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An engineering approach to the simulation of gross damage development in composite laminates Floyd, Anthony Michael 2004

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AN ENGINEERING APPROACH TO THE SIMULATION OF GROSS DAMAGE DEVELOPMENT IN COMPOSITE LAMINATES by Anthony Michael Floyd B.Sc. (Physics), Dalhousie University, 1994 B.Eng. (Civil Engineering), The Technical University of Nova Scotia, 1997 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Civil Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA MARCH 2004 ©Anthony Michael Floyd, 2004 ABSTRACT It is challenging to predict the development of gross damage in polymer matrix composite (PMC) laminates accurately. Existing tools are not suitable for analyses of this sort; in general they predict initial failure but do not consider the post-failure behaviour, or they treat it in a relatively simple fashion. The accuracy of these tools often depends on the load case and material considered. A plane-stress continuum damage mechanics (CDM) constitutive model for gross damage development in PMC laminates has been developed previously by Williams et al. The model, CODAM, was implemented into a finite element analysis code and applied with considerable success to modelling crack growth in tension tests, and damage development in non-penetrating impact events. While sufficient for those specific applications, the model had several shortcomings including sensitivity to the size of elements, restrictions on the shape of stress-strain curve, an insufficient treatment of the characterization of the post-peak stress behaviour, and no consideration of the energy dissipated by individual elements. This work addresses these shortcomings. The most significant change is the inclusion of a technique that explicitly accounts for the energy dissipated by single elements. This change addresses strain localization, reducing the element-size dependency of the model, and improves on the physically based interpretation of the model parameters. By introducing the available fracture energy per unit area (GF) as a model parameter, characterizing the material in the post-peak stress regime (i.e., in gross damage states) becomes easier to understand and implement. Additionally, the description of the damage development has been generalized such that all the mathematical relationships are piece-wise linear, which allows for more flexibility and by allowing for concurrent, independent damage modes, increases the physical nature of the inputs. Other improvements including a generalization to three dimensions and the development of algorithms to treat penetrating impact events have been implemented but are not addressed in this thesis. The model has been exercised and validated in a number of applications. Damage development in unnotched, single edge notched, double edge notched, over-height compact tension, centre notched, and centre hole tension tests has been examined and compared to experiments. Additionally, the development of damage in two bending applications has been studied. The performance of the model in simulating structures of different sizes has been a focus of the work, and although the structural size effect in unnotched applications (which is mainly due to statistical effects) is not captured, the results show that CODAM is able to predict structural size effect in all notched tension applications. TABLE OF CONTENTS Abstract ii Table of Contents iii List of Tables vii List of Figures x Nomenclature xxiii Acknowledgements xxvi Chapter 1 : Introduction 1 1.1. COMPOSITES IN INDUSTRIAL AND CIVIL APPLICATIONS 2 1.2. RESEARCH INTO T H E MECHANICS OF COMPOSITE MATERIALS 5 1.3. NUMERICAL MODELLING OF T H E MECHANICS OF COMPOSITE MATERIALS 7 1.4. RESEARCH OBJECTIVES 9 1.5. THESIS OUTLINE 10 Chapter 2 : A Review of Damage Mechanisms and Non-linear Modelling of Composites 12 2.1. DAMAGE DEVELOPMENT IN LAMINATED COMPOSITES 13 2.2. MODELLING FAILURE IN COMPOSITE MATERIALS 22 2.2.1. The World-Wide Failure Exercise 22 2.2.1.1. W W F E Format 23 2.2.1.2. WWFE Initial Data - Material Data and Test Cases 26 2.2.1.3. WWFE Exercise Participating Failure Theories 27 2.2.1.4. WWFE Summary 35 2.2.2. Additional Approaches 38 Chapter 3 : Development of Continuum Damage Models for Composites 43 3.1. INTRODUCTION TO CONTINUUM DAMAGE MECHANICS 44 3.1.1. Model Scale 45 3.1.2. State Variables 48 3.1.3. Effective Stress and Strain Equivalence 51 3.1.4. Damage Growth 52 3.1.5. Effect of Damage 5 5 3.2. REVIEW OF CDM MODELS FOR COMPOSITE MATERIALS 57 3.3. OVERVIEW OF T H E CODAM M O D E L 61 3.3.1. Original Formulation 61 3.3.2. Model Improvements 62 3.3.3. CODAM in context of the WWFE 63 3.4. DETAILS OF T H E CODAM M O D E L 65 3.4.1 Effective Strain 65 3.4.2. Damage State 67 3.4.3. Modulus Reduction 68 Table of Contents 3.4.4. Damage Growth Curve 68 3.4.5. Normalized Modulus Curve 69 3.4.6. CODAM Stress-Strain Relationship 70 3.5. P L A N E S T R E S S SPECIALIZATION OF CODAM 7 3 3.5 .1 Simplifications 73 3.5.2. Elastic Constants 75 3.5.3. Effective Strain Interaction Constants 75 3.5.4. Effective Strain Parameters 76 3.5.5. Damage Parameters 79 3.5.6. Modulus Reduction Parameters 80 3.6. T H E SIMPLE DAMAGE MODEL 82 3.7. EXPLICIT AND IMPLICIT ANALYSIS 83 Chapter 4 : An Investigation into Element Size and Localization Effects 85 4.1. INTRODUCTION 86 4.1.1. Mesh Effect 86 4.1.2. Crack Band Method 88 4.1.2.1. Background and Formulation 88 4.1.2.2. Applications of the Crack Band Method in the Literature 90 4.1.3. Equivalent Work Method 91 4.1.4. Other Approaches 94 4.2. PRELIMINARY M E S H E F F E C T INVESTIGATION 95 4.2.1. Preliminary CODAM Analyses 95 4.2.2. Preliminary SDM Analyses 100 4.3. SDM AND T H E C R A C K BAND METHOD 105 4.4. T H E M E S H E F F E C T AND CODAM 112 4.4.1. Implementing the Crack Band Method into CODAM 113 4.4.2. Determining the Crack Band Scaling Factor. 117 4.4.3. CODAM and the Crack Band Method 121 4.4.4. CODAM and the Crack Band Method: Refinements 125 4.4.5. Mesh Width Effects and Stress-Based Scaling 128 4.4.5.1. Preliminary SDM and CODAM SNT Results Revisited 134 4.4.6. Implementing the Equivalent Work Method in CODAM 136 4.4.7. Crack Band Scaling versus Equivalent Work Scaling 139 4.5. SUMMARY 142 Chapter 5 : A Review of Structural Size Effects in Composites 143 5.1. INTRODUCTION 144 5.2. BACKGROUND 145 5.3. WEIBULL'S STATISTICAL DISTRIBUTION 147 5.4. EVIDENCE OF THE SIZE E F F E C T IN COMPOSITES AND MODELLING EFFORTS 150 5.5. SUMMARY 157 Chapter 6 : Simulations of Unnotched Tension Tests 160 6.1. T E S T GEOMETRY 161 6.2. MATERIAL 162 6.2.7. Quasi-isotropic Lay-up 762 6.2.2. Cross-ply Lay-up 767 6.3. OTHER PREDICTIVE APPROACHES 171 6.4. SIMULATIONS 172 6.4.7. Mesh Size Effect 772 iv Table of Contents 6.4.2. Structural Size Effect 175 6.5. SUMMARY 179 Chapter 7 : Simulations of Single-Edge and Double-Edge Notched Tension Tests 182 7.1. SINGLE NOTCHED TENSION SIMULATIONS 183 7.1.1. Test Geometry 183 7.1.2. Material 184 7.1.3. Other Predictive Approaches 189 7.1.4. Simulations 190 7AAA. Mesh Size Effect 191 7.1.4.2. Structural Size Effect 193 7.2. DOUBLE NOTCHED TENSION SIMULATIONS 196 7.2.1 Test Geometry 196 7.2.2. Material 197 7.2.3. Simulations 200 7.2.3.1. Mesh Size Effect 201 7.2.3.2. Structural Size Effect 202 7.3. SUMMARY 210 Chapter 8 : Simulations of Notched Tension Tests: S/RFI Laminates 211 8.1. OVER-HEIGHT C O M P A C T TENSION SIMULATIONS 212 8.1.1. Test Geometry 212 8.1.2. Material 214 8.1.2.1. Physical Characteristics 214 8.1.2.2. Numerical Characterization 216 8.1.2.3. Material Anomalies 222 8.1.3. Simulations 223 8.1.3.1. Preliminary Simulations 224 8.1.3.2. Refined Simulations 230 8.1.3.3. Pin Load vs. CMOD 233 8.1.3.4. Virtual Line Analysis 235 8.1.3.5. Mesh Size Effect 240 8.1.4. Summary 243 8.2. L A R G E N O T C H TENSION AND O P E N H O L E TENSION SIMULATIONS 245 8.2.1. Test Geometry 245 8.2.1.1. LNT Geometry 245 8.2.1.2. OHT Geometry 246 8.2.2. Material 247 8.2.3. Simulations 248 8.2.3.1. LNT Simulations 248 8.2.3.2. OHT Simulations 251 8.2.4. Summary 258 8.3. S/RFI SIMULATION SUMMARY 259 Chapter 9 : Conclusions and Future Work Recommendations 260 9.1. CONCLUSIONS 260 9.2. FUTURE WORK RECOMMENDATIONS 265 References 267 Appendix A : Simulations of Bending Tests 281 v Table of Contents A. 1. ECCENTRIC LOADING OF A BEAM-COLUMN 282 A l l Test Geometry 282 A 1 2 . Material 283 A 1 3 . Simulations 284 A. 1.3.1. Explicit Analyses 285 A. 1.3.2. Implicit Analyses 289 A.2. FOUR-POINT BENDING T E S T SIMULATIONS 293 A2.1. Test Geometry 293 A.2.2. Material 294 A.2.2.1. Angle-ply Lay-up 295 A2.3. Simulations 299 A.2.3.1. Quasi-lsotropic Lay-Up 300 A.2.3.2. Angle-Ply Laminates 303 A.2.4. Analytical Modelling 304 A.2.4.1. Effect of Through-Thickness Discretization 305 A.2.4.2. Structural Size Effect 318 A.3. SUMMARY 324 Appendix B : Proof of the Independence of the Stress-Strain Relationship from the Damage Parameter 327 Appendix C : An Example LPT and Ply Discount Calculation 331 vi LIST OF TABLES Table 2.1. Summary of approaches and failure theories invited to participate in the WWFE. (Adapted from Hinton and Soden, 1998) 24 Table 2.2. List of invitees who did not participate in the WWFE. (Adapted from Hinton and Soden, 1998) 25 Table 2.3. Results of the qualitative assessment of participating theories in the W W F E by the organizers. (Based on an unnumbered table presented in Hinton et al., 2002) 36 Table 3.1. model. Parameters required to describe a general 2D C O D A M constitutive 74 Table 4.1. Specimen sizes for preliminary tension simulations 95 Table 4.2. Laminate properties for [+45/0/90]s AS4/3502 96 Table 4.3. CODAM parameters for the quasi-isotropic lay-up of AS4/3502 97 Table 4.4. £ u n required to keep GF constant for the SDM single-notch tension simulations 106 Table 4.5. 25 mm Scale factor k required to keep GF constant for C O D A M simulations of the wide single notch tension test simulations 121 Table 4.6. DCB geometric parameters used in the study by McClennan (2003)... 128 Table 4.7. Average stress concentration factors for the SDM SNT simulations 135 Table 4.8. Average stress concentration factors for the C O D A M SNT simulations 135 Table 6.1. Laminar properties for AS4/3502 162 Table 6.2. Laminate properties for [±45/0/90]s AS4/3502 163 Table 6.3. C O D A M parameters for the quasi-isotropic lay-up of AS4/3502 165 Table 6.4. Laminate properties for [90/0]2s AS4/3502 167 Table 6.5. C O D A M parameters for the cross-ply lay-up of AS4/3502 169 Table 7.1. SNT specimen dimensions 184 vii List of Tables Table 7.2. Laminar properties for IM7/8551-7A 184 Table 7.3. LPT values for [0/±45/90]s IM7/8551-7A 184 Table 7.4. CODAM parameters for quasi-isotropic lay-up of IM7/8551-7A 187 Table 7.5. SNT parameters for the BASEL 190 Table 7.6. Specimen dimensions for DNT simulations 197 Table 7.7. IM7/8551-7A [0/902]s laminate elastic properties 197 Table 7.8. C O D A M parameters for cross-ply lay-up of IM7/8551-7A 199 Table 7.9. Energy parameters for saturation strain study 207 Table 7.10. Energy parameters for fibre initiation strain study 208 Table 7.11. Degree of size effect for the experiments, LEFM analysis, and the CODAM simulations of the DNT specimens 209 Table 8.1. Geometric parameters for O C T specimens 212 Table 8.2. S/RFI laminate properties supplied by The Boeing Company 216 Table 8.3. AS/3501 lamina properties (Agarwal and Broutman, 1990) 216 Table 8.4. LPT values for [±45/02/90/02/±45] AS/3501 216 Table 8.5. C O D A M parameters for S/RFI material (preliminary data set) 219 Table 8.6. Comparison of G F values for the previous study ("BSS") and the estimates using the current procedure 223 Table 8.7. CODAM parameters for S/RFI material (data set OCT-A) 225 Table 8.8. Refined 0° C O D A M parameters for the S/RFI material ("OCT-B") 232 Table 8.9. Comparison of OCT-A, OCT-B, and BSS data sets for S/RFI CODAM parameters 233 Table 8.10. Nominal failure stresses and nominal failure strains for the LNT experiments and simulations. The experimental results have been averaged. 251 Table 9.1 Comparison of previous C O D A M version and updated CODAM formulation 264 Table A.2. Laminar properties for AS4/3502 283 Table A.3. Laminate properties for [±45/0/90]s AS4/3502 284 viii List of Tables Table A.4. CODAM parameters for the quasi-isotropic lay-up of AS4/3502 284 Table A.5. Geometric parameters for the four-point bend tests 294 Table A.6. Laminate properties for [+45/-45/+45/-45]s2 AS4/3502 295 Table A.7. C O D A M parameters for the angle-ply lay-up of AS4/3502 298 Table A.8. Crack-band scaling factors (k) for the quasi-isotropic lay-ups used in the four-point bend simulations 300 Table A.9. Crack-band scaling factors (k) for the angle-ply lay-ups used in the four-point bending simulations 300 Table A. 10. Maximum net section moments and corresponding upper chord strains for each of the four-point bend geometries, without using crack-band scaling. 32 through-thickness integration points were used with the trapezoidal integration rule 319 Table A.11. Maximum net section moments and corresponding upper chord strains for each of the four-point bend geometries, using proper crack-band scaling and 32 through-thickness integration points with the trapezoidal integration rule 320 Table A. 12. Maximum net section moments and corresponding upper chord strains for each of the four-point bend geometries, using CODAM, implicit analysis, proper crack-band scaling, and 32 through-thickness integration points with the trapezoidal integration rule 320 Table A. 13. Maximum net section moments and corresponding upper chord strains for each of the four-point bend geometries, using proper crack-band scaling and variable through-thickness integration points (to approximate sublaminate level scaling) with the trapezoidal integration rule 322 Table A.14. Normalized failure force and corresponding cross-head displacement at failure for four-point bend analyses using an elastic perfectly-plastic material model. Note the variation in the results for identical simulations 323 Table C.1. AS/3501 lamina properties (Agarwal and Broutman, 1990) 331 Table C.2. Elastic engineering constants for the example laminate (see also Table 8.4) 334 Table C.3. Partially discounted elastic engineering constants for the example laminate 335 Table C.4. Fully discounted elastic engineering constants for the example laminate... 336 ix LIST OF FIGURES Figure 1.1. Schematic of the solving routine in explicit finite element analysis (Adapted from Williams, 1998) 8 Figure 2.1. Illustration of transverse matrix cracks 14 Figure 2.2. Illustration of fibre-matrix debonding (based on Fig. 3.17, Agarwal and Broutman, 1990) 14 Figure 2.3. Types of in-plane damage (adapted from Williams, 1998) 15 Figure 2.4. Illustration of delamination in a composite laminate 19 Figure 2.5. Characteristic "staircase" repeating damage pattern observed in through-thickness loading of PMC laminates (Adapted from Delfosse, 1994) 20 Figure 2.6. W W F E scores in quantitative tests, sorted by total Grade A and B scores. 36 Figure 2.7. Illustration of the Whitney-Nuismer point-stress and average-stress failure criteria 41 Figure 3.1. Illustration of analysis techniques at various length scales 45 Figure 3.2. Illustration of a representative volume element (RVE) of a composite laminate 47 Figure 3.3. Illustration of effective resisting area (Adapted from Chow and Wang, 1987). 49 Figure 3.4. Typical stress-strain diagram for original C O D A M formulation 62 Figure 3.5. Representative stress-strain curve for the current formulation of CODAM. 63 Figure 3.6. Illustration of replacing lamina based analysis with sublaminate based analysis 65 Figure 3.7. Representative damage growth curve 69 Figure 3.8. Representative normalized modulus curve 70 Figure 3.9. Sample one dimensional stress-strain curve resulting from the previous example damage growth and normalized modulus curves 71 x List of Figures Figure 3.10. Example of normalized modulus as a function of effective strain, based on the example damage growth and normalized modulus curves 71 Figure 3.11. Representative volume element subjected to uniaxial displacement loading 76 Figure 3.12. RVE with matrix cracks 77 Figure 3.13. RVE with matrix cracks and delaminations 77 Figure 3.14. RVE with matrix cracks, delaminations, and fibre cracks 78 Figure 3.15. RVE with saturated damage. Matrix damage and fibre damage have grown across the full RVE 78 Figure 3.16. Stress-strain relationship defined by the Simple Damage Model 82 Figure 4.1. Schematic of one dimensional localization 87 Figure 4.2. Illustration of replacing an element of a characteristic height with an integer number of smaller elements 92 Figure 4.3. Illustration of equivalent work scaling, in load-displacement space. Note that the load has been normalized with respect to the peak load 93 Figure 4.4. Mesh used in the preliminary 6.35 mm wide un-notched tension simulation 96 Figure 4.5. (a) Normalized modulus curve and (b) Stress-strain curve for the quasi-isotropic lay-up of AS4/3502 97 Figure 4.6. C O D A M results for the unnotched tension specimen simulations with non-uniform mesh. The coupon widths are relative to the smallest specimen, and the nominal failure stress and strain at failure have been normalized with respect to the failure of the smallest specimen 98 Figure 4.7. Mesh and damage patterns for the 6.35 mm wide specimen at the end of the simulation 98 Figure 4.8. C O D A M results for uniform meshed unnotched tension simulations. The coupon widths are relative to the smallest specimen, and the nominal failure stress and strain at failure have been normalized with respect to the failure of the smallest specimen 99 Figure 4.9. Mesh and damage patterns for the (a) 6.35 mm and (b) 25.4 mm wide specimens at the end of the simulation (using 1.27 mm elements) 100 Figure 4.10. Normalized nominal stress - average strain results for the 1D axial tension simulation with the Simple Damage Model. The stresses have been normalized to the peak stress exhibited by the smallest specimen. The inset plot shows the stress-strain behaviour of a single element. The results are virtually identical for the 2, 4, 8, and 16 element cases 101 xi List of Figures Figure 4.11. Schematic of SNT specimen. Note a=w/5 102 Figure 4.12. Nominal stress - average strain results for the 25.4 mm wide specimen simulation with the SDM and varying mesh densities. The nominal stresses have been normalized with respect to the ultimate stress of the simulation using 0.508 mm elements (-510 MPa) 103 Figure 4.13. Trends in the nominal failure stress and average strain at failure for the 25 mm single notched tension simulations with the SDM and varying mesh densities. The nominal failure stresses and average strains at failure have been normalized with respect to the simulation with 0.508 mm elements 104 Figure 4.14. Application of the crack band method to the SDM. The absolute stress for the preliminary simulations is -1560 MPa 105 Figure 4.15. Nominal stress - average strain results for the 25 mm wide single-notch tension simulations with the SDM and the crack band method incorporated. Note that the nominal stresses have been normalized with respect to the peak stress exhibited by the analysis with 0.508 mm elements (-340 MPa) 107 Figure 4.16. Trends in the nominal failure stress and the average strain at failure for the 25 mm wide fracture-energy controlled SDM single-notch tension simulations. The nominal failure stresses and average strain to failure have been normalized with respect to the results of the analysis with 0.508 mm elements 107 Figure 4.17. Nominal failure stress as a function of eun for each element size over a wide range of eun for the SDM simulations of the 25.4 mm width single-notch (a=w/5) tension specimen. The sun values are absolute, no scaling (e.g., crack band method) was involved 108 Figure 4.18. eun predicted by the crack band (CB) method using each element size as the reference for the GF that predicts a nominal failure stress of 600 MPa 110 Figure 4.19. Nominal stress - average strain results for the 25.4 mm wide single-notch tension simulations using the C O D A M model. The nominal stresses have been normalized with respect to the ultimate nominal stress predicted by the analysis using 0.508 mm elements 112 Figure 4.20. Trends in the nominal failure stress and average strain at failure for the 25 mm single notched tension simulations with the C O D A M constitutive model and varying mesh densities. The results have been normalized to the results of the analysis containing 0.508 mm elements 113 Figure 4.21. Expected stress—strain curve for C O D A M with an implementation of the crack band method 114 Figure 4.22. Illustration of the process used to determine stress from an arbitrary post-peak-stress strain in an element using the crack band-based approach 115 Figure 4.23. Relationship between the effective strain and normalized modulus for C O D A M with the crack band method integrated 116 xii List of Figures Figure 4.24. Plots of the simplified C O D A M parameters used for demonstrating the calculation of the scaling factor, k. For illustrative purposes, coms =0.5, £m s=0.5. ; 118 Figure 4.25. Stress-strain relationship for the reference element with a characteristic height, and the smaller element, scaled according to the crack band method with /c=2.75 121 Figure 4.26. Results of the 25 mm wide single notch tension specimen simulations with CODAM and with the crack band modifications. The stresses have been normalized to the results of the analysis performed with 0.508 mm elements.. 122 Figure 4.27. Trends in the nominal failure stress and the average strain at failure for the 25 mm wide fracture-energy controlled C O D A M single-notch tension simulations. The results have been normalized to those from the analysis with 0.508 mm elements 123 Figure 4.28. Nominal failure stress as a function of the crack-band scaling factor, k, for each element size over a wide range of k for the C O D A M simulations of the 25 mm width single-notch tension specimen 124 Figure 4.29. Results of the 25 mm wide single notch tension specimen simulations with CODAM, the crack band modifications, and the removal of the shear strain effects. The stresses have been normalized to the results of the analysis with 0.508 mm elements 126 Figure 4.30. Trends of the nominal failure stress and average strain at failure for the C O D A M simulations. Each run has been normalized to the 0.508 mm element width in order to provide comparisons to the effectiveness of the approach 126 Figure 4.31. Plot of the local stresses in the direction of loading, a y , near the notch tip for the 25 mm wide single-notch tension specimen simulations. A linear-elastic constitutive model without damage was used in the analyses. The three largest element sizes are shown, along with the linear elastic fracture mechanics solution 127 Figure 4.32. Schematic of the double cantilever beam test. The out-of-plane width of the specimen, S, is not shown 128 Figure 4.33. Critical strain energy release rate (G,c) for DCB simulations performed using elements of varying width. Data from analyses performed by McClennan (McClennan, 2003) 129 Figure 4.34. Average stresses along the notch-plane in DCB simulations using elements of varying width. The size of the first element is superimposed to provide context to the average stress results. Note that under-integrated (i.e., single integration point) elements were used. The data is from analyses by McClennan (McClennan, 2003) 130 Figure 4.35. Illustration of the need to reduce failure stress for larger elements 131 xiii List of Figures Figure 4.36. Average stresses in the first element ahead of the notch in the DCB analyses. Shown are expected results based on the stress field descriptions from an LEFM analysis of a sharp notch and the stress field predicted by the FE analysis using 0.0625 mm wide elements (McClennan, 2003). The actual results from the FE analyses are also shown 132 Figure 4.37. Illustration on how stress-scaling could be implemented 133 Figure 4.38. Stress-strain curve resulting from the simplified C O D A M parameters assumed in the demonstration calculations for equivalent work scaling 138 Figure 4.39. Illustration of evaluating sbi based on a known Sc 139 Figure 4.40. Comparison of the energy dissipated per unit area (GF) for an example material with a characteristic size of 10 mm, simulated with crack band scaling and equivalent work scaling for elements of 1.25 mm in height 140 Figure 4.41. Comparison of the stress-strain responses for an example material with a characteristic size of 10 mm, analyzed with crack band scaling and equivalent work scaling of elements of 1.25 mm in size 140 Figure 5.1. Laboratory test of damage development in stitched resin film infused carbon fibre epoxy laminate at the centimetre scale (left) and a wingtip spar (right) at the metre scale 144 Figure 5.2. An illustration of da Vinci's size effect experiment (Lund and Byrne, 2000). 145 Figure 5.3. Illustration of the strength of individual links in a short chain. The strength of the chain in this case is 92% the average strength of the links 147 Figure 5.4. Illustration of the strength of individual links in a long chain. The strength of the chain in this case is 85% the average strength of the links 147 Figure 5.5. Illustration of the effect of the Weibull modulus on the distribution of failure strengths 149 Figure 5.6. Illustration of the zones of size effects (Adapted from Fig 2.2 in Bazant, 2002) 158 Figure 6.1. Unnotched tension test specimen configuration 161 Figure 6.2. (a) Normalized modulus curve and (b) Stress-strain curve for the quasi-isotropic lay-up of AS4/3502 166 Figure 6.3. (a) Normalized modulus curve and (b) Stress-strain curve for the cross-ply lay-up of AS4/3502 169 Figure 6.4. Nominal stresses vs. average strains of the UNT simulations of the quasi-isotropic lay-up, using a coarse and fine mesh, nc=5 mm 173 xiv List of Figures Figure 6.5. Nominal stresses vs. average strain of the UNT simulations of the quasi-isotropic lay-up, using a coarse and fine mesh, frc=10 mm 173 Figure 6.6. Nominal stresses vs. average strains of the UNT simulations of the cross-ply lay-up, using a coarse and fine mesh, hc=5 mm 174 Figure 6.7. Nominal stresses vs. average strains of the UNT simulations of the cross-ply lay-up, using a coarse and fine mesh, /7C=10 mm 174 Figure 6.8. Nominal stress - average strain curves for the UNT simulations of the quasi-isotropic lay-up ([±45/0/90]n) of AS4/3502 using a characteristic height of /7C=5 mm 175 Figure 6.9. Nominal stress - average strain curves for the UNT simulations of the quasi-isotropic lay-up ([±45/0/90]n) of AS4/3502 using a characteristic height of fic=10mm 176 Figure 6.10. Nominal failure stress vs. specimen size curves for the UNT simulations of the quasi-isotropic lay-up ([±45/0/90]n) of AS4/3502. Also shown are the results of analysis with the statistical model and fracture model used by Jackson etal 176 Figure 6.11. Nominal stress - average strain curves for the UNT simulations of the cross-ply lay-up ([90/0]2n) of AS4/3502 using a characteristic height of /Jc=5 mm. 177 Figure 6.12. Nominal stress - average strain curves for the UNT simulations of the cross-ply lay-up ([90/0]2n) of AS4/3502 using a characteristic height of /?c=10 mm. 177 Figure 6.13. Nominal failure stress vs. specimen size curves for the UNT simulations of the cross-ply lay-up ([90/0]2n) of AS4/3502. Also shown are the results of analysis with the statistical model and fracture model used by Jackson et al. ..178 Figure 6.14. Nominal stress-strain results for quasi-isotropic AS4/3502 UNT simulations with small initiator defects 180 Figure 7.1. Schematic of SNT specimen 183 Figure 7.2. Normalized modulus vs. effective strain curve for quasi-isotropic lay-up of IM7/8551-7A 187 Figure 7.3. Stress-strain curve resulting from selection of C O D A M parameters for quasi-isotropic lay-up of IM7/8551-7A 188 Figure 7.4. Comparison between fine mesh (0.254 mm) simulation and coarse mesh (0.508 mm) simulation for the 25.4 mm wide SNT specimen without the crack band scaling 191 xv List of Figures Figure 7.5. Simulations of the four SNT specimen sizes using the same inputs, keeping the notch width to element width ratio constant. Note that the slight variation in the results is due to the different data densities in the simulation output 192 Figure 7.6. Nominal failure stresses for C O D A M analyses of the SNT specimens at three fracture energy levels. Also shown are the results from an LEFM analysis, the BASEL analysis, and the experiments 193 Figure 7.7. Comparison of the degree of size effect, normalized to that predicted by the LEFM analysis, predicted by the various analysis techniques shown in Figure 7.6 194 Figure 7.8. Double notched tension specimen schematic 196 Figure 7.9. Normalized modulus vs. effective strain for cross-ply lay-up of IM7/8551-7A used in the DNT simulations, (a) 0° direction, (b) 90° direction 199 Figure 7.10. Stress-strain curves for cross-ply lay-up of IM7/8551-7A used in the DNT simulations, (a) 0° direction, (b) 90° direction 199 Figure 7.11. Results of the coarse mesh simulations of the DNT tests 201 Figure 7.12. Results of the finely meshed DNT simulations using the same scaling factor (k = 89) 202 Figure 7.13. Results from the finely meshed DNT simulations, with proper consideration of fracture energy 203 Figure 7.14. Comparison of the finely meshed DNT simulations with the experimental results 204 Figure 7.15. Nominal failure stresses for the DNT simulations analyzed with different characteristic heights (fracture energies). Also shown are an LEFM analysis and the experimental results 205 Figure 7.16. Comparison of ultimate stresses for DNT simulations with equal characteristic heights in both the x- and y- directions, and with approximately equal fracture energies in both directions (in which case, n c , y =10mm, and hcx = 5 mm.) Note that the two curves overlap completely 206 Figure 7.17. Effect of varying damage saturation strains on the ultimate stresses exhibited by the DNT simulations, while maintaining the fracture energy constant. 207 Figure 7.18. Effect of varying fibre damage initiation strains on the ultimate stresses exhibited by the DNT simulations, while maintaining the fracture energy constant. 208 Figure 8.1. Schematic of the O C T specimen (from Mitchell, 2002) 212 xvi List of Figures Figure 8.2. Schematic of the OCT experimental set-up (from Mitchell, 2002) 214 Figure 8.3. Schematic of specimen orientations for OCT tests. The 0° material direction is coincident with the stitching direction, indicated by the dashed lines in the figure 215 Figure 8.4. Normalized modulus vs. effective strain for S/RFI material used in the OCT simulations for (a) 0° direction and (b) 90° direction (preliminary data set). 220 Figure 8.5. Stress-strain curves for S/RFI material used in the OCT simulations for (a) 0° direction and (b) 90° direction (preliminary data set) 220 Figure 8.6. Stress-strain relationships for 5-stack S/RFI material used in a previous study (Starratt et al., 1999) for the (a) 0° and (b) 90° orientations. Uses the "BSS" data set 221 Figure 8.7. Damage evolution in (i) the 4-stack and 6-stack S/RFI specimens, and (ii) the 5-stack specimens (From Mitchell, 2002) 222 Figure 8.8 "Coarse" OCT mesh with 0.625 mm x 0.625 mm elements in the zone of interest 224 Figure 8.9. Normalized modulus vs. effective strain for S/RFI material used in the OCT simulations for (a) 0° direction and (b) 90° direction (OCT-A data set).... 225 Figure 8.10. Stress-strain curves for S/RFI material used in the OCT simulations for (a) 0° direction and (b) 90° direction (OCT-A data set) 225 Figure 8.11. Illustration of crack growth in the 5-stack 90° simulations 226 Figure 8.12. Pin load vs. CMOD for 90° orientation. The 4-, 5-, and 6-stack experimental data have been normalized to the 5-stack thickness. The inset figure indicates the direction of crack growth 227 Figure 8.13. Illustration of crack growth in the 5-stack 0° simulations 228 Figure 8.14. Pin load vs. CMOD for 5-stack 0° orientation simulation using the OCT-A data set as well as two experimental results. The inset picture indicates the direction of crack growth 228 Figure 8.15. Illustration of crack growth in the 4-stack 0° simulations 229 Figure 8.16. Pin load vs. CMOD for 4-stack 0° orientation simulation using the OCT-A data set as well as three experimental results. The inset picture indicates the direction of crack growth 230 Figure 8.17. 0° simulations showing progression of the reduction of damage initiation strains for (a) 4-stack laminates and (b) 5-stack laminates. OCT-A is the original data set 231 XVII List of Figures Figure 8.18. (a) Normalized modulus curve and (b) stress-strain curve for OCT-B C O D A M parameters 232 Figure 8.19. Pin load vs. C M O D for 4-stack 0° orientation simulations using the OCT-A and OCT-B data sets as well as three experimental results. The inset picture indicates the direction of crack growth 234 Figure 8.20. Pin load vs. C M O D for 5-stack 0° orientation simulations using the OCT-A and OCT-B data sets as well as two experimental results. The inset picture indicates the direction of crack growth 234 Figure 8.21. Numerical (OCT-A) and experimental line analysis for the 4-stack 0° O C T test. The solid markers (numerical) correspond to the open markers (experiment), and each marker type corresponds to a different C M O D as indicated in the inset figure 235 Figure 8.22. Numerical (OCT-B) and experimental line analysis for the 4-stack 0° O C T test. The solid markers (numerical) correspond to the open markers (experiment), and each marker type corresponds to a different C M O D as indicated in the inset figure 236 Figure 8.23. Numerical (OCT-A) and experimental line analysis for the 5-stack 90° O C T test. The solid markers (numerical) correspond to the open markers (experiment), and each marker type corresponds to a different C M O D as indicated in the inset figure 237 Figure 8.24. Numerical (OCT-A) and experimental line analysis for the 5-stack 0° O C T test. The solid markers (numerical) correspond to the open markers (experiment), and each marker type corresponds to a different C M O D as indicated in the inset figure 238 Figure 8.25. Numerical (OCT-B) and experimental line analysis for the 5-stack 0° O C T test. The solid markers (numerical) correspond to the open markers (experiment), and each marker type corresponds to a different C M O D as indicated in the inset figure 238 Figure 8.26. Comparison of the line analysis of the 4-stack and 5-stack simulations of the 0° tests, at the y = 10 mm inscribed line pair. The open symbols are the 5-stack simulation, and the closed symbols are the 4-stack simulation 240 Figure 8.27. Comparison of the load-displacement results for the 90° O C T simulations. The fine mesh uses square elements 0.625 mm on each edge, while the coarse mesh is twice the size, with 1.25 mm elements. The difference in peak force is about 11% 241 Figure 8.28. Comparison of the load-displacement results for 0° O C T simulations using a modified OCT-A data set (saturation strains are 4%) and (a) a lower-bound GF, and (b) an upper bound GF. In each case, the difference in peak load is approximately 10% 242 xviii List of Figures Figure 8.29. Plot of the axial (load direction) stresses for the O C T simulations shown in Figure 8.28(a). The difference in the peak stresses is 15% 242 Figure 8.30. Schematic of the LNT specimens. P is the applied load, t is the specimen thickness, H is the specimen height, W is the specimen width, and a is half the notch width 246 Figure 8.31. Schematic of the OHT specimen. P is the applied load, t is the specimen thickness, H is the specimen height, W is the specimen width, and d is centre hole diameter 247 Figure 8.32. Damage pattern at the end of simulating the LNT test (using the OCT-B data set) 249 Figure 8.33. LNT simulation load-displacement results (using the OCT-B data set). 249 Figure 8.34. Damage pattern at the end of simulation of the small LNT test (using the OCT-B data set) 250 Figure 8.35. Small LNT load-displacement results (using the OCT-B data set) 250 Figure 8.36. Damage patterns at the end of the simulation of the OHT specimens (using the OCT-B data set) 252 Figure 8.37. Stress distribution (ay) expressed as stress concentration factors for the fine (0.625 mm) element size and the coarse (1.25 mm) element size OHT simulations (in an undamaged state) 253 Figure 8.38. OHT load-displacement results 253 Figure 8.39. Stress distributions (oy) expressed as stress concentration factors (SCF) for the OHT simulation and an O C T simulation with the load applied as a distributed load along the specimen ends. The S C F in the element closest to the notch is 27.6 in the O C T simulation and 2.37 in the element closest to the hole edge in the OHT simulation 254 Figure 8.40. Plot of the stress concentration factors in the elements nearest the notch/hole edge and the theoretical S C F for (a) the O C T simulations and (b) the OHT simulations 256 Figure 8.41. Force-displacement results for the OHT simulation with an increased peak stress, compared to the original (OCT-B data set) OHT simulation and the experimental results 257 Figure A. 1. Schematics of the bend test set-up used by Jackson, (a) Load and specimen configuration and (b) details of the hinge assembly. (Adapted from Jackson, 1992) 282 Figure A.2. Schematic of the finite element representation of the bend tests 283 xix List of Figures Figure A.3. (a) Normalized modulus curve and (b) Stress-strain curve for the quasi-isotropic lay-up of AS4/3502 284 Figure A.4. Load-displacement results for explicit simulations of the 1/6 and 2/3 scale specimens. The load has been normalized to the Euler buckling load of the specimen, and the crosshead displacement has been normalized to the original specimen length 285 Figure A.5. A progression of deformed shapes showing failure of the bending specimen predicted using explicit analysis. Note the asymmetric failure shape, caused by momentum-related vibrations 286 Figure A.6. Global energy in the explicit beam-bending simulation, shown for the 1/6 scale specimen 287 Figure A.7. Plot of energy dissipated by loading the beam-columns to a slightly damaged state and then unloading. The energy dissipated has been normalized to the peak elastic strain energy (equivalent to the total work) in identical simulations using an elastic material model 289 Figure A.8. Load-displacement results for implicit simulations of the 1/6 and 2/3 scale specimens 290 Figure A.9. Normalized failure loads vs. relative specimen size for the implicit analysis of the eccentrically loaded beam-columns. Note that the range of the normalized failure load used in this plot is quite narrow, which exaggerates the size effect 291 Figure A. 10. Relative failure load vs. relative specimen size for the various fracture energies and the experiments for the implicit simulations of the eccentrically loaded beam-columns 292 Figure A.11. Schematic of the four-point bend test. The specimen width is 12.5n mm and the thickness is approximately 2n mm, where n = 1, 2, 4 293 Figure A. 12. Illustration of the deformed shape of the four-point bend simulation 294 Figure A. 13. Illustration of angled fibres completely spanning the damage zone 296 Figure A.14. Illustration of damage in an angle ply with no fibre scissoring. No fibre damage is present in this damage mode 297 Figure A. 15. (a) Normalized modulus curve and (b) stress-strain curve for the angle-ply lay-up of AS4/3502 298 Figure A. 16. Normalized displacement at ultimate load for the ply-level scaled four-point bending simulations of the quasi-isotropic laminates. Also shown are the experimental results and analytical predictions using LEFM and a Weibull-based approach (Johnson et al., 2000) 301 xx List of Figures Figure A. 17. Normalized displacement at ultimate load for the sublaminate scaled four-point bending simulations of the quasi-isotropic laminates. Also shown are the experimental results and analytical predictions using LEFM and a Weibull-based approach (Johnson et al., 2000) 302 Figure A. 18. Normalized displacement at ultimate load for the ply-level scaled four-point bending simulations of the angle-ply laminates. Also shown are the experimental results and analytical predictions using LEFM and a Weibull-based approach (Johnson et al., 2000) 303 Figure A. 19. Normalized displacement at ultimate load for the sublaminate scaled four-point bending simulations of the angle-ply laminates. Also shown are the experimental results and analytical predictions using LEFM and a Weibull-based approach (Johnson et al., 2000) 304 Figure A.20. Schematic of integration point locations for the Gaussian integration scheme and the trapezoidal integration scheme, for 8 through-thickness integration points 306 Figure A.21. Four-point bend simulations with varying degrees of through-thickness discretization. Except for those indicated as Gaussian, the trapezoidal integration rule was used. The n = 1 geometry is shown, with the quasi-isotropic inputs, and hc = 10 mm. For overlapping curves, the tip of the label leader indicates the end of the curve 307 Figure A.22. Plot of the net section moment as a function of the strain at the surface of the cross-section under a pure bending load. In this particular case, the inputs correspond to the quasi-isotropic lay-up of AS4/3502 and the n = 2 four-point bend geometry. No crack-band scaling has been used (k = 1) 309 Figure A.23. Plot of the net section moment as a function of the strain at the surface of the cross-section under a pure bending load. In this particular case, the inputs correspond to the quasi-isotropic lay-up of AS4/3502 and the n = 2 four-point bend geometry. Proper crack-band scaling has been employed, and for this example, k= 16.527 309 Figure A.24. Convergence of the maximum net section moment and strain at maximum net section moment for the n = 2 four-point bend geometry using the trapezoidal rule. No crack-band scaling has been used, k= 1 310 Figure A.25. Convergence of the maximum net section moment and strain at maximum net section moment for the n = 2 four-point bend geometry using the trapezoidal rule. Appropriate crack-band scaling has been used, k = 16.527.. 311 Figure A.26. Progression of damage through the cross-section of a beam in pure bending. In this case, 16 through-thickness integration points have been used with the trapezoidal integration rule, on the n = 2 geometry with the quasi-isotropic AS4/3502 inputs. Note that no crack-band scaling has been employed (k = 1). Figure A.26c is at the maximum section moment 312 xxi List of Figures Figure A.27. Progression of damage through the cross-section of a beam in pure bending. In this case, 16 through-thickness integration points have been used with the trapezoidal integration rule, on the n = 2 geometry with the quasi-isotropic AS4/3502 inputs, and k = 16.527. Figure A.27d is at the maximum section moment 314 Figure A.28. Progression of damage through the cross-section of a beam in pure bending. In this case, 16 through-thickness integration points have been used with the trapezoidal integration rule, on the n = 2 geometry with the quasi-isotropic AS4/3502 inputs, with an elastic-perfectly plastic constitutive model with a yield stress equivalent to the peak stress in the CODAM formulation (~566 MPa). Figure A.28f is at the peak net section moment 315 Figure A.29. Net section moment as a function of the outer chord strain for an elastic-perfectly plastic material. The n = 2 geometry is shown, analyzed with 32 through-thickness integration points 316 Figure A.30. Convergence of the maximum net section moment and strain at maximum net section moment for the n = 2 four-point bend geometry using the trapezoidal rule and an elastic-perfectly plastic constitutive model 317 Figure A.31. Illustration of stress distribution in pure bending 318 Figure A.32. Structural size effects in the four-point bend tests. The analytical and numerical C O D A M normalized outer chord strains are shown for 32 through-thickness integration points only. Also shown are the experimental results for both ply and sublaminate scaled cases, expressed as the normalized crosshead displacement (Johnson et al., 2000) 321 Figure A.33. Comparison of ply-level equivalent scaling and sublaminate-level equivalent scaling, using the analytical method 322 Figure B.1. Representative C O D A M parameter curves and normalized stress -average strain curve resulting from the illustrative assumptions 328 xxii NOMENCLATURE Superscripts Compression c Relating to a characteristic size e Relating to an element f Fracture o Relating to an original or initial state ult Ultimate Subscripts J Final (usually preceded by m or f) _/' Initial (usually preceded by m or f) s Saturated (usually preceded by m or f) 1, 2, 3 Laminate principal directions l 2 In Weibull analysis, the first and second sizes bi Relating to a bifurcated element c Relating to a characteristic size e Relating to an element e Relating to an elastic state f Relating to fibre f In the Simple Damage Model, the final state L Lateral (parallel to fibre direction) LB Lower bound m Relating to matrix m In Jackson's strength analysis, relating to the model max Relating to the maximum stress N Nominal n, n-1, etc. Relating to the nm, n-1m, etc., element o Relating to an original or initial state P Relating to a plastic state P In Jackson's strength analysis, relating to the prototype P In the Simple Damage Model, relating to the peak stress state peak Relating to the peak stress ret Reference size s Relating to the scaled state T Transverse (perpendicular to fibre direction) UB Upper bound u Ultimate ult Relating to the ultimate strain x, y, z Global co-ordinate system (load directions) Nomenclature Latin Symbols a Notch length a 0 In the average stress failure theory, the characteristic distance B Constant based on plastic limit analysis for Bazant Approximate Size Effect Law b Width cN Calibration coefficient for Bazant Approximate Size Effect Law C Stiffness matrix c In OCT tests, the distance between load pin centres D In BASEL, characteristic dimension D In thermodynamic formulations of CDM, the damage state variable D 0 Constant depending on fracture process zone and specimen size for Bazant Approximate Size Effect Law d In OCT tests, the diameter of the load pins d0 In the point stress failure theory, the characteristic distance 5 Normalized instantaneous modulus (normalized modulus for brevity) E Modulus, but when succeeded by subscripts (m_, f_, etc.) modulus reduction factor F Damage potential (effective strain) F(x) Generic function of x F\y Interaction term in Tsai failure criteria fu Material reference strength fH In OCT tests, the separation between inscribed lines Q Normalized instantaneous shear modulus GF Fracture energy per unit area G/c Critical strain energy release rate (mode I) H, h Height k Scaling factor required to scale fracture energy in CODAM K, L, M In CODAM, strain interaction constants I, I Length M Damage effect transformation, damage effect tensor M, M Moment, normalized moment m Weibull modulus n Number of plies h Unit vector P Load Q Lamina stiffness tensor R Size increase of the plastic yield surface r Isotropic hardening state variable S In Tsai failure criterion, shear strength S In the Weibull model, probability of survival S In Jackson's failure analysis, strength S In CDM, the cross-sectional area of an RVE S In CDM, the effective resisting area after loading S, T, U In CODAM, shear strain interaction constants T Temperature t Thickness V Volume W In OCT tests, the distance from the centre of the load pins to the compression edge We Elastic strain energy density w Crack opening displacement, for cohesive crack model w Specimen width X Y Longitudinal and transverse strength (Tsai failure criterion) V Damage strain energy release rate xxiv Nomenclature Greek Symbols CO, 0) p <J, & £, £ YF P P 5 A v In the BASEL, the relative structure size, In Jackson's strength analysis, the Weibull shape parameter Elongation Total elongation Strain, strain rate Fracture energy density In Jackson's strength analysis, the geometric scale factor Poisson's ratio Density Stress, stress rate Effective stress Damage parameter, damage parameter growth rate Helmholtz free energy Note, in this thesis, there are often three co-ordinate systems: the lamina co-ordinate system, the laminate co-ordinate system, and the global load co-ordinate system. For consistency and clarity, the subscripts L and T are used to refer to the lamina co-ordinate system, the longitudinal (parallel to fibre) direction and transverse (perpendicular to fibre) direction. The sub-scripts 1, and 2, traditionally used to denote the lamina directions, in this case refer to the laminate co-ordinate system. The 1 direction is parallel to the 0° lamina fibre direction, and the 2 direction is perpendicular to that in the plane of the laminate. The subscripts x and y refer to the global load coordinate system. In this manner, there is no need to refer to a "new" laminate each time a laminate is subjected to a load applied in a different direction. xxv ACKNOWLEDGEMENTS A thesis is never the work of just one person and this is no exception. In particular, I want to acknowledge the involvement of my supervisor of many years, Dr. Reza Vaziri. Reza has shepherded me through my life as a graduate student with many fascinating discussions, challenging me when needed, encouraging me when necessary, and providing wisdom, insight and patience at all times. Dr. Anoush Poursartip has also been particularly helpful, always providing insight firmly rooted in reality, confronting me when I wandered outside the realm of physics, and encouraging me with his excitement and passion for learning. The UBC Composites Group has been an unending source of interesting experiences. Group members past and present have astonished me with their knowledge and astounded me with their friendship. Many engaging and lively discussions have guided me along my path. In particular, Mr. Karim Kanji, Mr. Scott McClennan, and Mr. Ali Shahkarami have been invaluable in their assistance. Graduate students do not operate within a vacuum, and I was fortunate to work beneath the umbrella of both the Department of Civil Engineering and the Department of Metals and Materials Engineering. I would be remiss not to acknowledge the assistance of staff and faculty from both departments. Financial support from the Natural Sciences and Engineering Research Council (NSERC), the University of British Columbia, and the generous support of Dr. Anoush Poursartip and Dr. Reza Vaziri is truly appreciated. Finally, I need to acknowledge my anchor in stormy seas, my beacon in the darkness, the inspiration of my life, my loving and understanding wife, Gwendolyn. She has supported me through thick and thin; degrees should be given to spouses who have to endure the completion of a thesis. Non illigitamus carborundum. xxvi Chapter 1: Introduction This thesis addresses the prediction of gross damage development in laminated fibre reinforced polymer matrix composites. Gross damage development, in this context, means significant damage at the structural scale. Cracks, holes, and delaminations (the separation of laminated plies) in structural components are all examples of gross damage. Smaller manifestations of damage such as matrix cracking and small delaminations are considered in the discussion and analysis but are not, per se, the focus of the research. As this research concentrates on laminated polymer matrix composites (PMCs), for simplicity they will be hereafter called "composites" or "composite materials". Obviously, other sorts of composites exist: ceramic matrix composites and metal matrix composites are examples of some other traditional composite materials, and wood products, some reinforced cementitious materials, and some bio-materials can also be considered as composites. The research presented in this thesis applies explicitly only to PMCs, but some insight may be gained by those interested in other types of composite materials. 1 Chapter 1: Introduction 1.1. Composites in Industrial and Civil Applications As a class of materials, composites have had what has been seen by some as a slow adoption rate in industrial and civil applications. Compared to polymers, this is undeniable. History is often categorized by the type of materials used by the society of the time, from the Stone Age to the Age of Steel. In this context, the second half of the 20 t h century was certainly the Plastics Age. In structural applications however, unreinforced polymers are generally unsuitable and while plastics were widely adopted in manufactured goods, other materials such as aluminium and advanced metal alloys, ceramics, engineered wood products (EWPs), and polymer composites were changing the materials used in structures. The rate of adoption for these advanced materials has been slow compared to plastics, but much faster than other structural materials at any point in history. In the 21 s t century, the full potential of engineered materials will be realized. Each of these advanced materials has innate difficulties to overcome. Perhaps the materials with the fewest difficulties are the aluminiums and advanced metal alloys. These materials are in a form familiar to designers and manufacturers, and older design and manufacturing methods can be applied in working with them. The benefits of the advanced metals can be realized within the comfort of the existing design and manufacturing methodologies. The limitations of the materials (manufacturability, weldability, hardness, etc.) are inconveniences that have been generally addressed. Other advanced materials have had a more difficult path. Certainly, no material is ideal in all applications, but often advanced materials can be used in applications that they were not originally intended. EWPs, ceramics, and PMCs are all examples of advanced materials that have not been adopted as fully as they could be due to the changes that working with these materials require. These materials generally cannot be designed or manufactured using the well established procedures that have evolved for metals. Often, you cannot even separate design from manufacturing, as these materials are truly engineered materials. This reluctance to adopt these advanced materials has led to a chicken-and-egg problem. Research in these areas is slow and under-funded with a few notable exceptions: the defence and aerospace industries have been very active in the 2 1.1. Composites in Industrial and Civil Applications development and use of advanced materials. The nature of those industries has required that new and emerging material technologies be actively pursued. In mainstream industrial applications and in structural (civil) applications, research has been lacking. Due to the lack of research and technology transfer from the research institutions to industry, industrial and civil adoption has been slow. However, this slow industrial adoption leads to slow research and development, especially in today's atmosphere of the expectation of instant realization of investments and research funds. For composites, the perceived difficulties are well known: 1. The design of composites requires engineers to learn new techniques, use different methods, develop a deeper understanding of the material, and is generally harder than designing with metals. 2. Composite manufacturing is expensive, time consuming, labour intensive, and the product quality can be highly variable. 3. Composites are considered new materials and generally the public does not adopt new ideas quickly. Consider each of these points: 1. The design of composites requires engineers to learn new techniques, use different methods, develop a deeper understanding of the matehal, and is generally harder than designing with metals. This issue is perhaps the most challenging to address. If engineers are not comfortable designing, specifying, or using a particular class of material then its adoption into any industry is not going to occur. It falls to researchers to provide engineers with tools that make their work more efficient and easier while preserving safety in design. These tools should include codes and standards that allow for empirical design of composites, as well as more advanced analytical and numerical tools that allow for in-depth design. Unfortunately, the development of such tools is difficult. The understanding of the behaviour of PMCs within the research community is far from the understanding of the behaviour of metals. The elastic behaviour of composites is reasonably well understood, or would be if all composites could be manufactured to idealized tolerances. The inelastic behaviour: plasticity, fatigue, damage, etc., is much less understood. The mechanisms are mostly known, but scientific predictions of the conditions that cause the 3 1.1. Composites in Industrial and Civil Applications initiation and growth of the mechanisms lack accuracy outside the specific application for which a specific theory was developed. It will always be true that engineers with backgrounds in metals will have to learn different concepts when working with composites. Composites are not metals, and will never behave like metals. Similar to the different concepts required to design with traditional concrete and traditional wood materials, new concepts and design and analysis techniques will be required for the use of composites. However, once the basic concepts are learned there should be no reason why a sufficient "toolbox" of design aids could not be assembled for engineers to use in the design of structures using composite materials. 2. Composite manufacturing is expensive, time consuming, labour intensive, and the product quality can be highly variable. The perception may be there, but the reality is different. Raw material cost for composites have steadily declined and the trend will continue as more composites are used in industry. Additionally, improved manufacturing techniques are constantly being developed, lowering the manufacturing costs and time while increasing the product quality. 3. Composites are considered new materials and generally the public does not adopt new ideas quickly. This issue seems like one that could resolve itself given sufficient time, but perhaps should be addressed more directly. Several high-profile failures of composites in recent years (the vertical stabilizer failure of AA 587 over New York, the failure of the leading edge of the portside wing on the Space Shuttle Columbia) have tarnished the reputation of the material class in general. Proactive marketing of composites, especially in industries that the public can relate to, such as recreational, civil and bio-medical applications may be required to further the adoption of composites. 4 Chapter 1: Introduction 1.2. Research into the Mechanics of Composite Materials Research into the mechanical behaviour of composite materials can be divided into the three categories of research: experimental testing, analytical investigation, and numerical modelling. In this context, the distinction is drawn between analytical approaches and numerical approaches. Analytical modelling predicts the behaviour of composites through the development and use of a system of equations. These equations are useful in describing the behaviour of small representative volumes of material, and are often used to produce response surfaces (such as failure surfaces) that capture the behaviour of the material in a wide variety of stress or strain states. Numerical modelling is closely related to analytical modelling, in that a system of equations are developed and used to describe the behaviour of a small representative volume of material. The equations may be identical to the analytical equations, or significantly different, depending on the level of sophistication demanded by the model. What differentiates numerical modelling from analytical modelling is that numerical modelling often considers thousands of these small representative volumes of material simultaneously. This is achieved by using numerical approaches such as the finite element method. Numerical modelling allows for the consideration of a wide range of sizes, from the fibre and matrix level right to the complete structure scale. Significant research in composite materials has been experimental in nature, despite the expenses associated with manufacturing and testing. Analytical and numerical research in the structural mechanics of composites first addressed undamaged behaviour but has since moved to predicting failure and damage. Analytical and numerical research in other areas of interest, such as the manufacturing of composites, has been extensive but will not be considered in the context of this thesis. A thorough and robust research programme in the mechanical behaviour of composites should consist of a number of experiments to try to understand the physics of the mechanics, followed by analytical and numerical representations of the mechanisms. The analytical and numerical analyses should then be compared to the experiments and further refinement of the experiments or models may be required. Often, this sort of arrangement is difficult to achieve, usually due to the cost. Instead, experiments are performed on a particular application and published. The experiments are studied by 5 1.2. Research into the Mechanics of Composite Materials other researchers, and analytical or numerical theories are proposed and exercised against the small set of experimental data. Typically, the feedback from the analytical or numerical theories is not addressed by subsequent experimentation, nor are the theories exercised against other experimental data sets. One effort to assemble many of the analytical failure theories in composites and compare their performance to a wide number of applications and data sets was the "World Wide Failure Exercise". This impressive exercise is discussed in detail in § 2 . 2 . 1 . 6 Chapter 1: Introduction 1.3. Numerical Modelling of the Mechanics of Composite Materials Numerical modelling in the context of this thesis uses the finite element method to predict the mechanical behaviour of composites. This is achieved using the LS-DYNA finite element program, developed by Livermore Software Technology Corporation (LSTC). LS-DYNA is a derivative of DYNA3D, a hydro-code developed at the Lawrence Livermore National Lab (LLNL) and is primarily an explicit finite element solver that solves the finite element method equations stepping though time. LS-DYNA has recently included an implicit solver of the sort traditionally used by mainstream finite element codes such as A N S Y S or ABAQUS. LS-DYNA has been selected as the solver due to the nature of the applications examined through the development of the constitutive model described herein: crack growth and impact damage in materials that exhibit a strain-softening behaviour. The explicit solver allows for the detailed examination of dynamic events and permits material behaviours that are described by negative or zero tangential stiffnesses. These two conditions are not easily handled by implicit solvers. In explicit finite element analysis, the constitutive model is one of several modules used in determining the stress and strain state of an element of material. In general, the solving engine (in this case, LS-DYNA) passes the change in strain and strain rate for each integration point (typically there is one in-plane integration point per element) to the constitutive model. The constitutive model then uses this information as well as the past strain history of the integration point to determine the corresponding stress state. In elastic analysis, this might be as simple as multiplying the strains by the stiffness matrix. In damage analysis, the strain state and strain history are used to determine the state of damage and the effect of damage. The constitutive model passes the stress state back to the solving engine, and the solver determines the nodal accelerations. From the accelerations, the equations of motion are solved, nodal displacements and velocities are determined and the change in strain and strain rate are determined for the next time step. This process is illustrated in Figure 1.1 7 1.3. Numerical Modelling of the Mechanics of Composite Materials Figure 1.1. Schematic of the solving routine in explicit finite element analysis (Adapted from Williams, 1998). Numerical modelling can occur at many different scales. Some models have been developed that consider the constituents of composite materials, fibres, matrices and their interfaces, while other models have been developed that consider the material at a much larger scale, at the sublaminate or laminate level. The details of the modelling approaches will be detail later in the chapters that follow. 8 Chapter 1: Introduction 1.4. Research Objectives This thesis strives to improve a tool that contributes to an "engineer's toolbox" of aids for the design and analysis of composite laminates. This tool is in the form of a constitutive model for gross damage development in composite laminates, CODAM. The original C O D A M model (Williams, 1998; Williams et al., 2003) had a number of limitations (discussed in §3.3.1. ), that have been addressed so that the model is more versatile, more accurate, and easier to use and understand. In this context, the objectives of this thesis are to: • Examine the original C O D A M model, and: 1. Consider the literature to determine how C O D A M fits within the context of existing models, and what can be learned from other models or approaches to improve CODAM. 2. Generalize the model's formulation, allowing for analyses that can consider generalized damage development. 3. Make C O D A M more physically-based by providing guidelines for the selection of material parameters, and by considering the introduction of additional parameters as needed. • Using the improved model, demonstrate its suitability for modelling gross damage development in a variety of loading states, including: 1. Unnotched tension and bending applications. 2. Edge notched tension applications. 3. Centre notched tension applications. • Using the improved model, demonstrate its suitability for modelling gross damage development over a range of structural sizes. 9 Chapter 1: Introduction 1.5. Thesis Outline The thesis is basically organized into two sections. Chapters 2 - 5 address the background, development, and refinements of the model while Chapters 6 - 8 address the performance of the model in various applications. The remaining chapter and appendices provide closure to the study. The first section addresses diverse topics and is organized such that a literature review on the particular chapter topic introduces each chapter. Where appropriate, work performed in this thesis follows the literature review. The chapters in the second section do not begin with a literature survey; rather they introduce the application and present the new work performed in each case. This approach means that there is no single encompassing "literature review" chapter, and that each chapter can be read independently of the others. However, with this in mind, Chapter 2 and Chapter 5 consist only of literature reviews. Chapter 2 provides a general overview of the state of composite materials, and some detail on damage development in composites. It goes on to describe various approaches to modelling damage development in composites and concentrates in some detail on the World-Wide Failure Exercise. Chapter 3 continues the discussion on modelling damage in composite materials, but concentrates on continuum damage mechanics (CDM). The basics of CDM are discussed, as are some CDM models for composite materials. The CODAM constitutive model is then described in detail, along with simplifications typically adopted in the use of CODAM in plane stress applications. A simple damage model (SDM) is also described, and some discussion is devoted to explicit and implicit finite analyses using the CODAM and SDM models. Chapter 4 introduces the concept of strain localization resulting from the use of strain-softening material models in numerical analyses. The consequence of this strain localization is that the results of the numerical analyses depend strongly on the size of the elements used, the so-called "mesh effect". Two approaches for dealing with the mesh effect are outlined, the crack band method, and the equivalent work method. Both CODAM and the SDM are used to demonstrate the mesh effect, then both the 10 1.5. Thesis Outline equivalent work method and the crack band method are incorporated into the analyses, and it is shown that the mesh dependency is effectively removed. In Chapter 5, the structural size effect in composites is reviewed. In composites, small structures show different strength behaviours than large structures. This size effect is different than that expected by statistical analysis or analyses using linear elastic fracture mechanics. Experimental evidence for the structural size effect is reviewed, as are some approaches for capturing the size effect with analytical models. The application chapters (Chapters 6 through 8) are all structured in the same manner. An overview of the test is given, with the details of the test geometry. The material is then characterized for the CODAM model. Where appropriate, other predicted approaches are presented so that the C O D A M predictions can be considered in context. The simulations are then presented, along with the results and discussion on the results. Chapter 6 involves using C O D A M with unnotched tension tests of AS4/3501 carbon epoxy laminates. Chapter 7 focuses on single-edge notched and double-edge notched tension tests on IM7/8551-7A carbon epoxy. Chapter 8 examines over-height compact tension tests as well as large centre notched tension tests and open hole tension tests involving a stitched resin film infused carbon epoxy. In all the application chapters, an emphasis is placed on characterizing the materials for use with the constitutive model and examining the size effect predicted by the numerical analysis, compared to that observed experimentally. Chapter 9 summarizes the findings of the previous chapters and presents some conclusions on the effectiveness of the model and its improvements. A number of recommendations are made for improving the model. Finally, Appendix A considers bending applications, examining both an eccentrically loaded beam-column, and a four-point bending test. A variety of complications lead to unsatisfactory results in both bending applications, and these are discussed in detail in this appendix. Appendix B illustrates how the stiffness degradation predicted by the CODAM model is insensitive to the choice of the damage parameters attributed to the damage modes, and Appendix C provides an example of using laminate plate theory and the ply discount method. 11 Chapter 2: A Review of Damage Mechanisms and Non-linear ModellingotXorriposit^s This chapter reviews the damage mechanisms that occur in composite laminates, followed by a review of the non-linear modelling techniques used to capture these mechanisms. The so-called "World-Wide Failure Exercise" is presented in detail, examining the models that participated as well as the analyses of the participants and organizers. In addition to the models reviewed as part of the World-Wide Failure Exercise, some additional approaches are examined. 12 Chapter 2: A Review of Damage Mechanisms and Non-linear Modelling of Composites 2.1. Damage Development in Laminated Composites As a class of materials, composite materials have a number of unique characteristics that make their use very attractive in a variety of applications. In particular, due to their very high specific strength and specific stiffness, they are often used in applications where weight is an important design consideration (Kaw, 1997). Composites achieve their desirable properties by selectively placing reinforcement in the load directions. While this is beneficial from a mechanics perspective, very complicated damage modes can occur. Unlike traditional macroscopically homogeneous materials like metals, composites are so heterogeneous that the distinction between material and structure becomes blurred. In fact, composite materials often behave more like "structures" than "materials" at the traditional mechanical analysis scale, the macro-scale (Daniel and Ishai, 1994). These composite material "structures" typically consist of four or more phases: the reinforcement, the matrix, voids, and the various interfaces. In most cases, this is abstracted to two "materials", the matrix and the reinforcement. In undamaged, linear-elastic loading, the macro-scale behaviour of the composite material is complicated due to orthotropicity. When damage begins to accumulate, the complexity of the behaviour increases tremendously. The behaviour becomes anisotropic on a m/cro-scale. Damage develops in composite materials in ways particular to each of the phases. The matrix develops micro-cracks that coalesce into detectable matrix cracks (Agarwal and Broutman, 1990). Figure 2.1 illustrates matrix cracking in a laminate. Matrix cracking can develop due to stress concentrations caused by voids or phase changes (fibres, free-edges, etc.), local tension or shear stresses, strain waves from impact or shock loading, or other mechanisms. 13 2.1. Damage Development in Laminated Composites matrix cracks Figure 2.1. Illustration of transverse matrix cracks. The interface between the matrix and the fibre can fail, commonly known as debonding or pull-out. This sort of failure is usually caused by a poor bond between the matrix and reinforcement, but can also manifest as shear failure in the matrix surrounding the reinforcement, as the matrix is unable to transfer the load sustained by the reinforcement. See Figure 2.2 for an illustration of fibre-matrix debonding. Debonding can be complete, such that the reinforcement transfers no load to the matrix, or partial, where some load transfer still occurs across the interface. debonding Figure 2.2. Illustration of fibre-matrix debonding (based on Fig. 3.17, Agarwal and Broutman, 1990). The reinforcement can fracture in tension and is also susceptible to shear failure. Shear failure may be due to shear loading, or may be the result of compressive loading resulting in "kinking" of the reinforcement (see, for example, Figure 4.11 in Agarwal and Broutman, 1990). Figure 2.3 illustrates the progression of some in-plane damage modes under tensile loading of a coupon. 14 2.1. Damage Development in Laminated Composites Figure 2.3. Types of in-plane damage (adapted from Williams, 1998). Steif modelled fibre breakage in unidirectional laminated (Steif, 1984). In this work, Steif showed that for tensile loading of unidirectional composites that results in fibre breakage and fibre-matrix debonding, overall laminate stiffness reductions occur. Previously, it had been argued that fibre breakages do not in themselves lead to stiffness reductions since the matrix is still bonded to the fibres (Highsmith and Reifsnider, 1982). Steif demonstrated that if no fibre-matrix debonding accompanies fibre breakage then this is the case, but in the presence of fibre-matrix debonding, stiffness reductions are expected. These micro-scale damage modes are the fundamental damage modes present in composites, but the damage patterns become more complicated when laminates are considered rather than laminae. Additionally, the evolution of the damage is important to understand: how loading states lead to damage initiation and how the damage progresses until failure occurs. Of course, there is the issue of defining "failure" as well. In composite laminates, it is not sufficient to consider the number of plies of a particular orientation. Pagano and Pipes showed that the stacking sequence - the order that plies occur in a laminate - is an important factor in the performance of the material, despite the fact that standard stress analysis does not predict a difference in the stress states of laminates with the same number of plies of particular orientations (Pagano and Pipes, 1971). 15 2.1. Damage Development in Laminated Composites In 1980, an important symposium was sponsored by A S T M called "Damage in Composite Materials: Basic Mechanisms, Accumulation, Tolerance, and Characterization". In this symposium many interesting studies were brought forth. Masters and Reifsnider examined damage development in typical quasi-isotropic graphite epoxy laminates with lay-ups of [0/±45/90]s and [0/90/±45]s (Masters and Reifsnider, 1982). Despite the very similar lay-ups, damage evolved differently in each laminate. Transverse matrix cracks developed in the off-axis laminae (±45°) with a regular spacing, which was termed as the saturation spacing. Their data showed that the saturation spacing was unique for each lamina in each laminate, but that the spacing was reproducible between similar specimen. The crack spacing was not purely a characteristic of the lamina orientation however, as different characteristic spacings were observed in, for example, the 90° laminae in the two different lay-ups. They noted that laminate plate theory (LPT) does not predict any differences in the ply stresses between the two lay-ups. Highsmith and Reifsnider produced an often-referenced paper at the same symposium on stiffness reduction mechanisms in composites (Highsmith and Reifsnider, 1982). The study focused on transverse cracking (matrix cracking). Highsmith and Reifsnider pointed out that three important mechanical properties in a composite material (indeed, any material) are strength, stiffness and life. Measurement of strength and life during damage development in a material is not feasible, but measurement of stiffness is so they suggest that stiffness can be used as a non-destructive indicator for damage in a material. This concept has been adopted in a number of models (see Poursartip et al., 1982 for example). Using glass-epoxy laminates designed to encourage transverse cracking in off-axis plies ([0/903]s, [903/0]s, [0/90]s, [0/±45]s), they performed quasi-static tension tests and tension-tension fatigue tests. Stiffness was measured by recording the applied load and the openings of several clip gauges during unloading and reloading of the specimens several times over the duration of the test. Again, transverse cracks at even spacings were observed. Additionally, a strong correlation was reported between the crack density and stiffness reduction. While the relationship is not quite linear, Highsmith and Reifsnider show that the results are of the same shape as might be predicted with shear lag theory, but the absolute results vary in accuracy between the lay-ups. The differences are suggested to be the result of longitudinal splitting or some 16 2.1. Damage Development in Laminated Composites fibre rupture. A significant quantity of this mode of damage was observed in the [0/±45]s laminate, which also showed the poorest correlation with the shear-lag theory. Crossman and Wang discussed transverse cracking in angle-ply ([25/-25/90n]s and [252/-252/902]S) carbon-epoxy laminates (Crossman and Wang, 1982). They noted that in tension tests, the strain required to initiate matrix cracks in the 90° ply depended on its thickness. The damage progression also changed, with delamination occurring at different stages in the loading. They did not offer a theory to predicted the exact sequence of damage. Transverse matrix cracking continued to be a topic of considerable interest. Ogin et al. studied matrix cracking in glass fibre cross-ply laminates and observed linear reductions of the stiffness with crack density (Ogin et al., 1985b). A model was developed and its use compared well with crack-growth fatigue experiments (Ogin et al., 1985a). Talreja also studied the relationship between transverse cracking and stiffness reduction in laminates (Talreja, 1985b). Talreja reviewed many of the transverse cracking theories of the day, as well as some of the experimental data that had be reported in previous years. He proposed a system of classifying transverse cracking effects as a means to qualitatively assess the laminate behaviour following cracking. Using a damage mechanics formulation (see Chapter 3), the effects of transverse matrix cracks in cross-ply laminates were captured quite well. Talreja also observed significant changes in Poisson's ratio as a function of transverse matrix cracking, a topic that had not been addressed by previous research. Finally, the ply-discount method (see Appendix C) was found to over-estimate the modulus reductions for highly constrained laminates and under-estimate the modulus reductions for the low-constraint laminates. In a subsequent paper, Talreja addresses the combination of intra-laminar matrix cracks and inter-laminar matrix cracks (delamination) (Talreja, 1986). Both were found to affect stiffnesses, but only the intra-laminar cracks affected the orthotropic symmetry of the damaged laminate. Talreja showed how his C D M framework was able to consider these effects. Later, Talreja et al. tested and modelled cross-ply carbon-fibre laminates subjected to tensile loading (Talreja et al., 1992). They measured "S" shaped crack growth curves that initiated near 0.25% and saturated near 1.50% for several laminates. Significant 17 2.1. Damage Development in Laminated Composites degradation was observed in Poisson's ratio, between 40%-70% for the laminates tested. Talreja's C D M model was used to simulate the experiments and correlated quite well with the experimental results. Laws and Brockenbrough studied the effect of micro-cracks on brittle materials, ceramics in this case, and determined stiffness reduction relationships for a variety of crack shapes and orientations (Laws and Brockenbrough, 1987). They presented a number of interesting plots showing relationships between crack density and elastic moduli, shear moduli, and Poisson's ratio for different crack shapes, orientations, and distributions. They also presented analytical expressions to describe these relationships. The work was confined, however, to (initially) isotropic materials. Hoover et al. examined transverse cracking in multi-directional glass-epoxy laminates and confirmed many of the earlier investigations into matrix cracking and modulus reduction (Hoover et al., 1997). In their study, the laminate stiffness reduction was divided into three stages: pre-transverse cracking, linear modulus reduction, and final failure. In the pre-transverse cracking stage, although no cracks were observed, the modulus was shown to decrease as little as 0.03% and as much as 20%. The idea of sub-critical damage was put forward as a possible explanation of this decrease. Sub-critical damage is micro-cracking and micro-debonding, mechanisms that are undetectable using standard analysis techniques. Also suggested as a possible explanation for this modulus decrease is the presence of manufacturing related residual strains. The idea is that the residual strains may relax through some mechanism under small load, resulting in an apparent stiffness reduction. In a similar study, Stout et al. found a high initial stiffness reduction in bending test of [0/45/90/-45/90/45] carbon epoxy laminates (Stout et al., 1999). This large stiffness reduction at relatively low loads was also attributed to the development of micro-cracks. Stage 2 in the study by Hoover et al. is a region of linear stiffness reduction with increasing crack density. In Stage 3, the stiffness drops quite suddenly as other damage modes (debonding, fibre failure) are activated and transverse matrix cracking no longer is an effective indicator. Delamination is an important failure mode in laminated composites. Delamination is the interface failure between plies in a laminate, as illustrated in Figure 2.4. Delamination 18 2.1. Damage Development in Laminated Composites can result from direct through-thickness stresses from out-of-plane loading and bending, but it can also result from out-of-plane stresses created at free edges of in-plane loaded composites (Pipes and Pagano, 1970; Pagano and Pipes, 1971). Delamination is fundamentally an interface failure, but can also be considered a matrix failure. delamination Figure 2.4. Illustration of delamination in a composite laminate. Delaminations are not always "clean" interface failures. Often, fibres will bridge in across the delamination in varying degrees. The degree of fibre-bridging affects the resistance of the material to delamination. Further, the delamination is not smooth in a pure matrix failure, with some degree of waviness in the failure path. This allows for the opportunity for mechanical interlock in shear loaded applications. Further discussion about this can be found in Paris (Paris, 1998; Paris and Poursartip, 2003) and others. O'Brien discussed the onset and growth of delamination in graphite-epoxy laminates. The lay-up, [+ 301+ 30 / 90 / 9oJs was designed to delaminate at the edges under tensile loading due to the free-edge effect. As tensile load was applied, the 90° plies developed some matrix cracks, after which delaminations occurred at the specimen edges. The matrix cracks then significantly increased in number. As the delaminations grew, so did the matrix cracks. Through a rule of mixtures analysis, O'Brien showed that stiffness degraded linearly with delamination size, and used this as the basis for a method to determine the strain energy release rate, G, for the delamination growth. Delfosse et al. have shown that delaminations, fibre-breakage, and matrix cracks will often form repeating units through the thickness of an impacted composite laminate (Delfosse, 1994). This repeating unit corresponds to the sublaminate in a particular lay-up, and the repeating pattern forms a "staircase", as illustrated in Figure 2.5. 19 2.1. Damage Development in Laminated Composites Fibre breakage Delamination Figure 2.S. Characteristic "staircase" repeating damage pattern observed in through-thickness loading of PMC laminates (Adapted from Delfosse, 1994). Most of the work in damage development in composites has been done on tensile load applications or delaminations (double cantilever beam, etc.). The more complicated load states, like shear, compression, or triaxial loading have not received the same attention. There are several reasons for this: most composites are used in tensile applications, through-thickness loading is rare, and given our level of understanding (or lack thereof) of the simpler tensile case, progression to the complicated states is premature. In addition to the structural causes of damage in composites, they can be susceptible to numerous environmental threats. Moisture, ultra-violet radiation, and various chemicals can attack the constituents and degrade their mechanical properties and affect the interface strengths. Kriz and Stinchcomb in the 1980 A S T M symposium discussed how moisture in graphite epoxy quasi-isotropic laminates ([0/±45/90]s and [0/90/±45]s) affects the stress state and damage evolution in tension tests (Kriz and Stinchcomb, 1982). Their study showed that moisture resulted in lower loads for delamination initiation and higher loads for transverse crack initiation in the 90° plies. Despite the differences in the load conditions, the damage state seemed to be independent of the moisture content. 20 2.1. Damage Development in Laminated Composites Beaumont provided a good, introductory overview of failure mechanisms and some ways to determine when those failures occur (Beaumont, 1989). While that review touched on some approaches to modelling and predicting failure, the next section will address this subject in considerably more detail. 21 Chapter 2: A Review of Damage Mechanisms and Non-linear Modelling of Composites 2.2. Modelling Failure in Composite Materials The concept of numerically modelling failure and damage in materials, especially composite materials, is relatively new. The continuous exponential increase in computational power has provided the ability to perform very large and complex simulations in a short period, and many damage and failure theories have evolved due to this increased ability. As recent as 1990, a supercomputer was required to perform impact analyses using two-dimensional finite element analysis (Langlie et al., 1990). Although run-times are not specified in that particular paper, the runs likely took a significant length of time; a desktop computer today could perform a similar analysis in a matter of some few seconds. As a result of this history, most failure theories for composite materials are analytical, easily computed by a series of "hand" calculations. Many of the failure theories are initial failure theories. That is, they define load conditions that cause "failure", and are not concerned with what happens after that initial failure point. Composites, however, are capable of sustaining considerable loads following initial failures. In composite materials, one of the first to consider post-failure response were Hahn and Tsai (Hahn and Tsai, 1974). Using what is now a relatively crude approach, they use an arbitrary failure surface and then a progressive failure model to determine the behaviour beyond the initial failure point. This section will address some of the failure theories and constitutive models developed for composite materials. A landmark study in the effectiveness of failure theories and constitutive models, the so-called World-Wide Failure Exercise (WWFE), is reviewed first and then some theories that were not covered in the W W F E will be considered. 2.2.1. The World-Wide Failure Exercise In 1991, attendees at the workshop "Failure of Polymeric Composites and Structures: Mechanisms and Criteria for the Predictions of Performance" held in St. Albans, UK, agreed on two particular items (Hinton and Soden, 1998): • There is no universal definition of failure of a composite • There is a lack of faith in the failure criteria currently in use 22 2.2. Modelling Failure in Composite Materials As a result of these findings, Dr. Paul Hogg, an editor at the Advanced Composites Bulletin and Bryan Harris, an editor of the Composites Science and Technology (CST) journal, along with workshop attendees Dr. Clive Phillips and Dr. John Hart-Smith determined that the subject of current failure theories in composite materials deserved a special edition of CST. Accordingly, Dr. Michael Hinton from DERA (now QinetiQ) and Dr. Peter Soden (UMIST) volunteered to produce (Harris, 1998): " a special edition of CST with the aim of providing an authoritative source for designers and researchers by making it possible, for the first time, to obtain unbiased comparisons of the mehts and shortcomings of current failure criteria." 2.2.1.1. WWFE Format After deliberating on the matter, three goals for the exercise - eventually referred to as the "World-Wide Failure Exercise", and sometimes as the "Failure Olympics" owing to its competitive nature and international participants - were established by Hinton and Soden (Hinton et al., 2002): 1. Establish the current level of maturity of theories for predicting the failure response of FRP laminates. 2. Close the knowledge gap between theoreticians and design practitioners in the field. 3. Stimulate the composites community into providing design engineers with more robust and accurate failure prediction methods as well as the confidence to use them. To achieve these goals, Hinton and Soden decided that it was necessary to gather a comprehensive description of the current and foremost failure theories for FRP laminates, and with these theories, compare their predictive capabilities with each other and directly with experimental data. To do this, Hinton and Soden ('the organizers') laid out a plan for a unique blind study of failure theories for composite laminates. First, the organizers surveyed the literature and their colleagues and assembled a list of various implementations of state-of-the-art failure theories for composite laminates. Researchers associated with the failure theories were contacted and invited to participate in the upcoming exercise. The full list of invitees is shown in Table 2.1. 23 2.2. Modelling Failure in Composite Materials Slightly more than half of those invited to participate in the exercise declined. The list of decliners is shown in Table 2.2. Table 2.1. Summary of approaches and failure theories invited to participate in the WWFE. (Adapted from Hinton and Soden, 1998) Contributor Approach represented Chamis C C (USA) Micromechanics Hart-Smith J (USA) Generalized Tresca and maximum strain theories Eckold G C (UK) Design codes Edge E C (UK) Industry Ellis D (UK) ANSYS Grayley M E (UK) ESDU international Hall R L (USA) PDA/PATRAN Hallquist J (USA) DYNA3D Johnson A (Germany) PAMFISS/PAMCRASH Haug E (France) PAMSISS/PAMCRASH Stanton E (UK) MSA/NASTRAN Hibbitt D (USA) ABAQUS Allen D H, Talreja R, Nairn J, Reifsnider Damage models and cracking of composites K (USA), McCartney L (UK) Puck A, Schurmann H, Cuntze R G, Aoki Puck's theory R (Germany) Hashin Z and Rotem (Israel) Rotem and Hashin's theories Rosen B W (USA) MIL-HNDBK-17 Zinoviev P (Russia), Skudra A M (Latvia), General application and methods Chang F-K, Chou T W, Sun C T, Swanson S R (USA), Marom G (Israel), de Wilde P (Belgium), Surrel Y (France), Hansen J S (Canada) Tsai S W (USA) Interactive failure theory Uemura M (Japan) Statistical prediction of failure Sandhu R S and Wolfe W E (USA) Strain energy failure theory After assembling a list of failure theories, the organizers selected a number of test cases for the failure theories to examine. The test cases were limited to continuous fibre reinforced thermoset plastics, and specifically to two types of carbon fibres, two types of glass fibres (E-glass), and epoxy resin systems. The test cases were selected on the basis of encompassing a wide range of lay-ups, a wide range of loading conditions, a variety of damage types, and behaviours that were linear and non-linear. The organizers then distributed the data, test cases, and the requested analyses to the participants. The participants were to write a paper describing their particular theory, and present results for each of the test cases. The organizers, on receiving each participant's paper, assembled all the results and compared the theories to each other. 24 2.2. Modelling Failure in Composite Materials Table 2.2. List of invitees who did not participate in the WWFE. (Adapted from Hinton and Soden, 1998) Name Remarks Allen D H Declined invitation to join the exercise Chang F-K Declined Chou T W Declined Cuntze R G Declined in favour of Puck de Wilde P Declined because of different expertise Ellis D Accepted but later withdrew Grayley M E Accepted but later withdrew Hall R L Declined Hallquist J Declined Hansen J S Accepted but later withdrew Hashin Z Declined Hibbitt D Declined Johnson A Accepted but later withdrew due to pressure of work Marom G Declined because of different expertise Reifsnider K Declined Rosen B W Very late invitation. Declined Sandhu R S Declined but nominated Wolfe Stanton E Declined Swanson S R Accepted but later withdrew Talreja R Accepted but later withdrew This first part of the exercise is referred to as 'Part A' and was published in a special edition of Composites Science and Technology in 1998 (Volume 58, No. 7). After the publication of Part A, the participants were given the experimental results of the test cases that were simulated, and were asked to compare the results of their theories to the experimental data. The participants were welcome to revise their Part A contributions after receiving the experimental data. The participants were required to submit another paper, outlining the changes to the theory if any were made, and comparing the theory's predictions to the experimental data. After assembling all the contributions from the participants, the organizers prepared what was intended to be an objective and unbiased evaluation of the performance of each of the theories. The theories were qualitatively compared to each other and to the experimental results, and a quantitative comparison to the experimental data was made as well. The theories were then ranked according to their abilities in specific areas. This section part of the exercise was called "Part B". Part B of the WWFE was published in a special edition of Composites Science and Technology (Volume 62, No. 12-13) in 2002. 25 2.2. Modelling Failure in Composite Materials A further part of the exercise, "Part C", has yet to be published. Part C will consist of the "Part A" and "Part B" papers of several participants who elected to participate after the publication of the original Part A papers. These participants were required to submit their Part A predictions before the publication of the original Part B experimental data in order to preserve the "blindness" of the numerical predictions. These additional participants did not submit their Part B papers in time for the original Part B publication and so will be published in Part C. There was one exception to this, however. John Hart-Smith, who performed predictions with two other failure theories, submitted a third failure theory, his "10% Rule", and its Part A and Part B papers appear in the original Part B publication. 2.2.1.2. WWFE Initial Data - Material Data and Test Cases In the first paper appearing in the WWFE Part A publication, the organizers detail the information that was given to the participants at the outset of the exercise (Soden et al., 1998). The paper provides the material properties for each of the four material systems: AS4/3501-6, T300/914C, E-glass ("21xK43 Gevetex") / LY556 / HT907 / DY063, and E-glass ("Silenka 1200tex") / MY750 / HY917 / DY063. Thermal and mechanical properties were given for unidirectional laminae for each material system. Additionally, mechanical and thermal properties for the four fibres and matrices were given. The in-plane shear stress-strain data was given for each lamina, as was the transverse compressive stress-strain curve for the E-glass / MY750 / HY917 / DY063, and the AS4/3501-6, and longitudinal tensile stress-strain data for the AS4/3501-6. The test cases selected by the organizers were based on both unidirectional (UD) laminae and multidirectional (MD) laminates. The six balanced and symmetric lay-ups were: 1. 0° UD laminae, selected in order to evaluate the models at a basic level 2. [907±30°] s, selected as a non-quasi-isotropic lay-up with a complicated failure mode 3. [907±4570°] s, selected as a typical aerospace lay-up 4. [±55°]s, selected as a typical pipe lay-up 5. [0790°] s, selected in order to evaluate matrix cracking 6. [±45]s, selected to investigate biaxial tension and pure shear in a cross-ply laminate 26 2.2. Modelling Failure in Composite Materials For each lay-up, there were requirements to determine failure stress envelopes, both initial failure and final failure, and stress-strain curves for loading in specific directions, depending on the case. A total of 14 test cases were requested. 2.2.1.3. WWFE Exercise Participating Failure Theories (i) Chamis The theory identified as Chamis was presented by Gotsis, Chamis and Minnetyan (Gotsis et al., 1998; Gotsis et al., 2002). This "theory" involved two approaches, solved by two computer programs. The two approaches were (a) the generation of first-ply failure envelopes for multi-axial loading by using a micro-mechanical approach, and (b) the generation of laminate fracture envelopes and stress-strain diagrams by using progressive failure. The two computer programs cited by Gotsis et al. were Integrated Composite Analyzer (ICAN) and Composite Durability Structural Analyzer (CODSTRAN). The ICAN code contains what the authors term: a structured multi-scale formalism which is (1) 'upward integrated' (synthesis) from material behaviour space to structural analysis and (2) 'top-down traced' (decomposition) from structural response to material behaviour space." The authors indicate that this is a micro-mechanical approach, but the details of the approach are left to other references. The laminate properties are derived from the micro-mechanics by classical laminate theory. The failure stress analysis is based on two criteria, the first-ply failure based on maximum strength, and the first-ply failure based on fibre breakage. The second code used by the authors is CODSTRAN. The CODSTRAN code is used to predict the progressive failure of laminates and structures. The code considers a number of different failure modes and applies different strategies depending on the failure mode for degrading the material, or laminate, properties. In commenting on the effectiveness of the Chamis theory, the organizers noted that it provided a good description of the UD lamina failure envelope but it did not predict the maximum shear stresses under combined direct and shear load. The initial failure stress 27 2.2. Modelling Failure in Composite Materials predictions for MD laminates were lower than the experimental results and other theories, largely due to the full allowance of residual thermal stresses according to the organizers (Hinton et al., 2002). According to the authors however (Gotsis et al., 1998): "All the laminate failure envelopes were generated without accounting for lamination residual stresses." Some other factor must have caused the discrepancies between the simulations and the experiments. The organizers noted that the theory had a fundamental weakness in the prediction of laminate behaviour, and that it was unable to accurately simulate experiments that had a modest initial failure stress but a much larger final failure stress. Additionally, the theory was unable to adequately simulate the post-initial failure behaviour, was generally too stiff in its laminate predictions, and was unable to capture any non-linearity in the stress-strain curves. (ii) Eckold The theory presented by Eckold (Eckold, 1998; Eckold, 2002) was the approach detailed by the British design code BS4994.1987, which is intended for the design of glass-fibre tanks and pressure vessels. The design code uses a limited strain approach, and contains detailed procedures for calculation of allowable values to be used in design. The nature of the code meant that its applicability was limited in this exercise. Eckold only used the design code for the glass-fibre laminates. Additionally, since the design code provides allowable values rather than failure stresses, no distinction between initial and final failure could be made, although it was expected that all the predicted stresses would be conservative. Further, the code did not provide procedures for cases involving shear loading. The predictions of this theory was always conservative or very conservative when it the longitudinal and shear strains were limited to 0.4% and the transverse strains were limited to 0.1%. Due to the limited applicability of the approach, the organizers indicated that it was not sufficiently robust, flexible, or accurate for wide-spread use. 28 2.2. Modelling Failure in Composite Materials (iii) Edge The theory put forward by Edge (Edge, 1998; Edge, 2002) was the Grant-Sanders method that was developed at British Aerospace. The theory is a lamina stress criteria method, using classical laminate plate theory (LPT) to decompose stresses in laminates into stresses in the constituent laminae. The criteria are not interactive with the exception of shear-tension interaction for matrix failure and shear-compression interaction for fibre failure. After initial failure, the shear and transverse tension stiffnesses are reduced gradually as strain increases. In addition to these failure criteria, Edge chose to consider the thermoelastic residual stresses, with a temperature variation of 100°C from the stress-free state. After receiving the Part B experimental data, Edge modified his approach somewhat, removing the shear-longitudinal compression interaction for glass fibre laminates only. The justification for this change was the relative difference in transverse fibre modulus of glass fibre compared to carbon fibre. Given that glass fibre is much stiffer than carbon fibre in the transverse direction, Edge postulates that it requires less support in compression than the carbon fibre and is therefore less affected by a reduction in shear modulus (Edge, 2002). The organizers found that the Edge theory provided mixed comparisons to the experiments. There was moderate agreement of the failure envelopes for the UD lamina cases with the experimental data. The MD laminate initial failure strengths were low, however, likely due to the consideration of residual thermal stresses. The final failure strengths, however, were sometimes conservative but sometimes not. Further, the theory predicted non-linear stress-strain curves, but the shape of the curves did not match the experimental results. (iv) Hart-Smith(l) Hart-Smith contributed three theories to the exercise. The first of these theories Hart-Smith(1) is a generalized Tresca model (Hart-Smith, 1998a; Hart-Smith, 2002a). As the description implies, the theory is a generalization of the maximum shear-stress failure criterion normally attributed to Tresca. It is applied to composites based on an observation that the highest measurements of fibre-dominated in-plane shear strength of 29 2.2. Modelling Failure in Composite Materials a ±45° carbon epoxy laminate were half the measurements of the uniaxial tension or compression strength of corresponding 0790° laminates. The Hart-Smith(l) theory establishes failure envelopes for the fibres and then for the UD lamina constructed from the fibres. The failure envelopes are then modified to provide failure envelopes for embedded {in-situ) lamina rather than isolated lamina. The construction of the failure envelopes is described in terms of geometric constructions based on the material properties and various cut-offs. The failure envelopes are constructed in strain space rather than stress space. LPT is used to decompose laminate stresses and strains into ply-level strains. The results of the analyses using the Hart-Smith(l) theory were sometimes unconservative for both the UD laminae predictions and the MD laminate predictions. Additionally, the theory was not applied to all the test cases. (v) Hart-Smith(2) The second theory championed by Hart-Smith is the Truncated Maximum-Strain model (Hart-Smith, 1998b; Hart-Smith, 2002a). This theory is similar to Hart-Smith(l) but instead of using maximum shear strain as the limiting parameter, general strain component thresholds are used. Various cut-offs are introduced to account for damage failure modes such as intra-laminar matrix cracking. The Hart-Smith(2) approach fared similarly to the Hart-Smith(l) approach, sometimes showing unconservative tendencies. No initial or intermediate failures were predicted, nor were any stress-strain curves presented. (vi) Hart-Smith(3) Hart-Smith's third approach was added after the initial publication of Part A of the exercise. This approach is his 10% rule, an empirical rule-of-thumb for estimating laminate behaviour from very limited material data (Hart-Smith, 2002b; Hart-Smith, 2002a). The 10% Rule requires only three measured material properties: the longitudinal modulus, the longitudinal tensile strength, and the longitudinal compressive strength for a 0° lamina. The corresponding transverse modulus, transverse lamina strengths and uniaxial strength of ±45° plies is estimated. The in-plane shear modulus and shear 30 2.2. Modelling Failure in Composite Materials strength of a 0°, 90°, or 0790° laminate are derived. The in-plane shear strain can also be computed. Cut-offs are introduced for carbon-fibre laminates, but are not present for glass-fibre laminates. The original formulation was limited to laminates with 07±45790° lay-ups (or any combination of plies with those orientations), but the presented formulation was extended to include ±55° and 907±30° lay-ups as well. As with the other Hart-Smith theories, residual thermal stresses are ignored. The organizers found that Hart-Smith(3) did not do well for the UD laminae. For the MD laminates, however, the final failure envelopes were generally in good agreement. The organizers noted that there was no means of predicting initial or intermediate failure modes or stresses. The strength of the model in any case is the fact that it is a reasonably good approximation for MD laminates that is performed quickly and on the basis of limited information. (vii) McCartney The theory advanced by McCartney (McCartney, 1998; McCartney, 2002) is a damage mechanics approach based around the strain energy in cross-ply laminates with cracks in one orientation. The formulation is limited in applicability, but was extended to include angle-ply and quasi-isotropic laminates in Part B. The McCartney approach fully includes the residual thermal stresses. Of the two cases originally analyzed by McCartney, the shape of the stress-strain curve for the 0790° uniaxial tension curve compared well to the experiments, but the approach over-estimated the stiffness of the ±45° laminate under biaxial load. The original approach also did not provide a means of predicting the final failure. After the Part B refinements, the predictions were somewhat closer to the experiments, and the addition of a maximum fibre strain criterion provided an estimate of final failure. The organizers note that the theory has many limitations, and is too immature for use right now. The organizers also note, however, that the approach: " offers the future promise (after development) of providing a rigorous and flexible method for modelling the initial and post initial failure 31 2.2. Modelling Failure in Composite Materials response of laminates, which remains as a clear problem area for many of the competing theories." (viii) Puck The Puck approach was provided by Puck and Schurmann (Puck and Schurmann, 1998; Puck and Schurmann, 2002). This approach is a phenomenological theory that provides failure criteria for fibre failure (FF) and inter-fibre failure (IFF) modes. The approach evolved from experimental studies of mechanisms of failure in a lamina when subjected to biaxial loading. The IFF, or matrix cracking, criteria involves several ply cracking mechanisms and consideration of the angle of the cracking plane. Thermal stresses are considered in the typical linear-elastic manner. Degradation is accomplished by degrading fracture resistances by a weakening factor that is related to stress level. The degradation model allows for a continuous and gradual loss of stiffness after the crack initiation point. The results of the analyses for the WWFE were relatively good when compared to the experiments. The UD laminae failure envelopes were rated 'very good' by the organizers, and they remarked that the final failure envelopes and stress-strain curves for the MD laminate analyses were also in good agreement with the experiments. The Puck approach did have some problems with non-linearities, showing poor agreement with the experiments, and the failure envelopes for those cases (in particular, the ±55° lay-up) were not closed. The organizers questioned the importance of that flaw, noting that the need to design in the large displacement, non-linear region was "open to debate". Compared to the other theories, the Hinton and Soden note that the Puck approach is one of the best currently available. (ix) Rotem The Rotem approach (Rotem, 1998; Rotem, 2002) is failure theory based on three assumptions: 1. The failure of a laminate occurs either in the fibre or the matrix, and the onset of failure is a localized event. 2. The laminate has no free edges and only in-plane stresses are effective. This precludes inter-laminar stresses that lead to delamination. 32 2.2. Modelling Failure in Composite Materials 3. The matrix is weaker and softer that the fibres. The failure criterion is then based on a separation of the failure modes of the fibres and the matrix. Stresses in the laminate are reduced to stresses in the laminae using LPT. The organizers of the exercise determined that the Rotem approach performed well in the UD lamina predictions, as well as the initial MD laminate predictions. The final laminate predictions, though, compared quite poorly with the experiments showing extremely conservative results. The organizers conclude that the model does not properly discriminate between initial and final failures, and exhibits a fundamental weakness in its ability to model the post-failure response. (x) Sun (Linear, L) and (Non-linear, NL) Sun and Tao provided two similar theories to the exercise (Sun and Tao, 1998; Sun et al., 2002). The approaches incorporate the Hashin-Rotem failure criterion to predict lamina failures. The ply-discount method is used to degrade the laminate in the post-failure regime. Residual thermal stresses are considered in the standard elastic manner. The Sun (NL) approach includes material non-linearities in the predictions of the stress-strain curves for the various cases. Additional non-linearities are introduced by considering the transverse matrix cracking. Only the stress-strain curves were computed with the non-linear approach. Both the Sun (L) and (NL) performed similarly in relation to the competing theories. The UD lamina predictions were reasonable in their correlation to experiments, but they were neither the best nor the worst predictions. Similarly, the predictions of initial MD laminate failure were middle-of-the-pack, while the final laminate failure predictions were slightly better. The stress-strain curves compared well to the experiments for fibre-dominated lay-ups, but did not compare well in matrix dominated and shear loading cases. (xi) Tsai The Tsai approach (Liu and Tsai, 1998; Kuraishi et al., 2002) incorporates the classic Tsai-Wu interactive failure criteria with slight modifications. The Tsai approach is an interactive, quadratic failure criterion as shown in Equation 2.1. 33 2.2. Modelling Failure in Composite Materials { 2F x ycr xo- y XX' y/XX'YY' YY' S " 1 1" "1 1" crY + — — X X'. X y Y'_ (Ty =1 (2.1) where X, Y, and S are strengths in the longitudinal, transverse, and shear directions, a prime indicates compressive strength, and F'xy is an interaction term. The modifications to the classic Tsai-Wu approach include provisions for including the effects of micro-cracking and progressive failure. The Tsai simulations provided the best UD laminae predictions in the WWFE. The organizers noted though that the simulations predict a strength enhancement under biaxial compressive loading, which is a questionable result. A lack of data in that loading quadrant, however, meant that the prediction could not be dismissed out of hand. The MD laminate predictions were less successful. The initial failure envelopes did not provide good agreement with the experiments, but the finial failure envelopes were in closer agreement. In Part B (Kuraishi et al., 2002) modifications were made to the approach's post-initial failure behaviour that significantly improved the correlation between the simulation and experimental results. (xii) Wolfe The Wolfe approach (Wolfe and Butalia, 1998; Butalia and Wolfe, 2002) exercises a theory developed by R.S. Sandu at the Wright-Patterson AFB. The theory incorporates a laminate plate theory that includes an incremental constitutive law that accounts for non-linearities in the material behaviour, and a strain-energy based failure criterion. The failure criterion is based on findings by Sandu that the longitudinal, transverse and shear strain energies for a material orthotropic along its material axes are independent parameters. In Part B of the exercise, the Wolfe simulations were revisited. Modifications were made to empirical shape factors in the failure criterion based on comparisons of the Wolfe-A results to the experiments. The organizers found that the Wolfe approach performed reasonably well in its predictions of the UD lamina. The initial failure envelopes for the MD laminates also offered relatively good correlations to the experimental results although the organizers noted that none of the theories performed well in this regard. The final failure envelopes 34 2.2. Modelling Failure in Composite Materials predicted by Wolfe were less successful in providing good representations of the experiments. The predictions tended to be very conservative, leading the organizers to note that there was a poor distinction between the initial and final failures by the model, and its post-initial failure behaviour was not very satisfactory. (xiii) Zinoviev The final failure theory considered in the WWFE is that put forward by Zinoviev (Zinoviev et al., 1998; Zinoviev et al., 2002). The authors of the Zinoviev approach refer to it as a structural-phenomenological coupled deformation/failure model (DFM). Generally, the model assumes that the material progresses from an initial uncracked state to a final fully cracked state. Further distinctions are made between open cracks and closed cracks. Changes in the angles of the fibres in damaged plies are also considered, allowing for non-linearities. A set of non-interactive maximum-stress failure criteria is used to identify failure modes. The organizers have noted that Zinoviev makes less of an attempt to link the failure modes to physical mechanisms compared to Puck. The Zinoviev predictions of the UD laminae envelopes were reasonably good in comparison to the experimental results. The initial failure envelopes for MD laminates were the best of the Exercise, but when compared in isolation to the experiments, they were not particularly good. The final failure envelopes for MD laminates predicted by the model provided better correlation with the experiments. Overall, the organizers felt that the Zinoviev approach was one of the best. 2.2.1.4. WWFE Summary The WWFE organizers (Soden and Hinton) assessed the performance of the participating theories in three ways. Qualitative assessments of the correlation between the theories and the experiments were made on a test-by-test basis, qualitative assessments of the correlation between the theories and experiments were made on a theory-by-theory basis, and they performed a quantitative assessment of the performance of the theories using 125 tests that they developed in five categories: the biaxial strength of UD laminae, the initial biaxial strengths of MD laminates, the final strengths of MD laminates, the deformation (stress-strain curves) of MD laminates, and the ability of the theory to predict general trends. 35 2.2. Modelling Failure in Composite Materials The theories were separated into three groups as a result of the qualitative assessments made on a theory by theory basis. These groups are shown in Table 2.3. Table 2.3. Results of the qualitative assessment of participating theories in the WWFE by the organizers. (Based on an unnumbered table presented in Hinton et al., 2002) Group 1 Puck, Zinoviev, Sun (L) Good predictive capability, none or one fundamental weakness and many relatively minor weaknesses Group 2 Group 3 Edge, Chamis, Wolfe, A few or more significant and fundamental weaknesses Rotem, Hart-Smith(3) Eckold, McCartney, Hart- Clear limitations and many fundamental weaknesses Smith(1,2), Sun(NL) and/or test cases not solved I Grade A • Grade B • Grade C D N o Attempt 100% -j 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% CO CN E CM CQ _i 2 >. to £ I amis Rote mith( mith( rtney z E 3 Ecko :Cartm ffl to to cs to :Cartm e o i t: O s Ha • i Ha S Figure 2.6. WWFE scores in quantitative tests, sorted by total Grade A and B scores. In the quantitative assessments, theories were assigned letter grades for each test based on their level of correlation. An "A" was awarded for predictions within ±10%, a "B" for predictions between ±10% and ±50% of the experimental value, and a "C" for 36 2.2. Modelling Failure in Composite Materials predictions below 50% or above 150% of the experimental value. The theories were then ranked on the basis of the number of "A", "B", or "C" grades they achieved. These results are summarized in Figure 2.6. Of the theories that performed well, Zinoviev and Puck were phenomenological approaches, while the Tsai approach is an interactive failure criterion approach. Those theories that performed poorly consisted of Sun(NL), an interactive approach, Eckold, a design code approach, and McCartney, a damage mechanics approach. All three of these theories did not address all of the test cases, which significantly affected their ranking by the organizers. An honourable mention was given to Hart-Smith(3), the 10% Rule. Due to the limited amount of data required to perform an analysis, and the time required to do so, this method is a very easy and reasonably accurate way to perform initial analyses. The organizers recognized a number of unresolved issues within the Exercise. Within the scope of the Exercise and of interest to those using composites in pressure vessel applications, one of the unresolved issues was that of leakage as failure mode. It was noted that leakage does not begin at initial failure, or at final failure, but at some intermediate stage that none of the model was able to address. Another unresolved issue is that of the behaviour of an isolated lamina compared to that embedded in a laminate. A number of the WWFE participants adjusted the laminar properties when the laminae were in laminates. However, the participants did not agree on the approaches for making this adjustment, or even the need to do so. Two other unresolved issues are major issues. The first, the problem of accounting for residual thermal stresses caused by the manufacturing process, is generally recognized by the participants, but again the participants did not agree on the manner of dealing with the problem, or the need to do so. In fact, some of those approaches that considered residual stresses may have had better experimental correlation had they not considered the thermal stresses. The final outstanding issue, and one not addressed by the organizers or participants is the size effect. It has been shown that the strength of geometrically similar specimens depends on the absolute size of the specimen tested. This issue will be addressed further in Chapter 5. 37 2.2. Modelling Failure in Composite Materials The organizers developed a number of recommendations for designers on the basis of the results of the WWFE. They noted that for isolated laminae, the theories were fairly mature, and recommended using either Tsai or Puck for these types of analyses. In attempting to predict the initial strength of MD laminates, the theories were all fairly poor. In recommending approaches to predict initial failure, Zinoviev and Wolfe-B were recommended. A similarly dismal situation exists for predicting the final strength of MD laminates. Puck, Zinoviev, Tsai-B, and Hart-Smith(3) were recommended, but the organizers noted that the best a designer could hope for was ±50% accuracy. Deformation predictions were only reasonable accurate where fibre failure was the dominant mode of failure. Zinoviev and Puck were recommended. The organizers noted that in this context, "lack of robustness is a feature". For situations where other the prediction of other features, such as the mode or sequence of failure, is required, once again Puck and Zinoviev were recommended. In conclusion, the organizers noted that the concept of "failure" was still undefined, but the Exercise served to point out and highlight the gap between the theoreticians and the designers. They also note that designers want easy to use software tools to perform the analyses of composite laminates. In the same vein, theoreticians need to recognize the gulf between an interesting theory and a practical predictive tool. Finally, the organizers proudly observed that the Exercise renewed interest in predictive failure theories for composites, and as a result, eight of the leading theories had been improved. 2.2.2. Additional Approaches In addition to the CDM models, which are discussed in Chapter 3, other approaches were not covered by the WWFE. Notably, none of the commercial finite element codes participated in the study. This may be understandable, as the commercial codes are solving engines with material models often supplied by third-party engineers and scientists. As such, they are not necessarily attached to any one particular material model. For example, there are many material models in LS-DYNA intended for use with composite materials, and several of them address damage in composite laminates. The most popular composite damage model in LS-DYNA is based on work by Chang and Chang (1987), but others exist including a continuum damage mechanics model based on work by Matzenmiller et al. (1995) and even CODAM. 38 2.2. Modelling Failure in Composite Materials The Chang and Chang model is a progressive damage model originally intended for notched laminates subjected to tensile loading (Chang and Chang, 1987). The failure criteria are phenomenological, based on matrix cracking, fibre-matrix shearing, and fibre breakage damage modes. The failure criteria are very similar to Hashin's failure criteria (Hashin, 1980). After the assessment of failure, the material is degraded based on the failure mode detected. For matrix cracking, the transverse modulus and Poisson's ratio are completely reduced to zero and the longitudinal modulus is un-modified. For the fibre breakage or fibre-matrix shearing damage modes, the longitudinal modulus and shear modulus are degenerated according to a Weibull distribution based on the area of damage predicted by the model. The Chang and Chang model was later updated to specifically address shear loading (Shahid and Chang, 1995). An alternative numerical technique to finite element analysis is Smooth Particle Hydrodynamics (SPH). SPH is a meshless Lagrangian numerical method, also known as the Element Free Galerkin (EFG) method. In a nutshell, the analysis method involves a gridless collection of nodes, and each node bears a relationship to its neighbours similar to a shape function in finite element analysis. The EFG method also uses an averaging function to determine the relative influence of neighbouring nodes to the node considered. SPH in structural mechanics (its original roots are in astrophysics) has typically been used in hypervelocity impact simulations of homogeneous materials. The references (Beissel and Belytschko, 1996) are suggested for background in using SPH/EFG in computational mechanics. Some have extended the SPH method to composite materials. In particular, Chen and Medina (Chen and Medina, 1998; Medina and Chen, 2000) have used a SPH model with a constitutive model based on a one-parameter viscoplasticity model developed by Chen and Sun (Chen and Sun, 1993) to simulate high-velocity impacts on graphite epoxies. Damage is captured by employing the maximum stress criteria with individual fracture criteria for each stress component. When damage is detected, the stresses and shear stresses for the appropriate direction are reduced to zero, with the exception of compressive stress. The simulations were used to explore various design parameters such as face sheet type, position, and impact velocity. No comparisons were made with experiments. 39 2.2. Modelling Failure in Composite Materials Some models do not attempt to address the whole composite, but deal with specific damage modes. In particular, significant efforts have been made at modelling delamination. There have been three major approaches to delamination modelling, discrete element interface modelling, interface element modelling, and constitutive-model based modelling. In discrete element interface modelling, discrete one-dimensional elements such as springs and dashpots are employed to tie elements together across a crack (or potential crack) interface. Alternatively, tie-break contact surfaces are employed in the analysis code that effectively do the same thing. The discrete elements typically need to be described by a traction-displacement relationship that includes a pre-failure and post-failure behaviour. This approach has the disadvantage of requiring prior knowledge of the location of the crack and constraining the crack growth to a pre-determined path. The approach also suffers from mesh-density sensitivity, where the energy dissipated by the discrete elements in failure depends on the number of elements, which is related to the degree of mesh refinement. The interface element approach is similar to the discrete element interface modelling approach, except that finite elements are used to simulate the crack behaviour. These thin elements have constitutive models that damage in such a manner to allow crack opening or shear displacement behaviour. This approach shares the same problems that the discrete element approach has. These two methods are covered by so-called cohesive models. The cohesive crack method and the fictitious crack method are common approaches to modelling crack growth from an energetic or fracture mechanics point of view. For further details on these approaches to delamination modelling, the references (Backlund, 1980; de Borst, 2003; Planas et al., 2003; Llorca and Elices, 1991; Borg et al., 2002; Cornec et al., 2003; Jansson and Larsson, 2001) are suggested. The last approach is to incorporate delamination directly into the constitutive model of the material. This macro approach is appropriate for CDM-based models, but studies on delamination using CDM models typically involve interface elements at delamination prone areas. In this macro approach, the effect of the delamination is smeared into the overall behaviour of the element through the degradation of the appropriate stiffnesses. This approach has the advantage of not needing apriori knowledge of the crack path, but 40 2.2. Modelling Failure in Composite Materials characterization of the model such that delaminations are properly captured is challenging, and not specifically addressed in this thesis. Although numerical models are popular, analytical models continue to be developed. The Whitney-Nuismer point-stress and average stress linear-elastic fracture mechanics (LEFM) failure criteria (Whitney and Nuismer, 1974) continue to be popular. The relatively simple concept attempts to include a damage process zone ahead of a developing crack. This process zone causes a deviation from the infinite stress singularity predicted by traditional LEFM. Rather than predict the deviation in the stress field, the point stress failure criteria specifies that failure occurs when the stresses at a characteristic distance (d0) from the crack tip exceed a threshold. The average stress failure criteria is similar, except that in this case, the threshold is the average stress along a characteristic distance (a0) from the crack tip. Figure 2.7 illustrates this for a particular failure stress. This set of failure criteria allows for considerably larger applied stresses than would a typical stress-based failure theory. \ Distance from Crack Tip Figure 2.7. Illustration of the Whitney-Nuismer point-stress and average-stress failure criteria. 41 2.2. Modelling Failure in Composite Materials Chen et al. have used the Whitney-Nuismer criteria as the basis for a failure theory they call the "load-bearing ply-failure criterion" (Chen et al., 2001). This model is based on the failure of the load-direction plies in a laminate and a residual strength calculation after failure. Good correlation was obtained with tests of graphite-BMI (bismaleimide resin) open-hole tension specimens. Others, including (Boniface et al., 1991; Daniel, 1982; Chu, 1988; Belmonte et al., 2001) have used LEFM to model transverse crack growth with some success. In addition to traditional LEFM analyses, other energy-based models have been applied to composites. In particular, Spearing, Beaumont and Ashby at Cambridge have developed an energy-based method applied to composites in fatigue applications (Spearing et al., 1992). The model predicts the extent of damage growth using prior knowledge of the fatigue damage pattern, and does not predict the damage path. With this foreknowledge, the model was used to simulate fatigue crack growth in carbon fibre/epoxy laminates. There are several stress-based failure theories for composites other than those considered in the WWFE. Thorn reviewed many of these theories, describing their evolution over the past thirty years (Thorn, 1998). Thorn also reviews several experimental tests used to verify the strength models, but does not evaluate the models in the manner of the WWFE. This review of composite failure theories is not complete, but does provide a good overview of the types of models available to the designer. Details of several of the major theories have been presented, and references for the other theories have been provided. It is within the context of these theories that the CODAM model is developed. 42 Chapter 3: Development of Continuum Damage Models for Composites This chapter concentrates on continuum damage mechanics (CDM) models. In first half of the chapter, the background and literature review portion of the chapter, the first section presents an overview of the background and fundamentals of CDM models, and the next section describes some of the CDM models that have been developed for composite materials. In the second half of the chapter, an overview of the earlier CODAM model is presented next and then new developments to the model, simplifications and general usage guidelines and procedures are presented. A new simple damage model (SDM) is also developed. Finally, new developments in the use of the two models in both explicit and implicit numerical analysis are addressed. 43 Chapter 3: Development of Continuum Damage Models for Composites 3.1. Introduction to Continuum Damage Mechanics Continuum damage mechanics has been developing and maturing at a tremendous pace in the past thirty years. Rather than provide a comprehensive review of the efforts of the many researchers in this field, an overview of the highlights of the research work will be given. A number of authors have summarized the state of the art of CDM at various times, and the reader is directed to those for further background. These reviews include: Krajcinovic, 1984; LemaTtre, 1984; Bui and Stolz, 1985; Murakami, 1987; Chaboche, 1987; Krajcinovic, 1989; Krajcinovic, 1995; Krajcinovic, 2000; and de Borst, 2001. The acknowledged grandfathers of CDM are Kachanov and Rabotnov, two Russian researchers who applied the concept of smearing damage due to microcracking over a continuous medium and representing this damage by discrete and deterministic internal variables in their studies of creep in metals (Kachanov, 1958; Rabotnov, 1963). Although their work was not strictly what has become continuum damage mechanics, it certainly provided the conceptual basis from which CDM was derived. Continuum damage mechanics has had a remarkably rapid evolution since the work of Kachanov and Rabotnov. In order to discuss its history and development, the foundations of the theory will be established. Continuum damage mechanics is used to model damage evolution in brittle and quasi-brittle materials, such as rock, concrete, ceramics, and polymer matrix composites. These materials exhibit strain-softening and CDM is particularly well suited to model such behaviour. Creep in metals was the primary application for the early CDM-like models, but they are not addressed here. Other than creep analyses, the earliest applications of CDM to mechanical analyses are found in the work of Dougill (Dougill et al., 1977; Dougill and Rida, 1980), Krajcinovic (Krajcinovic and Fonseka, 1981; Krajcinovic and Silva, 1982; Krajcinovic, 1983; Krajcinovic, 1984), Resende and Martin (Resende and Martin, 1984), and the work of Talreja et al. who pioneered the application of CDM to composites (Talreja, 1985b; Talreja, 1985a). 44 3.1. Introduction to Continuum Damage Mechanics 3.1.1. Model Scale At its simplest, mechanical analysis relates external influences to the change in state of a body. External influences can include forces, pressures, impacts, changes in temperature, and so on. The body can include very large complicated structures like buildings or airplanes, but could also include small structures like laboratory test specimens, or even very small structures at the molecular scale. In an effort to solve problems at these very different length scales, different analysis techniques have been developed, tailored to be useful at particular length scales. Structural mechanics is used at the larger end of the scale spectrum, where mechanical behaviour is abstracted to the component level: beams, trusses, columns, plates, and connectors all have known behaviours, and the details of what happens within the particular component is often unimportant in the analysis. At the very small end of the scale spectrum, micro-mechanics has developed where the behaviour is abstracted to the level of material phases, interfaces, voids, and crystalline structures. The scale could be even smaller, the atomic scale, at which the interaction of individual molecules is considered. Molecular mechanics and statistical mechanics are analysis techniques employed at the atomic scale. Figure 3.1 illustrates the analysis techniques used at the various length scales. Micro-mechanics Macro-mechanics Molecular mechanics Meso-mechanics Structural mechanics 1 0 - 1 0 1 0 - 6 1 0 - 3 1 0 ' 1 1 0 ° 1 0 2 Length scale (m) Figure 3.1. Illustration of analysis techniques at various length scales. Between the extremes of molecular mechanics and structural mechanics is a length scale that is analyzed with macro-mechanics. This scale is the intermediate scale, where material behaviour is more abstract than the small details considered by micro-mechanics, but more detailed than the large assemblies of components considered by structural mechanics. Continuum mechanics has developed out of the need to understand what occurs at the macro-scale. Small structures are abstracted to consist of assemblies of small components which are considered to behave as if they were 45 3.1. Introduction to Continuum Damage Mechanics homogeneous. At the micro-scale, structures are piece-wise continuous, but not homogeneous. That is, discontinuities exist at the micro-scale, and they are considered in the mechanical analysis. At the macro-scale, however, the discontinuities are integrated into the description of the behaviour of an equivalent homogeneous material. As a consequence, a material model at the macro-scale may include the effect of micro-cracking, but it may not include the details of those micro-cracks. At the micro-scale, the details of those cracks are integral to the behaviour of the model. Krajcinovic notes that in the process of transiting from the atomic scale through to the macro scale (Krajcinovic, 1995): "Billions of ruptured bonds, dislocations, and atomic vacancies are reduced to the tens of thousands of microcracks and slip systems which are, in turn, replaced by a damage variable and the plastic strain tensor." The abstraction of the micro-mechanical scale to the macro-mechanical scale requires the introduction of a representative volume element (RVE). The RVE itself represents the scale to which homogenization has been applied. The effects of interactions and processes within the RVE are smeared, so that the RVE is the smallest unit that is considered in a macro-mechanical analysis. Stresses, strains, stiffnesses, and any other parameters that can be used to describe the state of an RVE are homogeneous (uniform) in the RVE. Within the context of composite materials, micro-mechanical analyses consider fibres, matrices, the interfaces between the fibres and matrices, and voids. The development of cracks, the growth of voids, the interaction of debonded fibres, and plastic deformation of the matrix are examples of damage processes that concern those performing micro-mechanical analyses. Macro-mechanical analyses are concerned about the effects of the same processes on an RVE of the same material. For example, matrix crack development may lead to a reduction of stiffness in the RVE. The details of the crack development are not considered, but the generalized effects are. Figure 3.2 illustrates a representative volume element of a composite laminate. 46 3.1. Introduction to Continuum Damage Mechanics Full laminate Section from laminate (3 sub-laminates) Sub-laminate based RVE Figure 3.2. Illustration of a representative volume element (RVE) of a composite laminate. Krajcinovic details two requirements for RVEs (Krajcinovic, 1995): An RVE is the smallest volume of material 1) that possesses the same material characteristics as the test specimen and, 2) for which continuum mechanics models lead to results of acceptable accuracy. Continuum damage mechanics in essence sacrifices the details of the smallest processes to provide an efficient method for predicting the behaviour of macro-scale structures. Ideally, analyses at the different structural scales feed into each other: the behaviour predicted at the atomic scale is incorporated in the models predicting behaviour at the micro-scale, the predictions of the micro-mechanical models are incorporated in the models predicting the macro-scale, and the macro-scale models are incorporated into the large-scale structural analyses. In reality, however, some disconnect exists between the models at the various scales, perhaps due to the innate disagreement of how accurate is accurate enough. 47 3.1. Introduction to Continuum Damage Mechanics 3.1.2. State Variables r With the establishment of a suitable RVE, the next step at forming a workable CDM framework is defining state variables. State variables represent the current state of the RVE beyond the usual material parameters, such as elastic stiffness, Poisson's ratio, etc., and the mechanical parameters, such as stress state, strain state, etc. Typically, this means defining damage as a quantifiable parameter. Damage is a natural choice as a state variable, since these sorts of analyses typically involve materials that form voids and cracks, easily identifiable as damage. In composites, damage is usually related to crack density in some manner. The definition of damage varies from model to model, as does the representation of damage. Chaboche discusses a number of types of damage measurements that could be incorporated into a CDM theory (Chaboche, 1988a). The list of possible damage measures includes the life ratio from fatigue analysis (the ratio of the number of cycles sustained by a structure to the number of cycles required to initiate failure), quantification of microstructural defects through direct measurement, and the indirect measurement of damage through the observation of physical parameters and the effective stress concept (discussed in §3.1.3. ). Employing this last method requires a definition of damage consistent with the parameter being observed. A common damage definition (Chow and Wang, 1987; Lemaitre, 1984; Chaboche, 1988a; Matzenmiller et al., 1995) is to consider the cross-sectional area of an RVE, S, at some orientation, h. This area becomes an effective resisting area after loading, S , due to damage caused by micro-cracks and void growth, as shown by Figure 3.3. 48 3.1. Introduction to Continuum Damage Mechanics Figure 3.3. Illustration of effective resisting area (Adapted from Chow and Wang, 1987). Damage in this direction can then be defined according to Equation 3.1: ® = 1 - - | (3-1) S If damage is uniformly distributed in all directions, then there is no directionality involved, and co becomes an isotropic damage parameter. Li et al. define damage according to the influence it has on the transverse stiffness (Li et al., 1998). They note that damage is related to the crack density but do not provide a direct relationship between crack density and damage. Instead, damage is defined according to Equation 3.2. (3.2) where E° and ET are the original and current transverse lamina stiffnesses. Note, in this thesis, there are often three co-ordinate systems: the lamina co-ordinate system, the laminate co-ordinate system, and the global load co-ordinate system. For consistency and clarity, the subscripts L and T are used to refer to the lamina co-ordinate system, the longitudinal (parallel to fibre) direction and transverse (perpendicular to fibre) direction. The sub-scripts 1, and 2, traditionally used to denote 49 3.1. Introduction to Continuum Damage Mechanics the lamina directions, in this case refer to the laminate co-ordinate system. The 1 direction is parallel to the 0° lamina fibre direction, and the 2 direction is perpendicular to that in the plane of the laminate. The subscripts x and y refer to the global load coordinate system. In this manner, there is no need to refer to a "new" laminate each time a laminate is subjected to a load applied in a different direction. Damage, co, can be represented as a scalar, where a single parameter describes the complete damage state of an RVE, a vector, where the overall damage is represented by components in each co-ordinate direction, or as higher-order tensors. Pickett et al. used a scalar damage parameter to describe damage in composite laminates, but noted that it was a simplification used in the interests of computational efficiency (Pickett et al., 1990). Li et al. also employed a scalar damage parameter, but unlike Pickett et al., attributed its influence on the elastic constants in a manner particular to each direction (Li et al., 1998). The model by Li et al. was also targeted at a lamina based RVE, whereas the model from Pickett et al. was targeted at a laminate based RVE. Scalar damage representations are appropriate for isotropic damage, where the damage is randomly distributed, or in cases that the void density is small (Murakami, 1987). To describe anisotropic damage, a higher-order representation of damage is required. To achieve a better description of damage, vectorial and tensorial forms of the damage parameter can be adopted. Vectorial representations separate the damage into vector components, with damage parameters in each of the coordinate directions. Tensorial representations of damage have also been used, in the form of second-order, fourth-order, and higher-order tensors. These tensor forms accommodate descriptions of damage that include damage magnitudes in many directions as well as interactions of damage parameters in the different directions. Tensorial representations become mathematically complicated, and an increasing number of parameters are required to characterize the damage with increasingly higher order representations. Yazdani incorporated a fourth-order damage tensor in his CDM model (Yazdani, 1993). On the subject of these higher order presentations, Krajcinovic notes (Krajcinovic, 1984) that: " tensors of fourth- or even eighth-order eliminate any hope for the preservation of the physical clarity of the model." 50 3.1. Introduction to Continuum Damage Mechanics 3.1.3. Effective Stress and Strain Equivalence Two concepts that are central to most CDM formulations are those of effective stress and strain equivalence. Stated simply, the concept of effective stress is that the stress state in a damaged material, a, can be represented by an effective stress state, &, in an undamaged material of the same geometry. In equation form (e.g., Williams and Vaziri, 2001): {&} = [M(<o)la} (3.3) where [/W(a>)] is some transformation that is a function of the damage parameter, co. Chow and Wang refer to M as a damage effect tensor (Chow and Wang, 1987), but the form of M obviously depends on the choice and form of the damage parameter. Strain equivalence takes the same idea one step further. Strain equivalence is the hypothesis that (Lemaitre, 1984): "Every strain behaviour of a damaged material is represented by constitutive equations of the undamaged material in the potential of which the stress is simply replaced by the effective stress." In linear elasticity, the relationship between strain and stress can be written as: M=M>} (3.4) where C° is the original undamaged stiffness matrix. Now, the strain equivalence hypothesis allows us to write the relationship for strain in a damaged material by incorporating the effective stress: M = [ C ° ] > } (3-5) Rearranging the terms of Equation 3.5 and substituting Equation 3.3, the relationship between stress and strain in a damaged material can be found: {o-} = [/W(<y)]-1[c°]{£} (3.6) 51 3.1. Introduction to Continuum Damage Mechanics Combining the damage transformation function with the stiffness matrix to have a stiffness matrix dependent on damage, [C(»], a general constitutive relationship for CDM models is found: {o-} = [c(*} (3.7) Note that Equation 3.7 is the simplest elastic form of the relationship, but it can be expanded to consider rate effects {s} (see Equation 3.8) and plasticity (Chaboche, 1988a; Simo and Ju, 1987; Lubliner et al., 1989; Armero and Oiler, 2000; Chaboche, 1988b). {d} = [C(a,oj)fc} + [c(a>,cb)ls} (3.8) 3.1.4. Damage Growth The relationship between the external influences and the state variable can now be defined. Loosely speaking, this is how damage grows or develops in the RVE. Many different approaches have been taken in defining this relationship, and these can be divided into three main categories (Murakami, 1987): (i) The thermodynamic theory of inelastic constitutive equations. (ii) Statistical approaches. (iii) Formal extension of uniaxial damage growth equations to stress or strain based damage surfaces. (i) Thermodynamic approach The thermodynamic approach became popular during the early development of CDM. It was a comfortable approach for those working in the analysis of metals since it easily incorporated the ideas of both damage and plasticity. In general, the concept of plasticity was extended to damage growth by considering damage as an internal state variable similar to those used in plasticity. Damage growth, an irreversible thermodynamic process, shares a number of parallels with plasticity, and Frantziskonis demonstrated the equivalence of plasticity and CDM with a scalar damage representation (Frantziskonis, 1994). 52 3.1. Introduction to Continuum Damage Mechanics As formulated by Chaboche, the Helmholtz free energy for a damaged solid is expressed in terms of the internal state variables, namely damage and plastic hardening parameters (Chaboche, 1988a): V = %(se,T,D) + Vp(T,r) (3.9) where W is the Helmholtz free energy, separated into an elastic component *Fe, and a plastic component, sg is the elastic strain tensor, T is temperature, and D and r are the state variables corresponding to damage and isotropic hardening. Note that the partitioning into an elastic damaged component and plastic hardening component is complete; the plastic component does not depend on damage at all, and vice versa. As proposed by Chaboche, the elastic component of the Helmholtz free energy (the damaged elastic behaviour) is: <F.=-L-(1-D)fo.a (3.10) where E is the conventional elastic stiffness matrix and p is the material density. Using the concepts of effective stress and strain equivalence, stress can be now expressed as: * = p ^ = ^-D)Ee9 (3.11) From this definition of the damage elastic behaviour, the size increase of the plastic yield surface, R, and its equivalent "damage strain energy release rate", Y, are the thermodynamic forces associated with r and D: Y = p — = --Eel (3.12) y 3D 2 e R = p— (3.13) dr Note that in this derivation, all the damage effects are represented by the damage variable D, and the material density p is constant. Now, considering the elastic strain energy density, We: 53 3.1. Introduction to Continuum Damage Mechanics dWe=odse (3.14) we can substitute the expression for stress and rearrange it: dWe=p^-dee={l-D)Esed£( (3.15) We=j(A-D)Es2e (3.16) (3.17) (1-0) Or simply: dWe (3.18) dD Equation 3.18 illustrates why Vis called the damage strain energy release rate, as it can be defined as the elastic strain energy released per unit increasing damage. The next restriction that the laws of thermodynamics place on the evolution of damage is the requirement that intrinsic dissipation remains positive. This leads to development of flow rules that incorporate both plasticity and damage. The development of these equations will not be presented here; rather, the references are suggested for further investigation of the many ways to formulate the necessary equations (Chaboche, 1988a; Chaboche, 1988b; Lacy et al., 1997; Krajcinovic and Fonseka, 1981; Krajcinovic, 1983; Krajcinovic, 1984; Krajcinovic, 1989; Lemaitre, 1984; Allix et al., 1989; Allix et al., 1991; Ladeveze et al., 1993; Lene, 1986). (ii) Statistical Approach Damage growth can also be related to statistical variations in the rupture strength of a material. Damage is related to the probability that a given stress or strain state is likely to exceed the strength of the RVE. This sort of formulation allows for freedom in selecting the probability distribution function that governs the damage development. 54 3.1. Introduction to Continuum Damage Mechanics A common statistical approach to describing variations in the strength of materials is Weibull's "statistical distribution function of wide applicability" (Weibull, 1951). This theory is described in some detail in Section 5.3. Weibull's probability distribution function has been implemented into a number of models (Krajcinovic and Silva, 1982; Xia and Wang, 1998; Matzenmiller et al., 1995; Williams and Vaziri, 2001). Damage in Matzenmiller's model is defined in one dimension as: where e is the exponential function, m is the Weibull modulus, and sf is the nominal failure strain. A major shortcoming of such models, however, is the difficulty in relating the statistical parameters to physical characteristics of the problem. Of particular difficulty with models based on Weibull's approach is the selection of the Weibull modulus, m, without a priori knowledge of the statistical variation of the nominal failure strain in the specific material examined. (Hi) Extension Of One Dimensional Growth Curves To Three Dimensions The clearest way to formulate damage growth curves is to explicitly define the relationships between the stress or strain state of the RVE and damage. The statistical approach discussed above is a subset of this method, with the relationship between strain and damage related by a probability distribution function. There is no requirement for this sort of relationship; any damage growth curve can be considered as long as it can be justified within the context of the material that the model is targeted and the definition of damage adopted by the model. The CDM model by Pickett et al. (Pickett et al., 1990) and the CODAM model both belong to this last category. 3.1.5. Effect of Damage With the definition of damage, and a method for describing how damage evolves, the remaining step is to define how damage affects the behaviour of the material. This is the specification of M(m), the damage effect tensor. In almost all the models, the effect of damage is incorporated into the definition of damage. By doing so, the material (3.19) 55 3.1. Introduction to Continuum Damage Mechanics parameters, (usually the stiffnesses), are degraded by a factor of (1-<y). The model developed by Matzenmiller et al. uses a damage effect tensor of the form (Matzenmiiler etal., 1995): M(a)) = 1 SYM 0 1 1-<y 22 0 1 1 - » 1 2 (3.20) The choice of the parameters to be degraded is tied directly to the model. Li et al., for example, degrade only the transverse stiffness, the shear stiffness, and the minor Poisson's ratio (Li et al., 1998). Models with higher-order damage tensors may have fully populated damage effect tensors. In addition to the stiffnesses, other material parameters may be affected by damage. The most obvious additional parameters are the Poisson's ratios. Talreja has investigated the effect of damage on the Poisson's ratios, and has suggested that damage has significant effect on the Poisson's ratios in laminated composites (Talreja et al., 1992). Zhao and Yu propose separate independent damage parameters for the Poisson's ratios (Zhao and Yu, 2000). Very few models address the degradation of Poisson's ratio in practice (Chow and Wang, 1987, and most others, for example). 56 Chapter 3: Development of Continuum Damage Models for Composites 3.2. Review of CDM Models for Composite Materials The quasi-brittle nature of composite laminates, and the strain-softening behaviour observed in experiments (for example, Dopker et al., 1994) makes CDM a natural tool to model the failure of composites. Talreja was one of the first to recognize the advantages of using CDM in modelling composites (Talreja, 1985a; Talreja, 1985b). Talreja's model assigned a damage parameter to the fibre direction and the matrix direction in a UD lamina. Although details were not provided on the damage growth functions, the model was used to predict the stiffness reduction in angle-ply laminates to good success. Talreja's model has recently been updated to include rate effects by incorporating linear viscoelasticity (Kumar and Talreja, 2003). The updated model was used to predict the viscoelastic behaviour of cross-ply laminates with transverse cracks. Pickett et al. developed a simple CDM model for sheet moulding compound (SMC) composites, which was computationally efficient, but made a number of simplifying assumptions that limited the applicability of the model (Pickett et al., 1990). Pickett's model incorporated linear damage development curves, and linear modulus reduction curves with a scalar representation of damage. Supporting this simplified relationship is a study by Aboudi who showed a modulus reduction with increasing crack density that could be approximated with a linear relationship (Aboudi, 1987), and a CDM model by Harris et al. that also used linear stiffness reductions with the damage parameter (Harris etal., 1988). Frantziskonis and Joshi developed a lamina based CDM model for laminated composites (Frantziskonis and Joshi, 1990). This model considered damage to Poisson's ratio, differences between tensile and compressive loading, and used phenomenological failure criteria to describe the onset of damage. An interesting innovation in Frantziskonis and Joshi's approach was their treatment of the damage parameter. A single damage parameter was used to characterize the damage, but this parameter was used in a damage effect tensor such that its influence on the RVE stiffnesses was unique to each component. Good agreement was found in comparisons between the model and experiments on simple tensile tests. 57 3.2. Review of CDM Models for Composite Materials Randies and Nemes addressed the issue of thick composite laminates with a CDM model that included a damage parameter to describe delamination (Randies and Nemes, 1992). Although delamination is a significant damage mode, even in the case of planar loading, it tends to be ignored by most CDM based composite models. Later development of the model by Nemes and Speciel incorporated rate effects (Nemes and Speciel, 1996). The model was used to simulate cantilever bend tests on graphite/epoxy at high strain rates. Renard et al. presented a CDM model based on a damage parameter that related the transverse ply thickness and the average crack spacing within that ply (Renard et al., 1993). This damage parameter reduced all the components of the stiffness tensor except the axial stiffness (Qn), by an exponential decay that was a function of the damage parameter. The model was compared to tensile tests of cross-ply and angle-ply lay-ups of T300/314 epoxy, and good agreement was found. Divergence between the model and notched tensile tests was attributed to the model's inability to capture delamination. Nguyen later extended the model to a general three-dimensional formulation, by using multi-planar volume finite elements, and achieved better agreement with the experiments (Nguyen, 1998). Matzenmiller et al. (1995) developed a lamina based CDM model that incorporated Weibull's probability distribution function to describe the statistical nature of the rupture strain. Additionally, the model used an independent shear damage parameter that was used to degrade shear stiffness of the lamina. This model, while straightforward and one of the few models to consider non-deterministic effects, required the specification of the Weibull modulus, which sets the shape of the probability distribution function. Currently, no guidelines are available for making the selection of the Weibull modulus other than comparing the simulation results to experimental results and calibrating the model. Several attempts at doing so have demonstrated that the Weibull modulus required for the same material used in different applications is not unique (Williams and Vaziri, 2001; van Hoof et al., 2000). c Zako et al developed a phenomenological CDM model, using Hoffman's failure criterion (Zako et al., 1996; Zako and Uetsuji, 2002). The model is abstracted at the laminar level, and interface elements are used to model inter-laminar failures. A standard modulus reduction (E= (1-co)£°) is employed, where damage is derived from a damage 58 3.2. Review of CDM Models for Composite Materials tensor determined by the various failure criteria. Zako et al. compared their numerical model to open-hole tension experiments with reasonable success. Li et al. (1998) developed a lamina based CDM model that considered only transverse matrix cracking. Only the transverse stiffness, shear stiffness, and the minor Poisson's ratio were degraded with accumulated damage. lannucci et al. considered the development of damage in woven glass fibre / epoxy laminates (lannucci et al., 2001). This CDM model separates each lamina into two, considering the effects of damage in the warp and weft individually. The lamina response is determined from a rule-of-mixtures based combination of the in-plane properties. No coupling is included, and the effect of crimp is neglected. An interesting innovation is including rate effects in the damage growth curves. The model was used to simulate a bird impact on a wing structure, but quantitative comparisons to the experiments were not included. In addition to the standard CDM models described above, researchers at LMT-Cachan have developed what they have called a damage mesomodel for laminates (DML). The DML considers laminated composites at the laminate scale, rather than the lamina scale. Using a combination of CDM and micro-mechanics, the DML considers damage from fibre breakage, matrix microcracking, fibre/matrix debonding, and delamination. Damage parameters exist for each ply and the interfaces are treated in a semi-discrete manner. A thermodynamic approach is taken with respect to damage in the laminae. For more details on the DML model, the reader is directed to the references (Dumont et al., 1987; Allix et al., 1989; Ladeveze et al., 1990; Ladeveze et al., 1993; Ladeveze et al., 2000; Hochard et al., 2001). Recently, attempts have been made to tie micro-mechanical models to the DML (Ladeveze and Lubineau, 2002; Ladeveze and Lubineau, 2003). Of all the CDM models available, none are able to provide everything that an engineer requires to confidently predict the behaviour of composites subjected to loads that introduce damage into the structure. The most striking evidence of this is the inclusion of only a single CDM model in the World-Wide Failure Exercise. The models are too complicated to understand, require too many parameters to be characterized, are unable to provide clear procedures for characterizing the model parameters, are too specialized 59 3.2. Review of CDM Models for Composite Materials to accommodate diverse loading applications, are unable to handle all the damage modes of interest (for example, delamination), are unable to accurately match experimental tests, require too much computational power, and so on. 60 Chapter 3: Development of Continuum Damage Models for Composites 3.3. Overview of the CODAM Model The Composite DAMage (CODAM) model is intended to provide a constitutive model for the simulation of polymer composite laminates that: 1. Is physically based. 2. Is easy to understand and easy to use. 3. Applies to cases where gross amounts of damage are developed in a composite and the post-initial damage behaviour of the composite is important. 4. Is computationally efficient. 5. Is able to include the effects of the major damage modes: matrix cracking, fibre-breakage and delamination. 3.3.1. Original Formulation CODAM was originally developed by Williams (Williams, 1998; Williams et al., 2003). The original version of CODAM (CODAM3Ds) had many of the characteristics of the current model developed as part of this thesis work (CODAM). The original model was a CDM formulation, abstracted at a sublaminate level. Damage growth was assumed to grow with strain state linearly in two zones, and the modulus reduction was assumed to be related to increasing damage linearly in two zones. The two zones were a region of matrix damage and a region of mixed matrix and fibre damage. Figure 3.4 shows a typical stress-strain curve resulting from this approach, with a third zone showing the initial elastic behaviour. CODAM3Ds, the Williams model, was formulated for shell elements and was demonstrated in two applications: simulations of crack growth in over-height compact tension specimens, and simulations of non-penetrating impact. 61 3.3. Overview of the CODAM Model 1.2 Zone I: Elastic Zone III: Combined matrix and fibre damage z 0.2 s 0 6 1 0.4 o 0 0 0.01 0.02 0.03 0.04 Average Strain (Al/I) Figure 3.4. Typical stress-strain diagram for original CODAM formulation. Williams identified a number of issues that needed to be addressed in further development of the model (Williams, 1998). The need to separate the damage growth functions into phase specific functions (e.g. separate matrix/delamination damage and fibre damage functions) was highlighted and improved characterization data or methods was identified as being important for successful use of the model. Mesh sensitivity was also acknowledged as an issue, with the nonlocal approach suggested as a possible avenue to solve this problem. Finally, the need for a fully three-dimensional formulation was recognized in order to address applications such as penetrating impact events. 3.3.2. Model Improvements The present work expands on Williams model in many aspects. The original three-zone model was reconsidered and generalized. Note that the original terminology employed by Williams was two-zone and bilinear. The current formulation treats the elastic zone the same as the damaging zones, so using this terminology, the Williams formulation was three-zone and trilinear. In the current formulation, the relationship between the strain state and the damage, and the damage and the modulus reduction, is piecewise linear. This allows for any number of linear segments to begin and end at any strain or damage state. One practical application of this approach is to allow matrix damage to 62 3.3. Overview of the CODAM Model saturate before fibre damage saturates. A resulting stress-strain curve could be as shown in Figure 3.5. </> </) 0) CO 7 3 Ci N "to E 1.2 1 0.8 0.6 0.4 0.2 0 Zone I: Elastic 0 Zone III: Combined matrix and fibre damage Zone IV: Fibre damage only Zone II Matrix damage only 0.06 Figure 3.5. 0.02 0.04 Average Strain (Al/l) Representative stress-strain curve for the current formulation of CODAM. Additionally, the model was expanded to include a three-dimensional brick formulation, and new element erosion algorithms were added in order to simulate penetration events. The full three dimensional formulation of the model is not addressed in this thesis, but the references are suggested for explanation of the features and applications of the model (Floyd et al., 1999; Floyd et al., 2001a). Lastly, but most significantly, the energy involved in the damage evolution has been incorporated into the model. This consideration of the energy dissipated is addressed in detail in Chapter 4. Briefly, by considering the energy the model becomes globally insensitive to localization. This also aids in the characterization of the material. These substantial modifications to the model resulted in the complete rewrite of the computer code underlying CODAM's implementation. 3.3.3. CODAM in context of the WWFE The CODAM model fits into a group of models that was significantly under-represented in the World-Wide Failure Exercise, the continuum damage mechanics models. The one 63 3.3. Overview of the CODAM Model CDM model examined, the McCartney model, was limited in its ability to address all the test cases in the Exercise, and did not perform very well in those cases that it did attempt. The poor performance of this model should not be considered as representative of CDM approaches as the organizers noted (see §2.2.1.3. (vii)). The CODAM model predicts the behaviour of composite laminates that are subjected to gross damage development. The model operates at a macro-mechanical scale, and is not intended to provide the details of the material behaviour at the scale considered by the participants of the WWFE. As such, the initial failure (damage) in a structure is not as significant to CODAM as the final failure and the post-failure behaviour. As in the WWFE, the definitions of failure are nebulous. For the CODAM model, initial failure is the strain state that initially causes some damage to manifest in the sublaminate. Final failure is the strain state at which damage has fully developed in the sublaminate. This final failure is well beyond the point of maximum stress that is typically called final failure by most failure theories. The CODAM approach fits well into the suggestions of the organizers of the WWFE. The WWFE showed that initial failures were difficult to predict by all of the leading theories. In the CODAM model, the analysis is relatively insensitive to the initial failure data. In terms of framework, the CODAM approach has been developed with the intention of providing a useful analysis tool. Its inputs have been designed to be physically realistic and conceptually easy to determine, although in its current completely general form, the number of parameters would be overwhelming to an analyst in industry. Furthermore, it has always been intended to be used in the context of a finite element analysis; the tool industry relies on for stress analysis. Its formulation is such that the FE analyses of composite structures subjected to loads that cause severe damage are possible. 64 •Chapter 3: Development of Continuum Damage Models for Composites 3.4. Details of the CODAM Model CODAM is essentially a set of maximum strain failure criteria that uses CDM to model the post-initial failure behaviour of a composite laminate. CODAM has been formulated as a macro-mechanical model that is based on the behaviour of sublaminates. Figure 3.6 illustrates replacing the analysis at a lamina level with an analysis at a sublaminate level. This differs from the traditional micro-mechanical approach to failure in composites that examines failure at the fibre and matrix level. The sublaminate based approach takes advantage of the repeating damage patterns evident in composites (Delfosse and Poursartip, 1997) (see Figure 2.5), allows for the capture of interlaminar effects, and makes its implementation into finite element analysis programs straight-forward and practical. Others (Chatterjee and Ramnath, 1988) have also used the sublaminate approach to modelling composites. tttasttstsosostotiwsi [0/90/0/90/0]4 20 laminae [0/90/0/90/0]4 4 sub-laminates Figure 3.6. Illustration of replacing lamina based analysis with sublaminate based analysis. 3.4.1. Effective Strain CODAM is based on the concept that the strain state of a representative volume of a sublaminate dictates the representative volume's damage state. Rather than directly relating the strain tensor to some equivalent damage parameter tensor, an effective strain is introduced. This effective strain, F, allows interactions between various strain components when determining a particular damage component. 65 3.4. Details of the CODAM Model The form of the effective strain function has been chosen such that it resembles a measure of strain energy. The effective strain is determined for each sublaminate direction (F,, /' = 1, 2, 3) according to Equation 3.21. * - tfj •(?!- -fefeHtfeMtMf H f J if (3,21) where £1t s2, £3, £12, £23, and £31 are the strain tensor components, K, L, and M, are interaction constants that may be different in tension or compression, S, T, and U are shear strain interaction constants, and the subscript /' refers to each of the principal sublaminate directions 1, 2, and 3. (Recall that the subscripts 1, 2, and 3 are in the sublaminate co-ordinate system, the subscripts x, y, and z are in the global load co-ordinate system, and the subscripts L and T refer to the lamina coordinate system.) The form of Equation 3.21 is the most general expression of effective strain, and 21 interaction constants are required to fully characterize it. In practice, the constants are chosen such that only the strain in the direction considered and the corresponding shear strain contribute to the effective strain. For example, for shell elements (plane stress) Fj might be expressed as: + £. 12 (3.22) where the other parameters are chosen to be sufficiently large as to make the contribution of the strains in the other directions negligible. The form of the effective strain expression is not integral to the concepts underlying CODAM. The current form is convenient in that it can be both very general and very simple depending on the choice of the interaction constants. It also has the familiar form, of strain energy which is often used as the driving force for damage in CDM formulations (for example, Lemaitre). However, due to the construction of the CODAM formulation, replacing this expression of effective strain by another measure is a perfectly feasible task. 66 3.4. Details of the CODAM Model 3.4.2. Damage State The damage state in the representative volume of the sublaminate is represented by the damage parameter, OJ. A separate damage parameter exists for each principal sublaminate direction (<y?, co2, co3), and a further parameter exists for each of the shear directions (OJI2, OJ23, co3i). The damage parameters are representations of the extent of damage in the sublaminate. These parameters can vary from 0, indicating no damage at all, to 1, indicating complete damage (i.e., no further damage can occur). The directionality of the damage parameters is representative of the effect of damage on the behaviour of the sublaminate in that particular direction. The need to express the damage state with a separate parameter comes from two sources. First, in typical continuum damage mechanics formulations, the effect of damage on a material is a stiffness reduction by a factor of (1-<w). Formulating CODAM in a similar manner aids in the understanding of the model. More importantly, however, expressing the damage state aids in the interpretation of the model. Appendix B presents an algebraic illustration of how the damage parameter does not affect the mechanics of the analysis, but aids the interpretation of the results. The shear damage parameters are derived parameters, unlike the principal direction damage parameters that are computed from the CODAM curves. The main reason for this distinction is based on the interpretation of the damage parameters. The directional damage parameters are interpreted to be representative of cracks with crack-face normals in the direction considered. For example, coi is interpreted to be an expression of the projection of all the cracks in a representative volume of the sublaminate in the 1-direction. The adoption of this interpretation relates the principal direction damage parameters to physical damage. All the physical damage can be accounted by the three principal directions, leaving no physical interpretation for the shear damage parameters. The shear damage parameters are necessary, though, to provide a measure by which the shear stiffnesses can be reduced. As such, the shear damage parameters are computed according to Equation 3.23. (3.23) 67 3.4. Details of the CODAM Model where ij are the principal sublaminate directions (1 2, 3). Chow and Wang employed a similar form for the shear components of their damage effect tensor (Chow and Wang, 1987), as have Chaboche et al. (Chaboche et al., 1995). 3.4.3. Modulus Reduction The consequence of damage in a representative volume of a composite sublaminate is a reduction in the structural stiffness. This is achieved by modifying the secant stiffnesses by a normalized instantaneous modulus, E. The normalized modulus ranges from 1, or the full initial stiffness, to 0, no stiffness at all. Each principal direction stiffness and shear stiffness has its own normalized modulus. The stiffness tensor of a damaged element is then constructed from the modified stiffnesses. No permanent deformation is included in the model, and secant stiffnesses are used throughout, so unloading of a damaged specimen proceeds at the current stiffness to an unstressed and unstrained state. 3.4.4. Damage Growth Curve The effective strain and damage parameters are related through damage growth curves. CODAM employs a piecewise linear relationship between the effective strain and the damage parameters, allowing for any number of segments to be independently defined for each of the material directions. Pickett et al. used a similar approach, but their damage growth relationship was bilinear, like that of the original CODAM formulation (Pickett et al., 1990). Ogihara et al. showed results from experiments on cross-ply and angle CFRP laminates that showed transverse crack density growth as a function of axial strain that could easily be represented by a piece-wise linear formulation (Ogihara et al., 2001). The multi-segment piecewise linear formulation allows more flexibility in the model, for example a relationship could be defined between the strain state and the matrix damage and an independent relationship could be defined between the strain state and the fibre damage. The combination of these two independent damage modes results in the overall system behaviour. This approach is typically adopted in the use of CODAM in this thesis, but the generality of the formulation allows for the introduction of other damage modes, such as delamination, if necessary. Figure 3.7 shows the construction of the damage growth curve from the individual damage mode growth curves. 68 3.4. Details of the CODAM Model 0.080 Figure 3.7. Representative damage growth curve. In CODAM, a damage growth curve is defined for each principal direction, in tension and compression. 3.4.5. Normalized Modulus Curve In a manner similar to the damage growth curves, the normalized moduli are related to the damage parameters through normalized modulus curves. The normalized modulus curves are also piecewise linear in nature, and for each damage mode specified in the damage growth curve, a segment of normalized modulus as a function of damage parameter is defined. The original CODAM formulation was bilinear in nature, similar to the approach taken by Pickett et al. (1990), but again the multi-segment piecewise linear approach allows for more flexibility. Linear normalized modulus curves can be defined for each damage mode, and these independent curves can be combined in the same way to create a system response. This is illustrated in Figure 3.8. 69 3.4. Details of the CODAM Model 1.00 0.75 Matrix • Fibre System lui 0.50 0.25 0.00 0 0.2 0.4 0.6 0.8 1 1.2 CD Figure 3.8. Representative normalized modulus curve. Despite the fact that the shear damage parameters are calculated from the principal direction damage parameters, a shear modulus reduction curve must be defined for each shear modulus. Owing to the calculated nature of the shear damage parameters, and their dependence on the form of the effective strain equation, the form of the normalized shear modulus curves is not easily defined. Additionally, the shape that the curve should take is not understood or easily intuited. As such, the normalized shear modulus curves have been generally simplified to be linear: 3.4.6. CODAM Stress-Strain Relationship A common way of visualizing the behaviour of constitutive models is to consider the resulting one dimensional stress-strain curve. The one dimensional stress-strain curve prescribed by the example curves presented above is illustrated by Figure 3.9. The curve is characterized by a linear segment, where no damage exists in the sublaminate, followed by a series of intersecting parabolas. Each parabola results from a segment in the damage growth curve and the normalized modulus curve, and since the elastic modulus in CODAM is a secant modulus and no permanent deformation is included, the parabolas all pass through the origin. The net result is some restrictions on the shape of the one dimensional stress-strain curve. (3.24) 70 3.4. Details of the CODAM Model 1.2 0 0.02 0.04 0.06 Ave rage S t ra in ( A \ I \ ) Figure 3.9. Sample one dimensional stress-strain curve resulting from the previous example damage growth and normalized modulus curves. Figure 3.10. Example of normalized modulus as a function of effective strain, based on the example damage growth and normalized modulus curves. 71 3.4. Details of the CODAM Model An alternate method of visualizing the effect of damage within the CODAM framework is to examine the relationship between the effective strain and the normalized modulus, as shown in Figure 3.10. This relationship is in effect the important one from the point of view of the computer code since the damage parameter is only used for interpretation purposes (see Appendix B). 72 Chapter 3: Development of Continuum Damage Models for Composites 3.5. Plane Stress Specialization of CODAM The development and use of CODAM in three dimensions, or more precisely, the development and use of CODAM in the brick element formulation, are not addressed in this thesis. Instead, the references are suggested for details of the three-dimensional formulation and examples of its usage (Floyd et al., 1999; Floyd et al., 2001a). In the current section, the typical simplifications and processes involved in using CODAM in two dimensional plane-stress will be addressed. 3.5.1. Simplifications Even with the simplification of working with two-dimensional shell elements, a tremendous amount of information is required to fully characterize the model. One of the most important simplifications is the selection of the number of segments in the damage growth and normalized modulus curves. Generally, it is assumed that there are three modes of behaviour: undamaged elastic, matrix damage, and fibre damage. Further, the assumption is made that the shear modulus degrades linearly with the shear damage parameter, as explained in §3.4.5. With these assumptions, Table 3.1 lists all the parameters required to satisfy a general formulation of CODAM. 73 3.5. Plane Stress Specialization of CODAM Table 3.1. Parameters required to describe a general 2D CODAM constitutive model. Parameter Number of constants Elastic constants Ei, E2, v12, Gn 4 Matrix damage mode Damage initiation effective strain, Fmi 4 x 2 directions (1, 2) x 2 (tension, compression) Damage saturation effective strain, Fms Damage parameter attributed to matrix damage saturation, a)ms Sublaminate modulus reduction due to saturated matrix damage, £ m s Fibre damage mode Damage initiation effective strain, F f i Damage saturation effective strain, Ffs Damage parameter attributed to fibre damage saturation, <% Sublaminate modulus reduction due to saturated fibre damage, E f e Effective strain interaction constants K, L 2 x 2 directions (1, 2) x 2 (tension, compression) S + 1 x 2 directions (1, 2) Total 38 2 x 2 directions (1, 2) x 2 (tension, compression) (^=1-<»ms, £ = 1 - E m s ) With the relatively simplified two-dimensional formulation, 38 independent parameters are required to characterize one sublaminate with CODAM. Although this allows for an impressive amount of flexibility, it is excessive for casual use. On closer examination however, there is only a little room for further simplification. For each of the damage modes, the assumption can be made that the tensile and compressive behaviour is the same. This simplification is crude, as differences have been observed in between the compressive and tensile damaged behaviour (Soden et al., 1998, for example), but for applications where tensile failure is the dominant mode and no global load reversal is present, it is acceptable. This assumption removes 12 constants from the requirements. The remaining area for simplification is in the interaction constants in the effective strain expression. Again, assuming that tensile and compressive behaviour is the same 74 3.5. Plane Stress Specialization of CODAM removes a few more constants. Further, as the form of the effective strain expression has not been studied in detail, it is assumed that only those strain components that are in the same direction as the effective strain being determined will be considered. Equation 3.22 in §3.4.1. demonstrates this assumption. In addition, it is assumed that the shear strain contributes equally to the effective strain in each direction, resulting in Si=S2-With these assumptions, the number of parameters required is reduced to 21. 3.5.2. Elastic Constants Material data for the sublaminate being examined is likely expressed in terms of fibre and matrix data or in terms of 0° unidirectional (UD) lamina data. Rarely, the sublaminate values will be directly available. In the case of only having the fibre and matrix data on hand, a standard rule-of-mixtures approach can be adopted to estimate the UD lamina data (see for example Kaw, 1997). The UD lamina data, either from this rule-of-mixtures calculation or from the published lamina properties, can then be used in a standard laminate plate theory (LPT) approach to estimate the multidirectional (MD) sublaminate data. The net result is the axial stiffnesses, Poisson's ratio, and shear stiffness of the sublaminate. 3.5.3. Effective Strain Interaction Constants With the simplifying assumptions mentioned in the previous section, the remaining constants in the effective strain interaction function are Ki, L2, and S. Choosing Ki = Z_2 = 1 and S = •» results in F, = &, and makes initial failure in CODAM equivalent to a non-interactive maximum strain failure criterion. This approach is regularly adopted in analyses performed with the current CODAM formulation as a further simplification of the problem. Alternatively, shear strain interaction has been included in some analyses, choosing S = 2 or some other value that reduces the influence of shear strain. No rule of thumb is offered as to which approach to take for any given problem. The former approach is simpler while the latter acknowledges the idea that shear strain will contribute to damage affecting the principal directions. In fact, for cases where S = °°, the model is completely insensitive to shear strain, and under pure shear loading it will not manifest any damage. 75 3.5. Plane Stress Specialization of CODAM 3.5.4. Effective Strain Parameters The CODAM parameters are those that describe the shape of the damage growth and the normalized modulus curves. These include the effective strains at damage initiation and saturation for each of the damage modes, the damage parameters associated with the saturation of damage in each of the modes, and the modulus reduction associated with each damage mode. These parameters could be determined by some experimental means, which has not been developed yet, or by estimating the values based on the constituent behaviour and engineering judgement. A procedure for performing this estimate is outlined below, based on a method suggested by Williams (Williams et al., 2003). Figure 3.11. Representative volume element subjected to uniaxial displacement loading. Estimating values for these parameters is currently a very subjective process. Consider the case where only fibre and matrix damage modes are modelled. In order to characterize the sublaminate in each principal direction, a thought experiment ("virtual experiment") is performed in each direction. A representative volume element (RVE) of the sublaminate is quasi-statically subjected to uniaxial displacement in the direction of interest, illustrated in Figure 3.11. At some point, the strain state in the RVE is such that matrix cracks, typically the first manifestation of damage in a sublaminate, begin to form. Figure 3.12 shows a cross-section of.the RVE with significant matrix cracking. 76 3.5. Plane Stress Specialization of CODAM Figure 3.12. RVE with matrix cracks. The strain at which these cracks form is related to the measured strain at failure for an isolated body of matrix material. In context of characterizing a sublaminate for use in CODAM, this "failure strain" of the matrix material is typically used as the matrix damage initiation strain state. Substituting the damage initiation strain into the expression for effective strain provides the value for Fmi. As the displacement increases, delaminations may form between the plies in the RVE. Whether delaminations occur depends on the lay-up of the sublaminate. Development of delaminations is illustrated in Figure 3.13. Delaminations Figure 3.13. RVE with matrix cracks and delaminations. Continuing the thought experiment, damage increases in the RVE with increasing uniaxial strain, with matrix cracks accumulating in various plies. At some later point, fibres will begin to develop cracks. In a uniaxial longitudinal tensile test of a UD lamina, this will occur at the ultimate failure strain of the UD lamina since the failure is unstable and once begun the whole lamina will fail. In a MD sublaminate, it is reasonable to assume that fibre cracks will begin in the 0° plies at (or near) the UD failure strain. Thus, Ffl is determined. 77 3.5. Plane Stress Specialization of CODAM Figure 3.14. RVE with matrix cracks, delaminations, and fibre cracks. Determination of the damage saturation strains is much more of a challenge. At a later time in the virtual uniaxial tension test of the RVE, matrix cracking reaches a point such that no further matrix cracks can develop. Finally, the fibre damage also saturates and the RVE can develop no more damage or sustain any load. Figure 3.15 illustrates the RVE with saturated damage. It may be that both these events occur simultaneously or independently. For simplicity, the assumption is often made that both components saturate at the same strain state. Figure 3.15. RVE with saturated damage. Matrix damage and fibre damage have grown across the full RVE. It should be noted that the strains developed after damage initiates are average strains in the RVE rather than actual material strains. As damage develops and cracks open, the average strain in the RVE may be much larger than the local strain at any given location in the sublaminate. Since the CODAM model captures the behaviour of the entire sublaminate, it is these larger average strains that should be considered when determining the strain states after damage initiation. 78 3.5. Plane Stress Specialization of CODAM The state of strain at damage saturation remains difficult to characterize. Currently, no experimental techniques are available to directly measure this value. Kongshavn and Poursartip have developed a technique for growing damage in a material in a controlled fashion (see discussion about the Overheight Compact Tension tests in Chapter 8) and have measured the nominal strain-to-failure of small specimens cut from damaged material on the order of 3%-4% (Kongshavn and Poursartip, 1999), exceeding the strain-to-failure of similar but undamaged material of about 1.5%. From the author's experience of using the CODAM model to simulate a variety of materials and lay-ups, it seems that for laminates dominated by 0° plies, the saturation strain for fibre damage should be relatively close to the fibre damage initiation strain. For angle ply and cross ply laminates, the damage saturation strain should be higher to account for the displacements caused by crack openings. Typical saturation strains that have been used with success in simulations range from 2% to 4%. For examples, refer to the application chapters, Chapters 6-8. 3.5.5. Damage Parameters In a simplified CODAM model where damage has been separated into fibre and matrix modes, only one damage parameter is required, either the value of the damage parameter associated with matrix damage saturation (coms) or fibre damage saturation (ojfS). Note that previous symbology (Williams, 2002, for example) used co'm to refer to coms. As previously discussed (see §3.4.2. and Appendix B), selection of the damage parameter does not affect the mechanics of the analysis, but is important in the interpretation of the values of the damage parameters in the analysis. The value of the damage parameter associated with matrix damage saturation can be estimated by examining the lay-up of the sublaminate. Returning to the thought experiment, 0° plies can be approximated as exhibiting no matrix damage. The 90° plies can be approximated as being fully damaged. Plies at other angles will exhibit matrix damage somewhere between fully damaged and not at all at damage saturation. 45° plies, for example, have been estimated to contribute 50% to the matrix damage parameter at saturation. 79 3.5. Plane Stress Specialization of CODAM The rule of mixtures can then be employed to estimate the damage parameter at saturation for the sublaminate. Using a [0/±45/90/±45/0] lay-up as an example, the amount of damage parameter associated with matrix damage saturation would be: 2 x 0 + 4 x 0 . 5 + 1x1 3 _ . „ ®ms* j = y = 0.429 Since the damage parameter at saturation must be equal to 1.0, the damage associated with fibre damage at saturation is simply: « f c = 1 - ^ m S =0.571 3.5.6. Modulus Reduction Parameters As with the damage parameters, when the simplification of separating the damage evolution into only two damage modes is made, only one modulus reduction parameter is required. Either the normalized modulus loss associated with matrix damage saturation (Em s) or fibre damage saturation ( E m f ) can be specified. The easiest method for determining these values is to employ the ply-discount method and laminated plate theory (LPT) to estimate the sublaminate stiffness at a particular damage state. At matrix damage saturation in the thought experiment, 0° plies effectively have no axial stiffness loss, 90° plies effectively have no axial stiffness at all, and angled plies have some degree of their original axial stiffness between none and full stiffness. Depending on the degree of sophistication of the estimate, the plies can be discounted fully, so that a damaged ply has no stiffness in any direction, or the individual stiffnesses can be discounted. For example, although the 90° plies have no stiffness at saturation in the 0° direction, they have effectively full stiffness in the 90° direction. Typical ply-discounting methods would completely dismiss the contribution of the 90° ply in all directions, but this full discounting is not required. Using this method, the lamina stiffnesses are required. Knowing these stiffnesses, the appropriate discounting method can be applied and LPT can be used to determine the stiffness of the damaged sublaminate. For example, using the traditional ply discount method, discounting the 90° and ±45° plies in a [0/±45/90/±45/0] laminate, the original 0° 80 3.5. Plane Stress Specialization of CODAM stiffness of a carbon/epoxy laminate is 50 GPa, and the discounted 0° stiffness is 33 GPa. The resulting normalized modulus loss due to matrix damage saturation is: E m s =1-—= 0.340 m s 50 Using a more sophisticated discounting where the stiffness of the 90° and ±45° plies in only the direction perpendicular to the fibres is discounted, the discounted stiffness is 42 GPa, and the normalized modulus loss due to matrix damage saturation would be: 42 Ems =1- —= 0.160 s 50 The normalized modulus loss due to fibre saturation is simply the remaining available normalized modulus. In the purely discounted case, this would be: E f e =1-0.340 = 0.660 Note that in previous versions of CODAM, the modulus reduction parameter was called R'E and that value is equivalent to E f t . 81 Chapter 3: Development of Continuum Damage Models for Composites 3.6. The Simple Damage Model In order to investigate the behaviour of strain-softening simulations without the additional complications of all the parameters required by the CODAM model, a new simple damage model (SDM) has been developed. This simplified constitutive model formulated within the continuum damage mechanics framework has a stress-strain relationship that is triangular, as shown in Figure 3.16. 0 0.01 0.02 0.03 0.04 0.05 Average Strain (Al/I) Figure 3.16. Stress-strain relationship defined by the Simple Damage Model. The SDM is defined by three parameters, the initial stiffness £°, the strain at which the peak stress occurs, £peak, and the strain at which the material has zero stiffness, eM. Each direction, including shear, is treated independently, and Poisson's ratio is not degraded. The model only represents a simple strain-softening formulation and is not intended to be representative of any particular material. 82 Chapter 3: Development of Continuum Damage Models for Composites 3.7. Explicit and Implicit Analysis In the past, strain-softening constitutive models have been typically implemented in only explicit finite element codes, and CODAM in particular had only been implemented in the explicit FE code LS-DYNA and its variants (e.g., VEC-DYNA). Many reasons existed for this preference, the largest of which was numerical difficulty. In typical implicit FE codes, the relationship between the stresses and strains is defined by the tangent stiffness matrix. For softening materials, the tangent stiffness matrix can be negative and may be zero at times. Many codes cannot handle negative (negative definite) tangent stiffness matrices, and none can easily handle zero tangent stiffness matrices (singular matrices) since they cannot be inverted. Explicit finite element solvers do not use the tangent stiffness matrix to determine the nodal displacements from the forces. Instead, nodal displacements are determined from nodal accelerations, which are in turn determined from the nodal forces and elemental stresses. The stiffness of the element is required only to relate the strains at the integration points (determined from the nodal displacements) to the stress at the integration points. The stiffness could be expressed either as a tangent stiffness or a secant stiffness; from a numerical perspective, it does not matter. Recently, an implicit solver has been incorporated into the LS-DYNA code. This solver, while still under development, interacts with constitutive models by requesting the tangent stiffness matrix from the constitutive model. However, the constitutive model can return the secant stiffness matrix instead of the tangent stiffness matrix. In doing so, achieving convergence at the next load step requires more iterations but the difficulties associated with returning negative or zero tangent stiffness matrices are completely avoided. Adding an interface between CODAM and LS-DYNA to ailow implicit analyses with CODAM was a relatively minor modification to the CODAM code. As soon as the normalized moduli are determined, the dsave() array is set to the updated secant stiffness tensor. LS-DYNA uses this array to estimate the nodal displacements at the next iteration. 83 3.7. Explicit and Implicit Analysis Implicit analyses involving CODAM are interesting in quasi-static simulations. In quasi-static simulations, momentum effects are negligible but in explicit analyses, the momentum is always considered. In explicit analyses, strain waves or other dynamic effects are often observed in quasi-static simulations, and their presence may affect the results. By performing these simulations implicitly, the dynamic events do not interfere with the analysis. However, implicit analyses are unable to capture sudden and large changes such as severe geometric changes due to catastrophic material failure, so their use in failure analysis may be somewhat limited. Implicit analyses can also require vast amounts of computer memory and may lead to large simulation run-times. Although explicit analyses may involve many more time steps than equivalent implicit analyses, less computational effort is required per time step in explicit analyses. In minimizing simulation run-times, there is a balance between small step sizes in explicit analyses and the number of degrees of freedom in implicit analyses. For this reason, implicit analyses usually involve significantly fewer degrees of freedom than explicit analyses. The ability to perform implicit analyses with CODAM is an additional tool in the efforts to simulate and understand the development of severe damage in laminated composite materials. 84 Chapter 4: An Investigation into Element Size and Localization Effects This chapter focuses on strain localization and the effects of choosing different element sizes in otherwise identical simulations. Background information on the so-called "mesh effect" is presented, followed by several approaches to deal with the issue. Among these approaches are the crack band method developed by Bazant and others, the equivalent work method, developed over the course of this thesis, and some alternatives. Following this background information, concrete examples of the mesh effect are presented in context of the CODAM and SDM constitutive models; then the incorporation of the crack band method and the equivalent work method into CODAM and the SDM is shown, and the usefulness of these approaches is demonstrated. 85 Chapter 4: An Investigation into Element Size and Localization Effects 4.1. Introduction Simulation of structures using the finite element method (FEM) and a strain-softening material model has been shown to lead to issues with localization and a loss of mesh objectivity. Localization is the condition where damage becomes concentrated in a single element, or a row of elements, and does not affect or influence the neighbouring elements. A lack of mesh objectivity in general refers to the inability of the analysis to achieve convergence with progressively refined discretization. Generally in finite element analyses (FEA), the more refined the mesh becomes, the closer the analysis comes to the analytical solution. FEA with strain-softening material models do not achieve this convergence without additional numerical techniques. The dependence of the FEA solution on the size of the elements is here referred to as the mesh effect. 4.1.1. Mesh Effect Belytschko et al. discussed these difficulties at some length (Belytschko et al., 1986). Using the example of a one-dimensional rod into which a wave was introduced, and another example of a strain wave introduced into a sphere, they examined strain-softening behaviour, localization, and use of a nonlocal approach. Their analyses showed significant dependence of the numerical analysis of the sphere example on the element size. Additionally, they noted that the energy dissipation vanishes with increasing element refinement. de Borst addresses the localization issue as well, in his review of issues in computational failure mechanics (de Borst, 2001). Using the examples of a bar subjected to an axial force, as well as a more complicated two-dimensional example of a SiC composite, de Borst illustrates how a softening material model results in an inability to converge to a solution with increasing mesh refinement. Localization has also been examined by others (Belytschko et al., 1986; Ladeveze et al., 1990; Nemes and Speciel, 1996). To demonstrate the localization problem, consider a sequence of four elements, with a progressively damaging material behaviour, attached in series and subjected to a tensile load. Figure 4.1a illustrates these elements with a stress that remains in the elastic regime. As the overall load is increased, the load is increased in each element, and as 86 4.1. Introduction long as the material remains elastic, each element has the same strain (Figure 4.1b). At some point however, one element becomes damaged before the others, and its stiffness decreases. In order to maintain the same axial force within the series of elements, the damaged element must undergo a greater axial strain than the other non-damaged elements (Figure 4.1c). The other elements relax, showing a reduction in strain. Figure 4.1d shows the stress-strain history for element 2 (top), the early damaging element 3 (middle), and the average stress-strain history for the complete structure (bottom). (a) (b) (c) (d) Figure 4.1. Schematic of one dimensional localization. In this case, the structure shows a certain ultimate strength and certain post-failure behaviour. In the case of fewer or greater number of elements modelling the same geometry, the post-failure behaviour will change, demonstrating the mesh effect. In one dimension, the changes in the global response between analyses using different mesh sizes are minimal. For multi-axial load states however, localization causes more severe problems, affecting global parameters such as the peak load and the strain distribution. 87 4.1. Introduction This will be demonstrated later in the chapter with the CODAM model as well as the Simple Damage Model. To address the issue of localization and the mesh effect, several techniques have been proposed in the literature. Early attempts at dealing with the mesh effect and the simulation of cracks using finite elements used a technique known as the cohesive crack model (Hillerborg et al., 1976). In this method, a crack's location is predetermined, and an interface is placed between opposing elements along the crack length. The interface that ties the opposing elements together has a softening behaviour that is calibrated to the energy release rate of the material. The method continues to be developed and used, and is a common approach to modelling crack growth from an energetic or fracture mechanics point of view. For more information on cohesive methods, the references (Backlund, 1980; de Borst, 2003; Planas et al., 2003; Llorca and Elices, 1991; Borg et al., 2002; Cornec et al., 2003; Jansson and Larsson, 2001) are suggested. This approach has several limitations. The major issue is that the crack location and path are predetermined. Often, simulations are run on problems for which no experimental data is available, and the expected location of the cracks is not available. In addition, the cohesive crack model does not allow for damaging materials, only a damaging interface. As a result, a damaging material with a cohesive crack interface would still exhibit localization and mesh sensitivity. 4.1.2. Crack Band Method 4.1.2.1. Background and Formulation Predating the cohesive crack model, yet addressing the same concerns, Rashid introduced a concept of smearing the effect of the discrete crack in a manner similar to that used in continuum damage mechanics (Rashid, 1968). In this approach, the material stiffness is reduced in the direction normal to the crack after the peak strength is achieved. In order to deal with the localization mesh effect problem that affects this approach and CDM approaches, Bazant et al. (Bazant and Oh, 1983; Bazant, 1984, Bazant and Oh, 1984) suggest using the fracture energy, G F , of the material as a guide to set the shape of the strain-softening curve. This approach has been called the crack band method, and Bazant covers its development quite extensively (Bazant and Planas, 1998). The basics of the approach will be summarized below. 88 4.1. Introduction A crack or a zone of cracking is represented by a band of distributed cracks that has a height, characteristic to the material being considered, of hc. After damage occurs, the strain in the band is called the fracture strain, or / The crack opening displacement, w, can then be given by Equation 4.1, which is identical to the relation found in the basic cohesive crack models. heef =w (4.1) Softening is introduced into the material once the maximum stress is realized. The type of softening is arbitrary, but typically, the secant modulus is reduced in some simplified manner. The area under the curve of the stress-strain relationship from the peak stress onward is the specific fracture energy, yF. The energy required to completely fracture a unit area of a single element can then be given by Equation 4.2. GeF=hcfF (4.2) where the superscript e indicates an element quantity. The useful provision in the crack band method is that in the numerical application of the method, the restriction on the element size to be equal to the characteristic height, hc, is removed. Instead, as long as the characteristic fracture energy GF is kept constant according to Equation 4.3, then elements larger and smaller than the characteristic height may be used. GF=hcrF=herF (4.3) The justification for this relaxation is that the elements in which a crack is being modelled will localize if a strain-softening constitutive relationship is involved. Since the localization occurs in a band of elements a single element high, as long as those elements dissipate the same fracture energy as the equivalent elements of height hc, the behaviour will be identical. Therefore, in a general implementation of the crack band method, one must use a softening constitutive relation and ensure that the characteristic fracture energy GCF is maintained. 89 4.1. Introduction Bazant postulates that GCF is a material constant, and can be experimentally determined. Several of the references demonstrate how this is accomplished with concrete (Bazant and Oh, 1983; Bazant and Oh, 1984; Bazant and Planas, 1998). 4.1.2.2. Applications of the Crack Band Method in the Literature The crack band method has been extensively applied to investigations involving quasi-brittle failure of concrete and rock. Bazant and Oh introduced the crack band method for the simulation of fractures in quasi-brittle materials in 1983 (Bazant and Oh, 1983). The approach was compared against quite a number of experiments that had previously been reported in the literature. Fracture involving unreinforced concrete was the focus of their investigation, and the approach was shown to represent the results of all the experiments quite satisfactorily. In a follow up to their previous paper, Bazant and Oh applied the crack band method to fracture in solid rock (Bazant and Oh, 1984). They note that LEFM does not correctly predict the trends in modelling rock fracture. In order to perform the simulations of the fractures, the crack band method was applied once again. In these simulations, a simple bi-linear strain-softening constitutive relationship (similar to the Simple Damage Model discussed in §3.6. ) was applied in conjunction with the crack band method. A number of different experiments were simulated, with very good agreement between the model and the experiments. Bazant and Oh note that the approach is very convenient for finite element simulations, and very simple. Around the same time that Bazant was developing the crack band method, others were exploring similar ideas. Pietruszczak and Mroz presented a model that was very similar to the crack band method, but it lacked a direct link to the fracture energy GF (Pietruszczak and Mroz, 1981). Their approach was related to a shear band within individual finite elements, and they showed that it scaled with respect to the element area. They also demonstrated that the approach removed the mesh dependence of the softening model. No comparisons to experiments were drawn, however. In 1984, Bazant continued exploring the crack band method and how it fit into the existing approaches to deal with size effect (Bazant, 1984). Bazant showed that the crack band method predicted a size effect different than that expected by linear elastic fracture mechanics as well as traditional Weibull-type analysis. Further, he showed that 90 4.1. Introduction the crack band method has a size effect in quasi-brittle strain softening materials such as rock and concrete similar to that exhibited by analyses on metals involving elastic-plastic fracture. In 1989 Oliver presented a very mathematical treatment of an approach very similar to the crack band method (Oliver, 1989). The paper discussed difficulties in allowing for different crack paths within elements, and showed methods for determining the characteristic length of a material by comparing numerical analyses to experimental results. Oliver showed that the approach was entirely element size insensitive for a generalized isotropic strain softening material. Rather than deal with the smeared crack approach by modifying the constitutive model, others have taken the approach of incorporating the ideas of the crack band method in the element formulations instead. Ortiz et al. presented such a model (Ortiz et al., 1987), and demonstrated that it effectively avoided the localization problems commonly observed in strain softening analyses. Dvorkin et al. also presented an element formulation that dealt with localizations (Dvorkin et al., 1990), based on the cohesive crack approach and the crack band method. Their approach, involving displacement interpolated localization lines within the elements also demonstrated mesh objectivity and predicted the proper energy dissipation in fracture applications. Their approach is able to handle mesh distortions and other non-uniformities in the mesh during the numerical analysis, a problem that typical crack band formulations are unable to properly model. 4.1.3. Equivalent Work Method Building on the work of Bazant and others with the crack band method, a new work-based method has been developed. The equivalent work method was developed as an alternative to the crack band method that is capable of matching the energy dissipated in damage, even in cases where the damage was not complete. In the crack band method, GF is the total energy dissipated per unit area, and due to the non-linearities in the stress-strain curves, it does not require elements in intermediate damage stages to dissipate the same energy as the characteristic element. §4.4.7. explores this in more detail. 91 4.1. Introduction The equivalent work method requires that the total work done by a force applied to an element of the same height as the material characteristic height (hc) must be the same as the work done by a force applied to a series of elements replacing the characteristic element. This concept is illustrated in Figure 4.2 and expressed in Equation 4.4. P . 8 hc/n 1 Si 6o hc /n l T Figure 4.2. Illustration of replacing an element of a characteristic height with an integer number of smaller elements. \PdS = X \PndSn (4.4) Localization occurs when strain-softening material models are used. In the replacement elements, one element will localize or bifurcate, while the others will not. The elements that do not localize (the n-1 elements) remain contained on the pre-peak stress portion of the stress-strain curve. The bifurcated element proceeds into the softening portion of the stress-strain curve. The force in the characteristic element is the same as the force in each of the n replacement elements, so Equation 4.4 can be rewritten as: Sc=Sbi+(n-i)Sn_, (4.5) where Sc is the elongation of the element of characteristic height, is the elongation of the bifurcated element, and is the elongation of each of the n-1 remaining elements. 92 4.1. Introduction 0.02 0.04 0.06 0.08 0.1 Normalized Displacement (Al/I) 0.12 Figure 4.3. Illustration of equivalent work scaling, in load-displacement space. Note that the load has been normalized with respect to the peak load. Figure 4.3 illustrates this concept in terms of force and displacement. Alternatively, Equation 4.5 can be written in terms of strain: •+-n-1 n 'n-1 (4.6) where the subscripts c, b/', and n-1 refer to the characteristic, bifurcated, and remaining n-1 elements respectively. Equations 4.5 and 4.6 describe compatibility between the characteristic element and the replacement elements. The equivalent work method requires that the work done by the replacement elements be equal to the work done by the characteristic element at all locations on the load-displacement curve of the replacement elements. This is somewhat different than the crack band method, which requires the work to be the same if the replacement elements are loaded to a completely damaged state. In the crack band method, the intermediate states (before complete damage) do not necessarily have the same work done. 93 4.1. Introduction AAA. Other Approaches In addition to the cohesive crack approach and the crack band method, there are a number of other ways to deal with the localization problem. These other approaches are typically referred to as nonlocal theories, and involve considering the effect of neighbouring elements or nodes in the determination of a local parameter, such as damage. These approaches will not be reviewed, rather the references (Bazant and Chang, 1987; Bazant and Jirasek, 2002; Bazant and Pijaudier-Cabot, 1988; Bazant and Ozbolt, 1990; Bazant and Novak, 2000a; Bazant and Novak, 2000b; Geers et al., 1998; Kennedy and Nahan, 1996; Nahan, 1998; Bazant and Planas, 1998) are suggested for further investigation. A particular type of nonlocal theory is the so-called smooth particle hydrodynamics (SPH). In this method, also referred to as the element free Galerkin (EFG) method, elements are not used, but a structure is abstracted as a collection of nodes, similar to finite difference nodes. The nodes are influenced by neighbouring nodes according to a relationship similar to a constitutive relationship, and the influence of neighbouring nodes is determined by using a spatially averaging function, which introduces a length scale or characteristic size. EFG (or SPH) has been used to simulate penetrating impacts and fracture growth in homogeneous materials, but the methods have difficulty with interfaces and orthotropic materials. The references (Beissel and Belytschko, 1996; Johnson et al., 1993; Johnson et al., 1996; Libersky and Petschek, 1990; Belytschko et al., 1994; Belytschko et al., 1996) are recommended for further details on SPH, and (Medina and Chen, 2000) for an example of using SPH to model laminated composites. 94 Chapter 4: An Investigation into Element Size and Localization Effects 4.2. Preliminary Mesh Effect Investigation To explore the mesh effect associated with strain-softening constitutive models, some analyses of simple applications were performed using the CODAM model. To reduce the complexity of the simulations and to isolate sources of interesting behaviour, the simple damage model (SDM) was later used to further the investigation. 4.2.1. Preliminary CODAM Analyses Initially, the CODAM model was used to simulate several unnotched tension tests, geometrically scaled in two dimensions. Four sizes of rectangular specimens were considered, with dimensions as shown in Table 4.1. Only half of each specimen was meshed, using a symmetric boundary condition at the middle of the specimen. Each specimen was meshed such that the mesh was fine along the symmetry plane and became coarser toward the end of the specimens. The specimens were meshed separately, resulting in unique meshes in each simulation. Figure 4.4 illustrates the types of meshes used, showing the mesh for the 6.35 mm wide specimen. Table 4.1. Specimen sizes for preliminary tension simulations w h (mm) (mm) 6.35 25.4 12.7 50.8 25.4 101.6 50.8 203.2 95 4.2. Preliminary Mesh Effect Investigation Figure 4.4. Mesh used in the preliminary 6.35 mm wide un-notched tension simulation. CODAM was used in the simulation of these specimens, and neither the elastic properties nor the CODAM parameters differed between each specimen size. Table 4.2 and Table 4.3 show the parameters used, and Figure 4.6 shows the resulting normalized modulus curve and stress strain curve. The details of the development of these parameters are provided in Chapter 5 when unnotched behaviour is investigated with CODAM. Table 4.2. Laminate properties for [±45/0/90]s AS4/3502. Property Value E1 46.9 GPa E2 46.9 GPa G12 17.9 GPa "tt 0.309 f larrinate 1.000 mm 96 4.2. Preliminary Mesh Effect Investigation Table 4.3. CODAM parameters for the quasi-isotropic lay-up of AS4/3502. Matrix Fibre 0.010 Ffi 0.012 Fms 0.030 Ffs 0.030 0.500 ®t* 0.500 E ms 0.352 Efs 0.648 I UJ 0. 0.04 (a) (b) Figure 4.5. (a) Normalized modulus curve and (b) Stress-strain curve for the quasi-isotropic lay-up of AS4/3502. The results of these preliminary simulations are shown in Figure 4.6. Although the element sizes along the mid-plane were consistent among the different specimen sizes, with square elements 0 . 2 0 mm on each side, failure of the specimens was occurring in the region of the mesh where the mesh was becoming coarser and the element sizes and shapes were quite variable. Figure 4.7 shows the mesh and damage patterns at the end of the simulation of the 6 . 3 5 mm wide specimen. 97 4.2. Preliminary Mesh Effect Investigation 10 Relative Coupon Size Relative Coupon Size Figure 4.6. CODAM results for the unnotched tension specimen simulations with non-uniform mesh. The coupon widths are relative to the smallest specimen, and the nominal failure stress and strain at failure have been normalized with respect to the failure of the smallest specimen. zones of severe damage Figure 4.7. Mesh and damage patterns for the 6.35 mm wide specimen at the end of the simulation. Figure 4.6 shows that the tension simulations showed a significant size effect. However, no specimen size effect should be present in unnotched tension simulations, since the stress fields within the specimens are uniform, and failure should occur as soon as any element achieves its ultimate stress. All the elements should therefore fail at the same 98 4.2. Preliminary Mesh Effect Investigation stress level. These stresses do not change with geometric scaling, so each size specimen should fail at the same nominal stress. To eliminate a further variable from the simulations, the specimens were remeshed, using a square mesh size of 1.27 mm throughout all the specimens, including the grip tabs, which for numerical efficiency were reduced to a single row of elements. Once again, the identical elastic and damage parameters were employed in the simulations. Figure 4.8. 2 4 6 Coupon Width CODAM results for uniform meshed unnotched tension simulations. The coupon widths are relative to the smallest specimen, and the nominal failure stress and strain at failure have been normalized with respect to the failure of the smallest specimen. The results in Figure 4.8 show that the a constant element size eliminates the apparent specimen size effect. With uniform square elements used throughout the meshes of each of the specimen sizes, the nominal stress - average strain results are identical. 99 4.2. Preliminary Mesh Effect Investigation (a) (b) Figure 4.9. Mesh and damage patterns for the (a) 6.35 mm and (b) 25.4 mm wide specimens at the end of the simulation (using 1.27 mm elements). This mesh sensitivity has some important consequences. The most obvious conclusion is that the mesh size effect (hereafter "mesh effect") can have a significant effect on the simulation results. Perhaps more importantly, however, it also means that the material and CODAM parameters that are specified and calibrated for a given element size are valid for only that element size. Elements of sizes different from the calibrated or reference size will not behave in the same manner. 4.2.2. Preliminary SDM Analyses The simple damage model was used to further simplify the analyses. The small number of parameters and the simple shape of the stress-strain curve in the SDM allow some confidence that the effects observed in the analyses are due to factors other than the shape and complexity of the constitutive model. Two applications were examined with the SDM. To begin, a simple 1D analysis was performed, and following that a more complex notched tension test was simulated. In each analysis, a single geometry was meshed with uniform elements at different levels of refinement, to examine the effect of mesh size on the results. 100 4.2. Preliminary Mesh Effect Investigation The 1D analysis consisted of a single column of rectangular elements in series subjected to a prescribed displacement (i.e., displacement control). The overall specimen geometry was fixed, but the number of elements along the length of the specimen was varied from 2 to 4, 8, and 16. Figure 4.10. 2,4,8,16 elements J 2000 J 1500 -I - 1000 -I in £ 500 0 0.02 0.04 0.06 Strain (mm/mm) A t\ An, ft r>,M 0.01 0.02 0.03 0.04 Average Strain (Al/I) 0.05 0.06 Normalized nominal stress - average strain results for the 1D axial tension simulation with the Simple Damage Model. The stresses have been normalized to the peak stress exhibited by the smallest specimen. The inset plot shows the stress-strain behaviour of a single element. The results are virtually identical for the 2,4, 8, and 16 element cases. Figure 4.10 shows that for each of the cases tested, the resulting nominal stress -average strain results are virtually identical. This result was not unexpected, given the localization problem. It has been suggested in several works (Bazant and Planas, 1998, for example) that a snap-back phenomenon should be observed, where the descending arm of the stress-strain curve angles back towards the origin, but in finite element analysis, this cannot occur. To demonstrate that snap-back cannot occur, consider the two possible cases, load-controlled loading and displacement-controlled loading. In 101 4.2. Preliminary Mesh Effect Investigation load-controlled tests, no restriction is placed on the displacement behaviour of the load heads. As such, as soon as one element in the series begins to soften, the displacement of the specimen must continue to increase such that the same axial force is present in each element in the specimen. This becomes unstable immediately since the force is always increasing, and the softening element cannot maintain a force (nominal stress) larger than that specified in its constitutive relation. In displacement-controlled tests, as the term suggests, the displacement is controlled. Typically this means that the displacement rate is controlled, with the displacement of the load points increasing steadily at a known rate. The average strain within the specimen (Al/I) will never be decreasing for this arrangement. The other test case used to evaluate the mesh sensitivity of the SDM model is the same geometry investigated by Bazant et al. (Bazant et al., 1994). The 25.4 mm wide single-notch specimen (with a notch length of 0.508 mm such that a=w/5) was chosen as a reasonably sized specimen, and five element sizes were selected for meshing the specimen (see §7.1. for a thorough background and analysis of this application). The test is shown schematically in Figure 4.11. p=P/w w vglass/epoxy grip tabs Figure 4.11. fTTTTTT Schematic of SNT specimen. Note a=w/5. The elements were square and sized as close to 0.1016 mm, 0.2032 mm, 0.508 mm, 1.016 mm, and 1.693 mm as possible, corresponding to scale factors of 1:2:5:10:16.67. Not all sizes were analyzed in every circumstance; often simulations with the 0.1016 mm elements were avoided due to the time required to complete such analyses. In cases where the notch length did not correspond to exactly a whole number of elements, those 102 4.2. Preliminary Mesh Effect Investigation elements on the notch side of the specimen (behind the notch tip) were sized slightly larger or smaller so as to get the proper notch length. This should not have affected the results, since this region near the notch is relatively stress-free, and away from the notch the stresses are in the elastic regime and therefore mesh size insensitive. Using the SDM as the constitutive relation, with £-^=0.02 and fu#=0.04, the specimens with the various mesh densities were subjected to displacement controlled tensile loading. 2 Average Strain (Al/I) Figure 4.12. Nominal stress - average strain results for the 25.4 mm wide specimen simulation with the SDM and varying mesh densities. The nominal stresses have been normalized with respect to the ultimate stress of the simulation using 0.508 mm elements (-510 MPa). Figure 4.12 shows that in this case, the simulations demonstrate significant mesh sensitivity. The simulations with the larger elements exhibit larger nominal stresses and average strains at failure than the simulations with smaller elements. Figure 4.13 shows 103 4.2. Preliminary Mesh Effect Investigation elements on the notch side of the specimen (behind the notch tip) were sized slightly larger or smaller so as to get the proper notch length. This should not have affected the results, since this region near the notch is relatively stress-free, and away from the notch the stresses are in the elastic regime and therefore mesh size insensitive. Using the SDM as the constitutive relation, with £peak=0.02 and £^=0.04, the specimens with the various mesh densities were subjected to displacement controlled tensile loading. 2 Average Strain (Al/I) Figure 4.12. Nominal stress - average strain results for the 25.4 mm wide specimen simulation with the SDM and varying mesh densities. The nominal stresses have been normalized with respect to the ultimate stress of the simulation using 0.508 mm elements (-510 MPa). Figure 4.12 shows that in this case, the simulations demonstrate significant mesh sensitivity. The simulations with the larger elements exhibit larger nominal stresses and average strains at failure than the simulations with smaller elements. Figure 4.13 shows 103 4.2. Preliminary Mesh Effect Investigation the trends for both the nominal failure stress and the average strain at failure for these simulations. 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 Relative Element Size Relative Element Size Figure 4.13. Trends in the nominal failure stress and average strain at failure for the 25 mm single notched tension simulations with the SDM and varying mesh densities. The nominal failure stresses and average strains at failure have been normalized with respect to the simulation with 0.508 mm elements. Having demonstrated that the 2D single-notch tension test simulations suffered from a significant mesh-size dependency, then next step was to reduce or remove that dependency by integrating the SDM with the crack band method. 104 Chapter 4: An Investigation into Element Size and Localization Effects 4.3. SDM and the Crack Band Method In its most general form, the crack band method requires a strain-softening constitutive relation and some mechanism to keep the energy required to completely fracture a band of material constant. For the SDM, these requirements are not particularly onerous. The model is strain softening, and in order to keep the fracture energy constant, a simple adjustment of euit is all that is required. Figure 4.14 demonstrates fracture energy matching with the SDM. The remaining implementation issue to be addressed is how to treat multi-axial strain states. The crack band method has been formulated in one dimension, and only cursory consideration has been given to multi-axial strain (or stress) states (Bazant and Oh, 1983). However given that the approach focuses only on Mode I cracking, multi-axial strain states modelled with the SDM will be treated independently. That is, each 105 4.3. SDM and the Crack Band Method principal direction will be treated completely independently with no interactive consideration given to the strain state and damage state of any other direction. This is a simplification, since this interaction is present in a physical material. For example, a matrix crack transfers no shear or normal stress across the crack, so both the shear and normal stiffnesses should be affected by the presence of the crack. In order for the crack band method to be tested with the SDM and in the absence of any physical correlation, one arbitrarily chosen element size was selected as the characteristic size for the material. That is, GCF was selected as the GF resulting from the model parameters associated with the 0.508 mm element width. The £u t t required to keep GF constant for the given element widths is shown in Table 4.4. Table 4.4. em, required to keep GF constant for the SDM single-notch tension simulations. Element height, he (mm) Suit 0.1016 0.20 0.2032 0.10 0.5080 0.04 1.016 0.02 The results of the single-notched tension simulations using the SDM and incorporating the crack band method are shown in Figure 4.15. Note that mesh with 1.693 mm elements was not analyzed, a decision made after considering the results of the analysis of the mesh with 1.016 mm elements. For the smaller element sizes, 0.1016 mm, 0.2032 mm, and the 0.508 mm benchmark, the nominal failure stress and the average strain at failure are very close. The larger element width, however, does not follow this trend and exhibits a nominal failure stress and average strain at failure that is larger than expected. Figure 4.16 shows the trend for these simulations. 106 4.3. SDM and the Crack Band Method 1.6 Average Strain (Al/I) Figure 4.15. Nominal stress - average strain results for the 25 mm wide single-notch tension simulations with the SDM and the crack band method incorporated. Note that the nominal stresses have been normalized with respect to the peak stress exhibited by the analysis with 0.508 mm elements (-340 MPa) 1 1.5 Relative Element Size 1 1.5 Relative Element Size Figure 4.16. Trends in the nominal failure stress and the average strain at failure for the 25 mm wide fracture-energy controlled SDM single-notch tension simulations. The nominal failure stresses and average strain to failure have been normalized with respect to the results of the analysis with 0.508 mm elements. 107 4.3. SDM and the Crack Band Method The curious behaviour of the large element size prompted further investigation into the behaviour of each of the meshes. Each mesh was subjected to scrutiny, varying the value of eM over an extensive range and recording the nominal failure stress for each simulation. In this manner, the behaviour of the model as a function of eM, which for a given element size is functionally equivalent to G F , can be compared across the range of element sizes. Only suit was varied for each of the simulations; the pre-peak portion of the model behaviour was constant in all cases. Figure 4.17 shows the results of this exhaustive study. z 100 : 0 0.2 0.4 0.6 0.8 1 Suit Figure 4.17. Nominal failure stress as a function of eult for each element size over a wide range of £u t t for the SDM simulations of the 25.4 mm width single-notch (a=w/5) tension specimen. The eM values are absolute, no scaling (e.g., crack band method) was involved. Figure 4.17 has some very interesting features. With the exception of the largest element size, 1.693 mm, changing Eun to a value less than the initial value of 0.04 caused no change in the nominal failure stress of the specimens. Section 4.4.5.1. discusses these plateaus (limiting stresses) in context of the element widths. Following the initial 108 4.3. SDM and the Crack Band Method plateau in nominal failure stress, each of the element sizes increased the nominal stress that they were able to resist. Given the direct relationship between sM and GF, this is hardly surprising. The consequence of this behaviour is significant. The figure shows that given a certain element size, the nominal failure stress will never fall below a certain level, regardless of the fracture energy contained within the constitutive relation. For example, suppose that the fracture energy had been chosen such that the mesh with 0.508 mm wide elements exhibited a nominal failure stress of 600 MPa (see Figure 4.17). For a mesh insensitive formulation, simulations with elements of other sizes should also predict a nominal failure stress of 600 MPa. Consulting Figure 4.17, one can determine that all the element sizes reviewed in this study should be capable of having their constitutive curves modified (by selecting an appropriate suit) such that they will indeed predict nominal failure stresses at 600 MPa. Another way of using the information presented in Figure 4.17 is as a tool to determine the £uit that would be required to maintain a certain nominal failure stress for different element sizes. These values read from the figure, can then be compared to the sM determined independently from the crack band scaling law. In the example of using 0.508 mm wide elements as the reference size and the nominal failure stress of 600 MPa, the figure indicates that the required sun is 0.149, and the resulting fracture energy per unit area is calculated to be 59.1 mJ/mm2. Using the crack band method, the elastic modulus, and Speak, the appropriate eun for each element size can be determined. However, using the 0.508 mm elements as the reference size is arbitrary, so any other size of element can be used as the reference size. Using the 0.2032 mm wide elements as the reference element size results in a suit of 0.447, and the associated fracture energy per unit area is 70.8 mJ/mm2, larger than that determined using the 0.508 mm wide elements simulation. In fact, sun for 0.508 mm elements to achieve this fracture energy is 0.179. This exercise demonstrates that SDM simulations do not completely agree with the crack band scaling method. To illustrate the correlation between the two methods, the Suit predicted by the crack band method taking each element size as the reference size is 109 4.3. SDM and the Crack Band Method compared to the sult read directly from Figure 4.17 in Figure 4.18. The reasonably small scatter shows that mesh dependence in the simulations is acceptably insignificant. 0 0.5 1 1.5 2 Element Size (mm) Figure 4.18. em, predicted by the crack band (CB) method using each element size as the reference for the G f that predicts a nominal failure stress of 600 MPa. Had the GF been chosen, however, so that the nominal failure stress was 400 MPa, then some difficulties would have arisen. At the 400 MPa level in Figure 4.17, the two larger element sizes, 1.016 mm and 1.693 mm, would not have been able to reach that nominal failure stress, regardless of the choice of Thus, it seems that there is an upper limit to the size of elements that can be used in these simulations. Figure 4.15, which shows that the larger elements do not behave as expected, demonstrates this difficulty. The absolute nominal failure stress for that figure is about 340 MPa, well below the threshold for using the larger element sizes 1.016 mm and 1.693 mm. A significant portion of these discrepancies can be explained by considering the mesh width effects near the notch tip. All the discussion and examination up to this point has 110 4.3. SDM and the Crack Band Method only considered mesh height effects. The mesh width effects are addressed in considerable detail in §4.4.5. 111 Chapter 4: An Investigation into Element Size and Localization Effects 4.4. The Mesh Effect and CODAM Having demonstrated the mesh effect with the SDM, and having shown that incorporating the principles of the crack band method significantly reduced the mesh sensitivity of the simulation, similar investigations were carried out involving the CODAM constitutive model. First, the effect of the mesh size on the 25.4 mm wide single-notch specimen modelled with CODAM without any modifications (i.e., without the crack band method) was examined. 1.6 Average Strain (Al/l) Figure 4.19. Nominal stress - average strain results for the 25.4 mm wide single-notch tension simulations using the CODAM model. The nominal stresses have been normalized with respect to the ultimate nominal stress predicted by the analysis using 0.508 mm elements. Figure 4.19 shows that CODAM, much like the SDM, shows significant mesh size sensitivity. In these simulations, the CODAM parameters were chosen such that the 112 4.4. The Mesh Effect and CODAM simulation with a mesh consisting of 0.508 mm high elements corresponded closely with the experimental test results. Once the simulation matched the experiment, the element widths (and heights) were changed without changing the model inputs. The trends for the nominal failure stress and the average strain at failure are shown in Figure 4.20. 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 Relative Element Size Relative Element Size Figure 4.20. Trends in the nominal failure stress and average strain at failure for the 25 mm single notched tension simulations with the CODAM constitutive model and varying mesh densities. The results have been normalized to the results of the analysis containing 0.508 mm elements. 4.4.1. Implementing the Crack Band Method into CODAM Given the success that the SDM exhibited in reducing the sensitivity of the simulations to mesh size effects by incorporating the crack band principles, a similar undertaking was pursued with the CODAM model. Due to the manner that stresses and strains are related in the CODAM model, scaling the post-peak behaviour is not a trivial process. 113 4.4. The Mesh Effect and CODAM 1.2 Average Strain (Al/I) Figure 4.21. Expected stress—strain curve for CODAM with an implementation of the crack band method. Figure 4.21 shows how one expects the stress—strain curve to appear when the CODAM model is used with the crack band method implemented. The unshaded area Yu is not considered in the calculation of GF. The darkly shaded area yF° shows the fracture energy density for an element of the characteristic size, hc, for the material. The lightly shaded area yF (which includes the darkly shaded area) is scaled in the strain dimension such that it obeys the rule of keeping G°F constant. To achieve this scaling, the following approach has been developed. For simplicity, consider a one-dimensional case, where an element smaller than the characteristic size (he < hc) is being analyzed. In the analysis code used throughout this thesis, the strain in the element is the known quantity (rather than the stress) therefore the strain, E, lies on the scaled curve (the "element" curve), E is related to a strain on the "master" (characteristic element) curve by a single scaling factor used for the whole curve, k. See Figure 4.22 for an illustration. 114 4.4. The Mesh Effect and CODAM 1.2 0 0.05 0.1 0.15 0.2 0.25 Strain (Al/I) Figure 4.22. Illustration of the process used to determine stress from an arbitrary post-peak-stress strain in an element using the crack band-based approach. The scaled strain (i.e., that on the master curve), £-s, is found from Equation 4.7: (4.7) where s is the current (element) strain, ss is the scaled strain (which falls on the master stress-strain curve), Epeak is the strain at maximum stress, and k is the crack band scaling factor. Once this scaled strain is determined, the CODAM parameters and the elastic properties can be used to determine the corresponding stress on the master curve, as shown in Equation 4.8: o- = E(es)E0es (4.8) This stress is then used as the equivalent stress generated by the actual element at a strain level of s. Once the relationship between the strain in the element and its 115 4.4. The Mesh Effect and CODAM corresponding stress is established, the relationship between the strain and the normalized modulus can be examined as well. The normalized modulus for the element of characteristic height and for an element with a height less than the characteristic is shown in Figure 4.23. 0 0.05 0.1 0.15 0.2 Effective Strain (mm/mm) Figure 4.23. Relationship between the effective strain and normalized modulus for CODAM with the crack band method integrated. Implementing the crack band scaling approach into CODAM raises an issue that was not a factor with the SDM. The problem is with the CODAM effective strain interaction terms. The effective strain equation includes the potential for all the other strain components to influence the effective strain of a particular direction. Equation 4.9 shows the general two-dimensional form of this equation. 2 2 + + (4.9) 116 4.4. The Mesh Effect and CODAM The crack band method focuses entirely on the stress—strain relationship, whereas the CODAM approach functions on an effective strain basis. As mentioned previously, the interaction constants can be chosen such that the effective strain reduces to the individual directional strains (as in the one-dimensional case), but in cases where interaction is required, the behaviour of the model as it pertains to the crack band method has not been investigated. For the sake of this study, the simplified approach is taken. The interactive terms are eliminated with the careful selection of the interaction constants, (e.g., in the 1-direction, K = 1, L = 0 ° , S = °°) and as in the SDM simulations, each direction is considered independently, such that each direction has a GCF that is maintained constant. 4.4.2. Determining the Crack Band Scaling Factor The following section demonstrates the procedure used to determine the scaling factor used in the crack band method (k). Due to the piece-wise linear nature of the CODAM approach, the equations governing the stresses, strains, and the fracture energy density YF are not simple. In practice, the scaling factor is numerically determined in a pre-processing step by a non-linear solver. However, in the case presented below a simplified form of the CODAM relationships is assumed in order to make the algebra tractable. The assumptions are: • No effective strain interactions (K1=L2='\, L1=M1=K2=M2=S=T=U=°°) • Matrix and fibre damage initiate immediately (Fm,=F,5=0) • Matrix and fibre damage saturate at the same strain (Fms=Ffe= sf) These assumptions result in CODAM parameters that are illustrated by Figure 4.24. 117 4.4. The Mesh Effect and CODAM 1.00 -•-Matrix •*• Fibre System 0.050 0.100 0 0.2 0.4 0.6 0.8 1 1.2 1.00 0.75 H lu j 0.50 0.25 0.00 0.000 0.050 0.100 0.02 0.04 0.06 0.08 Average Strain (Al/I) Figure 4.24. Plots of the simplified CODAM parameters used for demonstrating the calculation of the scaling factor, k. For illustrative purposes, fflms=0.5, Erm =0.5. ms These assumptions result in a linear degradation of stiffness with strain, as shown in Equation 4.10: (4.10) Stress is then expressed as: cr = E • E • e = EnE •0 c.2 (4.11) The strain at maximum stress can be found: 'peak (4.12) and the resulting maximum stress is: 4 (4.13) 118 4.4. The Mesh Effect and CODAM The crack band scaling approach requires the post-peak area underneath the stress-strain curve (yF) multiplied by the element height to be constant. The post-peak area for the characteristic sized element {YF°) is given by: 7 f 2 2 4 + t ° •r f s^ S V e i ) 48 0 ' (4.14) For elements with heights less than the characteristic height, scaling is employed. The scaling is applied to the effective strain, as given by: e 2_ for E >^-s 2 k 2 (4.15) The maximum scaled strain (sf, see Figure 4.22 ) can then be expressed as: £. = — + ' 2 k (4.16) *;=^ , (*+i ) (4.17) The pre-peak stress is given by Equation 4.11. The post-peak stress, however is: - E E2 cr = E0 E(ss)ss =E0SS (4.18) The resulting post-peak area under the stress strain curve for the element {yFe) is given by: e, t / F 16 2\ Es V <sf ds (4.19) (3 +4k) 48 E0sf 119 4.4. The Mesh Effect and CODAM Recalling the assumptions, the above expression for yF is valid only for a CODAM parameter set that has a single damage phase with linear damage growth and linear normalized modulus. The crack band method requires: rFhc=reFhe (4.20) Now, we will assume that the characteristic element is divided into n elements of equal heights. The formulation allows for elements of arbitrary heights, but in order to compare this method to that of the equivalent work scaling method, it is convenient to make this assumption. hc=nhe (4.21) Then solving Equation 4.20 for k, feo*?-nh.=&^0s?.h. (4.22) k = 7-^ (4.23) 4 Inserting the expression for k into Equations 4.15 and 4.18 results in the following non-linear expression for stress: ^ _ E0 {7nsf +ef- Bspne, - 7ef + 8s) ( 4 2 4 ) Aef (7n-3)2 As an example, for the case where n=2, k=2.75, and the resulting equation for stress reduces to: °=TIM105*'2+64£»S' -64 <^) ( 4 2 5 ) 484^ 120 4.4. The Mesh Effect and CODAM with sbi as defined in Equation 4.6. The characteristic and scaled stress-strain relationships for this example are shown in Figure 4.25. 10 V) CD k_ to •o <D N ra E *_ o z 1.2 1 0.8 0.6 0.4 0.2 0 _ -v / — e l e m e n t height / \ X (he=hJ2) /character is t ic—\\ - / height, h c \ 0 0.05 0.1 0.15 Average Strain (Al/I) 0.2 0.25 Figure 4.25. Stress-strain relationship for the reference element with a characteristic height, and the smaller element, scaled according to the crack band method with fc=2.75. 4.4.3. CODAM and the Crack Band Method The 25 mm wide single-notch tension simulations with the wide range of element sizes were examined once more, this time using the crack band modified CODAM model. The 0.508 mm element height was chosen as the reference characteristic element. Table 4.5 shows the parameters used in the simulations. Table 4.5. Scale factor k required to keep GF constant for CODAM simulations of the 25 mm wide single notch tension test simulations. Element height, he (mm) 0.1016 8.000 0.2032 3.625 0.5080 1.000 1.016 0.125 1.693 0.001 The results of these simulations are shown in Figure 4.26. The results show more mesh size sensitivity than the equivalent SDM simulations. Once again, the large element simulations do not show much change from their previous (without crack band) results. 121 4.4. The Mesh Effect and CODAM The smaller element simulations show a reduction in the difference from peak nominal stress and average strain at failure, but are not as mesh insensitive as the equivalent SDM results. Figure 4.27 shows the trends in the nominal failure stress and the average strain at failure for this series of simulations. 1.6 Average Strain (Al/I) Figure 4.26. Results of the 25 mm wide single notch tension specimen simulations with CODAM and with the crack band modifications. The stresses have been normalized to the results of the analysis performed with 0.508 mm elements. 122 4.4. The Mesh Effect and CODAM 16 . 1.6 z Q 1 0 } I 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 Relative Element Size Relative Element Size Figure 4.27. Trends in the nominal failure stress and the average strain at failure for the 25 mm wide fracture-energy controlled CODAM single-notch tension simulations. The results have been normalized to those from the analysis with 0.508 mm elements. In the same manner that the SDM was investigated for the effect of suit on the results of the simulations, the CODAM simulations were examined for the effect of the fracture energy scale factor, k, on the predicted nominal failure stress. The model inputs were kept constant in all the simulations with the exception of the scale factor, k. Each mesh size was simulated over a wide range of k values, and the nominal failure stress was recorded for each simulation. The results of this series of simulations are shown in Figure 4.28. 123 4.4. The Mesh Effect and CODAM 1200 0 ~l 1 1 — — i 1 — 0 5 10 15 k Figure 4.28. Nominal failure stress as a function of the crack-band scaling factor, k, for each element size over a wide range of k for the CODAM simulations of the 25 mm width single-notch tension specimen. Figure 4.28 has a number of interesting features. The behaviour of each of the mesh sizes is similar to their behaviour under the SDM. Each mesh size shows a plateau in the nominal failure stress for low values of k, and a gentle increase as k increases. Once again, it is demonstrated that each mesh size has a threshold minimum nominal failure strength. §4.4.5.1. addresses these threshold minimum nominal failure strengths in more detail. The simulations that had been calibrated in the fracture energy to the 0.508 mm element size simulation are shown on the figure as well. Had the results been as expected and complete mesh insensitivity observed, the results should have followed the open box series (horizontal line) in the figure. Given the behaviour of the mesh sizes, however, it is immediately clear that the meshes with the larger element sizes could never achieve the nominal failure stress near 500 MPa. The actual results of the simulations are plotted with the open circle series. 124 4.4. The Mesh Effect and CODAM Some reasons for the still-observed mesh effects are explored in §4.4.4. and §4.4.5. 4.4.4. CODAM and the Crack Band Method: Refinements One of the difficulties in incorporating the crack band method into CODAM that has not been addressed is the issue of shear. The crack band formulation affects the direct (normal) strains eh e2l and s3, but not the shear strains. It is not immediately clear how the shear strains contribute to the crack band method, but given that it is currently limited to Mode I crack openings, there is no theoretical basis for including shear strains. The CODAM model degrades shear stiffnesses on the basis of damage in the 1, 2, and 3 directions. Since the degradation is strain softening, then localization will occur. Since the crack band method does not take the shear strains into account, it will not limit this localization. Further, as there is no direct link in CODAM between shear strains and the normalized shear moduli, the same effective strain scaled approach cannot be taken. Once again, a simplifying assumption has been made. The simulations of the 25 mm wide single-notch tension test were performed once more, this time with the effect of shear minimized. This means that the shear strain contributions to the effective strain calculations were removed through the appropriate choice of the interaction constants, and the shear stiffnesses were not degraded at all. Figure 4.29 shows the results of the simulations with the shear effects removed. To provide a helpful comparison of the results, the trends for the nominal failure strain and the average strain at failure have been normalized to the 0.508 mm element width simulation. The results of the simulations without the crack band (CB) modifications, using the typical CODAM with crack band modifications, and using CODAM with the shear influences removed and the crack band modifications are shown in Figure 4.30. 125 4.4. The Mesh Effect and CODAM 1.4 1.2 co 1 "TO I 0.8 o % 0.6 CD N | 0.4 0.2 0.005 0.01 Average Strain (Al/I) 1.016 mm 0.508 mm 0.2032 mm 0.1016 mm 0.015 Figure 4.29. Results of the 25 mm wide single notch tension specimen simulations with CODAM, the crack band modifications, and the removal of the shear strain effects. The stresses have been normalized to the results of the analysis with 0.508 mm elements. 0 1 I 0 1 i I 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 Relative Element Size Relative Element Size Figure 4.30. Trends of the nominal failure stress and average strain at failure for the CODAM simulations. Each run has been normalized to the 0.508 mm element width in order to provide comparisons to the effectiveness of the approach. 126 4.4. The Mesh Effect and CODAM Figure 4.30 shows that the mesh sensitivity of the nominal failure stress and average strain at failure is significantly reduced when CODAM is used with the crack band modifications and the shear influence is completely removed, for elements equal to or smaller than 0.508 mm. The larger elements have been shown to be unable to predict the correct failure stress or strain, and are therefore inadmissible. A common reason for inadmissibility of large elements is a simple finite element analysis issue. In finite element analyses, one must use elements of sufficient size to allow convergence to the elastic solution. In this converged state, the stresses predicted by the simulation agree with the analytical solution to some satisfactory level of accuracy. This presents an upper limit to the size of elements that can be used in any analysis. With this in mind, the stresses near the notch of the 25 mm wide single-notch tension specimen were examined in the meshes with 0.508 mm, 1.016 mm, and 1.693 mm wide elements, using a linear-elastic constitutive model. In addition, the analytical linear elastic fracture mechanics prediction for this geometry was determined. The results are shown in Figure 4.31. 4 6 8 10 12 Location (mm) Figure 4.31. Plot of the local stresses in the direction of loading, oy, near the notch tip for the 25 mm wide single-notch tension specimen simulations. A linear-elastic constitutive model without damage was used in the analyses. The three largest element sizes are shown, along with the linear elastic fracture mechanics solution. Examining Figure 4.31, it is clear that the two coarse mesh densities, with elements greater than 1 mm, do not capture the (undamaged) stress distribution near the notch tip 127 4.4. The Mesh Effect and CODAM very well. The far-field stresses converge for the three meshes, but the behaviour of the specimen at failure will be dominated by the way the elements near the tip react to the loads placed upon them. Based on the scaling results and by examining the stress distributions, the 1.016 mm and 1.693 mm wide elements are inappropriate for this analysis. The effect of element width is discussed in the next section. 4.4.5. Mesh Width Effects and Stress-Based Scaling McClennan, working in the same research group as the author, has been pursuing parallel research that has included element width effects in double-cantilevered beam (DCB) simulations performed using CODAM (McClennan, 2003). A schematic of the test is shown in Figure 4.32 and the geometric parameters are given in Table 4.6. P i 5 V J>. -A. A . 1 X I Figure 4.32. Schematic of the double cantilever beam test. The out-of-plane width of the specimen, B, is not shown. Table 4.6. DCB geometric parameters used in the study by McClennan (2003). L 80 mm h 1.5 mm ao 20 mm B 6 mm McClennan has observed that the apparent strain energy release rate (G,c) calculated from the critical load, PcrA , determined in the analyses (according to Equation 4.26, below) does not correspond to the G, c expected from the material inputs. 128 4.4. The Mesh Effect and CODAM 3 P X G, c= r \ (4-26) where P c r i f is the peak force, S'\s the crack opening displacement, B is the DCB width, a is the crack length, and y is the area under the stress-strain curve for the material (the total available strain energy density). Following up on this issue, McClennan has noticed that as the width of the elements used in the simulations decreases, the energy dissipation approaches the expected value. Figure 4.33 shows the trend of the critical strain energy release rate (G/c) as a function of element width for the DCB analyses (McClennan, 2003). Figure 4.33. Critical strain energy release rate (GIC) for DCB simulations performed using elements of varying width. Data from analyses performed by McClennan (McClennan, 2003). Further investigation by McClennan shows that this effect is related to the discretization of the stress field near the notch in the DCB specimen. 129 4.4. The Mesh Effect and CODAM 70 "ro" Q. S 60 | 50 o> a> c 40 N •B 30 tf) (A d) * 20 CD O ) 5 10 CD > < 0 0 0.0625 mm element width 0.125 mm element width 0.250 mm element width S \ N 0.500 mm element width N. •C ^ ^ 0.1 0.2 0.3 0.4 Distance from notch (mm) 0.5 0.6 Figure 4.34. Average stresses along the notch-plane in DCB simulations using elements of varying width. The size of the first element is superimposed to provide context to the average stress results. Note that under-integrated (i.e., single integration point) elements were used. The data is from analyses by McClennan (McClennan, 2003). Elements of a large width do not capture the stress field close to the notch tip very well, particularly using under-integrated, single integration point elements such as those that McClennan employed. Consequently, the average stresses in those (larger) elements are much lower than the "actual" stresses near the notch. This has implications in failure analysis. Failure for the 0.0625 mm element width analysis occurs at some average stress, say 62 MPa for the sake of illustration (see Figure 4.34), which corresponds to some global applied load. If it were desirable for a larger element (again, for the sake of illustration, the 0.500 mm wide element) to fail at the same global load, the failure stress in that element would be required to be reduced to ~35 MPa. This is illustrated in Figure 4.35. 130 4.4. The Mesh Effect and CODAM 0.0625 mm element width At a certain global load, an average stress of -35 MPa in 0.5 mm wide elements corresponds to an average stress of -62 MPa in 0.0625 mm elements. If failure occurs at 62 MPa with the smaller elements, analyses with the larger elements would require the failure stress to be scaled to 35 MPa. 0.500 mm element width 0 0.1 0.2 0.3 0.4 0.5 0.6 Distance from notch (mm) Figure 4.35. illustration of the need to reduce failure stress for larger elements. The effect of discretization (element width) on the stress field can be calculated with relative ease. If the exact stress field for the application being analyzed is known a priori, then the average stress in elements of a given width, w, can be calculated according to Equation 4.27: A r+w crav=- \odr (4.27) w f where r is the location of the edge of the element closest to the notch. The difficulty is in determining the exact stress field. An alternative approach is to perform an elastic finite element analysis with very small elements and use the resulting stress field as a good approximation to the exact stress field. For the DCB analyses, this calculation has been performed for the average stress in the first element (ahead of the crack tip) using two stress field descriptions: 1) an LEFM solution for a notch in an infinite medium (McClennan, 2003) and, 2) the stress field resulting from the finite element analysis 131 4.4. The Mesh Effect and CODAM using 0.0625 mm elements. In both cases, the results have been normalized to the average stress in the 0.0625 mm wide element analysis. The results of these calculations are shown in Figure 4.36, along with the actual average stresses from the simulations. Figure 4.36. Average stresses in the first element ahead of the notch in the DCB analyses. Shown are expected results based on the stress field descriptions from an LEFM analysis of a sharp notch and the stress field predicted by the FE analysis using 0.0625 mm wide elements (McClennan, 2003). The actual results from the FE analyses are also shown. Figure 4.36 suggests that accounting for the element widths in this manner is encouraging, but does not completely explain the element width effect. McClennan's studies are ongoing. As a first order approach to removing this dependency on the element width, McClennan has suggested that the peak stress in the simulations should be scaled with respect to the element width, while the fracture energy per unit area (G F ) is maintained. The 132 4.4. The Mesh Effect and CODAM amount of scaling depends on the characteristics of the stress field involved (as discussed above). The implementation of this stress-based scaling has yet to be developed. McClennan has suggested that scaling could be accomplished by applying a scale factor to the characteristic stress-strain curve to reduce the peak stress as element widths become larger, and then employing the crack-band approach to scale the curve in strain to ensure that the fracture energy (GF) is conserved. An illustration of this approach is shown in Figure 4.37. 0 0.005 0.01 0.015 0.02 0.025 0.03 Strain (mm/mm) Figure 4.37. Illustration on how stress-scaling could be implemented. For simple stress-strain curves (i.e., one damage zone, resulting in a parabolic stress-strain curve), McClennan has demonstrated that this form of scaling does reduce the mesh width dependency. Unfortunately, the approach still has some details to address before it can be implemented into a constitutive model. If a scale factor is applied to the whole curve (e.g., CT' = KO-), then the initial (elastic) apparent stiffness changes (although the input material stiffness E does not change), as illustrated in Figure 4.37. 1.2 T3 CO 133 4.4. The Mesh Effect and CODAM 4.4.5.1. Preliminary SDM and CODAM SNT Results Revisited Recall that in Figure 4.17 and Figure 4.19, there were lower thresholds to the nominal failure stresses predicted by simulations of SNT geometries with the SDM and CODAM. These thresholds varied, depending on the element size used in the simulations. Defining the stress concentration factor (SCF) as the ratio of the stress at a point to the far-field stress, the SCF for stresses parallel to the applied load, along the notch plane (as in the SNT simulations) can be found from LEFM (Tada et al., 1985): a 1 (a 1 ( 1 + . j= U y[7 1 + 2r (4.28) where a0 is the far-field stress, a is the notch length, p is the radius of the notch tip, and r is the distance from the notch measured from a point p/2 inside the notch. Integrating this expression along a length, w, corresponding to the width of an element, and dividing by the width, one can calculate the average SCF for an element: SCF. avg.element W SCF. (4.29) avg.element [j2p + 4w + 2J2a ^2p + 4w (4.30) In the case of the 25.4 mm wide SNT simulations, the notch was infinitely sharp (p = 0), and the notch length, a, was 5.08 mm. The average SCF for each element size can be calculated by Equation 4.30. Table 4.7 shows the computed SCFs along side the numerical results for the SDM simulations, and the same SCF values (since the SCF depends only on the geometry) are shown in Table 4.8 with the numerical results for the CODAM simulations. In order to discuss the mesh width effects without considering a particular load, the SCFs can be normalized and in this case, the SCFs are divided by the average SCF for the 1.693 mm wide elements. 134 4.4. The Mesh Effect and CODAM As demonstrated in the previous section (and not surprisingly as it is the source of mesh sensitivity for most failure simulations), a particular global load results in higher stresses in the element closest to the notch for smaller element widths. Small elements have high stresses for a given global load and large elements have small stresses for the same global load. Given that the material models are formulated such that elements both large and small fail at the same stress, this means that smaller elements fail at lower global loads than large elements. This is exactly the trend observed in the SNT simulations. Normalizing the limiting stresses in the SDM (Figure 4.17) and CODAM simulations (Figure 4.28) with respect to the results of the 1.693 mm wide element simulations and then taking the inverse should be equivalent to the normalized average element stress concentration factors. That is, if the normalized SCFs correlate with the inverse of the normalized limiting stresses, then the limiting stresses are dictated only by the element width effects described in the previous section. The results of these calculations are shown in Table 4.7 and Table 4.8. Table 4.7. Average stress concentration factors for the SDM SNT simulations. Element width (mm) Stress concentration factor Normalized SCF Limiting stress (MPa) Normalized limiting stress Inverse normalized stress 1.693 3.45 1.00 584.7 1.00 1.00 1.016 4.16 1.21 460.4 0.79 1.27 0.508 5.47 1.59 338.0 0.58 1.73 0.2032 8.07 2.34 233.4 0.40 2.51 0.1016 11.0 3.19 147.3 0.25 3.97 Table 4.8. Average stress concentration factors for the CODAM SNT simulations. Element width (mm) Stress concentration factor Normalized SCF Limiting stress (MPa) Normalized limiting stress Inverse normalized stress 1.693 3.45 1.00 720.0 1.00 1.00 1.016 4.16 1.21 619.0 0.86 1.16 0.508 5.47 1.59 468.0 0.65 1.54 0.2032 8.07 2.34 335.6 0.47 2.15 0.1016 11.0 3.19 226.5 0.31 3.18 135 4.4. The Mesh Effect and CODAM The tables show that the limiting stresses observed in Figure 4.17 and Figure 4.28 are mostly dictated by the width of the element in the stress field. 4.4.6. Implementing the Equivalent Work Method in CODAM Unlike the crack band method where a scaling constant is predetermined for a given element size, the equivalent work scaling (EWS) method does not employ a scale factor. Instead, the strains are calculated directly from the compatibility equation (Equation 4.6). The piece-wise nature of CODAM makes these calculations difficult, and rather than an analytical approach, a numerical one needs to be taken. In a numerical computation, the material model (constitutive model) is provided with the strain in an element (or integration point) as its input. This strain is the strain in the potentially bifurcated element (sbi). The strain in a fictitious characteristic element must be determined so that the stress in the bifurcated element can be determined. Unfortunately, the strain in the n-1 elements is also unknown; therefore the compatibility equation has two unknowns, namely ec and en-i- Recall Equation 4.6: ebl n-1 n n From equilibrium, the stress in each element must be the same. The stress can be found from strains by the CODAM relationships, and the CODAM relationships can be solved to express strains in terms of stresses. Due to the nature of CODAM, however, the stress equations are a piece-wise combination of parabolas and the resulting solution of these equations in terms of strain is a number of conditional expressions. Algebraically, this quickly becomes a problem that is extremely difficult to manage. As CODAM potentially can have any number of linear pieces, the solution requires a large number of conditional statements to be evaluated, a number that grows exponentially with the number of phases. Prior to the peak stress, determining the strain components in the compatibility equation is trivial. No bifurcation has occurred, and all the strains are equal, i.e., etrEn-i and from Equation 4.6 it follows that £fc=Q»-136 4.4. The Mesh Effect and CODAM Beyond the peak stress, the solution of Equation 4.6 becomes much more involved. Generally, the task is to determine the stress in an element that is smaller than the characteristic height, given the strain in that element. The given strain is therefore the strain in the bifurcated element, sbi. The stress in the bifurcated element is the same as the stress in the characteristic element, and the stress can be computed once the strain in the characteristic element is known. In order to determine the strain in the characteristic element, the strain in the remaining n-1 elements needs to be determined. For any given stress, the strain in the non-bifurcated elements is found from solving Equation 4.11. Recalling Equation 4.11, for the special case of CODAM presented in Figure 4.24: a = E0s-^s2 The strain in the non-bifurcated elements becomes: F V sf (4.31) 2E°. The stress in all the elements is the same as the stress in the fictitious characteristic element; thus: * = E0ec-^s2c (4.32) Combining Equations 4.6, 4.31, and 4.32 and solving for <%: gfr+ gfM) (4.33) £ c 2n-1 Replacing the solution for Sc into Equation 4.32, stress equation for post-peak stresses becomes: °= l. % -1>' + "*») ( 4 3 4 ) (2n-1) s, 137 4.4. The Mesh Effect and CODAM Figure 4.38 shows the resulting stress as a function of strain in the bifurcated element. 0.25 Average Strain (51/1) Figure 4.38. Stress-strain curve resulting from the simplified CODAM parameters assumed in the demonstration calculations for equivalent work scaling. For the more complicated general problem, a numerical approach is used to avoid the problems associated with solving the complex system of equations. In a pre-processing stage, strain in the characteristic element can be assumed. The stress in the characteristic element, which is the same as the stresses in the replacement elements, can be easily evaluated through the CODAM relationships. Once the stress is evaluated, the pre-peak strain at that stress level can be evaluated, and this strain level is the strain that the n-1 elements will have. The strain in the bifurcated element can be evaluated through compatibility equation. There is a unique one-to-one relationship between the strain in the characteristic element and the bifurcated element. Having performed this pre-processing step, the strain in the characteristic element can be matched to any given bifurcated element strain. This approach is illustrated in Figure 4.39. 138 4.4. The Mesh Effect and CODAM 0.25 3. Wi th en-i compute 8 b i Average Strain (Al/I) Figure 4.39. Illustration of evaluating ebl based on a known Sc. This is precisely how the EWS method is implemented into CODAM. Given n, the number of elements that replace the characteristic element, the code computes the relationship between sc and sbi in a pre-processing step. Later, arbitrary sbi are matched to their equivalent sc, and the resulting stress is evaluated. 4.4.7. Crack Band Scaling versus Equivalent Work Scaling Given the two approaches to capturing the energy effects in the softening CDM model, it is interesting to compare the two methods. The crack band method matches the energy dissipated by a localized element that has become completely damaged. Only elements that have become completely damaged satisfy this energy balance. On the other hand, the equivalent work scaling method matches the energy dissipated in the localized element to the energy dissipated by a fictitious equivalent stack of elements exactly, at every strain state. An example of how the energy dissipated in these methods compare is shown in Figure 4.40 and Figure 4.41 shows the corresponding stress-strain response using the two methods. 139 4.4. The Mesh Effect and CODAM 180 0 0.2 0.4 0.6 0.8 1 1.2 Damage (co) Figure 4.40. Comparison of the energy dissipated per unit area (GF) for an example material with a characteristic size of 10 mm, simulated with crack band scaling and equivalent work scaling for elements of 1.25 mm in height. 1200 E (m/m) Figure 4.41. Comparison of the stress-strain responses for an example material with a characteristic size of 10 mm, analyzed with crack band scaling and equivalent work scaling of elements of 1.25 mm in size. 140 4.4. The Mesh Effect and CODAM While the result of two approaches differ somewhat, the crack band method is clearly more flexible and numerically more efficient. While the equivalent work scaling method requires elements that are an integer fraction of the characteristic size, the crack band method allows elements of arbitrary size. The crack band method determines the resulting equivalent strain through the use of a fixed scale factor where the equivalent work method requires a lookup table to perform the same task. For the applications examined later in this thesis, typically damage growth continues until complete damage saturation. The differences between the two approaches are minimal when the damage is saturated so for this reason, and the previously noted reasons of flexibility and numerical efficiency, the crack band method will be used in subsequent analyses. 141 Chapter 4: An Investigation into Element Size and Localization Effects 4.5. Summary Simulating the size effect with the CODAM model first requires addressing the issue of mesh sensitivity. It has been demonstrated that CODAM, as with any continuum damage mechanics based constitutive model, tends to localize damage in crack growth simulations. This localization changes the behaviour of the simulation depending on the size of elements involved in the analysis. To avoid the spurious mesh dependency, the element sizes can be kept constant in the analysis of specimens of different size, or some method of nonlocalization must be adopted. One method of nonlocalization that has been shown to be effective is the crack band method. The crack band method has been shown to reduce the effect that element size has on the results of the simulation when the CODAM constitutive model is involved, but the resulting analyses are not completely mesh size independent. It has also been shown that for analyses involving crack growth, care must be taken in the selection of element size in the crack zone. The elements must be able to reproduce the stress gradient around the notch tip before the crack grows in order to satisfactorily simulate the whole crack growth event. Elements that do not capture the (undamaged) stress distribution around the notch tip will not properly observe the mesh independence expected with nonlocal approaches like the crack band method. Use of the McClennan technique, which considers the effect of element widths in the presence of high gradient stress fields and modifies the stress-strain curve of the constitutive model to reduce the mesh width dependency, shows promise but, as yet, it is not fully developed. In general, consideration of the average stresses in elements near notches explains much of the mesh size effect that the crack band model cannot address. 142 Chapter 5: A Review of Structural Size Effects in Composites This chapter reviews the effects of changing structure size on structural performance (indicators such as nominal strength, nominal strain-to-failure, etc.). In the sections that follow, a background on statistical structural size effects is presented, followed by a review of experimental observations on structural size effects on composites. This chapter is completely a review of existing literature on size effects. In later chapters, the behaviour of the CODAM model will be examined in applications where the material system remains constant but the size of the specimen being simulated is scaled. In this way, CODAM's ability to predict size effects is evaluated for each of the applications considered. 143 Chapter 5: A Review of Structural Size Effects in Composites 5.1. Introduction One of the challenges in composites engineering research and development is that of the scaling of structural response. For reasons of convenience and expense, material and structural testing is preferably performed at a small scale. Typically, coupons and scaled models of structures are tested in the laboratory. Numerical simulations of experiments are based on the available data, which are the small-scale tests. In the field, however, actual structures tend to be much larger than those tested in the laboratory. Figure 5.1. Laboratory test of damage development in stitched resin film infused carbon fibre epoxy laminate at the centimetre scale (left) and a wingtip spar (right) at the metre scale. In the aerospace industry, for example, coupons on the order of 10 cm are tested in the laboratory to characterize the behaviour of structures that will be on the order of 10 m (see Figure 5.1). In progressing from testing on a small scale to design and use at a much larger scale, it has been observed that materials and structures do not perform to the same level at the larger scales. In general, larger structures exhibit lower strengths and strains-to-failure than smaller structures. This apparent change in behaviour is called the "size effect". The term "size effect" is very general, referring to any case where only a change in scale results in a change in behaviour, for any reason. As such, changes in scale in one, two, or three dimensions resulting in a change in strength, strain-to-failure, or some other defining characteristic is called a size effect. 144 Chapter 5: A Review of Structural Size Effects in Composites 5.2. Background A size effect in materials and structures has been observed for centuries. Leonardo da Vinci investigated the effect of the length of a wire supporting a bucket of sand on the strength of the wire, sometime around 1500 CE. In this experiment, illustrated in Figure 5.2, da Vinci discovered that a longer wire of the same gauge was unable to support as much sand as a shorter wire. This apparent change in strength is a size effect. Figure 5.2. An illustration of da Vinci's size effect experiment (Lund and Byrne, 2000). Size effects are typically observed in materials that are brittle in nature. These materials tend to be sensitive to flaws within the material, and increasing the volume of material under load increases the chance of a critical flaw being present. This line of reasoning is often referred to as the "weakest link" theory. The theory, detailed and refined by Weibull (Weibull, 1951), requires a bit of a thought experiment. A length of chains is imagined, with links of identical shapes and strengths. The chain is put into tension, and loaded until the chain breaks. If all the links had identical strengths, then all the links would break at once. This does not happen, though, instead the weakest link in the chain causes the whole chain to fail. With an increasing number of links in the chain, the chance of finding one with a significantly reduced strength is increased. Weibull's analytical theory based on this "weakest link" idea has been successfully applied to various brittle materials, notably brittle metals and ceramics. Whether a size effect exists in polymer matrix composite materials (composites) has been the subject of much investigation. A number of researchers have been performing experiments in order to quantify size effects in composite materials since the 1960s. In Ua8^ 3B>L 145 5.2. Background many cases where a size effect was claimed, others have responded by critically evaluating the test methods, pointing out systemic problems that contributed to a conclusion of a size effect in composite materials. 146 Chapter 5: A Review of Structural Size Effects in Composites 5.3. Weibull's Statistical Distribution Since Weibull's weakest link theory (Weibull, 1951) predates the investigation of size effects in composite materials, many researchers in composites use Weibull's theory as a test to determine if an analytically justifiable size effect is present. Weibull's theory is presented here, while other theories to describe size effects will be presented later in the chapter. Weibull's theory is based on simple statistics and some common-sense logic. Consider several poorly made links joined together in a chain. The links, on average, have a certain strength but each individual link may have a strength above or below the average. If a small number of links are fashioned together in a chain, as in Figure 5.3, the strength of the chain is dictated by the weakest link in the chain. As more links are added to the chain, the probability of getting an ever weaker link increases, as in Figure 5.4. In fact, the longer the chain is, the lower the strength of the chain is likely to be. 0 120% 01 104% 1 1 0 / 0 Q R 0 / •Hi! Figure 5.3. Illustration of the strength of individual links in a short chain. The strength of the chain in this case is 92% the average strength of the links. Figure 5.4. Illustration of the strength of individual links in a long chain. The strength of the chain in this case is 85% the average strength of the links. 147 5.3. Weibull's Statistical Distribution The probability of survival for a given number of 'weak links' is defined according to a distribution function, shown in Equation 5.1: where F(x) is the distribution function for some parameter x, x0 is the average "failure" value of the parameter, xu is some value at which the function vanishes, and m is the Weibull shape factor, that changes the shape of the distribution. Weibull demonstrated that although the chosen distribution function lacks a theoretical basis, it can be used with considerable accuracy to describe things as wide ranging as the yield strength of steel to the fibre strength of cotton and the breadth of beans of Phaseolus Vulgaris. Weibull does not explicitly present his theory in terms of predicting strength of materials based on the stressed volume and the probability of that volume including a fatal flaw, but this extension is relatively trivial. For example, see (Bullock, 1974; Wisnom, 1999). As expressed by Wisnom, the probability of survival, P(s), for a volume, V, subjected to i a stress cr, is: where o0 is the reference strength and m is the Weibull modulus, a parameter that governs the shape of the distribution function. The effect of the shape parameter is demonstrated in Figure 5.5. This figure shows the probability of failure predicted by the Weibull model, with an average failure stress of 150 MPa. The change in the Weibull modulus or shape parameter has a significant effect on the distribution. F(x) = 1 - e (5.1) (5.2) 148 5.3. Weibull's Statistical Distribution 1.2 0 50 100 150 200 250 300 350 400 Stress (MPa) Figure 5.5. Illustration of the effect of the Weibull modulus on the distribution of failure The change in failure strength due to a change in stressed volume can then be expressed by: From Equation 5.3 it can be seen that according to the Weibull theory, the relative strength of materials of different sizes, yet geometrically scaled, is affected by the relative volumes and the distribution shape factor, m. Similar equations can be developed based on the assumptions that only the length or the cross-sectional area results in a size effect. strengths. m (5.3) 149 Chapter 5: A Review of Structural Size Effects in Composites 5.4. Evidence of the Size Effect in Composites and Modelling Efforts One of the earliest investigations into size effects in composites was performed by Bullock at the General Dynamics Corporation in 1974 (Bullock, 1974). Bullock examined two composite systems, T300/5208 carbon epoxy and Modmor II/5208 carbon epoxy, and compared the strength of tensile coupons to the strengths determined from flexural tests. Bullock also examined the strength of epoxy-impregnated strands. Comparing the results to those predicted by applying Weibull's theory, Bullock concluded that a size effect as described by Weibull does exist. Wisnom (Wisnom, 1999) notes in his review of experimental tests on composite materials for size effect that the test methods described by Bullock may have lent themselves to a systemic size effect. This systemic problem is the stress concentrations created by the grips in tension tests. The stress concentrations have a greater effect on thicker laminates rather than individual strands, and arguments have been made that this contributed to the results that Bullock observed. In 1978, Hitchon and Phillips (Hitchon and Phillips, 1978) noted that Argon (Argon, 1974) argued on a micro-mechanical basis that the tensile strength of a unidirectional composite in the direction of reinforcement should be independent of length, yet dependent on cross-sectional area. Bullock had, of course, concluded that Weibull's theory applied to laminated composites, and by extension laminate strengths should be affected by length changes. Hitchon and Phillips performed a series of experiments using carbon epoxy composites. These experiments included waisted tensile tests, burst tests, and flexural tests. On the basis of these experiments, they concluded in one part of their study, the use of the Weibull model described the tension and burst tests, but did not correctly predict the trends for the flexural tests. In the second part of their study, in which a different material was used in burst tests involving cylinders of different sizes, the differences in strengths were less than those expected by the Weibull model. They note that the effective difference between the tension tests and the burst tests was the effective length in the fibre direction for the specimens. The burst tests had hoop direction lengths roughly 200 times larger than the tension coupons. This suggests a Weibull-type length effect. The discrepancies between the burst tests in the second part of their study, however, were a cause of concern, demonstrating some inconsistencies in 150 5.4. Evidence of the Size Effect in Composites and Modelling Efforts the approach. Based on the specimen geometries, they concluded that the Weibull theory could be applied in the case of length changes, but cross-sectional area changes required further investigation. In 1980, Whitney and Knight (Whitney and Knight, 1980) examined tensile and flexural tests of unidirectional carbon-epoxy laminates. They found a significantly larger variation in the tensile strength versus the flexural strength than could be predicted using the Weibull approach. Two systems were used, T300/5208 and AS/3501-5A, and similar results were obtained for each. Note that the T300/5280 was the same system examined by Bullock. In discussing the results, a number of concerns were raised involving the testing procedure; the perfect alignment of the tension specimens in the test rig was questioned, as was the load-nose size in the flexural test. It was suggested that these systemic problems contributed to the variation in the strengths. More recently, experimental research in the size effect in composites has been spearheaded by two groups of researchers. The University of Bristol has a research group headed by Wisnom that has been very actively investigating size effect in the past 15 years. In 1991, Wisnom (Wisnom, 1991) released a study that examined length scaling effects in carbon epoxy (XAS/913) laminates. The laminates were unidirectional, and the cross-sectional area was kept constant through the tests. Based on the results of the tests, Wisnom concludes that there is a size effect, but it is more pronounced than that predicted by the Weibull theory. Further, it was shown that the variability of the results for tests at the same length was much smaller than that predicted by the Weibull theory for the given size effect. Accordingly, Wisnom presented an equation that fits the data by including a term that increases the influence of length. tr, W ' ^ 2 ""I (5.4) In 1992, Wisnom expanded the ideas from the previous paper (Wisnom, 1992). In this work, a modified version of Weibull's theory is presented. Wisnom's approach was called the fibre bundle theory. In this theory, the unidirectional laminate is treated as a bundle of fibres with resin holding the fibres together, preventing buckling under compressive loading but splitting under tension. The probability of survival for the bundle is then expressed in terms of strain: 151 5.4. Evidence of the Size Effect in Composites and Modelling Efforts S = e U o j (5.5) This slight modification results in a prediction of the ratio of bending strength to tensile strength that is more in line with the experimental data acquired by Wisnom for unidirectional carbon epoxy laminates than the unmodified Weibull prediction. Additionally, the fibre bundle approach does not require a critical defect to cause catastrophic failure, reflecting the gradual failure often observed in flexural tests of composite laminates. The fibre bundle model is limited, however, to unidirectional composites. Continuing with his study of bending failures in carbon-epoxy laminates, Wisnom examined bending-induced tensile, compressive, and shear failures (Wisnom, 1993). No attempt to fit the results to a size effect model was made, but the results demonstrated significant size effects in the strength of the laminate for each of the failure modes when the specimens were geometrically scaled in all directions (3D scaling). Wisnom concluded that failure strength cannot be regarded as a material property. In 1997, Wisnom and Atkinson (Wisnom and Atkinson, 1997) examined glass fibre / epoxy unidirectional laminates in tension and bending. In the tensile tests, the specimens were laid-up with internal ply-drops so that the plies in the gauge section of the specimen were continuous along the whole section, while the tab areas of the specimens were built up in order to prevent grip failure. The specimens were not waisted, preventing broken fibres on their surfaces. The tension tests were length scaled while the bend tests were 3D scaled. In both cases, the Weibull model was fit to the data, and the same Weibull modulus (m) provided good agreement. Additional work in 1997 by Wisnom et al (Wisnom et al., 1997) examined compressive strain to failure in bend tests involving unidirectional carbon epoxy laminates (T800/924). The specimens were 3D scaled, and the results showed significant size effects: decreasing strain-to-failure with increasing specimen size. Additional tests were performed where the length and width of the specimens were changed, but the thickness of the specimens was kept constant. These tests showed similar stains-to-failure, leading the researchers to conclude that the size effect was largely due to the reduction in the strain gradient with the increasing specimen thickness. 152 5.4. Evidence of the Size Effect in Composites and Modelling Efforts Finally, in 1999, Wisnom presented a comprehensive review of size effects in the testing of composite materials (Wisnom, 1999). The review is limited to unnotched strength of continuous fibre reinforced composites, and concentrates on carbon epoxy and glass-fibre epoxy materials. The review discusses possible factors that would lead to size related effects in composite materials, and then reviews data presented in the literature for tensile failures (tension and flexural tests), compressive failures (end-loaded compression, flexural, and cylindrical tests), and matrix-dominated failures. There is some discussion on the applicability of Weibull's theory to describing the size effect, but the literature is inconclusive on the generality of applying this theory to all size effects in composites. Wisnom concludes that size effects are real in composite materials, and not simply an artefact of testing procedures. The other research group spearheading the investigation into size effects in composites is a collaboration between a number of researchers linked through the NASA Langley Research Center. In 1992, Kellas and Morton (Kellas and Morton, 1992) examined scaling effects in carbon epoxy (AS4/3502) laminates. The laminates were 3D scaled, with ply-level scaling in the thickness direction, and four lay-ups were examined. The specimens were all tested in tension. The test results generally showed that larger specimens exhibited lower tensile strengths, and the failure mode and damage development were affected by specimen size. Fibre dominated lay-ups exhibited less size sensitivity than the matrix dominated lay-ups. Further, they observed that transverse ply cracks seemed responsible for both strength and failure-mode variations in different sized specimen. The data was fit with both the Weibull model and a fracture mechanics approach, but the accuracy of each model was not consistent across all the lay-ups. In associated research, Jackson (Jackson, 1992) examined the same material and lay-ups in flexural tests. Similar to the previous study, the failure stresses and strains decreased as the size of the specimen increased. Jackson used the Weibull model and a fracture mechanics approach to model the strength size effect, and showed that the Weibull model fit the data set better than the LEFM approach, but neither correctly predicted the trend. Both studies are also presented in (Jackson et al., 1992). Research continued at NASA Langley, and in 1998, Johnson et al. published a paper (Johnson et al., 1998) examining carbon epoxy laminates, AS4/3502 and APC-2 (AS4 153 5.4. Evidence of the Size Effect in Composites and Modelling Efforts and a PEEK matrix), that were 3D scaled with through thickness scaling at the sublaminate level. The tension tests showed that, in general, the strength and strain-to-failure of fibre-dominated lay-ups that did not tend to delaminate were relatively insensitive to size changes. The strength and strain-to-failure of the matrix dominated lay-ups, however, were very size dependent. Interestingly, they noted that sublaminate scaled laminates increased strength with increasing specimen size. Similar specimens from the previous study (Kellas and Morton, 1992) scaled at the ply level showed the opposite trend. The change in behaviour was attributed to a change in failure modes exhibited by the different thickness scaling methods. Additionally, the PEEK matrix laminates (APC-2) showed less sensitivity to the specimen size. In 2000, Johnson et al. examined two lay-ups of carbon epoxy (AS4/3502), scaled both at the ply level and sublaminate level, tested in a 4-point bending configuration (Johnson et al., 2000). For those specimens scaled at the ply level, a significant size effect (larger size results in a smaller strength and strain-to-failure) was observed. The fracture mechanics approach to predicting these reductions worked well with this data set. The mode of failure in the largest specimen was very different from that in the smaller specimens, with more delaminations being evident in the large specimens. The sublaminate-scaled specimens, on the other hand, did not show any significant size effect. The Weibull model did not predict either series of experiments. Researchers other than those at the University of Bristol and NASA Langley have tackled the issue of size effects in composite materials. In 1994, Bazant et al. (Bazant et al., 1994) considered single and double notched tensile tests involving 2D scaled CFRP specimens. Two lay-ups were examined. Significant nominal strength size effect was found in both series of tests, and the results were not successfully described by either the LEFM approach or the Weibull approach. Instead, Bazant et al. showed that the Bazant approximate size effect law (BASEL) satisfactorily predicts the effect of size. This approximate size effect law was originally developed for concrete, but has been shown to be applicable to other quasi-brittle materials. The approximate size effect law is given in Equation 5.6. CTn =Bfu(. + /3)i (5.6) 154 5.4. Evidence of the Size Effect in Composites and Modelling Efforts where /? is the relative structure size D/D0, crw = CwP/oD = nominal strength of the structure, P = maximum load, D = characteristic dimension, b = width of structure, cN = calibration coefficient, D0 = constant depending on fracture process zone and specimen size, B = constant based on plastic limit analysis, Tu = reference strength of the material. Note that the form of Equation 5.6 is such that the strength is bounded by two limits: the first is the strength-based failure prediction fu, and the second is the fracture mechanics based square root strength reduction with increasing size. Bazant (Bazant et al., 1994) and others (Bazant and Kazemi, 1990) have used the size effect law when dealing with concrete, rock, and toughened ceramics to reasonable success. In 1998, Liu et al. (Liu et al., 1998) examined glass epoxy laminates subjected to drop-weight impacts. Their target specimens were scaled in both in-plane 2D and through-thickness. The results of their tests showed that the in-plane scaling did not have as significant an effect as the through-thickness scaling. Tabiei and Sun (Tabiei and Sun, 1999) presented a model in 1999, based on the Equal Load Sharing (ELS) Rule, a sequential multi-step failure of laminates, and the Weibull approach, which they used to describe the strength size effect associated with both length and thickness scaling in composite laminates. Bazant and Novak offered a series of papers (Bazant and Novak, 2000a; Bazant and Novak, 2000b) that integrated nonlocal theory, deterministic (energy based) size effect, and probabilistic size effect. Once again, they were applying the theory to rock and concrete. The theory required a strain-softening material behaviour, and demonstrated that for small specimens, the strength limit was approached, for large specimens, the probabilistic limit was approached, and for intermediate sizes, the theory agreed well with a number of data sets reported in the literature. Recently, Bazant et al. examined size effect in composite sandwich structures (Bazant et al., 2001). The laminated composite faces and the core foam were examined separately; experiments were performed on the foam and existing data in the literature was used as the basis of the analysis of the face sheets. The researchers concluded that both components separately exhibited size effects. Further, the size effects of the laminates seemed to include both a deterministic component and a statistical component, leading to the conclusion that a mix of the LEFM approach and the Weibull 155 5.4. Evidence of the Size Effect in Composites and Modelling Efforts approach is necessary to adequately describe the size effects. Some discussion on a method of doing this is presented in the references (Bazant and Kazemi, 1990; Bazant and Novak, 2000a; Bazant and Novak, 2000b). Most of the previous work has focused on the failure of strength based failure models to correctly predict the size effect. In 1994 however, Dopker et al. (Dopker et al., 1994) used a continuum damage mechanics based model (that they called a Damage Zone Model) to try to predict failure strength of composites with damage induced cracks and flaws. They applied their model to various sized specimens and noted that its results were better than those of typical failure theories, but required more work. Importantly, they also addressed the problem of "mesh effect". 156 Chapter 5: A Review of Structural Size Effects in Composites 5.5. Summary Size effects exist in composite materials. The sources of those effects are varied, debated, and poorly understood. In all likelihood, all the research is right, but the problem is so complicated that everyone is looking at a small subset of the issue. Statistical size effects are bound to be present in any composite. Defects will be present in any composite due to manufacturing, and depending on the manufacturing process, more or less defects may be present in a given structure. Additionally, variations of material quality within a particular manufacturing process will add to size effects of the statistical nature. These sorts of size effects may be captured by statistical models, like the Weibull model. Size effects due to the behaviour of brittle materials around cracks are also present in some composites. These cracks may be initiated by defects in manufacturing or some extrinsic damage process. These sort of size effects may be captured by fracture mechanics based models. Most composites, however, combine the effects of many damage drivers and suppressors. Notches will drive damage in one particular manner. Statistical small defects will drive damage in another manner. Lay-up, ply-drops, boundary conditions, and any number of other factors will serve to augment or suppress various damage modes. This "mixed source" size effect is the most difficult to predict. The best model suitable for capturing all these factors that contribute to a mixed source size effect would be one that combines all these factors, but until a greater understanding of the factors themselves is reached, a model containing all the details is unachievable. CODAM has some advantages at capturing the size effects. The CODAM approach smears the damage that develops in the sublaminates. The details of how that damage develops are unimportant to the model, rather-the strain states that generate damage and how that damage develops with increasing strain are important. The benefit is that once a material has been sufficiently characterized, CODAM should be applicable in diverse applications. Characterizing the material sufficiently, of course, is a non-trivial task. 157 5.5. Summary The finite element method captures LEFM effects quite well. Introducing a failure criterion (such as the point-stress or average-stress criteria, or CODAM) with a brittle response into a FEM analysis will result in a LEFM-type size effect. The CODAM damage model with a non-brittle response should capture the "mixed source" size effects, given appropriate material characterization. Figure 5.6 illustrates this concept. Other failure theories (plasticity, various maximum stress, or maximum strain theories, etc.) are typically strength-based failure theories that do not exhibit any size effect at all. This appears as the horizontal line in Figure 5.6. LEFM predicts a size effect as shown in the figure, predicting infinitely stronger structures with decreasing structure size. CODAM should capture both these behaviours, as well as the intermediate transition zone. Maximum stress \ \ failure theory \ \ Experimental — \ response / \ Linear elastic ' \ \ fracture mechanics \ \ Zone of maximum strength response (Plasticity, maximum stress theory, etc) Zone of mixed size effect sources ^ Zone of fracture mechanics based response \ iog(D/Dref) Figure 5.6. Illustration of the zones of size effects (Adapted from Fig 2.2 in Bazant, 2002). Where CODAM is currently inadequate, however, is in the prediction of statistical size effects. This is an area to which future development may be focused. Past usage of CODAM has focused on specific applications (Williams et al., 1998; Williams et al., 1999; Starratt et al., 1999; Floyd et al., 1999; Floyd et al., 2002; Floyd et al., 2001c; Floyd et al., 2001b). These applications tended to involve numerous tests on 158 5.5. Summary specimens of identical size and with a few exceptions, the same material system was not used in applications with significant size differences. Given this prior experience, the behaviour of CODAM in different size ranges was not understood, and its ability to capture size effects in composite laminates was unknown. The chapters that follow will investigate CODAM's ability to capture the structural size effect in laminated composite materials. 159 Chapter 6: Simulations of Unnotched Tension Tests The unnotched tension (UNT) test is a classic test that has been used to evaluate size effects in structures. Traditionally, the UNT test is related to Weibull's weakest-link theory (Weibull, 1951), and it is expected that larger specimens will fail at a smaller nominal stress than smaller specimens. To evaluate the size effect in composite UNT specimens, Kellas and Morton studied four lay-ups of CFRP each scaled to four sizes in three dimensions (Kellas and Morton, 1992). This study showed that the four lay-up sequences each exhibited a specimen size effect, but the degree of size effect depended on the lay-up sequence. 160 Chapter 6: Simulations of Unnotched Tension Tests 6.1. Test Geometry The UNT specimen is a simple, non-waisted tension specimen gripped by wedge grips at the specimen ends (approximated by grip tabs in the simulations). Four sizes (full size, 3/4, 1/2, 1/4) have been examined, scaled at the ply level. The full-scale specimen had a nominal length of 508 mm, 50.8 mm width, and 4 mm thickness. The gauge length for the full-size specimen was 356 mm. The smaller specimens were proportionally scaled in all three dimensions. A schematic of the test is shown in Figure 6.1 below. h w Figure 6.1. Unnotched tension test specimen configuration. 161 Chapter 6: Simulations of Unnotched Tension Tests 6.2. Material The material used in the study (Kellas and Morton, 1992) was AS4/3502. This is a graphite fibre embedded in a relatively brittle epoxy matrix system (G|C=160 J/m2, (Whitney et al., 1982)), and the single-ply properties are shown in Table 6.1. The laminar elastic properties are based on the data from Kellas and Morton, adjusted so that the structural stiffness in the simulations matches the structural stiffness reported in the test data. The study examined four lay-up sequences, two of which have been selected for analysis using the CODAM model. The two lay-ups selected for numerical analysis were a quasi-isotropic lay-up [±45°ri/0°r,/90on]s and a cross-ply lay-up [90on/0on]2s-The specimen sizes correspond to n=4, 3, 2, and 1 respectively. Table 6.1. Laminar properties for AS4/3502. Property Value EL 119 GPa ET 9.85 GPa 0.293 GLT 4.82 GPa tply 0.125 mm Selection of the CODAM parameters is based on a process that consists of evaluating the lay-up in a number of one-dimensional thought experiments and using a method similar to the ply-discount method to determine the appropriate uniaxial strains and stiffnesses at various failure levels. For simplicity, two damage phases have been considered: matrix damage and fibre damage. Further, although the assumption was made that each mechanism initiates at a unique strain level, in the interests of simplifying the problem and acknowledging that saturation strains are difficult to estimate, both mechanisms have been assumed to saturate at the same strain. 6.2.1. Quasi-isotropic Lay-up First, the elastic properties of the laminate are determined, using laminate plate theory. This is a relatively trivial exercise, and the LPT results for the [±45/0/90] s lay-up are presented in Table 6.2. Next, the CODAM parameters must be determined. Given the assumptions outlined above, damage initiation strains, damage saturation strains, 162 6.2. Material damage parameters assigned to each damage phase, and modulus reductions assigned to each damage phase are required. Table 6.2. Laminate properties for [±45/0/90]s AS4/3502. Property Value E1 46.9 GPa E2 46.9 GPa G 1 2 17.9 GPa "12 0.309 t larrinate 1.000 mm The thought experiment required to generate the CODAM parameters involves subjecting an imaginary representative volume element (RVE) of the laminate to imaginary uniaxial loads in the 0° and 90° directions. Note that in this case, the sublaminate and the laminate are the same; there is no repeating sequence of plies. Further, the laminate is quasi-isotropic, so it is sufficient to consider just one direction in the thought experiment. The evolution of damage in the thought experiment is as follows: • The first sign of damage in the RVE is cracking of the matrix in the 90° plies. These cracks develop since the stresses in the laminate are uniform across the cross-section, and the matrix has the lowest strength in the laminate. No fibres are damaged in any plies at this point. • Next, matrix cracks develop in the ±45° plies. After re-distributing the stresses following the damage in the 90° plies, the stresses in the ±45° plies reach a threshold that exceeds the strength of the matrix in those plies. Additionally, cracks from the 90° plies will cause stress concentrations near the 0° plies, and some matrix cracking may develop there. The fibres in the ±45° plies will begin to scissor - align themselves in the direction of the load. • Delaminations may develop between plies of the RVE. • Finally, fibre damage occurs. The 0° plies and the ±45° plies will have fibre damage, with the 0° plies beginning to damage at a nominal strain lower than the ±45° plies due to the scissoring of the fibres in the angled plies. 163 6.2. Material Translating this characterization of the damage progression in the RVE into CODAM parameters is an exercise in engineering judgement. To begin, quantifying the damage parameters is a relatively easy and numerically inconsequential task. The reason this is inconsequential is that the damage parameters "fall out" of the mathematics of the mechanics, as shown in Appendix A. These parameters are important to the interpretation of the results, however, and must be quantified. With the total damage partitioned into the two damage phases, the damage parameters can be assigned on the basis of projected area that has been damaged in the RVE. In the fully damage saturated RVE, the 90° plies have had complete matrix damage, and no fibre damage. The ±45° plies have had both matrix damage and fibre damage. The 0° plies effectively have no matrix damage. As such, the magnitude of the damage parameter that is assigned to matrix damage is 0.500, and an equal amount to the fibre damage parameter. Adamaged = 4 A 8 "total ° The modulus reduction associated with the matrix damage achieving saturation (Ems) can be approximated using the ply discount method. See Appendix C for an example on how the ply discount method is applied. The ply discount method neglects the stiffness contribution of the damaged plies in standard LPT. In this case, the 90° plies and the ±45° plies are discounted. £ _ ^ _ Ediscounted _ -j _ 30.4 _ Q 352 m s ~ F ~~ 46 Q ~ undamaged u ^ Data on how damage affects the shear modulus is not available, so for simplicity, the shear modulus is reduced linearly with increasing damage: Gj2=(1-«r2)G° 12 The remaining CODAM parameters are the strains at which each damage phase initiates and saturates, as well as the strain interaction parameters. For simplicity, the 'strain interaction parameters have been chosen such that only strains in the directions considered affect the damage parameters in that direction. For example, only F1(£)=EI 164 6.2. Material causes <vi. The interaction of shear strain and the strain in the orthogonal directions are neglected. This assumption is justified in the current application, due to the dominance of the stresses in the load direction, even in the presence of damage. In the absence of strain-to-failure data for the 3502 resin system, an initial failure strain of 1% was assumed, in line with the failure strains typical of other epoxy matrices. AS4 carbon fibres are reported to have a strain-to-failure of ~1.6% (Schwartz, 1997). Using this value in the CODAM parameters, however, leads to nominal failure stresses in simulations that far exceed those reported by Kellas and Morton in the experiments. A reduced value of 1.2% was assumed as the strain at which failure initiates in the fibres. The saturation strain is the most difficult value to estimate. Given the lay-up of this laminate, and the extent to which the imaginary RVE will distort through the failure process, the damage saturation strain is estimated to be 3%. This high value accounts for the scissoring of the fibres and the large amount of matrix damage expected in the presence of slight fibre damage. Recall that this is a nominal strain for the RVE and not the actual strain that the composite constituents will be experiencing at ultimate failure. The CODAM parameters used in the simulation of the quasi-isotropic specimens are shown in Table 6.3. Table 6.3. CODAM parameters for the quasi-isotropic lay-up of AS4/3502. Matrix Fibre 0.010 Ffi 0.012 F ms 0.030 F,s 0.030 *>ms 0.500 0.500 E ms 0.352 Efs 0.648 The material and CODAM parameters can be represented by a normalized modulus curve or a stress-strain curve, and both of these curves appear in Figure 6.2. 165 6.2. Material I in o. 0.04 (a) (b) Figure 6.2. (a) Normalized modulus curve and (b) Stress-strain curve for the quasi-isotropic lay-up of AS4/3502. Two final parameters must be defined before simulations can be performed. These parameters consider the fracture energy available in the RVE and must be specified so that localization can be addressed. In CODAM, the preferred way to manage the localization problem is to incorporate the crack band method, so the characteristic height of the RVE, hc, and the fracture energy available in the RVE, GF, are required. These two parameters are not independent quantities; they are related by Equation 4.3. If the stress-strain relationship for the RVE is known, then the fracture energy density, fF, is already known. As a result, only one of hc or GCF can be independently determined. Little is known about this material's fracture energy characteristics. The damage developed in quasi-isotropic carbon/epoxy laminates tested in uniaxial tension is known to be extensive, so the characteristic damage height is estimated to be hc=10 mm. The corresponding fracture energy with this choice of characteristic height and stress-strain relationship defined by the CODAM parameters is approximately 100 mJ/mm2, a value that is not unreasonable. Consider the fracture energy available for a lamina of IM6/937 as reported by Delfosse in the 0° (fibre) direction 150mJ/mm2 (Delfosse and Poursartip, 1997). The same study estimates the fracture energy in the orthogonal (90°) direction at ~5 mJ/mm2. Using these values as a rule of thumb, if the 0° and ±45° plies dissipated all of the 166 6.2. Material 150 mJ/mm2 available, and the remaining plies dissipated only 5 mJ/mm2, then the combined fracture energy would be: {Gf)UB * 075x150 + 0.25x5 *114mJ/mm2 This approach assumes that the ±45° plies completely fracture in both the matrix and the fibres, just like the 0° plies. Otherwise, a lower bound to this value would be to apportion only the energy available to the 90° direction to the ±45° plies. In that case, {GF)LB * 0.25 x 150 + 0.75 x 5 » 41 mJ/mm2 This approximately corresponds to choosing a characteristic height of 5 mm, which results in a fracture energy of -50 mJ/mm2 The analyses (discussed in §6.4. ) were performed with both of these characteristic heights in order to capture the bounds of the behaviour. 6.2.2. Cross-ply Lay-up The difference between the quasi-isotropic lay-up and the cross-ply lay-up is only in the stacking sequence. The elastic properties of the laminae are the same, so the same approach that was taken with the quasi-isotropic lay-up can be used in this case. A LPT analysis of the lay-up provides the elastic properties of the laminate, and the results are shown in Table 6.4. Table 6.4. Laminate properties for [90/0]2s AS4/3502. Property Value E1 64.75 GPa E2 64.75 GPa G12 4.85 GPa vn 0.045 t lamnate 1.000 mm A similar thought-experiment is performed on a RVE of this material. The cross-ply lay-up is balanced (that is, it has the same number of 0° and 90° plies) so the 0° parameters will be identical to the 90° parameters, simplifying the process. The evolution of damage in the thought experiment of the cross-ply laminate is as follows: 167 6.2. Material • Damage begins in the matrix of the 90° plies. • Additional matrix damage may develop in the 0° plies due to the stress concentrations caused by the cracks in the 90° plies, as well as stresses caused by Poisson's ratio effect. Furthermore, delaminations may be occur. • The 0° plies will develop fibre damage until saturation of damage in the whole laminate. This progression of damage is simpler than that in the quasi-isotropic laminate, and estimating the CODAM parameters is a somewhat more straightforward process. The damage attributable to matrix damage corresponds to damage only in the 90° plies: damaged = 4 = Ao,a, 8 The modulus reduction is also straightforward. Using the ply discount method to reduce the stiffness of the laminate by neglecting stiffness contribution of the 90° plies to the axial stiffness results in an axial stiffness of 60 GPa. The 90° plies do not contribute very much to the axial stiffness of the laminate. The modulus reduction associated with matrix damage is then: F Q Ems • = 1 - C d ' s c o " n t e d = 1 - -gg^L = 0.075 E undamaged 64.76 Again, in the absence of better information concerning the shear modulus, the shear modulus is reduced linearly with increasing damage: As in the quasi-isotropic lay-up, the strain interaction parameters have been chosen in the interests of simplicity such that only the strains in the directions considered contribute to the development of damage in those directions. The matrix and reinforcing materials are the same as in the quasi-isotropic lay-up, so the same reasoning applies when considering the strain thresholds for the development of damage. In particular, the damage initiation strains will be unchanged. Matrix cracking 168 6.2. Material will begin in the 90° plies, as in the quasi-isotropic lay-up, and fibre cracking will begin in the 0° plies. The saturation strains are expected to be different, however, since the damage mechanisms have changed. There are no ±45° plies in the cross-ply lay-up, so there is no scissoring of the fibres. This scissoring allowed for extension of the RVE without much additional damage creation. Experimentally, cross-ply laminates fracture much more "cleanly" than quasi-isotropic laminates, usually confining the damaged region to a smaller zone. For this reason, the saturation strains in the cross-ply laminate have been reduced to a strain which corresponds more to what would be expected to be the actual strain in the fibres at ultimate failure, 2%. The CODAM parameters for the cross-ply laminate are shown in Table 6.5: Table 6.5. CODAM parameters for the cross-ply lay-up of AS4/3502. Matrix Fibre 0.010 Fn 0.012 F ms 0.020 Ffs 0.020 °>ms 0.500 0.500 E ms 0.075 E,s 0.925 The material and CODAM parameters can be represented by a normalized modulus curve or a stress-strain curve, and both of these curves appear in Figure 6.3. I ui o. 0.03 0.04 Figure 6.3. (a) Normalized modulus curve and (b) Stress-strain curve for the cross-ply lay-up of AS4/3502. 169 6.2. Material Considering the fracture energy, a similar comparison to IM6/937 can be made. Assuming that ~150 mJ/mm2 of energy is available to the 0° plies and ~5 mJ/mm2 of energy is available to the 90° plies, in this application the combined fracture energy would be: GF * 0.5-150+ 0.5-5 *78mJ/mm2 Using a characteristic height of 10 mm with the CODAM parameters discussed above results in a fracture energy of ~83 mJ/mm2. Although a lower bound cannot be estimated using the same process, the simulations were also run using a characteristic height of 5 mm, corresponding to ~42 mJ/mm2. 170 Chapter 6: Simulations of Unnotched Tension Tests 6.3. Other Predictive Approaches Jackson et al. compared the experimental ultimate nominal stress results to two analytical approaches (Jackson et al., 1992). First, they used the Weibull theory based method outlined by Bullock (Bullock, 1974), which gave the relationship between ultimate strengths of geometrically similar models as: ult Sm ult Vff • (6.1) where the m and p subscripts indicate the model and the prototype, S"" is the ultimate strength, V is the volume of the structure, and /?is the Weibull shape parameter. In their analysis, /?was determined separately for each lay-up by fitting the data. Jackson et al. refer to this model as the "statistical" model. The other approach used by Jackson et al. was a LEFM analysis. They reduced the LEFM approach to a simple similitude scaling equation: OUft S p ""=^L- (6.2) where I is the geometric scale factor and the remaining variables are the same as those in Equation 6.1. This model is called the "fracture" model by Jackson et al. The predictions of both these examples will be shown along with the experimental results in the sections that follow. 171 Chapter 6: Simulations of Unnotched Tension Tests 6.4. Simulations With the appropriate material and CODAM parameters, an explicit analysis of the UNT tests was performed. The specimens were completely modelled without taking advantage of the initial geometric symmetry, due to the lack of symmetry after the initiation of damage in the specimen. Each of the four sizes was considered for each lay-up, and mesh-insensitivity was demonstrated using a coarse and fine mesh for the full-size specimen of both lay-ups. The cross-ply simulations were performed using brick elements whereas the quasi-isotropic simulations used shell elements. Shell elements were unstable with the low shear stiffness present in the cross-ply laminates, but the solid elements performed as expected. 6.4.1. Mesh Size Effect Demonstration of the mesh size effect involved running simulations of the largest specimen using a element size (using square elements) of 0.508 mm and a element size of 1.016 mm. The rest of the input parameters were identical. This was performed for both the cross-ply and quasi-isotropic laminates, at both fracture energy levels. The nominal stress results for the quasi-isotropic analyses are shown in Figure 6.4 and Figure 6.5. 172 6.4. Simulations 600 0.02 Average Strain (mm/mm) Figure 6.4. Nominal stresses vs. average strains of the UNT simulations of the quasi-isotropic lay-up, using a coarse and fine mesh, hc=5 mm. 600 0.02 Average Strain (mm/mm) Figure 6.5. Nominal stresses vs. average strain of the UNT simulations of the quasi-isotropic lay-up, using a coarse and fine mesh, /ic=10 mm. The cross-ply laminate results are shown in Figure 6.6 and Figure 6.7. 173 6.4. Simulations 800 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Average Strain (mm/mm) Figure 6.6. Nominal stresses vs. average strains of the UNT simulations of the cross ply lay-up, using a coarse and fine mesh, nc=5 mm. 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Average Strain (mm/mm) Figure 6.7. Nominal stresses vs. average strains of the UNT simulations of the cross-ply lay-up, using a coarse and fine mesh, nc=10 mm. These plots clearly illustrate that there is no mesh effect for this geometry, regardless of the material properties. This is unsurprising, since the stress field generated in the specimens is more or less uniform. There are some stress waves generated in the 174 6.4. Simulations specimens due to the loading method and the nature of the explicit code used to perform the analysis, but the factors are controlled such that the stress waves are inconsequential. As a result, the uniform stress field is captured quite well with elements of any size and failure will occur as soon as some random element happens to have a stress that exceeds the peak stress specified by the CODAM parameters. 6 . 4 . 2 . Structural Size Effect The four specimen sizes were simulated using the two lay-ups and using the two fracture energy levels for each lay-up. The nominal stress-nominal strain results are shown for the quasi-isotropic lay-up in Figure 6.8 and Figure 6.9. The results are compared to the experiments and the two analytical methods in Figure 6.10. Similar results are observed for simulations of the cross-ply lay-up. Figure 6.11 and Figure 6.12 show the nominal stress-nominal strain results for the cross-ply simulations, and Figure 6.13 compares the numerical results to the experiments and the analytical analyses. Figure 6.8. Nominal stress - average strain curves for the UNT simulations of the quasi-isotropic lay-up ([±45/0/90],,) of AS4/3502 using a characteristic height of hc=5 mm. 175 6.4. Simulations 600 « 500 UNT, quasi h c =10 mm 0.005 0.01 0.015 Average Strain (mm/mm) 0.02 Figure 6.9. Nominal stress - average strain curves for the UNT simulations of the quasi-isotropic lay-up ([±45/0790]n) of AS4/3502 using a characteristic height of /ic=10 mm. 0 1 2 3 4 5 Specimen Size (n) Figure 6.10. Nominal failure stress vs. specimen size curves for the UNT simulations of the quasi-isotropic lay-up ([±45/0/90],,) of AS4/3502. Also shown are the results of analysis with the statistical model and fracture model used by Jackson et al. 176 6.4. Simulations 0.015 Average Strain (mm/mm) Figure 6.11. Nominal stress - average strain curves for the UNT simulations of the cross-ply lay-up ([90/0]2n) of AS4/3502 using a characteristic height of 0 0.005 0.01 0.015 Average Strain (mm/mm) Figure 6.12. Nominal stress - average strain curves for the UNT simulations of the cross-ply lay-up ([90/0]2n) of AS4/3502 using a characteristic height of /ic=10 mm. 177 6.4. Simulations 1000 0 4 0 1 2 3 4 5 Specimen Size (n) Figure 6.13. Nominal failure stress vs. specimen size curves for the UNT simulations of the cross-ply lay-up ([90/0]2n) of AS4/3502. Also shown are the results of analysis with the statistical model and fracture model used by Jackson et al. The structural size effect analyses for both lay-ups and all fracture energy levels show no structural size effect when CODAM is employed. 178 Chapter 6: Simulations of Unnotched Tension Tests 6.5. Summary The unnotched tension test simulations do not exhibit any mesh size effect or structural size effect. The reason for this lack of sensitivity is simple: the stresses generated in the simulations are more or less constant throughout the whole test specimen. Since the stresses are constant, as soon as the nominal stresses in the specimen exceed the peak stresses in the stress-strain curve for the individual elements, failure occurs in the whole specimen. Ideally, all the elements would fail at exactly the same time, but in reality, the stress waves present in the specimen cause local regions of higher stresses and these high stress regions will fail first. Once a single element exceeds the peak stress, localization will occur and the failure will grow across the section. There are no statistical size effects built into the CODAM model. Neither are there any micro-defect effects captured in the model. It is the statistical distribution of micro-defects in the specimens that lead to the structural size effects present in the experimental tests. To simulate this sort of failure with the CODAM model, one or both of these approaches needs to be considered. The statistical approach could be integrated into the analysis, based on the size of elements. This approach could introduce variations in the CODAM parameters such that the peak strength of the element was a statistically variant quantity. The other approach is to introduce micro-defects into the mesh to simulate these effects. One possible way to do this could be to "disconnect" a single element in the specimen mesh. That is, a crack of a single element width could be introduced into the mesh in order to introduce a stress gradient at some arbitrary point. This initiator crack would need to be small enough to not affect the global stress field in the specimen while locally causing sufficient disturbance in the stress field to introduce localization. The maximum size of this defect is a subject that requires further research. A demonstration of the stress initiator concept can be discussed however. The quasi-isotropic, full scale (n=4) UNT specimen has been simulated once more, this time with a defect the size of a single element introduced into the mesh. The fine mesh and the coarse mesh have been considered. In the case of the fine mesh, one element width is 0.508 mm. The coarse mesh element width is 1.016 mm wide. An additional analysis of 179 6.5. Summary the fine mesh with a two element wide defect (1.016 mm wide) has been considered, so that the defect size is the same as that in the coarse mesh. The results of this stress-initiator are shown below. Figure 6.14 shows the nominal stress-nominal strain curves for the three cases. It can be seen that a mesh size effect now exists in the unnotched tension test analysis. This mesh size effect also indicates that a structural size effect will be present in the analyses of the various specimen sizes. The current simulations all use the same element size (0.508 mm) so introducing a single element width defect on the smallest specimen width would have a large effect than that on the widest specimen. Investigation of the specimen size effect of these small initiators and the size of initiator required to introduce a specimen size effect into the analyses is left for future consideration. Figure 6.14. Nominal stress-strain results for quasi-isotropic AS4/3502 UNT simulations with small initiator defects. Other applications that do not have stress concentrators or initiators will also suffer from the same complications observed in the simulations of the unnotched tension tests. Simulations of bending tests, in particular, have these same problems. Eccentrically loaded beam-column and four-point beam bending tests are examined in Appendix A, but the results of these simulations are unsatisfactory due to the lack of stress concentrations and initiators. Bending simulations also require special considerations of 180 6.5. Summary the number of through thickness integration points, and the through-thickness integration scheme employed in the analyses. All these issues are addressed in detail in Appendix A. 181 Chapter 7: Simulations of Single-Edge and Double-Edge Notched Tension Tests This chapter is the first of several that address the performance of the CODAM model in simulating notched tension applications. Simple edge-notched tension tests documented by Bazant et al. (Bazant et al., 1996) were selected for investigation with CODAM, in both single-edge notched and double-edge notched configurations. The Bazant study examined the structural size effects in tests of geometrically similar specimens, and these studies were used as an opportunity to explore the capacity of CODAM to address structural size effects. 182 Chapter 7: Simulations of Single-Edge and Double-Edge Notched Tension Tests 7.1. Single Notched Tension Simulations 7.1.1. Test Geometry The single notched tension (SNT) specimens are rectangular specimens with grip tabs fixed to both ends and a notch machined into one side of the specimen. The grip tabs were made from a glass/epoxy composite and are 38 mm long in all cases. The notch length is 20% of the width of the specimen (a=tv/5). Tension was introduced to the specimen and the load cell force and crosshead displacement were measured. The load was controlled with crack opening displacement control in the SNT experiments. The details of the SNT specimen geometries are given in Figure 7.1 and Table 7.1. Four sizes of specimens were examined, and the scaling involved in these experiments was two-dimensional, with geometric similitude in the plane of the specimen. The size of the specimens was chosen such that the ratio in specimen sizes was 1:2:4:8. The thickness of the specimens remained constant, at 8 plies, or 1.02 mm. glass/epoxy grip tabs h a w 1 Figure 7.1. Schematic of SNT specimen. 183 7.1. Single Notched Tension Simulations Table 7.1. SNT specimen dimensions. Specimen w h=4w a=tv/5 size (mm) (mm) (mm) 1 6.35 25.4 1.27 2 12.7 50.8 2.54 3 25.4 101.6 5.08 4 50.8 203.2 10.2 7.1.2. Material The material used in the SNT experiments was IM7/8551-7A, a tough carbon-epoxy composite material (G,c~500-600 J/m2, Delfosse, 1994; Hexcel Corporation, 1998) typically used in aerospace applications. The single-ply (lamina) properties as given by Bazant et al. are presented in Table 7.2. The laminates were quasi-isotropic, with a stacking sequence of [0/±45/90]s. Using LPT and the lamina properties cited in the paper, the laminate properties were determined, and are shown in Table 7.3. Table 7.2. Laminar properties for IM7/8551-7A. Parameter Value EL 169 GPa ET 9.4 G P a 0.3 GLT 6.4 GPa tply 0.127 mm Table 7.3. LPT values for [0/±45/90]s IM7/8551-7A. Parameter Value E1 64.75 GPa E2 64.75 GPa G12 24.9 GPa 0.3 t lairinate 1.016 mm To determine the CODAM parameters, several assumptions are convenient. First, it is assumed that the total laminate (once again, the sublaminate and the laminate are the same) damage can be described by two independent damage measures, matrix damage and fibre damage. The CODAM parameters can then be determined though the use of 184 7.1. Single Notched Tension Simulations a thought experiment. In this case, a representative volume element (RVE) of the laminate is imagined to be subjected to a uniaxial tension. In the case of the quasi-isotropic laminate, the damage progression in the RVE is: • The damage begins with the cracking of the matrix in the 90° plies, effectively removing their contribution to the axial stiffness. • Next, the ±45° plies become damaged, with cracks developing in the matrix. The fibres in the ±45° plies will then scissor, aligning themselves in the load direction. • Matrix cracks will then develop in the 0° plies. • Finally, fracture in the fibres of the ±45° and 0° plies occurs. To simplify the generation of the CODAM parameters, the critical events involving the fibre and matrix damage in the laminate are reduced to the initiation of damage in the matrix, initiation of damage in the fibre and the saturation of both damage mechanisms. The matrix damage initiation is therefore the beginning of matrix cracking in the 90° plies, and the saturation is at the final laminate failure. Fibre damage initiation occurs as fibres in the 0° ply begin to fracture and saturation occurs at the final laminate failure. The quantification of the damage parameters corresponding to these damage states is a relatively easy task. The value of the damage parameters corresponding to matrix saturation (coms) represents the amount of damage corresponding to the 90° plies being fully damaged, and the ±45° plies being partially damaged. The modulus reduction associated with the matrix damage reaching saturation (Ems) can be approximated using the ply discount method, neglecting the stiffness contribution of the damaged plies to the overall stiffness of the laminate. See Appendix C for an example on how the ply discount method is applied. A damaged - = 0.500 8 co, ms E =1 -^ '-ms ' r: undamaged discounted = 1- 42.25 64.75 = 0.35 185 7.1. Single Notched Tension Simulations The planar isotropy resulting from the lay-up means that the 1-direction and 2-direction inputs will be identical, further reducing the complexity of the problem. Data on how damage affects the shear modulus is not readily available, so for simplicity, the shear modulus is reduced linearly with increasing damage: The effective strain functions, F(s), require interaction constants to relate the contribution of the strain components to the development of the damage parameters in particular directions. As was the case for the UNT analyses, the strain interaction parameters have been chosen such that only strains in the directions considered affect the damage parameters in that direction. To complete the CODAM inputs for the material at the characteristic size, the effective strains at the initiation and the saturation of each phase must be determined. Bazant et al. report the failure strain in the 0° direction for the lamina to be 1.3%. Matrix damage is assumed to begin before the ultimate strain, so it has been assumed to begin at 1.0% strain. This is very close to the 90° lamina failure strain of 0.93% (Hexcel Corporation, 1998). IM7 fibres have been reported to have ultimate failure strains ranging from 1.7% to 2.0% depending on the source (Hexcel Corporation, 1998; Delfosse, 1994). The fibre damage initiation strain is assumed to be 1.6%. The saturation strain, however is not as simple as choosing a value consistent with the local strain in the fibres or matrix at complete damage saturation. Instead, a value consistent with the strain in an RVE must be chosen, and the RVE will have a greater nominal strain than the exact strain in each constituent. Unfortunately, a means to measure this saturation strain is not available, so other methods must be used to estimate an appropriate value. In another simplification, the matrix damage saturation strain and the fibre damage saturation strain are assumed to be the same value. In doing so, a single strain related parameter remains to be specified. In their analysis of a similar material system and lay-up, Williams et al. used a value of 4% as the saturation strain with success (Williams et al., 2003), and the same saturation strain is adopted for this analysis. 186 7.1. Single Notched Tension Simulations The CODAM parameters used in this simulation for this laminate are summarized in Table 7.4. Table 7.4. CODAM parameters for quasi-isotropic lay-up of IM7/8551-7A. Matrix Fibre 0.010 Fn 0.016 • ms 0.040 FfS 0.040 ®ms 0.500 0.500 Ems 0.350 E,s 0.650 The resulting normalized modulus vs. strain and stress vs. strain curves are shown in Figure 7.2 and Figure 7.3. 1.00 + 0.75 lUJ 0.50 0.25 0.00 0.000 -•- Matrix -*- Fibre System 0.010 0.020 0.030 F 0.040 Figure 7.2. Normalized modulus vs. effective strain curve for quasi-isotropic lay-up of IM7/8551-7A. 187 7.1. Single Notched Tension Simulations 1200 0.05 8 (mm/mm) Figure 7.3. Stress-strain curve resulting from selection of CODAM parameters for quasi-isotropic lay-up of IM7/85S1-7A. To complete the characterization of the material, the fracture energy in the laminate must be considered so that the localization of the numerical analysis is handled correctly. In the context of the CODAM model and the crack band method, this requires the consideration of two additional parameters, the characteristic height, hc, and the fracture energy, GCF, which are related through the fracture energy density, yF. The CODAM parameters result in a failure energy density of yF=2M mJ/mm3 One way of estimating the characteristic height is to examine the fracture energy density that is predicted by a particular stress-strain relationship, yF, and the characteristic height. This fracture energy results from two damage mechanisms, according to the assumptions presented above: fibre breakage and matrix cracking. The energy required to fracture an RVE of matrix is small, estimated by Delfosse (Delfosse and Poursartip, 1997) to be 5 mJ/mm2 for 3900-2, a tough resin system, and 0.8 mJ/mm2 for 937, a brittle resin system. Data for 8551-7A is not available, so 0.8 mJ/mm 2 will be used as a first-order estimate. The same study (Delfosse and Poursartip, 1997) estimates the fibre 188 7.1. Single Notched Tension Simulations fracture energy (GFD) at 150 mJ/mm2 for IM6, but does not provide an estimate for IM7 fibres. A review of the available literature does not show any estimates for fibre fracture energy for this fibre either. Rather, comparing the elastic moduli and strain-to-failure for the fibres (Hexcel Corporation, 1998), the ratio of elastic strain energy at failure for the two fibre types (IM6:IM7) is about 1:1.05. Using this as a guide, the GFD for IM7 is estimated to be 160 mJ/mm2 For this lay-up, the 0° plies clearly use all the available energy in the 0° direction, and the 90° plies use the energy determined from tests in the 90° direction. The ±45° plies, however are somewhat different. If cracks develop completely across the specimen, then all the fibres in the ±45° plies are fractured. An estimate for the fracture energy available for this lay-up could be: &)UB * 0 7 5 x 158 + 0.25x 1»119mJ/mm2 where the UB subscript indicates an upper bound. This (GF ) U B results in a characteristic height of about 5 mm. The study by Williams (Williams et al., 2003) used a similar reasoning to determine G°F for a similar lay-up, but assigned the ±45° plies the fracture energy of the 90° tests. Doing so in this case provides a lower bound to the GF estimate: (GCF ) L B * 0.5 • 158 + 0.5 • 1 * 80 mJ/mm2 where the LB subscript indicates a lower bound. This lower bound estimate requires a characteristic height of about 3.6 mm. Both these characteristic heights will be investigated in the numerical analysis. 7.1.3. Other Predictive Approaches The Bazant Approximate Size Effect Law (BASEL), as described in §5.4. , was fit to the experimental results (Bazant et al., 1996). Bazant et al. used regression analysis of the experimental data to determine the parameters for the BASEL, and the resulting parameters are shown in Table 7.5. 189 7.1. Single Notched Tension Simulations Table 7.5. SNT parameters for the BASEL. Parameter Value D0 Bfu Gf 30.9 mm 892 MPa 521 mJ/mm ,2 Cf 9.19 mm Additionally, a linear elastic fracture mechanics (LEFM) analysis of the data can be performed. For the SNT geometry, LEFM predicts the following response (Tada et al., 1985): In this case, the LEFM analysis was calibrated to match the experimental nominal failure stress for the 6.35 mm wide specimen. 7.1.4. Simulations A wide range of simulations was performed to explore the behaviour of the CODAM model in this application. The effect of the mesh size on the ultimate nominal stress was explored by simulating one of the SNT specimens with a coarse mesh and a fine mesh. Further, the mesh effect was demonstrated by simulating the four sizes of specimens without changing the crack band scaling factor, effectively neglecting the fracture energy. The structural size effect was studied by simulating the four specimen sizes with fracture energies of 110mJ/mm2, 78 mJ/mm2 (the upper and lower bounds discussed above), as well as 521 mJ/mm2, the fracture energy determined by Bazant in his analysis. These numerical results were compared to the experiments and several other techniques for predicting the structural size effect. Several supplemental simulations were also performed, to examine various areas of interest that developed in the course of the investigation. In all the SNT simulations, the complete specimen was modelled and the crack band method was incorporated into the analysis to manage the localization problem. Square elements were used, and the mesh density was constant throughout the whole specimen. The specimens were discretized such that the notch width to element size K 1.204 GPa4m (7.1) cr = 2.4343Va Va 190 7.1. Single Notched Tension Simulations ratio (a/he) was preserved as the physical size of the specimen changed, ensuring that the stress distribution around the notch tip was captured to the same degree in each simulation. 7.1.4.1. Mesh Size Effect In order to demonstrate the need to consider the localization problem and the fracture energy of the material, two types of simulations were performed. First, the 25.4 mm wide specimen was meshed using square elements of 0.254 mm and 0.508 mm in length, preserving the same alhe ratio as in the 50.8 mm wide simulations. The simulations were otherwise identical, and the crack band scaling factor was unchanged. The results of these simulations are shown in Figure 7.4. ro 1000 800 (8 600 CD CO "to c E o Figure 7.4. 400 200 0 UNT w=25.4 mm .y^^ine mesh ^ coarse mesh — 0.005 0.01 0.015 0.02 Nominal Strain (mm/mm) 0.025 Comparison between fine mesh (0.254 mm) simulation and coarse mesh (0.508 mm) simulation for the 25.4 mm wide SNT specimen without the crack band scaling. The second method of demonstrating the mesh effect is simulating all the specimen sizes without changing the crack band scaling factor. Since the element sizes are changing from specimen to specimen, and the scaling factor is not changing, differences in the simulations to those that properly employ crack band scaling is due to the mesh size. This is shown in Figure 7.5. 191 7.1. Single Notched Tension Simulations The SNT simulations are clearly sensitive to the size of the elements used in the simulation. The difference in the peak nominal stress in the simulations of the 25.4 mm wide specimen is solely due to the size of the elements. The interpretation of the results of the simulations run on the four specimen sizes with different element sizes and the same scaling factor requires more explanation but also proves the mesh size effect. Simulations with the same element size and no scaling (crack-band, equivalent work, etc.) will demonstrate a structural size effect, as shown in §7.2.3.1. The lack of a structural size effect in Figure 7.5, based on the observation that all of the simulations produce the same nominal stress-strain results, is a result of the element size effect balancing out the usual structural size effect (caused by the localization problem exhibited by strain-softening material models) that would have been observed. In fact, this shows that the size effect observed in simulations of notched tension tests run with the same element size, therefore different notch to element width ratios, is a result of mesh size effects rather than a structural size effects. All these mesh size effects are caused by the localization of softening in a single band of elements and the subsequent improper dissipation of energy. 200 £ 150 in <D ~ 100 C/D £ o S N T constant a/he constant scal ing factor (ie GF not constant) 0.001 0.002 0.003 Nominal Strain (mm/mm) 0.004 Figure 7.5. Simulations of the four SNT specimen sizes using the same inputs, keeping the notch width to element width ratio constant. Note that the slight variation in the results is due to the different data densities in the simulation output. 192 7.1. Single Notched Tension Simulations 7.1.4.2. Structural Size Effect The four specimen sizes were simulated at three energy levels: 78 mJ/mm2, corresponding to the lower bound estimate for fracture energy, 110mJ/mm2, corresponding to the most-reasonable estimate for fracture energy, and 521 mJ/mm2, corresponding to the fracture energy required to fit the BASEL analysis to the experimental data. Figure 7.6 shows the results of these simulations, with the BASEL analysis and LEFM analysis shown for comparison. The nominal failure stresses are expressed as the inverse of the square of the stresses, so that the LEFM analysis shows as a line. 0 i — , —r— — T — 0 10 20 30 40 50 60 Specimen Width (mm) Figure 7.6. Nominal failure stresses for CODAM analyses of the SNT specimens at three fracture energy levels. Also shown are the results from an LEFM analysis, the BASEL analysis, and the experiments. Several interesting observations can be made from these results. The LEFM analysis predicts significant size effect on the nominal failure stresses. The high fracture energy CODAM analysis (521 mJ/mm2) does a good job at predicting the magnitude of size effect but over-estimates the absolute failure stresses. With the estimated fracture energy (110mJ/mm2), analysis with CODAM predicts results that fall within the experimental error bars for the three smallest sizes, but does not capture the size effect trend accurately. Given the assumptions present in all the analyses and the estimates 193 7.1. Single Notched Tension Simulations required in selecting the CODAM parameters, these predictions are remarkably good. Time and effort could be spent 'tweaking' the CODAM parameters to reproduce the experimental results, but a better time investment would be in refining the techniques for determining those parameters and improving the quality of the available data. The most striking observation to be drawn from Figure 7.6 is the difference in the magnitude, or degree, of the size effect. If linear trendlines are drawn through the data points in this figure, the slopes of those lines could be interpreted as the degree of size effect predicted by the analysis. Figure 7.7 compares the degree of size effect for the different analysis techniques, normalized to the size effect predicted by LEFM. Note that the CODAM analyses intersect the experiments around 220 mJ/mm2, therefore CODAM analyses at this fracture energy level (which also corresponds to a characteristic height of ~10 mm) should closely match the experiments. 1.2 0 4 - - r - - r - - i i 0 100 200 300 400 500 600 Fracture Energy (mJ/mm2) Figure 7.7. Comparison of the degree of size effect, normalized to that predicted by the LEFM analysis, predicted by the various analysis techniques shown in Figure 7.6. The degree of size effect trend for the CODAM analysis makes sense. Tougher materials, those that have significant plasticity or other high-energy dissipation 194 7.1. Single Notched Tension Simulations mechanisms, should be less sensitive to the size of the structure. In a highly ductile material, for example, the nominal failure stress should be very close to the yield stress, regardless of the size of structure. At the other end of the spectrum, LEFM is known to be a good analysis technique for very brittle materials, and those materials should be very sensitive to the size of the structure. CODAM with the crack band method modifications follows this trend quite well, being less sensitive to the structure size with high fracture energies, and more sensitive to the structure size with low fracture energies. 195 Chapter 7: Simulations of Single-Edge and Double-Edge Notched Tension Tests 7.2. Double Notched Tension Simulations In the same study that examined the single notched tension specimens (Bazant et al., 1996), double notched tension specimens of a cross-ply laminate of the same material were examined. To further exercise the CODAM model, these cross-ply, double notched tension tests have also been simulated. 7.2.1. Test Geometry The double notched tension (DNT) specimens are very similar to the single notch tension specimens. A glass/epoxy grip tab 38 mm long is fixed to the top and bottom of a rectangular specimen. Notches are cut into the specimen on both sides to a length of a/2=w/16. Tension is introduced to the specimen and the load cell force and crosshead displacement are measured. Stroke (displacement) control was used in the DNT tests. Figure 7.8 shows a schematic of the specimens. p=P/w tttrtt a/2 a/2 w ^glass/epoxy grip tabs Figure 7.8. Double notched tension specimen schematic. Four sizes of specimens were examined, scaled in two-dimensions, keeping the thickness constant. The specimen sizes were chosen such that the ratio in specimen sizes was 1:2:4:8. The dimensions of the four specimens are shown in Table 7.6. 196 7.2. Double Notched Tension Simulations Table 7.6. Specimen dimensions for DNT simulations. Specimen Width, w Height, h Notch, a Size (mm) (mm) (mm) 1 6.35 25.4 0.794 2 12.7 50.8 1.588 4 25.4 10V.6 3.175 8 50.8 203.2 6.350 7.2.2. Material The material used in the DNT tests is identical to that used in the SNT tests, IM7/8551-7A. The laminates were cross-ply laminates, with a stacking sequence of [0/902]s. Using LPT and the lamina properties cited in the paper, the laminate properties were determined, and are shown in Table 7.7. Table 7.7. IM7/8551-7A [0/90JS laminate elastic properties. Parameter Value E1 62.85 GPa E2 116.25 GPa G12 6.4 GPa "12 0.024 tply 0.76 mm To determine the CODAM parameters, the same approach is taken. First, it is assumed that damage can be described by two independent damage drivers, matrix damage and fibre damage. Secondly, the assumption is made that in a tension test of the laminate in the 0° direction, damage occurs first in the 90° plies, eventually saturating such that the 90° plies do not contribute at all to the stiffness in the 0° direction. Fibres in the 0° plies carry the remaining load until they fracture. In a tension test in the 90° direction the same processes occur, but at different levels due to the differences in the number of fibre-dominated plies in the load direction. Using the ply-discount method to approximate the stiffness reduction due to the failure of the plies at 90° to the load, the value of the damage parameter attributed to matrix damage saturation and the modulus reduction due to matrix damage saturation can be determined: In the 0° direction: 197 7.2. Double Notched Tension Simulations damaged = 2 = Ao,a, ' 6 ^discounted _ <| _ ^ 13.18 _ Q -undamaged 116.25 In the 90° direction: damaged Yofa/ - = 0.667 6 -discounted -undamaged 56.6 62.85 0.10 Without any other knowledge about how damage affects the shear modulus, and for simplicity, the shear modulus is reduced linearly with damage: Gi2={\-C012)G°12 In an additional simplification, the effective strain values are assumed to be equal to the strain values through appropriate choices of the strain interaction constants. That is, FJ(E)=EJ and F2(z)=Z2 To complete the CODAM inputs for the material at the characteristic size, the effective strains at the initiations and saturation of each phase must be determined. Since the material is identical to that used in the SNT simulations, the damage initiation strains for the two phases should not be different. The lay-up will influence the saturation strain, though, and it has been reduced to 0.020 in recognition of the observation that cross-ply laminates tend to exhibit much more brittle-like failures than quasi-isotropic laminates. This brittle-like behaviour results from fewer cracks opening and less nominal strain in the region around the damage. The nominal strain of an RVE in that region would therefore be closer to the actual strain of the constituents, and the CODAM saturation strain should be closer to the expected failure strain of the constituents. The CODAM parameters used in this simulation for this material are summarized in Table 7.8. 198 7.2. Double Notched Tension Simulations Table 7.8. CODAM parameters for cross-ply lay-up of IM7/8551-7A. Matrix Fibre 0° dir Fm 0.010 Fn 0.016 F ms 0.020 FfS 0.020 <»ms 0.333 G>fs 0.667 E ms 0.026 Efs 0.974 90° dir Fm 0.010 Fn 0.016 F ms 0.020 Ffs 0.020 ms 0.667 0.333 Ems 0.100 Efs 0.900 The normalized modulus vs. effective strain and stress-strain curves resulting from the material parameters are shown in Figure 7.9 and Figure 7.10 1.00 0.75 lui 0.50 0.25 0.00 -*- Matrix • Fibre -*- System 1.00 0.00 0.000 0.010 0.020 0.030 0.040 F (a) 0.000 0.010 0.020 0.030 0.040 F (b) Figure 7.9. Normalized modulus vs. effective strain for cross-ply lay-up of IM7/8551-7A used in the DNT simulations, (a) 0° direction, (b) 90° direction. 0.01 0.02 0.03 s (mm/mm) (a) 0.04 0.05 0.01 0.02 0.03 8 (mm/mm) (b) 0.04 0.05 Figure 7.10. Stress-strain curves for cross-ply lay-up of IM7/8551-7A used in the DNT simulations, (a) 0° direction, (b) 90° direction. 199 7.2. Double Notched Tension Simulations The remaining task in characterizing the material for use with CODAM is determining the characteristic height of damage, hc. The CODAM parameters result in a failure energy-density of #=9-6 mJ/mm3 in the 0° direction and ^ =18.6 mJ/mm3 in the 90° direction. In the same manner as with the SNT simulations, the available fracture energy per unit area can be estimated using the lay-up and the fracture energies determined by Delfosse et al. (Delfosse and Poursartip, 1997). In the 0° material direction, the overall fracture energy can be estimated by estimating the energy required to fracture a unit area of the constituents: G| * 0.333-160 mJ/mm2 + 0.667 •1mJ/mm2 «54mJ/mm 2 This Gp results in a height of damage of about 5.5 mm. Given the approximations and assumptions used to arrive at this value, a characteristic height of 5 mm is assumed, but characteristic heights of both 10 mm and 20 mm are also examined to gain more insight into the effect of hc on the response of the system. These characteristic heights correspond to fracture energies of 48.0 mJ/mm2, 96.0 mJ/mm2 and 192.2 mJ/mm2, respectively. In the 90° direction, the loading and geometry are such that the influence of damage in that direction is significantly less than the loading direction. The same heights of damage as the 0° direction were assumed in the 90°- direction. The validity of this assumption was tested by performing simulations with a height of damage of 5 mm in the 90° direction, and the results were identical to those performed with a height of damage of 10 mm in the 90° direction. 7.2.3. Simulations A variety of simulations were performed to explore the behaviour of this geometry and the material system. The effects of the mesh size on the results were explored, as was the effect of changing the mesh size without changing the fracture energy by the crack band method. The structural size effect was examined by simulating the four sizes of specimens tested in the experimental study. Several supplemental simulations were performed to examine the sensitivity of the results to the saturation strain and the sensitivity of the results to the 90° direction parameters. 200 7.2. Double Notched Tension Simulations 7.2.3.1. Mesh Size Effect The specimens were meshed at two degrees of refinement. In the "coarsely" meshed simulations, approximately the same element sizes that were used in the SNT simulations were employed - 0.253 mm elements for the smallest sizes, 6.35 mm and 12.7 mm wide, and 0.505 mm elements for the largest sizes, 25.4 mm and 50.8 mm wide. The "finely" meshed simulations preserved the same ratio of element size to crack size, resulting in elements of 0.505 mm, 0.2525 mm, 0.12625 mm, and 0.063125 mm. Two mesh size effects are highlighted. The first is an elastic mesh size effect. Figure 7.11 shows the results of the coarse mesh simulations. Note that apparent stiffness of the simulations is different in each case. This is an indicator that the stresses around the notch tips are not being captured to sufficient detail. The solution is to mesh more finely near the notch tips. re o_ to co cu l_ (O "55 c £ o 0 0.005 0.01 0.015 Nominal Strain (mm/mm) 0.02 Figure 7.11. Results of the coarse mesh simulations of the DNT tests. The second mesh size effect is one that results from the localization of strain-softening material models. The DNT specimens were meshed such that the element size to crack length ratio was maintained for all the specimen sizes. This effectively preserves not 201 7.2. Double Notched Tension Simulations only geometric similitude in the scaling of the specimens, but also numerical similitude in the density of the mesh on each specimen. Now, if the fracture energy was not considered in the analysis of these specimens (i.e., no energy-based scaling) then the only difference in the simulations is the absolute size of the elements and the absolute size of the specimen. Rather than not using crack band scaling, the simulations were run with the same scaling factor (k), equivalent to neglecting the fracture energy. This is effectively the same as performing the simulations without scaling. Figure 7.12 shows the results of these simulations. The simulations of all the specimen sizes result in identical nominal failure stresses and strains. This is obviously not realistic and demonstrates the necessity for fracture energy-based scaling. 700 0.002 0.004 0.006 0.008 0.01 Nominal Strain (mm/mm) 0.012 Figure 7.12. Results of the finely meshed DNT simulations using the same scaling factor (* = 89). 7.2.3.2. Structural Size Effect Proper use of the crack band method removes the mesh size effect in simulations involving Mode I fracture events. Figure 7.13 shows the nominal stress-nominal strain results of the simulations involving the finely meshed specimens with the model inputs that consider the fracture energy consistent with the crack band method. Note that the 202 7.2. Double Notched Tension Simulations long "tail" present in the results of the simulation of the 6.35 mm wide specimen is a numerical artefact. The scaling factor required by that simulation is over 700, and as such the stiffness following the peak stress degrades very gradually, creating the tail that is evident. The other sizes do not exhibit the same tail since they localize much more abruptly. 0. (/> 0) k_ *•> w "en c I o 900 800 700 600 500 400 300 200 100 0 DNT hc=10 mm ' I / w=6.35 mm Fine meshes \ w=12.7mm — w=25.4 mm / w-50.8 m n ^ ^ ^ ^ ^ ^ — 0.005 0.01 0.015 0.02 0.025 Nominal Strain (mm/mm) 0.03 0.035 Figure 7.13. Results from the finely meshed DNT simulations, with proper consideration of fracture energy. These numerical results compare quite well with the experimental values. Figure 7.14 shows a comparison of the numerical ultimate stresses with the experimental results. 203 7.2. Double Notched Tension Simulations 4 3.5 3 «T 2.5 (0 Q. O 2 b 1.5 0.5 0 DNT hc=10 mm Fine meshes 0 10 simulations experiments 20 30 40 Specimen Width (mm) 50 60 Figure 7.14. Comparison of the finely meshed DNT simulations with the experimental results. The characteristic element size was adjusted, effectively changing the fracture energy (GF) dissipated by the material. Lowering the characteristic size to 5 mm resulted in a larger specimen size effect, and raising the characteristic size to 20 mm resulted in a small specimen size effect. These results are shown in Figure 7.15. This trend is the same as that observed in the SNT simulations. 204 7.2. Double Notched Tension Simulations 0 4— — i — — r - — i — — i — — i 1 0 10 20 30 40 50 60 Specimen Width (mm) Figure 7.15. Nominal failure stresses for the DNT simulations analyzed with different characteristic heights (fracture energies). Also shown are an LEFM analysis and the experimental results. Three variations on these simulations were run, in order to explore the sensitivity of the model. First, the characteristic height in the x-direction (perpendicular to the load) was changed to 5 mm, resulting in a fracture energy of 93 mJ/mm2, approximately the same as the fracture energy in the loading direction. Figure 7.16 shows the results of this simulation superimposed on the results from the simulation using the same characteristic height in each direction, 10 mm. The difference between the two simulations is negligible, showing that the simulation is very insensitive to the parameters in the x-direction. 205 7.2. Double Notched Tension Simulations 4 3.5 3 2.5 CO Q. O 2 b 1.5 1 0.5 0 DNT Fine meshes > c hc=10 mm, both directions hcy=10mm, h c = 5 mm Figure 7.16. 10 20 30 40 Specimen Width (mm) 50 60 Comparison of ultimate stresses for DNT simulations with equal characteristic heights in both the x- and y- directions, and with approximately equal fracture energies in both directions (in which case, hcy = 10mm, and hcx = 5 mm.) Note that the two curves overlap completely. The second variation to the simulations was to vary the saturation strain from 0.020 to 0.030 and 0.040. At the same time, the characteristic height was adjusted so that the fracture energy was maintained at ~96 mJ/mm2 These adjustments had minor influences on the peak stress in the characteristic stress-strain curves, less than a 10% increase, but obviously increased yF. The energy parameters are shown in Table 7.9. Running the simulations with these adjustments resulted in higher ultimate stresses, as shown in Figure 7.17 (shown as lower values in the figure since the inverse of the square of the nominal failure stress is plotted). 206 7.2. Double Notched Tension Simulations Table 7.9. Energy parameters for saturation strain study. Fms Ff s F s YF (mJ/mm3) CJmax (MPa) h c (mm) G F (mJ/mm2) 0.020 9.63 948 10.00 96.3 0.030 16.66 978 5.78 96.3 0.040 24.01 1029 4.01 96.3 Specimen Width (mm) Figure 7.17. Effect of varying damage saturation strains on the ultimate stresses exhibited by the DNT simulations, while maintaining the fracture energy constant. The remaining investigation focused on the fibre damage initiation strain, F f i. Fibre damage initiation strains were varied above (0.018) and below (0.014) the selected value of 0.016, while maintaining a constant GF. The energy parameters for this study are shown in Table 7.10, and the results of the simulations are shown in Figure 7.18. 207 7.2. Double Notched Tension Simulations Table 7.10. Energy parameters for fibre initiation strain study. Ff, YF (mJ/mm3) C m a x (MPa) hc (mm) G F (mJ/mm2) 0.014 8.83 847 10.91 96.3 0.016 9.63 948 10.00 96.3 0.018 10.47 1043 9.20 96.3 0.5 -0 - — i — — i — i i i 0 10 20 30 40 50 60 Specimen Width (mm) Figure 7.18. Effect of varying fibre damage initiation strains on the ultimate stresses exhibited by the DNT simulations, while maintaining the fracture energy constant. While these two explorations show that the ultimate nominal stresses increases with increasing fibre damage initiation strain and increasing damage saturation strain, they also show that the degree of size effect (i.e., the slope of the lines in Figure 7.17 and Figure 7.18) is affected only by increasing saturation strains. This is illustrated more clearly in Table 7.11. Linear regression analysis has been performed on each of the nominal ultimate stress (oN~2) vs. specimen width curves, and the slope of the lines has been used as a measure of the degree of size effect, as in §7.1.4.2. The degree of size effect is therefore related to the available fracture energy (GF) as well as the abruptness of the post-failure part of the stress-strain curve. For a given fracture energy, a more 208 7.2. Double Notched Tension Simulations abrupt post-failure stress-strain curve where the saturation strain is closer to the strain at peak stress, results in a greater size effect. A less abrupt post-failure stress-strain curve, with a greater saturation strain, results in less of a size effect. Table 7.11. Degree of size effect for the experiments, LEFM analysis, and the CODAM simulations of the DNT specimens. Degree of Size Effect Degree of Size Effect ( G P a 2 mm"1) (Normalized to Exp't) Ffi=0.014, Fs; Ffi=0.016, Fs; F„=0.018, F s : FtrO.016, Fs; Ffi=0.016, Fs; Experiments LEFM 0.020 0.039 1.00 0.246 6.26 0.032 0.81 0.020 0.033 0.85 0.020 0.033 0.83 0.030 0.027 0.69 0.040 0.025 0.63 209 Chapter 7: Simulations of Single-Edge and Double-Edge Notched Tension Tests 7.3. Summary The notched tension simulations clearly demonstrated the sensitivity of notched applications to the size of the elements used in the simulations and the localization problems that will occur when strain-softening material models are used to capture the behaviour of these events. With the use of CODAM and the crack-band method, it is possible to simulate the experiments and capture the behaviour without the spurious behaviour that has been exhibited in the past. CODAM with provisions for the crack band method and attention given to the fracture energy available to the material is able to reduce the significance of the size of the elements in the analyses. It has been shown that careful attention must be paid to the size of the elements so that the stresses based on undamaged linear elastic analysis are captured in sufficient detail, especially around the tip of the notch, but that is typical in any finite element analysis of notches and cracks. The characteristic height and fracture energy play very important roles in determining the behaviour of the simulations. The degree of the structural size effect predicted by CODAM heavily depends on the fracture energy. At the same time, the structural size effect also depends on the shape of the normalized modulus curve (or, alternatively, the resulting stress-strain curve) of the RVE. The DNT simulations demonstrated that simply by changing the saturation strain in the damage development curves, without significantly changing the peak stress or fracture energy, the magnitude of structural size effect predicted will be affected. It is these parameters, the saturation strains, and the available fracture energies about which the least is known. Better techniques need to be developed to estimate the saturation strains and fracture energies. 210 Chapter 8: Simulations of Notched Tension Tests: S/RFI Laminates The over-height compact tension test was developed by Kongshavn and Poursartip in the mid 1990s to grow large cracks in composite laminates in a controlled fashion (Kongshavn and Poursartip, 1999). The technique has been used extensively with carbon fibre laminates, and recently in stitched resin film infused (S/RFI) carbon/epoxy laminates. The testing has shown evidence of strain-softening behaviour in carbon fibre laminates and was essentially the motivation for the development of a strain-softening numerical model (i.e., CODAM). CODAM has been used, in various stages of the model's development, to simulate the crack growth in many OCT tests. The original version of CODAM was used to simulate Kongshavn's experiments (Williams, 1998), and more recent versions have been used to simulate the S/RFI material (Floyd et al., 2001c; Floyd et al., 2001b). In the chapter that follows, a rigorous approach to modelling the S/RFI material and the OCT test is taken. The OCT simulations have been compared to experiments performed by Mitchell (Mitchell, 2002). The same S/RFI material was used in other notched experiments, some featuring specimens considerably larger than the OCT specimens. These additional experiments included large notch tension (LNT) tests, and open hole tension (OHT) tests. The characterization of the S/RFI material developed for the OCT simulations is applied directly to the LNT and OHT simulations. These other notched experiments were performed by Integrated Technologies, Inc. (Intec) in Bothell, Washington, as part of a study commissioned by The Boeing Company (Starratt et al., 1999). 211 Chapter 8: Simulations of Notched Tension Tests: S/RFI Laminates 8.1. Over-height Compact Tension Simulations BAA. Test Geometry The OCT test is a pin-loaded, single-notched tension test, derived from the standard compact tension (CT) test. A schematic of the specimen is shown in Figure 8.1. width k » W H Figure 8.1. Schematic of the OCT specimen (from Mitchell, 2002). In Figure 8.1, a is the length of the notch, measured from the centre of the load pins, c is the distance between the centres of load pins in the load direction, d is the diameter of the load pins, fH is the separation between inscribed lines (see below), H is the overall height of the specimen, t is the thickness of the specimen, W is the distance from the centre of the load pins to the compression edge of the specimen, and width is the full width of the specimen. For the specimens examined, the dimensions are provided in Table 8.1. Table 8.1. Geometric parameters for OCT specimens. a 32.4 mm H 207 mm c 38.7 mm tply ~1.5 mm d 19.1 mm w 81 mm f H 2.5 mm width 107 mm 212 8.1. Over-height Compact Tension Simulations The OCT specimen is significantly taller than the standard CT specimen, which has a height equal to its width. The "extra" height is intended to prevent damage to the load application regions of the specimen while allowing damage to grow in a stable manner along the notch-plane. Further refinement of the technique was required for testing the S/RFI material. In initial tests with the S/RFI material, the specimen was observed to twist out-of-plane and crush near the loading pins. To control the specimen twisting, steel stiffeners coated with Teflon® were fastened at the compression side of the specimen. To control the material crushing at the loading pins, the pin diameter was doubled, reducing the stress concentration. A strain gauge extensometer was attached at the notch side of the specimen to measure the crack mouth opening displacement (CMOD) and the pin load was measured through the load cells of the universal testing machine. The specimens were loaded in tension using a displacement controlled loading rate of 0.508 fnm/min (0.02 in/min). In addition to the CMOD and load measurements, the local displacement field was measured in the region surrounding the notch mid-plane. This was achieved by inscribing a series of lines parallel to the notch plane on the un-notched side of the specimen. This series of lines was photographed through the course of the testing, and the displacement of the lines was determined using image analysis after the test was complete. The analysis of the local displacement field for each photograph was correlated with a CMOD measurement. This technique is called "line analysis" and is useful for determining the actual crack and process zone location during the test. A schematic of the experimental set-up is shown in Figure 8.2. More details on the experiments can be found in the references (Kongshavn, 1996; Kongshavn and Poursartip, 1999; Mitchell et al., 2001; Floyd et al., 2001b; Mitchell, 2002). 213 8.1. Over-height Compact Tension Simulations Applied Displacement, g A CMOD Gauge ToDAQ Guide Loading Pins Applied Displacement, s Y Figure 8.2. Schematic of the OCT experimental set-up (from Mitchell, 2002). 8.1.2. Material 8.1.2.1. Physical Characteristics Exact details on the material are not available, as it is classified as a commercial trade secret. What is known, however, is that the material consists of a mixture of standard and intermediate modulus carbon fibres embedded in a brittle epoxy (3501-6) matrix (G|C~90-120 mJ/mm2, O'Brien and Martin, 1993; Reeder, 1993). The laminates were manufactured by The Boeing Company using resin film infusement (RFI), in which a film of resin is drawn into dry fibres using a combination of heat, pressure, and vacuum. Some laminates were further reinforced with Kevlar® in the through-thickness direction. This through-thickness reinforcement is achieved by stitching the laminates in a manner similar to an industrial-sized sewing machine. Those laminates with this through-thickness stitching are called "stitched RFI" or S/RFI. The laminates without the stitching have been labelled "unstitched". Although experiments were performed on both types of laminates, the results were not significantly different. The S/RFI material was organized into laminates with a lay-up of [±45/02/90/02/±45]n and three thicknesses were tested, with n = 4, 5, and 6. In the references detailing the experiments, these thicknesses are also called 4-stack, 5-stack, and 6-stack. 214 8.1. Over-height Compact Tension Simulations The laminates were fabricated as large panels and the specimens were cut from the panels with a diamond saw. The notches were cut from the specimens using a band saw, resulting in a relatively blunt tip. Finally, the S/RFI was tested at two orientations; one where the loading was parallel to the principal material direction and another where the loading was perpendicular to the principal material direction. These two loadings were labelled 0° and 90° respectively, referring to the angle of the material's 0° plies with respect to the loading direction. The loading is shown schematically in Figure 8.3. 0" "0°" OCT o: Q: "90°" OCT Figure 8.3. Schematic of specimen orientations for OCT tests. The 0° material direction is coincident with the stitching direction, indicated by the dashed lines in the figure. The experimental studies showed that the 4-stack experiments and the 6-stack experiments scaled perfectly with thickness. The pin load divided by the specimen thickness was essentially the same in each case (allowing of course, for the statistical variations in experimental results). The 90° tests (with the applied load transverse to the material 0° direction) also scaled perfectly with thickness in all laminate thicknesses. This simplifies the simulations tremendously, requiring only simulations of either 4-stack or 6-stack in the 0°orientation, the 5-stack in the 0° orientation, and a simulation of the 90° orientation for any laminate thickness. As a result, three families of simulations were investigated: 4-stack 0° orientation ("4-0 ), 5-stack 0° orientation ("5-0 ), and 5-stack 90° orientation ("5-90 ). 215 8.1. Over-height Compact Tension Simulations V 8.1.2.2. Numerical Characterization Elastic material properties for the laminate were supplied by the manufacturer, and are presented in Table 8.2. The lamina properties were not supplied, but using typical properties for AS/3501 (Agarwal and Broutman, 1990), shown in Table 8.3, similar laminate properties were determined using LPT analysis (Table 8.4). Adopting these approximate lamina properties, the CODAM properties can be estimated. Table 8.2. S/RFI laminate properties supplied by The Boeing Company. Parameter Value E1 83.6 GPa E2 35.5 GPa G12 17.1 GPa vn 0.403 Table 8.3. AS/3501 lamina properties (Agarwal and Broutman, 1990). Parameter Value E i_ 138 GPa ET 8.96 GPa v. LT 0.3 GLT 7.1 G P a Table 8.4. LPT values for [±45/02/90/02/±45] AS/3501. Parameter Value E1 76.9 GPa E2 36.7 GPa G : 2 19.8 GPa v „ 0.385 Once again, the simplification of two damage modes, fibre damage and matrix damage, is adopted. The full laminate consists of n sublaminates, depending on the stack sequence considered. Accommodating the number of sublaminates is achieved by increasing the number of through-thickness integration points used in the shell formulations in the finite element analysis. Experience has shown, however, that no significant differences exist in the response of specimens analyzed with many (4, 5, 6) through-thickness integration points or few (1 or 2). Since the through-thickness 216 8.1. Over-height Compact Tension Simulations stresses are uniform, any other result would have been surprising. Therefore, in the interests of reducing the analysis times, only 2 through-thickness integration points were used for the majority of the simulations. The thought experiment used to determine the CODAM parameters for the material proceeds as follows: In the 0° direction (recall the lay-up is [±45/02/90/02/±45]): • The damage begins with matrix cracking in the centre 90° ply. • Next matrix cracks develop in the ±45° plies, allowing for some degree of scissoring of the fibres. • Matrix cracks may develop in the 0° plies, caused by constraints due to Poisson's effects and stress concentrations from the adjacent damaged ±45° and 90° plies. • Fibre damage develops in the 0° and ±45° plies, eventually leading to total fibre failure. Again, the fibre and matrix damage development is simplified into the initiation of matrix damage in the sublaminate, initiation of fibre damage, and finally, saturation of both damage modes. Determination of the CODAM parameters follows the same procedure as the other materials. First, the amount of damage corresponding to matrix damage saturation is estimated. With full matrix damage, the 90° ply is fully cracked, and the ±45° plies are partially damaged (for simplicity, the equivalent of half of the 45° plies - i.e., 2 - are assumed to be damaged): ^ damaged 3 / - . o o o OJmv = -— = — = 0.333 A 9 - "total v The normalized modulus is approximated using the ply discount method and the modulus reduction associated with the matrix damage reaching saturation is calculated. See Appendix C for an example on how the ply discount method is applied. In this case, the modulus reduction due to matrix saturation becomes: 217 8.1. Over-height Compact Tension Simulations F, = 1 — ^discounted _ <| _ 62.9 _ Q ^ g2 Eundamaged 76.9 In the 90° direction, the damage progression is similar, but the number of plies with 0° and 90° oriented fibres is different. As are result, the damage parameter and modulus reduction associated with matrix damage saturation are somewhat different. The damage at saturation becomes: ^damaged = &=QQQJ A q "total 3 And the modulus reduction becomes: F 1 *•> ft F, = 1 — discounted — "| _ _ Q gyg undamaged 0 0 • Damage affecting the shear modulus is handled in the same manner as the other carbon laminates, reducing the shear modulus linearly with damage: Gl2=C\-£t)l2)G°12 As in the cases for the UNT/SNT/DNT analyses, the effective strain functions F(s) have been chosen such that only the strain in the direction considered affects the damage parameter in that direction. The challenge is to determine the effective strains at the initiation and saturation of the two damage modes. The matrix in this material, 3501-6, has a low critical fracture toughness, G, c ~ 90-120 J/m2 (O'Brien and Martin, 1993; Reeder, 1993), yet also has a low strain to failure sult ~ 0.5%. Although the presence of the through-thickness stitches seems a likely cause for increasing the stress concentrations in the material, and perhaps lowering the damage initiation strain, experiments did not show any significant difference in the behaviour between the stitched and unstitched material in this application. While the exact fibre types are not known, the variation in fibre failure strain is not large, so a failure initiation strain similar to those used in the UNT/SNT/DNT applications is appropriate. 218 8.1. Over-height Compact Tension Simulations The saturation strains, once again assumed to be the same for fibre and matrix damage, should be different in the 0° and 90° directions, due to the number of plies with fibres in those directions. At damage saturation in the 0° direction, the strain in the RVE is assumed to be small, close to the actual strain in the constituents, due to the high percentage of 0° fibres present in the sublaminate. These 0° fibres constrain the opening of matrix cracks and the average strain in the RVE is similar to the strain in the fibres. The damage saturation strain is therefore close to the failure strain of the fibres. In the 90° direction however, the matrix cracks are assumed to open significantly since they are less constrained. This results in an average strain in the RVE that is larger than the actual strain in the constituents. Using this reasoning, the 90° direction saturation damage strains have been selected at values much larger than the 0° saturation damage strains. The CODAM parameters selected for the S/RFI material are shown in Table 8.5. These values have been labelled as "preliminary", since other sets of CODAM parameters will be generated later in the chapter. The resulting normalized modulus curves and stress-strain curves appear in Figure 8.4 and Figure 8.5 respectively. Table 8.5. CODAM parameters for S/RFI material (preliminary data set). Matrix Fibre 0.005 Ffj 0.016 0.020 Ffs 0.020 0.333 ars 0.667 0.182 E,s 0.818 Ffl 0.016 Ffs 0.040 fi>fe 0.333 Efs 0.421 0° dir Fni 0.005 F ms 0.040 ®ms 0.667 E ms 0.579 219 8.1. Over-height Compact Tension Simulations Figure 8.5. 1200 1000 _ 800 IB I 600 0.01 0.02 0.03 0.04 0.05 E (m/m) (a) 0.01 0.02 0.03 0.04 0.05 E (m/m) (b) Stress-strain curves for S/RFI material used in the OCT simulations for (a) 0° direction and (b) 90° direction (preliminary data set). Complicating matters somewhat is the foreknowledge of parameters that have adequately simulated the tests in past investigations (Starratt et al., 1999; Floyd et al., 2001b). These prior parameters were not developed using a rigorous procedure; instead, they were determined through a combination of engineering judgement and curve fitting. The results of this approach were parameters that produced stress-stain curves that were clearly non-physical, but still produced simulations that reasonably matched the experimental results. Figure 8.6 illustrates these parameters (henceforth "BSS") for the 5-stack S/RFI material. 220 8.1. Over-height Compact Tension Simulations Figure 8.6. 0.04 0.08 0.12 e (mm/mm) (a) 0.16 0.04 0.08 0.12 8 (mm/mm) (b) 0.16 Stress-strain relationships for 5-stack S/RFI material used in a previous study (Starratt et al., 1999) for the (a) 0° and (b) 90° orientations. Uses the "BSS" data set. These previously determined parameter sets are important. They were generated without any consideration of fracture energy, but since the results of the simulations were satisfactory, the energy dissipated by these elements does provide a guideline for estimating the proper energy parameters. The established procedure for estimating the fracture energy per unit area for the S/RFI sublaminates has been applied to get estimates for GF. Noting that there are four 0° plies, four ±45° plies, and one 90° ply, the upper bound estimate for the 0° orientation is: (GF)UB ~ 150 x ^  +150 x - + 5 x - * 134 mJ/mm2 9 9 9 The lower bound estimate is: / \ 4 4 1 \GF)LB * 1 5 0 x - + 5 x - + 5 x - « 6 9 m J / m m 2 9 9 9 The upper bound estimate for the 90° orientation is: {GF)UB * 150 x - +150 x - + 5 x - * 86mJ/mm2 v , U B 9 9 9 The lower bound estimate for the 90° orientation is: (GF)LB « 1 5 0 x l + 5 x - + 5 x - * 2 1 mJ/mm2 9 9 9 221 8.1. Over-height Compact Tension Simulations 8.1.2.3. Material Anomalies Further complicating the characterization of the material is the difference in the manufactured material quality. In the 4-stack and 6-stack material, the experimental behaviour was quite different from the behaviour of the 5-stack material. The 4- and 6-stack material had a damage mode characterized by a crack growing in the load direction, instead of along the notch mid-plane. This divergence in damage growth patterns is illustrated by Figure 8.7. Applied displacement^  4 4. 1 ii) Stitching (Parallel to load/applied displacement) Figure 8.7. Damage evolution in (i) the 4-stack and 6-stack S/RFI specimens, and (ii) the 5-stack specimens (From Mitchell, 2002). The reason for this difference in the damage development has been attributed to the quality of the stitching placement. In the 4- and 6-stack material, the stitches were observed to be "looser", with large loops of Kevlar evident on the backside of the material. In the 5-stack material, these stitching loops were much smaller, with less excess Kevlar. Mitchell theorized that the loose stitching impedes the damage development, since the Kevlar can be strained more as a result of the slack in the stitches. The 5-stack material had stitches that were effectively pre-strained and required very little additional energy to break the Kevlar fibres. The tight stitches were therefore less effective at constraining the damage development. Further details can be found in the references (Mitchell et al., 2001; Mitchell, 2002). 222 8.1. Over-height Compact Tension Simulations In the previous studies, this behaviour has been achieved in simulations by increasing the damage saturation strain to an extremely large value (e.g., 36%). In the present circumstance, the added damage toughness can be modelled by a simple increase in the work required fo cause complete facture. Using the previous study as a guideline, the fracture energy per unit area (GF) was increased from its lower bound of 69 mJ/mm2 (in the 0° direction) to 250 mJ/mm2. Table 8.6 shows the GF effectively used in the previous study compared to the estimated values determined above. These adjustments lead to a tendency to refer to the 4-stack material as "tough" and the 5-stack material as "brittle". Table 8.6. Comparison of GF values for the previous study ("BSS") and the estimates using the current procedure. "BSS" GF GFLB GFUB Specimen Orientation (mJ/mm2) (mJ/mm2) (mJ/mm2) 4-stack 0° 249 69 134 5-stack 0° 66 69 134 5-stack 90° 34 21 86 8.1.3. Simulations The OCT specimens were meshed with shell elements, with square elements near the notch tip and along the notch-plane. Two element sizes were used in this zone of interest, 0.625 mm and 1.25 mm. The finer mesh tended to be less stable, with significant hour-glassing evident in the preliminary analyses. As a consequence, fully-integrated elements were used in all the OCT simulations. The specimens were meshed completely, to allow damage to grow asymmetrically if needed. Figure 8.8 illustrates the coarse (1.25 mm) mesh. 223 8.1. Over-height Compact Tension Simulations 207 mm 59 mm notch (from edge, at mid-plane) 107 mm •lllllffllllllllllllllll il^fl^HHIL ^^^^MlllllBlllilll[llllllll|WM IIP zone of interest Figure 8.8 "Coarse" OCT mesh with 0.625 mm x 0.625 mm elements in the zone of interest. 8.1.3.1. Preliminary Simulations Past experience from simulating this material had demonstrated that the laminate stiffnesses provided by the manufacturer were not necessarily the same stiffnesses demonstrated by the experiments. To ensure that appropriate stiffnesses were used in the simulations, both the 0° and 90° orientations were simulated and the pin-load vs. crack mouth opening displacement (CMOD) results were compared. The elastic stiffnesses of the laminates were then calibrated to ensure that the elastic behaviour of the experiments and the simulations were comparable. This required adjusting the laminate stiffnesses such that E f = 65 GPa, and E2 = 31.4 GPa (compared to the original estimates of 76 GPa and 37 GPa, respectively). These calibrated stiffnesses compare well to the stiffnesses used in the previous studies. As a result of changing these stiffnesses, the CODAM parameters (the modulus reduction parameters) need to be changed to be consistent with the new stiffnesses. The same procedure outlined in 8.1.2.2. was followed, and the updated parameters (labelled as data set "OCT-A") appear in Table 8.7. Figure 8.9 and Figure 8.10 show the OCT-A data set parameters graphically. 224 8.1. Over-height Compact Tension Simulations Table 8.7. CODAM parameters for S/RFI material (data set OCT-A). Matrix Fibre 0° dir 0.005 Fn 0.016 • ms 0.020 Ffs 0.020 ms 0.333 ®fs 0.667 E ms 0.202 Efs 0.798 90° dir Fm 0.005 Ffi 0.016 F ms 0.040 Ffs 0.040 ms 0.667 0.333 Ems 0.620 Efs 0.380 0.00 0.000 Figure 8.9. 1.00 0.75 |UJ 0.50 0.25 0.00 0.010 0.020 0.030 0.040 0.000 0.010 0.020 0.030 0.040 F F (a) (b) Normalized modulus vs. effective strain for S/RFI material used in the OCT simulations for (a) 0° direction and (b) 90° direction (OCT-A data set). •*• Matrix -*• Fibre — Sys tem 0.01 0.02 0.03 0.04 0.05 s (m/m) (a) 0.01 0.02 0.03 0.04 0.05 E (m/m) (b) Figure 8.10. Stress-strain curves for S/RFI material used in the OCT simulations for (a) 0° direction and (b) 90° direction (OCT-A data set). The 90° orientation was simulated first. The lower bound estimate for GF (21 mJ/mm2) was used, but the peak force from the simulation was significantly lower than the 225 8.1. Over-height Compact Tension Simulations experimental failure force. Using the upper bound estimate resulted in a peak force significantly higher than the experimental value. Quickly it was discovered that using a GF = 29 mJ/mm2, a value between the upper and lower bound estimates and approaching the effective GF used in the BSS data set (34 mJ/mm2), satisfactorily simulated the 90° experiments. Figure 8.11 illustrates the crack growth by showing the zone of damage saturation (coloured red). Figure 8.12 shows the pin-load vs. CMOD results for the numerical simulation and all the 90° orientation experiments. The experiments were normalized to the 5-stack thickness for comparative purposes. The 90° simulations are generally insensitive to the 0° properties, so the following discussion does not affect the 90° simulations. Figure 8.11. Illustration of crack growth in the 5-stack 90° simulations. 226 8.1. Over-height Compact Tension Simulations C M O D (mm) Figure 8.12. Pin load vs. CMOD for 90° orientation. The 4-, 5-, and 6-stack experimental data have been normalized to the 5-stack thickness. The inset figure indicates the direction of crack growth. The 5-stack 0° simulations were run using the lower bound GF with good results. Figure 8.13 illustrates the crack growth in the 0° simulations. The pin force vs. CMOD (from now on, simply the force-displacement) results compared reasonably well to the experiments(shown in Figure 8.14). The peak force was approximately the same, but the post-peak behaviour was not; the experiments exhibited significant post-peak residual strength where the simulations failed abruptly. 227 8.1. Over-height Compact Tension Simulations Figure 8.13. Illustration of crack growth in the 5-stack 0° simulations. 50 CMOD (mm) Figure 8.14. Pin load vs. CMOD for 5-stack 0° orientation simulation using the OCT-A data set as well as two experimental results. The inset picture indicates the direction of crack growth. 228 8.1. Over-height Compact Tension Simulations The 4-stack 0° simulations were originally run at the upper bound GF, but the resulting damage mode was incorrect. Instead of growing damage in the direction of load as in the experiments, the damage grew along the notch mid-plane (like the 5-stack simulations). The GF was then increased to match that of the BSS data set, 250 mJ/mm2, but this was also insufficient to change the damage mode. Eventually, it was found that a GF of ~500 mJ/mm2 was required to change the damage modes to match the experiment. The final damage mode is illustrated in Figure 8.15. Having matched the damage mode, the load-displacement results (shown in Figure 8.16) provided a reasonable match with the experiment, exhibiting a somewhat larger peak force. Figure 8.15. Illustration of crack growth in the 4-stack 0° simulations. 229 8.1. Over-height Compact Tension Simulations 0 1 2 3 4 5 6 7 C M O D (mm) Figure 8.16. Pin load vs. CMOD for 4-stack 0° orientation simulation using the OCT-A data set as well as three experimental results. The inset picture indicates the direction of crack growth. 8.1.3.2. Refined Simulations The results of the preliminary simulations shared some characteristics of the experimental results, but were disappointing in some aspects. In the 90° simulations and the 5-stack 0° simulations, the post-peak force behaviour was not very similar to the experiments. In an effort to address this discrepancy, the CODAM parameters were revisited. The BSS data set used damage initiation strains that were significantly less than those determined in this work. In particular, the matrix damage initiation strain was taken to be 0.1% and the fibre damage initiation strain was taken to be 0.4%. The fibre damage initiation strain is smaller than what seems possible physically, however. Using these values as a guideline, a second set of CODAM parameters was created. First, the matrix damage initiation strain was lowered to 0.001. After running the 5-stack 0° simulation with this change, little difference was observed in the force-displacement results. Next, the fibre damage initiation strain was lowered to 0.010, while restoring the matrix damage initiation strain to its original value of 0.05. Running the 5-stack 0° 230 Chapter 8: Simulations of Notched Tension Tests: S/RFI Laminates 0 1 2 3 4 5 6 7 CMOD (mm) (a) 0 1 2 3 4 5 6 7 CMOD (mm) (b) Figure 8.17. 0° simulations showing progression of the reduction of damage initiation strains for (a) 4-stack laminates and (b) 5-stack laminates. OCT-A is the original data set. 231 8.1. Over-height Compact Tension Simulations simulation with this second change produced much more satisfactory results. The 4-stack 0° simulations were run with this new data set and once again, the damage growth was incorrectly along the notch mid-plane. GF was varied from 250 mJ/mm2 to 750 mJ/mm2, and it was only at this extreme value of GF that the damage mode changed. Figure 8.17 shows the load-displacement results for these iterations. When both damage initiation values were reduced, the analyses produced satisfactory results, and details of these results follow. The data set with both damage initiation values reduced ("OCT-B") are shown in Table 8.8. The normalized modulus and stress-strain curves for this data set are shown in Figure 8.18. Table 8.9 compares the OCT-A values, the OCT-B values and the previous study (BSS) values. Table 8.8. Refined 0° CODAM parameters for the S/RFI material ("OCT-B"). Matrix Fibre 0° dir Fni 0.001 Fn 0.010 Fns 0.020 Ffs 0.020 « m s 0.333 0.667 E ms 0.202 Efs 0.798 0.04 0.05 Figure 8.18. (a) Normalized modulus curve and (b) stress-strain curve for OCT-B CODAM parameters. 232 8.1. Over-height Compact Tension Simulations Table 8.9. Comparison of OCT-A, OCT-B, and BSS data sets for S/RFI CODAM parameters. Data set Specimen Orientation Fmi Fms Ffi F f . Peak stress (MPa) G F (mJ/mm2) OCT-A 4-stack 0° 0.005 0.020 0.016 0.020 892 500 5-stack 0° 0.005 0.020 0.016 0.020 892 69 5-stack 90° 0.005 0.040 0.016 0.040 431 29 OCT-B 4-stack 0° 0.001 0.020 0.010 0.020 588 250 5-stack 0° 0.001 0.020 0.010 0.020 588 69 BSS 4-stack 0° 0.012 0.348 0.004 0.029 980 249 5-stack 0° 0.001 0.028 0.004 0.114 691 66 5-stack 90° 0.013 0.032 0.019 0.065 440 34 The OCT-B data set changes only the damage initiation strains from the OCT-A data set and only in the 0° direction. Additionally, the fracture energy per unit area (GF) for the OCT-B data set has been changed to match the BSS data set in the case of the 4-stack 0° simulation. Note that the 90° analyses were revisited, but changing the "x" (or, perpendicular to the load) CODAM parameters to the OCT-B data set did not significantly change the results. The results for the 0° simulations using the OCT-A and OCT-B data sets are shown in the section that follows. 8.1.3.3. Pin Load vs. CMOD The loading pin forces as a function of CMOD have been examined for each specimen and orientation. The results for the 4-stack and 5-stack 0° simulations are shown in Figure 8.19 and Figure 8.20 respectively. 233 8.1. Over-height Compact Tension Simulations 0 1 2 3 4 5 6 7 CMOD (mm) Figure 8.19. Pin load vs. CMOD for 4-stack 0° orientation simulations using the OCT-A and OCT-B data sets as well as three experimental results. The inset picture indicates the direction of crack growth. 50 CMOD (mm) Figure 8.20. Pin load vs. CMOD for 5-stack 0° orientation simulations using the OCT-A and OCT-B data sets as well as two experimental results. The inset picture indicates the direction of crack growth. 234 8.1. Over-height Compact Tension Simulations In all cases, the OCT-B simulations do a better job at capturing the peak pin load and the shape of the post-peak curve. 8.1.3.4. Virtual Line Analysis Another method of examining the damage growth in the OCT specimens is by performing a line analysis. As described in §8.1.1. a line analysis is the measurement of the relative displacement of inscribed lines on the OCT specimen. This effectively measures the local displacement field during the crack growth. Experimentally, this is achieved by photographing the specimen during the test. Photographs are correlated to CMOD measurements so that the displacement field can be mapped to the pin load vs. CMOD curve. Numerically, line analyses are performed in the same manner. The displacement of nodes that correspond to the inscribed lines is measured at CMOD values matching the experimental photographs. In the analyses that follow, the numbers in the inset pin-load vs. CMOD plots correspond to the photograph number indicated in the experiments (Mitchell, 2002). Position ahead of notch tip (mm) Figure 8.21. Numerical (OCT-A) and experimental line analysis for the 4-stack 0° OCT test. The solid markers (numerical) correspond to the open markers (experiment), and each marker type corresponds to a different CMOD as indicated in the inset figure. 235 8.1. Over-height Compact Tension Simulations 1 0.9 0.8 Figure 8.22. Edge of specimen A1 I -5 0 5 10 15 20 25 30 35 40 45 50 Position ahead of notch tip (mm) Numerical (OCT-B) and experimental line analysis for the 4-stack 0° OCT test. The solid markers (numerical) correspond to the open markers (experiment), and each marker type corresponds to a different CMOD as indicated in the inset figure. The 4-stack simulations (with a crack that grows in the direction of load) do not show much difference in the line analyses when comparing the OCT-A (Figure 8.21) and OCT-B (Figure 8.22) data sets. Generally, the line analyses performed at CMODs in the elastic regime match the experimental analyses quite well, with the exception that the crossing-point of the lines is beyond the experimental crossing-point. This crossing-point is generally interpreted as the location of the leading edge of the damage zone. The single numerical line analysis at a significantly damaged state (solid squares) does not match the experimental line analysis (hollow squares) in either slope or crossing point. This indicates that the crack has grown further along the notch mid-plane in the experiment and that the crack (in this case, the vertical crack) has opened more than the simulation predicts. 236 8.1. Over-height Compact Tension Simulations -5 0 5 10 15 20 25 30 35 40 45 50 Position ahead of notch tip (mm) Figure 8.23. Numerical (OCT-A) and experimental line analysis for the 5-stack 90° OCT test. The solid markers (numerical) correspond to the open markers (experiment), and each marker type corresponds to a different CMOD as indicated in the inset figure. The line analyses of the simulations of the 5-stack, 90° orientation tests (Figure 8.23) match the experiments quite well. The triangular markers in the analysis correspond to a load drop in the numerical analysis but not in the experiments (photo 13). This is mirrored in the line analysis, where the crack has advanced more in the numerical simulation than in the experiment as determined by extrapolating the line analysis to the 0-displacement crossing point. 237 8.1. Over-height Compact Tension Simulations 5 10 15 20 25 30 35 40 Position ahead of notch tip (mm) 45 50 Figure 8.24. Numerical (OCT-A) and experimental line analysis for the 5-stack 0° OCT test. The solid markers (numerical) correspond to the open markers (experiment), and each marker type corresponds to a different CMOD as indicated in the inset figure. -5 0 5 10 15 20 25 30 35 40 45 50 Position ahead of notch tip (mm) Figure 8.25. Numerical (OCT-B) and experimental line analysis for the 5-stack 0° OCT test. The solid markers (numerical) correspond to the open markers (experiment), and each marker type corresponds to a different CMOD as indicated in the inset figure. 238 8.1. Over-height Compact Tension Simulations The line analyses of the 5-stack 0° tests show the clear advantage of the OCT-B (Figure 8.24) data set over the OCT-A (Figure 8.25) data set. In the OCT-A line analysis, with the exception of the single analysis in the elastic regime, the simulations do not correspond at all with the experimental analyses. In the OCT-B analyses however, the simulations are much more accurate than the OCT-A results. The analysis corresponding to photo 9 (in Figure 8.25) matches the experimental analysis very well, as does the analysis at the next load level (corresponding to photo 19). The remaining analyses capture the line crossing points but do not show the same slopes, indicating that the crack opened up more experimentally than predicted by the simulation. Of particular interest is the degree to which the simulations match the experiments, especially in demonstrating the same "kink" in the lines. Experimentally, this kink has been explained as being the manner of determining the location of the crack front and the leading edge of the damage zone. Extending the line formed from the data closer to the crack notch ("before" the kink) to the x-axis provides an estimate of the location of the crack tip while extending the line formed from the data closer to the specimen back-face ("after" the kink) provides an estimate of the location of the damage front. An interesting test of the value of line analysis is to examine the relative difference between the 4-stack, 0° direction line analysis and the 5-stack, 0° direction analysis. The two simulations predict very different damage growth, with the 4-stack exhibiting damage growth in the direction of load ("vertical" crack growth) and the 5-stack exhibiting crack growth in the notch-plane ("horizontal" crack growth). Using the simulations, the relative displacement of the same nodes can be compared between the two damage modes. In Figure 8.26, line analyses of the two simulations are shown for the y = ±10mm inscribed line pair. Each line represents a different CMOD, and the open symbols (5-stack) correspond to the closed symbols (4-stack). The figure shows that the analyses are very similar in the early stages of the simulation; this is expected since the material remains in the elastic regime. After damage initiation, the 4-stack analysis does not predict as much vertical displacement as the 5-stack, which can be interpreted as the crack not opening up as much in the 4-stack simulations. Again, this is expected behaviour, since a crack does not develop very far in the horizontal direction in the 4-stack simulations. An additional feature of the 4-stack analysis is the reversal in the progression of the lines as the CMOD increases. This is evident in the last two analyses, where the line marked by stars has less displacement than the line marked by 239 8.1. Over-height Compact Tension Simulations circles, despite occurring at a greater CMOD. The 5-stack progresses as expected, with the horizontal crack opening with increasing CMOD, but the horizontal (notch-plane) crack closes after a certain point in the 4-stack simulations with increasing CMOD. Position ahead of notch tip (mm) Figure 8.26. Comparison of the line analysis of the 4-stack and 5-stack simulations of the 0° tests, at the y = 10 mm inscribed line pair. The open symbols are the 5-stack simulation, and the closed symbols are the 4-stack simulation. Other than these trends, the line analyses of the two very different damage modes are quite similar. The magnitudes of displacement and the zero-crossing points of each of the lines are not very different despite the very different damage modes. Corresponding experimental line analysis of the same line set in the two damage modes is not available for comparison. 8.1.3.5. Mesh Size Effect The effect of mesh size has been investigated in some of the OCT simulations. Figure 8.27 shows the load-displacement results for the 90° OCT simulations, while Figure 8.28 illustrates the mesh size effect for 0° simulations using (a) the lower bound GF estimate and (b) the upper bound GF estimate. The figures show plots for the coarse and fine 240 8.1. Over-height Compact Tension Simulations meshes, using the properly scaled parameters, as well as the coarse mesh us