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Crack growth and damage modeling of fibre reinforced polymer composites McClennan, Scott Andrew 2004

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CRACK GROWTH AND DAMAGE MODELING OF FIBRE REINFORCED POLYMER COMPOSITES by Scott A n d r e w M c C l e n n a n B.S. ( A e r o n a u t i c a l and A s t r o n a u t i c a l E n g i n e e r i n g ) , U n i v e r s i t y o f I l l i n o i s , 2001  A THESIS S U B M I T T E D I N P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S FOR T H E D E G R E E OF M A S T E R OF A P P L I E D SCIENCE  in  T H E F A C U L T Y OF G R A D U A T E STUDIES (Department o f Metals and Materials E n g i n e e r i n g )  W e accept this thesis as c o n f o r m i n g to the required standard  T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A October 2004 © Scott A n d r e w M c C l e n n a n , 2004  THE UNIVERSITY OF BRITISH COLUMBIA  FACULTY OF GRADUATE STUDIES  Library Authorization  3  In presenting this thesis in partial fulfillment 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.  Name of Author (please print)  Date (dd/mm/yyyy)  Title of Thesis:  Degree:  t A c v ^ W r oV  Department of fAoi\<e.n The University of British Columbia Vancouver, BC Canada  grad.ubc.ca/forms/?formlD=THS  /\^V^1  ^ C T ^ ^ C Q -  Y  E  A  R  :  2.00^  o\<> ~€-  V  ',  page 1 of/I  .  .. . «;.•  . ,. . ' .  last updated:Q0-Jul-04-  Abstract  ABSTRACT N o t c h e d tensile strength and the development o f damage i n composite laminates are studied i n this thesis t h r o u g h experimental, n u m e r i c a l and analytical methods to develop simple models f o r p r e d i c t i n g notched strength u s i n g c o m p l e t e l y physical i n p u t parameters. A series o f experimental tensile fracture tests u s i n g the O v e r h e i g h t C o m p a c t Tension ( O C T ) specimen geometry, established b y ( K o n g s h a v n and Poursartip, 1999), were conducted to study the development o f the characteristic damage zone. The material used i n the tests was a quasi-isotropic carbon fibre / epoxy.  The specimens w e r e m o d i f i e d so that the n o t c h r o o t  radius v a r i e d w h i l e m a i n t a i n i n g a constant crack w i d t h to specimen section ratio.  Specimens  w i t h n o t c h root radii less than 16 m m display stable crack g r o w t h w i t h little n o t c h sensitivity. Specimens w i t h larger n o t c h root r a d i i are unstable and display m o r e n o t c h sensitivity.  For  all specimens, the height o f the damage zone converges to between 6 m m and 7 m m . The transition f r o m stable to unstable behaviour is explained u s i n g  fracture  mechanics  equations and a transition radius can be determined f r o m three material parameters; elastic m o d u l u s , specific strain energy and tensile strength. A  simple bilinear cohesive zone m o d e l called the simple damage m o d e l ( S D M )  developed as a material m o d e l i n the A B A Q U S  was  f i n i t e element code to be used as a  demonstrator f o r m o d e l i n g techniques that can be applied to all strain-softening material models. I n particular, the relationship between input parameters and the element w i d t h was studied. U s i n g the same i n p u t parameters as the analytical m o d e l and a s u f f i c i e n t l y r e f i n e d mesh, the n u m e r i c a l m o d e l predicts the peak loads from the series o f experimental O C T tests well.  A transition f r o m stable to unstable behaviour is predicted at the same radius as seen  experimentally.  A  m e t h o d o f d e t e r m i n i n g an appropriate element w i d t h to ensure an  accurate p r e d i c t i o n is presented. A m o d i f i e d v e r s i o n o f the S D M called the adaptive simple damage m o d e l ( A S D M ) was also developed, w h i c h a u t o m a t i c a l l y scales the input strength to account f o r the effect o f element w i d t h . T h i s m o d i f i c a t i o n a l l o w s larger elements to be used w h i l e still o b t a i n i n g an accurate s o l u t i o n , a useful feature i f large structures are b e i n g m o d e l e d .  ii  Table of Contents  T A B L E OF C O N T E N T S Abstract  ii  List of Tables vi List of figures vii Nomenclature xii Acknowledgements  xiv  Chapter 1  Introduction  1  Chapter 2  Background and literature review  3  2.1  Introduction  3  2.2  Fracture mechanics  3  2.3  Cohesive crack models  7  2.3.1  Cohesive crack models for intralaminar crack propagation  8  2.3.2  Cohesive zone models for interlaminar crack propagation  11  2.3.3  Mesh Sensitivity  2.4  N o t c h e d strength p r e d i c t i o n  13 15  2.4.1  Semi-empirical models  15  2.4.2  Cohesive models  17  2.4.3  Relation of the damage zone size to strength  20  2.5  A specimen f o r stable damage g r o w t h  21  2.6  Summary  22  Chapter 3  Experiments  28  3.1  Introduction  28  3.2  Specimens and procedures  28  3.2.1  Material  3.2.2  Test equipment  3.2.3  OCT tests  3.2.4  Four-point bend tests  3.3  3.3.1  Results and data r e d u c t i o n  OCT tests  28 29 29 32 33  33  Table of Contents 3.3.2 3.4  Four-point bend tests  40  Summary  Chapter 4  41  Numerical  61  4.1  Introduction  61  4.2  Elastic simulations  62  4.2.1  Stress concentrations  62  4.2.2  Stress contours  63  4.2.3  Element size required to resolve the notch tip stress concentration  4.3  D e v e l o p m e n t o f the cohesive zone models  4.3.1  Simple damage model (SDM)  4.3.2  Adaptive simple damage model (ASDM)  4.3.3  Algorithm for the SDM and ASDM.  4.3.4  How mesh insensitive is the ASDM?  4.4  O C T simulations  The virtual specimens  4.4.2  Parametric study on G and a k  64 65 68 72 73  4.4.3  SDM/ ASDM comparison  4.4.4  Simulation of experimental OCT tests  c  75  pea  Summary  Chapter 5  64  73  4.4.1  4.5  63  Discussion and analysis of results  75 77 83  104  5.1  Introduction  104  5.2  A n a l y t i c a l description o f n o t c h root radius sensitivity  104  5.2.1  Regime 1  104  5.2.2  Regime II.  106  5.2.3  Trans ition radius  5.2.4  Peak load solution for the OCT specimens  108 108  5.3  Damage zone size and shape  Ill  5.4  Significance o f the transition radius  112  5.5  Other material systems and geometries  113  5.6  C h o o s i n g an appropriate element w i d t h for strain-softening models  113  5.6.1  Element width case study  5.6.2  Guidelines for choosing an appropriate element size  114 775  iv  Table of Contents 5.7  Summary  Chapter 6  Conclusions and future work  118  128  6.1  Conclusions  128  6.2  Future w o r k  130  Appendix A  Load versus CMOD plots  132  Appendix B  Extra line analysis plots  141  Appendix C  Stress distribution and compliance plots  145  Appendix D  ASDM UMAT for ABAQUS  148  Appendix E  Transition radius and peak load prediction plot for an AS4/3501-6 DCB Specimen  160  E. 1  Specimen  160  E.2  Regime I  160  E.3  Regime I I  161  E.4  T r a n s i t i o n radius  161  E.5  Discussion  161  References  165  v  List of Tables  LIST OF T A B L E S  Table 3.1 O r i g i n a l elastic material properties f o r C F R P laminate ( M i t c h e l l , 2 0 0 2 )  43  Table 3.2 O C T tests and specimen measurements  44  T a b l e 3.3 F o u r - p o i n t bend tests and specimen measurements  45  Table 3.4 S u m m a r y o f O C T tests and analysis  46  Table 3.5 C r i t i c a l fracture energy release rates  47  Table 3.6 F o u r - p o i n t bend test results  48  T a b l e 4.1 M a t e r i a l properties f o r n u m e r i c a l O C T tests  85  Table 4.2 H e i g h t o f c r i t i c a l l y stressed material at peak load  85  Table 4.3 N u m e r i c a l tests  86  Table 4.4 A S D M v a l i d a t i o n tests  87  T a b l e 5.1 A p p r o x i m a t e material properties f o r a c r y l i c  120  Table 5.2 M e s h i n g guidelines based on n o t c h radius  120  Table E . l  163  A S 4 / 3 5 0 1 - 6 material properties  List of Figures  LIST OF FIGURES  F i g u r e 2.1 Coordinates at a crack t i p  24  F i g u r e 2.2 A slender n o t c h w i t h a b l u n t n o t c h t i p  24  F i g u r e 2.3 Cohesive crack m o d e l  24  F i g u r e 2.4 T r a c t i o n - displacement relationship f o r cohesive crack m o d e l  25  Figure 2.5 Cohesive crack models f o r a) ductile solids and b ) quasi-brittle solids ( f r o m (de Borst, 2 0 0 3 ) )  25  F i g u r e 2.6 D a m a g e localisation u s i n g a cohesive zone m o d e l  26  F i g u r e 2.7 D a m a g e d n o t c h specimen o f K o r t s c h o t and B e a u m o n t ( K o r t s c h o t and B e a u m o n t , 1990a)  27  F i g u r e 3.1 O C T specimen geometry (thickness B ranged f r o m 7.9 m m to 8.6 m m )  49  F i g u r e 3.2 O C T specimens w i t h v a r y i n g n o t c h root radii a) 0.5 m m , b ) 2.0 m m , c) 5.0 m m , d) 12.7 m m , e) 19.0 m m and f ) 27.0 m m  :  50  F i g u r e 3.3 Scribed lines f o r line analysis ( s h o w n f o r r d 4 - 2 )  51  F i g u r e 3.4 O C T test set-up ( s h o w n f o r r d 5 4 - 2 )  51  F i g u r e 3.5 Load-displacement plots f o r specimen r d 3 2 - 2 from the machine p o s i t i o n and f r o m the relative displacement o f the pins.  52  Figure 3.6 Schematic o f section taken t h r o u g h the damage zone ahead o f the i n i t i a l n o t c h tip. The dashed lines represent the 90° fibres i n the outer stacks. The measured damage height d i d not include any damage to the outside o f these lines F i g u r e 3.7 F o u r - p o i n t b e n d test geometry  52 53  F i g u r e 3.8 Specimen r d l 0 - 2 s h o w i n g orientation o f crack and extent o f surface damage (scribed lines are 2.5 m m apart)  53  Figure 3.9 L o a d versus crack m o u t h o p e n i n g displacement f o r specimens o f representative n o t c h root radius. S o l i d lines indicate continuous portions o f the curve w h i l e dashed segments indicate segments i n w h i c h the crack j u m p s and the load drops sharply  54  F i g u r e 3.10 Peak loads attained i n each O C T test versus n o t c h root radius p. Closed data points denote specimens d i s p l a y i n g a plateau on the load versus C M O D curve, open data points denote specimens that showed sharp load drops at the peak load F i g u r e 3.11 Profiles o f the damage zone along the crack plane  54 55  List of Figures F i g u r e 3.12 Section o f r d 2 - l a t 0.635 m m ahead o f i n i t i a l n o t c h tip...,  55  F i g u r e 3.13 Section o f rd54-2 at 0.635 m m ahead o f i n i t i a l n o t c h tip.r.  56  F i g u r e 3.14 Section o f r d 2 - l at 24.13 m m ahead o f i n i t i a l n o t c h t i p  56  Figure 3.15 Section o f r d 5 4 - 2 at 24.13 m m ahead o f i n i t i a l n o t c h t i p  57  F i g u r e 3.16 C O D p r o f i l e s determined from line analysis f o r r d 2 5 - 2  57  F i g u r e 3.17 L o a d versus p i n opening displacement f o r r d 2 5 - 2 s h o w i n g location o f photos f o r line analysis Figure 3.18 C O D profiles determined f r o m line analysis f o r r d 2 8 - 2  58 58  Figure 3.19 L o a d versus p i n opening displacement f o r r d 2 8 - 2 s h o w i n g l o c a t i o n o f photos f o r line analysis  59  F i g u r e 3.20 F i n a l crack lengths i n r d 2 - l and r d 5 4 - 2 f r o m sectioning and line analysis  59  F i g u r e 3.21 C r a c k resistance points f o r r d 2 5 - 2 and r d 2 8 - 2  60  F i g u r e 3.22 F o u r - p o i n t bend test load-displacement p l o t f o r b 8 5 - 9 0 - 2  60  Figure 4.1 Elastic stress fields ahead o f n o t c h t i p at P = 15.0 k N  88  Figure 4.2 Elastic stress field f o r p = 27 m m at e x p e r i m e n t a l l y determined peak load o f 17.8 k N  88  F i g u r e 4.3 Stress contours ahead o f n o t c h t i p ( s y m m e t r i c about the n o t c h plane) f o r a) p = 1 m m at 15 k N l o a d , b ) p = 4 m m at 15 k N load, c) p = 10.5 m m at 15 k N load, d) p = 16 m m at 16.9 k N load and e) p = 27 m m at 17.8 k N l o a d . Elements at the n o t c h t i p are 0.5 m m x 0.5 m m i n all cases. The l i g h t grey regions indicate a stress o f 4 6 0 M P a  89  F i g u r e 4.4 Stress concentrations on the n o t c h plane ahead o f a 4 m m radius n o t c h t i p i n an O C T specimen as resolved b y different sized constant stress (cs) and f u l l y integrated quadratic (quad) elements. Stresses are extrapolated to the nodes Figure 4.5 E f f e c t o f element w i d t h o n stress calculated at a n o t c h t i p  90 90  Figure 4.6 L o a d versus a r m displacement f o r D C B simulations s h o w i n g the effect o f element w i d t h o n the peak load p r e d i c t i o n  91  F i g u r e 4.7 Peak loads f o r D C B s i m u l a t i o n u s i n g element meshes o f v a r y i n g coarseness w i t h and w i t h o u t scaling the material strength t o m a t c h the 62.5 p m mesh size  91  F i g u r e 4.8 Strength scaling ratios to m a t c h the peak load i n the D C B s i m u l a t i o n u s i n g a mesh w i t h 62.5 p m constant stress elements at the crack t i p  92  Figure 4.9 Simple b i l i n e a r damage m o d e l  92  Figure 4.10 F l o w chart f o r S D M / A S D M a l g o r i t h m  93  F i g u r e 4.11 Integration p o i n t n u m b e r i n g  94 viii  List of Figures F i g u r e 4.12 Scaled stress-strain curve  94  Figure 4.13 D a m a g e zone size determined b y the cohesive zone m o d e l  95  Figure 4.14 E x a m p l e o f energy release determined u s i n g elements o f different size  95  Figure 4.15 V i r t u a l O C T specimen mesh f o r a) f u l l specimen and b ) at n o t c h plane  96  Figure 4.16 Parametric study o f G and a  96  c  Figure 4.17 R e d u c t i o n i n e *  pea  p e a  k f o r rd2  k at the integration points o f the n o t c h t i p element i n rd8  F i g u r e 4.18 Peak loads predicted u s i n g the A S D M (large d i a m o n d s ) p l o t t e d over the experimental results (small squares). O p e n symbols represent cases that w e r e unstable (i.e. n u m e r i c a l r u n crashed or d i d n o t display load plateau e x p e r i m e n t a l l y ) . N o t e that the ordinte scale begins at 10 k N to m a g n i f y the results ,  97  97  Figure 4.19 N o r m a l stress contours f o r rd2 a) at C M O D = 0 . 6 7 9 m m (before peak load), b ) at C M O D = 1.70 m m (peak load) and c) at C M O D = 3 . 2 1 m m (after peak load)  98  F i g u r e 4.20 N o r m a l stress contours f o r r d 5 4 a) at C M O D = 0 . 6 7 4 m m , b ) at C M O D = 1 . 7 0 m m and c) at C M O D = 2 . 4 8 m m (peak load)  99  F i g u r e 4.21 N o r m a l stress contours f o r r d 3 2 a) at C M O D = 0 . 6 7 4 m m (before peak load), b) at C M O D = 2 . 1 1 m m (peak load) and c) at C M O D = 3 . 2 1 m m (after peak load) F i g u r e 4.22 Progression o f the stress d i s t r i b u t i o n a l o n g the n o t c h plane f o r r d 2 . Each d i s t r i b u t i o n is at the C M O D value ( i n m m ) noted above it  100  101  F i g u r e 4.23 Progression o f the stress d i s t r i b u t i o n along the n o t c h plane f o r r d 2 8 . Each d i s t r i b u t i o n is at the C M O D value ( i n m m ) noted above it  101  F i g u r e 4.24 D a m a g e zone w i d t h determination f o r rd28 ( N o t e that the crack i n c l u d i n g the damage zone is 32.25 m m w i d e at this p o i n t and the C M O D is 3.9 m m ) . . . 102 F i g u r e 4.25 C o m p l i a n c e f o r rd28 before s h i f t i n g A S D M p r e d i c t i o n  102  Figure 4.26 C o m p l i a n c e f o r rd28 after s h i f t i n g A S D M p r e d i c t i o n b y 3.25 m m t o account f o r p a r t i a l l y damaged zone  103  F i g u r e 4.27 Progression o f the crack front i n selected v i r t u a l O C T specimens  103  Figure 5.1 Stress and energy failure criteria f o r fracture i n r e g i m e 1  121  Figure 5.2 Stress and energy failure criteria f o r fracture i n r e g i m e I I  121  F i g u r e 5.3 r"  122  1/2  singularity at the t i p o f a sharp n o t c h i n an O C T specimen  F i g u r e 5.4 M e s h o f quarter-point elements around a crack t i p ( f r o m ( C o o k , M a l k u s a n d P l e s h a , 1989))  122  Figure 5.5 A n a l y t i c a l p r e d i c t i o n o f peak loads f o r O C T specimens  123  F i g u r e 5.6 Cracks i n i t i a t i n g o f f the centerline i n a) r d 4 5 - l and b) r d 4 4 - 2  123 ix  List of Figures F i g u r e 5.7 A n ellipse w i t h a m a j o r axis o f 8 m m and a m i n o r axis o f 2 m m has the same radius o f curvature at the n o t c h t i p as a circle w i t h a radius o f 16 mm  124  F i g u r e 5.8 Stress d i s t r i b u t i o n ahead o f a n o t c h t i p s h o w i n g an element o f w i d t h w  e  that  is too large. M e s h must either be r e f i n e d or strength scaling used to achieve accurate results  124  Figure 5.9 L o a d versus C M O D results f o r a sharp-notched O C T specimen m o d e l e d w i t h different size elements  125  F i g u r e 5.10 N o t c h t i p stress fields f o r coarse and fine meshed O C T specimens w i t h a sharp n o t c h t i p at load equal to 8.3 k N Figure 5.11 E l e m e n t size contours f o r G = 80 k J / m  125 126  2  c  F i g u r e 5.12 L a r g e constant stress elements used to m o d e l the experimental 1 m m n o t c h root radius O C T specimen all give the same result F i g u r e 5.13 Element size contours f o r G = 20 k J / m  126 127  2  c  F i g u r e 5.14 W h e n G is reduced to 20 k J / m a difference is noted i n the peak loads predicted u s i n g different w i d t h elements c  127  F i g u r e A . 1 E x p e r i m e n t a l l o a d - C M O D results and n u m e r i c a l p r e d i c t i o n f o r specimen rd2  133  F i g u r e A . 2 E x p e r i m e n t a l l o a d - C M O D results and n u m e r i c a l p r e d i c t i o n f o r specimen rd8  133  F i g u r e A . 3 E x p e r i m e n t a l l o a d - C M O D results and n u m e r i c a l p r e d i c t i o n f o r specimen rd21  .'  134  Figure A . 4 E x p e r i m e n t a l l o a d - C M O D results and n u m e r i c a l p r e d i c t i o n f o r specimen rd28  134  F i g u r e A . 5 E x p e r i m e n t a l l o a d - C M O D results and n u m e r i c a l p r e d i c t i o n f o r specimen rd32  135  F i g u r e A . 6 E x p e r i m e n t a l l o a d - C M O D results and n u m e r i c a l p r e d i c t i o n f o r specimen rd38  135  F i g u r e A . 7 E x p e r i m e n t a l l o a d - C M O D results and n u m e r i c a l p r e d i c t i o n f o r specimen rd54  136  F i g u r e A . 8 E x p e r i m e n t a l l o a d - C M O D results f o r specimen r d l  136  Figure A . 9 E x p e r i m e n t a l l o a d - C M O D results f o r specimen r d 4  137  Figure A . 10 E x p e r i m e n t a l l o a d - C M O D results f o r specimen rd6  137  F i g u r e A . l 1 E x p e r i m e n t a l l o a d - C M O D results f o r specimen r d l O  138  F i g u r e A . 12 E x p e r i m e n t a l l o a d - C M O D results f o r specimen r d l 7  138  F i g u r e A . 13 E x p e r i m e n t a l l o a d - C M O D results f o r specimen rd25  139  x  List of Figures F i g u r e A . 14 E x p e r i m e n t a l l o a d - C M O D results f o r specimen rd35  139  F i g u r e A . 15 E x p e r i m e n t a l l o a d - C M O D results f o r specimen r d 4 4  140  F i g u r e B . l C O D p r o f i l e s determined from line analysis f o r r d 2 - l  142  F i g u r e B.2 C O D p r o f i l e s determined from line analysis f o r r d 2 - 2  142  F i g u r e B.3 C O D p r o f i l e s determined f r o m line analysis f o r r d 8 - 2  143  F i g u r e B.4 C O D p r o f i l e s determined from line analysis f o r r d 5 4 - l  143  F i g u r e B.5 C O D p r o f i l e s determined from line analysis f o r r d 5 4 - 2  144  F i g u r e C. 1 Progression o f the stress d i s t r i b u t i o n along the n o t c h plane f o r rd8. Each d i s t r i b u t i o n is at the C M O D value ( i n m m ) noted above it  146  F i g u r e C.2 Progression o f the stress d i s t r i b u t i o n along the n o t c h plane f o r r d 2 1 . Each d i s t r i b u t i o n is at the C M O D value ( i n m m ) noted above it Figure C.3 Progression o f the stress d i s t r i b u t i o n along the n o t c h plane f o r r d 3 2 . Each d i s t r i b u t i o n is at the C M O D value ( i n m m ) noted above it  146  147  Figure C.4 C o m p l i a n c e plots from location o f peak stress i n n u m e r i c a l simulations. 3.25 m m is subtracted from Aa to account f o r w i d t h o f damage zone  147  Figure E . l D C B specimen ( u n i d i r e c t i o n a l fibres oriented a l o n g length o f specimen)  164  F i g u r e E.2 Peak load p r e d i c t i o n f o r A S 4 / 3 5 0 1 - 6 D C B specimen  164  Nomenclature  NOMENCLATURE  Strain  S  Parametric element coordinate  a  Stress  e p  N o t c h root radius  A n g l e i n radians from n o t c h plane  s  Displacement Parametric element coordinate  V S*peak  Scaled peak strain  S*ult  Scaled u l t i m a t e strain  (7c  U n n o t c h e d laminate tensile strength  (7max  M a x i m u m stress ( i n stress d i s t r i b u t i o n )  SOpeak  I n i t i a l characteristic peak strain  Yc  C r i t i c a l specific strain energy  Yd  Dissipated specific strain energy  Sdam  D a m a g e zone strain  Speak  Current element peak strain  Smax  M a x i m u m strain i n an element ( f o r scaling)  <?N  N o t c h e d laminate tensile strength  Speak  Current peak strain  CTpeak  Peak stress  Yr  Residual specific strain energy  Sscale  M a x i m u m value o f strain field used f o r scaling  Suit  U l t i m a t e strain  a ao  C r a c k length M a t e r i a l system characteristic d i m e n s i o n (average stress m e t h o d )  ASDM  Adaptive Simple Damage M o d e l  B  Specimen thickness  C  Compliance  CMOD  Crack M o u t h O p e n i n g D i s p l a c e m e n t  D  D e p t h ( f o u r - p o i n t b e n d specimen)  do  M a t e r i a l system characteristic d i m e n s i o n ( p o i n t stress m e t h o d )  DCB  D o u b l e - c a n t i l e v e r b e a m (specimen)  E'  E f f e c t i v e anisotropic m o d u l u s  E E F  Initial modulus  0  Residual secant m o d u l u s  r  W o r k p e r f o r m e d b y external forces  G G G h c  Strain energy release rate c  C r i t i c a l fracture/ strain energy release rate  F  Fracture energy Characteristic stable height o f damage zone  Nomenclature hd h  e  H e i g h t o f damage zone i E l e m e n t height  h  H e i g h t o f c r i t i c a l l y stressed material  Ki  Stress intensity factor ( m o d e I )  Kf  Stress concentration factor  L  L o a d span ( f o u r - p o i n t b e n d specimen)  U  Support span ( f o u r - p o i n t bend specimen)  OCT  O v e r - h e i g h t C o m p a c t T e n s i o n (specimen)  P Pc POD  Load Peak load (structural strength) P i n O p e n i n g Displacement  R  Crack resistance energy  r  Distance f r o m n o t c h t i p  SDM  Simple D a m a g e M o d e l  T  Interface traction  t U  Simulation time  s  L  Elastic strain energy  W  Section w i d t h o f O C T specimen  Went VOdam  C r i t i c a l l y stressed w i d t h o f material ahead o f n o t c h t i p D a m a g e zone w i d t h  Wtotal  T o t a l w o r k done  X  Coordinate i n d i r e c t i o n o f crack  y  Coordinate perpendicular t o crack  Acknowledgements  ACKNOWLEDGEMENTS  First, m y deepest gratitude goes t o m y advisor, D r . A n o u s h Poursartip, f o r g i v i n g m e the o p p o r t u n i t y t o w o r k i n the Composites G r o u p at U B C and f o r the countless hours he has spent keeping m e focused i n the r i g h t direction. H e has the great a b i l i t y t o get t o the root o f every p r o b l e m that I b r i n g h i m . I a m also indebted t o m y co-advisor, D r . Reza V a z i r i , f o r his assistance and l o n g discussions o n the n u m e r i c a l side o f m y thesis. The w o r k that I have done here w o u l d n o t be possible w i t h o u t the w o r k o f numerous Composites G r o u p members that have come before m e , p a r t i c u l a r l y the w o r k o f J. Scott Ferguson, K e v i n W i l l i a m s , I n g r i d K o n g s h a v n , Jason M i t c h e l l , K a r i m K a n j i a n d A n t h o n y Floyd. I w o u l d l i k e t o thank all o f the group members that have made m y t i m e at U B C as enjoyable as i t w a s educational.  Roger Bennett deserves special r e c o g n i t i o n f o r p r e p a r i n g the  specimens, m o d i f y i n g the testing j i g s and assisting m e w i t h a n y equipment concerns that I had.  I a m also indebted t o A n t h o n y F l o y d f o r answering every n u m e r i c a l question I c o u l d  come up w i t h and t o K a r i m K a n j i f o r considerable assistance w i t h the experimental tests. A b i g thank y o u goes t o m y entire f a m i l y , especially b o t h o f m y parents, f o r supporting m e t h r o u g h o u t m y entire education n o matter w h a t I decided t o d o or w h a t part o f the continent I decided t o d o i t on. Last, thanks T r a c y , f o r b e i n g inspirational a n d f o r b e i n g w i t h m e throughout, even w h e n y o u ' r e thousands o f miles away.  xiv  Chapter I Introduction  CHAPTER 1 INTRODUCTION C o m p a r e d to most structural materials, fibre reinforced p o l y m e r ( F R P ) composites are v e r y n e w . T h e y have o n l y been used and studied f o r a little over h a l f a century, but i n that t i m e an enormous amount o f literature has been produced. N u m e r o u s models ( e m p i r i c a l , analytical, and since the rise o f the f i n i t e element m e t h o d , n u m e r i c a l ) have been devised to e x p l a i n every aspect o f their behaviour, f r o m processing to damage and fracture. W h i l e the science b e h i n d composite  materials  has been advanced i m m e n s e l y  by  this research and  the  development o f these material models, it presents a daunting task f o r an engineer or designer w h o wishes to use F R P composites i n a design.  H o w can the engineer n a r r o w d o w n the  n u m b e r o f m o d e l i n g choices and be certain that the m o d e l chosen f o r the design is the best one?  I n a recent exercise, several leading failure theories f o r composite laminates were  tested against fourteen c h a l l e n g i n g test cases ( H i n t o n , K a d d o u r and Soden, 2 0 0 2 b ; H i n t o n and Soden, 1998).  The exercise was a massive u n d e r t a k i n g and revealed that the different  models had different benefits, that even the best models c o u l d n o t describe all test cases, and that the most c o m p l i c a t e d models w e r e n o t always the best models ( H i n t o n , K a d d o u r and Soden, 2002a). T h e uncertainty i n design using composite materials has been apparent i n the aerospace industry. W h i l e the benefits o f u s i n g l i g h t w e i g h t , s t i f f and durable F R P composites that can be t a i l o r e d to meet the requirements o f each application are o b v i o u s , the uncertainty i n the b e h a v i o u r o f n e w FRPs has kept t h e m f r o m b e i n g used m o r e extensively.  A l t h o u g h FRPs  have been used i n r e l a t i v e l y small quantity on most c o m m e r c i a l aircraft i n the latter h a l f o f the last century, the c o m m e r c i a l aerospace industry has been cautious about u s i n g t h e m f o r m a j o r load bearing structures u n t i l v e r y recently, due to a lack o f understanding o f their behaviour. A h i g h n u m b e r o f f l i g h t hours w i t h r e l a t i v e l y little t i m e f o r inspection has made the uncertainty i n composite b e h a v i o u r too large a factor to overcome. F o r m i l i t a r y aircraft, i n w h i c h there is a m u c h higher ratio o f maintenance to f l i g h t hours, composites have always been used m o r e extensively. W i t h the composites k n o w l e d g e amassed over the past f e w decades h o w e v e r , the next generation passenger aircraft are using F R P composites i n larger quantity to realise large  1  Chapter  I  Introduction  gains in efficiency. In order to do so, the materials must go through extensive testing, which requires thousands of hours of experiment and modeling for each new material and application. To be competitive in the commercial aircraft market, the time between conception and rollout does not allow every newly developed model to be rigorously tested. Complicated models that can explain every micro-mechanical aspect of a material's response under specific conditions may be excellent laboratory tools but are impractical for real applications. Each additional material parameter or variable in a model adds uncertainty. At the end of the day, a designer needs to be confident in his understanding of the physics of the problem and needs to be certain that a design will not fail. Each parameter or variable that is added to a model that is not physically based or that is added for numerical convenience hurts the understanding of the physical processes taking place. The most useful tools for a design engineer are simple models containing few parameters, completely based on physical quantities. Such models can be used to quickly narrow down a list of eligible materials for a given application. They can also be used to quickly test the validity of more complicated models that might be necessary to investigate specialized aspects of a select few prospective materials or structures. Completely new models were not developed in this thesis. Instead, a new way of looking at models that have been around for many years is presented with the goal of providing a clear picture of the mechanisms of fracture in FRP composites. A large number of experimental tensilefracturetests were conducted, which provide insight into the physical processes of fracture. A simple, three-parameter, numerical, cohesive zone model for orthotropic composites is used to help explain the results of the experiments. Although several other researchers have proposed models similar to the one presented here, the interaction of mesh size and input parameters has often necessitated the use of non-physical parameters.  A better physical  understanding of the problem eliminates these issues. Finally, a simple analytical model, using the same three parameters as the numerical model, is devised from basicfracturemechanics equations. This model provides a designer with a simple tool for evaluating different materials. 2  Chapter  2 Background  and Literature  Review  CHAPTER 2 BACKGROUND AND LITERATURE REVIEW  2.1  Introduction  This thesis studies the behaviour of notched composite laminates under applied tensile loads by means of (a) experimental evidence, (b) the development of a numerical composite damage model and, (c) a simple analytical description. The aim is to develop a fully consistent, physically based explanation of the behaviourfromall three vantage points. There has been a considerable amount of work on damage and crack propagation in composites over the past several decades with numerous theories and models proposed and developed by a large number of researchers. Some of the models complement one another, some have been modified or improved upon by later researchers and others run parallel to each other.  Major contributions to the study of crack and damage propagation from  experimental evidence to closed-form solutions to numerical models are discussed here.  2.2  Fracture mechanics  The theory of crack propagation in a continuum originated well before the development of advancedfibre-reinforcedpolymer composites. Griffith established the energy criterion for crack growth in brittle materials (Griffith, 1921) in 1921. Considering the energy required to create the surfaces of a crack, he determined the amount of energy required for a crack to grow. A crack will grow if the energy released during an incremental growth of the crack by length da is greater than or equal to the energy needed for the crack to grow by that amount, expressed as 2.1  G>R.  G is called the strain energy release rate defined for unit thickness as the change in energy with crack extension such that  G=fcl, da  2.2  3  Chapter 2 Background and Literature Review  where F is the work performed on a plate with free ends by external forces, and U is the stored elastic energy in the plate. R is the "crack resistance" equal to the energy absorbed during crack growth. While this definition was developed for brittle, elastic materials, it is valid for materials with plasticity or damage at the crack tip, although the crack resistance term is then dominated by plastic or damage energy as opposed to only surface energy. Evaluating the elastic energy in terms of the compliance of the plate, a general expression for the energy release rate can be written in terms of load and the compliance, as given by (Broek, 1982) G-**.  2.3  IB da  Alternatively, the condition for fracture can be expressed in terms of the stress or strain field at the tip of a crack. The well-known function for the stressfieldimmediately around a crack tip is attributed to Irwin and given in many textbooks (e.g. (Broek, 1982)) as  -JlTtr  The stresses a , a , and z are defined in the directions given in Figure 2.1 in terms of the x  y  xy  cylindrical coordinates r and 6. Kj is the mode I stress intensity factor describing the 1  magnitude of the stress field. Along the crack plane ahead of the crack tip the function fy(0) is equal to unity. For a center crack in an infinite plate Ki is a function only of the stress in the plate and the length of the crack as given in Equation 2.5. Kj =<j4na~  For finite sized structures, K[ also depends on geometry.  2.5  Solutions for Kj have been  compiled for standard test geometries (e.g. (Broek, 1982), (Tada, Paris and Irwin, 1985)) or can be determined through finite element techniques. Again, these equations are elastic solutions, valid if the plastic or damage zone at the crack tip is small relative to other ' There are three modes of cracking. Mode I is crack opening in which the crack surfaces move perpendicular to the crack plane. Modes II and III are shear modes (sliding and tearing) in which the crack surfaces move parallel to the crack plane.  4  Chapter 2 Background and Literature Review specimen dimensions.  This means that the damaged material at the crack tip must be  constrained by the bulk elastic material of the structure so that stress, strain and crack opening displacements do not differ greatly from the purely elastic case. A critical value of the stress intensity factor Kj can be determined which corresponds to the stress at failure a c  c  as 2.6 Kic is called the fracture toughness of the material. If Kj is known, the far-field stress at c  which fracture will occur can be determined. In Linear Elastic Fracture Mechanics (LEFM), Kj is considered to be a material property. c  It seems that there are two independent approaches for determining the condition for crack initiation.  However, the two conditions, energy and stress intensity, are linked. Broek  (Broek, 1982) states,  "The energy criterion is a necessary criterion for crack extension. It need not be a sufficient criterion. Even if sufficient energy for crack propagation can be provided, the crack will not propagate unless the material at the crack tip is ready to fail: the material should be at the end of its capacity to take load and to undergo further straining. " A relationship can be defined between Ki and Gj, if G/ is written in terms of stress. The full derivation is given by Broek and is only summarised here. The strain energy release rate is the amount of energy released during an infinitesimal length of crack growth. Alternatively, it can be viewed as the energy required to close a crack over an infinitesimal distance given as(Broek, 1982) 2  G, =lim—  r y <J  v  dr  2.7  where S is the infinitesimal length of crack and v is the opening displacement. Substituting Equation 2.4 and crack opening displacement in terms of K/ and E into Equation 2.7 results  5  Chapter 2 Background and Literature Review 2  in the plane stress L E F M relationship between the stress intensity factor and the strain energy release rate as 2.8 Here E is the modulus for a homogeneous, isotropic material. It is often assumed that the stress criterion for fracture and the energy criterion for fracture are fulfilled simultaneously, e.g. (Broek, 1982; Kanninen and Popelar, 1985). If this is the case then  K  2  G =^. E  2.9  c  Indeed, Kanninen and Popelar (Kanninen and Popelar, 1985) write,  "When Ki, for example, attains its critical value, then G must also reach its critical valu and [Equation 2.8] implies [Equation 2.9]. Consequently, for linear elastic bodies, the stress intensity factor and the energy balance approaches to fracture are equivalent. " In L E F M , which assumes infinitely sharp crack tips, the simultaneous fulfilment of both conditions is assumed to be true and it is generally accepted for blunter crack tips as well. For blunt notch tips, holes and cut-outs the stress field around the stress concentrator is described by a stress concentration factor Kj instead of a stress intensity factor. While K/ describes the stress field as stresses approach infinity at a sharp crack tip, K is a factor that T  relates the finite stress at a notch tip to the far-field stress. The stress field normal to the crack plane around a blunt notch tip can be related to the stress field of an equivalent sharp crack tip a certain distance away from the tip by an equation given by (Creager and Paris, 1967): a =  K, 2 r' K  p { W'\ K, + , cos 2r\ 2 ) jj^?  . 9' . W\ 0' (, cos- 1 + s i n — s i n — 2 2 2)  I  2.10  This equation is valid for the case of plane stress. For plane strain the left side of the equation is multiplied by a (1-v ) term.  2  2  6  Chapter 2 Background and Literature Review The radius of curvature at the notch tip is p and r' is the distance from the effective notch tip as defined in Figure 2.2. In Equation 2.10, K\ is the stress intensity factor for a sharp crack tip located at a point p/2 behind the actual notch tip. A l o n g the x-axis in front of the notch tip (8' = 0) equation 2.10 reduces to  K, cr = . '  p  K,  .  2  .  1  2.11  A s mentioned above, the modulus in Equation 2.8 is valid for isotropic materials. Sih, Paris and Irwin defined an effective modulus E', given by Equation 2.12, in terms of the elastic constants that is valid for anisotropic materials such as F R P composites (Sih, Paris and Irwin, 1965).  E'=  r  \a  \a a u  22  22  V  For plane stress, au=\IEi, a2i=\IE2,  +  2a  v n a  l2  2.12  +a  66  2a  u  j  an=-vnlEi and a66=VGi2, where Eu E2,  V12 and Gn are  the orthotropic elastic constants.  2.3  Cohesive crack models  Another method o f predicting crack initiation and propagation, that is fundamentally different from L E F M , is the cohesive crack model.  The concept, originally proposed by  Dugdale (Dugdale, 1960) for metals, has given rise to numerous analytical and numerical models for crack growth and damage in composite materials. The cohesive crack model recognises that there is a region of material ahead of a crack tip that is not linear elastic. In F R P composite materials this "damage zone" consists of matrix cracking, fibre breakage, interface separation and fibre pullout. In the cohesive crack approach, a relationship between traction and displacement is defined along a potential crack surface, sometimes called a "fictitious crack", ahead of an actual crack or notch tip as shown schematically in Figure 2.3. This fictitious crack represents the damaged area lumped into a discrete plane. A typical separation relationship is shown in Figure 2.4. The interface is linear-elastic up to a specified critical stress. For strains larger  7  Chapter  2 Background  and Literature  Review  than the critical strain associated with this stress, the material is damaged. The damaged interface in Figure 2.4 obeys a linear relationship until the tractions on the interface are zero at some value of interface separation. The traction-displacement curve is often chosen to be linear for simplicity but is not required to be so. Dugdale used a cohesive crack approach to determine plastic zone sizes in steel panels containing edge slits. The concept was then developed further by Barenblatt (Barenblatt, 1962) to model crack propagation in metal sheets. Since that time, the cohesive approach has been used both as a continuum material model to predict damage growth and propagation, and also as strictly an interface model. When used as a continuum material model (called the cohesive zone model), the traction-displacement relationship becomes a stress-strain relationship. Important considerations have been brought to light from research in both areas and will be discussed in the following sections on interlaminar crack propagation and notched strength prediction. The link between fracture mechanics and cohesive models is the strain energy release rate. The area underneath the traction-displacement curve of a cohesive crack model is an energy release rate; essentially the amount of energy required to completely open the interface for a given increment of crack growth. The condition for complete crack separation in these models is the attainment of either a critical value of the energy release rate G or a critical c  interface displacement S . The two criteria are identical i f the bilinear constitutive curve is c  uniquely defined by one of these two values and by two of the three elastic parameters; EQ, apeak or Speak,  2.3.1  where  e k pea  is the strain at the peak stress on the curve.  Cohesive crack models for intralaminar crack propagation  In most cases cohesive crack models are implemented in finite element computer codes as zero thickness elements. A common method of implementing the cohesive law is to use a penalty stiffness scaled by a damage parameter co after the peak traction is reached, for example (Alfano and Crisfield, 2001). As illustrated in Figure 2.4 for mode I loading, the basic constitutive relation is V ,  ifS  i  <<5,  0  2.13 0  i/S,>S  k  8  Chapter  2 Background  and Literature  Review  Here, Tis the interface traction defined in the i' material direction, where /= 1,2,3. When the h  interface separation 8 is less than a threshold separation distance So, the interface is elastic. For 8 greater than So but less than a critical separation distance S , the interface is damaged c  by an amount co. After S the interface is fully damaged and the crack faces are completely c  separated. The damage parameter co e[0,l] is a monotonically increasing function, i.e. the amount of damage in the interface can never decrease. The damage parameter is a function of either interface separation as in (Camanho, Davila and de Moura, 2003), (Alfano and Crisfield, 2001), (Geubelle and Baylor, 1998), (Espinosa, Dwivendi and Lu, 2000) or a function of the energy release rate such as in (Borg, Nilsson and Simonsson, 2001; Borg, Nilsson and Simonsson, 2002), (Borg, Nilsson and Simonsson, 2002; Zou, Reid and L i , 2003). Another notable model is that of Xu and Needleman (Xu and Needleman, 1996; X u and Needleman, 1994). In their model the interface constitutive relations are based on a work potential. The tractions on an interface are determined by differentiating the work potential, which is an exponential function of both normal and shear interface separations. Typically, separate traction-separation relationships are defined for opening (mode I) and shear (mode II and III) modes.  Alfano and Crisfield ((Alfano and Crisfield, 2001))  investigated the problem of developing a coupled model for mixed-mode loading, wherein the critical strain energy release rate depends on the proportion of each mode of loading.  2.3.1.1  Shape of the traction-displacement curve  Camanho et al. (Camanho, Davila and de Moura, 2003), Alfano and Crisfield (Alfano and Crisfield, 2001) and Geubelle and Baylor (Geubelle and Baylor, 1998) have all used a bilinear traction-displacement relationship.  The formulations of Borg (Borg, Nilsson and  Simonsson, 2001), Corigliano (Corigliano, Mariani and Pandolfi, 2003) and X u and Needleman (Xu and Needleman, 1996; X u and Needleman, 1994) allow for the definition of different curve shapes by varying an exponential parameter in the constitutive relation. The shape of the constitutive curve can play an important role in the response of the model. A curve such as that shown in Figure 2.5 a) can better represent the response of a ductile material while that in Figure 2.5 b) is better for modeling a quasi-brittle material,  de Borst  ((de Borst, 2003)) observes that for ductile fracture, the most important parameters of the cohesive zone model are the tensile strength and the fracture energy while for quasi-brittle  9  Chapter  2 Background  and Literature  Review  materials, the shape of the stress-separation relation plays a larger role and can be even more important than the value of the tensile strength. He presents the example of a single edgenotched (SEN) tensile specimen modeled with different shaped linear and exponential constitutive softening curves but with the same input value of fracture energy. The resulting global peak loads differed by more than 15%. 2.3.1.2  Input parameter determination  For a simple bilinear cohesive crack or cohesive zone model, only a few parameters are needed to define the constitutive behaviour and these are related to physical quantities. A bilinear constitutive curve is completely defined by three parameters: •  The initial elastic modulus EQ  •  Either the peak stress a k or the strain at peak stress e k  •  Either the critical fracture energy G or the ultimate strain s u  pea  pea  c  u  In many of the models reviewed in the literature however, the input parameters are not arrived at independently.  Instead they are deduced by fitting the global response of the  numerical model, such as the load versus displacement of a notched tensile test, to the results of an experimental test for one or two notch sizes.  One of the reasons for this is the  dependence of the structural response on mesh discretization when the models are implemented in finite element codes. The peak tensile stress in particular is often adjusted to match experimental results. It is generally agreed that the peak stress does not have a large effect on crack growth (Zou, Reid and L i , 2003) but that proper selection of a k is pea  important for a good prediction of crack initiation (Alfano and Crisfield, 2001). Geubelle and Baylor noted in their study on impact-induced delamination of composites that the time and location of initial matrix damage depended strongly on a k (Geubelle and Baylor, pea  1998). Zou et al. (Zou, Reid and L i , 2003) and Alfano and Crisfield (Alfano and Crisfield, 2001) also noted that the value of the peak stress has a strong influence on the computational efficiency of the finite element simulation, in that the higher a k, the more refined the mesh pea  must be in the region of the delamination front. Usually, o- k is chosen to be reasonably close to the physical interlaminar strength of the pea  interface, however, depending on the element size used to mesh the crack plane, an incorrect  10  Chapter  2 Background  and Literature  Review  unstable solution is sometimes obtained when using this value for sometimes altered so that a stable solution is achieved.  Therefore, it is  <J h pea  Borg et al. (Borg, Nilsson and  Simonsson, 2001) noted that when modeling mixed mode delamination in carbon-fibre/ epoxy, if peak stresses equal to the failure stresses in uni-axial tensile tests were used, the peak stresses were of magnitudes that prevented the growth of delamination. Consequently, these researchers used peak stresses substantially lower than the interlaminar strengths of the material. Both X u and Needleman (Xu and Needleman, 1996; X u and Needleman, 1994) and Geubelle and Baylor (Geubelle and Baylor, 1998) chose  (j k pea  as a fraction of the elastic  modulus instead of using experimental values. Most of the models are implemented in explicit dynamic finite element codes and several of the models are used to investigate dynamic crack propagation.  For most of the models  though, the material parameters are independent of strain rate. To model dynamic crack propagation, Geubelle and Baylor (Geubelle and Baylor, 1998) pre-multiplied the input fracture toughness by a strain rate parameter.  Corigliano, Mariani and Pandolfi (Corigliano,  Mariani and Pandolfi, 2003) included a strain-rate dependent parameter as part of the mode I cohesive law in their model and compared it to a rate-independent model. They concluded that rate-dependency can greatly affect dynamic crack processes but that the effects of rate dependency on the test response during dynamic delamination growth decrease when inertial terms become dominant. 2.3.2  Cohesive zone models for interlaminar crack propagation  The concept of the cohesive crack model has also been expanded from an interfacial model to simulate the continuum response of a damaging material. Ffillerborg et al. (Hillerborg, Modeer and Petersson, 1976) appear to be the first to apply the concept of the cohesive crack method of Dugdale and Barrenblatt to a continuum finite element model. cohesive crack model to a solid continuum introduces certain problems.  Applying the  In the cohesive  crack model all damage is restricted to a discrete plane, but the cohesive zone model attempts to represent a volume of damaged material with elements that may not be of the same size as the material's characteristic damage size. The response of the model therefore depends on a length scale.  11  Chapter 2 Background and Literature Review In a cohesive zone model solid elements are used as opposed to zero-thickness interface elements. A stress-strain relation replaces the relationship between traction and displacement. The area beneath the stress-strain curve is a specific energy y (energy per unit volume). The specific energy is multiplied by the length scale to obtain a fracture energy release rate GF (i.e. energy per unit area) that can be related to fracture mechanics and known material properties. In finite element simulations damage tends to localise in a single layer or row of elements. Therefore the element height h is the length scale and GF is calculated as e  G =yh . F  2.14  e  Floyd (Floyd, 2004) has investigated localisation in a composite damage model and reviews the major work done in this area.  Damage localisation in a cohesive zone model is  demonstrated in Figure 2.6 for a simple three-element system. A l l three elements are defined using the same constitutive model. Initially there is zero strain in each element (a). A s the applied displacement is increased, the stress in each element increases along the slope defined by the initial modulus until the strain corresponding to the peak stress <j  max  is reached  (b). One of the elements will reach this critical strain slightly before the other two and will begin to soften. A s softening occurs in the middle element, its residual modulus is reduced, thereby reducing the stress in the entire structure. With the reduction in stress, the load in the other two elements decreases and they shed their elastic energy on to the softening element as they unload along the slope of the initial modulus (c). The unstable system quickly leads to complete damage in the softening element (d). A t this point there is zero strength in the softened element and the crack is considered to be fully developed at that location. Although only a three-element example is presented here, the effect is the same for stacks o f larger numbers o f elements.  The implication of localisation is that the damage height effectively  becomes the height o f a single element and the energy release rate of the material is dependent on that element height. A means of dealing with localisation and the mesh height effect was introduced by Bazant et al. (Bazant, 1984; Bazant and O h , 1983; Bazant and O h , 1984) for concrete.  In their  approach, termed the crack band model, the damaged zone is represented by a band of distributed cracks. The fracture energy release rate GF in a numerical model is a constant related to the height of a layer of elements and the area under the stress-strain curve in the  12  Chapter  2 Background  and Literature  Review  n u m e r i c a l m o d e l (as i n E q u a t i o n 2.14).  P h y s i c a l l y , i t is related t o the physical material  parameters: the characteristic height o f the damage zone h f o r the material system and the c  specific fracture energy y , therefore c  2.15  G =y -h =y -h . F  Numerically, G  F  e  e  c  c  can be h e l d constant i n a s i m u l a t i o n f o r v a r y i n g h o n l y b y adjusting y , e  e  since y and h are material parameters. F l o y d ( F l o y d , 2 0 0 4 ) has i m p l e m e n t e d a crack band c  c  scheme i n a composite damage m o d e l that scales the constitutive softening curve f o r v a r y i n g h w h i l e m a i n t a i n i n g GF constant. e  Cohesive zone models applied t o predict notched strength are discussed i n Section 2.4.  2.3.3  Mesh Sensitivity  One o f the m a i n drawbacks t o the cohesive crack m o d e l is that the path o f a propagating crack is mesh dependent. T h e crack is f o r c e d t o g r o w along the interface w h e r e the cohesive l a w has been defined. F o r continuous fibre r e i n f o r c e d p o l y m e r laminates, this p r o b l e m does n o t create large restrictions because delaminations are k n o w n t o f o r m i n the interlaminar layers. H o w e v e r , t o a l l o w f o r d e l a m i n a t i o n t o f o r m o n a n y interlaminar layer f o r a structure i n w h i c h the d e l a m i n a t i o n l o c a t i o n is n o t k n o w n a priori, be d e f i n e d between a l l layers o f the m o d e l .  cohesive interface elements should  I n a c o n t i n u u m cohesive zone m o d e l , the  cohesive l a w is defined w i t h i n a l l elements a l l o w i n g cracks t o initiate and develop at a n y location. I t w a s s h o w n i n Section 2.3.2 that f o r cohesive zone models, energy dissipation depends o n the height o f the elements used t o discretize the structure f o r cohesive zone models i n w h i c h v o l u m e t r i c elements are used.  Interface cohesive crack models do n o t have this d r a w b a c k  because the constitutive l a w is d e f i n e d i n terms o f length-independent quantities; traction and displacement. T h e solution f o r b o t h types o f models, h o w e v e r , is affected b y the spatial discretization i n the d i r e c t i o n parallel t o the crack.  A s a m i n i m u m requirement, the discretization must be fine  enough t o resolve the gross shape o f the stress field around the stress concentrator.  The  cohesive models account f o r damage at the n o t c h t i p so there is n o stress singularity i n the stress field as predicted b y L E F M .  13  Chapter  2 Background  and Literature  Review  D i f f e r e n t authors have f o u n d v a r y i n g degrees o f sensitivity to mesh size a l o n g the crack direction.  A t one extreme, Geubelle and B a y l o r (Geubelle and B a y l o r , 1998) conducted a  sensitivity study b y l o o k i n g at the convergence o f the speed o f a r a p i d l y propagating crack i n a pre-loaded P M M A  strip f o r various mesh sizes.  converged f o r elements w i t h an a p p r o x i m a t e l y  T h e y f o u n d that the fracture speed  7 pm  characteristic d i m e n s i o n .  They  concluded that the elements needed to be t w o or three times smaller than their estimate o f the cohesive zone size f o r convergence o f the scheme.  Conversely, B o r g et al. ( B o r g , N i l s s o n  and Simonsson, 2002) f o u n d that they c o u l d use a r e l a t i v e l y coarse mesh and achieve load versus displacement results comparable to those o f a fine mesh.  H o w e v e r , i n their m o d e l  they f o u n d it necessary t o decrease the input values o f strength and G  c  from  actual  experimental values b y 5 0 % or m o r e . A l f a n o and C r i s f i e l d ( A l f a n o and C r i s f i e l d , 2 0 0 1 ) also conducted a study on the effect o f v a r y i n g mesh size and tensile strength simultaneously. T h e y f o u n d that w h e n large tensile strength values (comparable to experimental values) were used, they required a v e r y r e f i n e d mesh to achieve g o o d load versus displacement results f o r a D C B specimen. T h e y also n o t e d that they c o u l d use a coarse mesh t o achieve g o o d results i f the m o d e l tensile strength was decreased ( b y up to a factor o f 3 0 ) . Xu  and N e e d l e m a n ( X u and N e e d l e m a n ,  1994) noted that w h e n u s i n g a  fine  mesh,  decohesion takes place i n a m o r e or less ductile manner w h i l e larger elements led t o a m o r e b r i t t l e response. Since most F R P composites have damage zones that are several m i l l i m e t r e s i n height, it is often necessary to use elements that are somewhat smaller than the damage zone height so that stress fields can be p r o p e r l y represented. The crack band scaling m e t h o d developed b y F l o y d takes care o f this aspect.  For the s i m u l a t i o n o f engineering structures h o w e v e r , too  fine a mesh can lead to enormous r u n times. A l s o i n a cohesive zone m o d e l , i n w h i c h  fibre  and m a t r i x properties are smeared into a c o n t i n u u m , it makes sense to use elements that are o n the order o f the size o f the damage zone. I f elements are smaller they b e g i n to represent v o l u m e s c o m p r i s i n g o n l y fibre or o n l y m a t r i x .  A m e t h o d o f scaling the material input  parameters f o r r e l a t i v e l y large element w i d t h s (on the order o f m i l l i m e t r e s or hundreds o f m i c r o n s as opposed to m i c r o n s ) is presented i n Chapter 4.  14  Chapter  2.4  2 Background  and Literature  Review  Notched strength prediction  N u m e r o u s models based o n strength o f materials, fracture mechanics and cohesive methods have been proposed over the years that attempt to predict the strength o f structures c o n t a i n i n g stress concentrators such as holes and notches.  2.4.1  Semi-empirical models  I n the 1970's a n d early 1980's i n particular, several models were proposed. A w e r b u c h and M a d h u k a r ( A w e r b u c h and M a d h u k a r , 1985) w r o t e a comprehensive r e v i e w o f the m a j o r models u p to 1985. T h e essential w o r k s discussed i n their r e v i e w are: 1) W a d d o u p s , Eisenmann a n d K a m i n s k i : L E F M approach 2)  W h i t n e y and N u i s m e r : Point stress and average stress methods  3) K a r l a k : A m o d i f i e d p o i n t stress m e t h o d 4)  Pipes, W e t h e r h o l d a n d Gillespie: A m o d i f i e d p o i n t stress m e t h o d  5) M a r and L i n : C o m p o s i t e stress d i s t r i b u t i o n approach M o s t studies have investigated the response o f plates c o n t a i n i n g either sharp slits o r circular holes where the n o t c h size a to specimen w i d t h W was variable. A f e w studies have l o o k e d at the effect o f v a r y i n g the n o t c h t i p radius w h i l e m a i n t a i n i n g a/W as a constant. Results are t y p i c a l l y presented as the ratio o f notched strength to u n n o t c h e d strength. T h e W a d d o u p s , Eisenmann and K a m i n s k i ( W E K ) ( W a d d o u p s , Eisenmann and K a m i n s k i , 1971) m o d e l is a s e m i - e m p i r i c a l m o d e l i n that i t uses L E F M equations a l o n g w i t h a characteristic material length that must be determined f r o m f i t t i n g curves to experimental data.  T h e characteristic length a  c  defines w h a t the authors call an intense energy r e g i o n  emanating f r o m the edge o f a hole i n a plate. T h i s r e g i o n is considered to be an equivalent crack so that a stress intensity factor f o r a crack emanating f r o m a hole can be d e f i n e d as  2.16  Here p is the hole radius and f o r a specimen c o n t a i n i n g n o hole the f u n c t i o n f(a /p) c  to unity.  T h e assumption is then made that f(a /p) c  is equal  is a constant equal to the ratio o f the  strength o f the control specimen to the strength o f a notched  specimen  By  15  Chapter 2 Background and Literature Review c o n d u c t i n g experimental tests w i t h a notched and an unnotched specimen, the m o d e l can be calibrated and used to predict strengths f o r other specimens o f s i m i l a r geometry. semi-empirical  nature, g o o d  agreement  with  experimental  results  D u e to its  was achieved.  In  independent tests b y A w e r b u c h a n d M a d h u k a r h o w e v e r , it was f o u n d that the characteristic length was dependent o n hole radius i n addition to material system, p l y orientation and stacking sequence. F o l l o w i n g the W E K m o d e l , W h i t n e y and N u i s m e r ( W h i t n e y and N u i s m e r , 1974) postulated t w o stress criteria f o r p r e d i c t i n g notched strength o f composite laminates that have received considerable attention.  T h e " p o i n t stress" and "average stress" models also incorporate a  characteristic distance at the n o t c h t i p . Failure is predicted to occur w h e n the stress at some distance f r o m the n o t c h t i p reaches the unnotched strength o f the material. Instead o f u s i n g an L E F M solution f o r the stress d i s t r i b u t i o n at a sharp n o t c h t i p , the models use the elastic solution f o r the stress d i s t r i b u t i o n at a hole. T h e theoretical basis o f their approach is that the material must be c r i t i c a l l y stressed over a characteristic d i m e n s i o n i n order to  find  a  sufficient f l a w size that w i l l initiate failure. I n the p o i n t stress m o d e l , the stress at a p o i n t a distance do f r o m the n o t c h t i p along the axis o f the n o t c h is compared to the u n n o t c h e d strength. T h e authors consider do to be a material property. T h e average stress m o d e l is the same as the p o i n t stress m o d e l except that the average stress over a distance ao f r o m the n o t c h t i p is compared to the unnotched strength. A s w i t h the W E K m o d e l , values f o r do and ao must be determined b y c o n d u c t i n g experiments and f i t t i n g the analytical curve to experimental data. dimensions  vary  I n so d o i n g they achieve g o o d results b u t find that the characteristic slightly  with  hole  size.  Subsequent  investigations  o f the  model,  e . g . ( A w e r b u c h and M a d h u k a r , 1985), ( K i m , K i m and Takeda, 1995), ( T a n , 1988), also f o u n d that do and ao depend o n material system and laminate geometry. Separate research b y K a r l a k and b y Pipes, W e t h e r h o l d and Gillespie  ( P W G ) (Pipes,  W e t h e r h o l d a n d Gillespie, 1979) f o u n d that do and ao depend o n the material system and also o n the n o t c h or hole size i n some cases. K a r l a k proposed a square-root relationship between hole radius and do using a n e w parameter kg.  T h e determination o f ko must be done  e x p e r i m e n t a l l y f o r each n e w material system that is m o d e l e d . T h i s adjustment i m p r o v e d the results i n most cases. Investigations b y P W G l e d to a m o r e general exponential relationship between R and do. 16  Chapter  2 Background  and Literature  Review  T a n ( T a n , 1988) developed the p o i n t stress and average stress models o f W h i t n e y and N u i s m e r further u s i n g classical laminated plate theory.  T h e models are constructed o n the  basis that fracture i n a composite laminate is fibre controlled. A n effective fibre strength i n the r e g i o n o f a n o t c h t i p c o n t a i n i n g heavy m a t r i x damage is calculated a n d failure is predicted t o occur w h e n the stress i n a n y p l y meets this effective strength.  T h e models  require the n o r m a l stress d i s t r i b u t i o n , l o n g i t u d i n a l and compressive strength parameters o f a l a m i n a a n d , again, a characteristic length that must be determined from an experimental notched specimen. A l l o f the models based o n the p o i n t stress and average stress models generally produce g o o d results, w h i c h is n o t unexpected as the models are s e m i - e m p i r i c a l and require experimental testing  t o determine  the characteristic  dimensions.  Additionally,  the characteristic  dimensions do n o t represent physical quantities and are therefore d i f f i c u l t t o j u s t i f y .  2.4.2  Cohesive models  M u c h o f the recent w o r k o n notched strength p r e d i c t i o n has incorporated the use o f cohesive models.  Cohesive models are generally m o r e p h y s i c a l l y based, t a k i n g into account the  development o f a damaged zone (or process zone) ahead o f a n o t c h tip. I n 1986, B a c k l u n d and A r o n s s o n ( B a c k l u n d and A r o n s s o n , 1986) presented a two-parameter damage zone m o d e l ( D Z M ) f o r notched composite laminates based on the cohesive zone m o d e l o f H i l l e r b o r g . I n a d d i t i o n t o d e t e r m i n i n g crack i n i t i a t i o n , the length o f damage zone extension is also obtained. The m o d e l is i m p l e m e n t e d as a material m o d e l i n a f i n i t e element code.  T h e constitutive curve is considered t o be r i g i d u n t i l crack i n i t i a t i o n so that the o n l y  parameters needed t o define the linear stress-displacement curve are the unnotched tensile strength and an apparent fracture energy displacement  curve.  Centre-notched  G *, c  w h i c h is the area beneath the stress-  quasi-isotropic  carbon/epoxy  laminate  specimens  c o n t a i n i n g circular and n o n - c i r c u l a r holes were tested e x p e r i m e n t a l l y b y B a c k l u n d and A r o n s s o n and predictions were made u s i n g the D Z M m o d e l w i t h 0.25 m m elements and the W E K model.  I t should be noted that G * c  predictions t o the average experimental determined independently. w i t h an average d e v i a t i o n  ( = 3 5 k J / m ) was chosen t o m a t c h n u m e r i c a l  fracture  load f o r five notched specimens and n o t  T h e D Z M p r o d u c e d s l i g h t l y better results than the W E K m o d e l from  the experimental peak loads o f about 3 . 3 % f o r t w e l v e  17  Chapter  2 Background  and Literature  Review  specimens o f t w o different n o t c h sizes.  A l t h o u g h it is n o t k n o w n h o w close the apparent  fracture energy is to the actual strain energy release rate o f the material, B a c k l u n d and A r o n s s o n noted that a v a r i a t i o n i n G * c  o f 1 0 % resulted i n a v a r i a t i o n i n the predicted peak  load o f about 3%. Interestingly, they also note that w h e n m o d e l i n g square holes that produce a singularity i n the stress field, values o f damage i n i t i a t i o n stress approach zero as element size is reduced but that the predicted peak load converges f o r element lengths o n the order o f 0.25 to 0.5 m m . R e c e n t l y Shin and W a n g ( S h i n and W a n g , 2004) have also used a cohesive zone m o d e l to predict notched failure i n open-hole tension tests. I m p r o v i n g u p o n the m e t h o d o f B a c k l u n d and A r o n s s o n they use a three-parameter constitutive curve w i t h the effective m o d u l u s f o r anisotropic materials as d e f i n e d b y Sih et al. ( S i h , Paris and I r w i n , 1965) i n a d d i t i o n to the c r i t i c a l fracture energy G  c  and material tensile strength cr . c  T h e y also p r o v i d e a g o o d  c o m p a r i s o n o f notched strength predictions from the cohesive zone m o d e l ( C Z M ) to p o i n t stress and average stress m o d e l predictions. The C Z M predicted failure loads s l i g h t l y closer to experimental results. The authors also attempt to measure the w i d t h o f the damage zone ahead o f the n o t c h t i p i n experimental specimens and correlate it to the length o f damaged material predicted b y the C Z M before failure.  T h e specimens w e r e center-notched tensile  specimens (210 m m x 25.4 m m ) w i t h f o u r hole diameters r a n g i n g from 3.0 to 12.0 m m . The material systems used were b o t h quasi-isotropic and cross-ply thermoplastic A S 4 / P E E K and thermosetting T 3 0 0 / 3 5 0 1 laminates.  U n f o r t u n a t e l y , the specimen g e o m e t r y they used is  unstable and they were unable to g r o w a f u l l y developed damage zone.  T h e y therefore  stopped the tests at 95 % o f the estimated failure strength and used radiographic  and  sectioning techniques t o determine damage zone w i d t h s i n f r o n t o f a 3 m m diameter n o t c h . F o r the quasi-isotropic laminates the w i d t h o f the damage zone was a p p r o x i m a t e l y 2 m m and f o r the cross-ply laminates it was a p p r o x i m a t e l y 0.55 m m .  F o r a constant stress l e v e l , the  w i d t h o f the damage zone is predicted to increase w i t h increasing hole diameter. Since most notched strength studies are conducted w i t h center-notched or edge-notched specimens c o n t a i n i n g circular cut-outs, the net section area o f a specimen changes w i t h c h a n g i n g hole radius.  T h e failure strength o f the specimen is often m o r e sensitive to a  r e d u c t i o n i n the net section than to a change i n the n o t c h radius.  T o isolate the effect o f  18  Chapter  2 Background  and Literature  Review  n o t c h radius on tensile strength o n l y a f e w researchers have m a i n t a i n e d a constant crack length a and constant a/W w h i l e changing the n o t c h tip radius p (e.g. ( H i t c h e n et al., 1994), ( H y a k u t a k e and H a g i o , 1990), ( K a m i y a and Sekine, 1997)). H i t c h e n et al. ( H i t c h e n et al., 1994) used centre-notched tension specimens (160 m m x 25 m m ) but kept a (and therefore a/W) constant.  N o t c h t i p r a d i i were varied b y d r i l l i n g holes  w i t h p o f 0.5 m m , 1.0 m m , 2.0 m m and 5.0 m m at each end o f a central slit so that 2a was m a i n t a i n e d at 10 m m .  H y a k u t a k e and H a g i o ( H y a k u t a k e and H a g i o , 1990) conducted  experiments w i t h double edge-notched bars where the n o t c h r a d i i were varied f r o m 0.08 m m t o 10 m m w h i l e m a i n t a i n i n g a constant a o f either 1 m m or 5 m m .  K a m i y a and Sekine  ( K a m i y a and Sekine, 1997) used a compact tension specimen g e o m e t r y w i t h values f o r p o f 0.25 m m , 2.0 m m , 5.0 m m and 10 m m . Generally speaking, a trend o f increased failure load w i t h increased n o t c h radius was observed.  Results presented b y K a m i y a and Sekine  c o n t r a d i c t i n g this trend are, as p o i n t e d out b y the authors, most p r o b a b l y due to holes that w e r e too large compared to the specimen size.  I n all cases the c o m b i n a t i o n o f specimen  g e o m e t r y and material system was unstable so that o n l y fracture loads c o u l d be determined and n o t the stable damage zone size. H y a k u t a k e and H a g i o ( H y a k u t a k e and H a g i o , 1990) introduced a concept they call " l i n e a r n o t c h mechanics".  T h e y propose that the strength o f a notched specimen is related to the  stress present at the n o t c h r o o t w h e n the failure load is reached. o'max.cip) t o be a f u n c t i o n o n l y o f the n o t c h root radius.  T h e y consider this stress  T h e n the n o m i n a l stress i n the  specimen at failure is  w h e r e Kj is the stress concentration factor.  The stress distributions i n the specimens at the  failure load were determined u s i n g a f i n i t e element analysis and several  experimental  notched tests are required to develop a characteristic curve f o r a material system.  This  m e t h o d o f notched strength p r e d i c t i o n seems questionable as it says the failure c r i t e r i o n f o r r e l a t i v e l y sharp notches is based solely on stress at the n o t c h t i p w i t h no consideration g i v e n t o the energy necessary f o r fracture.  19  Chapter  2.4.3  2 Background  and Literature  Review  Relation of the damage zone size to strength  Other models that attempt t o predict the strength o f notched specimens are based o n a c r i t i c a l height o f the damage zone. The critical height refers t o the height o f the damage zone w h e n the failure load is reached. T y p i c a l l y a shear-lag o r statistical analysis is used t o determine w h a t the damage zone height w i l l be a n d then a relationship i n v o l v i n g the damage zone height and the failure load is defined. Dharani  et al. ( D h a r a n i ,  Jones  a n d Goree,  1983) developed  strength  u n i d i r e c t i o n a l laminates c o n t a i n i n g slits or rectangular o r circular notches.  solutions f o r A shear-lag  analysis is used t o determine the remote stress w h e n the stress i n a fibre at the n o t c h t i p reaches the unnotched strength o f a fibre. K a m i y a a n d Sekine ( K a m i y a a n d Sekine, 1997) also used a m i c r o - m e c h a n i c a l analysis t o determine the stresses i n transverse fibres at a n o t c h t i p . B y m o d e l i n g a single fibre as a p u l l out p r o b l e m , they calculate the length o f fibre that w i l l p u l l o u t f r o m the s u r r o u n d i n g m a t r i x and then determine the p r o b a b i l i t y o f breakage o f a fibre o f that length. T h e upper b o u n d o f the length o f delaminated area perpendicular t o the crack is analogous t o the damage zone height. One o f the simplest p h y s i c a l l y based models f o r strength p r e d i c t i o n i n notched laminates is that o f K o r t s c h o t and B e a u m o n t ( K o r t s c h o t and B e a u m o n t , 1 9 9 1 ; K o r t s c h o t , B e a u m o n t a n d A s h b y , 1 9 9 1 ; K o r t s c h o t a n d B e a u m o n t , 1990a; K o r t s c h o t a n d B e a u m o n t , 1990b).  They  investigated the size and shape o f the sub-critical damage zone i n double-edge notched ( D E N ) tensile specimens and use fracture mechanics equations t o predict strength based o n the m a x i m u m stress experienced b y the 0 ° plies at the n o t c h t i p . U s i n g a radiographic technique, these authors were able t o measure the size and shape o f the damage zone d u r i n g l o a d i n g , p r i o r t o the onset o f fast fracture.  F o r the cross-ply graphite/epoxy specimens  studied, the damage zone consisted o f s p l i t t i n g i n the 0 ° plies, transverse cracks i n the 9 0 ° plies and a triangular d e l a m i n a t i o n area ahead o f the n o t c h t i p as s h o w n i n F i g u r e 2.7.  They  f o u n d that increasing damage zone height (i.e. split length) led t o higher specimen strength. D a m a g e height decreased w i t h increasing n o t c h w i d t h and therefore strength also decreased w i t h increasing n o t c h w i d t h . U s i n g a static, plane-stress f i n i t e element m o d e l i n c o r p o r a t i n g the orthotropic and layered properties o f the laminate to determine the stress concentration,  20  Chapter  2 Background  and Literature  Review  they determined a s e m i - e m p i r i c a l relationship f o r KT i n terms o f the damage zone height a n d n o t c h w i d t h . N o t c h e d specimen strength c o u l d then be related t o the tensile strength o f a 0 ° ply.  A f t e r n o t i c i n g a size effect i n specimens o f different w i d t h , K o r t s c h o t a n d B e a u m o n t  i n c l u d e d a W e i b u l l d i s t r i b u t i o n t e r m i n the m o d e l t o account f o r it. T h e y investigated the effect o f l a y - u p a n d n o t c h w i d t h o n the notched strength b u t d i d n o t consider n o t c h root radius. T h e i r unstable specimen g e o m e t r y prevented the development o f a stable damage zone.  2.5  A specimen for stable damage growth  A specimen g e o m e t r y f o r stable crack g r o w t h was developed b y K o n g s h a v n and Poursartip ( K o n g s h a v n a n d Poursartip, 1999). T h e y f o u n d that b y increasing the height o f a compact tension ( C T ) specimen they were able t o g r o w damage stably i n the specimen. T h e increased size o f the O v e r - h e i g h t C o m p a c t T e n s i o n ( O C T ) specimen ensured that the boundaries d i d not interfere w i t h the r e l a t i v e l y large damage zone i n the quasi-isotropic F R P laminate. The stability o f the system can be described i n terms o f fracture mechanics. T h e e q u i l i b r i u m state o f crack propagation w a s g i v e n b y the equality i n E q u a t i o n 2 . 1 .  Stability o f the  e q u i l i b r i u m state is determined b y the rate o f increase o f G w i t h respect t o a compared t o the rate o f increase o f R w i t h respect t o a. The three possibilities are: dG dR — <— da da  3G  95  — =— da da dG dR — >— da da  ,. stable +  . . . . . . . critical stability  2.18  unstable  I f i t is assumed that crack resistance is a material p r o p e r t y as i n L E F M then R = G is a c  constant and the stability c o n d i t i o n is  ^ < 0 . da  2.19  The change i n energy release rate w i t h crack g r o w t h can be obtained b y d i f f e r e n t i a t i n g E q u a t i o n 2.3 y i e l d i n g  21  Chapter  2 Background  and Literature  Review  8G P — = da IB 2  F o r C T and O C T specimens — da  8C -. da 2  2.20  2  is positive, although the increased height o f the O C T  specimen stiffens the arms, w h i c h decreases the change i n compliance w i t h crack g r o w t h . U n d e r displacement c o n t r o l the O C T can be stable i f the load drop w i t h a small increment o f crack g r o w t h is enough t o make G < R.  F o r t o u g h material systems and f o r sharp notches  under displacement c o n t r o l , load drops w i t h crack extension do b r i n g G b e l o w R, causing crack arrest.  F o r brittle material systems and other n o t c h geometries the specimen is no  longer stable as w i l l be discussed later.  K o n g s h a v n and Poursartip, and later M i t c h e l l  ( M i t c h e l l , 2 0 0 2 ) , used the O C T g e o m e t r y to study the size and shape o f the characteristic stable damage zone w i t h the intent o f d e v e l o p i n g a strain-softening damage m o d e l  for  composite laminates.  2.6  Summary •  I n fracture mechanics, the onset o f fracture i n a c o n t i n u u m can be predicted based o n an energy criterion (G > R) or a stress c r i t e r i o n (K/ > Ki ). c  F o r elastic materials, the  t w o criteria can be l i n k e d u s i n g E q u a t i o n 2.8. •  Cohesive crack models are often used to m o d e l g r o w t h o f fracture and damage i n composite materials.  These models are mesh sensitive.  D u e t o localisation and  issues w i t h energy dissipation, they do n o t give converged solutions w i t h increased mesh refinement. •  A l t h o u g h cohesive crack and cohesive zone models use simple, p h y s i c a l  input  parameters, interaction o f the input parameters w i t h mesh size is n o t w e l l understood and leads to parameters b e i n g adjusted to f i t experimental results. •  S i m p l e , s e m i - e m p i r i c a l models f o r p r e d i c t i n g n o t c h failure have also been proposed, n o t a b l y the p o i n t stress and average stress criterions, but these models also require the f i t t i n g o f the analytical curves to one or more sets o f experimental data.  •  All  experimental  investigations  ( K o n g s h a v n and Poursartip,  of  notched  strength,  with  the  exception  1999), that w e r e r e v i e w e d used unstable  of  specimen  22  Chapter  2 Background  and Literature  Review  geometries such as centre-notched tension, double edge-notched tension or compact tension tests. The researchers were therefore unable to make observations on crack growth or the stable height of the damage zone. Damage zone size at failure was predicted to depend on notch size and geometry.  23  Chapter  2 Background  and Literature  Figure  Review  2.3 Cohesive  crack  model  24  Chapter 2 Background and Literature Review T &max  1  do  8  C  5  Figure 2.4 Traction — displacement relationship for cohesive crack model  a)  b)  Figure 2.5 Cohesive crack models for a) ductile solids and b) quasi-brittle solids (from ( Borst, 2003))  25  Chapter 2 Background and Literature Review  (a)  (b)  (c)  (d)  Figure 2.6 Damage localisation using a cohesive zone model  26  Chapter 2 Background and Literature Review  Chapter  3  Experiments  CHAPTER 3  3.1  EXPERIMENTS  Introduction  T w o different specimen geometries were tested d u r i n g the course o f this study t o investigate the b e h a v i o u r and characteristics o f the damage zone i n a f i b r e - r e i n f o r c e d p o l y m e r composite and t o obtain material property inputs f o r the n u m e r i c a l m o d e l .  T h e t w o geometries tested  were a m o d i f i e d O v e r - h e i g h t C o m p a c t T e n s i o n ( O C T ) specimen and a f o u r - p o i n t b e n d specimen. T h e O C T specimens were m o d i f i e d t o include circular n o t c h tips o f v a r y i n g r a d i i . T h e p r i m a r y objectives i n testing the O C T specimens were t o : 1) O b t a i n a relationship between peak load and n o t c h t i p radius t o validate the n u m e r i c a l material m o d e l ' s a b i l i t y t o cope w i t h stress fields o f v a r y i n g magnitude. 2) Produce stable crack g r o w t h so that the energy release rate f o r the structure can be determined. 3)  Investigate the size and shape o f the process zone.  T h e f o u r - p o i n t bend specimens were tested p r i m a r i l y t o obtain the f l e x u r a l strength o f the material system.  3.2  Specimens and procedures  3.2.1  Material  The material used f o r b o t h the O C T tests and the f o u r - p o i n t b e n d tests w a s a quasi-isotropic carbon  fibre/  epoxy  laminate  designated  C o m p a n y , Seattle, W a s h i n g t o n , U S A .  BA-T-7  and m a n u f a c t u r e d  by The Boeing  I t is the same material as used b y M i t c h e l l ( M i t c h e l l ,  2 0 0 2 ) t o investigate the notched b e h a v i o u r o f stitched a n d unstitched R F I C F R P laminates. The laminates used i n this study are manufactured using a resin film i n f u s i o n ( R F I ) process and are u n r e i n f o r c e d (unstitched) t h r o u g h the thickness. T h e fibres are a m i x t u r e o f standard and intermediate m o d u l u s fibres and the e p o x y resin is 3501-6. [+45/-45/02/90/02/-45/+45J6.  T h e lay-up orientation is  T h e o r i g i n a l l y determined elastic properties f o r the material  system are g i v e n i n Table 3 . 1 .  28  Chapter 3 Experiments 3.2.2  Test equipment  A n M T S h y d r a u l i c u n i a x i a l testing machine w i t h an Instron controller ( M o d e l # 8500R.) was used f o r b o t h the O C T tests and the f o u r p o i n t b e n d tests. 6 6 1 . 2 1 A - 0 2 ) has a 50 k N capacity. used f o r each type o f test.  The load cell ( M T S M o d e l #  Separate testing j i g s , designed and b u i l t in-house, w e r e  M a c h i n e c o n t r o l and data acquisition were conducted w i t h a  dedicated PC u s i n g Instron W a v e m a k e r 7.0 software.  3.2.3  OCT tests  The O v e r - h e i g h t C o m p a c t T e n s i o n ( O C T ) specimen g e o m e t r y developed b y K o n g s h a v n and Poursartip ( K o n g s h a v n and Poursartip, 1999) was chosen to produce stable crack g r o w t h i n a specimen so that the composite damage zone c o u l d be investigated a n d t o a l l o w f o r v a l i d a t i o n o f the n u m e r i c a l m o d e l . K o n g s h a v n demonstrated that the O C T specimen w i t h a sharp n o t c h t i p is stable under displacement c o n t r o l a n d is large enough so that t h e boundaries d o n o t greatly affect the damage zone size o r shape.  I n this study the O C T  specimen is s l i g h t l y m o d i f i e d b y c u t t i n g circular notches w i t h v a r y i n g r a d i i at the n o t c h t i p . I t w i l l be demonstrated later that this has an effect o n the stability o f crack g r o w t h i n the specimen.  3.2.3.1  Specimens  The O C T specimen g e o m e t r y w i t h n o m i n a l measurements is s h o w n i n Figure 3 . 1 . T h e n o t c h root radius was varied f r o m a r e l a t i v e l y sharp radius o f 0.5 m m t o a v e r y large radius o f 27 mm.  Several o f the specimens are displayed i n F i g u r e 3.2 t o illustrate the difference i n the  size o f the notches.  The panel thickness B had a n o m i n a l value o f 8.5 m m b u t varied  between 7.9 m m and 9.0 m m due to inconsistencies i n the panel m a n u f a c t u r i n g . T h e O C T specimens were cut f r o m a large panel so that the material 0° d i r e c t i o n was parallel to t h e n o t c h as s h o w n i n F i g u r e 3 . 1 .  I t was i m p o r t a n t t o have p e r f e c t l y circular a n d  undamaged n o t c h tips. T h e y w e r e c u t at l o w speed w i t h d i a m o n d sintered saw blades t o reduce damage and no sanding o r f i n i s h i n g was required. A complete list o f the O C T tests conducted is g i v e n i n Table 3.2 along w i t h the i m p o r t a n t specimen dimensions. n o t c h t i p cut-out.  The specimens are n u m b e r e d corresponding t o the diameter o f the  Generally t w o tests w e r e conducted at each n o t c h root radius.  A l l tests  w e r e done under displacement c o n t r o l . T h e first test at each n o t c h root radius was displaced  29  Chapter  3  Experiments  at a constant rate t o an u l t i m a t e displacement and then unloaded. The second test o f each set i n c l u d e d unloads  at t w o  or three locations  a l o n g the softening p o r t i o n  o f the  load-  displacement curve. The tests that i n c l u d e d periodic unloads are noted i n Table 3.4.  3.2.3.2  OCT test set-up and procedure  The O C T test set-up is s h o w n i n Figure 3.4. T h e specimen is loaded i n tension t h r o u g h pins located above and b e l o w the notch.  T o prevent t w i s t i n g o f the specimen (as n o t e d b y  M i t c h e l l ( M i t c h e l l , 2 0 0 2 ) ) , a s t i f f e n i n g support was attached to the back edge o f the specimen. I t has little effect on the response o f the specimen and l o a d i n g was stopped before the crack g r e w into the r e g i o n covered b y the stiffener. F o r all tests an extensometer ( I n s t r o n 2620-825 w i t h ±5 m m travel) was f i x e d between the upper and l o w e r edges o f the n o t c h m o u t h as s h o w n i n Figure 3.4 to measure crack m o u t h opening displacement ( C M O D ) . S m a l l grooves were cut i n the specimen to prevent s l i p p i n g o f the knife-edges.  I t was realised after some i n i t i a l tests that c o m p l i a n c e i n the l o a d i n g j i g  created a difference i n the displacement o f the machine and the displacement at the l o a d i n g pins as can be seen i n Figure 3.5. procedure f o r d e t e r m i n i n g G . c  T h e precise displacement o f the pins is necessary i n the Therefore, f o r the remainder o f the tests, the relative  displacement o f the pins was also measured u s i n g an extensometer attached between grooves o n the l o a d i n g pins. The displacement rate d u r i n g load and u n l o a d f o r all tests was 0.508 m m / m i n (0.02 i n / m i n ) . The machine cross-head was displaced u n t i l a p p r o x i m a t e l y 4.5 m m o f C M O D and then displacement was reversed to the p o i n t o f zero load.  Since there was some residual  d e f o r m a t i o n , the p o i n t o f zero load d i d not coincide w i t h zero displacement. F o r certain tests (noted i n Table 3.2) partial unloads were conducted at intervals a l o n g the softening p o r t i o n o f the l o a d - C M O D curve. I n these cases the specimens were unloaded to 5 0 % o f the load at the i n i t i a t i o n o f the u n l o a d cycle. F o r all tests the values recorded d u r i n g testing w e r e load at the pins, machine p o s i t i o n and CMOD. frequency  F o r some tests the relative p i n displacement was also recorded.  T h e data output  was 1 0 s " . 1  Photographs o f the specimen were taken d u r i n g each test u s i n g a N i k o n d i g i t a l S L R D 1 0 0 camera w i t h a 1 0 5 m m macro N i k o r lens m o u n t e d on a stationary t r i p o d i n front o f the test  30  Chapter  3 Experiments  set-up. The photographs were taken i n large (6.1 m e g a - p i x e l ) .jpg format. Photographs were taken at regular intervals d u r i n g i n i t i a l l o a d i n g before damage, frequently near the peak load and f o l l o w i n g each incremental load drop d u r i n g crack propagation.  F o r each p h o t o g r a p h  the C M O D was noted and, i n a d d i t i o n , the internal c l o c k o n the camera was synchronised w i t h the data acquisition t i m i n g so that each p h o t o g r a p h c o u l d be placed on the l o a d - C M O D curve d u r i n g post-processing. F o r each specimen a p p r o x i m a t e l y 25 t o 40 photographs w e r e taken depending on the d u r a t i o n o f l o a d i n g .  3.2.3.3  Post-processing  3.2.3.3.1  Line analysis  The line analysis technique used b y K o n g s h a v n and Poursartip ( K o n g s h a v n and Poursartip, 1999) and M i t c h e l l ( M i t c h e l l , 2002) t o obtain the crack o p e n i n g displacement ( C O D ) p r o f i l e is also used i n this investigation.  B e f o r e testing, each specimen was prepared b y s c r i b i n g  h o r i z o n t a l lines 2.5 m m apart, parallel t o the n o t c h i n the region ahead o f the n o t c h t i p as s h o w n i n F i g u r e 3.3.  The lines are used t o p r o v i d e points o f reference f o r measuring the  crack opening p r o f i l e f r o m photographs taken d u r i n g the test. A p h o t o g r a p h o f the specimen at a g i v e n C M O D d u r i n g the test is compared t o a p h o t o g r a p h o f the specimen at the start o f the test. several  The relative displacements o f points along the scribed lines are measured. photographs  throughout  the test,  C O D profiles  at different  stages  Using  o f crack  propagation are obtained. Software developed in-house was used t o make the measurements. Results from the line analysis are presented i n Section 3.3.1.3.  3.2.3.3.2  Sectioning  Cross-sections were cut perpendicular t o the crack plane i n front o f the i n i t i a l n o t c h t i p f o r certain specimens t o produce a p r o f i l e o f the damage zone a l o n g the length o f the crack.  A  slow-speed d i a m o n d sintered c u t t i n g w h e e l w i t h a 0.32 m m (0.013 i n ) t h i c k blade was used to produce a v e r y s m o o t h surface that d i d n o t require p o l i s h i n g .  Sections were cut every  0.635 m m t o 1.905 m m and the r e m a i n i n g surface was photographed w i t h the d i g i t a l camera and a 105 m m macro lens u s i n g large . t i f f f o r m a t f o r v e r y h i g h resolution. A 6.35 m m (0.25 i n ) marker was placed i n each p h o t o g r a p h so that the images c o u l d be calibrated and the height o f the damaged area c o u l d be measured u s i n g image analysis software. T h e damage height h was measured from the t i p o f v i s i b l e damage o n one side o f the crack plane t o the c  t i p o f v i s i b l e damage o n the other side. D a m a g e i n the outer plies was n o t i n c l u d e d i n the  31  Chapter  3  Experiments  measurement o f damage height because i t c o u l d be either extensive o r not depending on the amount o f resin o n the surface.  F o r example, considerably m o r e surface damage w a s  apparent o n the top surface o f the laminate ( a w a y f r o m the resin f i l m i n the R F I process) than on the b o t t o m surface. F o r consistency, i t was decided n o t to include any damage i n the plies outside o f the 90° p l y i n the outer stacks as s h o w n in F i g u r e 3.6. Results f r o m sectioning are presented i n Section 3.3.1.2.  3.2.4  Four-point bend tests  F o u r p o i n t bend tests were conducted i n a d d i t i o n t o the O C T tests t o obtain a value o f material strength f o r input into the n u m e r i c a l material m o d e l .  The A S T M Standard Test  M e t h o d f o r F l e x u r a l Properties o f U n r e i n f o r c e d and R e i n f o r c e d Plastics a n d Electrical I n s u l a t i n g Materials b y F o u r - P o i n t B e n d i n g ( D 6272 - 00) was used as a guideline.  3.2.4.1 Specimens T h e bend specimens were cut from the same panel o f material used i n the O C T tests. S i x specimens o f three different depths were c u t i n each o f the 0 ° and 90° directions o f the material.  Testing details f o r each specimen are g i v e n i n Table 3.3.  The tests w e r e a l l  conducted i n the plane o f the laminate.  3.2.4.2  Test set-up and procedure  A schematic o f the test set-up is s h o w n i n Figure 3.7. A test j i g was designed and b u i l t i n house f o l l o w i n g the guidelines o u t l i n e d i n A S T M D 6272 - 00.  The t w o l o w e r noses are  r i g i d l y attached to the crosshead w h i l e the t w o upper noses are r i g i d l y connected b u t a l l o w e d to s w i v e l about the connection to the crosshead. T h i s set-up a l l o w s the j i g t o self-align as i t loads. The diameter o f the l o a d i n g noses is 5 m m , w h i c h is smaller than the r e c o m m e n d e d diameter g i v e n i n the standard.  F o r the 90° tests, a l l o f the specimens f a i l e d i n the r e g i o n  between the l o a d i n g noses i n d i c a t i n g tensile failure. V i s u a l inspection o f the l o a d i n g points revealed no damage due t o compression under the pins so the diameter o f the loading noses was not a factor i n the failure.  F o r a l l o f the 0 ° specimens h o w e v e r , compressive failure  occurred first under one o f the inner loading noses. Here the nose diameter was p r o b a b l y t o o small and v a l i d results w e r e not obtained. A n extensometer (Instron 2620-825 w i t h +5 m m travel) was fixed between a p o i n t r i g i d l y attached t o the base o f the test j i g and a p o i n t r i g i d l y attached t o the axis o f the s w i v e l j o i n t  32  Chapter 3 Experiments o n the t o p section o f the test j i g t o measure displacement.  T h e l o w e r crosshead was m o v e d  up t o load the specimen. The displacement rate depended o n the specimen dimensions a n d w a s determined u s i n g guidelines f r o m A S T M D 6 2 7 2 - 00. See Table 3.3. Photographs w e r e also taken at regular intervals d u r i n g each test.  H o w e v e r , failure w a s  always catastrophic and little useful i n f o r m a t i o n was obtained f r o m the photographs.  3.2.4.3  Post-processing  L o a d and displacement data were output at a frequency o f 10 s" and i m p o r t e d t o M i c r o s o f t 1  E x c e l t o create plots. T h e stress i n the outer fibres o f the beam at the p o i n t o f critical load was calculated u s i n g t h e linear, s m a l l - d e f l e c t i o n equation f o r stress g i v e n b y t h e A S T M standard ( A S T M , 2 0 0 0 ) as  ^ cnt a  =  Peril Ls V • BD  ~ ! 3.1  2  Results f o r the f o u r - p o i n t bend tests are presented i n Section 3.3.2.  3.3  Results and data reduction  3.3.1  OCT tests  Table 3.4 summarizes the analysis that was conducted o n each specimen. I n each specimen, a damage zone propagated from the n o t c h t i p r o u g h l y parallel t o the i n i t i a l n o t c h .  For all  tests, the crack g r e w parallel t o the material 0 ° d i r e c t i o n and there was n e g l i g i b l e damage at the l o a d i n g pins.  Surface damage at the e n d o f a test is s h o w n f o r a t y p i c a l specimen i n  F i g u r e 3.8.  3.3.1.1  Load - Displacement behaviour  F r o m the results o f notched tensile testing r e v i e w e d i n the literature, i t was expected that the peak load attained b y each specimen before failure w o u l d increase w i t h n o t c h root radius. H o w e v e r , v e r y little n o t c h sensitivity was observed f o r specimens c o n t a i n i n g n o t c h root r a d i i r a n g i n g from 0.5 m m a l l the w a y t o 12.7 m m . Plots o f load versus C M O D are presented i n A p p e n d i x A separately f o r each different n o t c h radius tested.  T h e results f r o m s i x  representative specimens are also p l o t t e d together i n F i g u r e 3.9 so that i m p o r t a n t features o f the responses can be compared. I n a l l o f the experimental load versus displacement plots i n  33  Chapter  3  Experiments  this thesis solid lines are used to s h o w continuous portions o f the curve and dashed segments are used to denote segments i n w h i c h the crack has j u m p e d or load has dropped sharply between s a m p l i n g points. I n all cases the i n i t i a l behaviour is linear elastic. laminate are reversible and no damage has occurred.  I n this phase, all o f the strains i n the A n increase i n the c o m p l i a n c e o f the  specimens is noticed as the radius o f the n o t c h root is increased. This is an effect o f u s i n g r e l a t i v e l y small specimens w i t h large n o t c h root r a d i i . A t some p o i n t before the peak load is reached the b e h a v i o u r becomes s l i g h t l y non-linear.  In  some cases, u s u a l l y the smaller n o t c h root radius specimens, (e.g. r d 2 - 2 , r d 8 - l and r d l O - 1 ) slight shifts a l o n g the C M O D axis are n o t i c e d i n d i c a t i n g v e r y small amounts o f damage are b e g i n n i n g to occur.  I n the larger n o t c h root radius specimens (e.g. r d 3 2 - 2 and r d 5 4 - 2 ) the  slight shifts are not noticeable but the curve softens s l i g h t l y f r o m a straight line.  The n o n -  linearities begin to be noticeable at around 11 k N o f load f o r all specimens. The most distinct feature d i f f e r e n t i a t i n g the responses o f the small and large n o t c h root radius cases is the b e h a v i o u r around peak load. I n F i g u r e 3.9, specimens r d 2 - 2 , r d 8 - l , r d l O - 1 and r d 2 5 - l  are all p l o t t e d u s i n g t h i n lines to distinguish their behaviour  specimens r d 3 2 - 2 and r d 5 4 - 2 .  from  that  of  The onset o f damage i n the smaller n o t c h radius specimens  occurs at r o u g h l y the peak load but does not result i n an i m m e d i a t e load drop.  Instead the  peak load is m a i n t a i n e d f o r a short t e r m w h i l e C M O D increases, resulting i n a plateau.  For  the t w o larger n o t c h root radius specimens, p l o t t e d here using t h i c k grey lines, the b e h a v i o u r is different. There is no r e g i o n o f stable damage g r o w t h . Instead the peak load occurs r i g h t at the onset o f damage f o l l o w e d i m m e d i a t e l y b y a large drop i n load. The smaller n o t c h root radius specimens appear to e x h i b i t stable fracture w h i l e the larger ones t e m p o r a r i l y e x h i b i t unstable  fracture.  The large load drop i n the large n o t c h radius  specimens is due t o a j u m p i n crack length as w i l l be s h o w n Chapter 4. T h e O C T specimens are large enough to contain the j u m p i n crack length. T h e sharp load drop i n the large n o t c h  In the literature, the maximum load attained by a specimen is usually called the failure load because the specimens tested are unstable and have no residual strength after the failure load is reached. In the tests done here, the geometry is stable and the specimens are able to sustain load after the peak on the load versus displacement plot. The "failure load" is therefore referred to as the "peak load" in this thesis. 3  34  Chapter  3  Experiments  root radius specimens reduces the stored energy i n the specimen to less than the fracture energy and r a p i d crack g r o w t h is arrested. A s seen i n F i g u r e 3.9, the effect o f the n o t c h root radius becomes n e g l i g i b l e as damage progresses. U p w a r d s o f 3.5 m m o f C M O D , the load versus C M O D curves f o r all specimens are a p p r o x i m a t e l y the same. I t is remarkable that as n o t c h root radius is increased, the transition between specimens e x h i b i t i n g stable damage g r o w t h at peak load and those e x h i b i t i n g unstable damage g r o w t h is not gradual, b u t rather appears to be d i s t i n c t l y defined. A l l o f the specimens w i t h n o t c h root r a d i i o f 14 m m or less always displayed a sizeable plateau at about the peak load, w h i l e all o f the specimens w i t h n o t c h root radii o f 16 m m or greater had load versus C M O D curves that dropped sharply at the onset o f damage.  I t appears that at around 16 m m there is a  transition i n behaviour. T h i s transition i n peak load at about a n o t c h root radius o f 16 m m is m o r e clearly d e f i n e d i n Figure 3.10 o f peak loads p l o t t e d against the n o t c h root radius p. The peak loads are not n o r m a l i s e d w i t h the thickness o f the specimen. I t is assumed that the tensile fracture load o f a continuous fibre r e i n f o r c e d composite laminate is p r e d o m i n a n t l y dependent on the b r e a k i n g o f fibres and not m a t r i x cracking.  A l l o f the specimens used i n  this study have the same n u m b e r and orientation o f plies and any differences i n thickness are due to the amount o f resin and the  fibre-volume  fraction. A n alternative is to normalise the  peak loads w i t h respect to the net section area o f each specimen. The stress at the n o t c h t i p o f a C T or O C T specimen is due to a c o m b i n a t i o n o f tension and b e n d i n g (Tada, Paris and I r w i n , 1985).  The failure strength o f a specimen is therefore dependent o n the net section  area b e i n g loaded.  The specimens were cut to a n o m i n a l w i d t h W o f 81 m m but actually  v a r i e d f r o m 80.3 m m to 82.5 m m .  The n o m i n a l stress i n the specimen is expressed as  follows  a N = a N Tension  N Bending  f P W-a 2P(2W [W -  6P\ a +  +  m W-a J  3.2  [W-af  + a) of  35  Chapter 3 Experiments W h i l e peak loads are d i r e c t l y related t o the dimensions a and W, n o r m a l i s a t i o n w i t h respect to (2W+a)l(W-a) d i d n o t change any o f the trends i n the peak load versus C M O D p l o t and i t 2  was decided t o leave the p l o t u n n o r m a l i s e d f o r clarity. A s s h o w n i n Figure 3.10, peak load is r o u g h l y constant at about 15.5 k N between n o t c h r a d i i o f 3.0 m m and 12.7 m m . B e t w e e n p equal t o 14.15 m m and 16 m m a transition occurs after w h i c h the peak loads range between about 16.5 and 18.5 k N .  3.3.1.1.1  Variability  Greater v a r i a b i l i t y is seen i n the peak loads o f the large n o t c h root radius tests as w o u l d be expected. The plateau e x h i b i t e d b y specimens w i t h smaller p indicates the development o f a damage zone ahead o f the n o t c h t i p before a discrete crack is f o r m e d .  T h i s damage zone  development absorbs the effects o f any small f l a w s that m a y affect the failure load o f m o r e " b r i t t l e " materials and geometries. F o r large p, s l o w development o f the damage zone does not occur and f l a w s are m o r e l i k e l y t o affect the peak load attained b y the specimen. A m o r e detailed discussion o f w h a t causes the differences i n b e h a v i o u r between specimens c o n t a i n i n g notches o f small root radius and those w i t h notches o f large root radius is g i v e n i n Chapter 5 after a l l o f the experimental and n u m e r i c a l results are presented.  3.3.1.2 The size and shape of the damage zone The damaged regions ahead o f the i n i t i a l n o t c h t i p o f three specimens were sectioned t o obtain a clearer picture o f the processes t a k i n g place at the n o t c h t i p d u r i n g  fracture.  A  r e l a t i v e l y sharp notched specimen, r d 2 - l , and the bluntest n o t c h t i p tested, r d 5 4 - 2 , w e r e chosen a l o n g w i t h a m i d - s i z e d n o t c h radius specimen, r d 8 - 2 .  Results o f damage height  versus distance from the i n i t i a l n o t c h t i p a l o n g the crack plane are presented i n F i g u r e 3 . 1 1 . I t is evident that the damage height at the n o t c h t i p is m u c h larger i n specimen r d 5 4 - 2 w h e n the crack initiates than i n specimen r d 2 - l . Photographs o f the sectioned material j u s t ahead o f the i n i t i a l n o t c h tips are s h o w n i n F i g u r e 3.12 and Figure 3.13. F o r the large p specimen, the damage is spread out over almost 2 0 m m and the actual crack location is n o t w e l l defined. F o r the small p specimen, damage is concentrated i n a v e r y small r e g i o n o n l y 2 t o 3 m m i n height. I n this case, a l l o f the fibre breakage occurs along a w e l l - d e f i n e d line t h r o u g h the thickness o f the sample.  36  Chapter  3  Experiments  The extent o f the damaged area i n specimen r d 5 4 - 2 q u i c k l y decreases u n t i l a stable damage height o f about 7.0 m m is reached 10 m m from the i n i t i a l n o t c h tip. I n specimen r d 2 - l the opposite trend is observed; the damage height increases from about 2.5 m m to about 6.0 m m b y the t i m e the crack has g r o w n about 10 m m from the i n i t i a l n o t c h t i p . A l t h o u g h specimen rd8-2 was not sectioned all the w a y to the back end o f the specimen, the damage zone was seen to commence w i t h a height o f a p p r o x i m a t e l y 5.0 m m and progress to around 6.0 m m at 10 m m from the i n i t i a l n o t c h tip. These trends indicate that w h e n fracture initiates from a notch o f r e l a t i v e l y small root radius, the damage zone g r o w s to its stable characteristic height as the damage zone progresses. Therefore, f o r small n o t c h root r a d i i , such as i n r d 2 - l , the stress concentration is reduced b y the f o r m a t i o n o f the damage zone.  A r e d u c t i o n i n stress concentration t o b e l o w its c r i t i c a l  value causes the crack to arrest. Conversely, f o r a specimen w i t h a r e l a t i v e l y large n o t c h root radius, the n o t c h e f f e c t i v e l y " s h a r p e n s " to the stable characteristic height o f the damage zone f o r the material system. The sharpening behaviour o f specimen r d 5 4 - 2 helps to e x p l a i n w h y the crack j u m p s unstably w h e n the peak load is reached.  A s the damage zone f o r m s at the n o t c h t i p , the stress  concentration g r o w s because the n o t c h root radius is e f f e c t i v e l y b e c o m i n g smaller. The small difference i n the stable characteristic height displayed b y specimens r d 2 - l and r d 5 4 - 2 is w i t h i n the experimental error o f the m e t h o d used f o r measuring the damage zone height.  W h i l e the characteristic height determined is o n l y an approximate value, the  i m p o r t a n t aspect o f this data is the sharpening versus b l u n t i n g behaviour that is observed f o r the large and small n o t c h r o o t radius specimens respectively. I n b o t h the small and large radius specimens the damage zone terminates a p p r o x i m a t e l y 36 m m from the i n i t i a l n o t c h t i p . T h i s corresponds to a C M O D o f 4.4 m m f o r r d 2 - l and 4.75 m m for rd54-2.  T h e difference is explained b y the difference i n compliance o f the t w o  specimens. Since a large amount o f material is cut out o f r d 5 4 - 2 it is m o r e c o m p l i a n t and has a larger C M O D . A s the damage zone tapers out to v i r g i n material, the height o f the damage zone decreases to zero. The distance over w h i c h this decrease takes place gives an estimate o f the w i d t h o f the damage zone. The height o f the damage zone begins to decrease at around 29 t o 30 m m f o r  37  Chapter  3  Experiments  b o t h r d 2 - l and r d 5 4 - 2 . mm.  The w i d t h o f the damage zone is therefore estimated to be about 6  The decrease i n the height o f the damage zone between 30 m m and 36 m m s h o w n i n  F i g u r e 3.11 comes f r o m the visual height o f damage i n the cross sections. I t becomes m o r e d i f f i c u l t to measure the damage height as damage becomes lighter. Therefore the tapering o f the damage height approaching v i r g i n material m a y n o t be the actual shape o f the damage zone. I t is m o r e l i k e l y that the height o f the damage zone remains closer to the characteristic damage height u n t i l c o m p l e t e l y undamaged material is reached.  Results f r o m d e p l y i n g b y  K o n g s h a v n and Poursartip ( K o n g s h a v n and Poursartip, 1999) and M i t c h e l l ( M i t c h e l l , 2 0 0 2 ) suggest that this is the case. T o determine the damage zone w i d t h m o r e r e l i a b l y , residual strength testing o f the sectioned material should be done.  N e g l i g i b l e strength w o u l d indicate a f u l l y separated crack w h i l e  increasing residual strength w o u l d indicate decreasing amounts o f damage u n t i l the strength o f the undamaged material is reached at the end o f the damage zone.  Q u a l i t a t i v e l y , it was  noted that the t h i n sections (less than 1 m m t h i c k ) became m o r e d i f f i c u l t to p u l l apart and contained some u n b r o k e n fibres at about 30 m m from the i n i t i a l n o t c h tip.  3.3.1.3  Determining the critical fracture energy G  c  One o f the p r i n c i p a l benefits o f o b t a i n i n g the stable g r o w t h o f a crack from a stable specimen g e o m e t r y is that it a l l o w s the determination o f the fracture energy o f the material s i m p l y b y measuring the area under the load-displacement curve. T h i s m e t h o d is not v a l i d f o r r a p i d , unstable crack g r o w t h because the energy release rate can be larger than the critical value G  c  and the area under the curve w i l l include energy dissipated t h r o u g h v i b r a t i o n , noise, heat, etc. F o r stable crack g r o w t h , the amount o f released energy is equal to G . c  G  c  A n average value o f  is therefore calculated as the total w o r k done at the pins d i v i d e d b y the area o f crack  g r o w t h as  T h i s m e t h o d assumes a constant crack resistance R, w h i c h is also assumed i n the n u m e r i c a l model.  T h i s is a g o o d assumption f o r this material since it appears that G  crack i n i t i a t i o n .  c  stabilizes after  A s seen i n Figure 3 . 1 1 , a stable zone o f self-similar crack g r o w t h was  observed f o r b o t h i n i t i a l l y b l u n t notches and sharp notches.  38  Chapter  3  Experiments  For calculation o f the w o r k done, it is i m p o r t a n t to measure the p i n o p e n i n g displacement ( P O D ) and not C M O D , since the load is measured at the pins. U n f o r t u n a t e l y , P O D was o n l y recorded i n the later tests and is not available f o r the m a j o r i t y o f the smaller n o t c h radius cases. Specimens r d 2 5 - 2 and r d 2 8 - 2 b o t h displayed a plateau o n the load versus P O D curves i n d i c a t i n g stable development o f the damage zone and they are used here t o determine  G  c  values. C r a c k lengths are determined f r o m the crack opening p r o f i l e s o f the damaged specimens.  To  obtain the crack o p e n i n g displacements as damage progressed i n the O C T specimens, the line analysis technique described i n Section 3.2.3.3 is used. The line analysis plots f o r specimens r d 2 5 - 2 and r d 2 8 - 2 are s h o w n i n Figure 3.16 and Figure 3.18 respectively. D u e to surface damage, it was n o t possible to use the line closest to the plane o f the n o t c h . T y p i c a l l y either the 3  r d  or 4  t h  lines on either side o f the n o t c h plane w e r e used t o make the  measurements. T h e relative separation o f the lines therefore includes b o t h the C O D and the straining o f 7.5 m m to 10.0 m m o f material on either side o f the crack plane leading t o a small error i n the measurements. A correction is made b y measuring the relative separation i n the scribed lines used i n the analysis j u s t p r i o r to the first n o n - l i n e a r i t y i n the l o a d - p i n displacement curve.  A s s u m i n g that no damage has occurred at this p o i n t , the relative  separation at the n o t c h t i p is a p p r o x i m a t e l y the amount o f elastic strain i n the material.  This  strain is noted i n the figures as a dashed line and the C O D s determined f r o m the analysis are measured from the intersection o f the C O D p r o f i l e w i t h the dashed line. L i n e analysis was also conducted f o r specimens r d 2 - l , r d 2 - 2 , r d 8 - 2 , r d 5 4 - l and r d 5 4 - 2 and the plots are presented i n A p p e n d i x B.  T h e final crack lengths determined b y line analysis  are compared to the values obtained from sectioning f o r specimens r d 2 - l and r d 5 4 - 2 i n F i g u r e 3.20.  I n b o t h specimens, the line analysis value underestimates the total length o f  damaged material obtained from sectioning. M i t c h e l l ( M i t c h e l l , 2 0 0 2 ) observed the opposite trend but i n c l u d e d the apparent crack length from elastic straining o f the material between the lines used i n the analysis.  W i t h o u t i n c l u d i n g the apparent crack length from elastic  strains, the line analysis gives an a p p r o x i m a t i o n to the crack length f a l l i n g somewhere i n the damage zone. F o r G determination, this a p p r o x i m a t i o n is reasonable. I n E q u a t i o n 3.3, a is c  39  Chapter  3  Experiments  an effective t h r o u g h - c r a c k representing the real crack composed o f regions o f f u l l damage and partial damage. Plots o f load versus P O D f o r specimens r d 2 5 - 2 and r d 2 8 - 2 are g i v e n i n Figure 3.17 and Figure 3.19 respectively.  The locations w h e r e line analysis was conducted are m a r k e d and  the change i n crack length is noted o n the plots. The amount o f w o r k done b y the system is calculated b y assuming an elastic u n l o a d i n g path back t h r o u g h the o r i g i n .  N o t e that the  o r i g i n has been shifted s l i g h t l y to account f o r the i n i t i a l take-up o f slack i n the system. A l s o , it can be seen f r o m the figures that the actual u n l o a d i n g path does not return to the o r i g i n but shows about 1.75 m m o f residual displacement. I t is assumed that this residual is not due to plastic d e f o r m a t i o n , but rather to the damaged surfaces not b e i n g able to f i t back together p r o p e r l y w i t h i n the damage zone. The w o r k done and the calculated values o f G are tabulated versus change i n crack length c  f o r b o t h specimens i n Table 3.5.  E x p e r i m e n t a l points on the crack resistance curve are  p l o t t e d i n Figure 3 . 2 1 . There is a large scatter i n the results due t o the error i n a p p r o x i m a t i n g the crack length u s i n g line analysis.  A t short lengths o f crack g r o w t h a small error i n Aa  used to calculate G has a large effect o n the solution. A t longer crack lengths the s o l u t i o n is c  less sensitive to small deviations i n Aa so that a m o r e reliable, although still approximate, solution is obtained. E v e n w i t h the uncertainty i n the results at small Aa, a trend o f i n i t i a l l y increasing crack resistance is apparent.  H o w e v e r , this is not true i?-curve behaviour.  The  l o w e r G values determined at small Aa correspond to the development o f the damage zone. c  A f t e r the damage zone height has stabilised at its characteristic height (as s h o w n i n Figure 3.11), G appears t o stabilise as w e l l . c  U s i n g the t w o largest Aa points o n the load versus P O D p l o t o f each specimen to calculate G , a value o f about 80 to 85 k J / m is determined. c  3.3.2  F o u r - p o i n t b e n d tests  F o u r - p o i n t bend tests were conducted t o determine the u n n o t c h e d tensile strength o f the material. Results f r o m the f o u r - p o i n t bend tests are presented i n Table 3.6. The load versus displacement p l o t f o r specimen b 8 5 - 9 0 - 2 is g i v e n i n F i g u r e 3.22 as a representative case. The rest o f the plots f o r the 90° orientation specimens are presented i n A p p e n d i x C.  40  Chapter  3  Experiments  Separate tests were done w i t h the specimens oriented i n both in-plane directions. I n the first case, the laminate 90° d i r e c t i o n was oriented a l o n g the axis o f the b e a m w i t h the stronger 0° d i r e c t i o n oriented i n the transverse direction. T h i s orientation corresponds t o the orientation o f the O C T specimens and is used to determine the y - d i r e c t i o n strength f o r the n u m e r i c a l m o d e l o f the O C T specimens.  Tests were also conducted w i t h the 0° fibres oriented a l o n g  the axis o f the b e a m to determine strength i n the x - d i r e c t i o n .  H o w e v e r , v a l i d results were  o n l y obtained f o r the 90° orientation, w h i c h is the i m p o r t a n t d i r e c t i o n f o r n u m e r i c a l m o d e l input. I n all o f the 90° orientation tests failure occurred i n the centre o f the span due to tension i n the outer fibres. F o r the specimens oriented i n the 0 ° d i r e c t i o n , failure occurred i n compression under the l o a d i n g pins and therefore gave incorrect tensile strength values. The strength values f o r the 0° orientation specimens are o n l y l o w e r bounds on the tensile strength. A s s h o w n b y the sharp load drop i n F i g u r e 3.22, failure was catastrophic, resulting i n a w e l l d e f i n e d peak load.  The stiffness o f the specimen, h o w e v e r , becomes non-linear at a p o i n t  before the peak load i n d i c a t i n g that a small amount o f damage has taken place before failure. T h e dashed line i n the f i g u r e illustrates that the curve first becomes non-linear at around 1 k N o f load. E q u a t i o n 3.1 is used to calculate the stress i n the outer fibres at peak load. A s stated i n the specification E q u a t i o n 3.1 is o n l y strictly v a l i d f o r materials that are linear up to the p o i n t o f rupture so a slight error w i l l be introduced i n using this equation. A n a p p r o x i m a t i o n to the tensile failure strength o f the material is then taken as the average o f the tests. F o r the 90° orientation the f l e x u r a l strength is taken f r o m the average o f five tests to be a p p r o x i m a t e l y 460 M P a . F o r the 0° orientation the m i n i m u m value o f the f l e x u r a l strength is taken as the m a x i m u m f l e x u r a l strength recorded f r o m five tests o f about 1000 M P a .  3.4  Summary •  A transition i n the load-displacement b e h a v i o u r o f the O C T specimens was n o t e d at a n o t c h root radius o f about 16 m m .  O C T specimens w i t h r a d i i less than this value  displayed stable crack g r o w t h w i t h little n o t c h sensitivity.  O C T specimens w i t h  larger r a d i i were unstable and showed m o r e n o t c h sensitivity.  41  Chapter •  3  Experiments  Post-test sectioning o f the O C T specimens reveals that the damaged height material is i n i t i a l l y dependent on the radius o f the n o t c h root.  of  A large i n i t i a l radius  creates a t a l l , b u t diffuse damage zone w h i l e a sharp n o t c h concentrates the damage in  a short  (height)  damage  zone.  After  stable  crack  growth  is achieved,  a  characteristic damage height o f 6 m m to 7 m m is p r o d u c e d f o r all i n i t i a l n o t c h sizes. •  A c r i t i c a l fracture energy value o f 80 to 85 k J / m  and a strength o f 4 6 0 M P a was  determined f o r the material used i n this thesis.  42  Chapter  3  Experiments  Table 3.1 Original elastic material properties for CFRP laminate (Mitchell, 2002) Property  Value  E  75 G P a  Ey  30 G P a  O  0.161  x  xy  Gy X  17.1 G P a  Chapter  3  Experiments  Table 3.2 OCT tests and specimen measurements. Specimen  Notch root  at, (mm)  ^(mm)  B (mm)  Identifier  radius (mm)  rdl-1  0.5  33.71  rd2-l  81.84  8.98  1  33.3  81.71  8.49  rd2-2  1  33.35  81.92  8.93  rd4-l  2  33.42  82.52  9.04  rd4-2.  2  33.43  80.53  8.32  rd6-l  3  33.51  82.17  8.55  rd6-2  3  33.07  80.90  8.71  rd6-3  3  48.1  81.90  8.75  rd8-l  4  33.35  81.91  8.89  rd8-2  4  33.45  82.10  8.85  r d l 0-1  5  33.03  80.70  8.55  r d l 0-2  5  33.2  81.75  8.57  r d l 7-1  8.5  33.33  81.00  8.43  rd21-l  10.5  33.43  81.20  8.13  rd25-l  12.7  33.99  81.91  8.07  rd25-2  12.7  34.1  80.50  8.00  rd28-l  14.15  34  80.50  8.10  rd28-2  14.15  33.8  80.30  8.50  rd32-l  15.9  33.71  81.93  8.70  rd32-2  15.8  34.2  81.20  8.60  rd32-3  15.8  34.18  81.32  7.91  rd35-l  17.5  33.8  81.24  7.96  rd35-2  17.5  34.03  81.41  8.30  rd38-l  19  33.64  81.96  8.26  rd38-2  19  33.76  81.42  8.35  rd44-l  22.25  33.46  82.02  8.56  rd44-2  22  33.92  81.06  8.32  rd54-l  27  32.85  81.3  8.83  rd54-2  27  33.16  81.12  8.14  44  Chapter  3  Experiments  Table 3.3 Four-point bend tests and specimen measurements Loading Specimen  Orient-  D  B  Identifier  ation  (mm)  b56-90-l b56-90-2 b85-90-l b85-90-2 bl50-90-l bl50-90-2 b56-0-l b56-0-2 b85-0-l b85-0-2 bl50-0-l bl50-0-2  90  Rate  (mm)  L (mm)  L (mm)  L /D  (mm/min)  5.92  7.79  135  45  22.8  5.70  90  5.81  7.81  90  30  15.5  2.58  90  8.70  8.21  135  45  15.5  3.88  90  8.65  8.18  135  45  15.6  3.90  90  15.20  8.18  135  45  8.9  2.22  90  15.23  7.8  135  45  8.9  2.21  0  6.00  8.05  90  30  15.0  2.50  0  6.05  8.03  90  30  14.9  2.48  0  8.64  8.07  135  45  15.6  3.90  0  8.67  8.03  135  45  15.6  3.89  0  15.18  8.13  135  45  8.9  2.22  0  15.14  8.06  135  45  8.9  2.23  s  L  s  45  Chapter  3  Experiments  Table 3.4 Summary of OCT tests and analysis Specimen  Load-  Load -  Periodic  Cyclic  Line  G c calc-  Identifier  CMOD  Pin  Unloads  Unloading  Analysis  ulation  Sections  Disp. rdl-1 rd2-l rd2-2 rd4-l rd4-2  >/  rd6-l  V  rd6-2  •  rd6-3 rd8-l •/  rd8-2 rdlO-1 r d l 0-2 r d l 7-1  V V  rd21-l rd25-l rd25-2 rd28-l rd28-2 rd32-l  s •/  rd32-2 rd32-3  s  rd35-l rd35-2 rd38-l  V Y  rd38-2 rd44-l rd44-2  Y S  S  rd54-l rd54-2  46  Chapter  3  Experiments  Table 3.5 Critical fracture energy release rates rd25-2 Aa  W(J)  G (kJ/m )  8.4  3.2  48.0  25.0  13.0  64.9  27.4  19.0  86.8  32.4  23.3  90.0  W{A)  G (kJ/m )  (mm)  2  c  rd28-2 Aa  (mm)  2  c  12.5  5.0  47.4  14.5  11.6  94.0  26.5  19.0  84.2  33.5  23.8  83.6  Chapter  3  Experiments  Table 3.6 Four-point bend test results Specimen  Maximum  Failure load  M a x . stress  displacement  (kN)  at f a i l u r e  (mm)  (MPa)  b56-90-l  NA  NA  NA  b56-90-2  4.981  1403.9  479.3  b85-90-l  7.237  2190.9  476.0  b85-90-2  6.530  1989.0  438.7  bl50-90-l  3.819  6394.3  456.8  b 150-90-2  3.881  6089.8  454.4  D56-0-1  3.962  3263.4  1013.5*  b56-0-2  4.087  3295.6  1009.1*  b85-0-l  5.395  4085.7  915.6* 815.2*  b85-0-2  5.855  3644.8  b150-0-1  NA  NA  NA  B150-0-2  3.467  10644.5  777.8*  * Strength values f o r the 0 ° specimens are the m i n i m u m b o u n d o f strength because failure occurred i n compression under the l o a d i n g pins instead o f tension between the pins.  48  Chapter  3  Experiments  106 m m  ^  ^  W= 80.6 m m c — 38.7 m m  Figure  3.1 OCT specimen  a = 33 m m  geometry  (thickness  B ranged from  7.9 mm to 8.6 mm)  49  Chapter  3  Experiments  Chapter 3 Experiments  Figure 3.3 Scribed lines for line analysis (shown for rd4-2)  Figure 3.4 OCT test set-up (shown for rd54-2)  51  Chapter  3  0  Experiments  0.5  1  1.5  2  2.5  3  3.5  4  Displacement (mm) Figure  3.5 Load-displacement  plots for specimen the relative  Figure  3.6 Schematic  rd32-2 from the machine position  displacement  and from  of the pins.  of section taken through the damage zone ahead of the initial notch tip.  The dashed lines represent  the 90 °fibres  did not include  in the outer stacks.  The measured  any damage to the outside of these  damage  height  lines.  52  Chapter 3 Experiments  Spec length  B = 8.5 m m .  ca  cC± L  L  u  Figure 3.7 Four-point bend test geometry  Figure 3.8 Specimen rdl 0-2 showing orientation of crack and extent of surface damage (scribed lines are 2.5 mm apart)  53  Chapter 3 Experiments  0  0.5  1  1.5  2  2.5  3  3.5  4  4.5  5  C M O D (mm)  Figure 3.9 Load versus crack mouth opening displacement for specimens of representativ notch root radius. Solid lines indicate continuous portions of the curve while dashed segments indicate segments in which the crack jumps and the load drops sharply.  O  o  O  o  O  t •  *  •  20  25  30  p (mm)  Figure 3.10 Peak loads attained in each OCT test versus notch root radius p. Closed da points denote specimens displaying a plateau on the load versus CMOD curve, open dat points denote specimens that showed sharp load drops at the peak load. 54  Chapter 3 Experiments 20 18  Figure 3.13  O |  16 14  ? E  sections  o• w  rd54-2  12  r  Figure 3.15  10  •a  X  0  Figure 3.12 6  rd8-2  Partial damage  4 2  Figure 3.14 Damage zone  Stable damage zone  development 0 10  15  20  25  30  35  x (mm)  Figure 3.11 Profiles of the damage zone along the crack plane  Figure 3.12 Section of rd2-lat 0.635 mm ahead of initial notch tip  55  Chapter 3 Experiments  Figure 3.13 Section of rd54-2 at 0.635 mm ahead of initial notch tip  Figure 3.14 Section of rd2-l at 24.13 mm ahead of initial notch tip  56  Chapter 3 Experiments  Figure 3.15 Section of rd54-2 at 24.13 mm ahead of initial notch tip  1.8  Distance from initial notch tip (mm)  Figure 3.16 COD profiles determinedfrom line analysis for rd25-2  57  Chapter  3  Experiments Aa =0 mm  P O D (mm) Figure  3.17 Load versus pin opening  displacement line  for rd25-2  showing  location  ofphotos  for  analysis  — - image38 POD=l .7 mm -«-image42 P0D=2.1 mm - • - image44 POD=2.5 mm - * - image48 POD=3.2 mm image51 POD=3.8 mm  -2  0  10  12  14  16  18 20  22 24  26  28  30  32  34 36 38  Distance from initial notch tip (mm) Figure  3.18 COD profiles  determinedfrom  line analysis for  rd28-2  58  Chapter  3  Experiments  POD (mm) Figure  3.19 Load versus pin opening  displacement line  for rd28-2  showing  location  ofphotos  30  34  for  analysis  • Sectioning (full damage height) • Sectioning (partial damage height)  0  2  4  6  8  10  12  14  16  18  20  22  24  26  28  32  36  38  Crack length (mm) Figure  3.20 Final  crack lengths in rd2-l  and rd54-2 from sectioning  and line  analysis  59  Chapter  3  Experiments  100 90 80 70 60  S  S C  |  50 40 30 20  • rd25-2 • rd28-2  10 0 10  15  20 Aa  Figure  3.21 Crack resistance  25  30  35  (mm)  points for rd25-2 and  rd28-2  2.5  2.0  1.5  «  T3  1.0  O  0.5  0.0  -0.5 3  4  P i n displacement (mm) Figure  3.22 Four-point  bend test load-displacement  plot for  b85-90-2  60  Chapter 4 Numerical  CHAPTER 4  4.1  NUMERICAL  Introduction  A n u m e r i c a l study o f damage development and propagation i n F R P composite laminates is undertaken i n this thesis.  I n particular, the over-height compact tension tests described i n  Chapter 3 are simulated u s i n g a three-parameter damage m o d e l . There are several goals to this endeavour: 1) T o c l a r i f y the state o f stress and strain at the n o t c h t i p at i n i t i a t i o n o f fracture and d u r i n g propagation u s i n g f i n i t e element n u m e r i c a l models. 2)  To  develop  a physically  based  cohesive  zone  model  that  is  described  by  independently determined material parameters capable o f m a t c h i n g experimental results. 3)  T o investigate the a b i l i t y o f the cohesive zone m o d e l to predict the transition i n structural behaviour w i t h n o t c h root radius observed i n the experimental O C T tests.  4)  T o investigate the dependence o f the m o d e l response on f i n i t e element mesh size and c o n f i g u r a t i o n and the interaction o f mesh size and material parameters.  Specifically,  the a b i l i t y to accurately m o d e l crack propagation o n a large scale u s i n g a coarse mesh is explored. The first goal can be p a r t l y achieved u s i n g elastic simulations o f the O C T tests at the p o i n t o f peak load.  Here, the size and shape o f the stress fields give clues to the response o f the  various sized notches i n the O C T tests. T o achieve the r e m a i n i n g three goals, t w o related user material models ( U M A T s ) have been developed and i m p l e m e n t e d i n the A B A Q U S f i n i t e element computer p r o g r a m .  The  first  m o d e l , called the simple damage m o d e l ( S D M ) , is a cohesive zone m o d e l r e q u i r i n g ( f o r each material direction) the i n i t i a l elastic m o d u l u s Eo, the strain at the onset o f damage e k and pea  the c r i t i c a l specific modification  fracture  energy  y  c  as input parameters.  The second m o d e l is a  o f the S D M . I t includes the a b i l i t y to adapt a u t o m a t i c a l l y the  constitutive relationship f o r each element based on mesh size and the strain  material  field.  61  Chapter 4 Numerical The development o f these models contributed significant insight into the observed response o f the experimental O C T tests.  4.2  Elastic simulations  The adaptive damage m o d e l presented i n this chapter makes use o f the strain d i s t r i b u t i o n w i t h i n an element. B e f o r e d e v e l o p i n g the m o d e l , elastic simulations o f the O C T specimens were carried o u t u s i n g the A B A Q U S f i n i t e element c o m p u t e r p r o g r a m t o g a i n a better understanding o f the stress a n d strain distributions at the n o t c h t i p . A n estimate o f the element size required t o resolve the stress or strain f i e l d is also made.  4.2.1  Stress concentrations  Elastic stress distributions along the n o t c h plane ahead o f the n o t c h t i p at the p o i n t o f peak load were obtained from static f i n i t e element analyses o f the O C T specimens u s i n g the orthotropic material properties g i v e n i n Table 4 . 1 . N o t e that the m o d u l u s E2 was increased slightly  from 3 0 GPa, as g i v e n b y the material supplier, t o 32 G P a t o better m a t c h the  compliance o f the experimental specimens.  O n l y the b o t t o m h a l f o f the specimens w e r e  m o d e l e d , w i t h s y m m e t r y conditions applied along the n o t c h plane. T w o - d i m e n s i o n a l , f u l l y integrated, 8-noded, plane stress elements were used w i t h 0.5 m m x 0.5 m m elements i n the r e g i o n o f the n o t c h plane and stresses were quadratically extrapolated t o the nodes. The c o m p u t e d stress fields f o r specimens w i t h n o t c h r a d i i o f 1, 4 , 10.5, 16 a n d 27 m m are presented i n F i g u r e 4.1 f o r a load o f 15.0 k N . T h i s load level is a p p r o x i m a t e l y the peak load f o u n d e x p e r i m e n t a l l y f o r specimens d i s p l a y i n g a stable plateau o n the load versus C M O D plots (i.e. p = 0.5 t o 14 m m ) .  T h e linear-elastic material m o d e l used does n o t a l l o w a n y  damage t o occur at the notch t i p . I t was seen from the experiments that damage does occur before the peak load is reached, as evidenced b y slight n o n - l i n e a r i t y i n the l o a d - C M O D curves. The elastic stress fields are therefore n o t c o m p l e t e l y accurate, h o w e v e r they do a l l o w a reasonable a p p r o x i m a t i o n o f the stress states f o r c o m p a r i s o n between n o t c h sizes. T h e material tensile strength o f 4 6 0 M P a that was determined f r o m the f o u r - p o i n t bend test and is used later i n the damage m o d e l is m a r k e d i n Figure 4.1 w i t h a dashed line. Clearly, a n o t c h root radius o f 27 m m creates a stress concentration at the notch t i p that is less than the tensile strength w h e n the load is at a critical level f o r smaller n o t c h sizes. A t a n o t c h root radius o f 16 m m , w h i c h is r o u g h l y the transition p o i n t between the stable a n d unstable  fracture  62  Chapter 4 Numerical initiation seen experimentally, the stress is approximately the same as the tensile strength. For the smaller notch root radii the stresses at the notch tip are well above the tensile strength of the material meaning that the stress criterion forfracturehas been met. Whether or not fracture initiates depends on whether sufficient energy is stored in the system. Increasing the load to the experimentally observed peak load of 17.8 kN for the 27 mm notch root radius specimen magnifies the elastic stress concentration at the notch tip to slightly above the tensile strength as shown in Figure 4.2. 4.2.2  Stress contours  From the elastic simulations it is also possible to estimate the extent of critically stressed material for a given load. The contour plots in Figure 4.3 show the elastic stressfieldsin the region of the notch tip.  Each notch size is shown at approximately its peak load, as  determinedfromthe experimental tests. The light grey areas around the notch tips are areas that experience a stress over 460 MPa at the peak load for that notch root radius. Generally, there is an elliptical shape to the stress contours with the critically stressed area extending by about one to two millimetres ahead of the notch tip, with a gradual decrease in the width with increasing notch root radius. Conversely, it can be seen that as the notch root radius is increased, the height of critically stressed material around the notch tip increases. Table 4.2 lists the approximate height of critically stressed material at the experimentally determined peak load for each of the specimens from the elastic analyses. Although the trend is increasing height with increasing notch root radius, the critically stressed height dips to 8.5 mm for p equal to 27 mmfrom9.5 mm for p equal to 16 mm suggesting that perhaps experimental specimens rd54-l and rd54-2 failed prematurely. The significance of the critically stressed height is discussed further in Chapter 5. 4.2.3  Element size required to resolve the notch tip stress concentration  It was found that relatively large, fully-integrated, 8-noded elements that have quadratic shape functions do a much better job of resolving the stress concentration ahead of a notch tip than smaller, constant stress elements containing a single integration point. Typically, explicit dynamic finite element codes that are used to model dynamic crack growth require  63  Chapter 4 Numerical the use of reduced integration, constant or linear stress elements for efficiency.  Static,  implicit codes on the other hand are often more efficient using higher order elements. The stress concentration at the tip of an OCT specimen with a four-millimetre radius notch is modeled in Figure 4.4 using four different width constant stress elements ranging from 0.25 mm to 2.0 mm and also using 0.25 mm and 0.5 mm wide quadratic elements. All of the elements match the stress field away from the notch tip well. The notch tip extrapolations of the stress are not very good for the constant stress elements but they do approximate the true solution fairly well at the location of the integration points (e.g. at 0.25 mm along the notch plane for the 0.5 mm wide constant stress element). The higher order elements do a better job of predicting the notch tip stress. There is little difference between the 0.25 mm and 0.5 mm solutions.  4.3  Development of the cohesive zone models  4.3.1  Simple damage model (SDM)  A three-parameter cohesive zone model for fracture of composite laminate materials was developed as a UMAT in ABAQUS. The model is intended as a simple representation of a more advanced composite damage model that has been developed by the Composites Group at the University of British Columbia called CODAM (Williams, Vaziri and Poursartip, 2003). CODAM (Composite DAMage Model) uses physical input parameters describing the onset and saturation of damage for fibre and matrix in a composite laminate. A linear relationship between a strain function that incorporates strain interactions from other material directions and a monotonically increasing damage parameter is defined for bothfibreand matrix in each direction. Total damage is defined as a combination of fibre and matrix damage depending onfibreand matrix properties and lay-up sequence. Material moduli are then reduced according to a linear relationship with total damage, resulting in a strainsoftening stress-strain curve similar to the curves used in cohesive zone models. The model developed here loses some of the generality of CODAM but is still physically-based and is used to demonstrate characteristics of softening models in a simple manner. The model is therefore termed the SDM (Simple Damage Model). It was decided to implement the model in ABAQUS to take advantage of the features of the implicit code. An implicit solution scheme uses an iterative technique such as the NewtonRaphson method to solve non-linear problems and is ideal for structural problems involving 64  Chapter 4 Numerical long time durations since the step size can be relatively large. Explicit, direct integration methods on the other hand require much less computation per time step but require relatively small time steps, making it ideal for wave propagation problems but not long duration structural dynamics problems. The unstable growth of a crack in a continuum would be best modeled using an explicit integration scheme since the crack travels at close to the wave speed of the material and inertia is important in the solution. However, the OCT tests that are modeled in this thesis are quasi-static tests and display, for the most part, stable crack growth. Therefore, an implicit code, which does not include inertia, can give an accurate solution much more efficiently than an explicit dynamic code. In addition, although COD AM is implemented in the LS-DYNA explicit finite element code, higher-order elements are not available in the code.  4  Higher-order elements containing a  quadratic strain field (available in ABAQUS) are required in the modified SDM, described below, in order to achieve a good description of the strainfieldin a single element.  4.3.2  Adaptive simple damage model (ASDM)  A modified version of the SDM will be described along with the development of the SDM. The modified version attempts to take into account the effect of element width that was 5  noted in the literature review. A coarse mesh in the width direction introduces error as demonstrated schematically in Figure 4.5. For a notch-tip element in which stress is calculated at the centre of the element, the stress at the calculation point is less than critical when the stress at the edge of the element has reached the critical level. This is due to the stress distribution at the notch tip. The wider the element, the lower the calculated stress. With the typically used Gauss integration scheme or a constant stress element, an element's calculation points are located at some point within the element and not on the edges. This means that the stress field at a notch tip will not be calculated at the tip, but rather a certain distance ahead of it. That distance depends on the size of the element and the integration scheme that is used. In higher  The reason for this is that higher-order elements yield higher maximum frequencies than lower-order elements. In explicit time integration schemes the required step size to achieve a correct solution decreases with the maximum frequency since the step size must be small enough so that information does not propagate across more than one element per time step. Width refers to the element dimension in the direction of crack growth.  4  5  65  Chapter 4 Numerical order elements error is reduced because the integration points are located closer to the element edge. The effect of sampling the stress field away from the notch tip is that damage initiation in the element is delayed. When the stress at the notch tip/ element edge is at the critical level for crack initiation, the stress at the calculation point will be lower than critical. When the stress at the calculation point reaches the critical level, the stress at the notch tip may be considerably higher than critical. For a large element there may be enough energy G in the system to propagate the crack, but it cannot initiate until the stress criterion is also met. In that case the applied load must be increased until the stress seen by the element reaches the material strength. In doing so, the stress at the notch tip will increase to above the material strength and G will surpass G . With the sudden initiation of the crack, the additional stored c  elastic energy is released into the crack causing it to propagate unstably until G <G . C  The difference between the actual stress at the notch tip when damage initiates and the critical stress depends on the stress concentration as well as the location of the integration point. From Figure 4.1, it can be seen that, using a finite sized element, a much larger error would be incurred in modeling an OCT specimen with a notch tip radius of 1 mm than a notch tip with a radius of 27 mm. The much steeper stress distribution in the sharper notch case means that the difference between the actual notch tip stress and the calculated stress at the integration point is very large. As a notch tip approaches the infinitely sharp case smaller and smaller elements should be required in order to minimize the difference between the actual and calculated stresses. Using the existing composite damage model (CODAM) the mesh width effect was seen in modeling a unidirectional carbon fibre/ epoxy double-cantilever beam (DCB) specimen. The input parameters in the direction perpendicular to the notch (out-of-plane) were determined to match those of a cohesive zone model with a three-parameter softening constitutive curve (initial modulus, strain at onset of damage and an ultimate strain at complete separation of the interface). The region ahead of the notch tip was discretized with four different constant stress element widths; 62.5 pm, 125 pm, 250 pm and 500 pm. The height of the elements was maintained at 250 pm. The arms of the virtual specimens were opened at a quasi-static loading rate past crack initiation. The effect of element width can be seen in Figure 4.6. The  66  Chapter 4 Numerical peak load was strongly dependent on the width of the element with the apparent strength of the specimen increasing with element size. Under the same loading conditions and for the same material properties, peak load varied from 19.5 N for 62.5 pm elements to 29.3 N for 500 pm elements. There are two ways of dealing with this situation. The first is to refine the mesh at the tip of a notch to very small elements to reduce the error.  A problem with this method is  immediately apparent. As mentioned in Chapter 2, mesh refinement when using a strainsoftening material model does not lead to a converged solution due to localisation and the dependence of GF on the height of the localised band of elements. Another option is to adapt the constitutive material model to take into account the distance of the integration point from the notch tip. In this manner, damage would initiate in an element when the stress at the element edge reaches the material strength. The stress-strain curve at the integration point would then need to be adjusted so that the critical stress level is lower while still maintaining the area beneath the curve for matching G . The second option is the solution technique that C  has been incorporated into the SDM. Since it involves adapting the constitutive model to the stress concentration and element size, it is called the Adaptive Simple Damage Model (ASDM). Returning to the problem of the D C B specimen modeled with different element widths, the method is demonstrated. When the problem was first modeled with a series of simulations using elements of different width, different global peak loads were obtained (light grey points in Figure 4.7).  Scaling the material strength of the larger sized elements against the  62.5 pm elements and re-running the solutions resulted in equivalent peak loads being predicted (black points in Figure 4.7).  To determine the scaling ratio for each element size,  an elastic simulation of the DCB was done with 62.5 pm elements in the region of the crack plane to obtain the stress distribution ahead of the crack tip as shown in Figure 4.8. The constitutive curves of the 125 pm, 250 pm and 500 pm element width specimens were then scaled by the ratio of stress at the element midpoints to the stress at the midpoint of the 62.5 pm element. In the A S D M , the procedure described above for the DCB specimen is done automatically in the material model. The strain distribution is determined across a second order element and  67  Chapter 4 Numerical  the strength at the integration point is scaled by the ratio of the strain at the integration point to the strain at the element edge. 4.3.3  Algorithm for the SDM and ASDM  The S D M and A S D M are implemented as U M A T s in A B A Q U S for use with plane stress two-dimensional solid elements.  The S D M can be used with elements of any order or  integration scheme but the A S D M requires fully-integrated, 8-noded elements to function properly. Similar to the more advanced composite damage model C O D A M , the S D M input parameters are derived from the properties of fibre and matrix smeared together to form the orthotropic material properties of each layer of a laminate. These layers are called sub-laminates. In the S D M , the material behaviour in each principal direction for a sub-laminate is defined by three parameters: initial modulus EQ, elemental specific fracture energy y and characteristic e  critical strain  So k pea  (strain at damage initiation).  size. The peak stress Suit  a k pea  These values are all invariant with element  (strength) is determined from  (strain at damage saturation) follows from <7  peak  ^peak  EQ  and  £o k pea  and the ultimate strain  and y . e  4.1  E() peak £  2y  4.2  e  peak  The softening behaviour described by these parameters is shown in Figure 4.9. Damage initiates when s is equal to s  k  pea  and a linear, negative tangent modulus reduces stress with  increasing strain until complete damage is reached at s it. Softening only occurs in tension. u  In compression the material is assumed to be undamaged (linear elastic). Currently there is no strain interaction between material directions so strains in a given direction only cause damage in the same given direction. At this stage the model is suitable for modeling Mode I crack growth. Further development is necessary to define the constitutive relationships for shear loading as would be necessary to model mode II or mode III crack growth. Before the onset of damage, the model is identical to an orthotropic elastic material model. The constitutive curve is defined separately in each material direction. The model is not limited to modeling only interfaces, but can be used as the material model for an entire structure,  68  Chapter 4 Numerical allowing the location of stress concentrations to determine the location of crack growth. As an elastic damage model there are no plastic strains before complete damage occurs and unloading takes place on a line through the origin. In this material model the criterion for crack initiation is strain equal to the damage initiation strain e k and the criterion for complete separation of the crack is the dissipated specific pea  fracture energy equal to the critical specific fracture energy y . c  Since damage is an  irreversible process, the amount of fracture energy dissipated yd is calculated at each step and the residual fracture energy y , equal to y - yd, is stored as a history variable by the code and r  c  used to keep track of the current state of the material. The current state is uniquely defined by y , the monotonically increasing current peak strain e k and the residual secant modulus 6  r  pea  E as shown in Figure 4.9. Initially s k is set equal to the strain at onset of damage eo k r  pea  pea  and the modulus and fracture energy are set equal to their undamaged values. £  —F peak  E =E r  Yd  0 peak  0  4.3  =7e  The algorithm for the S D M and A S D M is shown as a flowchart in Figure 4.10. Both models follow the same basic structure with the A S D M including some additional operations. A double box around a process indicates a calculation or subroutine that is only included in the ASDM.  The A S D M routine is briefly described here and the U M A T code is included in  Appendix D. The following steps are carried out at each time or load increment and iterated until equilibrium between the internal and external loads is achieved (i.e. convergence). Loop over each element: Step 1. Determine the strain profile in each element The strain profile at a notch tip or damage zone is used to carry out parameter scaling as opposed to the stress profile because, for a softening material model, the strain field is always increasing towards a notch tip while the stress field will begin to decrease towards the notch tip once softening has initiated. For an initially elastic material, as long as the crack is 6  This strain is referred to as the peak strain since it coincides with the peak stress on the constitutive curve.  69  Chapter 4 Numerical small compared to the other planar dimensions, the surrounding elastic material controls the strain in the damaged material at the notch tip. In a fully integrated, 8-noded, plane stress element there are nine integration points. Using a Gauss scheme the integration points are located as shown in Figure 4.11.  The x-direction  and y-direction strain fields across each row and column of integration points are determined by fitting a second order polynomial to the strains at each point. This is similar to using the element shape functions but allows the x - and y-strain fields to be described separately for each row or column using three polynomial coefficients. For the first time or load step the strains are zero everywhere and subsequently the strains are taken from a global storage matrix, which holds the integration point strains from the previous time step. Using a second order polynomial allows the strain function to closely approximate the r'  l/2  distribution found  near a sharp notch. In the current methodology, there is no provision to ensure continuity in the strain function at the edges of the element. This can lead to small errors in the magnitude of the strain field at the element edges.  To fix the problem the strains in neighbouring  elements would need to be taken into consideration, greatly increasing the complexity of the problem.  Loop over each integration point within the element: Step 2. Determine the scaling strain First, the integration point numbering sequence within the element is determined with respect to the local material axes to orient the element. There are six strain fields in the x-direction and six strain fields in the y-direction represented by separate polynomials in each element corresponding to the dashed segments shown in Figure 4.11.  Each integration point is at the intersection of two of these segments, one  segment in the x-direction and one segment in the y-direction. The maximum normal strains Smax, on each of the two segments extrapolated to the element edges, corresponding to the integration point are determined and become the x - and y-scaling strains for that integration point.  That is, the x-strain on the y-direction segment and the y-strain on the x-direction  segment.  If the element is within the strain field caused by a notch or other stress  concentrator then the maximum strains usually fall on the element edge. The element edge that has the highest strain is in the direction of the notch tip.  70  Chapter 4 Numerical Step 3. Decide whether or not to scale the constitutive curve The constitutive curve at an integration point is only scaled during the initial elastic loading of the element.  The objective of scaling is to modify the peak stress value so it is not  necessary to scale the strength after the peak stress has been reached and the element is softening. Also, to ensure the correct fracture energy, the constitutive curve is not scaled when reloading with a residual modulus. Therefore scaling takes place i f 0<£<  £p  A N D Speak < SOpeak-  eak  4.4  In the code a slight adjustment to this criterion was necessary so that scaling would not take place when s is close to s k- Since the strains used for scaling are from the previous time pea  steps, it could occur that the current strain had already passed the new strain at onset of damage, causing numerical errors. This adjustment does not change the effect of scaling in any way since the material is linear elastic before the damage onset strain is reached. If scaling does not take place then step 4 is skipped. Step 4. Determine the new constitutive curve If the element is still elastic and the current strain is less than the strain at onset of damage then scaling takes place. The new damage onset strain is calculated as (for each direction): £*  f  —£ peak  £ ^  0 peak V^max  4.5  7  Here s is the strain from the previous time step. A new ultimate strain is calculated so that the fracture energy remains constant: P *  LS  =  4 f.  E' 0e*peak n  C  The modified stress-strain curve after scaling for an integration point is shown in Figure 4.12. Step 5. Calculate stress The current peak strain is tracked throughout the solution and is updated in this step. If scaling took place then  71  Chapter  4  Numerical 4.7  otherwise, Speak — m a x (s, Epeak , £0peak )•  4.8  W i t h the stress-strain curve n o w d e f i n e d , e is s i m p l y mapped t o the curve t o determine the current stress cr.  Step 6 . Return the tangent stiffness tensor to the FE program A B A Q U S requires the tangent stiffness tensor and updated stresses t o obtain a converged solution. The tangent m o d u l i are therefore determined and the stiffness tensor is assembled and returned t o the code.  Step 7. Update history variables The increment o f fracture energy dissipated is calculated and the total dissipated fracture energy is updated. The residual secant m o d u l u s is then calculated from the residual fracture energy as  r m i n E,  2y  4.9  r  ult peak J  The strains at each integration p o i n t are also stored i n a g l o b a l storage array so that they can be used to determine the strain p r o f i l e s i n Step 1.  Continue integration point loop. Continue element loop. 4.3.4  How mesh insensitive is the ASDM?  The A S D M procedure is intended t o reduce the mesh sensitivity o f the m o d e l . T h e question o f h o w coarse the mesh can be and still return accurate results is discussed b r i e f l y . The damage zone size Wd predicted b y the n u m e r i c a l m o d e l is dependent on the values o f am  Speak and Suit that are used. I n the A S D M these values can also change t h r o u g h o u t the course o f the solution.  Figure 4.13 illustrates h o w t h e shape o f the strain field and the strain  parameters affect the size o f the damage zone.  I f the distance £d  am  is increased (i.e. s k is pea  72  Chapter  4  Numerical  decreased and/or s it is increased), the width of the damage zone increases, resulting in a u  tougher constitutive curve. If the element width used to discretize the area in front of a notch tip is increased, the A S D M scales Sp k, and consequently cr k, to a lower value. ea  energy, s it is increased. u  pea  T o maintain the critical fracture  A s element width is increased, Wd  am  also increases.  Thus the  material behaviour is slightly changed in using the scaling technique. Element widths should be less than the width of the damage zone in order to accurately represent it. The size of the damage zone in the numerical model is slightly altered by using the A S D M . Another question is whether large elements will be able to accurately represent the energy released in growing a crack part way through an element.  Using the example shown in  Figure 4.14, it is seen that the energy released in growing a crack a certain length will be constant regardless of the element width used. In Figure 4.14 a) a crack is grown for 2 mm through two 1 mm wide elements.  Each element has a constitutive curve as shown in the  figure, with a one-dimensional critical fracture energy of 100 mJ/mm. Since each element is 1 mm wide, the energy released per segment is 100 mJ and the total energy released for a 2 mm wide crack is 200 mJ. In Figure 4.14 b) a single 5 mm wide element is used to discretize the entire 5 mm width in front of the crack.  The element's constitutive curve also has a  fracture energy of 100 mJ/mm, but since it is 5 mm wide, the energy released per segment is 500 mJ. A crack growth o f 2 mm is represented by the release o f 40% o f the segment's critical fracture energy, equal to 200 mJ. Therefore the energy release is the same.  The  difference between these two situations is that there is a difference in the residual strength o f the interface at the location of the initial notch tip. It appears that the main limitation on the width of elements used to discretize the area ahead of a notch is the ability of the elements to resolve the stress and strain fields.  4.4  OCT simulations  4.4.1  T h e v i r t u a l specimens  Seven O C T specimens containing notches with root radii o f 1 mm, 4 mm, 10.5 mm, 14 mm, 16 mm, 19 mm and 27 mm were modeled. T o be consistent with the experimental tests, each  73  Chapter  4  Numerical  v i r t u a l specimen is referred to u s i n g its n o t c h diameter, f o r example rd2 f o r the specimen w i t h a n o t c h root radius o f 1 m m .  A s w i t h the experimental specimens, the n o m i n a l  dimensions o f the v i r t u a l specimens are the same as g i v e n i n F i g u r e 3.1 but the actual v i r t u a l dimensions v a r y s l i g h t l y to m a t c h the experimental dimensions o f each separate specimen. D e p e n d i n g o n the case b e i n g r u n , the mesh size and c o n f i g u r a t i o n varied.  Half-symmetry  versions and f u l l versions o f each specimen w e r e used depending on the test. I n all cases, the solutions w e r e f o r plane stress.  The f u l l version o f r d 2 1 is s h o w n i n F i g u r e 4.15 as a  representative mesh. F o r all specimens, a band o f rectangular, u n i f o r m size elements were used to mesh the area around the n o t c h plane ahead o f the n o t c h t i p .  E l e m e n t size was  increased a w a y f r o m the n o t c h plane. The dimensions and tests f o r each v i r t u a l specimen are tabulated i n Table 4.3. I n the n u m e r i c a l m o d e l the l o a d i n g pins are m o d e l e d w i t h an elastic material m o d e l u s i n g material properties f o r steel.  N o n - p e n e t r a t i n g contact is d e f i n e d between the pins and the  holes i n the O C T specimens. The s t i f f e n i n g brace along the back edge o f the specimen that was used i n the experiments is not i n c l u d e d i n the m o d e l . F o r all specimens the height o f elements i n the r e g i o n o f the n o t c h plane was kept constant at 0.5 m m . A s per F l o y d ( F l o y d , 2 0 0 4 ) the fracture energy is determined as the element height times the specific fracture energy. A constant element height was used so that the w i d t h w i s e discretization c o u l d be studied independently. A b r i e f note is made here o f an interesting p h e n o m e n o n .  W h e n constant stress elements  w e r e used, it was f o u n d that strain w o u l d localise to a band o f elements, a single element i n height, i n the same manner as noted b y F l o y d f o r the composite damage m o d e l i m p l e m e n t e d i n the e x p l i c i t f i n i t e element code L S - D Y N A .  I n the second-order elements used i n this  study h o w e v e r , there are nine integration points i n three r o w s o f three w i t h i n each element. I t was n o t certain whether strain w o u l d localise to an entire element (i.e. all three r o w s o f integration points) or to o n l y one or t w o r o w s w i t h i n the element. I t turned out that i n every case tested i n this thesis, strain always localised to t w o r o w s o f integration points m i d d l e r o w and one o f the edge r o w s . reason f o r this pattern.  the  Further investigation is necessary t o determine the  H o w e v e r , the height o f the localisation b a n d is therefore n o t the  entire element height, but rather a f r a c t i o n o f it. C o n s i d e r i n g the Gauss integration scheme,  74  Chapter  4  Numerical  each integration p o i n t represents a w e i g h t e d area o f the element. T h e integration p o i n t i n the m i d d l e is w e i g h t e d b y 8/18 and integration points o n the edges are w e i g h t e d b y 5/18. T h e element height therefore needs t o be m u l t i p l i e d b y 13/18 t o get the actual height o f the crack band.  4.4.2  Parametric study on G and a k c  pea  The S D M was used t o investigate the sensitivity o f the m o d e l t o the t w o i n p u t parameters c o n t r o l l i n g the fracture response.  G and a k were varied independently o n a halfc  pea  s y m m e t r y v e r s i o n o f r d 2 . Elements w i t h a w i d t h o f 0.5 m m w e r e used t o discretize the n o t c h plane region. T h e simulations used the basic material properties g i v e n i n Table 4 . 1 . I t w a s o n l y f o u n d necessary t o s l i g h t l y increase E2 t o 32 G P a ( f r o m the o r i g i n a l value o f 3 0 G P a f r o m the material supplier) t o m a t c h the elastic compliance o f the experimental tests. The load versus C M O D p l o t s h o w n i n F i g u r e 4.16 demonstrates that as G is increased, the c  magnitude o f the entire post-peak p o r t i o n o f the curve is increased. V a r y i n g Op  does n o t  eak  change the overall magnitude o f the curve b u t s l i g h t l y adjusts the peak load value that is attained. A n increase i n a k leads t o an increase i n peak load as w e l l . pea  The overall  sensitivity o f the solution t o these parameters is n o t large w i t h i n the range tested. T h e experimental results seem t o be b o u n d e d b y the 70 k J / m curve a n d the 85 k J / m . 2  2  I t was  therefore decided t o use the parameters corresponding t o the r o u g h experimental values determined i n Chapter 3 f o r the rest o f the simulations. T h a t is a strength o f 4 6 0 M P a and a r)  c r i t i c a l fracture energy o f 80 k J / m .  4.4.3  SDM/ ASDM comparison  T o test the effectiveness o f the A S D M , h a l f - s y m m e t r y simulations were conducted o f v i r t u a l specimen r d 8 u s i n g 0.5 m m and 1.0 m m w i d e elements i n the r e g i o n o f the n o t c h plane. I n b o t h cases the height o f the elements w a s 0.5 m m . T h e v a l i d a t i o n tests are listed i n Table 4.4. First, t o examine the effect o f u s i n g different sized elements, the S D M and constant stress elements were used f o r b o t h element w i d t h s w i t h the e x p e r i m e n t a l l y determined properties; G equal t o 80 k J / m c  2  and a  peak  equal t o 4 6 0 M P a . T h e l o w end o f the e x p e r i m e n t a l l y  determined G values w e r e f o u n d t o w o r k the best. A stable s o l u t i o n w a s obtained f o r b o t h c  mesh sizes and surprisingly n o difference i n response w a s observed between the coarse and  75  Chapter  4  Numerical  fine mesh cases. T h e load versus C M O D plots w e r e identical w i t h a peak load o f 16.1 k N i n b o t h cases.  F r o m the results o f the previous simulations o n D C B specimens this outcome  was n o t expected.  A s i m u l a t i o n u s i n g the same i n p u t parameters w a s p e r f o r m e d w i t h 0.5  m m w i d e second order elements. A g a i n the peak load w a s a p p r o x i m a t e l y the same, as n o t e d i n Table 4.4. T h e reason f o r this outcome can be explained s i m p l y .  R e f e r r i n g t o the elastic stress  d i s t r i b u t i o n f o r a 4 m m n o t c h radius O C T specimen i n F i g u r e 4 . 1 , the stresses near the n o t c h t i p are higher than the tensile strength to a distance o f almost 2 m m ahead o f the n o t c h t i p w h e n the load is 15 k N . T h e stress criterion f o r fracture is therefore satisfied i n a l l elements w i t h i n this d i m e n s i o n and the c o n d i t i o n f o r fracture to progress is the fracture energy equal to the critical fracture energy. Since all three meshes have the same critical fracture energies, there is v i r t u a l l y no difference i n their responses. I t appears that as l o n g as the elements are smaller ( i n w i d t h ) than the w i d t h o f c r i t i c a l l y stressed material at fracture, the mesh is adequately r e f i n e d and no scaling is necessary. T e s t i n g this theory further, the input G was changed to 10 k J / m w h i l e k e e p i n g a l l other c  parameters the same. T h e n u m e r i c a l solution became unstable at the peak load i n d i c a t i n g a transition i n b e h a v i o u r from one o f stable crack g r o w t h (plateau i n the load-displacement plots) to one o f t e m p o r a r y instability (sharp load d r o p ) . T h e peak load values still p e r m i t t e d a comparison to be made.  I n this case a difference i n peak l o a d w a s seen between the 0.5  m m w i d e a n d 1.0 m m w i d e constant stress elements. T h e difference was a p p r o x i m a t e l y 6 % w i t h peak loads o f 10.3 a n d 10.9 k N f o r the 0.5 m m a n d 1.0 m m cases respectively.  Using  quadratic elements w i t h the S D M , the peak loads w e r e closer together and between the constant stress values at 10.4 and 10.5 k N f o r 0.5 m m w i d e a n d 1.0 m m w i d e elements respectively. N e x t the A S D M was used t o m o d e l the 4 m m n o t c h root radius specimen w i t h G equal t o 10 c  k J / m . Figure 4.17 shows the r e d u c t i o n i n e* k i n the nine integration points o f the element 2  pea  situated o n the n o t c h plane i m m e d i a t e l y ahead o f the n o t c h t i p f o r a mesh o f 1.0 m m w i d e elements. N o t e that the m i d d l e c o l u m n o f integration points have a larger r e d u c t i o n i n e* k pea  because they represent a larger area o f the element. F o r b o t h 0.5 m m w i d e and 1.0 m m w i d e elements, the predicted peak load was 10.1 k N , a slight r e d u c t i o n from the S D M results b u t i t  76  Chapter  4 Numerical  is consistent, s h o w i n g that there is no difference i n u s i n g 0.5 m m or 1.0 m m w i d e elements w h e n the A S D M is used as the material m o d e l . I t turns out that i n the case o f the O C T specimens m o d e l e d i n this thesis, the t o u g h material makes scaling f o r element w i d t h v i r t u a l l y unnecessary i f elements are adequately r e f i n e d to m a t c h the stress d i s t r i b u t i o n . A n effect o f element w i d t h was o n l y n o t i c e d w h e n the critical fracture energy was reduced, m a k i n g the material m o r e brittle. E l e m e n t w i d t h also becomes m o r e i m p o r t a n t i f the n o t c h root radius is increased because the structural response becomes more "brittle".  T h e w i d t h o f c r i t i c a l l y stressed material is less i n larger n o t c h root r a d i i  specimens w h e n the energy criterion is met than i n smaller n o t c h root r a d i i specimens (see F i g u r e 4.1). F o r n o t c h root radii larger than the transition radius, the c r i t i c a l l y stressed w i d t h is close to zero so f i n i t e sized elements are always larger than the c r i t i c a l l y stressed w i d t h .  If  the onset o f fracture is c o n t r o l l e d b y the stress criterion as opposed to the energy c r i t e r i o n , element w i d t h is v e r y important.  Elements larger than the c r i t i c a l l y stressed w i d t h  material w h e n the energy c r i t e r i o n is met w i l l affect the structural response.  of  The effect is  m i t i g a t e d , h o w e v e r , b y the fact that the stress concentration becomes shallower as the n o t c h r o o t radius increases. Therefore, structures that are most sensitive to element w i d t h are ones made o f b r i t t l e material so that failure occurs due to satisfaction o f the stress c r i t e r i o n , but w i t h r e l a t i v e l y sharp notches so that there is a large stress concentration. T h i s observation is consistent w i t h literature. Guidelines f o r choosing an appropriate element size are discussed i n Chapter 5. B o t h the S D M and the A S D M have been s h o w n to w o r k w e l l f o r s i m u l a t i o n o f the O C T specimens and the A S D M is used to p e r f o r m the s i m u l a t i o n o f the experimental tests.  4.4.4  S i m u l a t i o n o f e x p e r i m e n t a l OCT  tests  The A S D M m o d e l was used t o simulate the experimental O C T tests u s i n g the seven v i r t u a l specimens listed i n Table 4.3.  A l l o f the specimens w e r e m o d e l e d i n f u l l as plane stress  p r o b l e m s w i t h u n i f o r m , 8-noded 0.5 m m x 0.5 m m elements i n the r e g i o n o f the n o t c h plane. The material and cohesive properties g i v e n i n Table 4.1 were used f o r all o f the specimens. F o r an element height o f 0.5 m m and localisation i n t w o r o w s o f integration points w i t h i n an element, the y corresponding t o a G o f 80 k J / m is 2.215 x 10 k J / m . e  c  77  Chapter  4  Numerical  I n i t i a l l y the simulations w e r e conducted u s i n g h a l f models o f the specimens w i t h s y m m e t r i c b o u n d a r y conditions applied a l o n g the plane o f the n o t c h . T h i s was f o u n d to be an adequate representation f o r most o f the specimens, b u t at large n o t c h r a d i i the crack w o u l d sometimes g r o w o f f o f the centre line causing small errors i n the peak load determined.  Simulations  w e r e next conducted on a f u l l mesh b y r e f l e c t i n g the mesh f o r the h a l f - s y m m e t r y m o d e l s about the n o t c h plane.  T h i s resulted i n more consistent predictions w i t h the crack either  propagating t h r o u g h the r o w o f elements i m m e d i a t e l y above the n o t c h plane or i m m e d i a t e l y b e l o w the n o t c h plane.  F o r v i r t u a l specimen rd2 h o w e v e r , the h i g h strain concentration i n  the r e l a t i v e l y large 0.5 m m h i g h elements at the n o t c h t i p caused localisation to occur simultaneously i n b o t h elements o n either side o f the n o t c h plane before propagating a l o n g o n l y one r o w .  T h i s caused an error i n the predicted peak load.  A l l o f the meshes were  therefore changed so that there was a r o w o f elements centred along the-notch plane ahead o f the n o t c h t i p . T h i s c o n f i g u r a t i o n resulted i n consistent predictions from all specimens. A couple o f notes to keep i n m i n d w h e n m e s h i n g w i t h a strain-softening material m o d e l : •  H a l f - s y m m e t r y should not be used so that the damage zone is free to propagate a l o n g the m i d - p l a n e or to branch to either side depending on the conditions  •  I f a r e l a t i v e l y sharp n o t c h is b e i n g m o d e l e d , the n o t c h t i p should n o t be located at a b o u n d a r y node ( m i d - s i d e nodes are f i n e ) since softening can then occur i n b o t h elements b o u n d i n g the n o t c h t i p resulting i n an energy release rate that is too h i g h  4.4.4.1  Load-CMOD behaviour  The load versus C M O D predictions f o r each o f the v i r t u a l specimens are p l o t t e d over the experimental results i n the plots i n A p p e n d i x A .  T h e peak loads predicted b y each v i r t u a l  test are p l o t t e d versus n o t c h root radius i n Figure 4.18. The n u m e r i c a l predictions m a t c h the experimental results r e m a r k a b l y w e l l at almost all n o t c h r a d i i . W h i l e the material parameters were determined f r o m the same set o f specimens as the tests that were m o d e l e d , the m o d e l was not calibrated b y f i t t i n g the load-displacement p l o t o f a s i m u l a t i o n at one n o t c h root radius to experiment and then u s i n g the resulting i n p u t parameters as is often the case i n the literature. C o m p a r i n g the n u m e r i c a l l y predicted peak loads to the experimental results, some trends are apparent. First, the sharpest n o t c h t i p specimens ( f o r p equal to 0.5 to 2 m m ) have the lowest  78  Chapter  4  Numerical  peak loads, increasing f r o m about 14.5 k N to 15.5 k N at p equal to 3 m m .  Between notch  root r a d i i o f 3 m m and 14.15 m m there is v e r y little increase i n peak load, w h i c h is matched v e r y closely b y the n u m e r i c a l predictions.  A t a p p r o x i m a t e l y p equal to 16 m m , there is  clearly a transition p o i n t at w h i c h peak loads b e g i n t o increase w i t h increasing p. w i l l be referred to as position-  This point  A g a i n , the n u m e r i c a l predictions also predict this transition.  There is a d e v i a t i o n i n the predicted peak load f r o m the experimental results at the largest n o t c h root r a d i i . F o r p equal to 27 m m , the predicted peak load is a p p r o x i m a t e l y 19.5 k N c o n t i n u i n g the t r e n d o f increasing peak load w i t h p, w h i l e the experimental result levels o f f at a p p r o x i m a t e l y 17.8 k N . N o t e that there are t w o experimental points f o r p equal to 27 m m located together. I n v e s t i g a t i n g the i n d i v i d u a l load versus C M O D plots m o r e closely, it can be seen that the n u m e r i c a l l y predicted shape o f the curves is similar to the experimental results.  A t small  n o t c h root r a d i i , the material begins to soften early, causing a w i d e , r o u n d e d peak o n the load versus C M O D plot.  T h i s rounded peak i n the n u m e r i c a l specimens represents the j a g g e d  plateau seen e x p e r i m e n t a l l y i n the small p specimens.  See f o r example Figure A . l  and  F i g u r e A . 2 . A s p increases, softening o f the material is delayed to higher loads, creating a m o r e linear response up to the peak load. pronounced.  A t large p the peak o f the curve is m o r e  F o r specimens rd21 and r d 2 8 , the peak is s l i g h t l y m o r e p r o n o u n c e d i n the  n u m e r i c a l specimens than seen experimentally.  Especially f o r rd21 (Figure A 3 ) , the  experimental specimen has a m o r e noticeable plateau w i t h a s l i g h t l y l o w e r peak l o a d , perhaps i n d i c a t i n g that the n o t c h t i p o f specimen r d 2 1 - 1 contained a small f l a w , causing it to damage prematurely. I n specimen r d 3 2 , the sharp load drop, caused b y a sudden j u m p i n the crack length after the peak l o a d , is w e l l represented b y a steep drop i n the n u m e r i c a l curve. For the t w o largest notch root r a d i i , 38 m m and 54 m m , s h o w n i n Figure A . 6 and F i g u r e A . 7 respectively, the experiments showed unstable crack g r o w t h f o l l o w i n g the peak load w i t h v e r y large load drops before the crack was arrested. The n u m e r i c a l simulations were unable t o handle this instability and the runs crashed at or j u s t after the peak loads were reached. I n the i m p l i c i t integration scheme used, the solution is stable as l o n g as the global stiffness m a t r i x is positive definite. Therefore the n u m e r i c a l m o d e l can handle localised softening but crashes i f there is global i n s t a b i l i t y as w e see at large p.  79  Chapter  4  4.4.4.2  Numerical  Stress fields  The stress field contours, f o r stress n o r m a l t o the crack, f o r the A S D M s i m u l a t i o n o f rd2 are s h o w n i n Figure 4.19. A s C M O D is increased a) a stress concentration appears at the t i p o f the n o t c h . B y the t i m e the peak load is reached b) the elements at the n o t c h t i p have begun softening and the stress concentration has detached f r o m the i n i t i a l n o t c h t i p . A considerable amount o f damage is predicted to occur before the peak load is reached. A t c) the C M O D is 3.21 m m and the damage zone has progressed r o u g h l y 3 0 m m . I n F i g u r e 4.19 b) and c) a " t a i l " o f h i g h stress b e h i n d the stress concentration is apparent. cohesive zone constitutive curve.  T h i s t a i l is due t o t h e  The stresses a l o n g the tail represent p a r t i a l l y damaged  material on the softening p o r t i o n o f the cohesive zone stress-strain curve. F o r c o m p a r i s o n , the predicted stress contours f o r r d 5 4 are p l o t t e d i n Figure 4.20. A t a) the C M O D o f 0.67 m m is a p p r o x i m a t e l y the same as i n F i g u r e 4.19 a) f o r r d 2 .  While rd2  already shows n o r m a l stresses at the n o t c h t i p equal t o the material strength, the n o t c h t i p stresses f o r r d 5 4 are w e l l b e l o w critical. E v e n at a C M O D o f 1.70 m m as s h o w n i n F i g u r e 4.19 b) and F i g u r e 4.20 b) f o r rd2 and r d 5 4 respectively, the stresses are b e l o w critical f o r r d 5 4 w h i l e rd2 is at its peak load. S h o r t l y after, r d 5 4 reaches its peak load at a C M O D equal to 2.48 m m , s h o w n i n F i g u r e 4.20 c). The stresses at the n o t c h t i p have j u s t reached the material tensile strength and the stress concentration has not yet detached from the i n i t i a l n o t c h tip.  The s i m u l a t i o n crashes at this point.  A s hypothesized f r o m the experimental  results, the specimen goes unstable as soon as the n o t c h t i p stresses reach the tensile strength. A l s o , as seen i n the elastic s i m u l a t i o n results, the height o f h i g h l y stressed material j u s t p r i o r to peak load is m u c h greater i n the larger n o t c h root radius specimen. A band a p p r o x i m a t e l y 20 m m h i g h is stressed at close t o the critical level w h e n v i r t u a l specimen r d 5 4 becomes unstable.  Conversely, i n rd2 the height o f material stressed at close t o the tensile strength  increases from o n l y a couple o f m i l l i m e t r e s at the n o t c h t i p to several m i l l i m e t r e s at the p o i n t o f peak load. I t appears that the height o f the c r i t i c a l stress zone either g r o w s t o the stable propagation height f o r sharp notches (stable b e h a v i o u r ) or decreases to the stable propagation height f o r b l u n t notches (unstable b e h a v i o u r ) . F o r t h e n o t c h root radius at the transition  from  stable t o unstable behaviour, the stress  contours should r e m a i n r o u g h l y the same size as the crack propagates. The stress contours at three C M O D values are p l o t t e d f o r specimen rd32 (16 m m n o t c h root radius) i n Figure 4 . 2 1 .  80  Chapter  4  Numerical  T h e magnitude o f the stress contours does i n fact reduce as the crack propagates. T h i s m a y be due t o there b e i n g a large cut-out o f material at the i n i t i a l n o t c h t i p that alters the gross stress d i s t r i b u t i o n a w a y from the n o t c h plane.  A s t h e crack t i p progresses a w a y from t h e  i n i t i a l n o t c h t i p , the effect o f the cut-out is d i m i n i s h e d . 4.4.4.3  Crack width  X Y - p l o t s o f the stress distributions i n the d a m a g i n g b a n d o f elements ( a l o n g the n o t c h plane) are u s e f u l f o r l o o k i n g at the propagation o f the crack front.  Stress d i s t r i b u t i o n p r o p a g a t i o n  plots f o r r d 2 a n d r d 2 8 are presented i n F i g u r e 4 . 2 2 a n d F i g u r e 4.23 respectively f o r increasing values o f C M O D . I t w a s n o t possible t o create one o f these plots f o r r d 5 4 because the r u n crashed after crack i n i t i a t i o n .  Plots f o r r d 8 , r d 2 1 and r d 3 2 are i n A p p e n d i x C . T h e  p o s i t i o n o f the crack front is determined from t h e location o f the peak stress f o r a g i v e n C M O D or time. or p a r t i a l l y .  T h i s p o s i t i o n marks the w i d t h o f material that has damaged, w h e t h e r f u l l y  T h e p o i n t at w h i c h the t r a i l i n g edge o f the stress curve goes t o zero m a r k s the  w i d t h o f f u l l y cracked material (interface w i t h zero residual strength).  T h e w i d t h o f the  damage zone i n t h e n u m e r i c a l m o d e l is t h e distance between these t w o points Wd ,eam  As  discussed i n Section 4.3.4, this damage w i d t h is n o t the same as the p h y s i c a l w i d t h o f the damage zone because its value is strongly dependent o n the shape o f the stress-strain curve and somewhat dependent o n mesh size. T h e shape o f the stress-strain curve determines the value o f s i f o r g i v e n y and e k inputs. u t  pea  T h e bilinear stress-strain relationship assumed n u m e r i c a l l y is o n l y an a p p r o x i m a t i o n o f the p h y s i c a l stress-strain relationship.  U n t i l this p o i n t , i t has been sufficient f o r p r o d u c i n g  accurate load-displacement predictions because the t w o factors that influence the predictions are G a n d a k, w h i c h are accurately captured b y the b i l i n e a r m o d e l . H o w e v e r , the w i d t h o f c  pea  the damage zone depends o n e i , w h i c h is n o t necessarily accurate i n the b i l i n e a r m o d e l . u t  I f i t is assumed, f o r t h e m o m e n t , that the actual constitutive b e h a v i o u r o f a characteristic v o l u m e o f the material is bilinear, as i n the n u m e r i c a l m o d e l , then the characteristic damage zone w i d t h o f the material Wd ,c can be determined from the characteristic s i and the strain am  u t  f i e l d as w a s s h o w n i n F i g u r e 4.13. A p r e d i c t i o n f o r Wd ,c is made u s i n g the ratio o f element height (localisation band) t o the am  characteristic damage height.  U s i n g E q u a t i o n 2.14 w i t h the e x p e r i m e n t a l l y  determined  81  Chapter  4  Numerical  characteristic damage height o f 6.5 m m and a G o f 80 m J / m m , y is a p p r o x i m a t e l y equal to 2  c  c  12.3 m J / m m . The elemental specific strain energy y is 221.54 m J / m m f o r the 0.5 m m h i g h 3  3  e  quadratic elements. U s i n g E q u a t i o n 4.2, the u l t i m a t e strain at failure i n the n u m e r i c a l m o d e l is then determined t o be 0.963.  T h e strain p r o f i l e f o r rd28 at C M O D equal t o 3.9 m m is  s h o w n i n F i g u r e 4.24. F o r a strength o f 460 M P a and i n i t i a l m o d u l u s o f 32 GPa, the location o f the peak and u l t i m a t e strains are also plotted. The n u m e r i c a l w i d t h o f the damage zone Wdame  is read as a p p r o x i m a t e l y 18 m m . The characteristic w i d t h o f the damage zone is m u c h  smaller.  I f a b i l i n e a r constitutive curve is used as the actual stress-strain response o f the  material, an u l t i m a t e strain o f 0.053 is obtained using E q u a t i o n 4.2. 4.24, this corresponds t o a  Wd ,c am  o f approximately 1 m m .  A s shown i n Figure  F r o m the specimens that w e r e  sectioned, this value seems t o o small. Sectioning suggests a damage zone w i d t h o f about 6 mm.  Therefore, the physical stress-strain softening b e h a v i o u r o f the material m u s t n o t be  linear.  4.4.4.4  Compliance plots  U s i n g the peak stress location t o determine the w i d t h o f cracked material, c o m p l i a n c e plots are generated f o r each specimen.  T h e n u m e r i c a l c o m p l i a n c e , as d e t e r m i n e d u s i n g the  A S D M , is plotted i n Figure 4.25 f o r rd28. The locations o f the peak stresses i n F i g u r e 4.22 and F i g u r e 4.23 g i v e the w i d t h o f material that contains any amount o f damage f o r a g i v e n CMOD.  The w i d t h o f material that is f u l l y cracked is less. T h i s can be seen i n F i g u r e 4.25  b y c o m p a r i n g the A S D M compliance p r e d i c t i o n t o the c o m p l i a n c e p l o t as determined from a series o f static, elastic simulations i n w h i c h the interface constraints were  successively  r e m o v e d . E x p e r i m e n t a l points f r o m the line analysis determination o f crack length are also i n c l u d e d on the plot. B y s h i f t i n g the A S D M c o m p l i a n c e p r e d i c t i o n b y 3.25 m m a l o n g the C M O D axis, i t is superposed o n the elastic p r e d i c t i o n , as s h o w n i n F i g u r e 4.26.  S i m i l a r t o w h a t was seen  e x p e r i m e n t a l l y w h e n the length o f damaged material f r o m sectioning was compared t o the crack length determined b y line, analysis, the length o f the crack i n c l u d i n g partial damage is a p p r o x i m a t e l y equivalent t o a shorter, b u t f u l l y separated crack.  Here, s h i f t i n g the crack  w i d t h b y 3.25 m m places the t i p o f the equivalent discrete crack i n the centre o f the a p p r o x i m a t e l y 6 m m w i d e damage zone that was determined f r o m sectioning.  82  Chapter 4.4.4.5  4  Numerical Crack  velocity  The static, i m p l i c i t m o d e l used does n o t a l l o w quantitative predictions o f crack v e l o c i t y to be made because inertia is n o t i n c l u d e d i n the solution. front  B y p l o t t i n g the p o s i t i o n o f the crack  versus a n o r m a l i s e d t e m p o r a l parameter, h o w e v e r , the sharp j u m p i n crack p o s i t i o n at  large n o t c h r a d i i can be seen compared to smaller n o t c h r a d i i . I n F i g u r e 4.27 the p o s i t i o n o f the crack front is plotted versus n o r m a l i s e d s i m u l a t i o n t i m e t/to. s i m u l a t i o n is to.  V i r t u a l specimen rd2 begins to  length w i t h t i m e is gradual.  fracture  The total duration o f the  early and the increase i n crack  A s n o t c h root radius is increased, the i n i t i a t i o n o f fracture is  delayed. A t p equal to 16 m m ( r d 3 2 ) , fracture initiates at a n o r m a l i s e d t i m e o f about 0.4 and the p o s i t i o n o f the crack front j u m p s f r o m 0 m m to 20 m m r a p i d l y .  B y about a n o r m a l i s e d  t i m e o f 0.6, the crack fronts f o r all o f the n o t c h r a d i i are almost coincident.  4.5  Summary •  Static, elastic simulations o f the O C T specimens help e x p l a i n the transition  in  b e h a v i o u r seen e x p e r i m e n t a l l y . F o r r e l a t i v e l y sharp notches, the n o r m a l stress at the n o t c h t i p attains the tensile material strength p r i o r to the onset o f structural softening. O n the other hand, the elastic stress f i e l d solutions reveal that the n o r m a l stress at the n o t c h t i p is less than critical f o r large r a d i i w h e n the peak load is attained f o r a sharp notch. •  A three-parameter cohesive m o d e l called the S D M was developed as a U M A T  in  A B A Q U S . The S D M was m o d i f i e d to a l l o w it to a u t o m a t i c a l l y adapt its constitutive b e h a v i o u r based on element size and the strain d i s t r i b u t i o n to account f o r the effect o f element w i d t h o n peak load p r e d i c t i o n . The m o d e l is called the A S D M . •  U s i n g the e x p e r i m e n t a l l y determined G  c  and a k, pea  the A S D M d i d a g o o d j o b  of  p r e d i c t i n g the experimental O C T results and also displayed a transition i n b e h a v i o u r at a n o t c h root radius o f about 16 m m . R a p i d crack g r o w t h i n v i r t u a l specimens w i t h n o t c h root r a d i i larger than the transition radius was unstable and the simulations stopped prematurely. •  Scaling to account f o r element w i d t h was f o u n d to be unnecessary f o r t o u g h materials as l o n g as the element w i d t h was less than the w i d t h o f c r i t i c a l l y stressed material at  83  Chapter  4  Numerical  the onset o f softening.  E l e m e n t w i d t h becomes increasingly i m p o r t a n t f o r b r i t t l e  materials and sharp n o t c h tips. •  The p h y s i c a l damage zone w i d t h cannot be predicted u s i n g the S D M  without  assuming a shape f o r the actual physical stress-strain relation. C r a c k length h o w e v e r can be r e l i a b l y predicted u s i n g the S D M b y p l o t t i n g the compliance o f the v i r t u a l specimen. •  Simulations o f the O C T specimens u s i n g the A S D M reveal that considerable damage and localised softening takes place ahead o f r e l a t i v e l y sharp n o t c h tips before discrete crack f o r m a t i o n and the onset o f structural softening. F o r specimens w i t h n o t c h root r a d i i larger than the transition radius h o w e v e r , v e r y little damage occurs p r i o r to discrete crack f o r m a t i o n and the onset o f unstable crack advance.  84  Chapter  4  Numerical  Table 4.1 Material properties for numerical OCT tests Property  Value  E  75 G P a  Ey  32 G P a  O  0.161  G  17.1 G P a  x  xy  xy  460 M P a  @peak  80 k J / m  G  F  2  Table 4.2 Height of critically stressed material at peak load Notch root radius . (mm)  Approximate peak load ( k N ) *  Height of critically stressed material (mm)  1  15  3.5  4  15  4.75  10.5  15  6  16  16.9  9.5  27  17.8  8.5  T a k e n f r o m O C T experiment results  Chapter  4  Numerical  T a b l e 4.3 N u m e r i c a l tests Virtual Specimen rd2  a  W  (mm)  (mm)  33.0  81.5  M e s h Size/  Tests  Configurations H a l f model:  1)  Parametric study o f G  2)  Gpeak Experiment simulation  0.5 m m quads Full model:  c  and  0.5, 1.0 m m quads rd8  33.0  81.8  H a l f model:  1)  S D M / A S D M comparison  0.5, 1.0 cs  2)  Experiment simulation  0.5, 1.0 quads Full model: 0.5 m m quads rd21  34.0  81.2  Full model: 0.5 m m quads  Experiment simulation  rd28  33.8  80.0  Full model:  Experiment simulation  0.5 m m quads rd32  34.0  81.5  Full model:  Experiment simulation  0.5 m m quads rd38  33.0  81.0  Full model:  rd54  33.0  81.2  Full model:  Experiment simulation  0.5 m m quads Experiment simulation  0.5 m m quads  86  Chapter  4  Numerical  Table 4 . 4 ASDM validation tests Sim#  Model  Gc (kJ/m2)  Element type  Element width (mm)  Peak Load (kN)  1  SDM  80  Const, stress  1.0  16,1  2  SDM  80  Const, stress  0.5  16.1  3  SDM  80  Quadratic  0.5  16.2  4  SDM  10  Const, stress  1.0  10.9  5  SDM  10  Const, stress  0.5  10.3  6  SDM  10  Quadratic  1.0  10.6  7  SDM  10  Quadratic  0.5  10.4  8  ASDM  10  Quadratic  1.0  10.1  9  ASDM  10  Quadratic  0.5  10.1  •  87  Chapter  4  Numerical  1400 1200  Notch plane  6  8  10  12  14  16  20  Distance along notch plane (mm) Figure  4.1 Elastic  stress fields ahead of notch tip at P — 15.0 kN  600 -, 550 500 -|  _Tensile strength  ^.450 400  g w  350  Vi  2 300 "  I  •3 1 I 200 Z  2 5 0  150 \ 100 50  J  0 6  0  8  10  12  14  16  20  Distance along notch plane (mm) Figure  4.2 Elastic  stress  fieldfor  p — 27 mm at experimentally kN  determined  peak load of 17.8  88  Chapter 4 Numerical  a) p = 1 mm  c) p= 10.5 mm  b) p =  4 mm  d)p= 16 mm  '22 +4. + 3. +3. + 2. +1. +7. -1.  (MPa) S00e+02 833e+0Z 067e+02 300e+0Z S33e+02 667e+01 5ZSe-0S 667e+01 S33e+02 300e+02 067e+02 833e+02 600e+02 988e+02  e)p = 27 mm Figure 4.3  Stress contours ahead of notch tip (symmetric about the notch plane) for a) p = 1 mm at 15 kN load, b) p = 4 mm at 15 kN load, c) p = 10.5 mm at 15 kN load, d) p = 16 m at 16.9 kN load and e) p = 27 mm at 17.8 kN load. Elements at the notch tip are 0.5 mm x mm in all cases. The light grey regions indicate a stress of460 MPa.  89  Chapter 4 Numerical  0 4 0  , 1  , 2  , 3  H  :  4  5  ,  , 6  H 7  , 8  , 9  , 10  Distance along notch plane (mm)  Figure 4.4 Stress concentrations on the notch plane ahead of a 4 mm radius notch tip in OCT specimen as resolved by different sized constant stress (cs) andfully integrated quadratic (quad) elements. Stresses are extrapolated to the nodes.  III  i 1  Figure 4.5 Effect of element width on stress calculated at a notch tip  90  Chapter 4 Numerical  0.0  0.2  0.4  0.6  0.8 8 (mm)  1.0  1.2  1.4  1.6  Figure 4.6 Load versus arm displacement for DCB simulations showing the effect of element width on the peak load prediction 35 n 30 25  0 •  •a 20  es o -J it « 0-  10  0  4  -I 0  ,  0.1  ,  0.2  ^  Without strength scaling  •  With strength scaling  ,  0.3  ,  0.4  ,  0.5  ,  0.6  Element w i d t h (mm)  Figure 4.7 Peak loads for DCB simulation using element meshes of varying coarseness with and without scaling the material strength to match the 62.5 pm mesh size  91  Chapter  4  Numerical  60 55  62.5 /um element (100% strength)  50 45  H {--\  125 /im element (84.5% strength)  40 250 jum element (64.2 % strength)  ^ 35 o-  S  30  b 25  500 /um element (42.2 % strength)  20 15 62.5 /um solution  10 5 0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.1  0.9  Distance ahead of initial crack tip (mm) Figure  4.8 Strength  scaling  ratios to match the peak load in the DCB simulation  mesh with 62.5 pm constant  using a  stress elements at the crack tip  Yc=Yr+ Yd  ^Opeak Figure  S  £peak 4.9 Simple bilinear  damage  model  92  Chapter  4  Numerical Read user input: ^01.  £<)2, G  ]  ,  2  V12,  ? c l , 7c2> ^peakb ^peak2  Initialize (for each direction):  Calculate initial s : M  £peak  —  ^Opeak  2r  E =E r  c  0  Yr=Yz  ^0 0 £  g  e t  s t a t e  v a r  i  a D  |  peak  e s  Determine the strain distribution in the element (initially all strains are zeroed)  Determine the maximum strain, £'ci„, v, across the element a  Yes  (  £*peak - min(£bp  No  OTfo  £  0  p  e  a  £  >  k  )  V ^elmax )  Update the current peak strain: s = max(£, mak  Calculate the current stress based on the current strain and the constitutive curve  Update the current peak strain: ^oeak  £  peak  Determine the tangent modulus necessary to arrive at current strain Assemble the tangent stiffness tensor Calculate amount of strain energy dissipated  Update parameters:  Yr=n- Yi  Loop over integration points  E, = m i n ( £  -—  n  Update history  variables: e  ,y E  peak  n  r  Store integration point strains in global matrix for determining element strain gradient  Figure  4.10 Flow  chart for  SDM/ASDM  algorithm  93  Chapter  4  Numerical  Figure  4.11 Integration  point  numbering  ^ — - ^ S c a l e d curve  £ peak  Sopeak Figure  4.12 Scaled stress-strain  curve  94  Chapter  4  Numerical  ^ult \-  am Speak am  x (mm)  Separated Figure  Figure  Damaged  4.13 Damage  4.14 Example  Elastic  zone size determined  of energy release determined  by the cohesive zone  model  using elements of different  size  95  Chapter 4 Numerical  a)  h)  Figure 4.15 Virtual OCT specimen mesh for a) full specimen and b) at notch plane  0  0.5  1  1.5  2  2.5  3.5  3  4  4  CMOD (mm) Figure 4.16 Parametric study of G and a kfor rd2 c  pea  96  Chapter 4 Numerical 0.016  0.015 ]  0.25  Figure 4.17 Reduction in £* k at the integration points of the notch tip element in rd8 pea  20  -i o  19 18  • O •  i  17  Z  16  •  •a « o 15  ct  • #  •  •  •  B  14  PH  13 12 , J  11 10 0  15  20  25  30  p (mm)  Figure 4.18 Peak loads predicted using the ASDM (large diamonds) plotted over the experimental results (small squares). Open symbols represent cases that were unstable numerical run crashed or did not display load plateau experimentally). Note that the ordinate scale begins at 10 kN to magnify the results. 97  Chapter  Figure  4  Numerical  4.19 Normal CMOD=  stress contours for rd2 a) at CMOD=0.679 1.70 mm (peak load) and c) at CMOD=3.21  mm (before peak load), b) at mm (after peak  load)  98  Chapter  Figure  4  Numerical  4.20 Normal  stress contours for rd54 a) at CMOD=0.674 mm and c) at CMOD=2.48  mm (peak  mm, b) at  CM0D=1.70  load)  99  Chapter  Figure  4  Numerical  4.21 Normal at CMOD=2.11  stress contours for rd32 a) at CMOD=0.674 mm (peak load) and c) at CMOD=3.21  mm (before peak load), mm (after peak  b)  load)  100  Chapter  4  Numerical  500 400 300 200  "3  E io  *  100 0-| 100 -200 20  15  25  30  35  40  x (mm) Figure  4.22 Progression distribution  of the stress distribution is at the CMOD  along the notch plane for rd2.  Each  value (in mm) noted above it.  3.90 |  4  43^  4  6 9  4.95  u CO  E u o Z  -200 15  20  25  30  35  40  45  x (mm) Figure  4.23 Progression distribution  of the stress distribution is at the CMOD  along the notch plane for rd28.  Each  value (in mm) noted above it.  101  Chapter  4  Numerical  x (mm) Figure  4.24 Damage  zone width determination  damage zone is.32.25  for rd28 (Note that the crack including  mm wide at this point and the CMOD  10  15  20  25  the  is 3.9 mm)  30  35  40  A a (mm) Figure  4.25 Compliance  for rd28 before shifting  ASDM  prediction  102  Chapter  4  Numerical  0.0018 -,  0.0000 -!  1  ,  ,  1  1  ,  1  ,  0  5  10  15  20  25  30  35  40  A a (mm) Figure  4.26 Compliance  for rd28 after shifting ASDM prediction partially damaged zone.  by 3.25 mm to account  for  45 -| 40 35 30 -  E 25 B, « 20 15 10 5 0 • 0  Figure  0.1  0.2  4.27 Progression  0.3  0.4  0.5 t/t„  0.6  0.7  of the crack front in selected virtual  0.8  OCT  0.9  1  specimens  103  Chapter  5 Discussion  and Analysis  of Results  C H A P T E R 5 DISCUSSION AND ANALYSIS OF RESULTS  5.1  Introduction  The experimental and n u m e r i c a l results presented i n Chapter 3 and Chapter 4 p r o v i d e evidence o f a transition i n the load versus displacement b e h a v i o u r o f a notched structure w i t h increasing n o t c h root radius. A l t h o u g h investigating notch sensitivity i n composites ( o r any material) is n o t unique, the tests done here have isolated the n o t c h r o o t radius f r o m the n o t c h length and section area i n a stable test specimen, w h i c h has p r o v i d e d a clear picture o f the mechanisms causing fracture g r o w t h . transition i n behaviour  A c o m p l i c a t e d m o d e l is n o t needed; i n fact the  can be explained  simply  using  fracture  mechanics  equations.  T r a d i t i o n a l l y , the onset o f fracture at a n o t c h t i p is determined u s i n g either the energy c r i t e r i o n f o r fracture or the stress criterion as described i n Chapter 2. W h i l e b o t h criteria must be satisfied f o r fracture t o occur, it is usually assumed that b o t h criteria are satisfied simultaneously  (Broek,  1982).  Under  certain  conditions  they  are  not  satisfied  simultaneously, leading t o t w o regimes o f fracture; energy d r i v e n fracture a n d stress d r i v e n fracture.  5.2  Analytical description of notch root radius sensitivity  5.2.1  Regime I  E n e r g y d r i v e n fracture w i l l be called r e g i m e I. I n this r e g i m e the stress c r i t e r i o n f o r fracture is satisfied (i.e. a > a ) before the energy criterion is satisfied (i.e. G > R). c  This typically  occurs f o r r e l a t i v e l y sharp notches, where h i g h stress concentrations are present.  Since the  stress at the t i p has already reached the material strength, the energy i n the system controls the onset and propagation o f fracture. U n d e r displacement c o n t r o l , fracture i n this r e g i m e is stable because fracture initiates as soon as it is energetically favourable. i n the system never exceeds the crack resistance energy.  T h e stored energy  Specimens r d l t h r o u g h rd28 f a l l  into this category. The schematic i n F i g u r e 5.1 helps e x p l a i n the behaviour o f structures i n regime I. T h e stress concentration at a n o t c h t i p increases w i t h applied load.  Since the n o t c h t i p is r e l a t i v e l y  sharp, stresses become critical i n a small r e g i o n at the n o t c h t i p at load P' before there is  104  Chapter 5 Discussion and Analysis of Results sufficient energy i n the g l o b a l system t o cause discrete crack g r o w t h . The h i g h local energy causes damage (or p l a s t i c i t y i n metals), b l u n t i n g the n o t c h tip. A s the n o t c h t i p blunts, the n o t c h t i p stress remains constant w i t h increasing load. W h e n G reaches G at P , the stress !  c  c  c r i t e r i o n and the energy criterion are simultaneously satisfied and crack g r o w t h initiates. W i t h the simultaneous satisfaction o f the stress and energy criteria, E q u a t i o n 2.8, repeated here as E q u a t i o n 5.1 u s i n g the orthotropic m o d u l u s , can be used t o determine the peak loads i n the O C T specimens analytically.  K) = E G,  5.1  F o r a k n o w n G , E q u a t i o n 5.1 is solved f o r Kj . T h e n a relationship between load and Kj can c  c  be used to determine the peak load f o r different geometries. A relationship f o r C T specimens is g i v e n b y ( S r a w l e y and Gross, 1972) u s i n g a coefficient F2 t o a l l o w f o r v a r y i n g specimen dimensions as i n E q u a t i o n 5.2.  5.2  K, = a JW-a-F (a/W) N  2  H e r e , <JN is the n o m i n a l stress at the n o t c h t i p , c o m p r i s i n g stress due t o tension and stress due to b e n d i n g as g i v e n i n E q u a t i o n 5.3 (Tada, Paris and I r w i n , 1985).  °N  =  N + °N Tension Bending A  f  6P a + c r  a  N  = ^ —  W-a  +  V  W-a^ t  5.3  J  (W-af  _ 2P(2W + a) ~  N  [W-af  The values o f F2 p r o v i d e d i n the reference f o r E q u a t i o n 5.2 are f o r isotropic materials w i t h standard specimen dimensions. T o determine an F2 value f o r the orthotropic O C T specimens tested i n this thesis, a n u m e r i c a l s i m u l a t i o n o f a sharp-notched O C T specimen was conducted in A B A Q U S  u s i n g an o r t h o t r o p i c  elastic material m o d e l  w i t h plane stress  quadratic  elements. T r i a n g u l a r elements (approx. 50pm) were used at the n o t c h t i p w i t h the m i d - s i d e nodes m o v e d t o the quarter-point o f the sides radiating f r o m the tip.  Such a c o n f i g u r a t i o n  1/2  creates an r"  singularity i n the stress and strain fields o f the element t o m a t c h the expected  stress and strain field at the tip o f a sharp n o t c h ( C o o k , M a l k u s and Plesha, 1989). The stress 105  Chapter field  5 Discussion  and Analysis  of Results  is displayed i n Figure 5.3. T h e mode I stress intensity factor Kj is determined u s i n g a  v i r t u a l crack closure technique ( V C C T ) f r o m the displacement o f nodes on the quarter-point elements at 9= ±n.  The equation f o r Ki is g i v e n b y ( C o o k , M a l k u s and Plesha, 1989) as  K,  =  2G K  f TC - V  + l <2lj  / 2  [(  - c )-(4v  4 v  v  2  B 2  s l  -v  c l  )].  5.4  where G is the shear m o d u l u s , i, B and C are g i v e n i n F i g u r e 5.4 and 3-v  5.5  K = -  1+ v  f o r plane stress. F o r a g i v e n l o a d , the value o f K/ f r o m E q u a t i o n 5.4 is substituted into E q u a t i o n 5.2 to determine F . 2  F o r the material and specimen geometry studied here a value o f 0.782 was  determined. C o m b i n i n g Equations 5 . 1 , 5.2 and 5.3 and rearranging, a solution f o r peak load f o r the O C T specimens i n m o d e I l o a d i n g is g i v e n as  _  pl  c  {W-a) 4WG-  ^  2  c  2(2W + a)slW -a-F (a/W) 2  The superscript / indicates r e g i m e I.  '  Since p does n o t enter into the equation, E q u a t i o n 5.6  predicts that the peak load is constant over a range o f n o t c h root r a d i i . The n o t c h r a d i i o f all specimens i n r e g i m e I are small enough that the stress at the n o t c h t i p is above the material strength before the energy criterion is satisfied. Therefore, the solution f o r a sharp n o t c h t i p should be equivalent to the solution f o r a blunter n o t c h t i p that is still smaller than the transition radius. 5.2.2  Regime I I  The second r e g i m e is the stress d r i v e n r e g i m e , w h i c h , f o r a g i v e n material system, comprises specimens w i t h n o t c h root r a d i i larger than those i n r e g i m e I (e.g. r d 3 2 t h r o u g h r d 5 4 ) .  For  v e r y b l u n t notches or unnotched specimens, the stress concentration is slight or non-existent. Therefore, it is possible to store excess strain energy i n a specimen before the stress at any p o i n t reaches the tensile material strength. Fracture initiates w h e n stress eventually reaches  106  Chapter  5 Discussion  the strength.  and Analysis  of Results  A s w i t h r e g i m e I, the stress and energy criteria are s h o w n schematically f o r  regime I I i n F i g u r e 5.2. D u e t o the b l u n t n o t c h t i p the stress concentration is less i n this r e g i m e and the n o t c h t i p stresses therefore increase less r a p i d l y w i t h load. T h e energy i n the system reaches the critical level f o r crack propagation at P"  before the stresses reach the  critical level. D a m a g e does n o t initiate at this point. T h e energy i n the system continues t o increase u n t i l F ^ w h e n the stress criterion is also satisfied. A discrete crack initiates at this 7  point. The release o f the excess energy w h e n c r a c k i n g initiates drives the crack f o r w a r d i n an unstable manner until the c o m b i n a t i o n o f a and P brings the energy release rate t o less than or equal t o the crack resistance. A t this point, the effective n o t c h root radius o f the crack has sharpened f r o m its i n i t i a l size to correspond to the height o f the m a t e r i a l ' s characteristic damage height.  A f t e r a crack j u m p o f several m i l l i m e t r e s i n the b l u n t notched O C T  specimens, the large drop i n load caused c r a c k i n g to arrest. Stable crack g r o w t h then ensued at the characteristic damage height. A n estimate o f the peak load i n regime I I f o r v a r y i n g n o t c h root r a d i i i n the O C T specimens is d e r i v e d as f o l l o w s . The n o r m a l stress d i s t r i b u t i o n equation o f (Creager and Paris, 1967) f o r b l u n t n o t c h tips is g i v e n i n E q u a t i o n 2.11 i n terms o f the sharp n o t c h stress intensity factor and repeated here as E q u a t i o n 5.7.  ^2nr' 2r  ^2nr'  The effective distance f r o m the n o t c h t i p is g i v e n b y  r'=r  5.8  +-p. 2  The m a x i m u m stress occurs at the n o t c h edge (i.e. r = 0) therefore,  ' - • ^ • (7max —  0  )  - ^ J i ( f c j  +  ^ f e  5.9  2K, np 107  Chapter  5 Discussion  and Analysis  of Results  Fracture i n i t i a t i o n occurs w h e n amax is equal to the material strength ac. T h e c r i t i c a l stress intensity factor K  Jc  is therefore  K  i c = —  •  5.10  F o r the O C T specimen, the relationship between Ki and P is k n o w n f r o m Equations 5.2 and 5.3 i n regime I such that  K  * = - £ zr^^W-aF,. (W - of  5.11  E q u a t i n g Equations 5.10 and 5.11 yields an estimate f o r the peak load i n regime I I as g i v e n i n E q u a t i o n 5.12.  u  p  CT J^p~{W-a)  ^  2  =  c  4(2W + ayjW-a  5  F  2  I n this regime the peak load is related t o the square root o f the n o t c h root radius.  5.2.3  Transition radius  The n o t c h root radius at w h i c h the specimens transition f r o m the stable regime I behaviour to the unstable regime I I behaviour is obtained b y equating Equations 5.6 and 5.12 y i e l d i n g  4E'G  C  rtransition  ?  *  7l(T  c  A s w i t h the n u m e r i c a l m o d e l , k n o w l e d g e o f o n l y three material parameters;  effective  m o d u l u s , c r i t i c a l fracture energy release rate and tensile strength, is enough t o determine the transition radius between stable and unstable behaviour.  5.2.4  Peak load solution for the OCT specimens  U s i n g the same parameters as were used f o r the S D M t o m o d e l the O C T specimens tested i n Chapter 3, the analytical peak load solution f o r v a r y i n g n o t c h root r a d i i is obtained. N o m i n a l specimen dimensions  are g i v e n i n F i g u r e  3.1.  However,  since the actual  specimen  dimensions varied slightly f r o m these n o m i n a l dimensions f o r each specimen (see Table 3.2), the solution is presented as a band encompassing the v a r i a t i o n i n actual dimensions. T h e effective m o d u l u s E' is calculated using E q u a t i o n 2.12 and the m o d u l i f r o m Table 4 . 1 . T h e  108  Chapter  5 Discussion  and Analysis  of  Results  tensile strength is taken to be the same as the peak stress f o r the S D M , 4 6 0 M P a , and G  c  is  also the same as the S D M , at 80 k J / m . 2  The a n a l y t i c a l l y predicted peak loads are presented i n Figure 5.5 on top o f the experimental and n u m e r i c a l results.  T h e analytical predictions match the experimental and n u m e r i c a l  results f a i r l y closely. I n regime I, the experimental and n u m e r i c a l results f o r the most part f a l l inside the analytical p r e d i c t i o n band, h o w e v e r , the l o w e r peak loads at the sharpest n o t c h tips ( f o r p equal to 0.5 m m to 2 m m ) are n o t captured b y the analytical p r e d i c t i o n .  T h e analytical m o d e l does n o t  e x p l a i n this n o t c h sensitivity at r e l a t i v e l y sharp n o t c h tips. T h e o r e t i c a l l y , the development o f a damage zone should eliminate n o t c h sensitivity f o r all r a d i i i n r e g i m e I. T h e trend is also predicted b y the n u m e r i c a l m o d e l , r e i n f o r c i n g the experimental results, but the exact reason f o r the trend has not been determined.  A s s h o w n i n F i g u r e 4.19, Figure 4.20 and F i g u r e  4 . 2 1 , the shape o f the n o r m a l stress field at a notch t i p is disrupted b y the presence o f a large radius n o t c h tip.  I t is speculated that the reason f o r the notch sensitivity at r e l a t i v e l y sharp  n o t c h tips is due to this difference i n the shape o f the stress fields, h o w e v e r  further  investigation is necessary to c l a r i f y the reason f o r the trend. The presence o f this small effect at sharp n o t c h tips does not affect the overall a i m o f this analytical m o d e l , w h i c h is to g i v e reasonably accurate predictions o f structural strength and to determine the transition w i t h n o t c h root radius from stable to unstable behaviour. E x p e r i m e n t a l l y and n u m e r i c a l l y , the transition p o i n t was f o u n d to be at a radius o f about 16 m m . A n a l y t i c a l l y , the transition n o t c h root radius is calculated to be s l i g h t l y larger at 17.67 m m . The r e g i m e I I p r e d i c t i o n and location o f p nsition is sensitive to the strength parameter. tra  Increasing the strength b y 4 % to 480 M P a yields a p ition tranS  experimental results. 2  o f 16.2 m m , closer to the  A l s o the r e g i m e I p r e d i c t i o n is v e r y sensitive to G . c  Decreasing  G  c  2  from 80 k J / m to 75 k J / m w h i l e k e e p i n g all else constant, changes position to 16.6 m m . A f t e r the transition point, the analytical peak load p r e d i c t i o n increases along the regime I I curve. Here, the p r e d i c t i o n o f peak load f o l l o w s m o r e closely the n u m e r i c a l p r e d i c t i o n u s i n g the A S D M than the experimental results, w h i c h actually saw a slight decrease i n peak load at the largest n o t c h root radius.  G o i n g back to the elastic stress field contours presented i n  F i g u r e 4.3, an unexpected result was seen. The height o f c r i t i c a l l y stressed material at the  109  Chapter  5 Discussion  and Analysis  of  Results  experimental peak load o f 17.8 k N f o r p equal to 27 m m was less than the height o f c r i t i c a l l y stressed material at peak load f o r p equal to 16 m m .  N o w , u s i n g the peak load  of  a p p r o x i m a t e l y 19.5 k N , predicted both n u m e r i c a l l y and a n a l y t i c a l l y f o r a 27 m m n o t c h root radius, the c r i t i c a l l y stressed height is 17 m m , considerably larger than the c r i t i c a l l y stressed height o f a 16 m m n o t c h r o o t radius. A statistical explanation f o r the experimental decrease i n structural strength at the largest n o t c h size is proposed. W e i b u l l theorized i n 1939 that a size effect on strength exists due to random flaws in a continuum.  I n regime I I , a structure w i l l f a i l w h e n the stress i n a small  element o f the structure reaches the critical strength.  F o r a g i v e n n o m i n a l stress, the  p r o b a b i l i t y o f a structure c o n t a i n i n g a small element w i t h a strength equal to the n o m i n a l stress is g i v e n b y the W e i b u l l d i s t r i b u t i o n . T h e p r o b a b i l i t y o f a structure c o n t a i n i n g a small element o f this strength increases w i t h the size o f the structure due to a larger stressed area. T h i s weakest l i n k explanation f o r size effect applies w e l l to tensile bars that see a u n i f o r m stress d i s t r i b u t i o n .  Bazant (Bazant, 2 0 0 2 ) , h o w e v e r , argues that W e i b u l l theory does not  h o l d f o r notched specimens. Increasing the size o f a notched specimen does not change the m a g n i t u d e o f the stress concentration at a n o t c h t i p i f the structure is s u f f i c i e n t l y large that the b o u n d a r y conditions do not interfere w i t h the stress concentration. A d d i t i o n a l l y , W e i b u l l theory does not h o l d f o r structures d i s p l a y i n g stable crack g r o w t h .  For  quasi-brittle  materials, the f o r m a t i o n o f a damage zone around a n o t c h t i p , p r i o r to peak load and the development o f a discrete crack, absorbs the effect o f any f l a w s i n the material. A different k i n d o f size effect occurs i n the O C T specimens t h o u g h .  That is the effect o f  increasing the size o f the n o t c h root radius w h i l e m a i n t a i n i n g the size o f the structure.  In  r e g i m e I, the arguments o f Bazant h o l d because o f the f o r m a t i o n o f a sizeable damage zone p r i o r to the development o f a discrete crack and the peak load. H o w e v e r , i n r e g i m e I I the specimens f a i l i n a b r i t t l e manner w i t h o u t the development o f a damage zone p r i o r to the peak load. Failure i n this regime is therefore subject to the occurrence o f f l a w s w i t h i n the r e g i o n o f c r i t i c a l l y stressed material ahead o f the n o t c h t i p .  A s the n o t c h root radius is  increased, this c r i t i c a l l y stressed r e g i o n is spread out over a larger height.  A s evidence o f  this, cracks i n the large n o t c h root radius specimens (rd44 and r d 5 4 ) were seen to initiate s l i g h t l y o f f centre as s h o w n i n Figure 5.6.  W i t h m o r e c r i t i c a l l y stressed area around the  110  Chapter  5 Discussion  and Analysis  of  Results  n o t c h t i p f o r larger radius notches, there is m o r e l i k e l i h o o d o f a f l a w b e i n g present, leading to l o w e r strengths than expected i n v e r y b l u n t notches. Further investigation is needed to v e r i f y this hypothesis.  5.3  D a m a g e z o n e size a n d s h a p e  I t has been s h o w n that a transition i n the behaviour o f the O C T specimens occurs at a circular n o t c h root radius o f a p p r o x i m a t e l y 16 m m . I t was also s h o w n that the characteristic damage height o f the material is about 7 m m . Is there a l i n k between these t w o values? The height o f the damage zone hd must be related to the extent o f c r i t i c a l l y stressed material at peak load. T h e c r i t i c a l l y stressed heights h f o r v a r y i n g n o t c h root r a d i i were presented i n s  Table 4.2.  These heights r o u g h l y correspond to the i n i t i a l hd that were determined b y  sectioning specimens r d 2 - l and r d 5 4 - 2 . F o r p equal to 1 m m , the c r i t i c a l l y stressed height f r o m the elastic s i m u l a t i o n was about 3.5 m m .  E x p e r i m e n t a l l y , hd was a p p r o x i m a t e l y 3.0  mm. A t the transition radius, h should be equal to hd, w h i c h should be equal to the characteristic s  stable damage zone height h , at the i n i t i a t i o n o f crack g r o w t h so that the n o t c h neither blunts c  nor sharpens w i t h the development o f the damage zone. equal to 16 m m (taken as p sition),  F r o m the elastic s i m u l a t i o n f o r p  the c r i t i c a l l y stressed height is a p p r o x i m a t e l y 9.5 m m ,  tran  s l i g h t l y larger than, but close t o , the stable damage zone height o f 7.0 m m d e t e r m i n e d  from  sectioning. Conversely, at the large n o t c h root radius, h and hd should i n i t i a l l y be m u c h higher. s  sectioning, the i n i t i a l damage height o f r d 5 4 - 2 was close to 20 m m .  From  A l t h o u g h at the peak  load determined e x p e r i m e n t a l l y , h was o n l y 8.5 m m , there is large v a r i a b i l i t y at this n o t c h s  root radius.  F r o m the n u m e r i c a l l y and a n a l y t i c a l l y determined peak load o f 19.5 k N f o r p  equal to 27 m m , h is greatly increased to 17 m m . s  O b v i o u s l y , the approximate values obtained here cannot be used to develop a quantitative relationship between hd and h , but they do s h o w a trend between the i n i t i a l notch size, the s  extent o f c r i t i c a l l y stressed material and the damage zone height. Regardless o f the i n i t i a l n o t c h root radius, as a crack progresses, the damage height stabilizes to the characteristic damage zone height h . c  Since a transition n o t c h root radius creates a  111  Chapter  5 Discussion  and Analysis  of  Results  stress d i s t r i b u t i o n that neither blunts nor sharpens w i t h the development and propagation o f a damage zone, the question arises o f w h y the p sUion is not equal to hJ2. lran  A s seen i n Figure  4.20, a large n o t c h root radius creates a stress f i e l d at the n o t c h t i p that is rather e l l i p t i c a l i n shape as opposed to circular.  I t is speculated then, that the shape o f the leading edge o f the  damage zone is also e l l i p t i c a l . I f this is the case, the radius o f curvature o f the characteristic e l l i p t i c a l damage zone should be r o u g h l y the same as ptransition- T h e radius o f curvature o f an ellipse is g i v e n b y  5.14  H e r e , A is the s e m i m a j o r axis and B is the s e m i m i n o r axis o f the ellipse. A s s h o w n i n Figure 5.7, an ellipse w i t h a radius o f curvature o f 16 m m , equal to the transition n o t c h root radius, can be constructed f r o m a m a j o r axis o f 8 m m and a m i n o r axis o f 2 m m . These dimensions correspond r o u g h l y to h and the w i d t h o f c r i t i c a l l y stressed material at peak load f o r a n o t c h c  root radius o f 16 m m (see F i g u r e 4.3 d). A n ellipse seems to approximate the shape o f the leading edge o f the damage zone f a i r l y w e l l a l t h o u g h n o t exactly. The radius o f curvature o f an ellipse, w i t h a m a j o r axis the same as the damage zone height, provides a relationship to the circular n o t c h root radius at the transition p o i n t between stable and unstable behaviour.  5.4  Significance of the transition radius  There is stronger evidence o f the l i n k between the stable damage height and the transition radius. The transition radius is the n o t c h root radius at w h i c h the stress c r i t e r i o n and energy criterion f o r fracture are satisfied at exactly the same t i m e . F o r a n o t c h w i t h radius equal to Ptranshion, the n o r m a l stress the n o t c h t i p j u s t reaches the material strength as G equal to G is c  also satisfied. Therefore, f o r all i n i t i a l n o t c h root r a d i i , the damage zone height converges t o the height that gives a n o t c h root radius equal to ptransition so that the stress criterion and energy criterion are satisfied simultaneously.  I n notches i n i t i a l l y blunter than ptransition this  results i n sharpening o f the n o t c h t i p w i t h g r o w t h o f the crack and f o r notches i n i t i a l l y sharper than position this leads to b l u n t i n g o f the n o t c h t i p as damage g r o w s .  112  Chapter  5.5  5 Discussion  and Analysis  of Results  Other material systems and geometries  T h r o u g h o u t this thesis, specimens w i t h notches sharper than T h e o r e t i c a l l y , this should be true f o r a l l materials.  have been called stable.  I n b r i t t l e materials, the size o f ptransitwn  decreases, so the l i k e l i h o o d o f stable fracture is reduced. measure o f toughness o f the material.  position  T h e transition radius is r e a l l y a  T h e larger the transition radius, the tougher the  material. The location o f  is, o f course, v e r y sensitive t o material properties as g i v e n i n  ptransition  E q u a t i o n 5.13. F o r b r i t t l e materials ( l o w e r G ) the transition radius moves t o the left o n c  F i g u r e 5.5. A l s o ,  p nsition tra  is related t o one over the strength squared s h o w i n g a strong  inverse relationship between strength and toughness. B e f o r e the significance o f  ptransition  w a s realised, f o u r experimental O C T tests were also  conducted u s i n g an isotropic acrylic material instead o f the composite. T h e n o t c h radii tested were 0.5 m m , 2 m m , 4 m m a n d 8.5 m m . T h i s material is m o r e b r i t t l e w i t h a p p r o x i m a t e material properties g i v e n i n Table 5 . 1 . Unstable, fast fracture occurred i n a l l cases. Using  these properties  and Equation  5.13,  position  is calculated  t o be 0 . 1 7 4 m m .  T h e o r e t i c a l l y , this material c o u l d display stable crack g r o w t h under displacement control i f the i n i t i a l n o t c h root radius was made less than 0.174 m m . The transition radius and peak load p r e d i c t i o n p l o t f o r a u n i d i r e c t i o n a l carbon f i b r e D C B specimen is d e r i v e d as an example i n A p p e n d i x E. This is an interesting case because i t gives insight into the a l l o w a b l e thickness o f a T e f l o n insert that can be used t o still achieve stable fracture.  5.6  Choosing an appropriate element width for strain-softening models  A n i m p r o v e d understanding o f the transition i n behaviour f r o m stable t o unstable crack g r o w t h i n notched specimens gives a better understanding o f h o w r e f i n e d a mesh should be in the direction o f a propagating crack.  A s m e n t i o n e d i n Section 4.4.3, f o r stable crack  g r o w t h behaviour, the w i d t h o f an element at a n o t c h t i p needs to be smaller than the w i d t h o f c r i t i c a l l y stressed material w  crit  i f n o strength scaling is used. Elements smaller than w , cri  w i l l b e g i n softening p r i o r t o the satisfaction o f the energy c r i t e r i o n leading t o stable crack g r o w t h . I f the n o t c h t i p element is w i d e r than w  crih  as s h o w n i n Figure 5.8, damage i n the  113  Chapter  5 Discussion  and Analysis  of Results  n o t c h t i p element w i l l be delayed u n t i l after f u l f i l m e n t o f the energy c r i t e r i o n , leading to an incorrect unstable crack g r o w t h behaviour..  I f the physical crack g r o w t h b e h a v i o u r o f the  structure b e i n g m o d e l e d is unstable, there w i l l be an effect o f element w i d t h regardless o f h o w w i d e the element is. I n this case it is i m p r a c t i c a l to refine the mesh to v e r y small w i d t h s and strength scaling should be used. 5.6.1  E l e m e n t w i d t h case s t u d y  A n example o f h o w a large element w i d t h , used w i t h o u t scaling, can lead to erroneous predictions  is presented.  I n the o r i g i n a l  simulations b y F l o y d ( F l o y d , 2 0 0 4 ) o f the  experimental O C T tests conducted b y M i t c h e l l ( M i t c h e l l , 2 0 0 2 ) , u s i n g the C O D A M material m o d e l i m p l e m e n t e d i n L S - D Y N A , a r e l a t i v e l y coarse mesh o f 1.25 m m b y 1.25 m m constant stress elements was used so that the crack b a n d m o d e l c o u l d be demonstrated. The specimen geometry, material and orientation was the same as f o r the tests done i n this thesis except that the laminate consisted o f f i v e stacks instead o f six so that thickness was reduced f r o m 8.5 m m t o 7.2 m m and there was loose through-thickness reinforcement i n the f o r m o f K e v l a r stitching. The n o t c h tip had a n e g l i g i b l e radius o f curvature. I n the previous w o r k , damage parameters were chosen to match the n u m e r i c a l s i m u l a t i o n to the e x p e r i m e n t a l l y determined peak load. A strength o f 4 2 0 M P a was used and it was f o u n d that i n order to m a t c h the peak l o a d , the input G needed to be 29 k J / m , considerably l o w e r 2  C  than the value o f 80 k J / m  d e t e r m i n e d i n the present w o r k .  f r o m the C O D A M s i m u l a t i o n is presented i n F i g u r e 5.9.  The load versus C M O D result A l t h o u g h a stable response was  seen e x p e r i m e n t a l l y , the n u m e r i c a l solution predicted a brittle, unstable response.  The  fracture energy dissipated d u r i n g the v i r t u a l test was determined b y measuring the area under the load versus P O D curve (not s h o w n ) .  D i v i d i n g b y the area o f crack g r o w t h gives GF  equal to over 40 k J / m , considerably larger than the input G . C  The same mesh and material and damage parameters were used w i t h the S D M m o d e l i n A B A Q U S and the results are also s h o w n i n F i g u r e 5.9.  A g a i n , the peak load reasonably  matches the experimental result, but an unstable specimen response is predicted.  The  p r e d i c t i o n looks w r o n g and t o v e r i f y , the analytical solution i n E q u a t i o n 5.6 is used w i t h the same i n p u t parameters as used n u m e r i c a l l y .  E q u a t i o n 5.6 predicts a peak load o f 8.3 k N ,  considerably different than the 13.3 k N peak load predicted n u m e r i c a l l y i n Figure 5.9.  The  114  Chapter  5 Discussion  and Analysis  of Results  reason f o r the error is made clear b y p l o t t i n g the stress d i s t r i b u t i o n ahead o f the crack t i p u s i n g the coarse mesh at the a n a l y t i c a l l y predicted peak load o f 8.3 k N . Figure 5.10 shows that the n u m e r i c a l l y determined stress at the n o t c h t i p has n o t even reached the tensile strength threshold because the coarse mesh does a p o o r j o b o f m a t c h i n g the real stress distribution.  N o t e that the d o w n t u r n near the n o t c h t i p i n the coarse mesh stress  field  p r e d i c t i o n i n F i g u r e 5.10 is n o t due t o damage b u t rather due t o the i n a b i l i t y o f the mesh t o m a t c h the real stress d i s t r i b u t i o n .  A n elastic solution o f w h a t the stress d i s t r i b u t i o n should  l o o k l i k e i f there w e r e no damage is also p l o t t e d i n the figure. 2  2  B y m a k i n g the fracture energy i n p u t 2 9 k J / m instead o f 80 k J / m , the transition radius has shifted t o the left. I n a d d i t i o n , the coarse mesh blunts the stress d i s t r i b u t i o n at the n o t c h t i p so that fracture becomes stress c o n t r o l l e d as opposed t o energy c o n t r o l l e d . correct p r e d i c t i o n there are t w o alternatives.  T o achieve a  T h e strength o f the constitutive m o d e l can be  scaled d o w n so that damage initiates p r o p e r l y . T h i s can be done as a pre-processing step o r a u t o m a t i c a l l y i n a material m o d e l such as the A S D M .  T h e other alternative is t o r e f i n e the  mesh. H e r e the mesh is r e f i n e d t o second-order 0.5 m m b y 0.5 m m elements as used i n the rest o f this thesis.  T h e s o l u t i o n f o r the " f i n e m e s h " is also g i v e n i n Figure 5.9. A stable  response is achieved w i t h a peak load o f about 8.3 k N as predicted analytically.  T h e stress  d i s t r i b u t i o n p l o t t e d at peak load i n F i g u r e 5.10 shows that damage has occurred i n elements up t o 3 m m i n f r o n t o f the i n i t i a l n o t c h before failure. N o w , i t is seen that a G input o f 2 9 c  kJ/m  2  is m u c h t o o l o w t o m a t c h the experimental results.  U s i n g the composite laminate  properties used i n the rest o f this thesis o f G equal t o 80 k J / m c  and cr equal t o 4 6 0 M P a , a c  solution v e r y close t o the experimental results is predicted as s h o w n i n Figure 5.9. T h e slight difference m a y , i n part, be due t o the through-thickness stitching that was n o t accounted f o r i n the m o d e l .  5.6.2  Guidelines for choosing an appropriate element size  The element size required t o achieve a correct solution w h e n u s i n g a strain-softening material m o d e l is dependent on b o t h material properties and the g e o m e t r y o f the structure b e i n g meshed. F o r fracture c o n t r o l l e d b y the energy c r i t e r i o n , the w i d t h o f elements used t o discretize the plane ahead o f a n o t c h can be larger than f o r unstable stress c o n t r o l l e d fracture.  115  Chapter  5 Discussion  and Analysis  of Results  First, considering r e g i m e I where fracture is c o n t r o l l e d b y satisfaction o f the energy c r i t e r i o n , the mesh size must be small enough such that the stress i n the n o t c h tip element cr is higher e  than the material strength <r w h e n the energy criterion is satisfied. T h i s is expressed as c  5.15  cr =aa . e  c  The factor a has a value greater than or equal to 1.0.  A s s u m i n g f o r the m o m e n t that the  n o t c h t i p is sharp and the element size is small enough to closely approximate the n o t c h t i p stress d i s t r i b u t i o n , E q u a t i o n 5.15 is w r i t t e n i n terms o f the stress intensity factor as 1  K, '  a  5.16  ^Inr  Here, r is replaced b y the distance to the calculation p o i n t o f the n o t c h t i p element. constant stress element that distance is h a l f o f the element w i d t h wJ2.  For a  A l s o , f o r r e g i m e I, the  stress intensity factor is replaced w i t h the critical fracture energy resulting i n  4^G  1  C  <T =-^J=^.  5.17  C  v  S o l v i n g f o r w at the critical stress gives e  EG w  e  = — T T na cr  5.18  c  U s i n g elements o f size w , the stress criterion w i l l be satisfied at the same t i m e as the energy e  c r i t e r i o n f o r a value o f a equal to 1.0.  T h i s is the minimum  requirement.  The factor  a  should be a n u m b e r larger than 1.0 t o a l l o w development o f the damage zone p r i o r to the onset o f fracture. A l s o , a should be increased to account f o r the error i n calculating the stress field w i t h a coarse mesh, although this error is small at the location o f the integration point.  A s was  s h o w n i n Figure 4.4 f o r a 4 m m n o t c h root radius, even f a i r l y coarse constant stress elements are able to match the true solution closely at the distance o f the first integration p o i n t near the n o t c h tip.  The small error i n the coarse mesh a p p r o x i m a t i o n  o f the stress  field  is  incorporated into a.  116  Chapter  5 Discussion  and Analysis  of  Results  The above element w i d t h s o l u t i o n is f o r a sharp n o t c h t i p . F o r a blunter n o t c h w i t h a radius still less than position, smaller elements than g i v e n b y E q u a t i o n 5.18 are required. F i g u r e 4.1 demonstrates that as p is increased, w  crit  decreases. A n estimate o f the element size needed  f o r blunter notches is made b y using the equation f o r the stress f i e l d at a b l u n t n o t c h t i p o f (Creager and Paris, 1967). F o r the element calculation p o i n t at w /2 E q u a t i o n 5.17 becomes e  f 5.19  I f the a p p r o x i m a t i o n is made that p/(w +p) « 1, then e  A s p is increased, element w i d t h should be made smaller.  A t a p o i n t equal to the transition  radius, w needs to be equal to zero to achieve the exact solution. e  A s stated i n Section 4.3.2, i n r e g i m e I I w h e n p is greater than ptransition and fracture is stress c o n t r o l l e d , the mesh either needs to be r e f i n e d to small element sizes to reduce the error or strength scaling should be used. O f course as p is increased, the stress concentration is reduced so that the size o f element needed to accurately m o d e l the stress f i e l d decreases.  A l s o , as p approaches i n f i n i t y , as is  the case f o r an unnotched specimen, the stress concentration approaches 1.0.  W h e n there is  u n i f o r m stress across the specimen, there is no error i n calculating the stress at any p o i n t w i t h i n the section. Guidelines f o r choosing an appropriate element w i d t h are g i v e n i n Table 5.2. E q u a t i o n 5.18 is p l o t t e d i n F i g u r e 5.11 f o r the material properties used to m o d e l the experimental O C T specimens (Table 4.1). value o f a.  Each contour i n the p l o t represents a different  A line representing the 460 M P a strength is also p l o t t e d on the f i g u r e , s h o w i n g  that f o r this t o u g h material, r e l a t i v e l y large elements can be used w i t h o u t resulting i n the stress criterion b e i n g satisfied after the energy criterion. R e - r u n n i n g v i r t u a l specimen rd2 (p =  1.0 m m ) w i t h 0.5 m m , 1.0 m m and 2.0 m m w i d e constant stress elements ( w h i l e  h  e  m a i n t a i n e d constant at 0.5 m m ) , m a t c h i n g stable solutions are achieved i n all three cases as  117  Chapter  5 Discussion  s h o w n i n Figure 5.12.  and Analysis  of  Results  I t is noted that the constant stress elements s l i g h t l y over-predict the  peak load compared to the o r i g i n a l second-order element s o l u t i o n .  T h i s m a y be due to the  a r t i f i c i a l stiffness that is n u m e r i c a l l y added to the constant stress elements to prevent zeroenergy hourglass modes.  E v e n the v e r y w i d e 2.0 m m elements t h o u g h g i v e a solution  s i m i l a r to the 0.5 m m w i d e elements. C o m p a r a t i v e l y , i f E q u a t i o n 5.18 is plotted f o r G equal to 20 k J / m as i n F i g u r e 5.13, the a 2  c  contours indicate that m u c h m o r e r e f i n e d elements are necessary.  T r a c i n g the 4 6 0 M P a  strength line o n the p l o t shows that about 0.28 m m w i d e constant stress elements w o u l d be required to achieve a g o o d solution ( f o r a = 2.0). The s i m u l a t i o n o f specimen rd2 was again carried out u s i n g 0.5, 1.0 and 2.0 m m w i d e constant stress elements w i t h no strength scaling to v e r i f y the element w i d t h effect. A s s h o w n i n F i g u r e 5.14, increasing values o f peak load are predicted w i t h increasing element w i d t h .  The r e f i n e d second-order element  mesh  produces a softer response w i t h damage o c c u r r i n g before the peak load is attained.  The  s i m u l a t i o n still crashed after the peak load was reached because the n u m e r i c a l m o d e l c o u l d not handle the sharp load drop. T h e guidelines presented i n Table 5.2 can be used to help determine an appropriate element size f o r a g i v e n situation.  The guidelines are based on the transition radius,  incorporates material properties and g e o m e t r y i n a single parameter.  which  I n r e g i m e I, E q u a t i o n  5.18 can be p l o t t e d to p r o v i d e a guideline f o r h o w large an element can be, but precise values o f a have not been determined. Setting a equal to 2.0 appears to result i n a s u f f i c i e n t l y small element f o r notches w i t h root r a d i i less than p nsitiontra  5.7  Summary •  T w o b e h a v i o u r regimes can be i d e n t i f i e d f o r structures w i t h v a r y i n g n o t c h root radii. I n the first r e g i m e , f o r sharp notches, the steep stress concentration at the n o t c h t i p ensures that the stress c r i t e r i o n is satisfied before the energy c r i t e r i o n .  Fracture is  therefore c o n t r o l l e d b y energy release and is stable under displacement c o n t r o l .  In  the second r e g i m e , f o r notches blunter than the transition radius, the stress criterion is satisfied after the energy criterion is f u l f i l l e d .  The stress c o n t r o l l e d fracture i n this  r e g i m e is unstable.  118  Chapter •  5 Discussion  and Analysis  of Results  A transition radius between the t w o regimes can be d e f i n e d as the intersection o f the energy and stress criteria. I t is a measure o f the toughness o f the material.  •  The height o f the characteristic damage zone f o r a material is related to the t r a n s i t i o n radius. I f the leading edge o f the damage zone is taken t o be r o u g h l y e l l i p t i c a l , it has the same radius o f curvature as the transition radius.  •  F o r notches i n i t i a l l y sharper than the transition radius, the n o t c h blunts w i t h damage g r o w t h u n t i l the stress c r i t e r i o n and the energy c r i t e r i o n are satisfied simultaneously, resulting i n the characteristic damage height.  F o r notches i n i t i a l l y blunter than the  transition radius, the n o t c h sharpens w i t h damage u n t i l the same c o n d i t i o n is met. •  S i m p l e guidelines based o n the transition radius w e r e presented i n Table 5.2.  For  notches w i t h r o o t r a d i i less than the transition radius, an element needs t o be smaller than the c r i t i c a l l y stressed w i d t h o f material at the n o t c h t i p .  F o r notches w i t h root  r a d i i larger than the transition radius, scaling o f the peak stress i n a strain-softening m o d e l should be conducted to a l l o w f o r larger element sizes.  119  Chapter  5 Discussion  and Analysis  of Results  Table 5.1 Approximate material properties for acrylic Property  Value  E  3.3 G P a 110 M P a 500 J / m  G  c  2  Table 5.2 Meshing guidelines based on notch radius Meshing approach  P .  ..  p  ^ Ptransition  Use E q u a t i o n 5.18 t o determine a l l o w a b l e element w i d t h w i t h 2.0 to a l l o w f o r pre-peak softening. A s p  — >  position,  a>  cx should be  increased. p  ~ Ptransition  Elements need t o be r e f i n e d t o accurately m o d e l the n o t c h t i p stress d i s t r i b u t i o n . Scaling can be used t o a l l o w f o r larger element sizes.  p  Ptransition  Use strength scaling t o a l l o w f o r larger element sizes o n large structures. E l e m e n t w i d t h is n o t important. N o strength scaling is necessary.  120  Chapter  5 Discussion  'G.  and Analysis  of Results  K{p\  Damage development/ notch blunting ^>  1.0  P' Figure  1.0  c  5.1 Stress and energy failure  criteria for fracture  s  >  1  5.2 Stress and energy failure  *  in regime  I  s  K.  '  P" Figure  1.0  P  Excess energy  G.  K.  1.0  P  criteria for fracture  e  in regime  II  121  Chapter  5 Discussion  and Analysis  of  Results  Chapter 5 Discussion and Analysis of Results  Stress Criterion (Regime II)  p (mm)  Figure 5.5 Analytical prediction ofpeak loads for OCT specimens  a)  b)  Figure 5.6 Cracks initiating off the centerline in a) rd45-l and b) rd44-2  123  Chapter 5 Discussion and Analysis ofResults 20  -20  J  -20  -15  -10  -5  0  5  x (mm) Figure 5.7 An ellipse with a major axis of 8 mm and a minor axis of 2 mm has the same radius of curvature at the notch tip as a circle with a radius of 16 mm.  a  Figure 5.8 Stress distribution ahead of a notch tip showing an element of width w that is too large. Mesh must either be refined or strength scaling used to achieve accurate results. e  124  Chapter 5 Discussion and Analysis of Results  0  1  0.5  1.5  2  3  2.5  3.5  4  CMOD (mm)  Figure 5.9 Load versus CMOD results for a sharp-notched OCT specimen modeled with different size elements  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18 19 20 21 22 23 24 25  x (mm) Figure 5.10 Notch tip stress fields for coarse andfine meshed OCT specimens with a sharp notch tip at load equal to 8.3 kN 125  Chapter 5 Discussion and Analysis of Results  0  0.5  1  1.5  2  2.5  3  3.5  4  4.5  5  C M O D (mm) Figure 5.12 Large constant stress elements used to model the experimental 1 mm notch root radius OCT specimen all give the same result  126  Chapter 5 Discussion and Analysis of Results  0  100  200  300  400  500  600  700  800  900  1000  o- (MPa) c  Figure 5.13 Element size contours for G = 20 kJ/m  2  c  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1  CMOD (mm)  Figure 5.14 When G is reduced to 20 kJ/m a difference is noted in the peak loads predicted using different width elements c  127  Chapter 6 Conclusions and Future Work  CHAPTER 6 CONCLUSIONS AND FUTURE WORK 6.1  Conclusions  Both fracture mechanics and cohesive zone methods have been used in this thesis. The two methods in fact complement each other, providing explanations for different aspects of the other's behaviour. This combined approach has led to a clearer picture of the mechanisms of fracture in FRP composites, as well as, perhaps, other materials. The numerical cohesive model has shown that the LEFM fracture criteria; stress and energy, are not satisfied simultaneously under all conditions. For sharp notch tips, the stress criterion is automatically satisfied for any applied load and the energy criterion drives fracture. For blunt notch tips, the delayed fulfillment of the stress criterion leads to excess energy storage and unstable crack growth. For its part, using the LEFM fracture criteria has provided a means of estimating the width of element necessary to accurately model fracture under different conditions. The development of a simple analytical model, using only three physical input parameters, provides a designer with a simple tool for evaluating candidate materials, designs or other more complicated models. Light has also been shed on the interaction between element size and input parameters for strain-softening finite element material models, which allows greater efficiency as well as certainty in predictions. The following conclusions are made: •  From experimental tests and numerical predictions it was found that structural strength is sensitive to notch root radius mainly at large notch root radii. Before a certain transition radius, which is a function only of material properties, the development of a damage zone absorbs the effect of the notch root radius.  •  For the tough material used for the OCT specimens, the transition radius was found to be about 16 mm.  For more brittle materials, the transition radius is less.  Theoretically, all materials have a transition radius between stable and unstable behaviour.  For very brittle materials in which notch sensitivity is apparent at  128  Chapter 6 Conclusions and Future Work relatively sharp notch tips (e.g. glass), the transition radius is very small and therefore not noticeable in most tests. •  The reason for the transition in behaviour is the separate fulfilment of the stress and energy criteria. Fracture can therefore be separated into two regimes. In regime I, the notch root radius is small and the normal stress at the notch tip reaches the material tensile strength prior to satisfaction of the energy criterion. Fracture occurs when the energy in the system is equal to the energy required to form a discrete crack.  In regime II, blunter notch tips create smaller stress concentrations.  Satisfaction of the stress criterion is therefore delayed until after the fulfilment of the energy criterion. The resulting excess energy in the system when the stress criterion is satisfied drives the crack forward unstably. •  A material's transition radius can be related to the characteristic height of the damage zone. If the front of the damage zone is taken to be roughly elliptical like the normal stress field contours, then the radius of curvature of the front of the damage zone is equal to the transition radius.  •  Sectioning analysis of experimental OCT specimens reveals that the height of damage along a crack plane stabilises to a characteristic height as the crack advances. For notches initially blunter than the transition radius, the damage height is initially larger than the characteristic height and decreases to the characteristic height over a few millimetres of crack growth. For notches initially sharper than the transition radius, the notch tip blunts with the development of the damage zone. Discrete crack formation does not occur until the characteristic damage zone height is fully developed.  •  A simple numerical cohesive model was developed as a user material model in the ABAQUS finite element code as a demonstrator for modeling techniques that can be applied to any strain-softening material model.  Using three experimentally  determined parameters, the model closely matches the experimental results. It was determined that the experimental input parameters give a good prediction as long as the element width used to mesh the area around the notch tip is sufficiently refined. In regime I, the element width must be smaller than the width of critically stressed 129  Chapter 6 Conclusions and Future Work  material at the notch tip when the energy criterion is satisfied. In regime II, the element width is of critical importance because fracture is driven by satisfaction of the stress criterion and there can be large differences in the stress calculated at an element integration point and the actual notch tip. •  A n addition was made to the simple damage model (SDM) to allow for larger element widths near the transition radius and in regime II.  The adaptive simple  damage model (ASDM), automatically scales the input tensile strength while maintaining the fracture energy so that softening initiates in an element when the stress at the element edge reaches the material tensile strength. •  Guidelines for selecting an appropriate element width based on the radius of the notch root relative to the transition radius of the material were given. In regime I, the fracture  mechanics stress criterion provides an equation for determining an  appropriate element size. In regime II, strength scaling should be used if the mesh cannot be adequately refined. As the notch root radius approaches infinity (i.e. an unnotched structure) the stress concentration drops to unity and strength scaling becomes less important.  6.2  Future work  There are several areas in which the work in this thesis can be advanced. •  First, only one material system, a tough quasi-isotropic composite laminate, was tested in this thesis. It is theorised that the transition radius model presented here is valid for most other materials as well. Validation tests need to be done to prove this. It would be interesting to test the theory further for a more brittle material such as acrylic and also to test it for metals that develop plastic zones before fracture. Testing on other types of specimens and composite laminates would also be useful.  •  Further investigation is necessary to determine the reason for the initial notch sensitivity in regime I at relatively sharp notch tips.  •  To be useful, the A S D M technique should be incorporated into a more robust strainsoftening composite damage code, such as C O D A M . Doing so would allow it to be tested under more varied conditions.  130  Chapter 6 Conclusions and Future Work •  This thesis focused on only mode I fracture. Notch sensitivity in shear mode loading (mode II and mode III) should also be studied.  A strain-softening model  incorporating interaction between material directions to allow for shear loading should be used to determine the effect of element size under shear loads.  It is  hypothesized that under shear loading the height of the element would become important as the width is important for mode I loading. •  Further investigation is necessary to determine ideal values of the factor a in the element width equation in regime I.  131  Appendix A Load versus CMOD Plots  APPENDIX A  L O A D VERSUS C M O D P L O T S  132  Appendix A Load versus CMOD  Plots  16 n  0.0  0.5  1.0  1.5  2.0  2.5  3.0  3.5  4.0  4.5  5.0  CMOD (mm) Figure A.l Experimental load-CMOD  16  results and numerical prediction for specimen rd2  i  0.0  0.5  1.0  1.5  2.0  2.5  3.0  3.5  4.0  4.5  5.0  CMOD (mm) Figure A.2 Experimental load-CMOD  results and numerical prediction for specimen rd8  133  Appendix A Load versus CMOD Plots  0.0  0.5  1.0  1.5  2.0  2.5  3.0  3.5  4.0  4.5  5.0  CMOD (mm) Figure A.3 Experimental load-CMOD results and numerical prediction for specimen rd21  0.0  0.5  1.0  1.5  2.0  2.5  3.0  3.5  4.0  4.5  5.0  CMOD (mm) Figure A.4 Experimental load-CMOD results and numerical prediction for specimen rd28  134  Appendix  A Load versus CMOD  0.0  0.5  1.0  Plots  1.5  2.0  2.5  3.0  3.5  4.0  4.5  5.0  CMOD (mm) Figure  A. 5 Experimental  0.0  0.5  load-CMOD  1.0  1.5  results and numerical  2.0  2.5  prediction  3.0  3.5  for specimen  4.0  rd32  4.5  5.0  CMOD (mm) Figure  A. 6 Experimental  load-CMOD  results and numerical  prediction  for specimen  rd38  135  Appendix A Load versus CMOD 20  Plots  n  0.0  0.5  1.0  1.5  2.0  2.5  3.0  3.5  4.0  4.5  5.0  C M O D (mm)  Figure A. 7 Experimental load-CMOD  16  results and numerical prediction for specimen rd54  i  0.0  0.5  1.0  1.5  2.0  2.5  3.0  3.5  4.0  4.5  5.0  C M O D (mm)  Figure A. 8 Experimental load-CMOD  results for specimen rdl  136  Appendix A Load versus CMOD Plots  rd4-l rd4-2  -a a o  Figure A. 9 Experimental load-CMOD results for specimen rd4  — rd6-l — rd6-2 — rd6-3  73  a o  Figure A. 10 Experimental load-CMOD results for specimen rd6  137  Appendix 16  A Load versus CMOD  Plots  i  0.0  0.5  1.0  1.5  2.0  2.5  3.0  3.5  4.0  4.5  5.0  4.5  5.0  C M O D (mm) Figure  A. 11 Experimental  load-CMOD  results for specimen  rdlO  16 ,  0.0  0.5  1.0  1.5  . 2.0  2.5  3.0  3.5  4.0  C M O D (mm) Figure  A. 12 Experimental  load-CMOD  results for specimen  rdl 7  138  Appendix  0.0  A Load versus CMOD  0.5  1.0  Plots  1.5  2.0  2.5  3.0  3.5  4.0  4.5  5.0  C M O D (mm) Figure  A. 13 Experimental  load-CMOD  results for specimen  rd25  C M O D (mm) Figure  A. 14 Experimental  load-CMOD  results for specimen  rd35  139  Appendix  A Load versus CMOD  Plots  C M O D (mm) Figure  A. 15 Experimental  load-CMOD  results for specimen  rd44  140  Appendix  B Extra Line Analysis  APPENDIX B  Plots  E X T R A L I N E ANALYSIS P L O T S  141  Appendix  B Extra Line Analysis  Figure  Plots  B.l COD profiles  determinedfrom  line analysis for  rd2-l  2.2 i  Distance from initial notch tip (mm) Figure  B.2 COD profiles  determined  from line analysis for rd2-2  142  Appendix  B Extra Line Analysis  Plots  1.6 n  Distance from initial notch tip (mm) Figure  2  B.3 COD profiles  determined from line analysis for  rd8-2  i  Distance from intial notch tip (mm) Figure  B.4 COD profiles  determined from line analysis for  rd54-l  143  Appendix  B Extra Line Analysis  Plots  144  Appendix  C Stress Distributions  APPENDIX C PLOTS  and Compliance  Plots  STRESS DISTRIBUTION AND C O M P L I A N C E  145  Appendix  C Stress Distributions  0  10  5  and Compliance  15  Plots  20  25  30  35  40  45  JC (mm) Figure  C1 Progression distribution  0  5  of the stress distribution is at the CMOD  10  15  along the notch plane for rd8.  Each  value (in mm) noted above it.  20  25  30  35  40  x (mm) Figure  C.2 Progression distribution  of the stress distribution is at the CMOD  along the notch plane for rd21.  Each  value (in mm) noted above it.  146  Appendix 500  C Stress Distributions  /  1.76 2.01  2.11  and Compliance  2.22  2.51  Plots  9  7  Q  4.36 3.05 3.32 3.58 | 4.10 1 3  8  4  4  6 2  400  ^300 PH  200 100 u o  Z, 100 -200 10  15  20  30  25  35  40  45  x (mm) Figure  C.3 Progression distribution  of the stress distribution is at the CMOD  along the notch plane for rd32.  Each  value (in mm) noted above it.  0.0020 0.0018 H 0.0016 0.0014 |  0.0012  |  0.0010 i  .2 a |  0.0008  CJ 0.0006 0.0004 -I 0.0002 0.0000  10  0  15  20 A  Figure  C.4 Compliance  plots from  is subtracted from  location  25  30  35  40  a (mm)  of peak stress in numerical  Aa to account for width of damage  simulations.  3.25 mm  zone.  147  Appendix  D ASDM  UMATfor  ABAQUS  APPENDIX D  c c c c c c c c c c c c c c c c c c c  ASDM UMAT FOR  ABAQUS  UBC A d a p t i v e Simple Damage Model f o r Abaqus v 4.3 E l a s t i c - l i n e a r s o f t e n i n g composite damage model S c o t t McClennan, J u l y 2004 (c) C o p y r i g h t The U n i v e r s i t y o f B r i t i s h Columbia, Composites Group 2004. A l l r i g h t s r e s e r v e d . T h i s o r t h o t r o p i c m a t e r i a l model i s f o r use w i t h second-order p l a n e s t r e s s elements. I t i s a c o h e s i v e zone model r e q u i r i n g t h e o r t h o t r o p i c e l a s t i c m a t e r i a l c o n s t a n t s and two a d d i t i o n a l damage parameters. I t i s i n t e n d e d t o model mode I c r a c k p r o p a g a t i o n i n l a m i n a t e c o m p o s i t e s . The model has t h e b u i l t - i n a b i l i t y t o adapt i t s c o n s t i t u t i v e b e h a v i o u r based on mesh s i z e and t h e s t r e s s - f i e l d t o a l l o w a c o a r s e r mesh than i s u s u a l l y r e q u i r e d by a c o h e s i v e zone model.  Q* *****************************************************  C  USER SUBROUTINE TO INITIALIZE STATE VARIABLES  Q* *********************************************************************  c c  ***************** *  SUBROUTINE SDVINI(STATEV,COORDS,NSTATV,NCRDS,NOEL, NPT, 1 LAYER,KSPT) INCLUDE 'ABA_PARAM.INC' DIMENSION STATEV(NSTATV),COORDS(NCRDS)  C  C c c c c c c c c c c c c c c c c c c Q 11  ******* STATE VARIABLES USED *************** STATEV(1)=currpeakepsnpt(1) STATEV(2)=currpeakepsnpt(2) STATEV(3)=currpeakepsnpt(3) STATEV(4)=Erl(1) STATEV(5)=Erl(2) STATEV(6)=Erl(3) STATEV(7)=gammadis(1) STATEV(8)=gammadis(2) STATEV(9)=gammadis(3) STATEV(10)=peakepsnpt(1) STATEV(11)=peakepsnpt(2) STATEV(12)=peakepsnpt(3) STATEV(13)=ultepsnpt(1) STATEV(14)=ultepsnpt(2) STATEV(15)=ultepsnpt(3) STATEV(16)=scale(1) STATEV(17)=scale(2) *******s^ATE VARIABLES************** do 11 i = l , 1 7 statev(i)=0.0 continue  148  Appendix  D ASDM  UMATfor  ABAQUS  c RETURN END c c 0  * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  C  STRAINFIELD SUBROUTINE  0  * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  c c c  t h i s subroutine c a l c u l a t e s the polynomial c o e f f i c i e n t s f o r the s t r e s s d i s t r i b u t i o n m a t r i x and s t o r e s them i n a r r a y s t f i e l d ( 6 , 2 , 3 )  c  c c c 0 c c c c  0  22 21 0  subroutine  strainfield(polycoeffs,global,noelmax,nptmax,NOEL)  double p r e c i s i o n p o l y c o e f f s , integer polynpts(6,3) integer i , j , k dimension  global  polycoeffs(6,2,3),global(noelmax,nptmax,2)  ********** POLYNOMIAL NPT LOOKUP ******************* Each i n t e g r a t i o n p o i n t i s a s s o c i a t e d w i t h a p o l y n o m i a l i n both t h e x - d i r e c t i o n and y - d i r e c t i o n **** SHELLS **** **** x—DIR **** j=l do 21 i = l , 3 do 22 k=l,3 polynpts(i,k)=j j=j+l continue continue **** Y—DIR **** polynpts(4,1)=1 polynpts(4,2)=4 polynpts(4,3)=7 polynpts(5,1)=2 polynpts(5,2)=5 polynpts(5,3)=8 polynpts(6,1)=3 polynpts(6,2)=6 polynpts(6,3)=9  c  ************** CALCULATE POLYNOMIAL COEFFS *********** S i m i l a r t o t h e element shape f u n c t i o n s , t h e 2nd o r d e r p o l y n o m i a l c o e f f i c i e n t s a r e c a l c u l a t e d based on t h e p o s i t i o n o f t h e i n t e g r a t i o n points  c  c c c c  1 1  do 61 i = l , 6 do 62 j = l , 2 polycoeffs(i,j,3)=GLOBAL(NOEL,polynpts(i,2),j) polycoeffs(i,j,2)=(GLOBAL(NOEL,polynpts(i,3),j) -GLOBAL(NOEL,polynpts(i,1),j))/(-2.0*Sqrt(0.6) ) polycoeffs(i,j,1)=(GLOBAL(NOEL,polynpts(i,1),j)-Sqrt(0.6) *polycoeffs(i,j,2)-polycoeffs(i,j,3))/0.6  c 149  Appendix  62 61 c c c  0 c Q  c c c c  D ASDM  UMATfor  ABAQUS  continue continue return end  ******************************************************** SCALESTRAIN SUBROUTINE  * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  t h i s s u b r o u t i n e c a l c u l a t e s t h e maximum v a l u e o f s t r a i n on t h e p o l y n o m i a l curve a s s o c i a t e d w i t h each i n t e g r a t i o n p o i n t f o r each principal direction  subroutine scalestrain(maxst,polycoeffs,global,orient,noelmax,NOEL,NPT) c c polynomial c o e f f i c i e n t s double p r e c i s i o n p o l y c o e f f s , g l o b a l , o r i e n t c current strains double p r e c i s i o n s t r a i n c i n t e g r a t i o n p o i n t lookup t a b l e double p r e c i s i o n n p t l u p ( 9 , 2 ) c parametric coordinates double p r e c i s i o n s i , s2 s3, s4, s5, s6, s7 c scaling s t r a i n array (polydirection, straindirection) double p r e c i s i o n maxst(2) c local variables i n t e g e r i,j,k,m,n double p r e c i s i o n k l , k 2 c dimension p o l y c o e f f s ( 6 , 2 , 3 ) , g l o b a l ( n o e l m a x , 9 , 2 ) , o r i e n t ( n o e l m a x , 3 ) c ************ NPT & EDGE COORDINATES ****************** s7=-1.0 s6=-Sqrt(0.6) s5=-4.0/9.0 s4=0.0 s3=4.0/9.0 s2=Sqrt(0.6) sl=1.0 c Q **************** LOOKUP TABLE ************************ c n p t l u p ( 9 , 2 ) -> v a l u e s a r e p o l y i d x , p o l y i d y f o r each npt c c c *** p o l y 1,2,3 *** k=l do 110 i = l , 7 , 3 do 111 j=0,2 nptlup(i+j,1)=k 111 continue k=k+l 110 continue c c *** p o l y 3,4,5 *** do 120 i = l , 7 , 3 do 121 j=0,2 c  150  Appendix  121 120 c c c c c c c c c c c c  D ASDM  UMATfor  ABAQUS  nptlup(i+j,2)=j+4 continue continue The o r i e n t a t i o n o f t h e element r e l a t i v e t o t h e g l o b a l c o o r d i n a t e s i s determined so t h a t t h e maximum normal s t r a i n s f o r the p o l y n o m i a l i n each t h e x- and y - d i r e c t i o n s can be determined f o r each i n t e g r a t i o n point m a x s t ( l ) = max x s t r a i n on y p o l y maxst(2)= max y s t r a i n on x p o l y ***** x - d i r max s t r e s s ***** if 1  c  c c c  (abs(orient(noel,3)-orient(noel,1)).gt. a b s ( o r i e n t ( n o e l , 2 ) - o r i e n t ( n o e l , 1 ) ) ) then m=2 n=l else m=l n=2 endif . kl=0.0 k2=0.0 i f (abs(polycoeffs(nptlup(npt,m),1,3)1 g l o b a l ( n o e l , n p t , 1 ) ) . l t . ( 0 . 0 0 0 0 1 * g l o b a l ( n o e l , n p t , 1 ) ) ) then kl=s3 k2=s5 elseif (abs(polycoeffs(nptlup(npt,m),1,1)*s2**2.0+ 1 polycoeffs(nptlup(npt,m),1,2)*s2+polycoeffs(nptlup(npt,m),1,3)1 g l o b a l ( n o e l , n p t , 1 ) ) . I t . (0.0001*global ( n o e l , n p t , 1 ) ) ) then kl=sl k2=s3 else kl=s5 k2=s7 endif maxst(1)=max(0.0, g l o b a l ( n o e l , n p t , 1 ) , 1 polycoeffs(nptlup(npt,m),1,1)*kl**2.0+ 1 polycoeffs(nptlup(npt,m),1,2)*kl+polycoeffs(nptlup(npt,m),1,3), 1 polycoeffs(nptlup(npt,m),1,1)*k2**2.0+ 1 polycoeffs(nptlup(npt,m),1,2)*k2+polycoeffs(nptlup(npt,m),1,3)) ***** y - d i r max s t r e s s ***** kl=0.0 k2 = 0 . 0 i f (abs(polycoeffs(nptlup(npt,n),2,3)1 g l o b a l ( n o e l , n p t , 2 ) ) . I t . ( 0 . 0 0 0 1 * g l o b a l ( n o e l , n p t , 2 ) ) ) then kl=s3 k2=s5 e l s e i f (abs(polycoeffs(nptlup(npt,n),2,l)*s2**2.0+ 1 polycoeffs(nptlup(npt,n),2,2)*s2+polycoeffs(nptlup(npt,n),2,3)1 g l o b a l ( n o e l , n p t , 2 ) ) . l t . ( 0 . 0 0 0 1 * g l o b a l ( n o e l , n p t , 2 ) ) ) then kl=sl 151  Appendix  D ASDM  UMAT for  ABAQUS  k2=s3 else kl=s5 k2=s7 endif c 1 1 1 1  maxst(2)=max(0.0, g l o b a l ( n o e l , n p t , 2 ) , polycoeffs(nptlup(npt,n),2,1)*kl**2.0+ polycoeffs(nptlup(npt,n),2,2)*kl+polycoeffs(nptlup(npt,n),2,3), polycoeffs(nptlup(npt,n),2,1)*k2**2.0+ polycoeffs(nptlup(npt,n),2,2)*k2+polycoeffs(nptlup(npt,n),2,3))  c c return end **********************************************************  c  c c  Q  C Q  The f o l l o w i n g u s e r - m a t e r i a l s u b r o u t i n e d e t e r m i n e s t h e damaging m a t e r i a l response o f an o r t h o t r o p i c m a t e r i a l . *******************  *.*  *********************************************  USER SUBROUTINE FOR MATERIALS * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  SUBROUTINE UMAT(STRESS,STATEV,DDSDDE,SSE, SPD,SCD, 1 RPL,DDSDDT,DRPLDE,DRPLDT,STRAN,DSTRAN, 2 TIME,DTIME,TEMP,DTEMP,PREDEF,DPRED,MATERL,NDI,NSHR,NTENS, 3 NSTATV,PROPS,NPROPS,COORDS,DROT,PNEWDT,CELENT, 4 DFGRDO,DFGRD1,NOEL,NPT,KSLAY,KSPT,KSTEP,KINC) c INCLUDE  'ABA_PARAM.INC  c parameter noelmax=700 parameter nptmax=9 c CHARACTER*80 MATERL C  DIMENSION STRESS(NTENS),STATEV(NSTATV), 1 DDSDDE(NTENS,NTENS),DDSDDT(NTENS),DRPLDE(NTENS), 2 STRAN(NTENS),DSTRAN(NTENS),TIME(2),PREDEF(1),DPRED(1), 3 PROPS(NPROPS),COORDS(3),DROT(3,3), 4 DFGRDO(3,3),DFGRD1(3,3) c c m a t e r i a l constants double p r e c i s i o n E0(3) double p r e c i s i o n nxy, nyx c compliance t e n s o r terms double p r e c i s i o n c, sn(2) c model parameters double p r e c i s i o n peakepsch(3) double p r e c i s i o n gammac(2) c d e t e r m i n e d parame t e r s double p r e c i s i o n STRANcurr(3) double p r e c i s i o n E t ( 3 ) double p r e c i s i o n E r l ( 3 ) double p r e c i s i o n peakepsnpt(3) double p r e c i s i o n c u r r p e a k e p s n p t double p r e c i s i o n u l t e p s c h ( 3 ) double p r e c i s i o n u l t e p s n p t ( 3 ) 152  Appendix  c  D ASDM  precision  gammarl(3)  double  precision  gammadis(3)  double  precision  fs  double  precision  dfs  double  precision  fscurr  double  precision  sigma(3)  strain  distribtion  local  strain  global  c c  variables  npt  npt  (x,y)  coordinates  storage storage  glcoords(noelmax,3)  and  flags debug  PRECISION  integer  current  locoords(3)  precision  DOUBLE  a l l  coordinates  orientation  local  c  for  maxeps(2)  precision  double  coeffs  glstrain(noelmax,nptmax,2)  strains  orientation  global  for  precision  double  polynomial  lostrain(9,2)  matrix  maximum e l e m e n t a l local  matrix  precision  double c  storage  strain  containing  stfield(6,2,3)  precision  double c  matrix  precision  double c  ABAQUS  double  double c  UMATfor  i , j , k  integer  step  integer  s c a l e (2)  c Q  **********  USER  INPUT  ***************************  c EO (1)  = p r o p s (1) ~-  EO(2)  = props(9)  nxy  = p r o p s (2) = props(3)  EO(3)  c  gammac(l)  = p r o p s (4)  gammac(2)  = props(12)  peakepsch(l)  = props(5)  peakepsch(2)  = props(13)  nyx=nxy nyx  =  nxy  * EO ( 2 ) / E O ( 1 )  c c c  **********  CALCULATE  CHARACTERISTIC  ULTIMATE  STRAIN  **********  C do  210  i  =  1,NDI  ultepsch(i) if  = (2.0*gammac(i))/(EO(i)*peakepsch  ( u l t e p s c h (i) w r i t e (*,*)  .lt.peakepsch(i))  then  'WARNING: u l t e p s ( ' , i ,  ')  (i))  < peakeps ( ' , i ,  ') '  endif 210  continue  c peakepsch(3)=min(peakepsch(1),peakepsch(2)) ultepsch(3)=max(ultepsch(1),ultepsch(2)) c c g a m m a c ( 3 ) = 0 . 5 * E 0 ( 3 ) * p e a k e p s c h ( 3 ) * u l t e p s c h (3) c c  write(6,*)  'end  calculate  ultimate  strain'  c Q c  **********  CURRENT  STRAINS  ************  153  Appendix  D ASDM  UMATfor  ABAQUS  do i = l , 3 STRANcurr(i)=STRAN(i)+DSTRAN(i) enddo fs=abs(STRAN(3)) dfs=abs(DSTRAN(3)) fscurr=fs+dfs c Q c  260 c c c c c c c  ********** GET STATE VARIABLES *********************** do 260 i = l , 3 i f ( S T R A N ( i ) . e q . 0 . 0 ) then Erl(i)=E0(i) gammarl(i)=gammac(i) currpeakepsnpt(i)=peakepsch(i) peakepsnpt(i)=peakepsch(i) ultepsnpt(i)=ultepsch (i) else currpeakepsnpt(i)=statev(i) Erl(i)=statev(i+3) gammadis(i)=statev(i+6) peakepsnpt(i)=statev(i+9) ultepsnpt(i)=statev(i+12) endif continue scale(1)=statev(16) scale(2)=statev(17) w r i t e ( 6 , * ) 'end s t a t e v a r i a b l e s ' ******* AUTOMATIC TIME INCREMENTATION **************** PNEWDT=0.5 ********* DETERMINE POLYNOMIAL COEFFICIENTS **********  C  c a l l strainfield(stfield,gistrain,noelmax,nptmax,NOEL) c c C c c c c Q  c c c  c  ********* CALCULATE SCALING STRAIN VALUE ************* call  scalestrain(maxeps,stfield,glstrain,glcoords,noelmax,NOEL,NPT)  write(6,*) ' s c a l e s t r a i n subroutine  finished'  * * * * * * * * * * * * * T O SCALE OR NOT TO SCALE? *************** Only s c a l e once d u r i n g t h e e l a s t i c l o a d i n g p o r t i o n o f t h e s t r e s s - s t r a i n curve  do 270 i=2,2 i f (maxeps(i) .ge.0.0.and.glstrain(NOEL, NPT, i ) .gt.0 . 0 . 1 and.(STRANcurr(i).gt.(0.3*peakepsnpt(i))). 1 and. (STRANcurr(i) . I t . (0.5*peakepsnpt ( i ) ) ) . 1 and.(DSTRAN(i).gt.0.0). 1 and. ( s c a l e (i) . ne . 1)) then scale(i)=1 w r i t e ( 6 , * ) ' s c a l i n g n o e l , n p t , d i r ' , NOEL, NPT, i w r i t e ( 6 , * ) ' g l s t r a i n , STRANcurr', g l s t r a i n ( N O E L , N P T , i ) ,  STRANcurr(i) 154  Appendix  D  ASDM  UMATfor  ABAQUS  if  270  ( m a x e p s ( i ) . I t . g l s t r a i n ( n o e l , n p t , i ) ) then w r i t e (6,*) 'WARNING: maxeps(',i,') i s l e s s than g l s t r a i n (', n o e l , ', ',npt, ', ' , i , ' ) ' else peakepsnpt(i)=min(peakepsch(i) , 1 peakepsch(i)*(glstrain(noel,npt,i)/maxeps(i))) endif i f (STRANcurr(i).gt.peakepsnpt(i)) then w r i t e ( * , * ) ' c u r r e n t s t r a i n i s l a r g e r than s c a l e d peak s t r a i n f o r 1 element, npt, d i r : ' , n o e l , npt, i w r i t e (*,*) 'DSTRAN=', DSTRAN(i) w r i t e (*,*) 'STRANcurr:', STRANcurr(i) w r i t e (*,*) 'peakepsnpt:', peakepsnpt ( i ) peakepsnpt(i)=STRANcurr(i) endif ultepsnpt(i) = (2.0*gammarl(i))/(Erl(i)*peakepsnpt (i)) currpeakepsnpt(i)=peakepsnpt (i) else currpeakepsnpt(i)=max(STRANcurr(i),currpeakepsnpt(i),peakepsnpt(i)) endif continue currpeakepsnpt(3)=max(fscurr,currpeakepsnpt(3),peakepsnpt(3))  c c c c c  w r i t e (6,*)  Q  *****************  'end  t o s c a l e or  poiSSON  not'  REDUCTIONS ******************  C  c c c c  c c c  s n ( l ) = s n l 2 , sn(2) = sn21 w r i t e ( 6 , * ) 'Update Poissons...' sn(l) = (Erl(1)/E0(1))*nxy sn(2) = ( E r l ( 2 ) / E 0 ( 2 ) ) * n y x c = (1. - s n ( 2 ) * s n ( 1 ) ) i f (c . l e . 0.0) then w r i t e ( * , * ) 'Warning: s i n g u l a r s t i f f n e s s w r i t e ( * , * ) s n ( l ) , sn(2) endif  tensor'  w r i t e ( 6 , * ) 'end P o i s s o n r e d u c t i o n s ' ********** DETERMINE UPDATED STRESSES ****************  C  sigma=0.0 c c  310 c c  do 310 i = l , 2 i f ( S T R A N c u r r ( i ) . g e . u l t e p s n p t ( i ) ) then sigma(i)=0.0 e l s e i f (STRANcurr(i) .ge.currpeakepsnpt ( i ) ) then sigma ( i ) = ( ( u l t e p s n p t ( i ) -^STRANcurr ( i ) ) / ( u l t e p s n p t ( i ) 1 peakepsnpt(i)))*(E0(i)*peakepsnpt(i)) endif continue write(6,*)  'End  s t r e s s update' 155  Appendix  c Q  D ASDM  UMAT for  *************  ABAQUS  DETERMINE TANGENT MODULI ******************  C  if  c  c c c c c c  ( S T R A N c u r r ( 1 ) . g e . u l t e p s n p t ( 1 ) ) then Et(l)=0.0 e l s e i f (STRANcurr(1).It.currpeakepsnpt(1).or.DSTRAN(1).le.0.0) then Et(1) = E r l (1) else Et (1) = (sigma (1) -STRESS (1) ) / ( (DSTRAN (1) +sn (2) * DSTRAN (2) ) /c) endif if  ( S T R A N c u r r ( 2 ) . g e . u l t e p s n p t ( 2 ) ) then Et(2)=0.0 e l s e i f (STRANcurr(2).It.currpeakepsnpt(2).or.DSTRAN(2).le.0.0) then Et(2) = E r l ( 2 ) else Et(2)=(sigma(2)-STRESS(2)-(sn(2)*Et(1)/c)*DSTRAN(1))/(DSTRAN(2)/c) endif Et(3)=E0 (3) write(6,*)  'End determine tangent m o d u l i '  ***********ASSEMBLE TANGENT STIFFNESS TENSOR**********  C  c  431 430 c  c c c Q c  w r i t e ( 6 , * ) 'Assemble s t i f f n e s s do 430 kl=l,NTENS do 431 k2=l,NTENS DDSDDE(kl,k2)=0.00 continue continue if  tensor...'  (c . l e . 0.0) then w r i t e (*,*) 'Warning: s i n g u l a r s t i f f n e s s w r i t e ( * , * ) s n ( l ) , sn(2) else DDSDDE(1,1) = E t (1) / C DDSDDE(1,2) = (sn(2) * E t ( l ) ) / c DDSDDE(2,1) = DDSDDE(1,2) DDSDDE(2,2) = E t (2) / c DDSDDE (3,3) = E t ( 3 ) DDSDDE(1,3) = 0 . 0 DDSDDE (2,3) = 0 . 0 DDSDDE (3,1) = 0 . 0 DDSDDE(3,2) = 0 . 0 endif write(6,*) **************  tensor'  'end-tangent s t i f f n e s s t e n s o r assembly' STRESS CALCULATION**********************  C  441  do 440 i=l,NTENS do 441 j=l,NTENS STRESS(i)=STRESS(i)+DDSDDE(i,j)*DSTRAN(j) continue 156  Appendix  440 c  444 c c c  D ASDM  UMATfor  ABAQUS  continue do 444 i = l , 2 i f (STRESS(i) .gt. ( E r l ( i ) * c u r r p e a k e p s n p t ( i ) ) ) then STRESS(i) = E r l ( i ) * S T R A N c u r r ( i ) endif continue w r i t e (6,*) 'end s t r e s s c a l c u l a t i o n ' ****** ELASTIC STRAIN ENERGY CALCULATION **************  C  450 c c  do 450 i=l,NTENS SSE=SSE+STRESS(i)*DSTRAN(i)12 .0 continue ******* DISSIPATED STRAIN ENERGY CALCULATION ************  c  455 c  do 455 i = l , 2 i f ( S T R A N ( i ) .ge.peakepsnpt ( i ) ) then gammadis(i)=min(gammac(i), 1 max(gammadis(i), ( 0 . 5 * E 0 ( i ) * p e a k e p s n p t ( i ) * * 2 . 0) + 1 (((E0(i)*peakepsnpt(i))+STRESS(i))12 . 0)*(STRANcurr(i) 1 peakepsnpt(i))-(0.5*STRESS(i)*STRANcurr(i)))) endif continue i f ( f s . g t . p e a k e p s n p t (3)) then gammadis(3)=min(gammac(3), 1 max(gammadis(3), (0.5*E0(3)*peakepsnpt(3)**2.0)+ 1 ( ( (E0 (3)*peakepsnpt(3))+abs(STRESS(3)))12 . 0 ) * 1 (abs(STRANcurr(3))-peakepsnpt ( 3 ) ) 1 (0.5*STRESS(3)*STRANcurr(3)))) endif  c c c Q c c c c  4 60 c c c 470 c c  w r i t e (6,*) 'end s t r a i n energy  calculation'  **************** UPDATE PARAMETERS ******************* ** c h a r a c t e r i s t i c parameters ** w r i t e (6,*) 'update parameters' do 460 i = l , 3 i f ( c u r r p e a k e p s n p t ( i ) . g e . u l t e p s n p t ( i ) ) then gammarl (i)=0 . 0 else gammarl(i)=min(gammac(i),gammac(i)-gammadis(i)) endif continue ** element parameters ** do 470 i = l , 2 Erl(i)=min(Erl(i),(2.0*gammarl(i))/(ultepsnpt(i)*currpeakepsnpt(i))) continue E r l ( 3 ) = m i n ( E r l ( 3 ) , (2 . 0*gammarl (3) ) / ( u l t e p s n p t ( 3 ) * c u r r p e a k e p s n p t (3)))  157  Appendix  c c c c c  D ASDM  UMATfor  ABAQUS  write(6,*) 'end update parameters' * * * * * * * * * * * * * UPDATE STATE VARIABLES write(6,*)  'update  *****************  statevariables'  STATEV(1)=currpeakepsnpt(1) STATEV(2)=currpeakepsnpt(2) STATEV(3)=currpeakepsnpt(3) STATEV(4)=Erl(1) S T A T E V ( 5 ) = E r l (2) STATEV(6)=Erl(3) STATEV(7)=gammadi s(1) STATEV(8)=gammadis(2) STATEV(9)=gammadis(3) STATEV(10)=peakepsnpt(1) STATEV(11)=peakepsnpt(2) STATEV(12)=peakepsnpt(3) STATEV(13)=ultepsnpt(1) STATEV(14)=ultepsnpt(2) STATEV(15)=ultepsnpt(3) STATEV(16)=scale(1) S T A T E V ( 1 7 ) = s c a l e (2) c c  ********************  ***********************  DEBUGGING  cc step=step+l (step write write write write  eq  1)  then  6 , *) 6 , *)  'el  number:',  6 , *) 6 , *)  ' locoords  1,2,3:',  locoords(l),  'glcoords  1,2,3:',  glcoords(noel,1),  'coords  NOEL,  x,y:',  ',  NPT',  COORDS ( 1 ) ,  NPT  C O O R D S (2) locoords(2),  locoords(3)  glcoords(noel,2)  ,  glcoords(noel,3) write  6 , *)  'peakepsch (1,2,3) ' ,  peakepsch (1),  peakepsch (2),  peakepsch(3) write  6, *)  'peakepsnpt(1,2,3)',  peakepsnpt(1),  peakepsnpt(2),  peakepsnpt(3) w r i t e ( 6 , *)  'currpeakepsnpt(1,2,3) currpeakepsnpt(2),  ',  currpeakepsnpt (1),  currpeakepsnpt(3)  w r i t e ( 6 , *) w r i t e ( 6 , *)  'ultepsch(1,2,3)  w r i t e ( 6 , *) w r i t e ( 6 , *)  'gammac(1,2,3)',  gammac(1),  gammac(2),  gammac(3)  'gammac(1,2,3)',  gammac(1),  gammac(2),  gammac(3)  w r i t e ( 6 , *) w r i t e ( 6 , *)  'gammadis(1,2,3)',  w r i t e ( 6 , *) w r i t e ( 6 , *)  'STRAN(1,2,3) ' ,  w r i t e ( 6 , *)  'DSTRAN(1,2,3)',  DSTRAN(1),  DSTRAN(2),  DSTRAN(3)  w r i t e ( 6 , *) w r i t e ( 6 , *)  'STRESS(1,2,3)',  STRESS(1),  STRESS(2),  STRESS(3)  w r i t e ( 6 , *) w r i t e ( 6 , *)  'sigma(1,2,3):',  sigma(l),  'sn(l) ,  sn(l) ,  w r i t e ( 6 , *) w r i t e ( 6 , *)  'Erl(1,2,3)  w r i t e ( 6 , *) w r i t e ( 6 , *)  'DDSDDE(1,1),(1,2):',  DDSDDE(1,1),  DDSDDE(1,2)  'DDSDDE(2,1),(2,2):',  DDSDDE(2,1),  DDSDDE(2,2)  ',  ultepsch(1),  'ultepsnpt(1,2,3)',  u l t e p s c h (2),  ultepsnpt(1),  ultepsch(3)  ultepsnpt(2),  ultepsnpt(3)  'gammarl(1,2,3)',  gammadis(1), gammarl(1),  STRAN(1),  'STRANcurr(1,2,3) ',  gammadis(2),  gammarl(2),  STRAN(2),  STRANcurr (1),  gammadis(3)  gammarl(3)  STRAN(3) STRANcurr(2),  S T R A N c u r r (3)  'fs',  fs sn(2) : '  'Et(l,2,3)  ', : ',  E r l ( l ) , Et(l),  sigma(2),  sigma(3)  sn(2)  E r l (2), Et(2),  Erl(3)  E t(3)  158  Appendix  D ASDM  UMAT for  write (6,* write (6,* write (6,* write(6,*) write (6,* write (6,* write (6,* write(6,*) step=0 endif  ) ) ) ) ) )  ABAQUS  ' D D S D D E ( 3 , 3 ) : ', D D S D D E ( 3 , 3 ) ' maxeps (1, 2)'., maxeps ( 1 ) , maxeps (2) ' g l s t r a i n ( n o e l , n p t , 1 ) ', g l s t r a i n ( n o e l , n p t , 1 ) 'glstrain(noel,npt,2)', glstrain(noel,npt,2) ' s t f i e l d ( 1 , 2 , 1 ) ', s t f i e l d ( 1 , 2 , 1 ) ' s t f i e l d ( 1 , 2 , 2 ) ', s t f i e l d ( 1 , 2 , 2 ) ' s t f i e l d ( l , 2 , 3 ) ', s t f i e l d ( 1 , 2 , 3 ) 'step', step  c c  *********** MATERIAL  MODEL E N D S H E R E  *****************************  C  Q c c c  *********** LOCAL store the strains temporary storage  S T R A I N STORAGE * * * * * * * * * * * * * * * * * * * * * i n x , y d i r e c t i o n f o r c u r r e n t NPT i n t h e array lostrain(nptmax,3)  610 i = l , 2 lostrain(npt,i)=STRANcurr(i) continue  do 610 c  if  (npt.eq.l) then locoords(1)=coords(1) elseif (npt.eq.7) then locoords(2)=coords(1) elseif (npt.eq.3) then locoords(3)=coords(1) endif c c c c  i f NPT = NPTMAX array. if  then  copy  the local  strains  to the global  strain  (npt.eq.nptmax) then 620 i=l,nptmax do 6 2 1 j = l , 2 glstrain (noel,i,j)=lostrain(i,j) continue continue do 6 2 2 i = l , 3 glcoords(noel,i)=locoords(i) continue  do  621 620  622 c  lostrain=0.0 locoords=0.0 c endif  c  write(6,*)  'end o f m a t e r i a l  loop'  c RETURN END  159  Appendix  E Transition  Radius and Peak Load Prediction  Plot for a DCB  Specimen  APPENDIX E TRANSITION RADIUS AND P E A K L O A D P R E D I C T I O N P L O T F O R A N AS4/3501-6 DCB S P E C I M E N  E.l  Specimen  T h e double-cantilever b e a m specimen ( D C B ) m o d e l e d here is s h o w n i n F i g u r e E . l w i t h dimensions. I t is m o d e l e d w i t h the material properties f o r u n i d i r e c t i o n a l A S 4 / 3 5 0 1 - 6 carbon fibre/epoxy  g i v e n i n Table E . l . T h i s is the same c o m b i n a t i o n o f specimen and material used  b y ( K a n j i , 2003).  E.2  Regime I  E n e r g y d r i v e n failure occurs f o r sharp notches. T h e energy failure c r i t e r i o n is developed i n the same w a y as f o r the O C T specimen. Since the stress c r i t e r i o n is satisfied first i n r e g i m e I, the relationship between the critical stress intensity factor and the critical energy release rate is g i v e n b y E q u a t i o n 2.9 repeated here as E.l  K =E'G . 2  IC  Jc  The stress intensity factor f o r a D C B specimen is g i v e n i n terms o f load a n d specimen dimensions b y (Tada, Paris and I r w i n , 1985) as  2V3~(%)b K,=C  E.2  y^-,  h  /2  where 1.0  plane stress  1  C =  F  v  2  T,  plane strain  C o m b i n i n g E q u a t i o n E. 1 and E q u a t i o n E.2 yields the p r e d i c t i o n f o r peak load i n regime I  , P  c  JE'G Bh' e  = -  T =  •  E  -  4  ClSa  F o r a g i v e n G , the peak load is invariant w i t h n o t c h root radius i n this regime. c  160  Appendix  E.3  E Transition  Radius and Peak Load Prediction  Plot for a DCB Specimen  Regime II  The stress failure c r i t e r i o n is used t o predict the peak load i n regime I I . A s was g i v e n i n E q u a t i o n 5.10, the sharp notch stress intensity factor can be related t o the critical stress i n a b l u n t notch t h r o u g h E q u a t i o n 5.9. The critical stress intensity factor is then  C o m b i n i n g the Ki-P  relation from E q u a t i o n E.2 w i t h E q u a t i o n E.5, the peak load i n regime  His  I Pf  =  ^  3/  .  H  E.  6  C4V3a A s w i t h the O C T specimen, the peak load is related t o the critical stress and the square root o f the notch root radius.  E.4  Transition  radius  The transition radius occurs at the p o i n t w h e n the peak load i n regime I is equal t o the peak load i n regime I I . C o m b i n i n g E q u a t i o n E.4 and Equation E.6, the transition radius is  4E'G  C  o  =  F7 2  r transition  "  ncj  c  Since  ptransition  is a material property, E q u a t i o n E.7 f o r a D C B specimen is exactly the same  as E q u a t i o n 5.13 f o r an O C T specimen.  E.5  Discussion  The peak load p r e d i c t i o n p l o t is presented i n F i g u r e E.2.  T h e peak load is predicted t o be  about 16.5 N f o r sharp notches and f o r this material the transition radius is predicted t o be 0.21 m m .  I t w o u l d actually be quite d i f f i c u l t t o obtain a n o t c h radius o f this size i n the  delamination specimen because it w o u l d mean that the resin layer between plies o n the delamination plane w o u l d have to be over 0.4 m m t h i c k at the notch tip. The t y p i c a l cured p l y thickness f o r this material (fibre and resin) is 0.132 m m ( H e x c e l C o r p o r a t i o n , 1998).  161  Appendix  E Transition  Radius and Peak Load Prediction  T h e interface strength plays a large role i n d e t e r m i n i n g stronger m o r e brittle m a t r i x is considered, is d o u b l e d to 200 M P a , The size o f  ptransition  ptransition  ptransition  Plot for a DCB ptransition-  Specimen  I f a material system w i t h a  w i l l greatly decrease. F o r example, i f cr  c  becomes 0.05 m m .  i n a D C B specimen is i m p o r t a n t because notches i n D C B specimens are  u s u a l l y not p e r f e c t l y sharp. A nonadhesive insert such as T e f l o n is usually inserted between plies d u r i n g processing t o create the i n i t i a l notch. A S T M provides guidelines o f h o w t h i c k the insert can be w i t h o u t affecting the results due t o n o t c h sensitivity.  I n A S T M D-5528  ( A S T M , 2002) the m a x i m u m thickness o f insert a l l o w a b l e is 0.013 m m . T h e transition radius provides an i n d i c a t i o n o f h o w t h i c k an insert can actually be w i t h o u t causing unstable fracture.  F o r u n i d i r e c t i o n a l A S 4 / 3 5 0 1 - 6 , the A S T M standard p r o v i d e s a safe value. F o r v e r y  strong, brittle interfaces, h o w e v e r , it becomes m o r e i m p o r t a n t t o use a v e r y t h i n nonadhesive insert t o create the i n i t i a l n o t c h . A theoretical brittle material w i t h an interlaminar strength o f 400 M P a a n d a strain energy release rate o f 6 0 J / m 2 w o u l d have a transition radius o f A  0.0064 m m , m e a n i n g that a 0.013 m m nonadhesive insert w o u l d be t h i c k enough t o make fracture  unstable.  162  Appendix Table E.l  E Transition  Radius  and Peak Load Prediction  Plot for  a DCB  Specimen  AS4/3501-6 material properties Parameter Ei  '  E  2  Vl2  G,  2  Gic  Value 102 G p a ' 9 Gpa  1  0.3' 7.1 G p a 125 J / m  1  2 1 1  100 M P a " 1  ' f r o m ( K a n j i , 2003) II  III  f r o m H e x c e l data sheets ( H e x c e l C o r p o r a t i o n , 1998) estimated f r o m values i n literature, e.g. (Jackson and M a r t i n , 1993; L i f s h i t z and Leber,  1998; W i s n o m , Jones and H i l l , 2001)  163  Appendix  E Transition  Radius and Peak Load Prediction  Plot for a DCB  Specimen  P  A  a  h  thicknessB,  Figure  E.l DCB specimen  a»h  (unidirectional  fibres  oriented  |  along length of  specimen)  Stress criterion  Energy criterion  \  —  Transition radius  /  I  0.05  0.1  0.15  0.2  0.25 p  Figure  E.2 Peak load prediction  0.3  0.35  0.4  0.45  0.5  (mm)  for AS4/3501-6  DCB  specimen  164  References  REFERENCES  A l f a n o , G and C r i s f i e l d , M A . 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