t r o p i c e l a s t o p l a s t i c b e h a v i o u r c a n be g a u g e d . The d i m e n s i o n s o f t h e s p e c i m e n , a p p l i e d l o a d s and t h e c o o r d i n a t e a x i s emp loyed a r e shown i n F i g . 5 . 7 . B e c a u s e o f t h e symmetry o f l o a d i n g and g e o m e t r y , a q u a r t e r o f t h e p l a t e ( s h a d e d r e g i o n o f F i g . 5 . 7 ) i s m o d e l l e d b y 96 f i n i t e elements. A t y p i c a l f i n i t e element mesh used i n the computations i s shown i n F i g . 5 .8. The mesh cons i s t s of 48 i soparametric elements with a t o t a l of 177 nodes. The mater ia l used i s an aluminum a l l o y 57S whose s t r e s s -s t r a i n curve and i t s b i l i n e a r approximation are shown i n F i g . 5 .9. The mater ia l constants are as fo l lows: E = 7000 kg/mmJ (9956 ks i ) v = 0.3 H' =225 kg/mm* (320 k s i ) o 0 = 24.3 kg/mm* (34.5 ks i ) The above data correspond to those used i n the experimental work of Theocaris and Marketos (1964). U t i l i z i n g b i r e f r i n g e n t coatings bonded on the surface of the specimen together with the e l e c t r i c a l analogy method and the p l a s t i c i n c o m p r e s s i b i l i t y assumption, they determined the complete e l a s t o p l a s t i c response of the per forated s t r i p . A body of l i t e r a t u r e containing f i n i t e element so lut ions of the same problem are a l so a v a i l a b l e ; notable among which are the works of Marcal and King (1967) and Zienkiewicz , V a l l i a p p a n and King (1969). These e a r l i e r approaches employed the constant s t r a i n t r i a n g u l a r element to d i s c r e t i z e the specimen and used the "tangent s t i f fnes s" and the " i n i t i a l s t r a i n / s t r e s s " schemes, r e s p e c t i v e l y , i n t h e i r f i n i t e element s o l u -t i o n a lgor i thms. In F i g . 5.10 the development of the maximum s t r a i n (at the Gauss i n t e g r a t i o n po int c loses t to the hole and the x -ax is ) i s compared with the experimental r e s u l t s of Theocaris and Marketos (1964) and the above referenced f i n i t e element s o l u t i o n s . In the f igure o stands for the mean ° mean appl i ed s tress along the x -ax i s which, for the p a r t i c u l a r geometry c o n s i d e r e d , takes the v a l u e 2 o . The r e s u l t s were obtained using a 3 x 3 97 Gauss i n t e g r a t i o n r u l e . We observe that i n general the numerical models p r e d i c t a curve somewhat lower ( i . e . more f l e x i b l e ) than the experimental one. This i s i n sp i t e of the fact that a l l the numerical models used a s t i f f representat ion of the mater ia l law. Y i e l d of the f i r s t Gauss point takes p l a c e a t a remote l o a d o f o = 0.23 o „ . This i s reasonable s ince for the case where the r a t i o of the p la te width to the diameter of the hole i s equal to 2 the e l a s t i c s tress concentrat ion factor i s approximately 4.3 (see Howland, 1930). The s ing l e step s o l u t i o n shown i n F i g . 5.10 i s evidence of the fac t that the r e s u l t s are i n s e n s i t i v e to the magnitude of the load i n c r e -ment, thus supporting the f indings of Zienkiewicz et a l . (1969). The e f f ec t of varying the number of Gauss points and refinement of the mesh were a l so inves t i ga ted . S p e c i f i c a l l y , two analyses were performed, one employing a 2 point Gauss r u l e with the same mesh as i n F i g . 5.8 and the other us ing a 2 po int Gauss r u l e with a f i n e r mesh cons i s t ing of 80 elements and 281 nodal p o i n t s . The d i f ferences i n the r e s u l t s were so small as to be ind i scernab le i n F i g . 5.10 and have not been e x p l i c i t l y inc luded i n the f igure for the sake of c l a r i t y . Figure 5.11 shows the development of the p l a s t i c zone around the ho le , as the app l i ed remote load i s increased . The c a l c u l a t i o n s were made i n f ive load increments ( in the p l a s t i c range) e s s e n t i a l l y c o i n c i d i n g with those of Theocaris and Marketos (1964). The d e p i c t i o n of p l a s t i c zone growth i s accomplished by p l a c i n g a f u l l square at each of the y i e l d e d Gauss p o i n t s . The p l a s t i c zone boundaries which were found by Theocaris and Marketos (1964) are a lso shown i n F i g . 5.11 by s o l i d curves. We note that the pat tern of the p l a s t i c zone and i t s evo lu t ion obtained i n the present work are s i m i l a r to those r e p o r t e d e a r l i e r with the exception of the loading case o o = 0.47 o 0 . 98 However, a c lo ser examination reveals that for a s l i g h t l y higher value of the load given by = 0.49 o 0 , y i e l d i n g i n i t i a t e s at the free edge (note that no experimental r e s u l t s were a v a i l a b l e at t h i s load l e v e l for d i r e c t comparison) and progresses inward i n much the same way as that observed by Theocaris and Marketos (1964) for oa = 0.47 o 0 . Contours of e f f e c t i v e s tress are presented i n F i g . 5.12. This f igure provides the l o c a t i o n of i n i t i a l y i e l d and the d i r e c t i o n of subsequent p l a s t i c flow. It can be seen that the contours are consis tent with the p l a s t i c zone patterns shown i n F i g . 5.11. F i n a l l y i n F i g . 5.13 we compare the s t r a i n (e .^) and s tress (o^) d i s t r i -butions at the net s ec t ion ( i . e . along the x axis) with the experimental v a l u e s f o r the a p p l i e d s tress o^ = 0.47 o 0 . I t can be seen that the trends exh ib i t ed by the f i n i t e element r e s u l t are cons is tent with the experiment. The g r e a t e s t d e v i a t i o n occur f o r the e s t r a i n d i s t r i b u t i o n near the hole y where the model overpredic ts the experiment. However, t h i s i s cons is tent with the previous numerical p r e d i c t i o n s . 5.2.4 Conclusions The numerical examples inves t iga ted i n t h i s sec t ion provide s u f f i c i e n t evidence of the accuracy of the present f i n i t e element code COMPLY i n ana lyz -ing var ious problems with i s o t r o p i c e l a s t o p l a s t i c mater ia l p r o p e r t i e s . I t i s now poss ib l e to apply the code with confidence to some a n i s o t r o p i c problems. This task i s undertaken i n the remainder of t h i s chapter. 5.3 Response P r e d i c t i o n of Laminated Composite Coupons In t h i s s ec t ion the e f fect iveness of the proposed c o n s t i t u t i v e model i s v e r i f i e d for a number of loading paths imposed on d i f f e r e n t types of 99 laminated FRM coupons. Throughout t h i s sec t ion cons iderat ion i s given to a m a t e r i a l po int remote from the edges of the coupon specimen, so that the s tress d i s t r i b u t i o n can be taken as constant everywhere. Accordingly a s ing l e element representat ion i s used i n the fo l lowing analyses . The objec t ive of t h i s sec t ion i s to examine the c a p a b i l i t y of the model to accurate ly reproduce a broad sample of experimental records , both u n i a x i a l and b i a x i a l , with monotonic and c y c l i c l oad ing , i n c l u d i n g p r o p o r t i o n a l and nonproport ional s tress paths. 5.3.1 U n i a x i a l Loading As a f i r s t t es t of the proposed model, the program COMPLY i s used to p r e d i c t the nonl inear response of t e n s i l e specimens. A s er i e s of laminates ,pf, .Boron/Epoxy (B/Ep) U/D composites for which experimental data had been obtained by P e t i t and Waddoups (1969) are examined numer ica l ly . Figures 5.14 to 5.16 show the three bas ic s t r e s s - s t r a i n curves ( i . e . l o n g i t u d i n a l t ens ion , transverse t ens ion , and in -p lane shear) for a s i n g l e layer of U/D B/Ep composite under i n v e s t i g a t i o n . To p r o p e r l y i d e n t i f y the mater ia l parameters required as input to the model, these bas ic s t r e s s - s t r a i n curves have been f i t t e d with b i l i n e a r curves (shown by s o l i d l i n e s i n F i g s . 5.14 to 5 .16) . The crosses i n these and subsequent curves i n d i c a t e the s tress and s t r a i n l e v e l at which u l t imate f a i l u r e occurs . The r e s u l t i n g mater ia l constants used as input to COMPLY are given i n Table 5 .3 . Figures 5.17 to 5.25 show pred ic ted and experimental r e s u l t s of t e n s i l e t e s t s on a v a r i e t y of B/Ep laminates. The r e s u l t s of a n a l y t i c a l models due to Hashin et a l . (1974) and P e t i t and Waddoups (1969) are a l so added for comparison. Hashin's model i s based on the deformation theory of p l a s t i c i t y while P e t i t ' s model i s nonl inear e l a s t i c (see Sect ion 2 . 3 . 3 ) . In F i g . 5.17 100 Table 5.3 Input M a t e r i a l Propert ies for a Single Layer of U/D B/Ep E l a s t i c (ksi) P l a s t i c (ksi) F a i l u r e (ksi) E . = 30000 E ^ = 26100 X = 200 l T i u E , = 3080 E„ = 2200 Y = 12.5 2 T 2 u = 0.3 (uni t less ) G T = 180 S = u 18.6 G = 1000 X 0 = 132.5 Y 0 = 9 S 0 = 10 the experimental s t r e s s - s t r a i n curve for a [ 0 ° / 9 0 ° ] s c r o s s - p l y B/Ep laminate i s compared to the r e s u l t s of the present a n a l y s i s . Though the ana lys i s requires only two d i s t i n c t l a y e r s , at l eas t four p l i e s would be required for symmetric layup. Both b r i t t l e and d u c t i l e modes of f a i l u r e were i n v e s t i g a -ted . I t appears that these extreme types of p o s t - f a i l u r e models have provided a good bound to the ac tua l behaviour a f t er t e n s i l e matrix cracking i n the 9 0 ° p l y . The ul t imate f a i l u r e i s assoc iated with f i b r e f rac ture i n the 0° p l y . The r e s u l t f o r the case o f a [ ± 4 5 ° ] s a n g l e - p l y laminate i s shown i n F i g . 5.18. As i s seen, the present model p r e d i c t s very c l o s e l y the ul t imate strength and general shape of the experimental curve. The h i g h l y nonl inear nature of the response i n t h i s case i s caused by the presence of a cons ider-able amount of shear s t r a i n i n the p l i e s of the laminate. It should be noted *In the nota t ion for laminate o r i e n t a t i o n used i n t h i s thes i s the p l y angles are separated by a s l a sh with the e n t i r e layup enclosed w i t h i n square brack-e t s . The 0° p l y has i t s f ibres along the loading d i r e c t i o n . Where there i s more than one p l y at any given angle , the number of p l i e s at that angle i s denoted by a numerical subscr ipt w i th in the brackets . Subscript s outside the brackets means that the layup i s symmetric about the raid-surface. 101 that for t h i s p a r t i c u l a r laminate and i n general a l l ang le -p ly laminates of the form [ ± 9 ] s u l t imate f a i l u r e co inc ides with the f a i l u r e of one l a y e r . F i g u r e 5 . 1 9 ( a ) p r e s e n t s the s t r e s s - s t r a i n curves f o r the [ ± 3 0 ° ] s laminate. I t can be seen that the H i l l ' s f a i l u r e c r i t e r i o n (Eq. (3.57)) underpredicts the u l t imate strength of t h i s laminate. However, by observing the s tress paths i n one of the p l i e s , say the +30° p l y , ( F i g . 5.19(b)) i t can be i n f e r r e d t h a t the u l t i m a t e f a i l u r e of the [ ± 3 0 ° ] laminate i s caused by compression i n the transverse d i r e c t i o n before f a i l u r e could occur i n the f i b r e d i r e c t i o n . Since H i l l ' s c r i t e r i o n does not account for the s trength d i f f e r e n t i a l between tens ion and compression, and that the transverse compressive s trength i s i n t h i s case about three times higher than the t e n s i l e s trength , the use of the maximum stress f a i l u r e c r i t e r i o n allows us to trace the response curve to the second cross i n d i c a t e d i n F i g . 5 .19(a) . The corresponding s tress paths are shown i n F i g . 5 .19(c) . I t should be noted that the H i l l ' s c r i t e r i o n was s t i l l used as the y i e l d c r i t e r i o n i n the analyses whether or not i t was used to i n d i c a t e f a i l u r e . The r e s u l t s o f the present computations for a [ ± 6 0 ° ] s laminate of B/Ep are compared to p r i o r a n a l y t i c a l and experimental r e s u l t s i n F i g . 5.20. The shapes of the curves are i n good agreement with the experimental curves showing a more nonl inear behaviour than e i t h e r of the a n a l y t i c a l r e s u l t s . The u l t imate f a i l u r e i n t h i s case occurred from tens ion i n the transverse d i r e c t i o n . F i g u r e 5 .21(a) d i s p l a y s the response r e s u l t s for a [ ± 2 0 ° ] s laminate of B / E p . I t can be observed again that there i s a r e l a t i v e l y large d i f ference between the measured ul t imate strength and the present p r e d i c t i o n s based on H i l l ' s f a i l u r e c r i t e r i o n . As can be seen from the s tress path diagram shown i n F i g . 5.21(b) the pred ic ted mode of f a i l u r e i s that of compression i n the 102 transverse d i r e c t i o n ( i . e . compressive matrix f a i l u r e ) . By adopting the maximum s tress f a i l u r e c r i t e r i o n (which accounts for the ac tua l compressive strength) the p r e d i c t e d mode of f a i l u r e s h i f t s to that of f i b r e f a i l u r e ( F i g . 5 .21(c ) ) . This i s i n fac t the type of f a i l u r e exh ib i t ed by the tes t specimen as reported i n the paper by P e t i t and Waddoups (1969). Figure 5.22 presents r e s u l t s for the case of a q u a s i - i s o t r o p i c * [ 0 ° / ± 4 5 ° / 9 0 ° ] l a m i n a t e of B / E p . A l l the r e s u l t s , i n c l u d i n g the present p r e d i c t i o n s , show a r e l a t i v e l y i n s i g n i f i c a n t amount of n o n l i n e a r i t y . At o = 60 k s i the 9 0 ° layer f a i l s i n transverse t ens ion . The ± 4 5 ° layers remain i n t a c t u n t i l the 0° layer f a i l s i n the f i b r e mode thereby causing the u l t imate f a i l u r e of the laminate. A very s i m i l a r r e s u l t i s presented i n F i g . 5.23 f o r the q u a s i - i s o t r o p i c laminate formed from the [ 0 ° / ± 6 0 ° ] layup. As s s e e n ' i n the f i g u r e , the experimental r e s u l t s and a n a l y t i c a l p r e d i c t i o n s are i n exce l l ent agreement. According to the a n a l y t i c a l r e s u l t s , u l t imate laminate f a i l u r e was due to f a i l u r e occurr ing almost s imultaneously i n the 0° p l y i n the f i b r e d i r e c t i o n and i n the ± 6 0 ° p l i e s i n the transverse d i r e c t i o n . I t i s worth not ing that for the two q u a s i - i s o t r o p i c laminates discussed above the r e s u l t s of the b r i t t l e p o s t - f a i l u r e model showed no p e r c e p t i b l e d i f f erence when compared to the d u c t i l e model and hence were omitted from the f i g u r e s . Figures 5.24 and 5.25 d i s p l a y the s t r e s s - s t r a i n curves for a [ 0 ° 3 / ± 4 5 ° ] g ( i . e . [ 0 o / 0 ° / 0 ° / ± 4 5 o ] s ) laminate tes ted at 0° and 65° to the 0° p l y , respec-t i v e l y . For the 0° t e s t ( F i g . 5.24) the numerical and experimental curves *Since we are only concerned with in -p lane loadings and a lso ignore the e f fec t s of in ter laminar s tresses a change of p l y s tacking sequence i s assumed not to cause a change i n the laminate response. This appl ies to a l l the laminates considered here . 103 a r e v e r y c l o s e e x h i b i t i n g good a g r e e m e n t . I n t h i s c a s e t h e l a m i n a t e u l t i m a t e f a i l u r e was c a u s e d b y f i b r e f a i l u r e i n t h e 0° p l i e s w h i l e no f a i l u r e s o c c u r r e d i n t h e ±45° p l i e s . F o r t h e l a m i n a t e t e s t e d a t 65° ( F i g . 5 . 25 ) t h e p r e s e n t p r e d i c t i o n s a g r e e r e a s o n a b l y w e l l w i t h t h e r e s u l t s o f H a s h i n 1 s t h e o r y . However , t h e d i s c r e p a n c y b e t w e e n t h e o b s e r v e d and p r e d i c t e d f a i l u r e s t r e s s i s q u i t e s u b s t a n t i a l . The c a u s e o f t h i s p r e m a t u r e f a i l u r e i s n o t known b u t may be a t t r i b u t e d t o i n t e r l a m i n a r e f f e c t s ( w h i c h have n o t b e e n i n c l u d e d i n t h e p r e s e n t a n a l y s i s ) . I t s h o u l d be p o i n t e d o u t t h a t t h e r e s u l t s shown a r e f o r t h e d u c t i l e p o s t - f a i l u r e mode l i n w h i c h c a s e t h e u l t i m a t e f a i l u r e (marked by f i b r e f a i l u r e i n t h e 20° p l y ) i s p r e c e d e d b y t e n s i l e m a t r i x f a i l u r e s i n t h e ±65° and -70° p l i e s . When t h e b r i t t l e p o s t - f a i l u r e mode l was emp loyed t h e f a i l u r e o c c u r r e d s i m u l t a n e o u s l y i n a l l p l i e s c a u s i n g t h e c o l l a p s e o f t h e l a m i n a t e a t a l o a d l e v e l o f 30 k s i . I t i s o f i n t e r e s t t o n o t e t h a t f o r l a m i n a t e s s h o w i n g e x t r e m e n o n l i n e a r -i t y d u e t o s h e a r ( s u c h a s [ ± 4 5 ° ] s a n d [ ± 6 0 ° ] s ) t h e r e s p o n s e c u r v e s a r e s e n s i t i v e t o t h e c h o i c e o f t h e b i l i n e a r f i t u s e d t o a p p r o x i m a t e t h e s h e a r s t r e s s - s t r a i n c u r v e . F o r t h e o t h e r l a m i n a t e c o n f i g u r a t i o n s , p a r t i c u l a r l y t h e ones h a v i n g f i b r e s i n s e v e r a l d i r e c t i o n s ( i n c l u d i n g t h e l o a d i n g d i r e c t i o n ) no s i g n i f i c a n t change i n r e s u l t s o c c u r s when t h e b i l i n e a r s h e a r s t r e s s - s t r a i n c u r v e i s a l t e r e d . G i v e n t h e e x t e n t o f e x p e r i m e n t a l e r r o r and t h e r e l a t i v e i n s e n s i t i v i t y o f t h e r e s u l t s t o t h e p r e c i s e f o r m o f t h e b a s i c s t r e s s - s t r a i n c u r v e s o f t h e i n d i v i d u a l p l i e s , i n s i s t i n g on p r e c i s e c u r v e f i t t i n g i s u n w a r r a n t e d . I n s t e a d , a good q u a l i t a t i v e agreement w i t h t h e d a t a s h o u l d be s o u g h t as a means o f t e s t i n g t h e v a l i d i t y o f t h e m o d e l . The r e s u l t s o b t a i n e d so f a r s e r v e t o i n d i c a t e t h a t t h e p r e s e n t mode l c a p t u r e s t h e most i m p o r t a n t t r e n d s o f l a m i n a t e d FRM b e h a v i o u r u n d e r m o n o t o n i c t e n s i l e l o a d i n g . I t s h o u l d be r e m a r k e d i n p a s s i n g t h a t a l t h o u g h t h e e x t e r n a l l o a d s i n t h e above examp les 104 grow p r o p o r t i o n a l l y , t h i s does not n e c e s s a r i l y imply that the i n t e r n a l s tress components i n a t y p i c a l layer a l so grow p r o p o r t i o n a l l y (see F i g s . 5.19(b,c) and 5 .21 (b , c ) ) . This ra i se s doubts concerning the v a l i d i t y of the deforma-t i o n theory o f fered by Hashin et a l . (1974), which n e c e s s a r i l y requires p r o p o r t i o n a l i t y of the s tress path . Figures 5.26 through 5.28 d i s p l a y the s t r e s s - s t r a i n curves for the case of a po lyes ter r e s i n matrix r e i n f o r c e d by var ious b i d i r e c t i o n a l woven glass f i b r e s . The three bas i c experimental s t r e s s - s t r a i n curves (extracted from MIL-HDBK-17 (1959)) and t h e i r b i l i n e a r representat ions are i l l u s t r a t e d i n F i g s . 5 .26(a) , 5.27(a) and 5.28(a) . The r e s u l t i n g mater ia l constants used as input data are tabulated i n Table 5.4. Table 5.4 Input M a t e r i a l Propert ies for a S ingle Layer of B/D Glass F a b r i c / P o l y e s t e r Resin Parameter 181 Glass Fabr ic 162 Glass Fabr ic 143 Glass F a b r i c (ksi) (ksi) (ksi) E l a s t i c * i 2740 2820 5770 2520 1740 1600 0 0 0 G 630 570 720 P l a s t i c \ 2100 600 5770 2000 730 430 6 T 190 260 155 x 0 34 32 90 25 24 32 S 0 5 4 4.8 F a i l u r e X 49.2 45 90 Y 45.3 29.4 11 u S 13.4 11.6 12 u 105 The u n i a x i a l t e n s i l e s t r e s s - s t r a i n responses at 4 5 ° to the f ibres are shown i n F i g s . 5 .26(b) , 5.27(b) and 5.28(b) . These are the output of the program COMPLY wherein the Puppo-Evensen y i e l d and f a i l u r e c r i t e r i a were used. Since here we are dea l ing with a s ing le layer of mater ia l the i n i t i a l and ul t imate f a i l u r e co inc ide . From the f igures i t can be observed that the experimental curves f a l l below the pred ic ted curves with the u l t imate s t r a i n l eve l s reached being under-pred ic ted . In sp i t e of such discrepancies i n the r e s u l t s the pred ic ted ul t imate s tress values are i n reasonably good agreement with the experimental va lues . Notice that the experimental curve i n F i g . 5.26(b) does not extend to the f a i l u r e l e v e l and i s terminated at s t r a i n l e v e l of 2%. It i s regretable that experimental data for laminates c o n s i s t -ing of var ious or iented layers of B/D FRM are not a v a i l a b l e i n the l i t e r a t u r e for comparison. This i s one area which could c e r t a i n l y benef i t from further experimental work. 5.3.2 B i a x i a l Loading To further v e r i f y the c a p a b i l i t y of the e l a s t i c - p l a s t i c - f a i l u r e model, a few tes t cases were inves t iga ted i n which the external loadings were b i a x i a l . In p a r t i c u l a r we consider the e f fec t s of pure i n t e r n a l pressure , and combined t o r s i o n / i n t e r n a l p r e s s u r e l o a d i n g on the response of [ 0 ° / ± 6 0 ° ] laminated s tube of U/D Graphite /Epoxy (Gr/Ep) m a t e r i a l . The corresponding experimental (and t h e o r e t i c a l ) work was c a r r i e d out by Tennyson et a l . (1980). As i n the preceding examples the three bas ic s t r e s s - s t r a i n curves were approximated by b i l i n e a r f i t s r e s u l t i n g i n the s ing l e p l y proper t i e s l i s t e d i n Table 5.5. Note that the l o n g i t u d i n a l and transverse s t r e s s - s t r a i n curves were taken to be l i n e a r l y e l a s t i c r i g h t up to f a i l u r e . Also the nonl inear shear e f fec t s appear to be n e g l i g i b l e for t h i s m a t e r i a l . 106 Table 5.5 Input M a t e r i a l Propert ies for a Single Layer of U/D Gr/Ep E l a s t i c (ksi) P l a s t i c (ksi) F a i l u r e (ksi) E : = 20500 E 2 = 1400 \ ) 1 2 = 0.26 (uni t less ) G = 600 E T = 20500 E T X = 1400 G T 2 = 380 X 0 = 185.6 Y 0 = 7.5 S 0 = 6.8 = 185.6 Y = 7.5 u S = 11.8 u A graph ica l p l o t of the p r e s s u r e - s t r a i n curve i s shown i n F i g . 5.29 for the case of i n t e r n a l pressure only . I t i s worth mentioning that the t h i c k -ness and the radius of the tes t tube were 1 i n and 0.0343 i n , r e s p e c t i v e l y . Both H i l l ' s and maximum s tress f a i l u r e c r i t e r i o n were employed i n the a n a l y s i s . According to the pred ic ted r e s u l t s the i n i t i a l f a i l u r e occurred i n the ± 6 0 ° p l i e s (from tens ion i n the transverse d i r e c t i o n ) at pressure l e v e l s of about 1.8 k s i and 1.9 k s i for H i l l ' s and the maximum stress c r i t e r i a r e s p e c t i v e l y . I t i s apparent from F i g . 5.29 that Tennyson's cubic s trength c r i t e r i o n o f fers a bet ter estimate of the u l t imate s tress value than e i t h e r of the two c r i t e r i a used i n the present a n a l y s i s . However, the shape of the pred ic ted response curve agrees more c l o s e l y with the experimental data than Tennyson's r e s u l t s . In view of the presence of higher order terms (and hence more bas i c s trength data ) , i t i s not s u r p r i s i n g that Tennyson's f a i l u r e c r i t e r i o n provides a more accurate estimate of the maximum load . The quest ion of whether or not the a d d i t i o n a l complexity (and cost) of evaluat ing the extra s trength parameters i s warranted depends on the a p p l i c a t i o n . The e f f ec t of a constant pre-torque (19.8 k s i ) and i n t e r n a l pressure loading on the response i s i l l u s t r a t e d i n F i g . 5.30. This provides a tes t 107 case i n which the external loading grows i n a nonproport ional manner. As can be seen from F i g . 5.30 the r e s u l t s of the present ana lys i s are i n reasonable agreement with the experimental data . Based on the present a n a l y t i c a l p r e d i c t i o n s t e n s i l e matrix f a i l u r e occurred f i r s t i n the - 6 0 ° p l y at a pressure l e v e l of 1.1 k s i , according to both the maximum s tress and H i l l ' s f a i l u r e c r i t e r i o n . Subsequent f a i l u r e occurred i n the + 6 0 ° p l y , at a pressure l e v e l of 2.02 k s i i f the H i l l ' s f a i l u r e c r i t e r i o n was used, and at 2.6 k s i i f the maximum stress c r i t e r i o n was used. The u l t imate f a i l u r e of the laminate occurred s h o r t l y afterwards when the 0° p l y f a i l e d i n the f i b r e d i r e c t i o n . The second example of nonproport ional load path i s that i n which a constant i n t e r n a l pressure (1.1 k s i ) was appl i ed while the tube was t o r s i o n -a l l y loaded to f a i l u r e . The r e s u l t s are i l l u s t r a t e d on a graph of app l i ed t o r q u e versus shear s t r a i n y i n F i g . 5.31. It appears that a l l the numer-i c a l models are over ly s t i f f i n comparison with the experimental r e s u l t s . I t can a l so be noted that the u l t imate f a i l u r e s tress pred ic ted by the H i l l ' s and the cubic c r i t e r i a are f a i r l y c lose and agree be t ter with the e x p e r i -mental data than the p r e d i c t i o n s of the maximum s tress f a i l u r e c r i t e r i o n . 5 . 3 . 3 C y c l i c Loading While the previous examples provide v e r i f i c a t i o n of the model under monotonic l oad ing , they do not i l l u s t r a t e the e f fec t s of c y c l i c loading on laminate behaviour. To demonstrate such e f fec t s we consider the u n i a x i a l c y c l i c response of some Boron/Aluminum (B/Al) laminates. The experimental data for comparison with the present r e s u l t s have been extracted from the report by Sova and Poe (1978). To evaluate the input mater ia l constants the l o n g i t u d i n a l and transverse t e n s i l e s t r e s s - s t r a i n responses of the u n i d i r e c -108 * t i o n a l l a m i n a t e [ 0 ° 6 ] were a p p r o x i m a t e d b y b i l i n e a r c u r v e s . S i n c e no e x p e r i m e n t a l d a t a were a v a i l a b l e f o r t h e s h e a r r e s p o n s e , t h e l a t t e r was d e r i v e d f r o m t h e u n i a x i a l t e n s i l e s t r e s s s t r a i n c u r v e o f [±45°] l a m i n a t e s T h i s t e c h n i q u e i s i n k e e p i n g w i t h t h e p r o c e d u r e o u t l i n e d b y R o s e n (1972) f o r d e t e r m i n a t i o n o f s h e a r m o d u l u s . A c c o r d i n g l y t h e s h e a r s t r e s s and t h e s h e a r s t r a i n i n t h e p l y c o o r d i n a t e s a r e g i v e n b y o = o / 2 6 x ( 5 . 3 ) e , = e - e 6 x y w h e r e o i s t h e t e n s i l e s t r e s s a p p l i e d t o t h e [±45°] l a m i n a t e i n t h e x x s d i r e c t i o n ( i . e . 0° d i r e c t i o n ) , and e , e a r e t h e c o r r e s p o n d i n g s t r a i n compon -x y e n t s . By c h o o s i n g a s u i t a b l e b i l i n e a r r e p r e s e n t a t i o n o f t h e t e n s i l e s t r e s s -s t r a i n c u r v e f o r [ ± 4 5 ° ] g l a m i n a t e o n e c a n d e t e r m i n e f r o m E q . ( 5 . 3 ) t h e n e c e s s a r y p a r a m e t e r s o f t h e s h e a r s t r e s s - s t r a i n c u r v e f o r a s i n g l e p l y . These a r e l i s t e d i n T a b l e 5 . 6 a l o n g w i t h t h e p a r a m e t e r s o b t a i n e d f r o m l o n g i t u d i n a l and t r a n s v e r s e s t r e s s - s t r a i n c u r v e s . T a b l e 5 . 6 I n p u t M a t e r i a l P r o p e r t i e s f o r a S i n g l e L a y e r o f U/D B / A l E l a s t i c (GPa) P l a s t i c (GPa) F a i l u r e (GPa) E 4 = 2 0 9 . 7 E j = 107 v 1 2 = 0 . 2 ( u n i t l e s s ) G = 3 2 E T = 2 0 2 . 7 E T ' = 2 4 . 3 G T 2 = 1 .5 X 0 = 1 .2 Y 0 = 0 . 0 9 S 0 = 0 . 0 4 5 X = 1 .7 u Y = 0 . 1 2 u S = 0 . 1 1 u * I t i s assumed t h a t t h e s t r e s s - s t r a i n r e s p o n s e o f a s i n g l e l a y e r o f t h e u n i d i r e c t i o n a l l a m i n a t e i s t h e same as t h e r e s p o n s e o f t h e t o t a l t e s t l a m i n a t e . 109 F i g u r e 5.32 p r e s e n t s the u n i a x i a l response p r e d i c t i o n of a [ 0 ° / ± 4 5 ° ] s B / A l laminate subjected to three load c y c l e s . As may be seen the model captures the re levant features of the response assoc iated with such c y c l i c l oad ing . Thus, for ins tance , the r e s i d u a l s t r a i n s are c o r r e c t l y pred ic ted and the hys teres i s loops pred ic ted by the model c l o s e l y fol low the observed p a t t e r n . The o r i g i n of the unloading h y s t e r e t i c loops i n F i g . 5.32 may be traced back to the ± 4 5 ° p l i e s which y i e l d i n compression upon unloading of the laminate from higher s t r a i n s . C a l c u l a t i o n s were a l so performed for a pure monotonic loading case and e s s e n t i a l l y the same f a i l u r e point (shown i n F i g . 5.32) was reached. This shows that the load cyc les d i d not a f f ec t the u l t imate s tress and s t r a i n l e v e l of the laminate, thus supporting the experimental f indings of Sova and Poe (1978). 5.3.4 Conclusions The foregoing numerical s imulat ions of various coupon t e s t s , each r e p r e -senting a t y p i c a l s tate of s t r e s s , i l l u s t r a t e how the proposed model adequately p r e d i c t s the bas ic mater ia l response c h a r a c t e r i s t i c s of laminated FRMS under a v a r i e t y of in -p lane loading cond i t i ons . Some cases were reported i n which experimental r e s u l t s d i d not compare very w e l l with the present p r e d i c t i o n s . These experimental data are , however, quite l i m i t e d and may be i n s u f f i c i e n t for drawing conclusions i n t h i s regard. A very a t t r a c -t i v e feature of the model i s the fac t that i t can represent nonl inear behaviour with only a few input mater ia l property values being r e q u i r e d . For complex loading s i t u a t i o n s such as nonproport ional loading and e s p e c i a l l y c y c l i c loadings the present incremental model i s superior to the deformation theory model of Hashin et a l . (1974). 110 5.4 Perforated Orthotrop ic Plates Subjected to Remote Uniform Tension A l l t h e a n a l y s e s p r e s e n t e d i n t h e p r e v i o u s s e c t i o n were a p p l i c a b l e t o l a r g e s h e e t s o f l a m i n a t e , c o n t a i n i n g no i m p e r f e c t i o n s and t h u s p r o v i d e v e r i -f i c a t i o n o f t h e mode l i n s i m u l a t i n g v a r i o u s u n n o t c h e d coupon t e s t s . To f u r t h e r c h a l l e n g e t h e mode l t h e p r e s e n t s e c t i o n i s d e v o t e d t o t h e a n a l y s i s o f l a m i n a t e s w i t h a c e n t r a l h o l e u n d e r remote t e n s i l e l o a d i n g . T h i s example e x h i b i t s many i m p o r t a n t e f f e c t s n o t t e s t e d i n t h e p r e v i o u s a p p l i c a t i o n s s u c h as m u l t i d i m e n s i o n a l s t r e s s s t a t e , s t r e s s g r a d i e n t s and s t r e s s c o n c e n t r a t i o n s . I n t h e f o l l o w i n g s e c t i o n , a n a l y s e s o f v a r i o u s o r t h o t r o p i c s h e e t s w i t h a c i r c u l a r h o l e a r e c o n d u c t e d f o r t h r e e d i s t i n c t l o a d i n g r e g i m e s . These a p p e a r u n d e r t h e s u b h e a d i n g s o f e l a s t i c , e l a s t i c - p l a s t i c and e l a s t i c - p l a s t i c - f a i l u r e a n a l y s e s . 5.4.1 E l a s t i c Ana lys i s An e l a s t i c a n a l y s i s o f t h e s h e e t w i t h a c i r c u l a r h o l e i s c o n d u c t e d f o r two o r t h o t r o p i c m a t e r i a l s , n a m e l y U/D l a y e r s o f B / A l and B / E p FRMs. The f o r m e r d e m o n s t r a t e s m i l d o r t h o t r o p y ( i n t h e e l a s t i c r a n g e ) , w h i l e t h e l a t t e r e x h i b i t s r a t h e r s t r o n g o r t h o t r o p y . Two t y p e s o f a n a l y s e s a r e c o n d u c t e d f o r e a c h m a t e r i a l , one w i t h t h e f i b r e d i r e c t i o n b e i n g o r i e n t e d a l o n g t h e x - a x i s ( i . e . n o r m a l t o t h e l o a d d i r e c t i o n ) , and t h e o t h e r w i t h i t a l o n g t h e y - a x i s ( i . e . p a r a l l e l t o t h e l o a d d i r e c t i o n ) . The t h e o r e t i c a l s o l u t i o n s u s e d h e r e f o r c o m p a r i s o n a r e v a l i d i f t h e s i z e o f t h e o p e n i n g i s s m a l l i n c o m p a r i s o n t o t h e e x t e r n a l d i m e n s i o n s ( w i d t h and l e n g t h ) o f t h e p l a t e . The o p e n i n g c a n be c o n s i d e r e d s m a l l i f t h e r a t i o o f t h e p l a t e w i d t h t o t h e d i a m e t e r o f t h e h o l e i s e q u a l t o o r g r e a t e r t h a n 4 ( G r e s z c z u k , 1 9 7 2 ) . To f a c i l i t a t e c o m p a r i s o n w i t h t h e o r e t i c a l s o l u t i o n s t h e l a t t e r r a t i o i s s e l e c t e d f o r t h e p l a t e s a n a l y z e d i n t h i s s e c t i o n . I l l A quadrant of the p la te i s modelled by f i n i t e elements, as shown i n F i g . 5.33, with 64 isoparametric elements and a t o t a l of 229 nodal p o i n t s . For future re ference , we designate the Gauss i n t e g r a t i o n point c loses t to the hole and the x -ax i s as point "A" and the po int at the l e f t corner of the upper edge of the modelled region as po int "B". Since l i n e a r e l a s t i c r e s u l t s are of i n t e r e s t here , l e t the mater ia l constants be l i s t e d as fo l lows: B / A l B/Ep 3 3 29.4 x 10 k s i 30.0 x 10 k s i 3 3 19.1 x 10 k s i 3.0 x 10 k s i 0.169 0.336 3 3 G 7.5 x 10 k s i 1.0 x 10 k s i A c c o r d i n g to the l i n e a r e l a s t i c t h e o r y ( G r e s z c z u k , 1972) the c i r c u m f e r e n t i a l s t r e s s , o Q , at the edge of the hole i s given by the fo l lowing o expression o Q (1 + C x ) d + C a ) d + C1 + l2 + 2cos29) cT = (1 + CJ + 2C 1 cos28) ( l + l\ + 2C 2cos29) ( 5 , 4 ) where { p 1 + P j } i " - 1 ( P i + P j ) 1 ' 2 + 1 (5.5) ( P i - p , ) 1 " - 1 { P l - p 2 } i " + 1 Si = 112 where for the case i n which the loading d i r e c t i o n (y) coincides with the f i b r e d i r e c t i o n (x x ) P i = 2G " v " (5.6) a E a Pa " fPi " Tj1'1 For the case of a p la te loaded transverse to the f i b r e d i r e c t i o n the subscr ipts 1 and 2 i n Eq . (5.6) must be interchanged. The present f i n i t e element so lut ions u t i l i z i n g 2 x 2 and 3 x 3 Gauss quadratures are compared with the above t h e o r e t i c a l p r e d i c t i o n s i n F i g s . 5.34 and 5.35, where the c i r c u m f e r e n t i a l s tress d i s t r i b u t i o n s at Gauss points c lose to the ho l e ' s edge, o^/o^, are p l o t t e d against angular l o c a t i o n 9. The f i n i t e element r e s u l t s are i n good agreement with the theory i n most l o c a -t i o n s . Note that the s tresses are c a l c u l a t e d at the Gauss s tat ions which do not co incide with the hole boundary. It can be seen that the s tress concentrat ion fac tor for B / A l i s s i m i l a r to the i s o t r o p i c mater ia l s ( i . e . = 3) which i s what one would expect s ince t h i s mater ia l i s only weakly or tho-t r o p i c . For the B/Ep mater ia l a s i g n i f i c a n t s tress concentrat ion i n the v i c i n i t y of the hole can be observed. This i s p a r t i c u l a r l y pronounced when the s t i f f f i b r e s are or iented along the load d i r e c t i o n . 5.4.2 E l a s t i c - P l a s t i c Ana lys i s [90°] L a y e r An experimental study of the e l a s t o p l a s t i c response of a U/D B / A l metal matrix composite s t r i p with a c i r c u l a r hole was conducted by R i z z i and reported i n the paper by R i z z i et a l . (1987). This p a r t i c u l a r experiment 113 invo lved loading a B / A l specimen as shown i n F i g . 5.36 with the f ibres or iented at 9 0 ° to the load . M u l t i p l e s t r a i n guages were mounted on the front and rear faces of the specimen. The f i n i t e element mesh used to analyze the tes t specimen i s the same as that shown i n F i g . 5.33 except that the dimensions were sca led to those of F i g . 5.36. The e l a s t i c mater ia l proper t i e s used here are those determined by Kenaga et a l . (1987), as E j = 29.4 x 10 3 k s i (203 GPa) E , = 19.1 x 10 3 k s i (132 GPa) v l a = 0.169 G = 7.49 x 10 3 k s i (52 GPa) The mater ia l proper t i e s governing the p l a s t i c i t y of the specimen can be obtained from b e s t - f i t b i l i n e a r representat ions of s t r e s s - s t r a i n curves. A s e n s i t i v i t y ana lys i s revealed that the key mater ia l constants a f f e c t i n g the p l a s t i c flow c h a r a c t e r i s t i c s were the y i e l d s tress and the tangent modulus t r a n s v e r s e to the f i b r e s (x 2 or y d i r e c t i o n ) , i . e . Y 0 and E T . Furthermore 1 2 the shear proper t i e s and the proper t i e s i n the f i b r e d i r e c t i o n were found to have very l i t t l e inf luence on the e l a s t i c - p l a s t i c r e s u l t s . To t h i s end s ince the shear s t r e s s - s t r a i n curve for t h i s mater ia l was not e x p l i c i t l y given i n the a v a i l a b l e l i t e r a t u r e , the values S 0 and G T were taken from Table 5.6, i . e . , G T = 6.5 k s i = 220 k s i (0.045 GPa) (1.5 GPa) 114 Along the f i b r e d i r e c t i o n the mater ia l behaviour was taken to be pure ly e l a s t i c up to f a i l u r e (Kenaga et a l . , 1987). Accord ing ly , (1.41 GPa) (203 GPa) Figure 5.37 shows the experimental ly determined s t r e s s - s t r a i n curve transverse to the f ibres (Kenaga et a l . , 1987). Since t h i s curve i s h i g h l y n o n l i n e a r and the parameter Y 0 and E _ n e c e s s a r y f o r m o d e l l i n g i t can d i r e c t l y a f f ec t the r e s u l t s one must f i r s t look at the s t r a i n range of i n t e r e s t . By examining the experimental r e s u l t s of R i z z i et a l . (1987) i t was o b s e r v e d t h a t the h i g h e s t s t r a i n (e ,^ or e 2 measured at the gauge point c loses t to the hole) l e v e l reached was of the order of 0.2%. Therefore i n order to simulate the same tes t r e s u l t s the b i l i n e a r representat ion shown i n F i g . 5.37 was se lec ted for the re levant p o r t i o n of the s t r e s s - s t r a i n curve. T h i s gave r i s e to the fo l lowing values for Y Q and E T (which i n c i d e n t l y are very s i m i l a r to the corresponding values given i n Table 5.6) Y 0 = 13.0 k s i (0.09 GPa) E = 4.0 x 10 3 k s i (27.6 GPa) 2 With the mater ia l proper t i e s e s tab l i shed above the problem was solved for the same load l e v e l s reported i n R i z z i et a l . ' s (1987) work using a 2x2 Gauss i n t e g r a t i o n p r o c e d u r e . The d i s t r i b u t i o n s of the e s t r a i n components along the x -ax i s of the specimen are shown i n F i g . 5.38 for a l l the load steps considered. Incipience of p l a s t i c deformation was found to occur at a remote s t r e s s o of a p p r o x i m a t e l y 4 k s i . I t can be seen t h a t the t e s t - t h e o r y X 0 = X = 204 k s i 0 u 3 E_, = E 1 = 29.4 x 10 k s i 1 I 115 agreement more t h a n a d e q u a t e l y c o n f i r m s t h e p r e d i c t i v e c a p a b i l i t y o f t h e COMPLY p r o g r a m f o r an o r t h o t r o p i c m a t e r i a l . The s p r e a d i n g o f t h e p l a s t i c z o n e s and c o n t o u r s o f n o n d i m e n s i o n a l e f f e c -t i v e s t r e s s o / k 0 a r e d e p i c t e d i n F i g s . 5 . 3 9 and 5 . 4 0 f o r a r e p r e s e n t a t i v e number o f l o a d s . The e v o l u t i o n o f p l a s t i c i t y d u r i n g t h e s u c c e s s i v e s t e p s o f l o a d i n g shows an i n i t i a l s p r e a d i n g a c r o s s t h e n e t s e c t i o n . A t t h e maximum r e m o t e s t r e s s o o = 1 0 . 7 k s i , t h e p l a s t i c z o n e a p p e a r s t o have p r o g r e s s e d a l m o s t c o m p l e t e l y a c r o s s t h e s p e c i m e n , f o r m i n g a c e r t a i n a n g l e t o t h e n e t s e c t i o n ( i . e . x - a x i s ) . The e x t e n t o f t h e p l a s t i c f r o n t seems t o be g r e a t e r away f r o m t h e n e t s e c t i o n c a u s i n g an e l a s t i c r e g i o n t o s t i l l r e m a i n c l o s e t o t h e s t r a i g h t b o u n d a r y o f t h e s p e c i m e n . T h e r e f o r e , i t i s r e a s o n a b l e t o assume t h a t l i t t l e o r no permanent s t r a i n w i l l be f o u n d a t t h e o u t e r s t r a i n gauge l o c a t i o n s . To s u b s t a n t i a t e s u c h f i n d i n g s , a d i r e c t s t u d y o f t h e p l a s t i c * s t r a i n s a r e i n o r d e r . T h i s c a n be a c h i e v e d b y u n l o a d i n g t h e s p e c i m e n f r o m a p p r o p r i a t e l o a d l e v e l s . The l o n g i t u d i n a l r e s i d u a l s t r a i n d i s t r i b u t i o n s f o l l o w i n g u n l o a d i n g f r o m t h r e e e l a s t i c - p l a s t i c remote s t r e s s e s (o = 6 , 7 . 7 and 1 0 . 7 k s i ) a r e p l o t t e d a l o n g w i t h t h e e x p e r i m e n t a l r e s u l t s i n F i g . 5 . 4 1 . The agreement b e t w e e n t h e p r e s e n t n u m e r i c a l s o l u t i o n ( s o l i d l i n e s ) and t h e e x p e r i m e n t a l r e s u l t s a p p e a r s f a i r . The l a c k o f any s u b s t a n t i a l r e s i d u a l s t r a i n s a t t h e o u t e r gauge l o c a t i o n s i s i n k e e p i n g w i t h t h e p l a s t i c zone v i s u a l i z a t i o n o f F i g . 5 . 3 9 . O v e r a l l , t h e r e s u l t s a r e i n d i c a t i v e o f t h e f a c t t h a t t h e p r o p o s e d m a t e r i a l model does a good j o b o f p r e d i c t i n g t h e m a g n i t u d e and d i s t r i b u t i o n o f pe rmanent d e f o r m a t i o n . N u m e r i c a l l y , t h e u n l o a d i n g i s p e r f o r m e d b y t a k i n g a s m a l l n e g a t i v e l o a d i n g s t e p , w h i c h a l l o w s t h e e l e m e n t s t o become e l a s t i c . The s m a l l u n l o a d i n g i n c r e m e n t a l s o p r o v i d e s an o p p o r t u n i t y f o r r e - a s s e m b l y o f t h e s t i f f n e s s m a t r i x . The r e m a i n d e r o f t h e l o a d i s t h e n removed i n t h e n e x t i n c r e m e n t b y t a k i n g one l a r g e u n l o a d i n g s t e p . 116 For the sake of completeness the r e s i d u a l s tress d i s t r i b u t i o n s along the net s ec t i on are i l l u s t r a t e d i n F i g . 5.42. It i s c l e a r from the r e s u l t s that the l o n g i t u d i n a l s tress component, o .^, demonstrates a s i g n i f i c a n t compressive (negative) r e s i d u a l value i n the v i c i n i t y of the hole while the other r e s i d u a l s tresses ( o x > a r e n e g l i g i b l e . Such large compressive s tresses can be accounted for by not ing that the mater ia l i n the proximity of the net s ec t ion i s permanently deformed, and upon load r e v e r s a l the surrounding e l a s t i c mater ia l tends to clamp i t down and produce compressive s tresses . The c u r v e s of a r e s i d u a l s tress d i s t r i b u t i o n show r e l a t i v e maxima that tend y to move away from the hole with the increase of p r i o r l oad ing . [ 0 ° / 9 0 ° ] Laminate s The purpose of t h i s study i s to u t i l i z e the p r e v i o u s l y developed computational too l s for i n v e s t i g a t i o n of the e l a s t i c - p l a s t i c behaviour of a [ 0 ° / 9 0 ° ] s l ayup o f B / A l p l a t e containing a c i r c u l a r ho le . The s ing le layer mater ia l p r o p e r t i e s , geometry of the specimen and the f i n i t e element mesh are taken to be the same as that of the preceding example. . The p la te was subjected to u n i a x i a l in -p lane t ens ion . The development of p l a s t i c zones i n the i n d i v i d u a l layers are shown i n F i g s . 5.43 and 5.44 for the 9 0 ° and 0° p l i e s r e s p e c t i v e l y . I t can be observed that as the loading continues the p l a s t i c zone i n the 9 0 ° layer spreads r a p i d l y across the specimen and almost completely covers i t at about 25 k s i . The c h a r a c t e r i s t i c shape and growth of the p l a s t i c zones i n the 0° layer (see F i g . 5.44) i s somewhat d i f f e r e n t i n that i t extends upward i n the loading d i r e c t i o n and remains constrained to the immediate v i c i n i t y of the ho l e . Th i s behaviour i s understandable, s ince the major s t r e s s component, o^, i s p a r a l l e l to the f i b r e s and i s not as l i k e l y to cause y i e l d i n g . I t i s found that at a s tress l e v e l of 44 k s i the 117 f i r s t f a i l u r e o f t h e f i b r e s i n t h e 0° l a y e r ( a t Gauss p o i n t A i n F i g . 5 .33 ) o c c u r s , i . e . t h e f i b r e s a t t a i n t h e i r u l t i m a t e t e n s i l e s t r e n g t h o f 204 k s i . F i g u r e s 5 . 4 5 and 5 . 4 6 p r e s e n t c o n t o u r s o f t h e n o r m a l i z e d e f f e c t i v e s t r e s s , o / k 0 , f o r t h e 90° a n d 0° p l i e s a t v a r i o u s l o a d l e v e l s . T h e s e c o n t o u r s p r o v i d e t h e n e c e s s a r y q u a n t i t a t i v e i n f o r m a t i o n r e g a r d i n g t h e e x t e n t o f y i e l d -i n g i n e a c h l a y e r . F o r e x a m p l e , c o n t o u r s o f v a l u e o / k 0 = 1 show t h e o u t l i n e o f t h e e l a s t i c - p l a s t i c b o u n d a r y a n d c o n t o u r s b e a r i n g v a l u e s o f o / k 0 > 1 r e p r e s e n t t h e w o r k - h a r d e n e d r e g i o n s . B a h e i - E l - D i n and D v o r a k (1980) have a l s o a n a l y z e d a s i m i l a r p r o b l e m f o r a F P / A 1 m e t a l m a t r i x c o m p o s i t e . The d e v e l o p m e n t o f p l a s t i c z o n e s f o r t h i s a n a l y s i s i s i l l u s t r a t e d i n F i g . 5 . 4 7 . Though a d i r e c t q u a n t i t a t i v e c o m p a r i -s o n c a n n o t be o f f e r e d due t o t h e d i f f e r e n t m a t e r i a l p r o p e r t i e s , g e o m e t r y and l o a d i n g , t h e c h a r a c t e r o f t h e r e s u l t s a r e t h e same. I t i s w o r t h r e c a l l i n g ( c f . C h a p t e r 2 , S e c t i o n 2 . 3 . 2 ) t h a t B a h e i - E l - D i n and D v o r a k ' s (1980) m a t e r i a l mode l was b a s e d on m i n i - m e c h a n i c a l c o n c e p t s as o p p o s e d t o t h e s i m p l e r m a c r o - m e c h a n c i a l a p p r o a c h a d o p t e d i n t h i s s t u d y . The o t h e r m a c r o - m e c h a n i c a l a n a l y s i s p e r f o r m e d b y Leewood (1985) ( see a l s o Leewood e t a l . , 1987) u t i l i z e s a m a t e r i a l mode l d e v e l o p e d b y Kenaga e t a l . (1987) w h i c h was b a s e d on a t r i a l a n d e r r o r a p p r o a c h t o d e t e r m i n e t h e a n i s o t r o p i c p a r a m e t e r s A ^ , t h a t b e s t f i t t e d t h e d a t a . The p r e s e n t m a t e r i a l m o d e l , h o w e v e r , i s f r e e f r o m s u c h e m p i r i c i s m and c a n be a p p l i e d t o a w i d e v a r i e t y o f m a t e r i a l s p r o v i d e d t h e t h r e e k e y s t r e s s - s t r a i n c u r v e s a r e a v a i l a b l e . F i g u r e 5 . 4 8 shows t h e s t r e s s d i s t r i b u t i o n a l o n g t h e x - a x i s (o r r a t h e r a t Gauss p o i n t s c l o s e s t t o t h i s a x i s ) f o r d i f f e r e n t remote l o a d l e v e l s . I t c a n be n o t e d t h a t t h e e l a s t i c a n a l y s i s u n d e r e s t i m a t e s t h e s t r e s s c o n c e n t r a -t i o n f a c t o r f o r t h e 0° l a y e r , and o v e r e s t i m a t e s i t f o r t h e 90° l a y e r . F i g u r e 5 . 4 9 s h o w s t h e l o n g i t u d i n a l s t r e s s o i n t h e 0° l a y e r a t t h e Gauss p o i n t A 118 (see F i g . 5.33) as a f u n c t i o n o f the a p p l i e d l o a d am. The inf luence of p l a s t i c deformation on the s tress concentrat ion factor and the n o n l i n e a r i t y of the curve due to p l a s t i c i t y are p a r t i c u l a r l y obvious from the f i g u r e . It can be i n f e r r e d that p l a s t i c i t y i n composite laminates cannot be expected to reduce the l e v e l of s tress concentrat ion at the free edge. This f ind ing may contrad ic t one's usual expectat ion of the reduct ion of s tress concentrat ion due to l o c a l i z e d y i e l d i n g such as that prevalent i n monol i th ic metals . The foregoing r e s u l t s support the f indings of B a h e i - E l - D i n and Dvorak (1980). F i g u r e 5.50 shows the o v e r a l l l o a d ( s t r e s s ) v e r s u s d e f l e c t i o n (v^) c u r v e , where v D r e f e r s to the d e f l e c t i o n at point B (see F i g . 5.33) i n the D d i r e c t i o n of l oad ing . Also shown i n the f igure i s the r e s i d u a l displacement obtained by unloading the laminate from a load l e v e l of 20 k s i . Unloading i n i t i a l l y occurred e l a s t i c a l l y , but before complete unloading to zero o v e r a l l tens ion the 9 0 ° layer rey i e lded ( in compression) at the s i x Gauss po ints c loses t to the hole perimeter . This accounts for the s l i g h t kink i n the unloading part of the response curve, a phenomenon which i s commonly observed i n exper iments on laminated metal-matrix composite p l a t e s . The r e s i d u a l o s tress d i s t r i b u t i o n s (along the x-axis ) due to unloading from o o = 20 k s i are i l l u s t r a t e d i n F i g . 5.51. Notice the s i g n i f i c a n t compressive component of the r e s i d u a l s tress i n the 9 0 ° l a y e r . This behaviour i s expected s ince the 9 0 ° layer experiences a s u b s t a n t i a l amount of permanent deformation. 5.4.3 E l a s t i c - P l a s t i c - F a i l u r e A n a l y s i s While the previous sec t ion treated the e l a s t o p l a s t i c behaviour of or tho-t r o p i c p la tes with a ho le , the a t t en t ion here i s focussed on the extension of the f i n i t e element ana lys i s to inc lude f a i l u r e . Of p r i n c i p a l concern i n the 119 fo l lowing analyses i s the p r e d i c t i o n of the u l t imate f a i l u r e loads and the progress ion of damage ( i . e . f i b r e and matrix crackings) p r i o r to the co l lapse of the laminated s t r u c t u r e s . Here again the dimensions of the p la te and i t s f i n i t e element d i s c r e t i z a t i o n are those of F i g . 5.33. Although numerical attempts at progress ive f a i l u r e analyses of laminated p la tes with geometric d i s c o n t i n u i t i e s have been made before (Chang and Chang, 1987; Sandhu et a l . , 1983; Lee, 1982), no experimental v e r i f i c a t i o n s of damage patterns were o f f e r e d . A l s o , due to the d i f f e r e n t geometries and lack of s u f f i c i e n t knowledge of mater ia l propert ie s ( in the f u l l nonl inear range) a d i r e c t comparison with these numerical r e s u l t s cannot be made. To t h i s end, present numerical p r e d i c t i o n s of the f a i l u r e patterns and co l lapse loads are made without the benef i t of comparison with other sources. The analyses presented here are meant to simulate l o a d - c o n t r o l l e d tes t s i t u a t i o n s . In l i g h t of the d i f f e r e n t mater ia l models for the p o s t - f a i l u r e regime ( b r i t t l e and d u c t i l e behaviour) the present study i s aimed at prov id ing bounds to the ac tua l behaviour of t es t specimens. The laminates considered i n the fo l lowing analyses are assumed to be made up of U/D layers of B/Ep with the mater ia l proper t i e s l i s t e d i n Table 5 . 3 . The a n a l y s e s are conducted f o r [ 9 0 ° ] , [ 0 ° ] , [ 0 ° / 9 0 ° ] , [±A5° ] and [ 0 ° / ± A 5 ° / 9 0 o ] s layups. [ 9 0 ° ] Layer As the f i r s t example, l e t us consider a s ing le layer with the strong d i r e c t i o n perpendicular to the load . The pred ic ted damage progress ion process at the Gauss points i s i l l u s t r a t e d i n F i g . 5.52 for the d u c t i l e p o s t - f a i l u r e model. I t i s seen from the f igure that matrix f a i l u r e i s 120 c o n f i n e d t o t h e a r e a s n e a r t h e s t r e s s c o n c e n t r a t i o n s . An a t t e m p t t o i n c r e a s e t h e a p p l i e d s t r e s s beyond 7 k s i r e s u l t e d i n a s i n g u l a r s t r u c t u r a l s t i f f n e s s m a t r i x . T h i s i n d i c a t e s t h a t t h e e n t i r e s t r u c t u r e fo rmed a c o l l a p s e mechan ism a t t h i s l o a d l e v e l . F o r t h e b r i t t l e f a i l u r e model t h e f i n i t e e l e m e n t p r o g r a m p r e d i c t s t h a t t h e s p e c i m e n f a i l s c a t a s t r o p h i c a l l y a t a l o a d l e v e l o f 6 k s i . I n o t h e r words t h e f i r s t Gauss p o i n t s t o f a i l p r e c i p i t a t e f i n a l f a i l u r e . [ 0 ° ] Layer I n t h i s c a s e , a s i n g l e l a y e r i s c o n s i d e r e d i n w h i c h t h e l o a d i s i n t h e f i b r e d i r e c t i o n . The s p r e a d i n g o f f a i l u r e z o n e s f o r t h e d u c t i l e mode l i s shown i n F i g . 5 . 5 3 where i t c a n be s e e n t h a t t h e f i r s t a p p e a r a n c e o f f i b r e f a i l u r e happens a t t h e r i m o f t h e h o l e and on t h e l o n g i t u d i n a l a x i s o f t h e s p e c i m e n . S i m i l a r t o t h e 90° l a y e r no s i g n i f i c a n t damage i s p r e d i c t e d p r i o r t o f i n a l f a i l u r e a t 70 k s i . Once a g a i n a b r i t t l e t y p e o f f a i l u r e model l e a d s t o a sudden c o l l a p s e o f t h e s t r u c t u r e a t i n c i p i e n c e o f f a i l u r e . [ 0 ° / 9 0 ° ] Laminate s V a r i o u s c o m b i n a t i o n s o f p o s t - f a i l u r e m o d e l s were c o n s i d e r e d h e r e f o r f i b r e and m a t r i x f a i l u r e . F o r t h e c a s e o f b r i t t l e f i b r e f a i l u r e t h e g r a d u a l , as o p p o s e d t o s u d d e n , s t r e s s r e l e a s e scheme was a d o p t e d . A c c o r d i n g l y , t h e f i b r e s t r e s s e s were r e l a x e d o v e r a number o f l o a d i n c r e m e n t s . To f a c i l i t a t e a c o n v e r g e n t s o l u t i o n v e r y s m a l l l o a d s t e p s had t o be u s e d a f t e r t h e f i r s t o c c u r r e n c e o f f i b r e f a i l u r e . D u r i n g e a c h l o a d s t e p t h e f a i l e d f i b r e s t r e s s e s were a r b i t r a r i l y r e d u c e d b y 25%. F i g u r e 5 . 5 4 shows t h e c o r r e s p o n d i n g s t r e s s p a t h , y i e l d s u r f a c e s and t h e f a i l u r e s u r f a c e , i n t h e o x / X 0 - o 2 / Y 0 p l a n e a t Gauss p o i n t A (see F i g . 5 . 33 ) f o r t h e 0° l a y e r . I t i s w o r t h r e c o r d i n g t h a t t h e n u m e r i c a l s t a b i l i t y o f b r i t t l e t y p e f a i l u r e s o l u t i o n p r o c e s s e s i s g e n e r -121 a l l y poor s ince on i n i t i a t i o n of f a i l u r e part or a l l of the e x i s t i n g stresses must be e l iminated by r e d i s t r i b u t i o n . The l o a d - d e f l e c t i o n curve evaluated at point B of the specimen (see F i g . 5.33) i s shown i n F i g . 5.55 for d i f f e r e n t mater ia l models i n the p o s t - f a i l u r e regime. The corresponding developments of damage zones i n the i n d i v i d u a l layers are depicted i n F i g s . 5.56 to 5.58, for representat ive load l eve l s up to the u l t imate f a i l u r e . As can be seen from F i g s . 5.56 and 5.57 for the d u c t i l e f i b r e f a i l u r e models, matrix cracking i n the 9 0 ° layer tends to p r o -pagate both towards the load and across the specimen width. F i b r e f rac ture p a t t e r n , however, appears to be discontinuous i n nature and remains confined to a narrow band near the ho l e . For the b r i t t l e f i b r e f a i l u r e model the damage pa t t ern i s markedly d i f f e r e n t as shown i n F i g . 5.58. In t h i s case, damage propagates h o r i z o n t a l l y leading to the u l t imate f a i l u r e of the s p e c i -men by t ear ing across the net s e c t i o n . I t i s important to note that here the elements near the f i r s t f rac ture l o c a t i o n are further loaded due to s tress r e d i s t r i b u t i o n . In order to c l a r i f y the f a i l u r e mechanism, the changes i n s tress d i s t r i b u t i o n for small v a r i a t i o n s of load l e v e l between the i n i t i a l and u l t imate f a i l u r e loads are shown i n F i g . 5.59. Note that the peak s tress changes i t s l o c a t i o n as the f a i l u r e propagates. [ ± A 5 ° ] Laminate s The f i n i t e element r e s u l t s i n d i c a t e that the predominant mode of f a i l u r e i n t h i s case i s shearing of the matr ix . The pred ic ted l o a d - d e f l e c t i o n curve i s shown i n F i g . 5.60 for both d u c t i l e and b r i t t l e matrix f a i l u r e . I t can be seen that the response i s extremely nonl inear owing to the nonl inear shear s t r e s s - s t r a i n behaviour i n each p l y (see F i g . 5.16). I n i t i a l f a i l u r e was found to occur at 19 .A k s i . In the b r i t t l e f a i l u r e case the shear s tress was 122 reduced by 50% at every load step fo l lowing the f i r s t f a i l u r e . A t o t a l of 14 load increments (each of magnitude 0.1 ks i ) were appl i ed between the i n i t i a l and f i n a l f a i l u r e l oad . The pred ic ted patterns of damage i n each layer are presented i n F i g s . 5.61 and 5.62 for d u c t i l e and b r i t t l e models, respec-t i v e l y . I t i s c l e a r from these f igures that the f rac ture of the elements s t a r t s from the hole and propagates d iagona l ly across the specimen forming a band at 45° to the x - a x i s . Ult imate f a i l u r e i s seen to be preceded by some scat tered f i b r e f rac ture i n the l a y e r s . [ 0 V ± 4 5 ° / 9 0 ° ] Laminate s The pred ic ted l o a d - d e f l e c t i o n curve for t h i s case i s shown i n F i g . 5.63 for various combinations of b r i t t l e and d u c t i l e p o s t - f a i l u r e models assigned to the f i b r e and matr ix . When f i b r e f a i l u r e i s assumed to be d u c t i l e , the u l t imate s tress l e v e l reached appears to be almost independent of the choice of matrix p o s t - f a i l u r e model. However, when the b r i t t l e matrix model i s invoked (corresponding to a sudden re lease of transverse and shear s t r e s s e s ) , the specimen e x h i b i t s a more nonl inear response. Since the f ibres i n the 0° layer c a r r y the major p o r t i o n of the l o a d , a b r i t t l e type f i b r e f a i l u r e leads to progress ive f a i l u r e of the laminate soon a f t er the f i r s t f i b r e f a i l u r e o c c u r r s . As a r e s u l t , the co l lapse of the laminate i s a t ta ined without not iceable increase i n l o a d - c a r r y i n g capac i ty above that of the i n i t i a l f i b r e f rac ture l oad . This accounts for the p lateau i n the l o a d - d e f l e c t i o n curve ( F i g . 5.63) which can be traced using very small load steps u n t i l the tangent ia l s t i f f n e s s matrix becomes s i n g u l a r . The spread of damage zones i n i n d i v i d u a l layers i s i l l u s t r a t e d i n F i g s . 5.64 to 5.66 for representat ive load l e v e l s . It i s c l e a r from these f igures that for the d u c t i l e f i b r e model the specimen f a i l s gradua l ly with cons ider-123 able damage occurr ing p r i o r to ul t imate f a i l u r e . Damage i n each layer i s seen to propagate across the specimen at a r a p i d rate (F igs . 5.64 and 5.65). For the b r i t t l e f i b r e model the c h a r a c t e r i s t i c l o c a l i z e d damage around the hole appears to p r e v a i l . The b r i t t l e f i b r e - b r i t t l e matrix model exh ib i ted a ca tas trophic f a i l u r e a f t er i n i t i a l matrix cracking i n the 9 0 ° l a y e r . This corresponds to the lower bound on co l lapse load shown i n F i g . 5.63. 5.4.4 Conclusions The or tho trop ic e l a s t i c c a p a b i l i t y of the present code, COMPLY, demon-s tra tes good agreement with the t h e o r e t i c a l s o l u t i o n s . Close agreements with the experimental r e s u l t s of R i z z i et a l . (1987) for a U/D B / A l composite provide further evidence of the adequacy of the proposed o r t h o t r o p i c e l a s t i c -p l a s t i c formulat ion and demonstrates the encouraging performance of the present f i n i t e element program i n t h i s regard . The e l a s t o p l a s t i c analyses revea l the strong dependence of p l a s t i c flow on the o r i e n t a t i o n of the p r i n c i p a l axes of orthotropy. The present e l a s t i c - p l a s t i c - f a i l u r e analyses with various p o s t - f a i l u r e options should prove to be use fu l i n prov id ing o v e r a l l bounds on the response h i s t o r y of laminates. The f i n i t e element ana lys i s a lso forms a v i a b l e procedure for p r e d i c t i n g the damage progress ion p r i o r to the u l t imate f a i l u r e of composite laminates with s tress concentra-t i o n s . Such c a p a b i l i t i e s are p a r t i c u l a r l y use fu l i n parametric studies leading to the design of laminates . 124 CHAPTER 6 SUMMARY AND CONCLUSIONS 6.1 Summary I n a l m o s t a l l o f t h e p r a c t i c a l s t r e n g t h a n a l y s e s o f l a m i n a t e d c o m p o s i t e s i n t h e l i t e r a t u r e , t h e s t r e s s e s u s e d f o r f a i l u r e c r i t e r i a have been d e t e r -m i n e d on t h e b a s i s o f e l a s t i c l a m i n a t e a n a l y s i s . Use o f t h e e l a s t i c t h e o r y w o u l d , i n g e n e r a l , l e a d t o o v e r l y s t i f f p r e d i c t i o n s , and may r e s u l t i n c o n s e r v a t i v e e s t i m a t e s o f t h e f a i l u r e l o a d s . The h e t e r o g e n e o u s n a t u r e o f FRMs i s s u c h t h a t a v a r i e t y o f p o s s i b l e damage modes e x i s t . T h u s , m a t r i x c r a c k i n g o r y i e l d i n g , f i b r e f r a c t u r e , d e b o n d i n g and o t h e r i n e l a s t i c e f f e c t s c a n a l l o c c u r i n l o c a l r e g i o n s a t r e l a t i v e l y l o w o v e r a l l s t r e s s l e v e l s . These n o n l i n e a r e f f e c t s g r e a t l y c o m p l i c a t e t h e p r o b l e m o f e s t a b l i s h i n g r e l i a b l e a n a l y s e s . I n t h e p r e s e n t s t u d y , t h e p r o b l e m o f n o n l i n e a r m a t e r i a l b e h a v i o u r o f l a m i n a t e d FRMs was i n v e s t i g a t e d . The p r i m a r y o b j e c t i v e was t o mode l t h e i n e l a s t i c b e h a v i o u r o f s u c h m a t e r i a l s and t o d e v e l o p a compute r p r o g r a m w h i c h c a n be u s e d as an e n g i n e e r i n g t o o l i n t h e d e s i g n a n d / o r a n a l y s i s o f f i b r e - r e i n f o r c e d c o m p o s i t e s t r u c t u r e s . I n p r e d i c t i n g t h e n o n l i n e a r s t r e s s - s t r a i n b e h a v i o u r o f FRMs, c o n s t i t u -t i v e e q u a t i o n s a r e g e n e r a l l y r e q u i r e d t o c o v e r t h e e n t i r e s t r e s s h i s t o r y . To t h i s e n d , a c o n t i n u u m m e c h a n i c s a p p r o a c h was u t i l i z e d h e r e i n t o d e v e l o p a r e l a t i v e l y s i m p l e o r t h o t r o p i c e l a s t i c - p l a s t i c - f a i l u r e c o n s t i t u t i v e model f o r s i n g l e l a y e r s o f FRM u n d e r g o i n g i s o t h e r m a l i n f i n i t e s i m a l d e f o r m a t i o n . The c o n s t i t u t i v e e q u a t i o n s so d e v e l o p e d , were t h e n combined u s i n g t h e c l a s s i c a l l a m i n a t i o n t h e o r y , t o a r r i v e a t t h e g o v e r n i n g r e s p o n s e r e l a t i o n s f o r m u l t i -l a y e r l a m i n a t e s . U n i d i r e c t i o n a l and b i d i r e c t i o n a l FRM l a y e r s were t r e a t e d w i t h i n t h e same g e n e r a l f ramework w i t h t h e e x c e p t i o n t h a t y i e l d i n g (and 125 f a i l u r e ) i n t h e s e l a y e r s was assumed t o be g o v e r n e d by d i f f e r e n t c r i t e r i a , n a m e l y , H i l l ' s and P u p p o - E v e n s e n ' s y i e l d (and f a i l u r e ) c r i t e r i a , r e s p e c -t i v e l y . The p r o p o s e d p l a s t i c i t y mode l a d o p t e d a 3 - p a r a m e t e r q u a d r a t i c y i e l d s u r f a c e and t h e a s s o c i a t e d f l o w r u l e o f t h e r a t e - i n d e p e n d e n t t h e o r y o f p l a s t i c i t y . The s u b s e q u e n t l o a d i n g s u r f a c e s were o b t a i n e d by a n o n - u n i f o r m e x p a n s i o n o f t h e i n i t i a l y i e l d s u r f a c e i n t h e s t r e s s s p a c e . T h i s was a c h i e v e d b y a l l o w i n g t h e p a r a m e t e r s i d e n t i f y i n g t h e i n i t i a l y i e l d f u n c t i o n t o v a r y i n a n o n - p r o p o r t i o n a l manner d u r i n g p l a s t i c f l o w , A 3 - p a r a m e t e r q u a d r a t i c f a i l u r e s u r f a c e s i m i l a r i n f o r m t o t h a t o f t h e i n i t i a l y i e l d s u r f a c e was d e f i n e d t o mark t h e u p p e r l i m i t o f p l a s t i c f l o w . Once f a i l u r e was r e a c h e d , i t was i d e n t i f i e d as f i b r e o r m a t r i x mode o f f a i l u r e d e p e n d i n g on t h e r e l a t i v e m a g n i t u d e o f v a r i o u s s t r e s s r a t i o t e r m s a p p e a r i n g i n t h e f a i l u r e c r i t e r i o n . I n t h e p o s t - f a i l u r e m o d e l l i n g , b o t h b r i t t l e and d u c t i l e t y p e o f b e h a v i o u r were c o n s i d e r e d i n t h e d i r e c t i o n o f t h e o f f e n d i n g s t r e s s . To c o m p l e t e l y q u a n t i f y t h e p r o p o s e d e l a s t i c - p l a s t i c - f a i l u r e mode l t h r e e p i e c e s o f e x p e r i m e n t a l s t r e s s - s t r a i n c u r v e s were r e q u i r e d , n a m e l y , t h e u n i a x i a l s t r e s s - s t r a i n c u r v e s a l o n g t h e two p r i n c i p a l a x e s o f o r t h o t r o p y , and t h e i n - p l a n e s h e a r s t r e s s - s t r a i n c u r v e . Once e s t a b l i s h e d , t h e s t r e s s - s t r a i n c u r v e s were r e p r e s e n t e d b y b i l i n e a r a p p r o x i m a t i o n s , t h u s c l e a r l y d e f i n i n g t h e k e y p a r a m e t e r s u n d e r t h e v a r i o u s l o a d i n g p r o g r a m s . No p r o v i s i o n s were made f o r t h e d i f f e r e n c e b e t w e e n t e n s i l e and c o m p r e s s i v e r e s p o n s e s . B a s e d on t h e p r o p o s e d m o d e l , c o n s t i t u t i v e e q u a t i o n s were p r o p e r l y f o r m u l a t e d . A n o n l i n e a r f i n i t e e l e m e n t code was s u b s e q u e n t l y d e v e l o p e d t o i n c o r p o r a t e t h e d e r i v e d c o n s t i t u t i v e e q u a t i o n s . The p r o g r a m named COMPLY, was b a s e d on t h e c o n v e n t i o n a l d i s p l a c e m e n t method f i n i t e e l e m e n t p r o c e d u r e u s i n g two d i m e n s i o n a l 8 - n o d e i s o p a r a m e t r i c e l e m e n t s . The n o n l i n e a r i t i e s i n t h e e q u i l i b r i u m e q u a t i o n s were h a n d l e d b y a m i x e d i n c r e m e n t a l and N e w t o n -126 Raphson i t e r a t i v e p r o c e d u r e . A n a l y s i s r e s t a r t and c y c l i c l o a d i n g c a p a b i l i -t i e s were a l s o i n c l u d e d t o expand t h e p r o g r a m ' s u s e f u l n e s s . The p e r f o r m a n c e o f t h e p r o g r a m and t h e e f f e c t i v e n e s s o f t h e mode l were v e r i f i e d f o r a number o f i n - p l a n e l o a d i n g p a t h s i m p o s e d on a w i d e v a r i e t y o f l a m i n a t e d FRMs w i t h and w i t h o u t g e o m e t r i c d i s c o n t i n u i t i e s . The r e s u l t s o f n u m e r i c a l s i m u l a t i o n s were compared w i t h t h e e x p e r i m e n t a l d a t a a v a i l a b l e i n t h e l i t e r a t u r e . 6.2 Concluding Remarks The f a v o u r a b l e agreement b e t w e e n t h e p r e s e n t n u m e r i c a l p r e d i c t i o n s and e x p e r i m e n t a l r e s u l t s i l l u s t r a t e t h e a c c u r a c y and v e r s a t i l i t y o f t h e p r o p o s e d e l a s t i c - p l a s t i c - f a i l u r e mode l f o r l a m i n a t e d c o m p o s i t e s . The mode l i s t y p i c a l o f c o n v e n t i o n a l c o n t i n u u m m e c h a n i c s t h e o r i e s i n t h a t i t c a p t u r e s t h e e s s e n c e o f t h e m a t e r i a l b e h a v i o u r r a t h e r t h a n i t s d e t a i l . I n l i g h t o f t h i s , t h e model must be a p p l i e d c a r e f u l l y w i t h t h e k n o w l e d g e t h a t t h e n o n l i n e a r i t i e s may n o t a c t u a l l y be due t o p l a s t i c y i e l d i n g b u t t o some c o m b i n a t i o n o f p l a s t i c i t y , m a t r i x o r f i b r e c r a c k i n g , f i b r e p u l l - o u t , e t c . The r e l a t i v e e a s e w i t h w h i c h t h e p r o p o s e d a p p r o a c h c a n h a n d l e n o n l i n e a r b e h a v i o u r i s a d i s t i n c t a d v a n t a g e o v e r c u r r e n t a n a l y t i c a l p r o c e d u r e s . I n d e e d t h e method i s s u f f i c i -e n t l y g e n e r a l t h a t i t may be a p p l i e d t o a v a r i e t y o f l a m i n a t e d s t r u c t u r e s p r o v i d e d e i t h e r a n a l y t i c a l o r e x p e r i m e n t a l d a t a a r e a v a i l a b l e t o d e s c r i b e t h e s t r e s s - s t r a i n c u r v e s f o r t h e c o n s t i t u e n t p l i e s . As d e m o n s t r a t e d i n s e v e r a l e x a m p l e s i n t h i s t h e s i s , t h e p r e s e n t t h e o r y r e a s o n a b l y p r e d i c t s t h e phenome-n o l o g i c a l b e h a v i o u r o f c o m p o s i t e s u n d e r m o n o t o n i c and c y c l i c l o a d i n g , i n c l u d i n g p r o p o r t i o n a l and n o n p r o p o r t i o n a l s t r e s s - p a t h s . The s t u d i e s c o n d u c t e d h e r e i n p o i n t t o t h e f a c t t h a t t h e g l o b a l l a m i n a t e r e s p o n s e i s a n o n l i n e a r s u p e r p o s i t i o n o f i n d i v i d u a l l a y e r m a t e r i a l p r o p e r -127 t i e s . T h i s c o m p l e x r e s p o n s e p r o v i d e s j u s t i f i c a t i o n f o r n o t a n a l y z i n g t h e l a m i n a t e as a w h o l e . I n t h i s r e g a r d , t h e p r e s e n t c o m p r e h e n s i v e c o n s t i t u t i v e mode l a p p l i e d t o e a c h l a y e r o f a l a m i n a t e and t h e s u b s e q u e n t s u p e r p o s i t i o n o f l a y e r r e s p o n s e s v i a l a m i n a t i o n t h e o r y a p p e a r s t o o f f e r t h e most s y s t e m a t i c a p p r o a c h . C o n s i d e r i n g t h e f a c t t h a t t h e p r o p o s e d m a t e r i a l mode l l e n d s i t s e l f t o a s t r a i g h t f o r w a r d c o m p u t a t i o n a l i m p l e m e n t a t i o n i t may be a v i a b l e o p t i o n f o r i n c o r p o r a t i o n i n t o g e n e r a l p u r p o s e f i n i t e e l e m e n t codes f o r s t u d y i n g t h e d e t a i l e d b e h a v i o u r o f a n i s o t r o p i c s t r u c t u r e s . I n t h e m e a n t i m e , t h e p r e s e n t p l a n a r f i n i t e e l e m e n t p r o g r a m , COMPLY, c a n be a u s e f u l t o o l i n p a r a m e t r i c s t u d i e s o f l a m i n a t e d FRMs as p a r t o f t h e d e s i g n p r o c e s s . 6 . 3 Further Areas of Research The t o p i c p r e s e n t s many i n t e r e s t i n g a r e a s f o r f u r t h e r d e t a i l e d i n v e s t i -g a t i o n s . The f o l l o w i n g b r i e f l y o u t l i n e s a few o f t h e s e a r e a s o f r e s e a r c h t h a t a r e needed t o i m p r o v e o r e x t e n d t h e e x i s t i n g t h e o r e t i c a l p r o c e d u r e . L e t us o b s e r v e t h a t b e c a u s e o f t h e s i m p l i c i t y i n te rms o f t h e n e c e s s a r y i n p u t i n f o r m a t i o n f o r i m p l e m e n t a t i o n , i t i s p o s s i b l e t h a t t h e mode l w i l l n o t s a t i s f a c t o r i l y d e s c r i b e i n a l l d e t a i l s some c o m p l e x l o a d i n g h i s t o r y r e s p o n s e s w h i c h may be e n v i s i o n e d . F o r e x a m p l e , i t i s c o n c e i v a b l e t h a t i n c e r t a i n c a s e s a b i l i n e a r r e p r e s e n t a t i o n o f t h e s t r e s s - s t r a i n c u r v e s w i l l n o t be s u f f i c i e n t t o c a p t u r e t h e n o n l i n e a r i t i e s i n v o l v e d . A g r e a t e r f l e x i b l i t y o f m o d e l l i n g c a n be a c h i e v e d by e x t e n d i n g t h e t h e o r y t o i n c o r p o r a t e m u l t i l i n e a r s t r e s s - s t r a i n c u r v e s . One b a s i c f e a t u r e l a c k i n g f r o m t h e p r e s e n t f o r m u l a t i o n i s t h e d e s c r i p t i o n o f s t r e n g t h and s t i f f n e s s d i f f e r e n t i a l b e t w e e n t e n s i l e and c o m p r e s s i v e r e s p o n s e s . Such d i f f e r e n c e s a r e o f i m p o r t a n c e f o r c e r t a i n t y p e s o f f i b r e c o m p o s i t e s , p a r t i c u l a r l y t h e ones w i t h c a r b o n c o n t e n t s ( e . g . G r a p h i t e / E p o x i e s ) . An o b v i o u s e x t e n s i o n o f t h e mode l w o u l d be t o i n c l u d e 128 i n d e p e n d e n t d e s c r i p t i o n s o f t e n s i l e and c o m p r e s s i v e b e h a v i o u r b o t h i n i t i a l l y and s u b s e q u e n t l y ( i . e . B a u s c h i n g e r e f f e c t ) . The n e x t l o g i c a l s t e p i s t o a l l o w f o r t h e c o m p r e s s i b i l i t y o f t h e p l a s t i c f l o w , s i n c e p l a s t i c vo lume changes c a n be i m p o r t a n t i n some t y p e s o f f i b r e - r e i n f o r c e d m a t e r i a l s . A d d i -t i o n a l p o s t - f a i l u r e s o f t e n i n g schemes and p l y f a i l u r e c r i t e r i a s h o u l d be i n v e s t i g a t e d . T h i s s h o u l d be comb ined w i t h c a r e f u l e x p e r i m e n t a l o b s e r v a t i o n s o f t h e f a i l u r e p r o c e s s . One o f t h e m a j o r c a u s e s o f f a i l u r e w h i c h i s p a r t i c u l a r l y o p e r a t i v e i n n o t c h e d l a m i n a t e s , i s d e l a m i n a t i o n . T h i s t y p e o f f a i l u r e i s o f t e n p r e c i p i t a -t e d b y h i g h i n t e r l a m i n a r s t r e s s e s t h a t e x i s t w i t h i n a b o u n d a r y l a y e r c l o s e t o t h e f r e e edge r e g i o n o f t h e l a m i n a t e ( i . e . h o l e o r n o t c h b o u n d a r y ) . These s t r e s s e s i n f l u e n c e t h e s t r e s s c o n c e n t r a t i o n a r o u n d t h e h o l e and may p l a y an i m p o r t a n t r o l e i n t h e y i e l d i n g a n d / o r f a i l u r e o f l a m i n a t e s . The p r e s e n t f i n i t e e l e m e n t p r o g r a m s h o u l d be e x t e n d e d t o t a k e d e l a m i n a t i o n s i n t o a c c o u n t . T h i s i s u s u a l l y a c h i e v e d b y e x t e n d i n g t h e p r o g r a m t o t h r e e d i m e n s i o n s . However , r e l i a b l e t h r e e - d i m e n s i o n a l f a i l u r e t h e o r i e s a r e s t i l l r e q u i r e d t o p r e d i c t t h e o n s e t and s u b s e q u e n t g r o w t h o f d e l a m i n a t i o n . A n o t h e r u s e f u l a p p l i c a t i o n o f t h e mode l w o u l d be t o s t u d y t h e n o n l i n e a r b e h a v i o u r o f p l a t e s u n d e r b e n d i n g . 129 R E F E R E N C E S A b o u d i , J . ( 1 9 8 4 ) , " E f f e c t i v e B e h a v i o r o f I n e l a s t i c F i b r e - R e i n f o r c e d C o m p o s i t e s , " I n t . J . Engng . S c i . , 2 2 , p p . 4 3 9 - 4 4 9 . A b o u d i , J . ( 1 9 8 6 ) , " E l a s t o p l a s t i c i t y T h e o r y f o r C o m p o s i t e M a t e r i a l s , " S o l i d M e c h a n i c s A r c h i v e s , 1 1 / 3 , p p . 1 4 1 - 1 8 3 . Adams, D . F . ( 1 9 7 0 ) , " I n e l a s t i c A n a l y s i s o f a U n i d i r e c t i o n a l C o m p o s i t e S u b j e c t e d t o T r a n s v e r s e N o r m a l L o a d i n g , " J . Comp. M a t s . , 4 , p . 3 1 0 . Adams, D . F . ( 1 9 7 4 ) , " E l a s t o p l a s t i c B e h a v i o u r o f C o m p o s i t e s , " i n M e c h a n i c s o f C o m p o s i t e M a t e r i a l s , V o l . 2 o f C o m p o s i t e M a t e r i a l s ( E d i t e d b y G . R . S e n d e c k y j ) , A c a d e m i c P r e s s , N . Y . , p p . 1 6 9 - 2 0 8 . A d k i n s , J . E . , and R i v l i n , R . S . ( 1 9 5 5 ) , " L a r g e E l a s t i c D e f o r m a t i o n s o f I s o t r o p i c M a t e r i a l s , X . R e i n f o r c e m e n t b y I n e x t e n s i b l e C o r d s , " P h i l o s o p h i c a l T r a n s a c t i o n s , 2 4 8 , A - 9 4 4 , p . 2 0 1 . A s h k e n a z i , E . K . , ( 1 9 6 5 ) , " P r o b l e m s o f t h e A n i s o t r o p y o f S t r e n g t h , " M e k h a n i k a P o l i m e r o v , _1, p p . 7 9 - 9 2 , ( P o l y m e r M e c h a n i c s , JL, p p . 6 0 - 7 0 ) . A x e l s s o n , K . and S a m u e l s s o n , A . ( 1 9 7 9 ) , " F i n i t e E l e m e n t A n a l y s i s o f E l a s t i c - P l a s t i c M a t e r i a l s D i s p l a y i n g M i x e d H a r d e n i n g , " I n t . J . Numer. Methods E n g . , 14 , p p , 2 1 1 - 2 2 5 . A z z i , V . D . and T s a i , S . W . , ( 1 9 6 5 ) , " A n i s o t r o p i c S t r e n g t h o f C o m p o s i t e s " , E x p . M e c h . , 5 , p p . 2 8 3 - 2 8 8 . B a h a e i - E l - D i n , Y . A . and D v o r a k , G . J . , ( 1 9 8 0 ) , " P l a s t i c D e f o r m a t i o n o f a L a m i n a t e d P l a t e w i t h a H o l e " , T r a n s . ASME, J . A p p l . M e c h . , 4 7 , p p . 8 2 7 - 8 3 2 . B a h a e i - E l - D i n , Y . A . , and D v o r a k , G . J . ( 1 9 8 2 ) , " P l a s t i c i t y A n a l y s i s o f L a m i n a t e d C o m p o s i t e P l a t e s , " T r a n s . ASME, J . A p p l . M e c h . , 4 9 , p p . 7 4 0 - 7 4 6 . B a l t o v , A . and Sawczuk , A . ( 1 9 6 5 ) , " A R u l e o f A n i s o t r o p i c H a r d e n i n g , " A c t a M e c h . , 1 / 2 , p p . 8 1 - 9 2 . B l a n d , D.R. ( 1 9 5 7 ) , "The A s s o c i a t e d F l o w R u l e o f P l a s t i c i t y , " J . Mech . P h y s . S o l i d s , 6 , p p . 7 1 - 7 8 . B u s h n e l l , D. , ( 1 9 7 7 ) , " A S t r a t e g y f o r t h e S o l u t i o n o f P r o b l e m s I n v o l v i n g L a r g e D e f l e c t i o n s , P l a s t i c i t y and C r e e p , " I n t . J . Numer. M e t h . E n g . , 1 1 , p p . 6 8 3 - 7 0 8 . C h a m i s , C . C . and S e n d e c k y j , G . P . , ( 1 9 6 8 ) , " C r i t i q u e on T h e o r i e s P r e d i c t i n g T h e r m o e l a s t i c P r o p e r t i e s o f F i b r o u s C o m p o s i t e s " , J . Comp. M a t s . , 2 , p p . 3 3 2 - 3 5 8 . C h a n g , F . K . , and C h a n g , K . Y . ( 1 9 8 7 ) , " A P r o g r e s s i v e Damage Mode l f o r L a m i n a t e d C o m p o s i t e s C o n t a i n i n g S t r e s s C o n c e n t r a t i o n s , " J . Comp. M a t s . , 2 1 , p p . 8 3 4 - 8 5 5 . 130 C h i u , K . D . ( 1 9 6 9 ) , " U l t i m a t e S t r e n g t h s o f L a m i n a t e d C o m p o s i t e s , " J . Comp. M a t s . , 3 , p p . 5 7 8 - 5 8 2 . C h r i s t e n s e n , R . M . ( 1 9 7 9 ) , M e c h a n i c s o f C o m p o s i t e M a t e r i a l s , J o h n W i l e y & S o n s . C h r i s t e n s e n , R . M . ( 1 9 8 5 ) , " F i b r e R e i n f o r c e d C o m p o s i t e M a t e r i a l s , " A p p l . Mech . R e v . , 3 8 , p p . 1 2 6 7 - 1 2 7 0 . C r a d d o c k , J . N . and Champagne, D . J . ( 1 9 8 5 ) , "A C o m p a r i s o n o f F a i l u r e C r i t e r i a f o r L a m i n a t e d C o m p o s i t e M a t e r i a l s , " P r o c . 2 3 r d S t r u c t u r a l Dynamics and M a t e r i a l s C o n f e r e n c e , O r l a n d o , F l o r i d a , A p r i l 1985 , p p . 2 6 8 - 2 7 8 . Dodds , R . H . , ( 1 9 8 7 ) , " N u m e r i c a l T e c h n i q u e s f o r P l a s t i c i t y C o m p u t a t i o n s i n F i n i t e E l e m e n t A n a l y s i s " , Comput. & S t r u c t . , 26 ( 5 ) , p p . 7 6 7 - 7 7 9 . D r u c k e r , D . C . , ( 1 9 7 5 ) , " Y i e l d i n g , F l o w and F a i l u r e " , i n I n e l a s t i c B e h a v i o u r o f C o m p o s i t e M a t e r i a l s ( E d i t e d b y C. H e r a k o v i c h ) , p p . 1 - 1 5 . Dubey , R. and H i l l i e r , M . J . , ( 1 9 7 2 ) , " Y i e l d C r i t e r i a and t h e B a u s c h i n g e r E f f e c t f o r a P l a s t i c S o l i d " , T r a n s . ASME, J . B a s i c E n g . , 9 4 , p p . 2 2 8 - 2 3 0 . D v o r a k , G . J . , R a o , M . S . M . , and T a r n , J . Q . ( 1 9 7 3 ) , " Y i e l d i n g i n U n i d i r e c t i o n a l C o m p o s i t e s Under E x t e r n a l Loads and T e m p e r a t u r e C h a n g e s , " J . Comp. M a t s . , 7 , p p . 1 9 4 - 2 1 7 . D v o r a k , G . J . , R a o , M . S . M . , and T a r n , J . Q . ( 1 9 7 4 ) , " G e n e r a l i z e d I n i t i a l Y i e l d S u r f a c e s f o r U n i d i r e c t i o n a l C o m p o s i t e s , " T r a n s . ASME, J . A p p l . M e c h . , 4 1 , p p . 2 4 9 - 2 5 3 . D v o r a k , G . J . and B a h e i - E l - D i n , Y . A . ( 1 9 7 9 ) , " E l a s t i c - P l a s t i c B e h a v i o u r o f F i b r o u s C o m p o s i t e s , " J . Mech . P h y s . S o l i d s , 2 7 , p p . 5 1 - 7 2 . D v o r a k , G . J . and B a h e i - E l - D i n , Y . A . ( 1 9 8 2 ) , " P l a s t i c i t y A n a l y s i s o f F i b r o u s C o m p o s i t e s , " T r a n s . ASME, J . A p p l . M e c h . , 4 9 , p p . 3 2 7 - 3 3 5 . D v o r a k , G . J . and J o h n s o n , W . S . , ( 1 9 8 0 ) , " F a t i g u e o f M e t a l / M a t r i x C o m p o s i t e s " , I n t . J . F r a c , 16 , p . 5 8 5 . F a n , W.X. ( 1 9 8 7 ) , "On P h e n o m e n o l o g i c a l A n i s o t r o p i c F a i l u r e C r i t e r i a , " C o m p o s i t e s S c i . T e c h . , 2 8 , p p . 2 6 9 - 2 7 8 . F o y e , R . L . ( 1 9 7 3 ) , " T h e o r e t i c a l P o s t - Y i e l d i n g B e h a v i o u r o f C o m p o s i t e L a m i n a t e s , P a r t I - I n e l a s t i c M i c r o m e c h a n i c s , " J . Comp. M a t s . , 7 , p p . 1 7 8 - 1 9 3 . F o y e , R . L , and B a k e r , D . J . ( 1 9 7 1 ) , " D e s i g n / A n a l y s i s Methods f o r A d v a n c e d C o m p o s i t e S t r u c t u r e s , " A F M L - T R - 7 0 - 2 9 9 , V o l . I , A n a l y s i s ; V o l . I I , Computer P r o g r a m s . F r a n c h i , A . and G e n n a , F . , ( 1 9 8 7 ) , " A N u m e r i c a l Scheme f o r I n t e g r a t i n g t h e R a t e P l a s t i c i t y E q u a t i o n s w i t h an A P r i o r i E r r o r C o n t r o l " , Comput. M e t h s . A p p l . M e c h . E n g . , 6 0 , p p . 3 1 7 - 3 4 2 . 131 F r a n c i s , P . H . and B e r t , C.W. ( 1 9 7 5 ) , " C o m p o s i t e M a t e r i a l M e c h a n i c s : I n e l a s t i c i t y and F a i l u r e " , F i b r e S c i . T e c h . , 8 , p p . 1 - 1 9 . G o t o h , M. ( 1 9 7 7 ) , " A T h e o r y o f P l a s t i c A n i s o t r o p y B a s e d on a Y i e l d F u n c t i o n o f F o u r t h O r d e r ( P l a n e S t r e s s S t a t e ) , " I n t . J . M e c h . S c i . , 1 9 , p p . 5 0 5 - 5 2 0 . G r e s z c z u k , L . B . (1972) , " S t r e s s C o n c e n t r a t i o n s and F a i l u r e C r i t e r i a f o r O r t h o t r o p i c and A n i s o t r o p i c P l a t e s w i t h C i r c u l a r O p e n i n g s " , C o m p o s i t e M a t e r i a l s : T e s t i n g and D e s i g n , 2nd C o n f e r e n c e , ASTM STP 4 9 7 , p p . 3 6 3 - 3 8 1 . G r i f f i n , O . H . ( 1 9 8 2 ) , " E v a l u a t i o n o f F i n i t e E l e m e n t S o f t w a r e P a c k a g e s f o r S t r e s s A n a l y s i s o f L a m i n a t e d C o m p o s i t e s , " C o m p o s i t e s T e c h n o l o g y R e v i e w , 4 , W i n t e r 1982 , p p . 1 3 6 - 1 4 1 . G r i f f i n , O . H . , Kamat , M . P . , and H e r a k o v i c h , C . T . ( 1 9 8 1 ) , " T h r e e - d i m e n s i o n a l I n e l a s t i c F i n i t e E l e m e n t A n a l y s i s o f L a m i n a t e d C o m p o s i t e s , " J . Comp. M a t s . , 1 5 , p p . 5 4 3 - 5 6 0 . Hahn , H . T . ( 1 9 7 3 ) , " N o n l i n e a r B e h a v i o r o f L a m i n a t e d C o m p o s i t e s , " J . Comp. M a t s . , 7 , p p . 2 5 7 - 2 7 1 . H a h n , H . T . a n d T s a i , S .W. ( 1 9 7 3 ) , " N o n l i n e a r E l a s t i c B e h a v i o r o f U n i d i r e c t i o n a l C o m p o s i t e L a m i n a t e s , " J . Comp. M a t s . , 7 , p p . 1 0 2 - 1 1 8 . H a s h i n , Z . ( 1 9 8 0 ) , " F a i l u r e C r i t e r i a f o r U n i d i r e c t i o n a l F i b r e C o m p o s i t e s , " T r a n s . ASME, J . A p p l . M e c h . , 4 7 , p p . 3 2 9 - 3 3 4 . H a s h i n , Z . ( 1 9 8 3 ) , " A n a l y s i s o f C o m p o s i t e M a t e r i a l s - A S u r v e y , " T r a n s . ASME, J . A p p l . M e c h . , 5 0 , p p . 4 8 1 - 5 0 5 . H a s h i n , Z . ( 1 9 8 5 ) , " A n a l y s i s o f C r a c k e d L a m i n a t e s : A V a r i a t i o n a l A p p r o a c h , " M e c h . M a t e r . , 4 , p p . 1 2 1 - 1 3 6 . H a s h i n , Z . ( 1 9 8 6 ) , " A n a l y s i s o f S t i f f n e s s R e d u c t i o n o f C r a c k e d C r o s s - P l y L a m i n a t e s , " E n g n g . F r a c . M e c h . , 2 5 , p p . 7 7 1 - 7 7 8 . • H a s h i n , Z . ( 1 9 8 7 ) , " A n a l y s i s o f O r t h o g o n a l l y C r a c k e d L a m i n a t e s Under T e n s i o n , " T r a n s . ASME, J . A p p l . M e c h . , 5 4 , p p . 8 7 2 - 8 7 9 . H a s h i n , Z . , B a g c h i , D. and R o s e n , B.W. ( 1 9 7 4 ) , " N o n l i n e a r B e h a v i o r o f F i b r e C o m p o s i t e L a m i n a t e s , " NASA R e p t . No . C R - 2 3 1 3 . H i g h s m i t h , A . L . and R e i f s n i d e r , K . L . ( 1 9 8 2 ) , " S t i f f n e s s - R e d u c t i o n Mechan isms i n C o m p o s i t e L a m i n a t e s , " Damage i n C o m p o s i t e M a t e r i a l s ( E d i t e d b y K . L . R e i f s n i d e r ) , ASTM STP 7 7 5 , p p . 1 0 3 - 1 1 7 . H i l l , R. ( 1 9 5 0 ) , The M a t h e m a t i c a l T h e o r y o f P l a s t i c i t y , O x f o r d U n i v e r s i t y P r e s s , O x f o r d . H i l l , R. ( 1 9 6 4 ) , " T h e o r y o f M e c h a n i c a l P r o p e r t i e s o f F i b r e - S t r e n g t h e n e d M a t e r i a l s : I . E l a s t i c B e h a v i o r , I I . I n e l a s t i c B e h a v i o r , " J . Mech . P h y s . S o l i d s , 12 , p p . 1 9 9 - 2 1 8 . 132 H i l l , R. ( 1 9 6 5 ) , " T h e o r y o f M e c h a n i c a l P r o p e r t i e s o f F i b r e - S t r e n g t h e n e d M a t e r i a l s : I I I . S e l f - C o n s i s t e n t M o d e l , " J . Mech . P h y s . S o l i d s , 13 , p p . 1 8 9 - 1 9 8 . Hodge , P . G . and W h i t e , G . N . , ( 1 9 5 0 ) , " A Q u a n t i t a t i v e C o m p a r i s o n o f F l o w and D e f o r m a t i o n T h e o r i e s o f P l a s t i c i t y " , T r a n s . ASME, J . A p p l . M e c h . , 1 7 , p p . 1 8 0 - 1 8 4 . H o f f m a n , 0 . ( 1 9 6 7 ) , "The B r i t t l e S t r e n g t h o f O r t h o t r o p i c M a t e r i a l s , " J . Comp. M a t s . , 1 , p p . 2 0 0 - 2 0 6 . H o f f m a n , 0 . ( 1 9 7 9 ) , " A C o n t i n u u m M o d e l f o r t h e E n g i n e e r i n g A n a l y s i s o f M e t a l M a t r i x C o m p o s i t e s , " Modern Deve lopments i n C o m p o s i t e M a t e r i a l s and S t r u c t u r e s ( E d i t e d b y J . R . V i n s o n ) , A m e r i c a n S o c i e t y o f M e c h a n i c a l E n g i n e e r s , p p . 1 0 1 - 1 0 7 . H o w l a n d , R . C . J . ( 1 9 3 0 ) , "On t h e S t r e s s e s i n t h e N e i g h b o u r h o o d o f a C i r c u l a r H o l e i n a S t r i p Under T e n s i o n " , P h i l . T r a n s . R o y . S o c , A 2 2 9 , p p . 4 9 - 8 6 . H u , L.W. ( 1 9 5 6 ) , " S t u d i e s on P l a s t i c F l o w o f A n i s o t r o p i c M e t a l s , " T r a n s . ASME, J . A p p l . M e c h . , 2 3 , p p . 4 4 4 - 4 5 0 . Hu , L.W. ( 1 9 5 8 ) , " M o d i f i e d T e s c a ' s Y i e l d C o n d i t i o n and A s s o c i a t e d F l o w R u l e s f o r A n i s o t r o p i c M a t e r i a l s and A p p l i c a t i o n s , " J . F r a n k l i n I n s t i t u t e , 2 6 5 , p p . 1 8 7 - 2 0 4 . Huang , W.C. ( 1 9 7 1 ) , " P l a s t i c B e h a v i o u r o f Some C o m p o s i t e M a t e r i a l s , " J . Comp. M a t s . , 5 , p p . 3 2 0 - 3 3 8 . H i i t t e r , U . , S c h e l l i n g , H. , and K r a u s s , H. ( 1 9 7 4 ) , "An E x p e r i m e n t a l s t u d y t o D e t e r m i n e F a i l u r e E n v e l o p e o f C o m p o s i t e M a t e r i a l s w i t h T u b u l a r Spec imens Under Combined Loads and C o m p a r i s o n Between G e n e r a l C l a s s i c a l C r i t e r i a , " NATO-AGARD C o n f e r e n c e P r o c e e d i n g s No . 163 , 3 9 t h M e e t i n g o f t h e S t r u c t u r e s and M a t e r i a l s P a n e l i n M u n i c h , Germany, 13 -19 O c t o b e r 1974 , P a p e r No . 3 , p p . 3 - 1 1 . J e n s e n , W . R . , F a l b y , W .E . and P r i n c e , N . ( 1 9 6 6 ) , " M a t r i x A n a l y s i s Methods f o r A n i s o t r o p i c I n e l a s t i c S t r u c t u r e s , " A F F D L - T R - 6 5 - 2 2 0 , A i r F o r c e F l i g h t Dynamic L a b o r a t o r y , W r i g h t - P a t t e r s o n A i r F o r c e B a s e , O h i o , A p r i l 1 9 6 6 . . J o n e s , R . M . ( 1 9 7 5 ) , M e c h a n i c s o f C o m p o s i t e M a t e r i a l s , M c G r a w - H i l l , New Y o r k . J o n e s , R . M . and M o r g a n , H . S . ( 1 9 7 7 ) , " A n a l y s i s o f N o n l i n e a r S t r e s s - S t r a i n B e h a v i o r o f F i b e r - R e i n f o r c e d M a t e r i a l s , " A IAA J . , 1 5 , p p . 1 6 6 9 - 1 6 7 6 . K a c h a n o v , L . M . ( 1 9 7 1 ) , F o u n d a t i o n o f t h e T h e o r y o f P l a s t i c i t y , N o r t h - H o l l a n d , Amste rdam. K e n a g a , D . , D o y l e , J . F . and S u n , C . T . ( 1 9 8 7 ) , "The C h a r a c t e r i z a t i o n o f B o r o n / A l u m i n u m C o m p o s i t e i n t h e N o n l i n e a r R a n g e a s a n O r t h o t r o p i c E l a s t i c - P l a s t i c M a t e r i a l , " J . Comp. M a t s . , 2 1 , p p . 5 1 6 - 5 3 1 . K r i e g , R . D . and K r i e g , D . B . , ( 1 9 7 7 ) , " A c c u r a c i e s o f N u m e r i c a l S o l u t i o n Methods f o r t h e E l a s t i c - P e r f e c t l y P l a s t i c M o d e l " , J . o f P r e s s u r e V e s s e l T e c h . , 99 ( 4 ) , p p . 5 1 0 - 5 1 5 . 133 L a b o s s i e r e , P . and N e a l e , K.W. , ( 1 9 8 7 a ) , " M a c r o s c o p i c F a i l u r e C r i t e r i a f o r F i b r e - R e i n f o r c e d C o m p o s i t e M a t e r i a l s " , S o l i d M e c h a n i c s A r c h i v e s 1 2 / 2 , p p . 4 3 9 - 4 5 0 . L a b o s s i e r e , P . and N e a l e , K.W. ( 1 9 8 7 b ) , "On t h e D e t e r m i n a t i o n o f t h e S t r e n g t h P a r a m e t e r s i n t h e T e n s o r P o l y n o m i a l F a i l u r e C r i t e r i o n , " J . S t r a i n A n a l y s i s , 2 2 , p p . 1 5 5 - 1 6 1 . L a n c e , R . H . and R o b i n s o n , D . N . ( 1 9 7 1 ) , " A Maximum Shear S t r e s s T h e o r y o f P l a s t i c F a i l u r e o f F i b r e - R e i n f o r c e d M a t e r i a l s , " J . Mech . P h y s . S o l i d s , 19 , p p . 4 9 - 6 0 . L a w s , N . , D v o r a k , G . J . and H e j a z i , M. ( 1 9 8 3 ) , " S t i f f n e s s Changes i n U n i d i r e c -t i o n a l C o m p o s i t e s C a u s e d by C r a c k S y s t e m s , " M e c h . M a t e r . , 2 , p p . 1 2 3 - 1 3 7 . L e e , J . D . ( 1 9 8 2 ) , " T h r e e D i m e n s i o n a l F i n i t e E l e m e n t A n a l y s i s o f Damage A c c u m u l a t i o n i n C o m p o s i t e L a m i n a t e s , " Computers & S t r u c t u r e s , 15 , p p . 3 3 5 - 3 5 0 . Leewood , A . R . , ( 1 9 8 5 ) , " N u m e r i c a l S t u d i e s o f P r o b l e m s i n A n i s o t r o p i c P l a s t i c i t y " , P h . D . T h e s i s , S c h o o l o f A e r o n a u t i c s and A s t r o n a u t i c s , Pu rdue U n i v e r s i t y . Leewood , A . R . , D o y l e , J . F . and S u n , C . T . ( 1 9 8 7 ) , " F i n i t e E l e m e n t P r o g r a m f o r A n a l y s i s o f L a m i n a t e d A n i s o t r o p i c E l a s t o p l a s t i c M a t e r i a l s , " Computers &. S t r u c t u r e s , 2 5 , p p . 7 4 9 - 7 5 8 . L i n , T . H . , S a l i n a s , D. and I t o , Y . M . ( 1 9 7 2 ) , " I n i t i a l Y i e l d S u r f a c e o f a U n i d i r e c t i o n a l l y R e i n f o r c e d C o m p o s i t e , " T r a n s . ASME, J . A p p l . M e c h . , 3 9 , p p . 3 2 1 - 3 2 6 . M a r c a l , P . V . and K i n g , I . P . , ( 1 9 6 7 ) , " E l a s t i c - P l a s t i c A n a l y s i s o f Two D i m e n s i o n a l S t r e s s Sys tems b y t h e F i n i t e E l e m e n t M e t h o d " , I n t . J . Mech . S c i . , 9 , p p . 1 4 3 - 1 5 5 . M e n d e l s o n , A . ( 1 9 6 8 ) , P l a s t i c i t y : T h e o r y and A p p l i c a t i o n , M a c M i l l a n , L o n d o n . M I L - H D B K - 1 7 , P l a s t i c s f o r F l i g h t V e h i c l e s , P a r t I , R e i n f o r c e d P l a s t i c s , U . S . Government P r i n t i n g O f f i c e , W a s h i n g t o n , D . C . , Nov . 5 , 1959. M u l h e r n , J . F . , R o g e r s , T . G . and S p e n c e r , A . J . M . ( 1 9 6 7 ) , " A C o n t i n u u m M o d e l f o r F i b r e - R e i n f o r c e d P l a s t i c M a t e r i a l s , " P r o c . Roy . S o c , A . 3 0 1 , p p . 4 7 3 - 4 9 2 . M u l h e r n , J . F . , R o g e r s , T . G . and S p e n c e r , A . J . M . ( 1 9 6 9 ) , "A C o n t i n u u m T h e o r y o f a P l a s t i c - E l a s t i c F i b r e - R e i n f o r c e d M a t e r i a l , " I n t . J . E n g n g . , S c i . , 7 , p p . 1 2 9 - 1 5 2 . N a g h d i , P . M . ( 1 9 6 0 ) , " S t r e s s - S t r a i n R e l a t i o n s i n P l a s t i c i t y and Thermo -p l a s t i c i t y , " i n P l a s t i c i t y , ONR S t r u c t u r a l M e c h a n i c s S e r i e s , P r o c . o f t h e 2nd Symposium on N a v a l S t r u c t u r a l M e c h a n i c s ( E d i t e d b y E . H . Lee and P . S . Symonds) , Pe rgamon , E l m s f o r d , N . Y . , p p . 1 2 1 - 1 6 9 . 134 N a g t e g a a l , J . C . , P a r k s , D.M. and R i c e , J . R . , ( 1 9 7 4 ) , "On N u m e r i c a l l y A c c u r a t e F i n i t e E l e m e n t S o l u t i o n s i n t h e F u l l y P l a s t i c R a n g e " , Comput. M e t h s . A p p l . M e c h . E n g n g . , 4 , p p . 1 5 3 - 1 7 7 . N a h a s , M . N . ( 1 9 8 6 ) , " S u r v e y o f F a i l u r e and P o s t - F a i l u r e T h e o r i e s o f L a m i n a t e d F i b e r - R e i n f o r c e d C o m p o s i t e s , " J . C o m p o s i t e s T e c h n o l o g y & R e s e a r c h , 8 , p p . 1 3 8 - 1 5 3 . N a h a s , M . N . ( 1 9 8 4 ) , " A n a l y s i s o f N o n - L i n e a r S t r e s s - S t r a i n Response o f L a m i n a t e d F i b r e - R e i n f o r c e d C o m p o s i t e s , " F i b r e S c i . T e c h . , 2 0 , p p . 2 9 7 - 3 1 3 . N a y a k , G . C . and Z i e n k i e w i c z , O . C . , ( 1 9 7 2 ) , " E l a s t o - P l a s t i c S t r e s s A n a l y s i s : A G e n e r a l i z a t i o n f o r V a r i o u s C o n s t i t u t i v e R e l a t i o n s I n c l u d i n g S t r a i n S o f t e n i n g " , I n t . J . Numer. M e t h . E n g n g . , 1 5 , p p . 1 1 3 - 1 3 5 . N y s s e n , C , ( 1 9 8 1 ) , " A n E f f i c i e n t and A c c u r a t e I t e r a t i v e Method A l l o w i n g L a r g e I n c r e m e n t a l S t e p s t o S o l v e E l a s t o - P l a s t i c P r o b l e m s , " Comput. o f S t r u c t . , 13_, p p . 6 3 - 7 1 . O c h o a , 0 . 0 . and E n g b l o m , J . J . ( 1 9 8 7 ) , " A n a l y s i s o f P r o g r e s s i v e F a i l u r e i n C o m p o s i t e s , " C o m p o s i t e s S c i . T e c h . , 2 8 , p p . 8 7 - 1 0 2 . O l s o n , M.D. and A n d e r s o n , D . L . , ( 1 9 8 8 ) , " M o d e l l i n g S t r u c t u r a l R e s p o n s e o f F i b r e - R e i n f o r c e d S t r u c t u r e s t o A i r B l a s t " , DRES C o n t r a c t R e p o r t , D e p t . o f C i v i l E n g . , UBC. O r t i z , M. and P o p o v , E . P . , ( 1 9 8 5 ) , " A c c u r a c y and S t a b i l i t y o f I n t e g r a t i o n A l g o r i t h m s f o r E l a s t o p l a s t i c C o n s t i t u t i v e R e l a t i o n s " , I n t . J . Numer. M e t h . E n g . , 2 1 , p p . 1 5 6 1 - 1 5 7 6 . Owen, D . R . J , and H i n t o n , E . , ( 1 9 8 0 ) , " F i n i t e E l e m e n t s i n P l a s t i c i t y " , McGraw-H i l l , New Y o r k . P e t i t , P . H . and Waddoups, M . E . ( 1 9 6 9 ) , "A Method o f P r e d i c t i n g t h e N o n l i n e a r B e h a v i o r o f L a m i n a t e d C o m p o s i t e s , " J . Comp. M a t s . , 3, p p . 2 - 1 9 . P i n d e r a , M . J . and H e r a k o v i c h , C . T . ( 1 9 8 3 ) , "An E n d o c h r o n i c Mode l f o r t h e Response o f U n i d i r e c t i o n a l C o m p o s i t e s Under O f f - A x i s T e n s i l e L o a d , " i n M e c h a n i c s o f C o m p o s i t e M a t e r i a l s : R e c e n t A d v a n c e s , P r o c e e d i n g s IUTAM Symposium on M e c h a n i c s o f C o m p o s i t e M a t e r i a l s ( E d i t e d by Z . H a s h i n and C . T . H e r a k o v i c h ) , Pergamon P r e s s , p p . 3 6 7 - 3 8 1 . P i p e s , R . B . , and P a g a n o , N . J . ( 1 9 7 0 ) , " I n t e r l a m i n a r S t r e s s e s i n C o m p o s i t e L a m i n a t e s Under U n i f o r m A x i a l E x t e n s i o n , " J . Comp. M a t s . , 4 , p p . 5 3 8 - 5 4 8 . P r a g e r , W . , ( 1 9 4 9 ) , " R e c e n t D e v e l o p m e n t s i n t h e M a t h e m a t i c a l T h e o r y o f P l a s t i c i t y " , J . A p p l . P h y s . , 2 0 , p p . 2 3 5 - 2 4 1 . P r a g e r , W. ( 1 9 5 5 ) , "The T h e o r y o f P l a s t i c i t y : A S u r v e y o f R e c e n t A c h i e v e -m e n t s , " James C l a y t o n L e c t u r e , P r o c . I n s t n . Mech . E n g r s . ( L o n d o n ) , 169 , p p . 4 1 - 5 0 . P r a g e r , W. ( 1 9 6 9 ) , " P l a s t i c F a i l u r e o f F i b e r - R e i n f o r c e d M a t e r i a l s , " T r a n s . ASME, J . A p p l . M e c h . , 3 6 , p p . 7 7 0 - 7 7 3 . 135 P r a g e r , W. and Hodge, P . G . , ( 1 9 5 1 ) , " T h e o r y o f P e r f e c t l y P l a s t i c S o l i d s " , W i l e y , New Y o r k . Puppo , A . H . and E v e n s e n , H .A . ( 1 9 7 2 ) , " S t r e n g t h o f A n i s o t r o p i c M a t e r i a l s Under Combined S t r e s s e s , " A IAA J . , 10 , p p . 4 6 8 - 4 7 4 . Ramberg , W. and O s g o o d , W.B . ( 1 9 4 3 ) , " D e s c r i p t i o n o f S t r e s s - S t r a i n C u r v e s b y T h r e e P a r a m e t e r s , " N A S A - T N - 9 0 2 . R e d d y , J . N . and P a n d e y , A . K . ( 1 9 8 7 ) , "A F i r s t - P l y F a i l u r e A n a l y s i s o f C o m p o s i t e L a m i n a t e s , " Computers &. S t r u c t u r e s , 2 5 , p p . 3 7 1 - 3 9 3 . R e e s , D.W.A. ( 1 9 8 4 ) , "An E x a m i n a t i o n o f Y i e l d S u r f a c e D i s t o r t i o n and T r a n s l a t i o n " , A c t a M e c h a n i c a , 52, p p . 1 5 - 4 0 . R i v l i n , R . S . ( 1 9 8 1 ) , "Some Comments on t h e E n d o c h r o n i c T h e o r y o f P l a s t i c i t y , " I n t . J . S o l i d s S t r u c t . , 17 , p p . 2 3 1 - 2 4 8 . R i z z i , S . A . , Leewood , A . R . , D o y l e , J . F . and S u n , C . T . , ( 1 9 8 7 ) , " E l a s t i c -P l a s t i c A n a l y s i s o f B o r o n / A l u m i n u m C o m p o s i t e Under C o n s t r a i n e d P l a s t i c i t y C o n d i t i o n s " , J . Comp. M a t s . , 2 1 , p p . 7 3 4 - 7 4 9 . R o s e n , B .W . , ( 1 9 7 2 ) , "A S i m p l e P r o c e d u r e f o r E x p e r i m e n t a l D e t e r m i n a t i o n o f t h e L o n g i t u d i n a l Shea r Modu lus o f U n i d i r e c t i o n a l C o m p o s i t e s " , J . Comp. M a t s . , 6 , p p . 5 5 5 - 5 5 7 . R o w l a n d s , R . E . ( 1 9 8 5 ) , " S t r e n g t h ( F a i l u r e ) T h e o r i e s and t h e i r E x p e r i m e n t a l C o r r e l a t i o n , " i n Handbook o f C o m p o s i t e s , V o l . 3 : F a i l u r e M e c h a n i c s o f C o m p o s i t e s ( E d i t e d b y G . C . S i h and A . M . S k u d r a ) , E l s e v i e r S c i e n c e P u b l i s h e r s , Amste rdam, p p . 7 1 - 1 2 5 . S a n d h u , R . S . ( 1 9 7 6 ) , " N o n - L i n e a r B e h a v i o u r o f U n i d i r e c t i o n a l and A n g l e - P l y L a m i n a t e s , " J . A i r c r a f t , 13 , p p . 1 0 4 - 1 1 1 . S a n d h u , R . S . , S e n d e c k y j , G . P . and G a l l o , R . L . ( 1 9 8 3 ) , " M o d e l l i n g o f t h e F a i l u r e P r o c e s s i n N o t c h e d L a m i n a t e s , " i n M e c h a n i c s o f C o m p o s i t e M a t e r i a l s : R e c e n t A d v a n c e s , P r o c e e d i n g s IUTAM Symposium on M e c h a n i c s o f C o m p o s i t e M a t e r -i a l s ( E d i t e d b y Z . H a s h i n and C . T . H e r a k o v i c h ) , Pergamon P r e s s , p p . 1 7 9 - 1 8 9 . S c h a p e r y , R . A . ( 1 9 7 4 ) , " V i s c o e l a s t i c B e h a v i o u r and A n a l y s i s o f C o m p o s i t e M a t e r i a l s , " i n M e c h a n i c s o f C o m p o s i t e M a t e r i a l s , V o l . 2 o f C o m p o s i t e M a t e r i a l s ( E d i t e d b y G . P . S a n d e c k y j ) , A c a d e m i c P r e s s , New Y o r k , p p . 8 4 - 1 6 8 . S c h r e y e r , H . L . , K u l a k , R . F . and K r a m e r , M.M. ( 1 9 7 9 ) , " A c c u r a t e N u m e r i c a l S o l u t i o n s o f E l a s t i c - P l a s t i c M o d e l s " , T r a n s . ASME, J . P r e s s . V e s s e l T e c h . , 1 0 1 , p p . 2 2 6 - 2 3 4 . S h i h , C . F . and L e e , D. ( 1 9 7 8 ) , " F u r t h e r D e v e l o p m e n t s i n A n i s o t r o p i c P l a s t i c i t y , " T r a n s . ASME, J . E n g r g . M a t . and T e c h . , 100 , p p . 2 9 4 - 3 0 2 . S m i t h , C . S . and C h a l m e r s , D.W. , ( 1 9 8 6 ) , " D e s i g n o f S h i p S u p e r s t r u c t u r e s i n F i b r e - R e i n f o r c e d P l a s t i c " , RINA S p r i n g M e e t i n g s , P a p e r No . 3 , p p . 1 - 1 2 . S o k o l n i k o f f , I . S . ( 1 9 5 6 ) , M a t h e m a t i c a l T h e o r y o f E l a s t i c i t y , M c G r a w - H i l l , New Y o r k . 136 S o v a , J . A . and P o e , C . C . , ( 1 9 7 8 ) , " T e n s i l e S t r e s s - S t r a i n B e h a v i o u r o f B o r o n / A l u m i n u m L a m i n a t e s " , N A S A - T P - 1 1 1 7 . Swanson , S . R . and C h r i s t o f o r o u , A . P . ( 1 9 8 7 ) , " P r o g r e s s i v e F a i l u r e i n C a r b o n / E p o x y L a m i n a t e s Under B i a x i a l S t r e s s , " T r a n s . ASME, J . E n g r g . M a t . and T e c h . , 109 , p p . 1 2 - 1 6 . T a k a h a s h i , K . and C h o u , T.W. ( 1 9 8 7 ) , " N o n - L i n e a r D e f o r m a t i o n and F a i l u r e B e h a v i o u r o f C a r b o n / G l a s s H y b r i d L a m i n a t e s , " J . Comp. M a t s . , 2_1, p p . 3 9 6 - 4 2 0 . T a l r e j a , R. ( 1 9 8 5 ) , " A C o n t i n u u m M e c h a n i c s C h a r a c t e r i z a t i o n o f Damage i n C o m p o s i t e M a t e r i a l s , " P r o c . Roy . S o c . L o n d . , A . 3 9 9 , p p . 1 9 5 - 2 1 6 . T a l r e j a , R. ( 1 9 8 6 ) , " S t i f f n e s s P r o p e r t i e s o f C o m p o s i t e L a m i n a t e s w i t h M a t r i x C r a c k i n g and I n t e r i o r D e l a m i n a t i o n , " E n g r g . F r a c . M e c h . , 2 5 , p p . 7 5 1 - 7 6 2 . T e n n y s o n , R . C . , M a c D o n a l d , D. and N a n y a r o , A . P . ( 1 9 7 8 ) , " E v a l u a t i o n o f t h e T e n s o r P o l y n o m i a l F a i l u r e C r i t e r i o n f o r C o m p o s i t e M a t e r i a l s , " J . Comp. M a t s . , 12, p p . 6 3 - 7 5 . T e n n y s o n , R . C . , N a n y a r o , A . P . and Wharrara, G . E . , ( 1 9 8 0 ) , " A p p l i c a t i o n o f C u b i c P o l y n o m i a l S t r e n g t h C r i t e r i o n t o t h e F a i l u r e A n a l y s i s o f C o m p o s i t e M a t e r i a l s " , J . Comp. M a t s . S u p p l e m e n t , 14 . T h e o c a r i s , P . S . and M a r k e t o s , E . , ( 1 9 6 4 ) , " E l a s t i c - P l a s t i c A n a l y s i s o f P e r f o r a t e d T h i n S t r i p s o f a S t r a i n - H a r d e n i n g M a t e r i a l " , J . M e c h . P h y s . S o l i d s , 1 2 , p p . 3 7 7 - 3 9 0 . T s a i , S.W. and Hahn , H .T . ( 1 9 7 5 ) , " F a i l u r e A n a l y s i s o f C o m p o s i t e M a t e r i a l s , " i n I n e l a s t i c B e h a v i o r o f C o m p o s i t e M a t e r i a l s ( E d i t e d b y C . T . H e r a k o v i c h ) , V o l . 1 3 , ASME, New Y o r k , p p . 7 3 - 9 6 . T s a i , S.W. ( 1 9 6 5 ) , " S t r e n g t h C h a r a c t e r i s t i c s o f C o m p o s i t e M a t e r i a l s , " NASA C R - 2 2 4 . T s a i , S.W. and Wu, E . M . ( 1 9 7 1 ) , "A G e n e r a l T h e o r y o f S t r e n g t h f o r A n i s o t r o p i c M a t e r i a l s , " J . Comp. M a t s . , 5 , p p . 5 8 - 8 1 . V a l a n i s , K . C . ( 1 9 7 1 ) , " A T h e o r y o f V i s c o p l a s t i c i t y W i t h o u t a Y i e l d S u r f a c e , I : G e n e r a l T h e o r y ; I I : A p p l i c a t i o n t o M e c h a n i c a l B e h a v i o r o f M e t a l s , " A r c h . M e c h . , 2 3 , p p . 5 1 7 - 5 5 1 . V a l l i a p p a n , S . , ( 1 9 7 1 ) , " E l a s t i c - P l a s t i c A n a l y s i s o f A n i s o t r o p i c W o r k - H a r d e n i n g M a t e r i a l s " , U n i v e r s i t y o f New S o u t h W a l e s , K e n s i n g t o n , N . S . W . , A u s t r a l i a , R e p o r t No . R - 7 0 . Whang, B. ( 1 9 6 9 ) , " E l a s t o - P l a s t i c O r t h o t r o p i c P l a t e s and S h e l l s , " P r o c . Symp. on A p p l i c a t i o n o f F i n i t e E l e m e n t Method i n C i v i l E n g i n e e r i n g , V a n d e r b i l t U n i v e r s i t y T e n n e s s e e , p p . 4 8 1 - 5 1 5 . Wu, E . M . ( 1 9 7 4 ) , " P h e n o m e n o l o g i c a l A n i s o t r o p i c F a i l u r e C r i t e r i o n , " i n M e c h a n i c s o f C o m p o s i t e M a t e r i a l s , V o l . 2 o f C o m p o s i t e M a t e r i a l s ( E d i t e d b y G . P . S e n d e c k y j ) , A c a d e m i c P r e s s , New Y o r k , p p . 3 5 3 - 4 3 1 . 137 Z i e g l e r , H. ( 1 9 5 9 ) , "A M o d i f i c a t i o n o f P r a g e r ' s H a r d e n i n g R u l e , " Q u a r t . A p p l . M a t h . , 17> PP* 5 5 - 5 6 . Z i e n k i e w i c z , O . C . , ( 1 9 7 7 ) , "The F i n i t e E l e m e n t M e t h o d " , 3 r d E d i t i o n , McGraw H i l l . Z i e n k i e w i c z , O . C , V a l l i a p p a n , S . and K i n g , I . P . , ( 1 9 6 9 ) , " E l a s t o - P l a s t i c S o l u t i o n s o f E n g i n e e r i n g P r o b l e m s ' I n i t i a l S t r e s s ' , F i n i t e E l e m e n t A p p r o a c h " , I n t . J . Numer. M e t h . E n g . , 1 , p p . 7 5 - 1 0 0 . 138 APPENDIX A DETERMINATION OF THE ANISOTROPIC PARAMETERS OF THE YIELD FUNCTION I n o r d e r t o d e f i n e t h e c o m p l e t e y i e l d s u r f a c e o f a n i s o t r o p i c m a t e r i a l s , i t i s r e q u i r e d t o know t h e p a r a m e t e r s A ^ . , and k t h a t d e t e r m i n e i t s s h a p e , o r i g i n and s i z e , r e s p e c t i v e l y . F o r t h e y i e l d f u n c t i o n f E A . . ( o . - a . ) ( o . - a . ) - k* = 0 ( A . D i j i 1 J J a p h y s i c a l i n t e r p r e t a t i o n o f t h e A „ , and k may be made i n t h e f o l l o w i n g manner . L e t T j , T 2 and T3 d e n o t e t h e t e n s i l e y i e l d s t r e s s e s i n t h e p r i n c i p a l m a t e r i a l d i r e c t i o n s x x , x 3 , and x 3 , r e s p e c t i v e l y . S i m i l a r l y , l e t t h e a b s o l u t e v a l u e s o f t h e c o m p r e s s i v e y i e l d s t r e s s e s a l o n g t h e same a x e s be d e n o t e d b y r i t T2 and T 3 . S i n c e a d i r e c t e v a l u a t i o n o f f r o m u n i a x i a l t e s t d a t a i s r a t h e r cumbersome, i t i s more c o n v e n i e n t t o e x p r e s s t h e y i e l d f u n c t i o n as ( c f . S h i h and L e e , 1978) f E A . . a.a. - L.a. - X = 0 (A .2 ) i j 1 j x i where b y c o m p a r i s o n w i t h Eq . ( A . l ) , L^ and x a r e L . = 2 A . . a . (A .3 ) 139 The e x p e r i m e n t a l m e a s u r e m e n t s c a n now be r e l a t e d t o and t h r o u g h E q . ( A . 2 ) , a n d d e t e r m i n e d f r o m E q . ( A ,3) , F o r s i m p l e u n i a x i a l t e n s i o n (and t c o m p r e s s i o n ) t e s t s i n x x d i r e c t i o n , s a y , we have o x = T1 (and a1 = - r i ) so t h a t 2 A 1 1 T 1 - L 1 T 1 = x ( t e n s i o n ) (A . 4 ) I 2 I A 1 1 T 1 + L X T X = x ( c o m p r e s s i o n ) S o l v i n g t h e above e q u a t i o n s s i m u l t a n e o u s l y , we o b t a i n A n r r 1 1 1 1 (A.5) L i = x ( — ) r r 1 1 1 1 S i m i l a r l y , t h r o u g h u n i a x i a l t e n s i l e and c o m p r e s s i v e t e s t s a l o n g t h e x 2 and x 3 a x e s , we o b t a i n A j 2 ^ r • I'2 X( t ) r r r r I 2 I 2 1 2 1 2 (A.6) A 3 3 = —L-7 I L 3 = xi^T ~ —) r 3 r 3 r 3 r 3 By i m p o s i n g p u r e s h e a r i n t h e 3 o r t h o g o n a l p l a n e s , we c a n r e l a t e t h e A ^ . ' s t o t h e s h e a r y i e l d s t r e s s e s , I \ , T 5 and T. as A = X— . A = X . A = X— ( A 7^ A A 2 ' A 5 5 2 » FT6 6 2 r* r 5 r 6 140 N o t e t h a t b y a r b i t r a r i l y p r e s c r i b i n g one o f t h e a n i s o t r o p i c p a r a m e t e r s , t h e r e m a i n i n g p a r a m e t e r s c a n be s c a l e d w i t h r e s p e c t t o t h e p r e s c r i b e d v a l u e . I f f o r example A X 1 = 1 t h e n i t f o l l o w s t h a t x = and k = A ^ o t ^ a . . + x -The r e m a i n i n g o f f - d i a g o n a l t e r m s i n t h e m a t r i x [A] c a n be o b t a i n e d f r o m any b i a x i a l l o a d i n g c o n d i t i o n : t e n s i o n - t e n s i o n , c o m p r e s s i o n - c o m p r e s s i o n , t e n s i o n - c o m p r e s s i o n . However , i t i s n o t e v i d e n t t h a t t h e v a l u e s o f A 1 3 , A 2 3 , and A 1 2 d e t e r m i n e d b y t h e s e d i f f e r e n t t e s t s w i l l be u n i q u e . One remedy t o t h e s e p r o b l e m s i s t o assume t h a t t h e p l a s t i c v o l u m e t r i c s t r a i n i s z e r o , i . e . d e P + de?j + d e P = 0 . I t i s t h e n p o s s i b l e t o e x p r e s s t h e o f f - d i a g o n a l t e r m s o f [A] ( i . e . A ^ . f o r i f j ) as f u n c t i o n s o f t h e l e a d i n g d i a g o n a l t e r m s , A ^ , i n t h e f o l l o w i n g way A 1 2 2 ^ A L 1 + A J 2 A S 3 ^ A i 3 = " \ ( A X 1 - A 2 2 + A 3 3 ) ( A . 8 a ) A j 3 = ~ 2 ( ~ A n + A a 2 + A 3 3 ) A l s o , L x + L 2 + L 3 = 0 (A .8b ) A s a c o n s e q u e n c e o f E q . ( A . 8 a , b ) , t h e A „ and p a r a m e t e r s a r e n o t a l l i n d e p e n d e n t and t h e k n o w l e d g e o f p r i n c i p a l components A^^ g i v e n b y E q s . (A .5 ) t o (A .7 ) and any two o f L x , L 2 , and L 3 i s s u f f i c i e n t t o c o m p l e t e l y d e s c r i b e t h e s t a t e o f p l a s t i c d e f o r m a t i o n . The a s s u m p t i o n o f i n c o m p r e s s i b i l i t y o f p l a s t i c s t r a i n s i s i m p l i c i t i n H i l l ' s (1950) f o r m u l a t i o n o f a n i s o t r o p i c y i e l d c r i t e r i o n . I t i s i n t e r e s t i n g t o n o t e t h a t t h e f u n c t i o n f r e d u c e s t o t h e v o n 141 M i s e s y i e l d c r i t e r i o n w h e n t h e A „ a n d p a r a m e t e r s assume t h e f o l l o w i n g v a l u e s A n Aj 2 " A 3 3 = 1 A i , " A 1 3 A 2 3 = -1/2 (A.9) A , , " A 5 5 = A6 6 = 3 L , = L , = L 3 = 0 I n t h e a b s e n c e o f any c o n s t r a i n t s s u c h as E q . (A.8), one has t o r e s o r t t o b i a x i a l t e s t s i n o r d e r t o d e t e r m i n e t h e i n t e r a c t i o n p a r a m e t e r s A 1 2 , A 2 3 and A 1 3 . Many a u t h o r s recommend t h e u s e o f 4 5 -degree o f f - a x i s s p e c i m e n f o r t h e d e t e r m i n a t i o n o f s u c h p a r a m e t e r s . T h i s c a n be done b y l e t t i n g : O j = o , = o 6 = U / 2 , o 3 = o A = o s = 0 (A.10) where U i s t h e t e n s i l e s t r e n g t h o f a 4 5 -degree o f f - a x i s s p e c i m e n . N o t e t h a t t h e comb ined s t r e s s e s i n E q . (A. 10) a r e a p p l i e d t o t h e symmetry a x e s o f an o r t h o t r o p i c m a t e r i a l . T h i s s t a t e o f s t r e s s i s e q u i v a l e n t t o a u n i a x i a l t e n s i l e s t r e s s a p p l i e d t o a r e f e r e n c e c o o r d i n a t e s y s t e m r o t a t e d 45 d e g r e e s f r o m t h e m a t e r i a l symmetry a x e s . By i n t r o d u c i n g E q . (A.10) i n t o E q . (A.2), we have ( A l l + A 2 2 + 2A 1 3 + A 6 6 ) - \ (L 4 + L 2 ) - X = 0 so t h a t A 1 2 c a n be w r i t t e n i n t e r m s o f t h e o t h e r known p a r a m e t e r s as 142 '1 2 = I i + 2 I P U ( L , + L a ) - j ( A l t + A , A 6 6 ) (A .11 ) S i m i l a r t e s t s c a n be p e r f o r m e d i n o r d e r t o o b t a i n A 1 3 and A 2 3 . I n a l l t h e s e c a s e s , h o w e v e r , t h e c o n d i t i o n s f o r c l o s u r e o f t h e y i e l d s u r f a c e (see E q . ( 3 . 4 9 ) ) p l a c e s a s e v e r e r e s t r i c t i o n on t h e a l l o w a b l e v a l u e s o f t h e s e s o -c a l l e d i n t e r a c t i o n p a r a m e t e r s . I n v i e w o f t h e s e d i f f i c u l t i e s , L a b o s s i e r e and N e a l e (1987) o f f e r a l t e r n a t i v e methods f o r d e t e r m i n i n g t h e A . . p a r a m e t e r s . 143 APPENDIX B VARIATION OF ANISOTROPIC PARAMETERS WITH STRAIN-HARDENING I t has b e e n shown i n A p p e n d i x A t h a t t h e a n i s o t r o p i c p a r a m e t e r s o f t h e y i e l d f u n c t i o n c a n be d e t e r m i n e d f rom s i m p l e t e s t s . However , f o r a c o m p l e t e d e s c r i p t i o n o f p l a s t i c f l o w b e h a v i o u r o f a n i s o t r o p i c m a t e r i a l s , i t i s n e c e s s a r y t o know t h e v a r i a t i o n o f t h e s e p a r a m e t e r s w i t h p l a s t i c s t r a i n . To b e c o n s i s t e n t w i t h t h e d i s c u s s i o n i n C h a p t e r 3 t h e p a r a m e t e r ( w h i c h a c c o u n t s f o r b o t h t h e i n i t i a l s t r e n g t h d i f f e r e n t i a l and B a u s c h i n g e r e f f e c t ) w i l l h e n c e f o r t h be i g n o r e d and c o n s i d e r a t i o n w i l l be g i v e n t o t h e v a r i a t i o n s o f A . . and k o n l y . Hu ( 1 9 5 6 ) a s s u m e d t h a t t h e A ^ p a r a m e t e r s r e m a i n e d c o n s t a n t d u r i n g p l a s t i c d e f o r m a t i o n , w h i l e J e n s e n , F a l b y and P r i n c e (1966) and l a t e r Whang ( 1 9 6 9 ) p o i n t e d o u t t h a t f o r s t r a i n - h a r d e n i n g m a t e r i a l s A „ s h o u l d v a r y . The o b j e c t i v e h e r e i s t o d e t e r m i n e t h e A „ p a r a m e t e r s i n s u c h a way t h a t a l l t h e s t r e s s - s t r a i n d i a g r a m s i n t h e p r i n c i p a l m a t e r i a l d i r e c t i o n s c a n be r e p r o d u c e d c o r r e c t l y w h e n m a p p e d o n t o a n a r b i t r a r y e f f e c t i v e s t r e s s (o) - e f f e c t i v e p l a s t i c s t r a i n ( e P ) d i a g r a m . To a c c o m p l i s h t h i s mapping we a d o p t t h e method o r i g i n a l l y d e v e l o p e d b y J e n s e n e t a l ( 1 9 6 6 ) . The b a s i c a s s u m p t i o n u n d e r l y i n g t h e l a t t e r m e t h o d i s t h a t f o r t h e same amount o f p l a s t i c work (W P) p r o d u c e d d u r i n g p l a s t i c l o a d i n g i n any o f t h e p r i n c i p a l m a t e r i a l d i r e c t i o n s t h e e f f e c t i v e y i e l d s t r e s s (k) r e a c h e d w i l l be t h e same i r r e s p e c t i v e o f t h e d i r e c t i o n o f l o a d i n g . When t h e s t r e s s - s t r a i n d i a g r a m s c a n be r e p r e s e n t e d i n b i l i n e a r f o rm as shown i n F i g . B . l t h e n t h e p l a s t i c work W P may be e x p r e s s e d i n c l o s e d fo rm as (see F i g . B .2 ) 144 WP = JdWP = (r! - l l ± ) ( i = 1,2 6) (B . l ) P i where Tn. and T. are the i n i t i a l and subsequent y i e l d values and E i s the 0 1 l p . p l a s t i c modulus , a l l r e f e r r e d to the ° ^ - e ^ s t r e s s - s t r a i n d i a g r a m . The p r e s e n t f o r m u l a t i o n a l l o w s one of the s t r e s s - s t r a i n diagrams to be a r b i t r a r i l y chosen as the e f f e c t i v e s t r e s s - e f f e c t i v e s t r a i n (o-e) diagram, while the remaining s t r e s s - s t r a i n curves are then normalized with respect to the prescr ibed curve. Equating the p l a s t i c work given by Eq . (B . l ) to the p l a s t i c work done by the e f f e c t i v e y i e l d s t r e s s , I 2 2 1 2 2 2 H 7 ( k ~ k o ) = 2E~ ( r i " r ° i ) P i or E p r ! = (k s - kJ) + r ' . ( i = 1,2 6) (B.2) where H 1 i s the hardening modulus def ined by Eq . (3.23), and k 0 i s the i n i t i a l e f f e c t i v e y i e l d s t r e s s . The p l a s t i c moduli E can be determined i n p i t e r m s o f t h e e l a s t i c and tangent m o d u l i (E^ and E T , r e s p e c t i v e l y ) as i fo l lows . Accepting the bas i c assumption that de. = de? + de? (B.3) i l l then s ince for an increment of s tress do. l 145 d o . d £ i = K t . 1 do . j e 1 d e i = ET de? = do . ( B . 4 ) one o b t a i n s F = F E ( B - 5 ) E p . E T . E i * l 1 U s i n g E q s . ( A . 5 ) t o ( A . 7 ) ( n o t i n g t h a t x = k* and H = I \ ) and ( B . 4 ) , t h e a n i s o t r o p i c p a r a m e t e r s A ^ , ^ a t a n y s t a t e o f p l a s t i c d e f o r m a t i o n d e s c r i b e d b y k, c a n be o b t a i n e d a s : o A . . f o r 0 £ k <; k„ ( B . 6 ) A. . (k) = ^ = W 1 1 r . l E P i » i r 1 ( k k J + r f f o r k £ k„ T h e v a r i a t i o n o f A ^ w i t h k i s shown s c h e m a t i c a l l y i n F i g . B . 3 . I t i s n o t e d t h a t d e p e n d i n g on t h e v a l u e o f H' ( i . e . d e p e n d i n g on t h e c h o i c e o f n o r m a l i z i n g d i r e c t i o n ) A ^ may i n c r e a s e as k i n c r e a s e s , and when k g e t s l a r g e t h e A . . d e p e n d o n l y on E and n o t r „ . . F o l l o w i n g t h e a rguments t h a t l e d t o i i p^ l t h e f o r m o f E q . ( A . 1 1 ) , one c a n w r i t e t N o t e t h a t r e p e a t e d i n d i c e s do n o t i m p l y summat ion h e r e . 146 0 ^ 0 2 1 0 0 A 1 2 = 2 ( y - ) - ± (1 + A 2 2 + A 6 6 ) , f o r 0 i k S k o ( B . 7 ) A 1 3 (k) = 2 (^ ) J - ^ (1 + A 2 2 (k) + A 6 6 ( k ) ) , f o r k i k 0 i n w h i c h 2 y4S° 2 2 2 U = -5=^- (k - k 0 ) + U„ T h u s f a r t h e p a r a m e t e r s A . . w e r e a l l o w e d t o v a r y w i t h t h e amount o f s t r a i n - h a r d e n i n g , marked b y t h e v a l u e o f k . However , i t i s o f t e n c o n v e n i e n t , as w e l l as c o m p u t a t i o n a l l y e c o n o m i c a l t o k e e p t h e s e p a r a m e t e r s c o n s t a n t a t o t h e i r i n i t i a l v a l u e s , A ^ . . To f a c i l i t a t e t h e l a t t e r and y e t s a t i s f y t h e r e q u i r e m e n t o f e q u a l p l a s t i c work g i v e n b y E q . ( B . 2 ) we must have 3A. . — ^ = 0 8 k o r m a k i n g u s e o f E q . ( B . 6 ) H I ^ 0 2 H = (=—) ( i = 1 , 2 6) ( B . 8 ) E v r 0 . p . ° i The above p l a c e s c o n s t r a i n t s on t h e c h o i c e o f b i l i n e a r f i t u s e d t o a p p r o x i m a t e t h e v a r i o u s v s e^ c u r v e s . S i n c e E q . ( B . 8 ) does n o t e s t a b l i s h u n i q u e v a l u e s o f t h e y i e l d s t r e s s e s r o . , and p l a s t i c m o d u l i E , we assume P.; 147 t h a t one o f t h e s t r e s s - s t r a i n c u r v e s i s a r b i t r a r i l y f i t t e d w i t h t h e b e s t b i l i n e a r r e p r e s e n t a t i o n . F o r t h e s a k e o f argument l e t t h i s be t h e o^ v s e ^ c u r v e , where k c a n t a k e one o f t h e v a l u e s 1 t o 6 . T h e r e f o r e , t h e q u a n t i t i e s E, , r o , a n d E a r e assumed t o be known. To u n i q u e l y d e f i n e t h e r e m a i n i n g k k p k s t r e s s - s t r a i n c u r v e s i t i s o n l y n e c e s s a r y t o s p e c i f y t h r e e p i e c e s o f i n f o r m a t i o n f o r e a c h c u r v e . We have h e r e assumed t h a t a l l t h e e l a s t i c m o d u l i E . ( i = 1 , 2 6) and t h e l o c a t i o n o f t h e u l t i m a t e p o i n t (T , e ) on t h e i i r e m a i n i n g i n d i v i d u a l o . v s e . c u r v e s a r e d e f i n e d . W i t h t h e above i n f o r m a t i o n & i i we c a n e s t a b l i s h t h e f o l l o w i n g e x p r e s s i o n s f o r t h e t a n g e n t m o d u l i E^ u . - 0 l : u . - r n _ . /E_ . ( i = 1 , 2 , . . . , 6) ( B . 9 ) ' i l E q s . ( B . 8 ) and (B .9 ) comb ined w i t h ( B . 5 ) c a n be s o l v e d f o r t h e unknowns T n . and E _ . The r e s u l t i s : ° i T . l where r 0 . = r . r o k ( i f k) E„ = s . E . T. i i l (B .10 ) 1 r . = [ -1 + VI + 4 b . c ] 1 Z D . 1 1 1 ( B . l l ) s . = 1 / l E . l + 1 r . p. l r k ( i * k ) r 148 i n w h i c h " u . E . 1 1 ( i f k ) (B.12) u . c i = T. ( i * k ) T h e c o n s t a n t A . . mode l d e s c r i b e d above has b e e n a p p l i e d t o s e v e r a l l a m i n a t e d FRM coupons and t h e r e s u l t s a r e documented i n t h e i n t e r n a l r e p o r t b y O l s o n and A n d e r s o n ( 1 9 8 8 ) . T h i s mode l w i l l n o t be p u r s u e d f u r t h e r i n t h i s t h e s i s . 149 APPENDIX C DERIVATION OF THE EFFECTIVE PLASTIC STRAIN INCREMENT d e P . Here we d r i v e an e x p r e s s i o n f o r t h e e f f e c t i v e p l a s t i c s t r a i n i n c r e m e n t d e P as a f u n c t i o n o f p l a s t i c s t r a i n i n c r e m e n t s d e ? . F o r a body d e f o r m i n g p l a s t i c a l l y , t h e i n c r e m e n t o f p l a s t i c work p e r u n i t vo lume i s dW p = a . de? ( i = 1 , 2 6) ( C . l ) i i U s i n g t h e a s s o c i a t e d f l o w r u l e , E q . ' ( 3 . 1 6 ) , i n c o n j u n c t i o n w i t h E q s . ( 3 . 1 7 ) and ( 3 .18 ) y i e l d s de? = dX • 2 A . . o . (C .2 ) S u b s t i t u t i n g de? f r o m E q . (C .2 ) i n t o E q . ( C . l ) dW p = 2dX a2 = 2 dX k* (C .3 ) D e f i n i n g t h e i n c r e m e n t o f e f f e c t i v e p l a s t i c s t r a i n d e P as dW p = k d e p (C .4 ) t h e q u a n t i t y dX i s f o u n d f r o m E q s . (C .3 ) and (C .4 ) t o be 150 A s s u m i n g t h a t l A ^ . . | ± 0 , t h e n t h e m a t r i x o f a n i s o t r o p i c p a r a m e t e r s [A] h a s an i n v e r s e [ A * ] ; and E q . (C .2 ) c a n be i n v e r t e d t o g i v e a . = A * . d e . P - r ^r = A* . de? — (C .6 ) j i j 1 2dX i j l d - p where t h e l a s t s t e p f o l l o w s f r o m E q . ( C . 5 ) . U s i n g t h e above s t r e s s - p l a s t i c s t r a i n i n c r e m e n t r e l a t i o n we c a n r e w r i t e o as °* - A i j ° i ° j = A i j ( A £i d£P ( A I j d £ ? ) ( C ' 7 ) 2 = k w h i c h a f t e r some t e n s o r m a n i p u l a t i o n r e s u l t s i n ( d e P ) 2 = A * . d e P de? (C .8 ) I n t h e p l a n e s t r e s s c a s e , t h e e x p r e s s i o n f o r d e P c a n be w r i t t e n e x p l i c i t l y as ( d i P ) J = ( A A l-A ») [ A » ( d e ^ ) 2 +-A» ( d £ ? ) J (C .9 ) - 2 A 1 2 d e P d e p ] + ( d e P ) 2 A 6 6 151 I t c a n be shown i n t h e t h r e e d i m e n s i o n a l c a s e t h a t t h e a s s u m p t i o n o f z e r o p l a s t i c v o l u m e t r i c s t r a i n ( i . e . d e p + d e p + d e p = 0) i m p l i e s a s i n g u l a r [A] m a t r i x ( c f . S h i h and L e e , 1 9 7 8 ) . Under t h e s e c i r c u m s t a n c e s t h e above p r o c e d u r e f o r o b t a i n i n g d e p f a i l s , a n d r e s o r t must be made t o a d i f f e r e n t t e c h n i q u e ( c f . Hu , 1 9 5 6 ) . 152 Post - failure (Ducti le) Post-failure (Brittle) strain Fig. 3.1 - Idealized stress - strain curve showing different stages of the proposed elastic-plastic-failure model P l y c o o r d i n a t e s y s t e m Fig. 3.2 - Transverse matrix cracking in a single layer of U/D FRM 153 Fig 3.4 - Puppo - Evensen yield surfaces in the a6 - 0 plane for bi-directional layers with X = Y and various values of the parameter A 155 Fig. 3.5 - Orientation of layer coordinate axes with respect to laminate coordinates 156 Fig. 3.6 - Illustration of an n-layered laminate along with a typical in—plane deformed geometry 157 Fig. 4.1 - Quadratic isoparametric element 159 { 160 u = 0 Fig. 5.1 - Finite element mesh for the analysis of an isotropic cylinder under elastoplastic internal pressure o d o I N o I o V) w S o d o •o 85 0) "5 . CL < o 6 o d • J / / U° • Exact P resent ana lys is a 0.0 0.4 0.1 0.2 0.3 Radial d i sp lacement - m m Fig. 5.2 - Pressure P versus inner and outer wall displacements ua and ub for the problem of Fig. 5.1 161 Fig. 5.3 - Progression of yielding for the problem of Fig. 5.1 162 163 T i o B A O Fig. 5.5 - Numerical model and the stress path for the analysis of an isotropic thin - walled tube subjected to combined tension and torsion n 1 I I | I 0.00 0.02 0.04 0.06 0.08 0.10 Shear Strain Fig. 5.6 - Strain path for an isotropic thin tube subjected to combined tension and torsion y I o • X F/g. 5.7 - Nomenclature for the perforated plate problem Fig. 5.8 w = 10 mm Finite element mesh used to analyze a quadrant of Fig. 5.7 <?> 166 Fig. 5.9 - The stress - strain curve in pure tension for Aluminum alloy 57S [ Theocaris and Marketos . 1964] Fig. 5.10 - Nondimensional graph of mean stress against maximum strain for the isotropic perforated plate shown in Fig. 5.8 Fig. 5.11 - Comparison of the computed and experimentally determined plastic zone growth for the isotropic perforated plate subjected to elastoplastic loading CO Fig. 5.11 - continued Fig. 5.12 - continued KJ O <0 o iri o d o iri q d 172 • • Exper imenta l • P resent Ana lys is - single s tep • • i . m j 0.5 0.6 0.9 0.7 0.8 X / W Fig. 5.13 a - Strain profile at the net section for the isotropic perforated plate subjected to = 0.47 a0 1.0 q d. 8 q iri o W) 9 M £ I b 9 o q in q d • • J L L - - - ^ Exper imenta l • Present ana lys is -• s ingle s tep 0.5 0.6 0.7 0.8 0.9 i. X / w Fig. 5.13 b - Stress profile at the net section for the isotropic perforated plate subjected to a . = 0.47 aQ 173 co co CO LxJ or r -CO slope E T , Experimental Bilinear Fit 0.0 2.0 4.0 6.0 8.0 STRAIN e, x 10 3 f/g. 5.14 - Longitudinal tensile stress - strain curve for a single layer of U/D Boron/Epoxy o slope E j 2 Experimental Bilinear Fit 1 i 1 I 1 i 1 i 1 0.0 1.0 2.0 3.0 4.0 5.0 STRAIN £ 2 x 1CT Fig. 5.15 - Transverse tensile stress - strain curve for a single layer of U/D Boron/Epoxy 174 co to b CO CO Ld or o CM O lO * o CO -< bJ I CO q i r i q ci Su s lope G T s 0 — 7 ^ / / Exper imenta l R i l i n e n r Fit / s lope G 0.00 0.02 0.04 SHEAR STRAIN e 6 - rods 0.06 Fig. 5.16 - Shear stress - strain curve for a single layer of U/D Boron/Epoxy o CM q 6 co w I o co 6 CO <° Ld or Lo ° o o CM q 6 ^^^^^^^ ** *" sis Exper imenta l Syr .._ Hashin P r e s e n t A n n l y s i s - D u c t i l e Present Ana lys is - Britt le 0.0 2.0 4.0 6.0 Fig. 5, STRAIN x 10" 17 - Tensile stress - strain curve for [0/90] Boron/Epoxy laminate 8.0 175 o Exper imenta l Hash in P r e s e n t Ana lys is •p 0.01 0.02 0.03 0.04 0.05 STRAIN Fig 5.18 - Tensile stress - strain curve for [+45/-A5] Boron/Epoxy laminate o d. Max s t ress Ult Exper imenta l Hash in P resent Analys is o.o 4.0 8.0 12.0 16.0 20.0 STRAIN x 1CT Fig. 5.19 a - Tensile stress - strain curve for [+30/-30] Boron/Epoxy laminate Fig. 5.19 b - Stress paths in the + 3 0 deg layer during uniaxial loading of [+30/-30] B/Ep laminate : Hiirs failure criterion Fig. 5.19 c — Stress paths in the +30 deg layer during uniaxial loading of [+30/-30] B/Ep laminate : Maximum stress failure criterion q d CM 178 . . . X Experimental Petit and Waddoups Present Analysis T — 8.0 10.0 STRAIN x 1CT Fig. 5.20 - Tensile stress - strain curve for [+60/-60] Boron/Epoxy laminate o d o O Max stress Ult Experimental Petit and Waddoups Present Analysis T — .8.0 12.0 STRAIN x 10 J Fig. 5.21 a - Tensile stress - strain curve for [+20/-20] Boron/Epoxy laminate 179 _ 2.0 Initial yield surface Failure surface Fig. 5.21 b - Stress paths in the +20 deg layer during uniaxial loading of [+20/-20] B/Ep laminate : Hiirs failure criterion 180 Fig. 5.21 c - Stress paths in the +20 deg layer during uniaxial loading of [+20/-20] B/Ep laminate : Maximum stress failure criterion 181 o Fig. 5.22 - Tensile stress - strain curve for [0/+45/-45/90] Boron/Epoxy laminate STRAIN x 10 3 Fig. 5.23 - Tensile stress - strain curve for [0/+60/-60] Boron/Epoxy laminate o 1 8 2 Fig. 5.24 - Tensile stress - strain curve for [Oj /45/-45] Boron/Epoxy laminate o 10.0 STRAIN x 103 Fig. 5.25 - Tensile stress - strain curve for [65j /2Q/-70] Boron/Epoxy laminate 183 o in CO °. ^ o CO CO UJ o £ ° o d o d a 2 - e 2 Exper imental Present Bi l inear Fit 0.00 0.01 I 0.04 0.02 0.03 STRAIN F/g. 5 . 2 6 a - B a s / c Stress - Strain curves for a B/D layer made of 181 glass fabric and polyester resin o 0.05 Failure © 2 2 . 9 3 ksi Exper imenta l P resent Analys is 0.00 0.01 0.02 0.03 STRAIN Fig. 5.26 b - Tensile Stress - Strain curve at 45 deg to the fibre directions for the material of Fig. 5.26 a 184 o 0.00 0.01 Experimental Present Bilinear Fit 0.02 0.03 STRAIN 0.04 Fig. 5.27 a - Basic Stress - Strain curves for a B/D layer made of 162 glass fabric and polyester resin 0.05 o m CN o-' o CO in-CO ~~ LJ or CO ° o • Experimental Present Analysis o.oo 0.01 0.02 STRAIN 0.03 Fig. 5.27 b - Tensile Stress - Strain curve at 45 deg to the fibre directions for the material of Fig. 5.27 a 185 Experimental Present Bilinear Fit 0.00 0.01 0.02 0.03 STRAIN 0.04 0.05 Fig. 5.28 a - Basic Stress - Strain curves for a B/D layer made of 143 glass fabric and polyester resin o CM . . X Experimental Present Analysis 0.00 0.01 0.02 0.03 STRAIN Fig. 5.28 b - Tensile Stress - Strain curve at 45 deg to the fibre directions for the material of Fig. 5.28 a 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 STRAIN x 103 Fig. 5.29 - Pressure - strain curve for [0/60/'-60] Gr/Ep tube under internal pressure 00 Fig. 5.30 - Pressure - strain curve for [0/60/-60] Gr/Ep tube under combined internal pressure and pre-torque o O 6-137 o w «• a Ix 1 o LxJ <c-ZD O or ?5 o O d. Max stress Ult Cubic Ult Hiirs Ult Pre-Pressure = 1.1 ksi • Experiment Tennyson Present Analysis - Ductile i— 8.0 —i 1 1 14.0 16.0 18.0 SHEAR STRAIN - yxy x 103 Fig. 5.31 - Torque - shear strain curve for [0/60/-60] Gr/Ep tube with pre-load of internal pressure o d o to Failure @ 630 Mpa o- Failure @ 580 Mpa in ' • o O o -CL * 1 5 CO o-CO "> bJ ^ 9 CO o. o CM O d o Experimental Present Analysis o.o - i — 4.0 - 1 — 6.0 - 1 8.0 STRAIN x 10" Fig. 5.32 - Tensile stress - strain curve for [0/45/-45] B/Al laminate under three load cycles 188 Fig. 5.33 - Geometry and the finite element model for a quadrant of an orthotropic perforated sheet 189 Fig. 5.34 - Elastic circumferential stress distribution around the hole for a U/D B/Al layer : a - Fibres perpendicular to the load direction b - Fibres along the load direction q r o q CN 190 q d \ o b q CN q o o 0 i i i i i i m i \ i 10.0 20.0 30.0 40.0 50.0 60.0 Q» 7 0 . 0 \ 80.0 • o \ ° Annlytirn! # 3 x 3 Gauss rule 0 2 x 2 Gau ss rule | ; (a) b q id o o o ro q C N o d : : \ < » • O Analytical 3 x 3 G a u s s rule \ • \° 2 x 2 Gauss rule \ • j o ; 0.0 10.0 20.0 30.0 60.0 70.0 80.0 40.0 50.0 6 - deg (6) Fig. 5.35 - Elastic circumferential stress distribution around the hole for a U/D B/Ep layer : a - Fibres perpendicular to the load direction b - Fibres along the load direction 90.0 191 5.36 - Geometry of the test specimen used by [Rizzi et al, 1987] for experimental determination of the stress—strain behaviour of a perforated 90-deg U/D B/Al layer 192 o STRAIN c 2 x 1 0 3 F/g. 5.37 - Transverse tensile stress - strain curve for a single layer of U/D B/Al [ Kenaga et al, 1987 ] 193 O (0 O O o m m CM o o 6 Coo = 1.71 ksi Prp<?pnt A n n l y c s k 9 Experiment • 0.375 0.750 1.125 x - in 1.500 O o o o I f ) I f ) CM o o d 0*00 = 4.3 ksi • j '. • : : 0.375 0.750 1.125 1.500 x - in K) O (0 o o. I f ) o m m CM o o 0.375 CTQO =6 .0 ks/ 4 ! r?. i 0.750 1.125 1.500 x - in Fig. 5.38 - Longitudinal strain distribution along the net section for various remote load levels imposed on a perforated 90 - deg layer of U/D B/Al a-- = 6.0 ksi 0*00 = 7.7 ksi 195 Fig. 5.39 - Development of plastic zones for a perforated 90-deg U/D B/Al layer subjected to elastoplastic loading a m = 6.0 ksi °"oo = 7.7 ksi 196 F/g. 5.40 - Nondimensional effective stress contours for a perforated 90-deg U/D B/Al layer subjected to elastoplastic loading CN m d o d q d o q d • j Prior 0*00 = 6.0 ksi P r e ^ n t Annly<?i<; # Experiment *~m— > 0.375 0.750 1.125 1.500 X - in to d CN d o d . Prior cToo = 7.7 /<s/ j • * ' —i "t»— • 0.375 0.750 1.125 1.5 x — in o m m CN o q d . • \ , Prior 0*00 = 10.7 ksi \ < > • > • 0.375 0.750 1.125 1.500 x - in Fig. 5.41 - Residual longitudinal strain distribution along the net section of a perforated 90-deg layer of U/D B/Al after unloading from various load levels Fig. 5.42 - Distribution of residual stress components along the net section of a perforated 90-deg layer of U/D B/Al after unloading from various load levels 199 = 6.0 ksi Coo = 10.0 ksi a m = 15.0 ksi • • • • _ • I I ! • * • (Too = 17.0 ksi • • • a . ' 5.43 - Development of plastic zones in the 90-deg layer of a perforated [0/90] B/Al laminate subjected to elastoplastic loading = 25.0 ksi <r„ = 30.0 ksi 200 • • O B • • • - "» • . . . •>••_ ^ • • • • • • • • • ^#!!!:::: CT„ = 35.0 ksi • • a „ = 40.0 ksi • • • L a • • ' * — Fig. 5.43 - continued Oo, = 6.0 ksi a « = 10.0 ksi 201 a„ = 15.0 ksi a r _ = 17.0 ksi Fig. 5.44 - Development of plastic zones in the O-deg layer of a perforated [0/90] B/Al laminate subjected to elastoplastic loading a., - 25.0 ksi a.,, » 30.0 ksi 202 • • • Fig. 5.44 — continued 203 a M = 6.0 ksi a«> = 10.0 ksi 5.45 - Nondimensional effective stress contours for the 90-deg layer of a perforated [0/90] B/Al laminate subjected to elastoplastic loading Fig. 5.45 - continued <7„ = 6.0 ksi On = 10.0 ksi 205 Fig. 5.46 - Nondimensional effective stress contours for the 0-deg layer of a perforated [0/90] B/Al laminate subjected to elastoplastic loading a„ = 25.0 ksi CToo = 30.0 ksi Fig. 5.46 — continued 5.47 - Development of plastic zones in a [0/90] Fp/AI laminate with a hole [Bahaei-EI-Din and Dvorak , 1980] Fig. 5.48 - Longitudinal stress distribution along the net section for each layer of a perforated [0/90] B/Al laminate 209 o d _ m~ CN o 0.0 10.0 20.0 30.0 40.0 50.0 CToo - ksi Fig. 5.49 - Longitudinal stress at point A versus the applied load for the 0 deg layer of a perforated [0/90] B/Al laminate o d vB x 10 - in Fig. 5.50 - Load versus deflection at point B for a perforated [0/90] B/Al laminate 2 1 0 o i r i o iri CM I Fig. 5.51 - Longitudinal residual stress distribution along the net section for each layer of a perforated [0/90] B/Al laminate due to unloading from a^ = 20 ksi a-- = 4.0 ksi c r * , = 6.0 ksi 9 Fibre fai lure A Matrix fai lure x Fibre and Matrix failure 5.52 - Predicted damage progression for a perforated 90-deg U/D B/Ep layer - Ductile Matrix a,-, = 55.0 ksi = 60.0 ksi a, F ibre failure A Matrix fai lure x F ibre and Matrix fai lure = 70.0 ksi 5.53 - Predicted damage progression for a perforated 0-deg U/D B/Ep layer - Ductile Fibre Fig. 5.54 - Stress path at point A for the 0-deg layer of a perforated [0/90] B/Ep laminate o vB x 1 0 3 - in Fig. 5.55 - Load versus deflection at point B for a perforated [0/90] B/Ep laminate a- = 30.0 ksi = 50.0 ksi # Fibre failure A Matrix failure x Fibre and Matrix failure 0*00 = 70.0 ksi A A A A A A A A A A A A A A * A A A A A A A * A A A A A A A A - A * * * A * A A * * A * * a„ -= 77.0 fcs/ A A A A A A A A A A A A A A A A A A A A A A A * A A A A A * A A A A . A A A A A A A A ^ 4 * A A A A B i t A A A A A A A 5.56 o - Predicted damage progression for the 90-deg layer of a [0/90] B/Ep laminate - Ductile Fibre and Ductile Matrix 215 a „ = 30.0 ksi 0 F ibre failure A Matrix failure x F ibre and Matrix fai lure a*, » 50.0 ksi om = 70.0 ksi l — i - i l i » t f CTm = 77.0 ksi • • • • • • • • •• • • J I • • • • Fig. 5.56 b - Predicted damage progression for the 0-deg layer of a [0/90] B/Ep laminate - Ductile Fibre and Ductile Matrix a M = 30.0 ksi CTo_ = 50.0 ksi * Fibre failure A Matrix failure x Fibre and Matrix failure a,*, = 62.0 ksi A A A A A A A A A A A A A A A A A A ** *A ' * A A A A A A A A 4 A A A A A A A .iVs*1::--A A A A A A A A A A tT-o = 67.0 ksi A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A * A A A A * A A 4 \ ^ A A A A A A A A A A \ A A A A H A AA A A A A A 5.57 a - Predicted damage progression for the 90-deg layer of a [0/90] B/Ep laminate - Ductile Fibre and Brittle Matrix Go,, = 30.0 ksi Coo = 50.0 ksi 0 Fibre failure A Matrix failure x Fibre and Matrix failure 5.57 b - Predicted damage progression for the 0-deg layer of a [0/90] B/Ep laminate - Ductile Fibre and Brittle Matrix a - = 30.00 ksi a , . - 31.50 ksi # Fibre failure A Matrix failure x Fibre and Matrix failure 5.58 a — Predicted damage progression for the 90-deg layer of a [0/90] B/Ep laminate - Brittle Fibre and Ductile Matrix CToo = 30.00 ksi » 31.50 ksi 9 Fibre failure A Matrix failure x Fibre and Matrix failure 5.58 b - Predicted damage progression for the O-deg layer of a [0/90] B/Ep laminate - Brittle Fibre and Ductile Matrix 220 co q d O O CN O 10 o CN q d co q d q d = 32.2 ksi (Ultimate failure) - 7 / A a,* = 31.7 ksi (Initial fibre failure) a-, = 32.1 ksi f 0.375 I 0.750 1.125 1.500 X - in f/'g. 5 . 5 9 - Change in stress distribution along the net section during the process of brittle fibre failure in the 0-deg layer of a perforated [0/90] B/Ep laminate CO 0.000 0.025 Ductile Matrix _ Brittle Matrix 0.050 VB - in 0.075 Fig. 5.60 - Load versus deflection at point B for a perforated [45/—45] B/Ep laminate 221 o~,= 21.0 ksi # Fibre failure A Matrix failure x Fibre and Matrix failure a* = 26.0 ksi At* _AA_A a M = 28.0 ksi A A * * * * * * \ t i * * * cr_ = 29.5 /cs/ * A * * * A A ^ A * Fig. 5.61 a - Predicted damage progression for the +45-deg layer of a [45/-45] B/Ep laminate - Ductile Matrix ax = 21.0 ksi , Fibre failure A Matrix failure x Fibre and Matrix failure i A _A_ 222 ao, = 26.0 ksi _AJk_A a 28.0 ksi * * A * A A A A -A^_A < 7 M = 29.5 ksi A A * * A » A X A * A A A A ^ A A ! »* F/g. 5.6/ b - Predicted damage progression for the -45-deg layer of a [45/-45] B/Ep laminate - Ductile Matrix a,,, = 19.6 ksi a* = 20.1 ksi 4t Fibre failure A Matrix failure x Fibre and Matrix failure or = 20.4 ksi a „ = 20.8 ksi A * A * A A A lit*' • 4 A 5.62 a - Predicted damage progression for the -t-45-deg layer of a [45/ B/Ep laminate - Brittle Matrix 224 a,*, = 19.6 ksi CToo = 20.1 ksi # Fibre failure 4 Matrix failure x Fibre and Matrix failure Fig. 5.62 b - Predicted damage progression for the -45-deg layer of a [45/-45] B/Ep laminate - Brittle Matrix 225 o b co o 6 o Ductile Fibre , Ductile Matrix Ductile Fibre , Brittle Matrix Brittle Fibre , Ductile Matrix Brittle Fibre , Brittle Matrix 0.0 10.0 20.0 30.0 VB x 103 - in 40.0 50.0 5.63 - Load versus deflection at point B for a perforated [0/45/-45/90] B/Ep laminate 226 a „ = 40.0 ksi 9 Fibre failure A Matrix failure x Fibre and Matrix failure •Mi ill = 60.0 ksi A A A A A * A A , * A A * * A A • A A * * A A A A * * * \ * A A * A A A A * * * I A A tt A A A A A A A A A A A A • A * A A a oo = 70.0 ksi A A A A A A A A A A * A A A A A A A A * A A * A A . * 4 A A A A A . A A A L A > A A A * A A ( T o o = 74.0 fes/ A A A A A * A A A * A A A 4 ^ A A. * A * A ~ " * A A A  A A A A 4 ~ A A A A A A 4 4 * A A A A rig. 5.64 a - Predicted damage progression for the 90-deg layer of a [0/45/-45/90] B/Ep laminate - Ductile Fibre Ductile Matrix 227 am = 40.0 ksi CTM = 60.0 ksi # Fibre failure . Matrix failure x Fibre and Matrix failure Si 12::' a M = 70.0 ks/ ffoo = 74.0 ksi • • • • • • • - • • • • » i • • • • • • • • • • • • • " S i » 5 : : : Fig. 5.64 b - Predicted damage progression for the 0-deg layer of a [0/45/-45/90] B/Ep laminate - Ductile Fibre Ductile Matrix On = 40.0 ksi c7oo = 60.0 ksi 228 t Fibre failure A Matrix failure x Fibre and Matrix failure it*'1:: ( r„ = 70.0 • 4 >¥K « X XA 1 A A * CToo = 74.0 ks/ .64 c - Predicted damage progression for the +45-deg layer of a [0/45/-45/90] B/Ep laminate - Ductile Fibre Ductile Matrix a„ = 40.0 ksi , Fibre failure A Matrix failure x Fibre and Matrix failure 229 a . = 60.0 ksi ^ A A A * <7 M = 70.0 ksi A A a M = 74.0 fcs/ A 4 A X A * 4 A A 4. A A 4 *A 4 A X A A A 4 A x ^ A A J x x " \ l * X * * X X X A !I A A A X A X A A Fig. 5.64 d - Predicted damage progression for the -45-deg layer of a [0/45/-45/90] B/Ep laminate - Ductile Fibre Ductile Matrix 230 o-oo = 40.0 ksi # Fibre failure A Matrix failure x Fibre and Matrix failure , A A A a,,,, = 60.0 ks; A A A A A A A A A A A A A * A A A A A 4 A A A A A A4 " A 1 A * A A * *A A A 4 A A A V * i * A A * * A — = 70.0 ks/ ac = 72.0 ks/ A A A A A A A A A A * '* A t * A 4 * A A A A 4 4 * * A A A AA 4 * A A A * AA A A X X 4 A . *A A X A A A *A X 4 A 4 A A A A A A A A A A A A A A A j A A 4 A A * A A A A A* 4 4 4 * A , 4 A A » A A *A X A '4 A 4 X A 4 X . X * X 4 X A A X X A A A A 4i AA *' A A A A A A A A A A Fig. 5.65 a - Predicted damage progression for the 90-deg layer of a [0/45/-45/90] B/Ep laminate - Ductile Fibre Brittle Matrix 231 a,,, = 40.0 ksi * Fibre failure A Matrix failure x Fibre and Matrix failure s, c7_ - 60.0 ksi • • • NISI; a „ = 70.0 ks/ • • • • • • * * • • k \ X X X ? • * \ l « * X * _ * • a „ = 72.0 ks/ • • • • • • • • • • • • • • • • „ X X • • • • X * - _ • X • • • X X •••A * X • X I T X X • _ X « X X * • X « • • • X X • Fig. 5.65 b - Predicted damage progression for the O-deg layer of a [0/45/—45/90] B/Ep laminate - Ductile Fibre Brittle Matrix 232 am = 40.0 ksi c Fibre failure m Matrix failure x Fibre and Matrix failure a- = 60.0 ksi &XA-A. a,,, = 70.0 ks/ • *> % 4 * * X X 1 A A * v A X X X A A era, = 72.0 ks/' • A A • 4 * X • A . X O 4 4 A ^ A * ^ • X A A g S j x x X A A A A A A 5.65 c - Predicted damage progression for the -r-45-deg layer of a [0/45/-45/90] B/Ep laminate - Ductile Fibre Brittle Matrix ffm = 40.0 ksi 233 <Xoo = 60.0 ksi 9 Fibre failure A Matrix failure x Fibre and Matrix failure » » A A A A A = 70.0 ksi A A A 4 A 4 *• A A 4 A 4 4 *A A 4 A A * 1 * A w X X X ' \1 * x X * v X * •55* " x Si! X A X X 4 4 4 a,,, = 72.0 ks/ 4 4 4 4 \ A X A A ^ A A « A A „ A A A X X * A * A X Ax x * X X X K X „ * A X *A A A X A i A A x 4. •* A x X * A X * „ X " x X 1* x x * * X X *8 X A * X X A X X A 5.65 d - Predicted damage progression for the -45-deg layer of a [0/45/-45/90] B/Ep laminate - Ductile Fibre Brittle Matrix 234 o"~ = 32.0 ksi cr„ = 34.0 ks; t> Fibre failure 4 Matrix failure x Fibre and Matrix failure » 4 A A <7m = 36.0 fcs/ = 37.1 ksi S i l l AA 4 at? 4 A X * 4 " x * 4 A * " v « * 4 ft:;*-"*«» 4 Fig. 5.66 a - Predicted damage progression for the 90-deg layer of a [0/45/-45/90] B/Ep laminate - Brittle Fibre Ductile Matrix or„ = 32.0 ksi 235 C o o = 34.0 ksi 0 Fibre failure A Matrix failure x Fibre and Matrix failure Fig. 5.66 b - Predicted damage progression for the 0-deg layer of a [0/45/-45/90] B/Ep laminate - Brittle Fibre Ductile Matrix a ~ = 32.0 ksi 236 CToo = 34.0 ksi 9 Fibre failure A Matrix failure x Fibre and Matrix failure an,, = 36.0 ks/' Coo = 37.7 ksi 1 XX X X X A A X X * X • A x x * * A * * Stt X X X X X X A X X 5.66 c - Predicted damage progression for the +45-deg layer of a [0/45/-45/90] B/Ep laminate - Brittle Fibre Ductile Matrix a - = 32.0 ksi 237 <*"oo = 34.0 ksi « Fibre failure 4 Matrix failure x Fibre and Matrix failure A AX A X * L « X 0^ = 36.0 ksi o^ = 37.1 ksi A A * X X A. A X A A x ' ^ X A A X A X 5.66 d — Predicted damage progression for the -45-deg layer of a [0/45/-45/90] B/Ep laminate - Brittle Fibre Ductile Matrix Fig. B.2 - Bilinear stress - plastic strain curve 239 Fig. B.3 — Variation of the principal anisotropic strength parameters with the effective stress 

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