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A physically-based continuum damage mechanics model for numerical prediction of damage growth in laminated… Williams, Kevin V. 1998

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A P H Y S I C A L L Y - B A S E D CONTINUUM D A M A G E MECHANICS M O D E L FOR N U M E R I C A L PREDICTION OF D A M A G E GROWTH IN L A M I N A T E D COMPOSITE PLATES by K E V I N V. WILLIAMS B.Eng., Carleton University, 1991 M.Eng., Carleton University, 1994 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES Department of Metals and Materials Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A December 1998 © Kevin V . Williams, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mg.*t<»As ^-cC^€.r\cSp B^i^eer!v\j The University of British Columbia Vancouver, Canada Date KJ avX? DE-6 (2/88) Abstract Rapid growth in use of composite materials in structural applications drives the need for a more detailed understanding of damage tolerant and damage resistant design. Current analytical techniques provide sufficient understanding and predictive capabilities for application in preliminary design, but current numerical models applicable to composites are few and far between and their development into well tested, rigorous material models is currently one of the most challenging fields in composite materials. The present work focuses on the development, implementation, and verification of a plane-stress continuum damage mechanics based model for composite materials. A physical treatment of damage growth based on the extensive body of experimental literature on the subject is combined with the mathematical rigour of a continuum damage mechanics description to form the foundation of the model. The model has been implemented in the LS-DYNA3D commercial finite element hydrocode and the results of the application of the model are shown to be physically meaningful and accurate. Furthermore it is demonstrated that the material characterization parameters can be extracted from the results of standard test methodologies for which a large body of published data already exists for many materials. Two case studies are undertaken to verify the model by comparison with measured experimental data. The first series of analyses demonstrate the ability of the model to predict the extent and growth of damage in T800/3900-2 carbon fibre reinforced polymer (CFRP) plates subjected to normal impacts over a range of impact energy levels. The predicted force-time and force-displacement response of the panels compare well with experimental measurements. The damage growth and stiffness reduction properties of the T800/3900-2 CFRP are derived using ii published data from a variety of sources without the need for parametric studies. To further demonstrate the physical nature of the model, a IM6/937 CFRP with a more brittle matrix system than 3900-2 is also analysed. Results of analyses performed under the same impact conditions do not compare as well quantitatively with measurements but the results are still promising and qualitative differences between the T800/3900-2 and IM6/937 are accurately captured. Finally, to further demonstrate the capability of the model, the response of a notched CFRP plate under quasi-static tensile loading is simulated and compared to experimental measurements. Of particular significance is the fact that the experimental test modelled in this case is uniquely suited to the characterization of the strain softening phenomenon observed in FRP laminates. Results of this virtual experiment compare very favourably with the measured damage growth and force-displacement curves. i i i Table of Contents Page ABSTRACT ii TABLE OF CONTENTS iv LIST OF TABLES vii LIST OF FIGURES viii NOMENCLATURE xvi ACKNOWLEDGEMENTS xix CHAPTER 1. INTRODUCTION 1 1.1 Low Velocity and Ballistic Impacts on Composite Structures 1 1.2 Research Methodologies 2 1.3 Constitutive Modelling 3 1.4 Research Objectives and Thesis Outline 5 CHAPTER 2. BACKGROUND 11 2.1 Damage in Composite materials 11 2.1.1 Damage Mechanisms 11 2.1.2 Effects of Damage on Laminate Response 14 2.1.3 Impact Damage 17 2.2 Analysis of Dynamically Loaded Composite Structures 19 2.2.1 Model Scale 19 2.2.2 Analytical Models 20 2.2.3 Numerical Models 21 2.3 Constitutive Modelling of FRP Composite Laminates 22 2.3.1 Traditional Approach 23 2.3.2 Damage Initiation 24 2.3.3 Effects of Damage on Material Behaviour 25 2.4 Continuum Damage Mechanics 27 2.4.1 Introduction to Continuum Damage Mechanics 27 2.4.2 CDM Models for Laminated Composite Materials 30 2.4.3 Numerical Problems Associated with CDM Models 34 iv 2.4.4 CDM Summary 36 2.5 Application of FEM to the Prediction of Damage in FRP Laminates 37 2.6 Summary 41 CHAPTER 3. THEORETICAL MODEL DEVELOPMENT 50 3.1 Introduction 50 3.2 Preliminary Work 51 3.3 Model Formulation 53 3.3.1 Sub-Structuring 54 3.3.2 Selection of Damage Parameters 55 3.3.3 Damage Growth Law 58 3.3.4 Damage Potential Function 60 3.3.5 Effect of Damage on Material Properties 63 3.3.6 Predicted Constitutive Relationship 67 3.3.7 Predicted ID Response 68 3.4 Computer Implementation 71 3.5 Material Characterization 72 3.5.1 Elastic and Strength Constants 73 3.5.2 Stiffness Reduction Functions 73 3.5.3 Damage Growth Functions 80 3.5.4 Effective Strain Functions 84 3.6 Post-Processing 86 3.7 Conclusion 88 CHAPTER 4. NUMERICAL CASE STUDIES 106 4.1 Case Study 1: Non-Penetrating Impact on CFRP Plates 106 4.1.1 Experimental Measurements 106 4.1.2 Objective 109 4.1.3 FEM Model 109 4.1.4 Interpretation of Results 112 4.1.5 T800H/3900-2, a Tough Matrix System 114 4.1.6 Parametric Studies 132 4.1.7IM6/937, a Brittle Matrix System 137 4.1.8 Case Study 1 Summary 144 4.2 Case Study 2: Prediction of Force-Displacement Response and Damage Growth in an OCT Specimen 145 4.2.1 Experimental Measurements 145 4.2.2 Objective 149 4.2.3 FEM Model 150 4.2.4 Material Characterization 152 V 7 4.2.5 Model Predictions 153 4.2.6 Case Study 2 Summary 158 4.3 Summary 159 CHAPTER 5. CONCLUSIONS A N D F U T U R E W O R K 234 5.1 Conclusions 234 5.2 Future Work 235 REFERENCES 238 APPENDIX A. A C D M M O D E L APPLICABLE T O THIN L A M I N A T E D STRUCTURES 249 A . l Outline of Model 249 A.l.I Selection of Damage Variables 250 A. 1.2 Effective Stress-Strain Relationship 251 A. 1.3 Damage Thresholds 252 A.l.4 Damage Growth Law 254 A.2 Application and Results 255 A.2.1 Material Driver Tests 255 A.2.2 Comparison with Experimental Data 256 A.2.3 Mesh Sensitivity 259 A.2.4 Rate sensitivity 260 A. 3 Summary and Conclusion 261 APPENDIX B. UBC-CODAM3DS M O D E L R E F E R E N C E 272 B. l LS-DYNA3D UMAT47 Input Deck Description 272 B.2 UBC-CODAM3Ds Supplementary Data File Format 273 B.3 History Variable Output 274 B.4 Damage Summary File 275 vi List of Tables Page Table 2.1 Comparison of methodologies of damage modelling 43 Table 4.1 Elastic properties for steel 162 Table 4.2 Characterization data for the T800H73900-2 plates 162 Table 4.3 UBC-CODAM3Ds damage model characterization for T800H/3900-2 163 Table 4.4 Characterization data for the IM6/937 plates 164 Table 4.5 UBC-CODAM3Ds damage model characterization for IM6/937 164 Table 4.6 Overheight compact tension (OCT) specimen dimensions 165 Table 4.7 Characterization data for T3007F593 165 Table 4.8 Elastic properties for T3007F593 from Engels (1996) 166 Table 4.9 UBC-CODAM3Ds damage model characterization for T300H/F593 166 Table B. 1 LS-DYNA3D User Material Model definition constants for the CODAM3Ds composite damage model 277 Table B.2 CODAM3Ds User Material Model input deck parameters 278 Table B.3 CODM3Ds User Material Model supplementary data file (matin.dat) input format reference 279 Table B .4 CODAM3Ds User Material Model element history variable list 282 vii List of Figures Page Figure 1.1 Material representations 9 Figure 1.2 Solution routine for an explicit finite element method 9 Figure 1.3 Comparison of (a) traditional elastic-brittle and (b) observed damaged material response characteristic of laminated FRPs 10 Figure 2.1 Intralaminar matrix cracking in a cross-ply laminate 44 Figure 2.2 Intralaminar damage modes in FRP composites showing effect of fibre-matrix interface strength on fibre failure mode 45 Figure 2.3 Interlaminar damage growth in FRP composites 46 Figure 2.4 Initiation and growth of a delamination from an intralaminar matrix crack in an angle-ply laminate 47 Figure 2.5 Characteristic pattern of delamination and fibre failure in an angle-ply laminate subjected to non-penetrating through-thickness static or dynamic loading 47 Figure 2.6 Comparative scale of various modelling approaches 48 Figure 2.7 Applicable integration techniques as a function of strain-rate 49 Figure 3.1 Hypothesis of stress and strain equivalence 90 Figure 3.2 Model development approach 91 Figure 3.3 Relative merits of modelling approaches 91 Figure 3.4 Shell representation of a laminated composite using a lamina and sublaminate based constitutive model 92 Figure 3.5 Construction of the bilinear damage growth law (c) based on the combined contributions of (a) matrix and delamination damage and (b) fibre breakage 93 Figure 3.6 Modulus variation predicted by (a) linear and (b) bilinear normalized residual stiffness functions 94 Figure 3.7 Bilinear residual stiffness curve showing modulus loss (EI E°) versus damage (co) relationship showing (a) generalized function with a residual modulus and (b) special case with saturation of damage at rupture (com - 1 and Em = 0) 94 Figure 3.8 Predicted one-dimensional stress-strain response based on bilinear damage growth and residual stiffness functions 95 Figure 3.9 Examples of one-dimensional predicted stress-strain response for the proposed composite damage model 96 Figure 3.10 Effect of changing the damage associated with saturation of matrix cracking on the residual stiffness and damage growth functions 97 viii Figure 3.11 COD A M solution algorithm 98 Figure 3.12 Experimentally measured and predicted normalized residual stiffness for (a) a [903/0]s GFRP and (b) a [45/90/-45/90/45/90/-45/90/90]s CFRP (Talreja, 1985b). Calculated values are based on a continuum damage mechanics approach developed by Talreja (1985a) 99 Figure 3.13 Experimentally measured normalized residual stiffness in fatigue loaded quasi-isotropic T300/5208 CFRP laminates. The results in (a) from Bakis and Stinchcomb (1986) show the effect of compression (solid line) and tension (dashed line) dominated loading on the stiffness loss in a [0/45/90/-45]4s plate, (b) from Kress and Stinchcomb (1985) shows the effect of material lay-up 100 Figure 3.14 Measured in-plane normalized residual stiffness with delamination size for a [45/90/-45/0]s CFRP loaded in tension-tension fatigue. Also shown is the good agreement with an empirical rule of mixtures based stiffness reduction rule 101 Figure 3.15 Strain for onset of matrix cracking ( £ F R P ) as a function of number of 90° plies for a [04/90Js CFRP 101 Figure 3.16 Measured crack density as a function of (a) applied load for a [0 /±45 /90 ] s CFRP from Reifsnider and Highsmith (1981) and (b) applied strain for a [60/90/-60/90/60/90/-60/90/90]s AS4/3502 CFRP from Talreja (1985b) 102 Figure 3.17 O C T specimen used by Kongshavn and Poursartip showing damage growth region and tensile specimens used to characterize properties of damaged material 103 Figure 3.18 O C T tensile specimen results showing (a) undamaged and damaged stress-strain curves and (b) predicted softening behaviour from the locus of failure points for two CFRP systems 104 Figure 3.19 Portion of damage attributed to matrix cracking/delamination (0Jm) and to fibre breakage (CO j) for a given damage state (CO) 105 Figure 4.1 Impact test specimen geometry 167 Figure 4.2 Post test C-scan measurements of delamination growth for (a) a 30 J high mass impact on a T800/3900-2 tough resin CFRP panel and (b) a 33 J low mass impact on an IM6/937 brittle resin CFRP panel, (a) illustrates the method of calculating the projected area of delamination from the image and (b) highlights the characteristic spiral staircase pattern of delaminations between neighbouring plies 168 Figure 4.3 Example of a deplied lamina showing the fibre breakage area measurement technique 168 Figure 4.4 Exploded view of the F E M model of the CAI impact specimen with the assembled mesh . The 960 element target mesh, corresponding to a 3.175x3.175 mm element size, is shown here 169 Figure 4.5 Experimental measurements of fibre breakage area as a function of total delamination from quasi-static indentation and non-penetrating impacts of (a) T800/3900-2 and (b) IM6/937 CFRP coupons 170 ix Figure 4.6 T800H/3900-2 (a) damage growth function and (b) normalized residual stiffness function 171 Figure 4.7 Predicted one-dimensional stress-strain response for T800H73900-2 171 Figure 4.8 Comparison of experimentally measured and predicted total delamination area as a function of incident energy for a [45/90/-45/0]3S T800H/3900-2 CFRP plate 172 Figure 4.9 Comparison of experimentally measured and predicted total fibre breakage as a function of incident energy for a [45/907-45/0]3S T80OH/3900-2 CFRP plate 172 Figure 4.10 Comparison of experimentally measured and predicted total delamination area as a function of impactor energy loss for a [45/90/-45/0]3S T800H/3900-2 CFRP plate 173 Figure 4.11 Comparison of experimentally measured and fibre breakage as a function of impactor energy loss for a [45/90/-45/0]3S T800H/3900-2 CFRP plate 173 Figure 4.12 Comparison of experimentally measured and predicted energy loss as a function of impact energy for a [45/90/-45/0]3S T800H73900-2 CFRP plate 174 Figure 4.13 Prediction of total delamination area as a function of projectile energy loss from a single F E M ran with an incident energy of 60 J (high and low mass events). Results shown are for a [45/90/-45/0]3S T800H/3900-2 CFRP plate 175 Figure 4.14 Prediction of total fibre breakage as a function of projectile energy loss from a single F E M run with an incident energy of 60 J (high and low mass events). Results shown are for a [45/90/-45/0]3S T800H/3900-2 CFRP plate 175 Figure 4.15 Total system energy as a function of impact energy for a [45/90/-45/0]3s T800H/3900-2 CFRP plate. The numerical results of energy absorbed in matrix and fibre damage (Ematrix and Efib r e , respectively) are based on the F E M damage area predictions and experimental measurements of energy absorption for each type of damage 176 Figure 4.16 Comparison of experimentally measured and predicted projected delamination size as a function of impact energy for a [45/90/-45/0]3S T800H/3900-2 CFRP plate 177 Figure 4.17 Relationship between the experimentally measured ratio of the total delamination area to projected delamination sizes (A t o t a i /A p r 0 jected) and impact energy. The ratios were obtained from micrographical analyses (total area) and C-scans (projected area) of the T800/3900-2 and IM6/937 impact results of Delfosse (1994) 177 Figure 4.18 Comparison of experimentally measured and predicted total fibre breakage as a function of delamination area for a [45/90/-45/0]3S T800H/3900-2 CFRP plate 178 Figure 4.19 Predicted projected average matrix/delamination damage distributions for impact energies of (a) 30 J and (b) 33 J and (c) 35 J. Results shown are for low mass (high velocity) impacts on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 179 Figure 4.20 Predicted damage state for a high mass 46.2 J (v = 3.82 m/s, m = 6330 g) impact on a [45/90/-45/0]3S T800H/3900-2 CFRP plate showing the variation in the predicted averaged damage, (ft)] + 6)2)/^ t n r o u g h t n e thickness of the plate 180 x Figure 4.21 Comparison of predicted projected matrix/delamination damage and experimental C-scan images for low mass impact events on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. Results presented are for (a) 9.4 J (v = 7.74 m/s, m = 314 g), (b) 22.0 J (v =11.84 m/s, m = 314 g), (c) 33.4 J (v =14.59 m/s, m = 314 g), and (d) 56.4 J (v = 18.97 m/s, m = 314 g) impacts 181 Figure 4.22 Comparison of predicted projected matrix/delamination damage and experimental C-scan images for high mass impact events on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. Results presented are for (a) 9.5 J (v = 1.76 m/s, m = 6141 g), (d) 46.2 J (v = 3.82 m/s, m = 6330 g), (c) 58.2 J (v = 4.29 m/s, m = 6330 g) impacts 182 Figure 4.23 Comparison of predicted and measured force-time histories for high mass, low velocity (drop weight) events on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 183 Figure 4.24 Comparison of predicted and measured force-displacement histories for high mass, low velocity (drop weight) events on a[45/90/-45/0]3S T800H/3900-2 CFRP plate 184 Figure 4.25 Comparison of predicted and measured energy-time histories for high mass, low velocity (drop weight) events on a [4590/-45/0]3S T800H/3900-2 CFRP plate 185 Figure 4.26 Comparison of predicted and measured force-time histories for low mass, high velocity (gas gun) events on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 186 Figure 4.27 Comparison of predicted and measured force-displacement histories for low mass, high velocity (gas gun) events on a[4590/-45/0]3S T800H/3900-2 CFRP plate 187 Figure 4.28 Comparison of predicted and measured energy-time histories for low mass, high velocity (gas gun) events on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 188 Figure 4.29 Comparison of predicted and measured force-time histories for a low mass 9.5 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 189 Figure 4.30 Comparison of predicted and measured energy-time histories for a low mass 9.5 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 189 Figure 4.31 Comparison of predicted and measured force-displacement histories for a low mass 9.5 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 190 Figure 4.32 Comparison of predicted and measured force-time histories for a low mass 22.0 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 190 Figure 4.33 Comparison of predicted and measured energy-time histories for a low mass 22.0 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 191 Figure 4.34 Comparison of predicted and measured force-displacement histories for a low mass 22.0 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 191 Figure 4.35 Comparison of predicted and measured force-time histories for a low mass 33.4 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 192 Figure 4.36 Comparison of predicted and measured energy-time histories for a low mass 33.4 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 192 xi Figure 4.37 Comparison of predicted and measured force-displacement histories for a low mass 33.4 J event on a [45/907-45/0]3S T800H/3900-2 CFRP plate 193 Figure 4.38 Comparison of predicted and measured force-time histories for a low mass 56.5 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 193 Figure 4.39 Comparison of predicted and measured energy-time histories for a low mass 56.5 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 194 Figure 4.40 Comparison of predicted and measured force-displacement histories for a low mass 56.5 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 194 Figure 4.41 Comparison of predicted and measured force-time histories for a low mass 84.4 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 195 Figure 4.42 Comparison of predicted and measured energy-time histories for a low mass 84.4 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 195 Figure 4.43 Comparison of predicted and measured force-displacement histories for a low mass 84.4 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 196 Figure 4.44 Comparison of predicted and measured force-time histories for a high mass 34.5 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 196 Figure 4.45 Comparison of predicted and measured energy-time histories for a high mass 34.5 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 197 Figure 4.46 Comparison of predicted and measured force-displacement histories for a high mass 34.5 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 197 Figure 4.47 Comparison of predicted and measured force-time histories for a high mass 58.2 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 198 Figure 4.48 Comparison of predicted and measured energy-time histories for a high mass 58.2 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 198 Figure 4.49 Comparison of predicted and measured force-displacement histories for a high mass 58.2 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 199 Figure 4.50 Effect of variations in the damage at saturation of matrix cracking ((o'm) on the predicted damage growth versus energy relationships for a normally impacted [45/90/-45/0]3S T800H/3900-2 CFRP plate 200 Figure 4.51 Effect of variations in the modulus at matrix damage saturation (R'm) on the predicted damage growth versus energy relationships for a normally impacted [45/90/-45/0]3S T800H/3900-2 CFRP plate 201 Figure 4.52 Effect of variations in the damage potential at onset of matrix cracking ( F 1 ) on the predicted damage growth versus energy relationships for a normally impacted [45/90/-45/0]3S T800H/3900-2 CFRP plate 202 Figure 4.53 Effect of variations in the damage potential at onset of fibre breakage ( F n ) on the predicted damage growth versus energy relationships for a normally impacted [45/90/-45/0]3S T800H/3900-2 CFRP plate 203 xii Figure 4.54 Effect of variations in the damage potential at rupture (F ) on the predicted damage growth versus energy relationships for a normally impacted [45/907-45/0]3s T800H/3900-2 CFRP plate 204 Figure 4.55 Effect of variations in the damage potential function in-plane shear strain scale factor (S) on the predicted damage growth versus energy relationships for a normally impacted [45/90/-45/0]3S T800H/3900-2 CFRP plate 205 Figure 4.56 Effect of variations in the mesh density on the predicted damage growth versus energy relationships for a normally impacted [45/90/-45/0]3S T800H/3900-2 CFRP plate 206 Figure 4.57 Effect of the damage growth function assumptions on the damage growth and energy loss of a normally impacted [45/90/-45/0]3S T800H73900-2 CFRP plate 207 Figure 4.58 TM6/937 (a) damage growth function and (b) normalized residual stiffness function 208 Figure 4.59 Predicted one-dimensional stress-strain response for IM6/937 208 Figure 4.60 Comparison of experimentally measured and predicted total delamination area as a function of incident energy for a [45/0/-45/90]3S IM6/937 CFRP plate 209 Figure 4.61 Comparison of experimentally measured and predicted total fibre breakage as a function of incident energy for a [45/0/-45/90]3S EVI6/937 CFRP plate 209 Figure 4.62 Comparison of experimentally measured and predicted total fibre breakage as a function of delamination area for a [45/0/-45/90]3S IM6/937 CFRP plate 210 Figure 4.63 Comparison of experimentally measured and predicted total delamination area as a function of impactor energy loss for a [45/0/-45/90]3S IM6/937 CFRP plate 210 Figure 4.64 Comparison of experimentally measured and fibre breakage as a function of impactor energy loss for a [45/0/-45/90]3S IM6/937 CFRP plate 211 Figure 4.65 Comparison of experimentally measured and predicted energy loss as a function of impact energy for a [45/0/-45/90]3S IM6/937 CFRP plate 211 Figure 4.66 Total system energy as a function of impact energy for a [45/0/-45/90]3S LM6/937 CFRP plate 212 Figure 4.67 Comparison of predicted projected damage and experimental C-scan images for low mass impact events on a [45/0/-45/90]3S IM6/937 CFRP plate. Results presented are for (a) 10.1 J (v = 8.01 m/s, m = 314 g), (b) 22.6 J (v = 12.00 m/s, m = 314 g), (c) 33.8 J (v = 14.67 m/s, m = 314 g), and (d) 43.6 J (v = 16.66 m/s, m = 314 g) impacts 213 Figure 4.68 Comparison of predicted projected damage and experimental C-scan images for high mass impact events on a [45/0/-45/90]3S IM6/937 CFRP plate. Results presented are for (a) 11.0 J (v = 1.89 m/s, m = 6141 g), (b) 29.6 J (v = 3.10 m/s, m = 6141 g), (c) 33.7 J (v = 3.31 m/s, m = 6141 g), impacts 214 Figure 4.69 Comparison of predicted and measured force-time histories for high mass, low velocity (drop weight impact) events on a [45/0/-45/90]3S M6/937 CFRP plate 215 xiii Figure 4.70 Comparison of predicted and measured force-displacement histories for high mass, low velocity (drop weight impact) events on a[45/0/-45/90]3S EVI6/937 CFRP plate 216 Figure 4.71 Comparison of predicted and measured energy-time histories for high mass, low velocity (drop weight impact) events on a [45/0/-45/90]3S IM6/937 CFRP plate 217 Figure 4.72 Comparison of predicted and measured force-time histories for low mass, high velocity (gas gun) events on a [45/0/-45/90]3S IM6/937 CFRP plate 218 Figure 4.73 Comparison of predicted and measured force-displacement histories for low mass, high velocity (gas gun) events on a[45/0/-45/90]3S IM6/937 CFRP plate 219 Figure 4.74 Comparison of predicted and measured energy-time histories for low mass, high velocity (gas gun) events on a [45/0/-45/90]3S EVI6/937 CFRP plate 220 Figure 4.75 Projected delamination size as a function of impact energy showing the difference in damage zone size between the brittle (IM6/937) and tough (T800H/3900-2) CFRP systems. Numerical predictions for each system are also shown 221 Figure 4.76 Overheight compact tension (OCT) specimen geometry. Detailed dimensions for specimen A3 and A4 are listed in Table 4.6 222 Figure 4.77 OCT specimen in loading fixture showing (a) the location of the C M O D gauge and the out-of-plane displacement constaint and (b) the lines inscribed on the specimen surface for the displacement field analysis 223 Figure 4.78 O C T pinhole force-displacement curve and corresponding extent of damage growth for the A3 specimen. Note that no experimental measurements were available for the damage zone size at point 5 224 Figure 4.79 Experimental measurements of the displacement of the inscribed line 4 showing the measurements corresponding to the points labelled in Figure 4.78 225 Figure 4.80 Schematic of the O C T finite element model 226 Figure 4.81 T300/593 (a) damage growth function and (b) normalized residual stiffness function 227 Figure 4.82 Predicted one-dimensional stress-strain response for T300/593 227 Figure 4.83 Comparison of experimentally measured force-displacement curve with the corresponding UBC-CODAM3Ds prediction for (a) the cross head and (b) the clip and C M O D gauges of the A4 specimen 228 Figure 4.84 Comparison of experimentally measured force-displacement history with the numerical prediction for the A3 specimen geometry 229 Figure 4.85 Predicted line displacements for (a) line 1 and (b) line 4 230 Figure 4.86 F E M predictions of damage growth corresponding to the line analyses shown in Figure 4.85 231 Figure 4.87 Experimentally measured and predicted force-displacement history for the A3 specimen showing the reference points used to compare the measured and predicted displacements of the line 4 analysis 232 xiv Figure 4.88 Comparison of predicted line 4 displacements with experimental measurements 233 Figure A . l Idealized damage state in a unidirectional fibre-reinforced lamina 262 Figure A.2 Multi-surface loading criteria in the space of effective stress 262 Figure A.3 Solution algorithm for the C D M model by Matzenmiller et al. (1995) 263 Figure A.4 Predicted stress-strain response of a material modelled using the C D M approach due to Matzenmiller et al. (1995) 263 Figure A.5 Effect of variations in the exponent, m„ on the longitudinal stress-strain behaviour predicted by the M L T model for a [45/90/-45/0]3S T800H/3900-2 CFRP plate 264 Figure A.6 Comparison of the uniaxial stress-strain responses predicted by the M L T (m = 2 and m = 10) and Chang and Chang models 264 Figure A.7 Comparison of predicted and measured force-time histories for a 9.4 J (v = 7.7 m/s, m = 314 g) impact on a [45/90/-45/0] 3 S T800H/3900-2 CFRP plate 265 Figure A.8 Comparison of predicted and measured force-time histories for a 22.0 J (v = 11.8 m/s, m = 314 g) impact on a [45/90/-45/0] 3 S T800H/3900-2 CFRP plate 265 Figure A.9 Comparison of predicted and measured force-time histories for a 33.4 J (v = 14.6 m/s, m = 314 g) impact on a [45/90/-45/0] 3 S T800H/3900-2 CFRP plate 266 Figure A. 10 Comparison of predicted and measured force-displacement histories for a 33.4 J (v = 14.6 m/s, m = 314 g) impact on a [45/90/-45/0] 3 S T800H/3900-2 CFRP plate 266 Figure A . l 1 Comparison of predicted back-face fibre damage for a 33.4 J (v = 14.6 m/s, m = 314 g) impact on a [45/90/-45/0]3s T800H/3900-2 CFRP plate showing results for (a) the Chang and Chang model and (b) the M L T model with m = 2 and (c) m = 10 267 Figure A. 12 Comparison of (a) and (b) experimentally observed and (c) numerically predicted back-face fibre damage for an impact energy of 33.4 J (v = 14.6 m/s, m = 314 g). The numerical prediction was made using the M L T model with m = 10 268 Figure A. 13 Comparison of predicted and measured force-time histories for a 34.5 J (v = 3.3 m/s, m = 6330 g) impact on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 269 Figure A. 14 Comparison of predicted and measured force-displacement histories for a 34.5 J (v = 3.3 m/s, m = 6330 g) impact on a [45/90/-45/0]3S T800H/3900-2 CFRP plate 269 Figure A.15 Comparison of predicted back-face fibre damage for a 34.5 J (v = 3.3 m/s, m = 6330 g) impact on a [45/90/-45/0] 3 S T800H/3900-2 CFRP plate showing results for (a) the Chang and Chang model and (b) the M L T model with m = 2 and (c)ro= 10 270 Figure A. 16 Comparison of predicted and measured force-time histories for a 58.2 J (v = 4.3 m/s, m = 6330 g) impact on a [45/90/-45/0] 3 S T800H/3900-2 CFRP plate 271 Figure A. 17 Comparison of predicted and measured force-displacement histories for a 58.2 J (v = 4.3 m/s, m = 6330 g) impact on a [45/90/-45/0] 3 S T800H/3900-2 CFRP plate 271 xv Nomenclature Variable and Constant Names A area Ai cross-sectional area of plies with lay-up angle / (e.g. Ag0 and A 4 5 ) [C] stiffness tensor Cy components of the stiffness tensor d displacement E, Et elastic modulus E1, E11, Em normalized residual stiffnesses (EI E°) defining the three points on the bilinear curve of the residual modulus function (see Figure 3.7 and Equation 3.24) F, Fi damage potential as a function of strain and strain rate (F = / (e , e)) F 1 , Fu, Fm damage potential thresholds defining the three points on the bilinear curve of the damage growth function (see Figure 3.5 and Equation 3.14) G, Gtj shear modulus [H] compliance tensor kij rate of reduction of the / h elastic constant with the i t h damage parameter Ki, Lt scale factors applied to the strain components in the damage potential functions (e.g. Kx and Ly, refer to Equations 3.18 and 3.19) [M] transformation tensor from damaged stress (strain) to effective stress (strain) nt number of plies with lay-up angle i (e.g. n 9 0 and n 4 5 ) Rx normalized residual modulus for elastic modulus X (e.g. REi and RVn ) R'x normalized residual modulus for elastic modulus X associated with saturation of matrix cracking and delamination (i.e. at co = co'm) Si, T{, Ui scale factors applied to the shear strain components in the damage potential functions (e.g. Sx and Ty, refer to Equations 3.18 and 3.19) t thickness ti thickness of plies with lay-up angle i (e.g. t90 and r 4 5 ) V velocity xvi strain effective strain strain rate components of the strain tensor in contracted notation vector representation of e, lamina failure strain associated with direction i shear strain Poisson's ration ply angle stress effective stress components of the stress tensor components of the stress tensor in contracted notation vector representation of cr, shear stress internal state variables representing damage portion of damage attributed to matrix cracking and delamination portion of damage attributed to fibre breakage portion of damage at rupture associated with saturation of matrix cracking and delamination portion of damage at rupture associated with fibre breakage damage parameter values defining the three points on the bilinear curve of the damage growth function and the corresponding point on the residual modulus function (see Figures 3.5 and 3.7 and Equations 3.14 and 3.24) xvii Variables specific to the M L T model described in Section 3.2 and Appendix A failure surface in stress space (lamina co-ordinate system) Ft tensorial representation of failure criteria in stress space (lamina co-ordinate system) gi failure surface in strain space (lamina co-ordinate system) G ( tensorial representation of failure criteria in strain space (lamina co-ordinate system) m Weibull function exponent rt damage thresholds X, Y, S lamina strengths in the fibre and matrix directions and in shear, respectively Superscripts and Subscripts 0 value relates to the undamaged state (e.g. E®, v®2, and G ^ ) otherwise the parameter is the effective (damaged) value 1 time state (e.g. co1 or e / + 1 ) 0' 45' 90 value is associated with plies with lay-up this lay-up angle (e.g. f 4 5 and Ag) !, 2 , 3 lamina co-ordinate system (e.g. El, v 1 2 , and rjj) c , t compression and tension respectively (e.g. Xt, and e^) piy ply value or property l a m laminate value or property s in-plane shear x , y , z laminate co-ordinate system (e.g. Ex, v , and crx) xviii Acknowledgements I would like to express my sincerest gratitude to Dr. Reza Vaziri and Dr. Anoush Poursartip for their wisdom, encouragement, and patience and for the many interesting conversations and discussions, work related and otherwise, which we have had over the last few years. I am indebted to Dennis Nandlall and Gilles Pageau of the Defence Research Establishment Valcartier (DREV) for their continued support of this project and for many interesting and fruitful discussions. I am also grateful to Dr. Larry Hcewicz for the many interesting and enlightening discussions on the strain-softening phenomenon in composites. I would like to acknowledge the financial support of Natural Sciences and Engineering Research Council of Canada (NSERC), the Defence Research Establishment Valcartier, and the University of British Columbia. I would also like to thank INTEC Inc., Bothel W A and The Boeing Company, Seattle W A for their support of this research project. I would like to express my sincerest thanks to all the members of the U B C Composites Group with whom I have had the honor and privilege of working over the past four and a half years. I have made many friends. Without them this Ph.D. would have been a year and a half of hard labour. Thank you to the staff and faculty of the Department of Metals and Materials Engineering at the University of British Columbia. Last on the page but by far first in my heart, I would like to thank my parents for their endless love and support and for their unerring faith and confidence in me as I have pursued my studies. This thesis is dedicated to them. xix Chapter 1 Introduction 1. Introduction Composite materials are used in a wide range of applications. The driving factors for the replacement of traditional materials by composites are the high specific properties (e.g. strength and stiffness to weight ratios) which can be achieved in these materials, coupled with the ability to tailor the material response by varying the constituents (the matrix and the reinforcing medium) of the composite. The rapid growth in this field drives the need for a more detailed understanding of this class of materials. Hashin (1983) provides a survey of the range of analysis techniques which are available for studying the characteristics of composite materials. Current analytical techniques provide sufficient understanding and predictive capabilities for application in preliminary design but many applications require extensive experimental studies in order to prove a design methodology. Numerical techniques fill this gap in applications of traditional materials. However, the current numerical models applicable to composites are few and their development into well tested, rigorous material models is currently one of the most challenging fields in composite materials research. In particular, the prediction of damage (plasticity or cracking) in composites is not well understood. 1.1 Low Velocity and Ballistic Impacts on Composite Structures The susceptibility of laminated composites to damage caused by normal impact loading is well known, as evidenced in survey articles by Abrate (1991 and 1994). There are two consequences of this. In structural applications damage significantly reduces the residual strength of the composite and can hamper the ability of the composite to carry the in-plane design loads (see Poon et al., 1990 and Xiong et al., 1995 for example). The problem is compounded by the fact 1 Chapter 1 Introduction that many damage modes are not visible and can only be detected using expensive and time consuming techniques such as ultrasonic scans. Aircraft skin panels are an example of an application where accidental tool drops during manufacturing or hail strikes during flight could potentially pose a serious threat to the structural integrity if damage tolerance is not considered during design. There are applications, however, where the energy absorbing qualities of the damage growth in composites are beneficial. Vehicle and personal protection (e.g. Abbott, 1993) and automotive structures for crash-worthiness (e.g. Hull, 1985 and Haugetal., 1997) are examples of uses of composites where the ability of the material to absorb large amounts of energy through damage and deformation can be used to its full advantage. Whether damage tolerance or damage resistance is desired, efficient design requires an understanding of the damage mechanisms and the ability to accurately predict the amount and effect of damage. 1.2 Research Methodologies There are three representations which can be used to describe the response of a composite: micromechanical, macromechanical, and mathematical (refer to Figure 1.1). To some degree these classes also define the boundaries between much of the current research on composites. The more fundamental representation attempts to quantify the material behaviour through the micromechanical characterization of the volume fraction of fibre, fibre geometry, and fibre and matrix behaviour (e.g. elastic, rate dependent elastic-plastic etc.) for example. The macromechanical level, however, looks at a more smeared representation of the material as an anisotropic continuum which can be described by a number of quantities or material properties. Examples of the qualitative and quantitative relationships between the micro and 2 Chapter 1 Introduction macromechanical understanding are rule of mixtures predictions of modulus and the use of shear lag theory to predict the degradation in stiffness due to transverse crack growth. Both take quantitative micromechanical observations and develop a relationship which can be described by one or more macroscale properties. Typically, a mathematical description (e.g. an analytical or numerical model) of the material response smears micromechanical considerations in favour of the simplicity and generality of a macroscale description. Here we can consider theories such as simple laminated plate theory and macromechanical failure theories. Work which straddles the mathematical and physical areas is much less common and, at best, is qualitative. Structural analysis is the application of the resulting body of research and involves the selection of an appropriate emphasis that needs to be placed on each representation. This, in turn, is a matter of achieving a balance between the accuracy of the approach (selecting the appropriate physical basis, a combination of micro and macromechanical) and the simplicity and efficiency of the model (the mathematical representation). Herein lies the role of constitutive modelling. 1.3 Constitutive Modelling The main component of a numerical code is the routine that incorporates the constitutive model (Figure 1.2). A significant amount of research has gone into developing elastic, visco-elastic, and elastic-plastic constitutive models for laminated fibre reinforced polymers (FRPs). The differences in behaviour predicted by each of these responses can be important for structural design in the service loading regime where changes in the stress-strain response will have an effect on the predicted deformational capability of the structure. Beyond the design loads, however, the effect of damage becomes the overriding energy absorption mechanism and hence dominates the material response. 3 Chapter 1 Introduction Unlike the loading response, the prediction of the structural behaviour beyond the design loads is generally treated in a relatively simple manner. This is not to say that it has received any less attention. A number of 'failure' theories have been proposed from simple maximum stress criteria to more complex polynomial functions. What has, until recently, received little attention is the manner in which the failure is treated. Most approaches treat the material on a ply-by-ply basis as an ideally brittle material (see Figure 1.3a). Tensile test coupons certainly do not show the same yielding or (relatively) gradual failure observed in ductile metals and it is easy to see why composites have been assumed to be brittle in nature. The loss of stiffness associated with the failure modes is usually incorporated using a ply-discount method whereby the appropriate material stiffnesses in the 'failed' plies are set to zero, as dictated by the active failure mechanism. This leads to a sudden unloading that is characteristic of brittle fracture. The attempts to describe the behaviour of laminated composites in this manner leads to some misleading conclusions. The most significant is the failure to recognize the influence of damage growth on the behaviour. While it is true that in tensile tests, composites are susceptible to unstable damage growth which leads to brittle fracture, the same damage growth under the more complex loading conditions observed in a multi-angle lay-up does not necessarily lead to complete fracture of the material. Rather, there is stable growth of a damage zone which softens the material response (Dopker et al., 1994 and Kongshavn and Poursartip, 1997). Therefore, the use of the term 'failure' to describe the final phase of the response is incorrect as failure really refers to the ultimate fracture of the material. The final phase is a damaged response. A more realistic description of the deformation behaviour of a FRP composite laminate is shown in Figure 1.3b. The stress-strain response incorporates any non-linear behaviour of the 4 Chapter 1 Introduction constituent materials (i.e. plasticity and/or non-linearity of the constituent material responses), the stiffness reduction due to damage, and ultimately a gradual softening behaviour resulting from the damage growth. An attempt to address the issue of damage growth has been made by an ever increasing number of researchers through the use of continuum damage mechanics (CDM). Simply stated, C D M attempts to predict the effect of microscale defects and damage at a macroscale by making assumptions about the nature of the damage and its effect on the macroscale properties (e.g. modulus) of the material. C D M has its roots, not only in FRP composites but also in ceramics and other materials in which void and crack growth play a role in the response (Krajcinovic, 1984). In fact it was originally conceived by Kachanov (1958) as a means of determining the brittle creep of metals. A more complete description of the development of C D M as an analysis technique is provided in Chapter 2. Although a number of C D M based composite damage models have been proposed, the literature on applications of these models is somewhat limited. What has been shown, however, is that an accurate prediction of damage progression rather than simply the prediction of ply failure plays a significant role in successfully predicting the overall response of laminates (Talreja, 1986, Randies and Nemes, 1992, Dopker et al., 1994, and Williams and Vaziri, 1998). 1.4 Research Objectives and Thesis Outline Experimental evidence (e.g. Kongshavn and Poursartip, 1997) has shown that the progressive damage behaviour is a characteristic of the system and not a material response. The existing C D M models, most of which have been developed at the ply level, fail to address the issue of damage growth at the level of the laminate. This may be at the root of a common problem 5 Chapter 1 Introduction associated with C D M models, the difficulty in characterizing the numerous material parameters required by the models. The material properties required are frequently lamina properties but a lamina does not exhibit the progressive damaging behaviour which must be characterized. The objective of this thesis is to address these weaknesses through the development of a physically-based C D M model applicable to FRP laminates. A plane stress C D M based model is developed for application to problems involving in-plane and out-of-plane loading of thin FRPs. Of particular importance during the model formulation are the implications of experimental observations of damage growth in laminated FRPs. The need for a clear physical-basis for the model has been used as a foundation throughout the conceptualization and development of the model. As such, an attempt is made at bridging the gap between the experimental work on damage growth and stiffness reduction (e.g. Camponeschi and Stinchcomb, 1982, Reifsnider and Highsmith, 1982, Poursartip et al., 1986a, and Dvorak and Laws, 1987) and the theoretical modelling work on C D M (e.g. Talreja, 1985a, Allen and Harris, 1986, Chaboche, 1988a, Randies and Nemes, 1992, and Matzenmiller et al., 1995). While individually contributing much to our understanding of damage growth and its effects on material behaviour, these two fields have seen less interaction than might be expected (or desired). This process of bringing the two together involves compromises on both the strict interpretation of the experimental observations and the fundamental thermodynamic and mathematical descriptions, perhaps a reason why few researchers have treaded into this middle ground, but the result is a physically-based constitutive model which is easy to characterize and which is credible. This model will hopefully spur more cross-disciplinary research into damage growth and the prediction of its effect on the mechanical behaviour of FRP composites. 6 Chapter 1 Introduction In the next chapter, Chapter 2, a brief review of the literature is presented, focusing on numerical modelling of FRPs and, in particular, continuum damage mechanics applied to laminated FRPs. Attention is also given to the literature on experimentally observed damage growth and the measurement of the resulting stiffness loss in laminated composites although the relevant work in this area is referred to in more detail later in Chapter 3. Chapter 3 presents the theoretical development of the proposed formulation and the basis for experimental calibration of the input parameters required by the model. As an additional source of background information, an accompanying appendix (Appendix A) discusses an initial study (Williams and Vaziri, 1995 and 1998) which was carried out to evaluate a C D M based composite damage model originally developed by Matzenmiller et al. (1995). The results of the application of the model are compared to a widely used ply-discount based composite failure model and to experiments. The comparisons serve to demonstrate the potential of C D M in the analysis of dynamically loaded FRPs while highlighting some of the weakness of many current C D M models, weaknesses that are addressed by the model developed in this thesis. An overview of the theoretical development of the so-called M L T model (after the developers A. Matzenmiller, J. Lubliner, and R.L. Taylor) is also provided in the discussion in Chapter 3 and Appendix A . The constitutive model has been implemented in the LS-DYNA3D commercial finite element hydrocode. Chapter 4 describes two case studies which were undertaken to evaluate the capabilities of the model developed in this thesis. The first half of Chapter 4 discusses the application of the model to the prediction of damage growth in CFRP laminates subjected to non-penetrating impacts. Extensive use of experimental data is made to validate the results. Predictions made using two existing composite damage/failure models are also included for 7 Chapter 1 Introduction comparison. In the second part of the chapter, the model is applied to the prediction of the damage growth and force-displacement curve of an Oversized Compact Tension (OCT) specimen, a problem involving in-plane loading of an FRP composite plate. Finally, Chapter 5 summarizes the results of the case studies and discusses the conclusions which can be drawn from the work. Further improvements to the models are also proposed to address issues raised during the development and application of the model. 8 Chapter 1 Introduction Model Figure 1.1 Material representations. Figure 1.2 Solution routine for an explicit finite element method. 9 Chapter 1 Introduction Linear Elastic Loading Catastrophic Failure (Elastic Moduli Set to 0) Compressive Failure (Hydrostatic Stress) At=O(100 time steps) for Numerical Stability' Tensile Failure (No Residual Strength) (a) Linear Elastic Loading a A Strain Softening Permanent Set (Plasticity) Reduced Moduli (Damage) (b) Figure 1.3 Comparison of (a) traditional elastic-brittle and (b) observed damaged material response characteristic of laminated FRPs. 10 Chapter 2 Background 2. Background There is a large body of literature on the subject of the damage growth and the effects of damage on the response of FRP composite structures. While it is beyond the scope of this study to provide a complete review of the work that has been done, the following sections will outline some examples related to damage growth, different approaches which have been adapted for modelling damage in composites, and some examples of applications of numerical models relating specifically to impact problems. For more in-depth coverage of the literature available on a variety of impact related issues the reader is directed to comprehensive review articles by Cantwell and Morton (1991) and Abrate (1991 and 1994). First, let us consider the damage mechanisms which develop in composites and the effects of damage on the material behaviour. 2.1 Damage in Composite materials 2.1.1 Damage Mechanisms Damage in laminated FRP composites can be grouped into two broad categories: intralaminar or intraply and interlaminar or interply damage. The discussion which follows is based on conventional understanding of FRP laminate failure mechanisms. For more details refer to texts by Jones (1975) and Hull (1992), for example. Intralaminar Failure Intralaminar damage is characterized by two forms of cracking, one parallel to the fibre direction and the other normal to the fibres. Laminae are particularly prone to matrix cracking when loaded transverse to the fibre direction or under in-plane shear. In a cross-ply laminate, the plies 11 Chapter 2 Background with fibres transverse to the loading direction will be the first to exhibit this type of damage. Under uniform tensile loading, a network of roughly evenly spaced cracks will develop (refer to Figure 2.1). As the load increases, the density of these cracks increases until the damage approximately saturates at a crack spacing approximately equal to the ply thickness. Not only is the response dominated by the low strength of the matrix but there are a number of other factors which contribute to the growth of damage under these loading conditions. These include the interface bond strength between the fibre and the matrix, voids in the matrix, incomplete bonding between fibre and matrix, and stress concentrations induced by the fibres all of which act as initiators for matrix cracking. When loaded in the fibre direction, a network of small matrix cracks develop between the fibres in a lamina. These matrix cracks also interact with the fibres leading to other damage modes. If the interface bonding between the fibre and the matrix is strong, the matrix crack may drive cracks directly through the fibres (see Figure 2.2). This form of failure is called planar fracture and is an unstable brittle fracture mode. In general, however, the interface between the fibres and the matrix is weak, relative to the fibre and matrix strengths. Matrix cracks which have grown to the interface propagate along the fibre leading to a failure mode called fibre-matrix debonding (see Figure 2.2). The resulting load redistribution increases the load in the fibres and eventually leads to fibre failure but the overall laminate response of the two relatively brittle constituents resembles that of a tough material. The fibres effectively blunt cracks growing perpendicular to the fibres and the broken fibres bridge the matrix cracks and still carry load though frictional contact between the fibre and matrix. Fibre pull-out is the damage mode associated with the fibres being drawn out of the matrix. 12 Chapter 2 Background Laminae subjected to compressive loading in the fibre direction fail by buckling and kinking of the fibres. The failure is strongly dependent on the matrix and interface properties and void content as the matrix provides support for the fibres in this mode of failure. The fibre alignment also plays a major role. Interlaminar Failure Jxiterlaminar damage is characterized by debonding of individual plies in an angle-ply laminate. Strong normal or transverse shear stresses resulting from bending loads, geometric discontinuities, and/or intralaminar matrix cracking drive microcracks between plies of dissimilar orientation. As the microcracks grow and coalesce they form a delamination site. The pattern of delaminations that develop in a laminate is strongly dependent on the lay-up because of the mismatch between the material properties of neighbouring plies with different orientations. The larger the angle mismatch, the more prone the lay-up is to delamination. Generally, delamination within a laminate will be initiated by intralaminar matrix cracking which has grown to the ply interfaces (see Figure 2.3). The crack cannot grow directly into the adjacent layer because of the orientation of the neighbouring fibres. Instead, the crack grows between the plies. If the crack front reaches a favourable angle relative to the undamaged ply, i.e. parallel to the fibres, it will initiate intralaminar cracking in that ply (see Figure 2.4). Damage in a laminate is characterized by the presence of one or more of the lamina failure modes mentioned above. The nature of the loading (e.g. dynamic or static, in-plane or out-of-plane) determines the extent and history of damage which develops. In Section 2.1.3, the specific case of dynamic out-of-plane loading of FRPs, which is relevant to the present study, will be discussed in more detail. 13 Chapter 2 Background 2.1.2 Effects of Damage on Laminate Response Intralaminar Damage The effects of matrix cracking on the laminate stiffness has been the subject of a number of studies including work by Parvizi and Bailey (1978), Masters and Reifsnider (1982), Poursartip etal. (1986a), Laws et al. (1983), Ogin et al. (1985), Talreja (1985b and 1986), Hashin (1986), and Laws and Brockenbrough (1987). Essentially, the stiffness reduction is due to nonuniform strains introduced into neighbouring undamaged plies by crack opening and local load redistribution which increases the stresses in the plies adjacent to the cracks (Talreja, 1985b). The contribution of fibre breakage to laminate stiffness loss has received less attention but Steif (1984) investigated a detailed micromechanics approach and made mention of more simplistic approaches to the incorporation of fibre damage in stiffness reduction predictions taken prior to that point in time. Supporting the experimental work on in-plane stiffness reduction due to cracking, there are predictive techniques which have been proposed including the work by Laws et al. (1983), Laws and Dvorak (1987), Talreja (1985a), Hashin (1987), and Hahn et al. (1992). Much of the analytical work in this area has concentrated on simple [0n/90m]s laminates although the approach by Talreja, for example, has been applied to more general angle-ply lay-ups (Talreja, 1985b). Numerical techniques have the potential to be applied to more complex cases although the results presented in Hahn et al. (1992) were still for a simple [0/90]s lay-up. Interlaminar Damage The subject of delamination in general has garnered a significant amount of attention, even more so than intralaminar matrix cracking or fibre breakage. While a number of methods have been 14 Chapter 2 Background proposed for predicting the onset and growth of delamination including fracture mechanics analyses based on energy release rate and stress-based failure criteria, the effects of delamination on the stiffness of laminates are not as well investigated as, say, the effects of intralaminar matrix cracking. Lu and Vaziri (1994) provide an overview of the various approaches taken. O'Brien (1982) used a simple rule of mixtures approach in conjunction with laminated plate theory to predict stiffness loss due to delamination. The resulting prediction, a linear function of delamination area, agreed extremely well with experimental measurements. Poursartip et al. (1986a) also applied this technique to a number of angle-ply laminates. While the agreement was not quite as good as that presented by O'Brien, the results supported the theory. Strain softening Considerable effort has gone into understanding and even predicting the various damage modes observed in FRP laminates but work has tended to concentrate on simple geometries and isolated damage modes (Ochoa and Reddy, 1992). There are few quantitative, even qualitative, investigations of the effect of the overall damage state (fibre breakage, matrix cracking, fibre-matrix debonding, ... etc.) on the behaviour of FRP laminates (Poursartip, 1990). In particular, detailed investigations of the material response after initiation of significant amounts of damage, the so-called post-peak stress-strain behaviour, are rare. Softening behaviour has been observed in other brittle media such as cementitious (e.g. concrete and ceramics) geomaterials (e.g. Read and Hegemier, 1984). In a composite laminate, this softening is the result of the progressive failure of laminae constrained by adjacent undamaged material (see, for example, Kongshavn and Poursartip, 1997 which is based on the thesis work of 15 Chapter 2 Background Kongshavn, 1996). Further evidence which supports the presence of strain softening comes in the form of measurements of stable crack growth. Work by Dcewicz et al. (1993) and Dopker et al. (1994) on damage tolerant design of fuselage structures measured fatigue damage growth at the tip of a centre through-crack in rectangular composite coupons. Using an FEM model of the tests, they were able to show that the effects of different sizes and crack lengths on the material response could be accurately predicted using a strain softening material law. The results represented a significant improvement over a linear elastic fracture mechanics (LEFM) approach which was also applied to the problem. Until recently, characterization of this behaviour did not seem to be possible as, using current test methods, catastrophic failure would result soon after the failure of a lamina. However, recent work by Kongshavn and Poursartip (1997) demonstrates an experimental technique with which it is possible to get stable damage growth in monotonically loaded laminated composites. Using an Oversize Compact Tension Specimen (OCT), they measured the development of a process zone in front of the growing macro-crack. This process zone contains a distribution of micro and macro cracks leaving both damaged and undamaged plies. Kongshavn and Poursartip also performed a series of tensile tests on specimens cut from the process zones of the OCT specimens. Not only do these tensile tests show the stiffness loss associated with various stages of damage growth but they also demonstrate that the damaged material is capable of carrying loads at strains far exceeding the uniaxial fracture strain of the laminate. Through these tests Kongshavn and Poursartip were able to obtain a preliminary strain softening curve. There has been much debate on the subject of strain softening focusing on the acceptance of the behaviour as either a material property or structural property (Read and Hegemier, 1984). The 16 Chapter 2 Background discussions are perhaps best summarized by Nemes and Speciel (1996) who state "arguments as to whether strain softening is a true material property or simply a manifestation of the developing cracks are irrelevant since what is desired from an engineering perspective is a material constitutive model that describes the behaviour in a continuum sense." Evidence that composites exhibit strain softening on a structural level exists and therefore is a valid and necessary tool for the prediction of the structural response. 2.1.3 Impact Damage As one might expect, the issue of damage generated by the out-of-plate dynamic loading of composites is an important one given the applications of these advanced materials in critical structural applications where they may be subjected to impacts either by accident or by design. As a result, this subject is well covered in the literature. What follows is a very brief overview of the subject which is based on an excellent review article by Abrate (1991). The type and extent of damage which is generated by the dynamic out-of-plane loading of an FRP plate depends on a number of parameters including impactor mass, velocity, plate properties, and boundary conditions. In general, the sequence of damage development is the same. High flexural and/or shear stresses initiate matrix cracking. Matrix cracks serve as initiation sites for delamination and subsequently, more matrix cracking (Figure 2.4). As the damage progresses the bending stiffness of the plate decreases resulting in higher and higher bending strains at the back-face. Eventually these tensile strains are large enough to initiate back-face fibre failure. Higher impact energies will result in more plies exhibiting fibre failure. The delamination and fibre failure pattern which results from a through-thickness load (static or dynamic) is a characteristic for the lay-up. Increasing the impact energy simply increases the 17 Chapter 2 Background diameter of the damage zone). In an angle-ply laminate this characteristic shape is one of a spiral staircase or helical pattern as shown schematically in Figure 2.5. The overall through-thickness damage zone is a truncated cone. Typically the initial stages of damage (matrix cracking and delamination) occur within the laminate and the matrix cracking is not visible at the surface. Only a slight indentation is detectable through visual examination and even then it may be hard to detect. It is for this reason that subcritical damage in laminated composites can be particularly dangerous. With no visible defects it is difficult to identify damaged components but the resulting stiffness and strength degradation of the laminate can significantly reduce the load carrying capability of the structure. The plate geometry, boundary conditions, elastic properties, and impactor mass influence the impact energy required to initiate each damage mode, the initiation point in the target, and the extent of the damage growth. For example, if the overall plate response is stiff, the deformation will be more localized and matrix damage will initiate in the upper surface plies, a result of the large contact stresses in this region. Examples of this situation (i.e. where bending deformation is minimal) are a thick target, a stiff material, constrained boundary conditions (e.g. small planar size or clamped edges), or a high impact velocity. A more compliant or thinner plate, a large span and/or a lower impact velocity leads to more global deformation of the plate and higher bending stresses driving matrix damage towards the back face. This change from local to global deformation was highlighted in the work by Cantwell and Morton (1989) who studied the differences between low and high velocity impacts. A comprehensive study of the effects of impact velocity and energy, target material, indentor geometry, and plate thickness is summarized in a report by Delfosse (1994), highlights of which 18 Chapter 2 Background are published in Delfosse and Poursartip (1995 and 1997) and Delfosse et al. (1995). 2.2 Analysis of Dynamically Loaded Composite Structures 2.2.1 Model Scale There are two broad categories of analysis techniques used for the prediction of the dynamic response of FRPs: analytical and numerical. These two fields of analysis can be further subdivided into micro-mechanical and macro-mechanical scales (refer to Figure 2.6). Micro-mechanical or physical models build up the response of the material from the dynamics and mechanical and geometric characteristics of the individual constituents of the material (the matrix, the fibre, and the interface zone between the fibre and matrix) at a microscopic level (see Xu and Reifsnider, 1993 and Pecknold and Rahman, 1994, for example). The detail with which these models approach the material response carries with it the need to include a large number of parameters and a fairly complex mathematical formulation. This, in turn, complicates the implementation of a model, requires extensive computational facilities to run the model, and necessitates a wide range of characterization experiments. Macroscopic or phenomenological models treat the material as a continuum and seek to represent the material behaviour in terms of the average stress and strain. By smearing any microscale variations in properties and material behaviour over a continuum element, a significantly simpler and more efficient constitutive model is obtained. The majority of numerical material models are of this type and the remainder of the thesis will focus on phenomenological approaches to modelling composite materials. Other types of analysis which deserve mention are intermediate meso or mini-mechanical models, empirical modelling, and fracture mechanics. While trying to strike a balance between 19 Chapter 2 Background the detail of micro-mechanics and the efficiency of continuum mechanics (e.g. the lamina and interface damage model developed by Ladeveze et al., 1990), mini-mechanical models are frequently difficult to implement numerically. Empirical models, which require extensive test regimes on representative structures, are arguably the most accurate but they are also the least general and therefore the most demanding in terms of resources (tests, personnel, and time). Finally, fracture mechanics is the most obvious choice for attempting to describe the growth of macro-scale damage (i.e. dominant cracks) such as delamination but it requires advance knowledge of the location of the crack and the description of a field of cracks, as would be present in a large structure (e.g. Whitcomb, 1990 and Jih and Sun, 1993). This requirement quickly makes the approach untenable for application to structural problems even with current computational capabilities. The relative merits and disadvantages of each approach are summarized in Table 2.1. 2.2.2 Analytical Models Analytical approaches make use of a smaller number of basic characterization tests than empirical models and use rigorous mathematical formulations to investigate the material response. Olsson (1989) presented a model based on a one parameter differential equation which was developed for predicting the force-time history of impact on orthotropic composite plates. A number of problem geometries and boundary conditions were investigated and the results were compared to experimental and other analytical and numerical work. The work presented by Olsson, his own and the results included by other authors, has served as a benchmark for many analytical and numerical models. Other examples of analytical modelling can be found in Prasad et al. (1994), Cairns and Lagace (1987), and Pierson (1994) where the effects of damage 20 Chapter 2 Background were included in the analyses. An analysis of this nature is frequently formulated in such a manner as to allow the investigation of a number of loading conditions and material properties but the extent to which the results or models can be applied to different geometries and boundary conditions is often limited. This is not to say that analytical methods are not useful as they provide a quick tool for investigating relatively simple geometries and test cases. However, for analysis of large structures with the flexibility to simulate a wide range of geometries, load cases, and materials, one must turn to numerical methods. 2.2.3 Numerical Models In the numerical analysis of dynamic problems, such as impact events, both space and time domains must be discretized. It is customary to use a finite element scheme for the spatial discretization and a finite difference scheme for the time integration. The choice of time integration scheme generally depends on the particular velocity regime of interest, or more precisely on the strain-rates involved. Figure 2.7 presents the strain-rates observed in typical problems and provides the range of rates over which each type of F E M technique applies. The implicit integration scheme has been successfully applied to the analysis of low velocity impacts by a number of authors. Montemurro et al. (1993) for example investigated the effects of boundary conditions and failure criteria on the predictions of an 80 degree of freedom implicit F E M code and found good agreement both with experimental data and with the numerical predictions of other authors. Work by Quan (1998) outlines a combined analytical and numerical approach to the prediction of non-penetrating and penetrating impact. The numerical approach is 21 Chapter 2 Background used to predict the global deformation behaviour of a normally impacted FRP while an analytical approach is used to model the local mechanisms caused by contact and penetration. For a recent review of implicit numerical methods applied to the impact analysis of composite structures refer to Vazirietal. (1996). While useful for structural analysis, the implicit method is limited in the range of velocities over which it can be used. Explicit methods are better suited to the analysis of short duration and intense impact events. The computer codes used tend to be commercial hydrocodes or wavecodes used as testbeds for material model development. Comprehensive reviews of the various explicit computer codes used can be found in Zukas et al. (1982) and Vaziri et al. (1989) and a similar review was recently given by Hamouda and Hashmi (1996). Vaziri et al. ranked the available codes on their performance in impact and penetration problems. Based on a comprehensive scoring system, the L S D Y N A family of codes (Hallquist, 1990, 1993, and 1994) were found to be the best suited to this type of problem. 2.3 Constitutive Modelling of FRP Composite Laminates In all numerical methods, be it implicit or explicit, the predictions are heavily influenced by the constitutive models used to describe the material's response. In light of this, let us first consider the variety of constitutive models that have been developed for FRP composites. Here again, the volume of literature on the subject of constitutive modelling of composites is enormous. A rather substantial and comprehensive review of the subject was recently assembled by Lu and Vaziri (1994) as a precursor to the current work. 22 Chapter 2 Background 2.3.1 Traditional Approach The most common approach to modelling composites has been to model the material response as linear elastic up to the point of failure initiation at which point some form of failure criteria and unloading routine is invoked. The basis for this approach is the assumption of an essentially brittle response. The application of linear elasticity to FRPs is well developed in laminated plate theory and can be found in any textbook on the mechanics of composites (e.g. Tsai, 1988 and Herakovich, 1997). It is well known, however, that many composite materials show a non-linear or even plastic response in one or more principal material directions. Composites which incorporate polymeric matrix materials for example, exhibit significant non-linearity in shear and, to a lesser extent, in transverse loading (see for example Hahn and Tsai, 1973). There are a number of fibre materials including polyethylene (e.g. Spectra™ or Dyneema™) which possess non-linear characteristics as well. Attempts to address this issue have been made by the implementation of non-linear elasticity (e.g. the incremental scheme by Petit and Waddoups, 1969), viscoelasticity (e.g. the combined creep and plasticity model by Krempl and Hong, 1989) and plasticity theory (e.g. the orthotropic elastic-plastic models developed by Vaziri, 1989 and Chen and Sun, 1993). Non-linear effects are important for structural design in the service loading regime where differences in the material response due to viscous and/or plastic behaviour will have an effect on the load carrying capability of the structure. In impact problems, the energy absorbed due to plasticity and other non-linear effects is relatively small when compared to the energy absorbed by various damage mechanisms. As a result the non-linearity due to plasticity is commonly ignored in favour of more detailed descriptions of the damage and failure. A number of researchers have published work on micromechanical, empirical, and analytical 23 Chapter 2 Background approaches to modelling damage in impact loaded composite plates. Although these models are based on rigorous analyses and provide insight into the mechanisms of damage initiation and growth, their application in a more general model suitable for implementation in a numerical code is limited by the complexity of the model (computationally or in the number of inputs required by the model) or the inability of the model to be applied to a general geometry (typical of analytical and empirical approaches). 2.3.2 Damage Initiation The phenomenon of the development and progression of damage is observed in a wide range of composite materials. This is particularly true of fibre reinforced composites with non-ductile matrices where the microscale response is typically elastic-brittle and is marked by the formation of microcrack fields associated with matrix cracking, delamination and fibre breakage. Although this type of damage is irreversible, much like plasticity, there is virtually no residual stress or deformation after unloading until the material is close to failure. The study of this type of behaviour falls into the field of damage mechanics. The first modelling approaches to damage in composites involved the use of so-called 'failure' criteria. The term failure is used in many contexts in composites. It can mean initiation of cracks, loss of load carrying capability of one or more plies in a laminate, or complete rupture of the structure. Strictly speaking, the latter definition is more appropriate. The first two uses really refer to damage initiation and growth. A wide range of failure criteria can be found in the literature. Generally they have been developed as analytical design criteria but have found their way into constitutive models as criteria for the onset of brittle fracture in one or more characteristic modes (e.g. lamina matrix or 24 Chapter 2 Background fibre failure). Most criteria are phenomenological and are in the form of a polynomial function in terms of stresses. A general approach, based on the yield criteria of plasticity theory, was first proposed by Tsai (1965). Later, Tsai and Wu (1971) proposed a more generalized polynomial failure criterion which has served as the basis for many other specialized criteria, such as the Maximum Stress, Maximum Strain, and Hashin's criteria. Although very elegant, the generalized polynomial criteria, among other things, lack the ability to predict the failure mode. Hashin (1980) addressed this shortcoming by proposing a set of quadratic equations, with independent failure criteria for compressive and tensile failure in the fibre and matrix directions. Various forms of these criteria have been implemented by a number of authors including Chang and Chang (1987b) and Langlie and Cheng (1989). A recent comparative evaluation of failure analysis methods by Sun et al. (1996) provides a detailed examination of the relative merits of the many laminated composite failure criteria which have been proposed over the past three decades. Other survey articles on the subject include Wu (1974), Tsai and Hahn (1975) and Labossiere and Neale (1987). 2.3.3 Effects of Damage on Material Behaviour By far the simplest approach to damage is to assume instantaneous failure as the post-failure degradation rule. In its simplest form, the failure criteria discussed in Section 2.3.2 are used to predict first ply failure. The laminate behaviour is unchanged until fibre failure occurs in a ply oriented closest to the principal loading direction at which point a complete loss of load carrying capability is predicted. The effects of other modes of failure are assumed to be insignificant, an unrealistic assumption that does not match experimental observations. A more common method, called the ply-discount method, involves the degradation of only some 25 Chapter 2 Background of the properties depending on the failure mode identified by the failure criterion (see, for example, applications by Chang and Chang, 1987b and Murray, 1989). The method is based on the assumption that a failed ply cannot carry any load in the dominant stress direction associated with the failure mode. A matrix failure, for example, would result in the following modification to the constitutive matrix (two-dimensional plane stress case): E \ E2VU o l - v 1 2 v 2 1 l - v 1 2 v 2 1 Elv21 E2 Q l - v i 2 v 2 1 l - v 1 2 v 2 1 0 0 G 1 2 This type of approach is also sometimes called the parallel spring model (Sun et al., 1996). A variation on this approach, used by Vaziri (1989) and Vaziri et al. (1992), involves the division of the failure into ductile and brittle modes. If the failure is determined to be ductile, a matrix dominated mode, the material retains its load carrying capacity but its stiffness is set to zero (i.e. a perfectly plastic response). A brittle, fibre dominated failure is characterized by a loss of both stiffness and load carrying capability. The ply-discount method has been used in a number of numerical models but several authors including Talreja (1985b) and Murray (1989) have indicated that this approach can lead to incorrect predictions of the stiffness reductions. In particular, an over-prediction of the reduction in longitudinal modulus for cross-ply laminates and poor agreement for shear modulus reduction were cited. While widely used, models based on these failure criteria and stiffness reduction schemes all suffer from a lack of a physical basis. As mentioned previously, experimental measurements do Fj 0 0 0 0 0 0 0 0 (2.1) 26 Chapter 2 Background not support the ply-discount method of describing damage growth. Recently, efforts have been focused on the field of continuum damage mechanics as a means of describing the progressive damage observed in laminated composites. 2.4 Continuum Damage Mechanics The field of continuum damage mechanics (CDM) was initiated by the work of Kachanov (1958) and later by Rabotnov (1963). Although their work was not in the field of C D M , the successful introduction of a separate field variable to account for the microdefect density in the macroscopic response of metals to creep sparked the development of the concepts and theories which were to become C D M (Chaboche, 1988a and Krajcinovic, 1984 and 1989). The key concept in C D M is the assumption that a micromechanical process (microcrack growth) can be treated at a macro level by homogenizing the damage over a representative volume. Another macromechanics approach which has achieved a significant degree of success is plasticity theory. Here again a micromechanism, dislocation movement, which is inhomogeneous on the microscale is successfully averaged or homogenized over a volume of material and described by macroscopic state variables, the plastic strain tensor (Krajcinovic, 1984). 2.4.1 Introduction to Continuum Damage Mechanics The concept used to describe the damage state in a material is the damage variable. The damage variable relates the state of microdamage to the macroscopic response of the material. As such, the accuracy of the prediction made by a C D M approach is largely dependent on the level of simplification used to derive the damage variable or variables. In many cases, the variable is chosen for mathematical simplicity or based on the intuition of the researcher. A more rigorous approach, however, is to base the internal state variables on a thermodynamic analysis (e.g. 27 Chapter 2 Background Allen et al., 1987a and 1987b). The probabilistic nature of the damage initiation mechanisms can lead to a statistical definition of damage as well (e.g. Krajcinovic and Silva, 1982 and Matzenmiller et al., 1995). Early work in the field was based on a scalar damage variable. The rationale for such a simple representation of damage was that the damage manifested itself as a distribution of voids in the material. The damage is then a function of the void density. Damage in the form of microcracks, however, requires a more complex representation to account for the directionality of the effect of cracking. First and second order damage tensors were proposed to account for the anisotropy of the damage (e.g. the vector representations by Krajcinovic and Fonseka, 1981 and Talreja, 1985a). Each damage variable, commonly denoted by co(-, represents a specific type of damage. The terms in the tensor account for the effect of one or more damage parameters (i.e. damage modes such as matrix cracking parallel to the fibres, fibre breakage, and fibre matrix debonding) on the material response in the particular direction associated with that term. More formal mathematical analyses led to fourth and even eighth order tensor representations (discussed briefly in Murakami, 1987) which can account for the effects of crack shape and orientation. It is unclear if the added complication of using a higher order tensor representation of damage is feasible in a numerical approach where the simplicity of the vector representation, both in implementation and physical clarity, is desirable (Krajcinovic, 1984). As with many analysis methodologies, the problem is one of experimentally determining the large number of constants or material parameters required by the model. A fourth order damage tensor for a plane-stress model can require up to 21 independent variables to characterize the damage state. It is difficult to summarize the main equations or relations in the field of C D M because of the 28 Chapter 2 Background many approaches and end applications which have been studied, each of which has its own notation and basic assumptions. As mentioned above, there is a wide array of different approaches to the choice of damage parameter, some based on intuition and some with a more physical interpretation. This diversity in methodology is to be expected in a relatively new field. Perhaps the easiest way to outline the major components of a C D M approach is to draw parallels with plasticity theory. Frantziskonis (1994), in fact, demonstrates the equivalence of isotropic damage mechanics and plasticity in a research note on the subject. The four key components of a numerical continuum damage model are: • A set of internal state variables (the damage parameters) which are used to relate the state of micro-crack growth within the material to the macroscopic material response. • An initial failure criterion or damage threshold which bounds a region in strain (or stress) space within which the material will sustain no damage with loading. • A set of damage growth functions which define the manner in which the damage threshold changes once damage has initiated in the material. • A corresponding set of associated damage rules which relate the development of the damage parameters to the strain (or stress) past initiation. The analogous components of a plasticity model are: • An internal state variable (e.g. dislocation density) which can be used to relate the amount of dislocation movement within the microstructure of the material to the macroscopic material response. • An initial yield surface which bounds the region in stress space within which deformation will be elastic. • A hardening rule which defines the manner in which the yield surface changes during plastic deformation. • A flow rule which relates the plastic strain increments to the stress state at each point in the loading history. A brief presentation of a C D M theory for composites by Matzenmiller et al. (1995) (originally published as Matzenmiller et al., 1991) which provides more detail on the constituents of a C D M 29 Chapter 2 Background model formulated for implementation in a F E M code can be found in Appendix A . For more comprehensive studies of C D M the reader is directed to Lemaitre (1984), Chaboche (1988a and 1988b), Murakami (1987), and Krajcinovic (1989). 2.4.2 CDM Models for Laminated Composite Materials Perhaps one of the first applications of C D M to laminated composites was made by Talreja (1985a). The lamina model proposed two damage variables (first order tensorial representation of damage), each associated with a principal direction in the material (i.e. matrix and fibre). No growth functions or damage thresholds were presented but a number of specific damage states were investigated analytically. In subsequent work, Talreja (1985b) applied the model to the prediction of stiffness reduction in a number of angle-ply laminates. The model showed good agreement with experimentally measured modulus loss. Ladeveze et al. (1990) presents the final development of a meso scale shell model for laminated composites. The laminate is broken down into a series of anisotropic plies, which are homogeneous through the thickness, and zero-thickness interface layers. Each ply, in turn, is analyzed at the fibre, matrix, and interface levels thereby providing "qualitative" information about the fibre and matrix properties to the laminate (Ladeveze et al., 1990). Damage is based on two independent modes, one representing matrix cracking and fibre pull-out while the second is associated with the transverse brittle failure of the fibre-matrix interface. Notable in the model development is the inclusion of plasticity in the matrix cracking damage mode to account for the non-linear effect of friction during fibre pull-out. The interface model incorporates three additional damage parameters, each associated with a through-thickness stress (one normal and two shears). While predictions of bending response were good, mesh sensitivity was identified 30 Chapter 2 Background as the most critical problem of the model. Pickett et al. (1990) proposed a very elegant and simple isotropic plane stress damage model applicable to short fibre sheet moulding compound (SMC) composites. Although two damage modes are identified, one associated with volumetric strain and the other with the deviatoric shear strain, they are added and the total is used to degrade all the stiffness properties. Of particular interest was the choice of bilinear curves to represent a two phase process of damage growth and stiffness. This was quite a departure from the previous strictly mathematical representation of these two functions. The model was implemented in the commercial explicit F E M code, P A M - C R A S H and applied to the impact of a SMC disk by a hemispherical punch. Comparisons with experimental force-time data and measurements of the extent of damage were good. The P A M - C R A S H implementation is more general than the original model presented in Pickett et al. (1990) and is applied to a bi-phase material model. The bi-phase model treats the fibre and matrix separately with a rule of mixtures approach used to model the overall lamina behaviour. The two damage modes mentioned above are then applied individually to the matrix (volumetric and shear modes added together) and the fibre (volumetric mode only). A more detailed description of the approach can be found in Haug et al. (1991). Some additional applications of the general composite damage model are described below in Section 2.5. Frantziskonis and Joshi (1990) took a slightly different approach to damage growth by proposing a single scalar internal variable to characterize the microstructural changes in a volume of material. The anisotropy of the effect of damage on the material properties is then introduced through a tensor. This is in contrast to the usual description of the anisotropy of damage through the use of a number of different damage variables each representing a different damage mode in 31 Chapter 2 Background addition to a tensorial representation of the effect of damage on the constitutive tensor. The damage threshold and growth function take the form of a polynomial in strain which Frantziskonis and Joshi (1990) acknowledge is similar to the Tsai-Hill failure criteria. Characterization of the model is in the form of a series of tensile and compression tests on unidirectional laminae. Quantitative predictions of these simple tensile tests showed good results as did qualitative predictions using angle-ply laminates. Randies and Nemes (1992) developed a C D M model applicable to thick composites by incorporating a delamination damage variable in addition to the in-plane isotropic matrix damage variable. Fibre breakage was described using a maximum strain criteria. Notable in the model, and its later development by Nemes and Speciel (1996), is the incorporation of rate dependence in the damage growth laws. Not only does this account for rate dependence in some composite materials but it had the added benefit of making the model mesh insensitive (see Section 2.4.3 below). Predictions of the model compared well with data from high rate cantilever bend tests on graphite epoxy . This is despite the fact that the test specimens were uniaxial while the model assumes a homogeneous in-plane damage. Also demonstrated was the models ability to predict spallation (delamination) failure although limited experimental data was available for comparisons. Matzenmiller et al. (1995) built on the work of Talreja (1985a) and implemented a C D M lamina model based on three damage parameters. In contrast to Talreja and many of the other models proposed up to that time, a third damage parameter was added to describe the so-called shear damage. Usually, the degradation of the in-plane shear modulus is based on a combination of the damage parameters associated with the matrix and fibre modes. Matzenmiller et al.'s approach 32 Chapter 2 Background to damage growth was a statistical one and the damage growth law introduced was based on a Weibull distribution of strengths, commonly associated with the strength of fibre bundles. A full presentation of the model development, hereafter referred to as the M L T model, is given in Appendix A. Engblom and Yang (1995) provide another development of the approach by Talreja (1985a). The model, developed to predict the effects of intralaminar damage (matrix cracking only) on the stiffness properties of angle-ply laminates, provides a straightforward application of C D M . Although presented as a 'sublaminate damage model' in the paper title, the implementation is really ply-by-ply. The damaged ply stiffness, however, are found by 'deconstructing' the damaged laminate elastic constants using laminated plate theory. Perhaps the most interesting aspect of the work is the ease with which experimental observations of stiffness loss and crack initiation are incorporated into the characterization of the model, a rare feature of a C D M model. This is a direct result of the development of the damage model at the laminate scale and the use of simple linear functions for the stiffness reduction. While the concept is similar to that used by Pickett et al. (1990), the model is only valid for one damage phase: matrix cracking. Yazdchi et al. (1996) have developed a comprehensive anisotropic damage model. Although not specifically designed for laminated composites it is flexible enough that such an extension would be possible. Two damage variables are introduced to account for the anisotropic nature of the damage. Both power law and Weibull distribution based growth functions (based on a threshold volume averaged Cauchy stress) were investigated. Other notable C D M model developments include work by Murakami (1988) for general 3D anisotropic solids and a very rigorous thermomechanical theory for laminated composites by 33 Chapter 2 Background Allen et al. (1987a and 1987b). The models presented above provide examples of the broad spectrum of applications of C D M to composites and the discussion is by no means a comprehensive review of all applications of C D M to composites. 2.4.3 Numerical Problems Associated with CDM Models The implementation of a C D M approach addresses some of the more fundamental aspects of modelling damage growth and fracture in FRP laminates. Quite apart from the level of complication associated with these approaches, there are two particular problems that arise when C D M models are applied to practical problems: localization and characterization. Localization The most significant obstacle associated with the application of C D M in F E M models is localization (Bazant, 1986, Ladeveze et al., 1990, and Nemes and Speciel, 1996). This problem is shared by any failure model and is characterized by a dependence of the numerical solution on the mesh density, usually without convergence to a unique solution. The most widely accepted approach to the problem has been nonlocal theories such as those put forward by Bazant et al. (1984), Bazant and Pijaudier-Cabot (1988), and Valanis (1990) for example. Another discussion of localization limiters can be found in Belytschko and Lasry (1989). Nonlocal theory modifies the damage growth function by making it a function of an averaged strain. The averaged strain, in turn, is a function of the strain in neighbouring elements and a characteristic material length. The purpose of the characteristic length is to bring in the microstructural aspect of the particular damage being modelled (Bazant and Pijaudier-Cabot, 1988). An example of the application of this approach to composites is the work by Kennedy and Nahan (1997) on the prediction of failure in notched laminates. 34 Chapter 2 Background An alternate approach was taken by Nemes and Speciel (1996) who introduced a rate-sensitive term into the damage growth law and were able to show mesh independence and convergence of the solution, albeit for a simple one-dimensional stress wave propagation problem. What is significant about this approach is that it is has the potential to be more computationally efficient than the nonlocal theory as it does not require the additional evaluation of an averaging function nor does it require modification of the solution modules of a numerical code. The latter is particularly important when developing models for implementation in commercial codes where access to source code is limited. In general one finds that work done in this field focused specifically on composites is limited with much of the discussions aimed at ceramics and concrete, essentially isotropic materials. Characterization of CDM Models The second problem with C D M models tends to be one of characterization. Many papers on model development include descriptions of standard, and sometime non-standard, uniaxial tests on uniaxial laminates which can be used to derive the various material constants required by the models (e.g. the procedure presented in Frantziskonis and Joshi, 1990). While these tests can yield values for the elastic and, arguably, strength constants, C D M models require additional parameters to characterize the damage growth and stiffness reduction relationships. The experimental data on stiffness loss and strain softening behaviour indicates that the damage growth is a property, not only of the matrices and fibres used (i.e. the laminae or unidirectional laminate), but of the entire material system including the stacking sequence (see the work by Dost et al., 1991 and Kortschot and Beaumont, 1991, for example). Therefore, it is unclear if the use of simple uniaxial tests to characterize the overall material behaviour is valid. Hart-35 Chapter 2 Background Smith (1990), for example, cites the effect of lay-up on the laminate response and advocates the use of certain cross-ply laminates rather than uniaxial coupons to better measure the in-plane tensile strengths of FRPs. Damage models based on statistical damage growth laws are particularly problematic as noted by Agaram et al. (1997) and Williams and Vaziri (1995). While Matzenmiller et al. (1995) simply state that the required material parameters can be determined from uniaxial tests, no details are given. Yazdchi et al. (1996), who also used a growth law of this form, does describe a method for characterizing the Weibull constants but the material discussed was aluminium, not a highly orthotropic one. Engblom and Yang (1995) is an example of a more realistic approach to the characterization of a damage model. Engblom and Yang rely entirely on experimental observations of laminate stiffness loss due to matrix cracking (e.g. from Highsmith and Reifsnider, 1982) to determine the constants required by their model. Even the damage threshold, a function of the stress state and lamina strengths, were determined from experimental measurements of stress at the onset of damage. The close predictions of laminate stiffness loss provided by the model were a result of this reliance on experimental data and did not require artificial correction factors although an adjustment of the undamaged moduli was required in order to match the experimentally measured initial laminate modulus. 2.4.4 CDM Summary The discussion above has focused on the application of C D M to laminated composites. This brief review only scratches the surface of the body of literature available on C D M but the vast majority of the published work focuses on more homogeneous materials such as concrete and 36 Chapter 2 Background ceramics. There, the determination of the creep and fracture characteristics is the impetus for the application of C D M . The models described in Section 2.4.2 show the major characteristics of the various approaches taken to applying C D M to laminates. There are a number of other models which, while having unique features, tend to be variations on the approaches discussed above; perhaps having different damage growth criteria or a different selection of damage parameters for example. Regardless of the approach, there are two significant weaknesses associated with C D M models: localization (shared with all models that predict failure) and difficulties in experimental characterization. 2.5 Application of FEM to the Prediction of Damage in FRP Laminates This section summarizes the recent work on numerical modelling of composite structures. The discussion will focus on applications of the DYNA2D/3D codes but some other examples will also be mentioned, notably applications of the damage model by Pickett et al. (1990) in the P A M - C R A S H code. The majority of the work published on the application of D Y N A 3 D to impact has focused on the failure model based on the work of Chang and Chang (1987a). The model uses a variety of criteria to predict failure due to fibre breakage, matrix cracking and matrix compression failure. Murray (1989) carried out a number of verification tests using a shell element representation of a laminate. The results were mixed and number of recommendations for improvements to the model were put forward. Most of the suggestions were aimed at improvements to the post-failure degradation rules which are based on the ply-discount method. Another study carried out by Murray and Schwer (1993) focused on verifying the elastic prediction of the orthotropic shell implementation in DYNA3D. In perhaps one of the few analyses which do not use the Chang 37 Chapter 2 Background and Chang composite failure model, Sturt and Dallard (1989) use a quadratic damage function based on the work by Tsai-Hill. Unfortunately the accuracy of the predictions were not discussed. Other applications of the Chang and Chang model include work by Madsen and Thomsen (1994), Edlund (1993), Murphy (1994), and Majeed et al. (1994). Haug et al. (1991), Haug et al. (1997), and Haug and de Rouvray (1993) describe a number of applications of the C D M model developed by Pickett et al. (1990) in P A M - C R A S H leading up to a fairly detailed analysis of the crash-worthiness of composite car body structures. In a similar application, Agaram et al. (1997) applied the C D M damage model by Matzenmiller et al. (1995) in D Y N A 3 D to the front end impact of a car. This work was supported by more fundamental work on calibrating the model with dynamic crush tests on composite tubes. Other applications of this C D M model to simpler structures include the work by Bilkhu et al. (1997) who looked at the failure of normally impacted rectangular and circular composite plates. While reasonable agreement with experiments was obtained, the difficulty in obtaining the material constants required for the model was raised. This particular model, the M L T model, was used in studies which made up the initial part of this thesis (initially published as Williams and Vaziri, 1995). The application, which is discussed in detail in Appendix A , found similarly good agreement with experimental measurements on normally impacted CFRP plates although the same reservations are raised regarding the determination of the material input. Examples of F E M techniques applied to the prediction of ballistic impacts on composites are less common in the open literature. One example is provided by Langlie and Cheng (1989) and Langlie et al. (1990) who investigated the application of the LS-DYNA2D code to the prediction of ballistic impacts of fragment simulating projectiles (FSPs) on S-2 Glass and Kevlar™ targets. 38 Chapter 2 Background Their work also involved the implementation of a composite failure model into the code. Maximum stress criteria were used to predict punching shear failure, in-plane tensile failure and delamination while the ply-discount method was used as the post-failure degradation rule. Another example can be found in Karaoglan and Springer (1993) who developed their own 2D axisymmetric code for predicting penetration of composites. The M L T model has also been applied to the predictions of the ballistic response of GFRP plates by Nandlall et al. (1998). A curve fitting procedure was used to evaluate the required material constants based on the predicted perforation velocity for one plate thickness. These material constants were then used successfully to predict the perforation velocity of a second thicker plate. Despite the good agreement, the calibration procedure required was seen as a weakness of the model; the same issue raised in applications of the 3D versions of the model. Applications of DYNA3D to ballistic impacts on composites include a fairly substantial investigation which was undertaken by Blanas (1991) to study the impact of FSPs on thin composite targets. A full 3-D model of the target and projectile was developed and predictions of the ballistic limit were compared to experiments with varying success. Along a similar line is the work by Nandlall (1995) who used a 3D numerical model of FSP impacts to develop and verify an equivalent 2D axisymmetric FSP with the goal of using the more efficient DYAN2D code to predict ballistic limits. van Hoof et al. (1996) also investigated the use of DYNA3D to predict the penetration response of ballistic head protection subject to impacts by FSPs. Not only was the in-plane damage considered, but a tie-break interface was implemented between a layer of the through-thickness bricks in order to investigate delamination. More recently, the C D M model by 39 Chapter 2 Background Matzenmiller et al. (1995) was adapted for use with brick elements by van Hoof et al. (1997). Implemented as a user material in D Y N A 3 D , the model was used to investigate the penetration and back-face deformation of a composite helmet impacted by an FSP. Recently Frissen et al. (1996) looked at the penetration behaviour of thin polyethylene fibre-reinforced laminates using the M A R C F E M package. Here thick shell elements were used in conjunction with a stiffness degradation scheme based on a Weibull curve to predict back-face deformation. Some of the most compelling evidence offered to date for the incorporation of a constitutive model which includes the type of laminate response resulting from progressive damage comes from the application of the ABAQUS Implicit code to notched crack growth in FRP laminates. The work by Hcewicz et al. (1993) and Dopker et al. (1994), mentioned previously in Section 2.1.2, demonstrated the effective use of a strain softening material F E M model to damage tolerant design analysis of composite fuselage structures. In follow-on work to the investigation by Kongshavn and Poursartip (1997) (see Section 2.1.2 above), Williams et al. (1996) used the same type of F E M analysis to perform numerical simulations with the goal of predicting the force-displacement histories and damage growth patterns of the OCT tests. Through a series of studies carried out using A B A Q U S Implicit, they found that neither an instantaneous failure nor a perfectly plastic lamina response could predict the trends observed by Kongshavn and Poursartip. In fact, both of these limiting cases bounded the experimental measurements. Only through the use of a strain softening material law, albeit a simple one, were they able to obtain the experimental trends. 40 Chapter 2 Background 2.6 Summary The field of damage growth, and more specifically impact induced damage, in" FRP composites has received wide attention in the open literature. Whether the effects of damage are viewed in a negative (e.g. critical load bearing applications such as aircraft structure) or positive (e.g. applications involving energy absorption such as armour) light, it is nonetheless an integral aspect of the use of composites and effective design requires analysis tools which correctly predict the effects of damage growth on material response. The continued demands placed on engineers for designs involving more and more complex parts coupled with the advent of more powerful computers have placed numerical models at the forefront in terms of analysis tools largely because of their flexibility. In turn, this is driving the need for more refined and more accurate constitutive models with which to predict the material response. The enormous volume of literature available on modelling the damage growth and impact behaviour indicates one thing; that there is no generally agreed upon approach. It seems clear that continuum damage mechanics is a powerful tool for predicting the macroscale (continuum) behaviour of composites resulting from microscale damage. Previous theories, failure criteria, and stiffness degradation rules do not provide sufficiently robust descriptions of these effects. There are a number of C D M models which have been proposed, all have their particular strengths and weaknesses. What is most noticeable is the gap between experimental observations of damage growth (e.g. lay-up dependence) and the more mathematical development of the damage models themselves. In between these two 'camps' is a middle ground which should take the essence of the C D M approach and build into it a more rigorous reliance on the experimental observations. The benefit of this type of approach is a physically based structural model which can be easily 41 Chapter 2 Background characterized and which is applicable to a wide range of material systems and structural geometries. The challenge becomes one of maintaining integrity with one field while trying to please proponents of the other. This thesis represents a first step into this middle ground. 42 Chapter 2 Background Table 2.1 Comparison of methodologies of damage modelling. (Adapted from a discussion in Lu and Vaziri, 1994) Approach Description Advantages Disadvantages Empirical Empirical relationships are developed from an experimentally measured data The most direct approach Restricted to the specific specimen geometries and stacking sequences. Fracture Mechanics Dominant cracks are identified in the material and crack growth (i.e. damage growth) is predicted using fracture mechanics. Allows a detailed investigation of the interaction of the cracks and the effect of this interaction on damage growth. Restricted to large cracks (e.g. delamination). The complexity of the analysis which results from the crack interactions makes this approach impractical for application to microcrack fields which are typical of many forms of damage in composites. Micromechanical A local analysis is performed at the level of the fibres and matrix. The results are then linked to the overall composite behaviour. Allows for the identification of the different damage mechanisms and provides insight into microcrack mechanisms. Crack interaction remains an issue making the application of this type of model too cumbersome for the analysis of any real structure. Mesomechanical Instead of considering the fibre and matrix separately (i.e. a micromechanical approach), the lamina is modelled as an elementary single layer and a zero thickness interface layer. The approach is formulated as an optimal balance between simplicity of the computations and considerations for local effects on the damage characteristics. Formulation does not lend itself to implementation in a numerical code which is based on a continuum representation of the material (i.e. the next order of magnitude in scale) Macromechanical (Continuum) The lamina is represented as a transversely isotropic medium. The analysis of the lamina is carried out using continuum theories. Damage progression is modelled through the use of internal state variables which are incorporated in the constitutive relationships. Allows for the effect of stacking sequence. The incorporation of the damage parameters in the constitutive formulation allows a relatively simple computational routine which can easily be implemented in a finite element code. Assumption of homogenized damage state restricts application to macroscale cracks (e.g. delamination) 43 Chapter 2 Background Applied load 1 ( \ ( } / 90° 0° 90° • 0° Increasing 90° load , 1 S I S I I I 1 i I \ / i ( i < / / i \ i i i t ' * S < ; \ S J 1 \ I I I n i t i i i I M U / -H K-Figure 2.1 Intralaminar matrix cracking in a cross-ply laminate. 44 Chapter 2 Background Crack grows along fibre-matrix interface Fibres. Applied load f i l l < c * Matrix ft\ cracks H H Crack grows through fibre Fibre pull-out Frictional force still applied to fibre by matrix Fibres bridge matrix cracks Brittle fracture Figure 2.2 Intralaminar damage modes in FRP composites showing effect of fibre-matrix interface strength on fibre failure mode. 45 Chapter 2 Background Matrix cracks Figure 2.3 Interlaminar damage growth in FRP composites, (a) Applied load initiates matrix cracks in plies transverse to the loading direction. As the matrix cracks reach the ply interfaces (b) they initiate microdelaminations. These interlaminar cracks coalesce (c) to form delaminations. 46 Chapter 2 Background Matrix crack grows to ply interface Delamination grows from matrix crack at ply interface Matrix crack initiates in next ply Figure 2.4 Initiation and growth of a delamination from an intralaminar matrix crack in an angle-ply laminate. The delamination initiates from the matrix crack at the ply interface. A matrix crack is initiated in the neighbouring ply when the delamination crack front is parallel to the fibre direction. Shear induced Delamination F i b r e breakage Figure 2.5 Characteristic pattern of delamination and fibre failure in an angle-ply laminate subjected to non-penetrating through-thickness static or dynamic loading. (Adapted from Delfosse, 1994) 47 Chapter 2 Background Structure Finite Element 7777 ////// Sublaminate Model Classical Laminate Plate Theory Fibre Idealized Lamina Material/Integration/Gauss Point Equivalent ^ Homogeneous Material Macro-Mechanical Homogeneous Orthotropic Lamina Matrix Mini-Mechanical Matrix / Interface Micro-Mechanical Fibre Figure 2.6 Comparative scale of various modelling approaches. (Adapted from Rahman and Pecknold, 1992) 48 Chapter 2 Background Implicit Integration Explicit Integration DYNA3D | A 10"8 10"6 10^  10"2 10° 102 104 1 06 108 Strain Rate Js"1) V | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Strain vs Time or Creep Rate Recorded Constant Strain Rate Test • i Mechanical i Resonance ] in Specimen | and Machine Elastic-Plastic Wave Propagation Shock Wave Propagation Dynamic Considerations in Testing Inertial Forces Neglected II Inertial Forces Important < » « * Isothermal II Adiabatic Figure 2.7 Applicable integration techniques as a function of strain-rate. (Adapted from Vaziri et al., 1989) 49 Chapter 3 Theoretical Model Development 3. Theoretical Model Development 3.1 Introduction The basis for the stress-strain relationship in many C D M approaches is the concept of stress (or strain) equivalence (Lemaitre and Chaboche, 1978 among others). The hypothesis of stress equivalence proposes that a material containing damage, subjected to a strain, e, and under a state of stress, <J, can be represented as an equivalent undamaged material subject to the same strain, £ , but under an effective stress state, a (refer to Figure 3.1). Mathematically this can be expressed by the relation: {d} = [M{co)]{cj} (3.1) where is a transformation tensor which is a function of the damage state, 0), and relates the stress (strain), {<7}({e}), in the damaged material to the effective stress (effective strain), {cr} ({£}), in an equivalent undamaged material. Substituting for the effective stress in Equation 3.1, we obtain: {cr} = [M]- 7 [c°]{ £ } (3.2) where [ c ° j is the constitutive or stiffness tensor of the undamaged material. Combining [M] and we obtain the damaged or effective stiffness tensor such that: {cr} = [C(co)]{£} (3.3) When written in expanded form for a plane stress, orthotropic solid, Equation 3.3 is: -> — y 0 C12 0 o c « (3.4) 50 Chapter 3 Theoretical Model Development where the Cy are the elastic coefficients which are functions of the undamaged or initial material elastic constants and the damage state. The role of a C D M model is to provide a mathematical description of the dependence of these elastic coefficients on the damage state (i.e. the residual stiffness functions) and of the change in the damage state, co, with load state; e.g.: co = f(e,e) (3.5) where e and e are the strain and strain-rate respectively. The key to the success of all C D M models is to maintain a coherent link with the physical observations of damage growth and material response. 3.2 Preliminary Work As a preliminary step in the development of a new C D M model, an existing composite damage model due to Matzenmiller et al. (1995) was implemented in the LS-DYNA3D code. The selection of this model over other models presented in the literature is not a reflection of any particular advantage of this model, other than the fact that the mathematical formulation is particularly well suited for implementation in an explicit F E M code. The so-called M L T model shares many features with other C D M models. Equally, the approach makes some unique assumptions about the damage growth. The purpose here is not to judge this model against other C D M theories but rather to compare the predictions made by a C D M approach with a more traditional composite failure model. The M L T model is based on the response of a FRP lamina with the laminate response built up using simple laminated plate theory. Matzenmiller et al. (1995) introduce three damage parameters (C0i,C02, and cos) to track the damage growth. The damage state of the lamina is divided into fibre dominated longitudinal damage (tracked by (Oi), transverse matrix cracking 51 Chapter 3 Theoretical Model Development (co2), and a less physical matrix shear damage (cos). As such, the damage parameters are uniquely associated with the principal lamina directions and lamina shear. Damage growth is postulated to follow a Weibull curve, a function often associated with the statistical description of strength in materials containing initial defects or microcracks (e.g. a bundle of loose fibres). The resulting dependence of damage, co, on the strain state, e, is of the form (for a ID case): j _ me (3.6) (0 = 1-e where X/E° is the material strength normalized to the undamaged elastic modulus and m is the Weibull function exponent, a constant used to fit the shape of the predicted stress-strain curve to the desired response. The threshold for damage growth is based on a maximum stress criterion with interactive terms assumed for the matrix dominated damage modes (transverse and shear damage). The equations used are similar to Hashin's criteria (Hashin, 1980). A linear stiffness reduction with damage is assumed such that the final form of the constitutive matrix is: [c] = 1 S Y M v\2 1 {\-(02)E2 0 1 ( 1 - 0 ) 1 2 ) G 1 2 - l (3.7) A more detailed discussion of the mathematical development used by Matzenmiller et al. (1995) can be found in Appendix A. The evaluation of the model was based on a series of runs performed to predict the response of T800/3900-2 CFRP composite plates subject to non-penetrating normal impacts at various 52 Chapter 3 Theoretical Model Development impact energies. The results of the analyses, outlined in detail in Williams and Vaziri (1995 and 1998), have been summarized in Appendix A . The implementation of the C D M based composite damage model demonstrated significant improvements in the prediction of damage growth and force and energy-time histories when compared to an existing composite failure model in LS-DYNA3D. While results of the application of this model show that C D M provides a useful framework for further development of damage models for laminated structures, a number of issues were raised during the investigation which should be addressed by a new damage model. These include: • the physical significance of the choice of damage parameter • ease of material characterization • stacking sequence or lay-up dependence of the damage growth in laminated structures • rate dependence in non-carbon fibre based systems • mesh size dependence of the predicted damage growth The first three items are all related and will be dealt with in some detail later in this chapter. The latter two items, which were discussed in Chapter 2, were deemed to be beyond the scope of the present work. This is not to say, however, that their importance is being downplayed. Rather, a reasonable structure for the model must first be developed before they can be adequately addressed. 3.3 Model Formulation In the development of a new C D M composite damage model, the approach that has been taken divides the problem of obtaining a complete 3D damage model into two components; one concentrating on the in-plane response (the subject of this thesis) and the other on the out-of-plane or through-thickness response (Figure 3.2). Not only does this simplify the formulation but 53 Chapter 3 Theoretical Model Development the model development is significantly quicker as one does not have the computational overhead of running full 3D models (Figure 3.3) during the iterative model development phase. This work represents the first stage in the ongoing development of a family of C D M based models, under the title U B C - C O D A M (University of British Columbia - COmposite DAmage Model), for application to the prediction of the damaged response of laminated FRP composite structures. The remainder of this section is devoted to the formulation of the UBC-CODAM3Ds plane-stress (i.e. shell element based) C D M composite damage model. 3.3.1 Sub-Structuring The first feature of our approach is to treat the material as an orthotropic medium made up of a series of repeating units through the thickness. Typically, constitutive models applicable to composite materials have treated the composite as a stack of perfectly bonded unidirectional laminae (Hallquist et al., 1994, Matzenmiller et al., 1995, and Pickett et al., 1990). These models are based on the lamina behaviour and the response of the bulk material is assumed to follow the 1s t order shear deformable laminated plate theory (constant shear through the thickness resulting in limited interaction between layers). Experimental observations, supported by analytical and numerical modelling, have clearly shown that the failure mode, and hence the response of laminated composite materials is closely linked to the stacking sequence and lamina interactions (for example Pagano and Pipes, 1971, Lagace, 1986, and Dost et al., 1991). Generally, a laminate is made up of a number of repeating units through the thickness (e.g. [0/±45/90]gs where the repeating unit is [0/+45/-45/90]). The lamina interactions caused by the stacking sequence are largely controlled by this sublaminate. As a result, a model constructed at this scale has the potential to incorporate these interactions. 54 Chapter 3 Theoretical Model Development From an implementation point of view the differences between the lamina and sublaminate approaches are relatively minor. Plane stress shell elements use through-thickness integration points to predict bending stresses (shown schematically in Figure 3.4). These integration points are used in lamina models by associating the appropriate local material angle with each integration point (e.g. a [0/45/-45/90]gs laminate would be modelled by a shell element with 32 integration points through the thickness with material angles of 0°, 45°, -45°, 90°, 0°, 45°.. . etc.). The constitutive model is evaluated using the global shell strains transformed into this local shell co-ordinate system. The stresses calculated by evaluating the constitutive model at each of these integration points are then transformed back into the global shell co-ordinates before the through-thickness integration is performed, usually following a trapezoidal integration scheme. The resulting laminate response prediction is analogous to that obtained from a laminated plate theory type analysis. The sublaminate approach uses the same technique although in most cases no material angle is required. Typically a minimum of one integration point would be used for each sublaminate. The exception is a thin laminate where the minimum number of integration points required to accurately capture the bending response may determine the number of points used. Overall, fewer integration points are required, hence increasing the efficiency of the model, and the interactions of the laminae are incorporated in the sublaminate constitutive behaviour, thus improving the accuracy of the prediction. 3.3.2 Selection of Damage Parameters Traditionally damage parameters are associated with cracking (e.g. reduction in the cross sectional area of the load bearing material) in the principal lamina directions, parallel and perpendicular to the fibre direction, corresponding to matrix and fibre dominated damage modes. At the sublaminate scale there is no clearly defined directionality of the damage modes in 55 Chapter 3 Theoretical Model Development arbitrary lay-ups. The problem is simplified if we consider only symmetric laminates ([0/±45/90]ns, [±60]ns and [0/90n]s for example), where there exists at least a material symmetry at the sublaminate level. By defining two damage parameters, cox and coy , aligned with the sublaminate co-ordinate system it is possible to characterize the damage state as projections of the crack densities normalized to the saturation crack density in the principal directions in the material. While adequately describing the effect of matrix cracking and fibre failure on Ex and Ey in simple cross-ply laminates (i.e. [0n/90m]ns), there is a need to define an additional damage term to model the interactive effect of the two damage parameters on shear modulus in more general symmetric lay-ups (e.g. [0/±45/90]ns). A survey of approaches taken to this problem in other C D M models shows a somewhat arbitrary treatment. Chow and Wang (1987) use a multiplicative form: G 1 2 = ( l - f l ) 1 ) ( l - f i ) 2 ) G 1 ° 2 (3.8) while Yazdchi et al. (1996) use a more complex interaction function for the shear modulus: _ 2(1-co,)2(I-co2)2 Q G i 2 ~ 7 72—; 72 G i 2 (3-9) ( l - f l > i ) 2 + ( l-fl> 2) 2 Note that both of these are lamina based (1 and 2 refer to the principal material directions in the lamina). Matzenmiller et al. (1995) take a somewhat unconventional approach of introducing a completely separate 'shear damage parameter', cos, with an associated damage threshold and growth law. Although less physical, this approach avoids the difficulty associated with defining an interactive term. The problem is perhaps best summarized by Randies and Nemes (1992) who, when defining the residual stiffness functions for their C D M model, noted that the 56 Chapter 3 Theoretical Model Development particular functional dependence "...is quite arbitrary.." and "..should be determined by corresponding experimental evidence... ." In their through-thickness damage formulation they chose a function of the form: G31=(l-co23)(l-kcol)G^1 (3.10) where 1 and 3 refer to the in-plane and through-thickness directions respectively. The second order effect of the damage parameters on the modulus is characteristic of all the residual stiffness functions used in their model and was proposed to account for the more rapid degradation of the elastic properties as the damage approaches saturation. In the current analysis, the simple form of Equations 3.8 has been used as a basis for the combination of the two principal damage parameters. Strictly speaking, the form of cos from Equations 3.8 is: cos =cox +coy -cox(Oy (3.11) however, at intermediate levels of damage, this function predicts a more significant effect of the damage state on shear than on either of the principal directions. For example, with cox = 0.5 and coy =0.5 one would expect cos to be on the order of 0.5 but Equation 3.11 gives cos = 0.75, a significantly higher value. By modifying Equation 3.11 slightly such that: cos = -yjcol +0)2y- coxcoy (3.12) is applied as the damage parameter for the residual shear modulus function, one gets the more reasonable result of cos =0.50. Similarly, for cox =0.3 and coy =0.7, cos =0.61 rather than 0.79 from Equation 3.11. 57 Chapter 3 Theoretical Model Development 3.3.3 Damage Growth Law One of our main concerns with the M L T approach is the use of a single mathematical expression to describe the damage evolution as a function of strain over the entire load range, an approach which is characteristic of many C D M models. This results not only in pre- and post-peak material responses which are linked but also in a response that has no elastic region, neither of which have a strong physical basis. There are three behaviour regimes or zones in the material response which must be considered: elastic, damage phase 1 (e.g. matrix/delamination only), and damage phase 2 (e.g. fibre and matrix/delamination). Each is characterized by a different relationship between the damage variables (<5),-) and the load state (characterized by the function F). The simplest and most flexible approach to capture the changes in damage growth is to assume a linear relationship within each of these regions (Pickett et al., 1990). First consider the matrix damage. We define F 1 as the damage threshold which marks the end of the elastic response and the onset of matrix failure (co1 =0 at F 1 ) . The assumption of decoupled fibre and matrix damage growth means that, regardless of any fibre damage which may develop, the matrix damage will continue to grow linearly with damage up to rupture defined by the damage threshold F m where com = 1 (see Figure 3.5). The amount of damage attributed to matrix cracking and delamination, co'm, will be some value, less than 1.0, determined by the lay-up of the sublaminate. Following the same argument for fibre breakage, we define F n as the threshold for onset of fibre damage and co'j -1 - co'm as the proportion of the total damage which is attributed to fibre failure. The net damage growth function is the summation of these two curves as shown in Figure 3.5. Note that the definition of the damage state <9n at the transition between the two 58 Chapter 3 Theoretical Model Development phases of damage growth is not an independent parameter but is rather a consequence of the values of F 1 , F n , F m , and co'm (coy being a function of co'm) such that: CO = CO, ( F n _ F n (3.13) The resulting piece-wise linear relation between CO and F is shown in Figure 3.5. co, as a function of the load state is: co = < co1 0 F-F1 ^ KF"-F*j cou+(l-cou) f f - F11 ^ K F m - F U j 0 < F < F 1 F 1 < F < F11 F 1 1 < F < Fm F > F in (3.14) It is important to note that damage is an irreversible process such that: co = max(eo', co' 1) (3.15) where co' is the damage calculated from Equation 3.14 for the current load state, and co' 1 is the previous state of damage. In the above discussion, the second phase of damage is assumed to incorporate both matrix damage and fibre damage. There are three other possibilities which should be identified: (1) the first mode of damage saturates at or before the onset of the second, (2) the first mode saturates after the onset of the second but before rupture, and (3) the second mode of damage saturates before rupture. The first two cases can be used to describe the response of a [0n/90m]s laminate, 59 Chapter 3 Theoretical Model Development for example. Matrix/delamination damage in brittle cross-plies can saturate before the onset of fibre failure while fibre breakage will typically initiate before matrix cracking saturation in a FRP incorporating a tough matrix system. The limit being the case considered in this thesis where matrix/delamination continues to grow until rupture. The behaviour described by the third case represents a material where fibre damage is arrested, either by a physical mechanism (a toughening of the response) or by a change in the load state resulting from the softening of the sublaminate and/or the decoupling of the plies. For example, growth of a significant amount of delamination damage can allow individual plies to rotate, aligning the fibre directions with the principal loading direction without driving more fibre damage. These three additional damage growth behaviours are characterized by a three phase (i.e. trilinear) response. For the purposes of this thesis we are assuming that the two phase damage growth model applies. The extension of the model to cover these more complex (i.e. mathematically complex not physically) material responses is largely a coding issue as the theoretical foundation of the model remains the same. 3.3.4 Damage Potential Function In general the damage 'potential' function, F , will be a function of the strains (or stresses) and of the strain-rates (stress-rates) such that F = f(e,e) thereby incorporating any rate sensitivity of the material being modelled. Note that the term potential is used to indicate a driving force for damage growth and should not be confused with the potential surfaces or functions that are customarily used in plasticity theory where you have flow perpendicular to those surfaces. The selection of the damage potential is the least physical component of the present model. A 60 Chapter 3 Theoretical Model Development number of approaches can be taken including the use of strength based failure criteria such as those presented by Tsai and Wu (1971) or Hashin (1980). In the current analysis, the growth function is assumed to be an effective strain of the general form: (3.16) where rate effects are ignored for simplicity. This limits the initial application of the model to rate insensitive materials such as CFRP and, to a lesser degree, GFRP. The modification of the model for application to strain-rate sensitive materials such as Kevlar™ and Specta™ is left for future work. Equation 3.16 is loosely based on the distortional energy failure criterion with the two additional shear terms added so as to incorporate all the strains which can drive damage growth. Here the constants K, L, S, T, and U are not used as measures of strength but rather as scalars to provide a measure of the relative contribution of each strain component to the driving force for damage growth. The incorporation of through-thickness shear terms accounts for the effects of the yyz and y^ strains on the initiation of delamination damage in a plate subject to bending loads. These strains, along with the through-thickness strain, ez, are also responsible for edge effects in laminated composites. Delamination growth can be initiated by an incompatibility of the through-thickness strain (stress) field between neighbouring plies of mismatched angle at a discontinuity such as the edge of a laminated plate subjected to bending and/or in-plane loading. However, by smearing the through-thickness anisotropy of the sublaminates, the model becomes insensitive to these effects under purely in-plane loading. The through-thickness strain has not 61 Chapter 3 Theoretical Model Development been included in the potential function because in a shell formulation ez is merely a scaled summation of the in-plane strains: £z =-(vxz£x+Vyz£y) (3.17) which are already included in Equation 3.16. However, the e z term becomes very important in 2D and 3D formulations where it is no longer a simple function of the in-plane strains. Local tensile and compressive through-thickness strains serve to promote and inhibit, respectively, delamination growth. These strains result from contact forces for example. The relative effect of tensile and compressive loading on the damage growth is also introduced through the damage potential functions. The damage growth laws as functions of the damage potentials are fixed relations between the potential and the damage and therefore do not change with loading condition. By introducing a separate set of constants for tension (K t and Lt) and compression (^"cand Lc) one accounts for the relative contribution of each to the potential function. The appropriate constant is selected depending on the sign of the associated strain (e.g. Kt if £j >0 and Kc if £] <0). Here again, the dominant loading mode (e.g. a tensile strain, £j) is selected as the reference and the other constants are set accordingly. In the current model, two threshold functions are required, one for cox: Fx = V Lxc,t + Lxc,t + fy \ 2 f r_xy_ V Sx J + ' yz + y U x J (3.18) and the other for CO, \Kyc,t) V \Kyc,t) \Lyc.t j + + xy \sy J 2 („ \ 2 + 7 yz T v 1y J + ZX vuyj (3.19) 62 Chapter 3 Theoretical Model Development 3.3.5 Effect of Damage on Material Properties The amount of damage sustained by the sub-laminate will affect each of the material moduli but to varying degrees. Many lamina based C D M models developed for composites (e.g. the model by Matzenmiller et al., 1995 discussed in Section 3.2) assumed that the loss of modulus is linear with damage (refer to Figure 3.6a): REi = - § - = (1-*,•©,•) (3.20) where RE. is the normalized residual modulus in the ith direction and kt (kt < 1) is a matrix of constants that defines the rate of modulus loss with the damage, G ) ( . This simple relationship is possible because the lamina moduli are only dependent on the damage parameter associated with the appropriate principal material direction. That is, the longitudinal stiffness (Ex) is dominated by fibre breakage (defined by cox) and the transverse stiffness (E2) by fibre breakage matrix cracking ( f t ) 2 ) . However, in a sublaminate, the modulus reduction is a function of both matrix and fibre damage, a biphase damage growth (refer to the discussion above in Section 3.3.3). One would expect the rate of stiffness loss associated with matrix cracking/delamination and fibre breakage to be different. Therefore, the modulus loss is assumed to be a bilinear function of the damage parameter (Figure 3.6b). A bilinear representation of the residual modulus function was also used by Pickett et al. (1990) although it was not applied at this scale nor did the damage parameters have the same meaning. This approach is supported by the work of Poursartip et al. (1986a), for example, which showed that the stiffness loss due to matrix damage and delamination was linear with damage density in a [45/90/-45/0]s laminate subjected to fatigue loading (i.e. stable crack growth). After a certain level of damage was reached, corresponding to a 35% stiffness loss in the particular CFRP 63 Chapter 3 Theoretical Model Development system investigated, a change in mechanism was observed. Poursartip attributed this to the initiation of fibre dominated failure and, based on the work by Steif (1984) on stiffness reduction due to fibre breakage, proposed that a second 'leg' could be added to the overall stiffness reduction function to reflect the additional effect of fibre failure. Steif (1984) has shown further that the relationship between stiffness loss and fibre breakage density is a linear function. Following this approach we define a general bilinear curve to describe the modulus loss with damage as shown in Figure 3.7. (ttn was defined in the discussion above in Section 3.3.3, however, the appropriate stiffness loss to associate with this level of damage, E11, has yet to be defined. What can be measured fairly directly, though, is the normalized reduced stiffness at saturation of the matrix cracking, R'E. By definition, R'E is associated with the level of damage a>'m, the portion of damage at rupture attributed to matrix and delamination damage (also defined in Section 3.3.3). Plotting this point on Figure 3.7 defines the slope of the residual stiffness curve associated with matrix and delamination damage. cou defines the onset of fibre damage where the behaviour deviates from this curve with the normalized residual stiffness at rupture ( co = 1) equal to Em. From Figure 3.7: n Eu=l-{\-R'E) (3.21) m A normalized residual stiffness function must be defined for each material stiffness: and Ex = REx(o)x)E°x Ey=REy(coy)E°y (3.22) 64 Chapter 3 Theoretical Model Development where Rx{o)) takes the general form (see Figure 3.7): RE = En+(Em-EUj CO c? rco-co^ II 0<co<co co11 <co<\ (3.23) Note that Equation 3.23 allows for a non-zero residual modulus (i.e. Em * 0 ) . In general, complete damage will result in total stiffness loss (i.e. Em = 0) and it is possible to further simplify the residual modulus-damage relationships as follows: l + (Eu-l)-^r 0<co<coI v ; CO Eu-Eu co-cou^ V l - G > j (3.24) fi>n < CO < 1 Potential differences in stiffness loss under tensile and compressive loading are incorporated by allowing independent definitions of the normalized residual modulus functions for the Young's moduli for each loading condition. Poisson's ratio v has not been included in the discussions leading to Equation 3.24 because the degradation of this interaction term is not independent of the functions chosen for the moduli Ex and Ey. The major symmetry of the constitutive tensor must be retained both for an undamaged and damaged material, and as a result: yx and v° v° y xy _ Y yx (3.25) therefore: 65 Chapter 3 Theoretical Model Development ^ . K ' ^ y K y _ RVyx(cox,G)y)v' REx{cox)E°x " REy(coy)E°y o yx (3.26) Using the second part of Equation 3.25: ) (3.27) This is uniquely satisfied if: Kv(o>x,o)y) = REx{(Ox) and RVyx (cox,coy) = REy (coy) (3.28) The modulus degradation functions applied to v and v must be the same functions of cox damage on Poisson's ratio is rare, possibly because of the difficulties associated with measuring in specimens with progressive damage development. However, Camponeschi and Stinchcomb (1982) provide some evidence that the reduction in v ^ measured during fatigue loading of quasi-isotropic CFRP laminates is of the same order and follows the same trends as the reduction in Ex. In the present approach the effect of delamination is included in the overall effect of matrix damage on the in-plane elastic constants (e.g. a reduction of E corresponding to a reduction in bending stiffness and the reduction of G and v corresponding to decoupling or a release of the constraint between neighbouring laminae within the sublaminate). The implementation of softening of the through-thickness shear moduli was considered during the model development as a means of explicitly modelling the effect of delamination on the bending stiffness of a plate. The effects of changes in Gxz and Gyz on the predicted force-time and deformation response of plates subjected to dynamic through-thickness loading were found to be small even for fairly and coy that are applied to Ex and Ey, respectively. Experimental evidence for the effects of 66 Chapter 3 Theoretical Model Development large reductions (up to 70%). Above a certain threshold (over =75% reduction) zero-energy deformation modes (hourglassing) were introduced into the finite element mesh. 3.3.6 Predicted Constitutive Relationship Having defined the normalized residual stiffness functions, we can now return to the constitutive equation and derive the final form of the constitutive tensor. Recall Equation 3.4: {a} = [C]{e} 'Cn C12 0 C21 C22 0 0 0 c66 £y > (3.29) 7 xy) where the coefficients of the constitutive tensor, C y , are functions of the damage state, RE[a>x,C0y^ derived above in Section 3.3.5, and the undamaged moduli, E°, G ° , and v ° . It is convenient to approach the problem of deriving these coefficients using the compliance tensor, [H]: {e} = [cr ;{o-} = [//]{o-} (3.30) where: [H] = V xy v yx 0 J _ Ey 0 0 0 1 'xy (3.31) Substituting for the damaged elastic moduli from Equation 3.22, and using the relationships developed for Rv^ and Rv in Equation 3.28 we obtain: 67 Chapter 3 Theoretical Model Development [H] = 1 R, v xy R E E x 0 R E x E x 1 R E E y SYM 0 1 RGxyGxy xy r e e 0 R E y E y 0 0 SYM (3.32) The final form of the constitutive tensor is found by inverting Equation 3.32: [c] R E E x REX R E y Eyvxy {\-REREv%v°yx) (\-RExREv%v%) Rf E 0 (I-*exRe,v%v%) 0 0 S Y M RG„Gxy (3.33) Note that the functions REi vary between 1 and 0. As a result, the constitutive tensor is positive definite over the entire range of damage. 3.3.7 Predicted ID Response To better illustrate the features of the model let us consider the predicted response for a one-dimensional case. First, let us construct the stress-strain relationships predicted for each of the three zones: elastic (stage 1), onset of the first damage phase (stage 2), and onset of the second damage phase (stage 3). In the one-dimensional case, the damage growth function becomes: F(e) = ^ 2 - £ (3.34) 68 Chapter 3 Theoretical Model Development Substituting into Equation 3.14 one obtains: co = < ft)1 0 < £ — F 1 ^ V F » - F ' y ft)n+(l-ft)n) v F i n - F n y 0 < e < F F 1 < e < F 1 1 F n < e < F m F 1 1 1 < e (3.35) In stage 1, the response is elastic: <j = E°e 0<e<Fl (3.36) For stage 2, from Equation 3.24: rj = Fe E°e (3.37) which after substitution for co from Equation 3.35 in the range F 1 < e < F 1 1 yields: a = ^ F n - F 1 1 F 1 ^ F°e + ' Fn - 1 A v F n -Fl j E°e2 Fl<e<F11 (3.38) Similarly for stage 3: a = v F m - F n y F ° e -f E11 ^ v F i n - F n y F V F 1 1 < e < F m (3.39) For e > F m , ft) = 1.0 and hence F = 0 and a = 0. The predicted response is shown in Figure 3.8. The flexibility of the bilinear model is 69 Chapter 3 Theoretical Model Development demonstrated in Figure 3.9. By varying the damage growth and residual stiffness curves it is possible to obtain a material response that mimics the response predicted by the traditional instantaneous failure models (Figure 3.9a), a material response that approaches the statistically based C D M models where the response is a single continuous function (Figure 3.9b), or responses in-between. Figure 3.9c shows a response with an elastic region followed by the onset of damage leading to catastrophic failure. Figure 3.9d shows the complete response (similar to Figure 3.8) with a linear-elastic region, the onset of damage, and finally a softening response. It is interesting to note that the resulting constitutive law is independent of a>n. Consider Equation 3.3 and 3.5: {cr} = [C(ft))]{e} and co = f{e,e) The role of the model has been to define the terms C(co), Equation 3.24, and co(e,e), Equation 3.14. In fact, what is implied in the model development is the decomposition of the relationship between the stiffness change and the strain state: M = M . ^ (3.40) de dco de where dE/dco and dco/de are described explicitly by Equations 3.24 and 3.14, respectively, co is, in fact, an intermediate value which, in itself is convenient as it allows a further interpretation of the resulting degradation of the material properties in terms of something more physical and more easily observed, damage. COU or rather co'm, through Equations 3.13 and 3.21, is the link between the two functions dE/dco and dco/de. The effect of changing co'm is shown in Figure 3.10. Note that, although the slopes of both the damage growth and residual stiffness functions change, the value of En at 70 Chapter 3 Theoretical Model Development F = F , and hence the relationship between E and F (i.e. e), remains the same in both segments of the two functions. The result is the effect observed above, i.e. the independence of the predicted response with co'm (or &) n). While its value may not directly influence the predicted response, co'm does provide a physical interpretation of the two characterization curves and allows a more direct application of the data available in the literature on stiffness reduction to the characterization of the model. 3.4 Computer Implementation The development of a finite element code from first principles was not a goal of this thesis. There are a number of widely used commercial and research oriented explicit finite element codes available. Some of the most widely used in the area of high-strain-rate and impact loading are the L S - D Y N A family of hydrocodes (LS-DYNA2D and LS-DYNA3D), originally developed at the Lawrence Livermore National Laboratory and now marketed by the Livermore Software Technology Corporation (LSTC) in Livermore, California (Hallquist, 1990, 1993, and 1994). These codes have been used extensively by government research laboratories and industry in applications as varied as sheet metal forming, ballistics, and car crash simulations. LS-DYNA3D features the characteristics desired for the current investigation including, notably, the ability to easily add user material subroutines and constitutive models. The material model described in Section 3.3 has been implemented as UMAT47, a user material (UMAT) module, in the LS-DYNA3D code. Sample F O R T R A N source code for simple UMATs was provided by LSTC and was used as a template for the coding of the C D M model. A schematic of the solution algorithm used in the constitutive routine is shown in Figure 3.11. Descriptions of the U M A T model definition parameters required by LS-DYNA3D, the UMAT47 71 Chapter 3 Theoretical Model Development material model input deck format, and the model outputs are provided in Appendix B. The large number of data needed to describe the piece-wise linear damage growth laws and modulus loss functions necessitated the use of a separate material input file to supplement the material input decks provided by the L S - D Y N A code which allow a maximum of 47 material parameters. The format for this file is also provided in Appendix B. 3.5 Material Characterization One of the most difficult tasks associated with applying many of the available C D M models has been the measurement of the material parameters required. Generally, C D M models have used a simple predefined residual stiffness function (e.g. EI E° = (1 - kco) where k is typically assumed to be 1) in conjunction with a more complex damage growth law (e.g. the Weibull function used by the M L T model outlined above in Section 3.2). The damage laws are frequently abstract and the means of determining the parameters required are often not clear. In the current approach, the damage growth function has been carefully chosen to be representative of currently available and published experimental observations. Equally, the residual stiffness functions, although more complex perhaps than many other C D M models, are also representative of experimental observations as will be discussed in the next sections. There are four sets of material parameters which are required by the model. The first set contains standard material properties such as density and modulus which are required by all models: elastic, elasto-plastic, and C D M . The second, third, and fourth sets are specific requirements of the model developed here. These sets consist of the points that define the damage growth curves (Equation 3.14), the normalized residual stiffness curves (Equation 3.24), and the constants required by the effective strain functions (Equations 3.18 and 3.19). 72 Chapter 3 Theoretical Model Development 3.5.1 Elastic and Strength Constants The experimental methods for measuring material elastic and strength constants for FRPs are widely accepted, and standard methodologies exist (see, for example, A S T M , 1997). While relatively simple to perform, the complete series of tests required to characterize a given material system are expensive both in terms of time and resources. With over four decades of research in the field of composites there exists an extensive body of published characterization data available for many of the widely used material systems (see for example Tsai, 1988). These data can be supplemented by material suppliers and larger material end-users who frequently have libraries of material characterization data for use in design calculations. Material property data is usually quoted for unidirectional lamina/laminates or for the individual fibre and matrix systems used. Deriving the effective laminate/sub-laminate elastic properties is a relatively simple exercise and PC based software such as P C - L A M I N A T E (Radford, 1989) and Cambridge Composites Designer (Xin et al., 1997) exist to simplify the task. Details of the methodology for deriving equivalent laminate elastic constants (lamination theory or laminated plate theory) from the lamina and constituent material properties can be found in most textbooks and some other more detailed publications on the subject (e.g. Herakovich, 1997 and Sun and L i , 1988). 3.5.2 Residual Stiffness Functions The characterization of the effect of damage growth in a laminated composite requires the measurement of the material behaviour under stable crack growth. Traditional testing techniques such as tensile tests result in unstable crack growth soon after damage initiation. The measurement of stiffness loss over an appreciable loading range is therefore not possible using these techniques. Experimentally there are two ways to achieve stable crack growth: 73 Chapter 3 Theoretical Model Development • grow the damage very slowly (e.g. cyclic or fatigue loading) • grow the damage in a very constrained manner (e.g. compact tension specimen) Most literature on the subject concentrates on the first method, fatigue crack growth, as a means of characterizing the susceptibility of composites to subcritical damage growth: initial flaws and in-service damage. As a result little work has been done on damage past saturation of matrix cracking. Typically, the initiation of any substantial fibre damage is considered to be outside the acceptable range of loading. Some notable examples of fatigue crack growth investigations include the work by Highsmith and Reifsnider (1982), Kress and Stinchcomb (1985), Ogin et al. (1985), and Talreja (1985b) to name but a few. A significant outcome of this body of work was the idea that changes in moduli observed during the fatigue testing of coupons provide a direct quantitative measure of damage (Camponeschi and Stinchcomb, 1982 and Poursartip et al., 1986a). Therefore, the converse is true; a physical measure of damage can be used to quantitatively predict changes in modulus. Herein lies the physical basis for the residual stiffness functions presented above in Section 3.3.5. This relationship between damage and modulus also allows the characterization of modulus loss based on the available body of literature on fatigue damage growth. The first segment of the residual stiffness curves proposed in Section 3.3.5 relates to the development of matrix cracking and delamination while the second represents the stiffness loss due either to a mixed mode of failure (matrix and fibre together) or fibre dominated failure. As discussed in Section 3.3.5, the first segment of the residual stiffness curve is most easily characterized by defining the residual stiffness at matrix crack and delamination saturation (R'E at co = co'm). The transition point between the two mechanisms, co11, is a direct consequence of the value of co'm and the damage thresholds discussed in Section 3.5.3 below. 74 Chapter 3 Theoretical Model Development Mathematical Approach A simple mathematical approach which can be used to estimate the stiffness loss at saturation of matrix damage is the ply-discount method. Under a ply-discount scheme, cracked plies are assumed not to contribute to the stiffness matrix in the dominant directions (i.e. transverse modulus, E, and shear modulus, G) and the load is assumed to be redistributed to neighbouring, undamaged, plies. Consider a simple example, a [0/904]2s with lamina moduli El and E2 in the fibre and transverse directions respectively. The laminate stiffness in the 0° direction is therefore: o _ ElAq + E2Ag0 Elam ~ " (3.41) Alam where Aq and AgQ are the cross sectional areas of the plies oriented at 0° and 90° respectively. Assuming that the laminae thickness, t0 and t90, are equal, we obtain: 0 _ E^qHq + E2tgon90 TqHq + f9Q«90 E\n0 + E2n90 no + "90 (3.42) E1 -4 + £ 2 1 6 4 + 16 Ei +4E2 If the laminate is loaded in tension along the 0° plies, matrix cracks will develop in the 90° plies. According to the ply-discount method, the 90° plies will no longer contribute to the stiffness. The damaged material stiffness in the 0° direction, Elam, will now be: 75 Chapter 3 Theoretical Model Development lam £i"o + E2n90 n0 + "90 $ -4 + (0)-16 4 + 16 (3.43) E 5 The normalized residual stiffness in the 0° direction is: E 5 E\ E\IE2 (3.44) E2 +4E2 5 El+4E2 EjE2+4 If we assume a longitudinal to transverse stiffness ratio, Ex/E2, of 15 (a typical value for a CFRP system), the normalized residual stiffness would be predicted to be 0.789. Consider now a [0/90]5s laminate. The same analysis predicts the stiffness loss due to matrix cracking in the 90° plies to be 0.062 (i.e. R'E = 0.938). The analysis becomes more complex with angle-ply laminates where Poisson's ratio and the shear modulus must be included and where the relationship between the matrix and fibre properties and R'E and R'G is more involved but the procedure is the same. Here the composite laminate analysis software packages mentioned in Section 3.5.1 can be used to facilitate the analysis. Implicit in the ply-discount method is the assumption that the crack growth is unconstrained. Results by Talreja (1985b) among others have shown that ply-discount predictions can either over- or under-predict the stiffness loss due to matrix damage depending on the ply lay-up. In response to this, various analytical methods have been proposed including a modified ply-discount method by Highsmith and Reifsnider (1982), a damage mechanics approach by Talreja (1985a and 1985b) and a more detailed micromechanical analysis by Laws and Brockenbrough (1987). While providing more insight into the effects of changing crack density 76 Chapter 3 Theoretical Model Development and crack geometry on stiffness loss, it is unclear if the added complication of these approaches is warranted in the current analysis where only the stiffness loss at saturation of matrix cracking is desired. While errors of five or ten percent in the prediction of the overall stiffness reduction due to matrix damage may be significant in some applications, results of the application of the model presented here show that predictions are not very sensitive to such variations in R'E (see Section 4.1.6 below). The definition of the level of damage at which the matrix damage saturates, co'm (not to be confused with the value of ft)n), is a consequence of the representation of the damage as the volume or projected area of damage in the sublaminate. Consider the same two examples discussed above, namely a [0/90J5S and a [0/904J2S laminate. In both cases, initiation of fibre failure in a given direction, say along the 0° plies (the 1-direction), will occur when matrix damage has saturated in the cross plies, the 90° laminae. The effective projected area of the damage in the 1-direction, assuming complete failure of the 90° plies, will be proportional to the relative total thickness of the 0° and 90° laminae: a>m=l_ANet_= "90 ( 3 4 5 ) ATotal "0 + "90 where ATotal = n0 • tpty + n90 • tply is the total cross-sectional area and ANet = n0 • tp[y is the cross-sectional area of the remaining load bearing material. For a [0/90J5S laminate, at saturation of damage in the 90° plies, co'm will be 0.5 ( n 0 = n 9 0 =10). By contrast, co'm will be 0.8 for the 1-direction of a [0/904J2S (nQ=4 and n 9 0=16). In the 2-direction (the fibre direction of the 90° plies), the corresponding values for co'm will be 0.5 and 0.2 for the [0/90]ss and [0/904J2S laminates, respectively. 77 Chapter 3 Theoretical Model Development The discussion above assumes that damage or cracking in the failed plies at saturation of matrix damage is complete. That is, the cracks are complete through-cracks. It also assumes that there is no stacking sequence effect. This is not strictly accurate. A [0/±45/90]ns lay-up will be more prone to delamination than a [0/45/90/-45] „s> for example. However, the approach provides a first approximation of co'm. Experimental Observations While providing guidance on appropriate values for R'E and co'm, the methods described above are intended for cases where experimental results are not available. As mentioned in the previous section, the ply-discount method is not a completely accurate predictor of stiffness loss due to damage. Figure 3.12a and Figure 3.12b from Talreja (1985a), for example, show experimental results for normalized stiffness reduction as a function of applied stress for a [903/0] s GFRP and a [45/90/-45/90/45/90/-45/90/90]s CFRP. Also plotted are the ply-discount predictions for each laminate. In Figure 3.12a the experimentally measured normalized stiffness at saturation of matrix cracking is 0.6 for the [903/0]s GFRP . The corresponding ply-discount prediction is approximately 0.53, an error of 12%. Figure 3.12b shows a measured E/E° of 0.72 with a predicted value of 0.82, an error of 14%. While correctly predicting the trend, the larger constraint offered by the [45/90/-45/90/45/90/-45/90/90]s compared to that of the [903/0] s, the ply-discount method over-predicts the stiffness reduction. Figure 3.13 shows more experimental results, in this case for quasi-isotropic CFRP laminates where normalized residual stiffnesses in the range 0.60 to 0.80 were observed. The results from Bakis and Stinchcomb (1986) show the effects of loading condition on stiffness reduction while Kress and Stinchcomb (1985) highlight the effects of material lay-up. In the preceding discussions of stiffness reduction due to matrix cracking, no mention has been 78 Chapter 3 Theoretical Model Development made of delamination. While a significant mode of failure in laminated composites, the difficulties associated with modelling delamination in a plane-stress analysis has often lead to the omission of an explicit description of its effect on the material response. In the analyses presented here, delamination and matrix damage have been treated synonymously. A l l discussion of the effect of matrix damage on stiffness predicted by the normalized residual stiffness functions, RE, apply equally to delamination. This implicit combination of the intralaminar and interlaminar matrix failure modes is based on work which has shown that the two damage modes generally occur together. There are a few exceptions such as delamination that grows from a free surface such as the edge of a plate or a hole in the absence of intralaminar matrix cracking (e.g. Hsu and Herakovich, 1977 and O'Brien, 1982) and matrix cracking that can occur in internally pressurised composite cylinders without causing significant delamination (e.g. Hull et al., 1978). Li most cases, however, matrix cracking drives delamination between plies of dissimilar lay-up angle which, in turn, drive more matrix cracking when the delamination front reaches the fibre direction of the neighbouring ply (see Figure 2.4). Further, work by a number of authors has shown that the in-plane stiffness loss in angle-ply laminates is linear with delamination size (Poursartip et al., 1986a). The linear relationship is shown in Figure 3.14 as is a simple rule of mixtures prediction of the stiffness loss based on the laminate stiffnesses of the sublaminates formed by the delaminations. Experimental measurements of stiffness reduction due to matrix cracking already include the effects of delamination and, as the reduction is linear with size of damage (i.e. co), the linear residual stiffness function presented in Equation 3.24 is still valid. 79 Chapter 3 Theoretical Model Development 3.5.3 Damage Growth Functions Characterization of the two damage growth functions is simply a matter of defining the thresholds for initiation of the two damage modes, matrix cracking/delamination and fibre failure. The other pieces of information required, the values for co'm and co'f (co'j = 1 -co'm), have been defined in the discussion of stiffness reduction above in Section 3.5.2. Mathematical Approach Characterizing the value of the constants F1, F n , and Fm may seem difficult due to the interaction between the normal and shear strains in the damage threshold functions Equations 3.18 and 3.19. However, this task is greatly simplified if one notes that in the case of a uniaxial strain, the damage growth functions simply become a function of the applied strain. From Equation 3.35 F1, F n , and Fm are the applied uniaxial strains at which each failure mode (matrix cracking/delamination, fibre failure, and final rupture) initiates. Therefore, a first approximation of these thresholds can be made based on the results of uniaxial tensile tests on laminate coupons. Consider first the value of F(e) at initiation of matrix cracking, F 1 . To simplify the analysis, it is assumed that matrix failure in a ply will be driven by the resolved strain normal to the fibre direction. F 1 is then the applied strain which, when resolved into its component normal to the ±0 ply, is equal to the transverse lamina failure strain. The strain transformation equation from laminate to lamina strains for a ply at an angle 0 to the global x-y laminate co-ordinates is 9 9 ~ cos 0 sin 0 -s in0cos0 £y - = 9 9 sin 0 cos 0 sin 0 cos 0 £y (3.46) 7l2. Y xy 9 9 2 sin 0 cos 0 -2 sin 0 cos 0 cos 0 - s i n 0 y xy where ex, £y, and y ^ are the applied laminate strains and £,,£2, and y 1 2 are the lamina 80 Chapter 3 Theoretical Model Development strains. The value of interest is e 2 = e 2 , where e 2 is the lamina transverse (i.e. matrix) failure strain. We are considering a uniaxial applied strain, ex = Fx and e = y = 0, therefore: From Equation 3.48, it is clear that the minimum value of Fx , over all plies, corresponds to the largest value of 0. Hence, matrix cracking will initiate in the ply which is oriented closest to normal to the applied strain, as one would intuitively expect. Following the same procedure, Ff (fibre failure threshold) is the applied strain which corresponds to a lamina strain in the 1-direction equal to the longitudinal failure strain of the lamina: The value of Fx is the minimum value found when Equation 3.49 is applied to each lamina. To illustrate, consider some examples. In a [0n/90m]s or [0/±45/90]ns laminate subjected to a unidirectional strain, ex, along the 0° plies (the x-direction) , the plies subject to the maximum matrix strain will be the 90° plies. The ply strain normal to the fibres in those cross plies will be equal to the applied strain. Putting 9 = 90° in Equation 3.48 we can write F x ! = e 2 , the lamina transverse failure strain. Because of the symmetry in the x-and y-directions, the matrix damage threshold will be the same for both damage growth functions, Fx = Fy. Applying the same analysis to determine F^ and F ° , the fibre thresholds, one finds that Ff = F^ = e{. For more general lay-ups, the values of F 1 and F n will depend on the stacking sequence and the £ 2 = e 2 = ex s in 2 6 (3.47) or (3.48) (3.49) 81 Chapter 3 Theoretical Model Development direction. Consider a [0/±45] n S lay-up. If the applied strain is along the 0° ply (in the x-direction), matrix failure will initiate in the ±45° plies and from Equation 3.48 for 8 = 45° f \ - 2e'2. For an uniaxial applied strain in the y-direction, the first matrix cracking would appear in the 0° plies, hence the value of F j would be equal toe 2 . Similarly for a uniaxially applied strain in the x-direction the fibre damage threshold is F^ = e[/cos2 0° = e{ and the TT / 9 corresponding value in the y-direction is F y u =e{ /cosM5° = 2e1'. Experimental Approach The mathematical approach described above presented a simple method for estimating the parameters F 1 and F n using published or easily obtained material data. One effect which is not taken into account is the constraint offered by neighbouring undamaged plies. The strain to onset of matrix cracking is observed to increase as the degree of constraint increases as shown in Figure 3.15. For example, F 1 = e'2 in the 0° ply direction for a [0/90]4s is observed to be higher than the corresponding value for a [0/903]s. For other angle-ply laminates the effect of the constraint depends on the lay-up and is by no means intuitive. Talreja (1985b) observed almost no difference between the strain to initiation of matrix cracking in a [60/90/-60/90/60/90/-60/90]s and that of an unconstrained 90° ply. If an increased level of accuracy is desired, one must turn to experimental measurements of the strain at onset of each phase of damage. The same stiffness reduction as a function of applied load curves used above to estimate R'E (e.g. Figure 3.16a), can also provide information on F 1 and F 1 1 . The initiation of matrix failure is clearly visible in Figure 3.16a as a reduction in stiffness which occurs at an applied stress of approximately 50MPa. Given the laminate modulus, the laminate strain at matrix damage initiation can be estimated. F 1 can be evaluated more directly from Figure 3.16b where the measured matrix crack density has been plotted as a 82 Chapter 3 Theoretical Model Development function of the applied strain for a CFRP system (F1 = 0.005). The laminate strain at fibre breakage initiation, F n , can also be estimated from the same experimental results if fibre failure is assumed to initiate at saturation of matrix cracking. From Figure 3.16a F n would be estimated to be 275 MPa divided by the laminate modulus. Note that in this case, the damaged or reduced laminate modulus should be used (i.e. E = 0.54- E°). Figure 3.16 again provides a more direct measure with Fn equal to 0.026. Rupture Threshold The parameter which is the least intuitive is the threshold for rupture, Fm . Here one must rely exclusively on experimental data. There is no simple analytical method available for estimating this value as it depends strongly on the lay-up and on interactions between neighbouring plies, both damaged and undamaged. Work by Kongshavn (1996) and Kongshavn and Poursartip (1997) provides some insight into an experimental technique which can be used to measure this value. Kongshavn grew damage in a composite laminate by loading a prenotched Oversized Compact Tension (OCT) specimen (see Figure 3.17). The damage zone ahead of the notch was observed to develop to a stable size as it progressed across the specimen, as shown in Figure 3.17. The stiffness loss and strain-to-failure of the damaged material in the softening zone were measured by cutting small tensile specimens from the damaged and undamaged zones (refer to Figure 3.17) and loading them to failure. Specimens cut from the undamaged region showed stiffnesses similar to the laminate stiffness and failure strains on the order of the fibre failure strain (refer to Figure 3.18). These results would be expected from laminate tensile tests. By contrast, specimens cut from within the damaged zone showed very low stiffness but high strain 83 Chapter 3 Theoretical Model Development to failure. Although only a few specimens could be cut from the specimens tested, the locus of the failure points obtained from the tests define the preliminary shape of a characteristic strain softening stress-strain curve for the two CFRP systems investigated, a [-45/90/45/0]2s T300/F593 and a sandwich panel with [F0/-45/90/45/0/45/90/-45/F0]T AS4/8552 face sheets (the layers denoted by F0 are woven [0/90] outer plies). From these results, shown in Figure 3.18, the value of Fm for these materials can be estimated to be between 0.03 and 0.04. These results are significant because they are much higher than the fibre failure strains (on the order of 1.5% or 0.015), the strain which one would normally associate with rupture of the laminate in a conventional unnotched tensile test. Kongshavn's experimental technique offers a robust and physically meaningful methodology for measuring this characteristic and the results of a statistically significant number of specimens should yield an accurate measure of Fm. 3.5.4 Effective Strain Functions The selection of the damage potential function presented in Section 3.3.4 above is the least physically based aspect of the model development. As a result, a detailed methodology for characterizing the required parameters (K, L, S, T, and U) is not currently possible. However, it is possible to argue from a purely intuitive point of view what some of the constants in Equations 3.18 and 3.19 should be. In keeping with the examples discussed above, consider again the case of a [0n/90m]s laminate. To a first approximation, damage growth in the 0°-direction (cox) will be uncoupled from ey. That is, strains transverse to the fibre direction will cause matrix cracking and ultimately fibre breakage in the transverse 90° plies (i.e. in the layers oriented parallel to the applied load) but 84 Chapter 3 Theoretical Model Development will have little effect on the plies oriented perpendicular to the applied load. Similarly, ex will have little influence on coy . Equations 3.18 and 3.19 should therefore be of the form (assuming equal contributions in tension and compression): Fx = v i J + fy \ 2 I xy V Sx J + fy \ 2 I yz v Tx j + f \ 2 I zx and V 1 j + xy + (y A 2 ' yz T v y ) + ZX Kuyj (3.50) (3.51) Note that the contribution by a given strain can be removed by setting the scaling constant to a large number. Arguments as to the relative contributions of the shear strains are more difficult to make. Typically, the through-thickness shear strains in a shell element will be smaller than the in-plane strains and, as a result, these can be treated as having a secondary effect on the damage growth with the associated scaling factors, T and U, set to 1. On the other hand the in-plane shear strain can be quite large especially when the in-plane shear modulus is degraded as is the case in this model. The correct value for S will be one of the parameters that is looked at in the actual application of the model later in Chapter 4. For the [0n/90m]s laminate considered above, a first approximation of the damage potential functions would be: F -x y fy \ 2 I xy V Sx J (3.52) 85 Chapter 3 Theoretical Model Development and 7 (ryzf+irzxf (3.53) + + 3.6 Post-Processing The constitutive model returns the element stress state to the F E M code which in turn updates nodal forces, contact forces, displacements, velocities etc. These values are typically available from the F E M code as standard outputs for either contour plots or time histories. The dedicated post-processor for the LS-DYNA3D code, LS-TAURUS, is no different in this respect. What is unique in the UBC-CODAM3Ds damage model is the ability to predict a damage zone in a manner which is consistent with experimental observations. Experimental measurements of matrix damage in composite laminates generally come from pulse-echo ultrasonics (PEUS), C-scan, or other non-destructive measurement techniques, and section micrographs (see the discussion by Girshovich etal., 1992 for example). In both cases, the results are used to create damage maps. C-scans generally give projected delamination area while micrographs allow a more detailed ply-by-ply measurement of the variation of delamination size through the thickness. Fibre damage is usually measured using a technique called deplying whereby the majority of the matrix material is burnt off in an oven leaving the individual plies intact like stacked sheets of paper. Carefully separating the laminae allows a visual inspection of fibre damage which can then be quantified by measuring the total length of fibre breakage, perpendicular to the fibre direction, in each ply. Multiplying the result of these measurements by the ply thickness gives a fibre breakage area. An example of the application of this method can be found in Delfosse and Poursartip (1997). A method of predicting matrix and fibre damage based on the interpretation of the damage state 86 Chapter 3 Theoretical Model Development with reference to the damage growth functions has been incorporated in the model. If the current state of damage at an integration point in an element, co, falls on the first portion of the damage growth function, it can be entirely attributed to matrix cracking and delamination (co'm - co). The relative amount of matrix cracking is determined by the proportion of the damage at saturation which is made up of matrix cracks and delaminations, co'm in Figure 3.19a: % matrix damage = ^ - x l 0 0 % (3.54) If the damage state falls on the second segment of the curve, the total damage is made up of a portion of fibre breakage and a portion of matrix damage/delamination. Again, the relative amounts of each can be determined using the damage growth curves as shown in Figure 3.19b: % matrix damage = - ^ L x l O O % (3.55) % fibre damage ^^L=(0~C0'n X 100% (3.56) co'f \-co'm The representation of these percentages as an actual total area of damage involves an interpretation of experimental observations of the area of matrix cracking and fibre breakage in a sublaminate at saturation of damage. At the saturation of damage in a four ply sublaminate the staircase or helical pattern of delamination and associated matrix cracking results in a total area of matrix cracking approximately equal to the in-plane area of the sublaminate. Consider an example. Assume that each integration point represents a sublaminate, and that the planar area of an element is 10 mm . Further assume that the lay-up is quasi-isotropic such that the area of matrix damage at saturation in a sublaminate is also equal to 10 mm 2. If the percentage of matrix damage for an integration point (i.e. sublaminate) were predicted to be 84% then the area of matrix damage in that sublaminate would be calculated to be 8.4 mm 2. Now 87 Chapter 3 Theoretical Model Development consider the entire finite element mesh, if the percentages of matrix damage at each integration point are summed over the entire mesh, the result is a total percentage of matrix damage relative to the saturation damage at an integration point (i.e. a fraction of element sized sublaminates that are matrix damaged to saturation). While the same calculations apply to fibre damage, the area of fibre breakage associated with saturation of damage in a sublaminate is not as simple to define. For a quasi-isotropic sublaminate, a first approximation may be four times the cross sectional area of a ply, or simply the cross sectional area of the sublaminate, times the fibre volume fraction (i.e. roughly the total cross-sectional area of fibres) but experimental evidence shows that the amount of fibre damage is not simply a function of the lay-up. Not only does fibre damage tend to be more localized than matrix damage but the material system (e.g. brittle versus tough matrix) plays a role as well (Delfosse et al., 1995). 3.7 Conclusion A mathematical model has been developed to predict damage growth and its effects on the response of laminated FRP composites. The formulation is based on the sublaminate response, recognition of the fact that the laminate response is driven not only by the lamina properties but also by the ply interactions through the stacking sequence and damage growth (i.e. it is a system response). The resulting application of C D M at the sublaminate level within the framework of a F E M model is unique and leads to a formulation that is both physically meaningful and mathematically simple. Throughout the model development, the implications of experimental observations of damage growth and of the effects of damage on the material response have been used to derive the 88 Chapter 3 Theoretical Model Development mathematical formulation. The model is based on a two phase response, the first dominated by matrix cracking and delamination and the second by combined matrix and fibre damage growth. The bilinear damage growth and residual stiffness functions used to describe this behaviour have been constructed based on a simple interpretation of widely available experimental observations of damage growth and associated degradation in material properties. Methodologies for determining the material constants required by the model have also been discussed. The most significant aspect of the methodologies is that they are based entirely on a physical interpretation of damage in laminated FRPs. Most properties can be gathered from the literature provided a well-established base of research exists for the desired composite system. Where experimental data is not available, simple analytical techniques have been proposed to provide order-of-magnitude estimates of many of the model input parameters. The physical basis of the model also allows a more direct interpretation of the damage predictions than is possible in other C D M models such as the M L T model discussed at the beginning of this chapter. The significance of this will be demonstrated in the next chapter where two applications of the model are presented. The problems chosen as the basis for the case studies are widely different. The first being a dynamic analysis of normally impacted CFRP plates, one with a brittle and the other with a tough matrix system. The second case study is that of a quasi-static analysis of crack growth in a prenotched CFRP panel. 89 Chapter 3 Theoretical Model Development [M] o 0 • . 1 - 1 --of • t 1 ""**" « 1 > • , 1 ' « 1 > [ft - 1 — • * • — p * - U l 1 1 U -[M] 7 e e (a) stress equivalence (b) strain equivalence Figure 3.1 Hypothesis of stress and strain equivalence. (Adapted from Simo and Ju, 1987) 90 Chapter 3 Theoretical Model Development Observed Damage Growth Experiments/Literature Theoretical ^ Development/ Characterization Implementation/ Verification Brick General 3D Damage Model Figure 3.2 Model development approach. 3D Shell 2D Axisymmetric 3D Brick Global Local Global and Local Computationally Efficient Detailed In-Plane Predictions but Smeared Through-Thickness Computationally Efficient Smeared In-Plane Predictions but Detailed Through-Thickness Computationally Expensive Detailed In-plane and Through-Thickness Predictions Figure 3.3 Relative merits of modelling approaches. 91 Chapter 3 Theoretical Model Development Node 4 Sublaminate Based Constitutive Model Figure 3.4 Shell representation of a laminated composite using a lamina and sublaminate based constitutive model. 92 Chapter 3 Theoretical Model Development Figure 3.5 Construction of the bilinear damage growth law (c) based on the combined contributions of (a) matrix and delamination damage and (b) fibre breakage. 93 Chapter 3 Theoretical Model Development E/E° Single Damage Mode Damage E/E° Phase 1 r* • Damage Phases 1 + 2 < H CO (a) (b) Figure 3.6 Modulus variation predicted by (a) linear and (b) bilinear normalized residual stiffness functions. (a) (b) Figure 3.7 Bilinear residual stiffness curve showing modulus loss (EI E ) versus damage (co) relationship showing (a) generalized function with a residual modulus and (b) special case with saturation of damage at rupture (com - 1 and Em = 0). 94 Chapter 3 Theoretical Model Development Damage Phase 1 (matrix/delamination) 0.00 0.01 0.02 0.03 0.04 0.05 Strain (nun/mm) Figure 3.8 Predicted one-dimensional stress-strain response based on bilinear damage growth and residual stiffness functions. 95 Chapter 3 Theoretical Model Development Damage Growth Stiffness Reduction u> 0.5 0.000 0.025 0.050 0.00 F 800 0.50 1.00 0.00 0.01 0.02 0.03 e (mm/mm) 0.04 Damage Growth Stiffness Reduction O) 0.5 0.000 0.025 0.050 0.00 0.50 1.00 0.00 0.01 0.02 0.03 0.04 e (mm/mm) (a) Instantaneous failure (b) Classical CDM progressive damage Damage Growth Stiffness Reduction Damage Growth Stiffness Reduction 0) 0.5 + 0.00 0.01 0.02 0.03 e (mm/mm) 0.04 0.00 0.01 0.02 0.03 e (mm/mm) 0.04 (c) Non-linear loading with brittle fracture (d) Linear-elastic, onset of damage, and softening Figure 3.9 Examples of one-dimensional predicted stress-strain response for the proposed composite damage model. 96 Chapter 3 Theoretical Model Development Figure 3.10 Effect of changing the damage associated with saturation of matrix cracking on the residual stiffness and damage growth functions. Shifting co'm (keeping E'm constant) results in a shift in tu1 1, defined by the damage growth function. This shift in co11, when plotted on the residual stiffness curves results in no change in £ n . As a result, £ n corresponding to F n is the same and the rate of stiffness loss with strain ( £ n ) is unchanged. 97 Chapter 3 Theoretical Model Development Constitutive Law e° de 00° e = 8° + de co=/(e) C =/(C°,co) a=/(C,e) I a co co E_ E° F(e) co Figure 3.11 C O D A M solution algorithm. 98 Chapter 3 Theoretical Model Development Gl./Ep. (903.0)s o calculated » measured (a) Gr./Ep. (45.90.-45.90.45.90.-45.90ls O calculated x measured ply discount prediction 200 6, MPa (b) Figure 3.12 Experimentally measured and predicted normalized residual stiffness for (a) a [9O3/OJS GFRP and (b) a [45/90/-45/90/45/90/-45/90/90]s CFRP (Talreja, 1985b). Calculated values are based on a continuum damage mechanics approach developed by Talreja (1985a). 99 Chapter 3 Theoretical Model Development oc o z 0.2 Low Level Tension Stiffness High Level Tension Stiffness 0.2 _L 0.4 0.6 NORMALIZED LIFE 0.8 1.0 (a) to UJ £ 9 0 8 0 S 70 o t 60 o SI 5 0 i B - S P E C I M E N A - S P E C I M E N 1.0 2.0 3.0 4 .0 L O G C Y C L E S 5.0 6.0 (b) Figure 3.13 Experimentally measured normalized residual stiffness in fatigue loaded quasi-isotropic T300/5208 CFRP laminates. The results in (a) from Bakis and Stinchcomb (1986) show the effect of compression (solid line) and tension (dashed line) dominated loading on the stiffness loss in a [0/45/90/-45]4s plate, (b) from Kress and Stinchcomb (1985) shows the effect of material lay-up. Specimen A is a [0/90/±45]s lay-up with a higher fatigue strength than specimen B which is a [45/90/-45/0]s of the same material. 100 Chapter 3 Theoretical Model Development ^ 1 R o UJ 0.9-to _l DU 0.8-o Q 0.7-UJ CO 1 MA 0-6-rr o 2 4 * V A 'RULE-OF-MIXTURESl - . X PREDICTION A B18 • V A R I O U S CFC (45/90/-45/0 ) s 1— I , I L D E L A M I N A T E D A R E A ( A / A Q ) Figure 3.14 Measured in-plane normalized residual stiffness with delamination size for a [45/90/-45/0]s CFRP loaded in tension-tension fatigue. Also shown is the good agreement with an empirical rule of mixtures based stiffness reduction rule. (Poursartip et al., 1986a) £ FPF V. 1.00 -0.75 0.50-0.25 0 0 2.0 4 0 6.0 8.0 2 n , number of 90° pl ies Figure 3.15 Strain for onset of matrix cracking (eFRp) as a function of number of 90° plies for a [O4/9OJS CFRP. The lettered regions are associated with decreasing levels of constraint. (A) no constraint on cracking with a single fracture, (B) low constraint with multiple fractures, (C) high constraint with multiple cracking and (D) highly constrained single fibre dominated failure. (Talreja, 1985b) 101 Chapter 3 Theoretical Model Development APPLIED STRESS (MPo) (a) (b) Figure 3.16 Measured crack density as a function of (a) applied load for a [0/±45/90]s CFRP from Reifsnider and Highsmith 1981) and (b) applied strain for a [60/907-60790/60790/-60/90/90]s AS4/3502 CFRP from Talreja (1985b). 102 Chapter 3 Theoretical Model Development =100 mm W = 0.30 =200 mm 7 mm p 25 mm I I Fully damaged materia] c^ 7 5 mm > Process zone (partially damaged material) o o ^ Tensile specimens cut from damaged and undamaged material Applied displacement, d Clip gauge t 9 Damaged material Figure 3.17 OCT specimen used by Kongshavn and Poursartip showing damage growth region and tensile specimens used to characterize properties of damaged material. (Adapted from Kongshavn and Poursartip, 1997) 103 Chapter 3 Theoretical Model Development Figure 3.18 OCT tensile specimen results showing (a) undamaged and damaged stress-strain curves and (b) predicted softening behaviour from the locus of failure points for two CFRP systems. (Adapted from Kongshavn, 1996) 104 Chapter 3 Theoretical Model Development Figure 3.19 Portion of damage attributed to matrix cracking/delamination (0Jm) and to fibre breakage (COf) for a given damage state (CO). 105 Chapter 4 Numerical Case Studies 4. Numerical Case Studies 4.1 Case Study 1: Non-Penetrating Impact on CFRP Plates Typical structural problems involve dynamic loading and detailed predictions of the various damage modes are of interest. To investigate the application of the model developed in this thesis to these types of problems, an extensive investigation of dynamic out-of-plane loading of composite test coupons by Delfosse (1994) has been used as the basis for the first case study. 4.1.1 Experimental Measurements The experimental work reported below is presented in detail in Delfosse (1994). Further synthesis and analysis of the data has been presented in Delfosse and Poursartip (1995 and 1997) and Delfosse et al. (1995) all of which have been used to guide the application of the model and the discussion of the predictions. The experimental work investigates the influence of a number of material and target related parameters on the normal impact response of polymeric composites. These parameters include target thickness, material system (fibre and matrix), indentor shape, impact energy, and target size (boundary condition effect) to name a few. For the purposes of the current study, two parameters were selected as variables for evaluating the performance of the COD A M model; target material and available energy. The work by Delfosse and Poursartip describes the effects of normal impact on a variety of laminated composite materials. Of particular interest to the current investigation is the work on two CRFP systems, T800H/3900-2 and TM6/937. While both materials have similar elastic properties, the 937 resin used in the latter is more brittle providing a contrast with the tougher 106 Chapter 4 Numerical Case Studies 3900-2 matrix system. Four velocity regimes were investigated experimentally, quasi-static, low velocity (high mass), high velocity (low mass) and ballistic. Results for the quasi-static deflection tests were included for comparisons with the numerical results, but were not modelled. Likewise, the ballistic tests involving penetration are not investigated here. The numerical model has no means of accurately predicting the penetration response of the composite. Therefore, simulation of this type of test is not currently within the capabilities of the COD A M model. Test Set-ups The target geometry used predominantly in the experimental investigation is that of a 101.6 mm by 152.4 mm (4 inch by 6 inch) coupon clamped on to an aluminium backing plate with a 76.2 mm by 127.0 mm (3 inch by 5 inch) rectangular opening (see Figure 4.1). The plate geometry is consistent with the Boeing and S A C M A Compression After Impact (CAT) Standards (Boeing, 1988 and S A C M A , 1988). The specimen is fixed to the test frame using four adjustable rubber tipped toggle clamps arranged two per side along the length of the coupon. Under these conditions, the coupon can essentially be considered as simply supported. In all the tests discussed in this section, a 12.7 mm (0.5 inch) hemispherical tipped hardened steel indentor was used. The quasi-static tests were performed using a universal testing machine. The indentor was fixed to the cross head of the machine. The tests were displacement controlled with a cross head displacement rate of 1.3 mm/min. Force-time data were gathered using a load cell mounted between the indentor and the cross head while the displacement was measured using a linear voltage displacement transducer (LVDT) mounted on the cross head. The low velocity impact testing was performed on two drop weight towers, one located at INTEC 107 Chapter 4 Numerical Case Studies Corporation in Bothell, W A and the other at the Department of Metals and Materials Engineering at The University of British Columbia. Apart from minor variations in the data acquisition systems used, the only significant difference was the drop head mass. INTEC's system uses a 6.141 kg head while the U B C systems drop weight has a mass of 6.330 kg. Again, the force and displacement time histories of the events were recorded. A gas gun, which launches a 0.314 kg projectile at velocities up to 50 m/s, was used to perform the high velocity impact tests. The bulk of the projectile mass is the 0.280 kg slider body with the remainder made up of the screw-on indentor. The projectile is instrumented with a piezoelectric load cell allowing the continuous measurement of the force-time history during the impact. The initial or impact velocity of the projectile is measured using three pairs of optical sensors located at the end of the gun barrel, just before the point of impact. Post-processing software used the data gathered to generate the force and displacement-time history of the event. The post-processing software also allowed the selective filtering of the measured response to remove oscillations introduced by the resonance of the indentor. A more detailed discussion of the filtering technique and overall set-up of the gas gun can be found in Delfosse et al. (1993). Experimental Measurements In addition to the force and displacement-time histories, detailed post-test damage characterizations were carried out. Of interest was the measurement of the relative energy absorption provided by the different damage mechanisms. For this, quantitative measurements of matrix damage, characterized by delaminations, and of fibre breakage were made. Pulse-echo ultrasonic (PEUS) and C-scans were used to measure internal delaminations. These measurements provide mappings of the location and projected area of delaminations (see Figure 4.2). However, the projected area is not an accurate measure of the total area of 108 Chapter 4 Numerical Case Studies delamination. Destructive examinations (sectioning and micrographs) of the specimens were used in conjunction with the PEUS measurements to determine the total delamination area. The measurement of fibre breakage requires the deplying of the impacted coupons. This is done by burning off the resin in an oven. The individual plies can then be peeled off allowing a visual inspection of each layer. Fibre breakage was determined directly from each ply by measuring the extent of fibre fracture perpendicular to the fibre direction (see Figure 4.3). Multiplying the width of fibre breakage by the ply thickness and summing over all the plies results in a total fibre breakage area. 4.1.2 Objective The goal of the numerical work is to quantitatively predict the various experimental measurements made during the test series. These comparisons include force and displacement-time histories and total fibre and matrix damage, size of fibre and matrix damage zones, and energy loss, all as a function of available energy. The most significant test is in predicting the differences in behaviour between two material systems, one characteristic of a more brittle response, the other of a tougher response. If the differences in the material model input can be argued from a physical point of view, or from experimental observations (i.e. not using calibration constants) then the predictive capability of the model can be used with a greater degree of confidence. 4.1.3 FEM Model In order to make the F E M model of the CAI test more efficient, a number of simplifications have been made. The first comes in the omission of a detailed model of the aluminium test frame. The test coupon is modelled as a simply supported 76.2 mm by 127 mm (3 inch by 5 inch) plate. 109 Chapter 4 Numerical Case Studies The hardness of the steel and the relative stiffness of the indentor compared to the through-thickness stiffness of the composite plates also make the assumption of a non-deformable indentor (modelled as a rigid material, MAT20, in LS-DYNA3D) reasonable. As a result, only the profile of the impactor has been included in the F E M model (see Figure 4.4). Previous work, reported in Williams and Vaziri (1995 and 1998) and Pierson and Vaziri (1996), has demonstrated that this simplified model is sufficient to provide good comparisons with experimental measurements. Four noded Belytschko-Tsai quadrilateral elements were used throughout the model. For the runs using the C O D A M model, the symmetry of the problem and the in-plane isotropy of the material would have allowed the use of two fold symmetry (quarter plate symmetry) to reduce the size of the model. However, two other lamina based composite damage models were used to provide comparisons with the existing capabilities of L S - D Y N A 3 D and as a result, a full model was required. Twelve integration points were used through the thickness, two per sublaminate. A minimum of six could have been used, one for each [0/45/90/-45] sublaminate. However, the problem involves a significant amount of bending deformation and thus the additional discretization for capturing the through-thickness behaviour is warranted. As mentioned above, the coupon was assumed to be simply supported and appropriate displacement boundary conditions (constrained out-of-plane, free in-plane) were applied along the edges of the plate F E M mesh. Rotational and translational constraints were applied to the rigid indentor so as to constrain movement in any direction other than normal to the plate. An appropriate initial velocity was applied to the indentor. Hourglass control was achieved using the standard LS-DYNA3D viscous method with the 110 Chapter 4 Numerical Case Studies default coefficients. The default hourglass control parameters were also found to be sufficient and no zero energy modes of deformation were observed during the runs. Contact between the indentor and plate was modelled using the penalty stiffness method incorporated in LS-DYNA3D. A rigid material model with the density of steel (see Table 4.1) was applied to the indentor mesh. An elastic modulus and Poisson's ratio (also shown in Table 4.1) are also required by the rigid material model but these values are only used by the contact algorithm to predict an appropriate contact stiffness. The use of a simple shell representation of the indentor does not provide the correct kinematic properties because the mass of the shell structure is significantly lower than that of the actual projectile. In fact the mass of the FEM indentor provided by the shell elements is only 10 g. One possible solution would have been to artificially increase the density of the rigid material. However, this adversely affects the critical time step for explicit time integration and is therefore unacceptable. As an alternative, a discrete mass was associated with the central node of the indentor mesh. By setting the value of this lumped mass to an appropriate value (e.g. 304 g for the gas gun projectile) the correct mass of the indentor can be modelled. Then: 304 g + 10 g = 314 g lumped mass shell mass corrected mass Two sets of numerical analyses were carried out for each velocity range. The first set consisted of tests at velocities corresponding to impact energies of 5 to 601 of energy, in increments of 5 J. Additional runs at 12, 17, and 22 J were used to provide more resolution in the results near the onset of damage, identified as 18 I for matrix/delamination initiation and 22 J for fibre breakage from the experimental results. The second set of runs corresponded to the experimental low and high mass tests. All the low mass tests were run with an impactor mass of 314 g. The majority 111 Chapter 4 Numerical Case Studies of the high mass impacts used an impactor mass of 6141 g but three of the experimental tests used a heavier 6330 g mass. 4.1.4 Interpretation of Results The quantities which are required for comparison with the experimental results are the force-time, force-displacement, and energy-time histories, total damage (matrix/delamination and fibre), spatial distributions of damage, impact energy, and energy absorbed by the target. Output available directly from LS-DYNA3D includes impactor contact force, displacement, and kinetic energy histories. The impact energy is set by the incident velocity and the energy absorption in the plate can be calculated from the residual impactor energy. Damage predictions from the numerical model come in two forms. The element integration point values of cox and coy are made available by the model through the LS-DYNA3D history variables and can therefore be used to generate fringe plots of damage at each sublaminate through the thickness of the plate. However, a more detailed measure of the predictive capability of the model comes from the total damage histories (refer to Appendix B). Output from the model is in the form of a total matrix damage and element fibre damage for each damage variable (i.e. cox and coy) as a percentage of damage at saturation (refer to Section 3.6), summed over the entire FEM mesh (i.e. over all the integration points). With two integration points per sublaminate (Section 4.1.3), halving the total damage gives us the total percentage of sublaminate damage which is attributed to matrix cracking/delamination and fibre breakage. As the lay-up is quasi-isotropic, an average of the damage in the two directions (x and y) provides a reasonable measure of the overall sublaminate damage. In order to convert these percentages into a physical measure of damage area, the matrix/delamination and 112 Chapter 4 Numerical Case Studies the fibre damage areas that are associated with saturation of the respective types of damage must be known. Saturation of delamination damage in a sublaminate will result in a staircase pattern of delamination damage shown previously in Figure 2.5. The total area of these delaminations is equal to the projected area of the staircase which, in an element sized sublaminate, is the element area. As a result, the total matrix damage area, which corresponds to the experimental measurement of total delamination area as a measure of matrix cracking, is: integration 2 ^ OJ ^ CO m y . _ points . rA n\ A total - X "-element (4.Z) delamination n where co^/co'^ and comy jco'my are the percentages of matrix damage relative to the saturation level of damage (refer to Section 3.6), Aelement is the planar area of an element, and n is the number of integration points per sublaminate in the model (in the CAI model n-2). Fibre breakage is more difficult to associate with the same type of calculation. While delamination is easily viewed as an area with respect to an element, fibre breakage is more of a linear damage measure (see Figure 4.3) even though it is presented experimentally as an area by multiplying it by the ply thickness. As a result, the experimental observation that fibre breakage area and delamination size are, to a first approximation, linearly related (e.g. the experimental trend line in Figure 4.5) has been used to derive total fibre breakage as a function of element area (i.e. delamination size). If we use the same calculation shown in Equation 4.2 but instead base the damage area on the predicted total percentage of fibre breakage (cOfjco'f) through this scaling factor, we arrive at a fairly realistic measure of fibre breakage area: 113 Chapter 4 Numerical Case Studies integration \ fx points pprntinn 2 CO fY CO A, fyj A A A. 1 fibre breakage (4.3) 1 total fibre breakage n ^delamination ^element For the T800H/3900-2 system, results reported in Delfosse (1994) (Figure 4.5a) show that the ratio of fibre breakage to delamination area, assuming a linear relationship, is roughly 0.03. From Figure 4.5b for the IM6/937, the ratio of fibre breakage to delamination are is found to be 0.0063, less than that of the T800H/3900-2, although scatter in the data makes this estimation more difficult. 4.1.5 T800H/3900-2, a Tough Matrix System Material Characterization Lamina elastic properties for the T800H/3900-2 CFRP are shown in Table 4.2. The PC-LAMINATE software (Radford, 1989) mentioned in Section 3.5.1 was used to derive the effective sublaminate properties. Also shown in Table 4.2 are the lamina strength properties required by both the Chang and Chang failure model and the M L T model. The Weibull function constant, m, required by the M L T model was found in a previous investigation, discussed in more detail in Appendix A and in Williams and Vaziri (1995 and 1998). The constants used to characterize the damage growth and residual stiffness functions of the COD A M model are summarized in Table 4.3. As the material has a quasi-isotropic lay-up, the in-plane constants for both directions, x and y, are the same. The only exceptions are the scale factors Ki and L t , on ex and ey respectively, where K l c t = Z ^ , and K l c t = L X c t . To simplify the discussion of the derivation of these material constants, let us associate the x and y-directions with the 0° and 90° ply directions, respectively. As was discussed in Section 3.3.7, the response predicted by the ID C O D A M model is 114 Chapter 4 Numerical Case Studies independent of co'm. As this remains to be proven in the multidimensional implementation of the model, an appropriate value must be selected. Using a simple volumetric analysis and making the assumption that the matrix cracking at saturation associated with a given direction in the laminate will result in complete cracking of all off-axis plies (i.e. the ±45° and 90° plies, for the x-direction), one obtains: CO'm = A d a m a ^ e d = "~45 + "45 + "90 = 3 S Q g { 4 A ) Atotal "0+"-45+"45+"90 4 In this calculation, the material system and the stacking sequence are assumed to be insignificant factors in determining co'm. Only the lay-up (or fibre orientation) play a role. The residual stiffness at saturation of matrix cracking, R'E, is estimated using the ply-discount method. The same assumptions noted above for co'm apply here. Further, in the application of the model a significant amount of delamination is expected and so it is possible to assume that the off-axis plies do not contribute to the stiffness once a significant amount of matrix cracking has been induced in the material (i.e. a completely decoupled response). Under these conditions, according to the ply-discount method, the damage stiffness of the laminate is Elam = 129.1/4 = 32.28 GPa and since E?am = 48.37 GPa (Table 4.2), the residual stiffness becomes R'E = Elam/E?am = 0.67 = 0.65. Depending on the assumptions made, the ply-discount method can give a wide range of predictions from as low as 0.67 (given the simplifications made above) to as high as 0.89 (a completely coupled response with perfect bonding between the plies such that the ±45° plies continue to contribute to the stiffness). To substantiate the selection of R'E we must turn to the literature. Experimental work by Kress and Stinchcomb (1985) on fatigue damage growth in T300/5208 CFRP ([0/45/90/-45]s)4 laminates shows stiffness reduction factors as high as 40%. 115 Chapter 4 Numerical Case Studies Lafarie-Frenot and Riviere (1988) also shows stiffness reductions on the order of 30-40% for quasi-isotropic CFRP lay-ups as do Poursartip et al. (1986a). Based on these observations, the ply-discount prediction of R'E = 0.65 is reasonable. Bakis and Stinchcomb (1986) observed that, under fatigue loading, the stiffness reduction in tension was less than that in compression in a quasi-isotropic T300/5208 CFRP with the residual compressive modulus being closer to 75%. In the current analysis, the difference between tensile and compressive stiffness loss has been ignored. Using the same assumptions, the ply-discount method predicts a residual stiffness of 0.05 for R'G, the characterization constant which is required to determine G11. This is clearly unreasonable and again we must turn to the literature for guidance. Data on shear modulus reduction due to damage is much less common than the published work available on E. O'Brien and Reifsnider (1981) noted reductions in Gxy of 13-14% in fatigue loaded [0/90/±45]s boron/epoxy laminates compared to 15-16% in Ex in the same tests. However, Camponeschi and Stinchcomb (1982) observed that the reduction in Gxy was almost twice that of Ex in [0/90/±45]s CFRP laminates. They also concluded that a stacking sequence that avoids ±45° sequences (e.g. the T800H/3900-2 [0/45/90/-45]3S studies in this section) would be less susceptible to a reduction in Gxy. Although this data yields no precise value to be used for R'G, the assumption that R'G = R'E seems reasonable as a first approximation. Appropriate values for most of the scaling constants on the strain terms in the damage growth potential functions, Equations 3.18 and 4.5, can be argued from intuition. For example, Kxt and Lyt should be 1 as £ x and £ y are the primary drivers of damage in the x and y-directions respectively. However, appropriate values for K t and Lxt are less obvious. While the effect of 116 Chapter 4 Numerical Case Studies £ y on damage in the x-direction will obviously be less than that of ex, the transverse strain still contributes through the ±45° plies and to a lesser extent through cracking induced in the 90° plies of a [45/90/-45/0]s lay-up. A value of Lxt - 1 is too low while Lxt = 2 (i.e. 0.5ey contributes to cox) is too high. Lacking a more physical argument than this, Lxt (and Kyt by the same argument) is set to 4/3 (i.e. 0J5ey contributes to cox). The thresholds for damage growth in compression are generally higher than those in tension (i.e. higher compressive strengths) and values of Kxc = Lyc = 1.2 and Kyc = Lxc = 1.4 are used to reflect this. From the discussion in Section 3.3.4, the out-of-plane shear actions are assumed to have a secondary effect on damage growth and so Tx and Ux (and Ty and U y by symmetry) are set to 1. The final parameters, Sx and Sy are more difficult to choose. The degradation of the in-plane shear modulus is expected to result in large shear strains. As a result, a low value for S could cause an instability in the predicted damage growth . On the other hand, in-plane shear certainly plays a role in damage growth and removing y' from the damage growth potential functions would be an over-simplification. Results of initial runs of the C O D A M model in the plate impact problem supported these observations. A value of S = 1 resulted in much too strong a sensitivity to shear stiffness reduction. A value of 4 was found to be a more reasonable choice. Through Equations 3.18 and 4.6, the thresholds F1 ,FU, and Fm are associated with the strain states at the onset of matrix cracking and delamination, fibre breakage, and rupture respectively. Typically, a strain of 0.5-0.7% is associated with first ply failure in unidirectional test coupons of CFRP laminates. Here a value of 1% has been assumed with F1 = 0.01. F n is the threshold for fibre failure. Again, in uniaxial tensile tests, fibre failure strains of approximately 1.2-1.7% are observed for quasi-isotropic CFRP laminates. F n has been set to 0.016. 117 Chapter 4 Numerical Case Studies Work by Kongshavn and Poursartip (1997) demonstrated rupture strains as high as 3 % in T300/593 CFRP laminates. The 593 matrix is considered to be brittle and it is assumed here that in a material with a tougher matrix, such as the 3900-2, it may be possible to achieve stable damage growth at even higher strains. Lacking any more substantial data on which to base Fm, a value of 0.04 was chosen for the T800/3900-2 material. Having set co'm and F 1 , F n , and Fm, co11 can be determined from Equation 4.7: = 0.16 (4.8) ft)n = co'm (Fu-F1^i n o ^0.016-0.010^ v0.040 - 0.010y And given co'm, R'E, G'M, and cou we can apply Equation 3.21 to determine E11 and G n : £ n = G n = 1 - (1 - RE)^j- = 1 - (1 - 0.65) • — = 0.93 (4.9) co m 0.80 From Equation 3.12 we can assume that co% = co^ . In the limiting case cox = coE at onset of fibre breakage. When coy = 0 , cos = coE but cos = COQ at the onset of fibre breakage and therefore O)Q = coE . And so, with R'E = R'G, GN = EU = 0.93 . Note that the assumption that COQ = coE holds for cases where either cox = 0 or coy = 0, and for cox = coy but is not true for any general damage state. The damage growth and residual stiffness functions defined by the parameters derived above are shown in Figure 4.6a and b respectively while the predicted ID stress-strain response for the COD A M model of T800H/3900-2 is shown in Figure 4.7. Predictions of Damage Growth Figure 4.8 through Figure 4.12 summarize the damage growth and energy loss predictions and demonstrate the extremely good correlation between the F E M results and the experimental 118 Chapter 4 Numerical Case Studies observations of Delfosse and Poursartip. Figure 4.8 shows the predictions of total matrix damage in the form of delamination size, as a function of incident or available energy while Figure 4.9 shows fibre breakage as a function of incident energy. In both cases the results match very closely with the experimental trends. Figure 4.10 and Figure 4.11 show the same damage measurements, delamination size and fibre breakage respectively, as a function of impactor energy loss (i.e. energy absorbed). Although the correlation is not quite as good as the damage as a function of incident energy, the predictions do pick up the overall trends. Figure 4.12 shows the energy absorbed for a given impact energy. Note that the experimental energy loss attributed to indentation and system losses (an average of approximately 2.5 J) has been added to the predicted energy absorption. At low and intermediate impact energies, the predictions are very close. At high energies, however, there is a marked change in the F E M prediction of the rate of energy loss with increasing impact energy. Figure 4.13 and Figure 4.14 show predictions of damage growth as a time history from single high energy impacts. Results are shown for both low and high mass impacts and the predicted trends follow the experimental observations. The following energy balance should hold for any of the impact events: Eimpact ~ Esystem loss * Eelastic * Ematrix damage * Efibre damage * Eprojectile KE * Eplate KE (4-10) where E ^ ^ ^ represents any system losses from indentation and heating for example, Eelastic is the elastic energy stored in the plate, Eprojecti[e K E is the kinetic energy of the impactor, and Eplate KE * s t n e kinetic energy of the plate. Consider the final state of the event where the impactor has rebounded from the plate (i.e. EplateKE =0). The residual kinetic energy in the projectile is equal to the maximum elastic energy stored in the plate (assuming no damage is 119 Chapter 4 Numerical Case Studies generated after the peak displacement occurs): ^absorbed ~ matrix damage Efibre damage ) -^ .ry.rtem losses — F — F — '-'impact projectile residual KE (4.11) From the experimental results in Figure 4.10, 5.0kJ/m 2 of energy is absorbed by delamination (the inverse slope of the initial segment of the curve). Delfosse and Poursartip (1997) measured the energy to break the fibres to be 160 kJ/m 2. If these values are multiplied by the measured or predicted damage areas, one should arrive at the total energy absorbed by damage (Ematrixdamage and Ejibredamage). Figure 4.15 is a graphical representation of Equation 4.10. The results in Figure 4.15a are very noisy due in large part to the prediction of elastic energy recovered as residual kinetic energy in the projectile. If these results are plotted using the experimental elastic energy stored (Figure 4.15b), the agreement is much better. This is to be expected because of the close agreement between the damage areas from the F E M model with experiments. Note that the total energy prediction is more reasonable as well. In Figure 4.15a the energy absorbed is predicted to be greater than the available energy while in Figure 4.15b, the total is slightly less. In the previous figures, only the experimental quasi-static and drop weight (i.e. high mass) results were used for comparison. Figure 4.16, projected delamination area as a function of impact energy, demonstrates why. This figure is included to show the difference in behaviour between the experimental high and low mass impacts. Low mass (high velocity) impact tests show a smaller delamination zone size. This is consistent with a more localized response resulting from indentation. In a shell element, the only mechanism which allows for a relative normal displacement of two contacting bodies is the contact algorithm. However, the contact stiffness and the 'penetration' depth of nodes belonging to one surface past the nodes of the other surface are artificial and involve a number of mathematical manipulations of these stiffness and 120 Chapter 4 Numerical Case Studies 'indentation; depths to maintain numerical stability of the contact. As a result there is no physically realistic mechanism for indentation to occur in the numerical model and the F E M results, both high and low mass, correlate well with the more global deformation and damage zone measured in the high mass and quasi-static tests. Note that projected area was derived from the total delamination size predicted using a relation that was determined experimentally. Delfosse and Poursartip observed a relationship between these two matrix damage measures. Plotting the ratio of the measured total and projected delamination size (from micrographical analyses and C-scans, respectively) as a function of impact energy results in a linear relationship with the ratio varying from 3.8 to 4.2 depending on the incident energy (refer to Figure 4.17). The value was important experimentally because it allowed projected delamination size data to be converted to total delamination size, a quantity that can be related to total energy absorption. Projected delamination size can be easily measured from C-scan images while quantifying the total delamination size requires a much more time consuming measurement technique involving micrographs of sectioned specimens. Here we use the appropriate scaling factor, determined from Figure 4.17, to go the other way, from total predicted delamination area to projected delamination size. While the results are extremely good, especially given the simplifying assumptions that have been made in the material characterization, there are deviations from the observed trends. For example, the prediction of delamination damage in Figure 4.8 is excellent for impact energies over 20 J but below that level the F E M model predicts damage where none was observed experimentally. Damage initiation is shown to occur at just 5 J, much lower than the measured 18 J. Even the damage histories for a single 60 J impact in Figure 4.13 and Figure 4.14 show the 121 Chapter 4 Numerical Case Studies same trend although, interestingly, the initial jump in damage occur earlier in the high mass impact than in the low mass event. The same over-prediction of damage at energies less than 20 J is observed in Figure 4.9, a result of the implicit link between matrix and fibre damage through the damage growth potential. It is also interesting to note that the fibre breakage prediction is worse at energies between 20 and 45 J but improves at higher energies. This over-prediction of fibre damage can be explained by Figure 4.18, which is a plot of total fibre damage versus total delamination area. Experimentally, there is a threshold of 950 mm 2 of delamination before fibre damage initiates. Numerically, however, fibre damage appears to be closely coupled to delamination damage. This could possibly result from too low a value for F n but if the interpretation of the threshold for fibre breakage is to be linked with the strain to fibre failure, values over 1.7 or 1.8% are unreasonable. The results of parametric studies, discussed below in Section 4.1.6, also show that higher values of F n are undesirable. While resulting in a small change in the fibre damage growth at intermediate impact energies, the decrease in fibre breakage is significantly more pronounced at high energies where the current predictions match the experiments quite well. Rather, the source of the discrepancy can be traced back to the inherent link between the fibre and matrix damage in the damage growth function. For example, fibre damage is driven primarily by strains in the fibre directions. Off-axis shear strains and strains normal to the fibre direction have little effect. This is in sharp contrast with matrix damage which is largely influenced by these strains. The correct prediction of energy absorption per unit area of damage for each type of damage is an important and desirable feature of a damage model. While the general trends in Figure 4.10 are in agreement with the experimental observations, there is a significant deviation from the observed behaviour at higher values of energy loss. Initially, the numerical prediction of the 122 Chapter 4 Numerical Case Studies toughness associated with matrix damage is quite good. Although the initial slope of the first segment of the curve is too steep (i.e. below 500 mm 2 of damage), the trend seems to approach the 5 kJ/m 2 of the experimental results. From 10 J to 25 J of energy loss, the predictions are very good. These two segments correspond to impact energies of 20 to 35 J, the same range over which we see good agreement between energy absorption and energy loss in Figure 4.12. Above 35 J of impact energy, however, the toughness is under-predicted. This region is associated with matrix/delamination damage and fibre damage so the source of error could come from the fibre energy absorption predictions. From Figure 4.9, we note that between 20 and 35 J, the rate at which fibre damage is introduced as a function of incident energy (i.e. the rate at which energy is absorbed by fibre damage) agrees well with the slope of the experimental results, even though there is a shift associated with too much fibre breakage. However, above 35 J we observe a decrease in the incremental amount of area of fibre breakage created for a given incremental increase in the impact energy. Less fibre damage created means a lower energy absorption (assuming that the numerical interpretation of energy absorption associated with an increment of fibre damage is correct) and hence, for a given impact energy, we would expect a lower predicted impactor energy loss as is shown Figure 4.12. By the same argument, the energy absorbed for a given delamination size should be less than expected. The deviation from the predicted trend over 25 J of energy loss in Figure 4.10 is therefore not an over-prediction of delamination damage (this is supported by Figure 4.8) but rather an under-prediction in the energy absorbed for a given a delamination size. The predicted rate of growth of fibre damage as a function of impact energy absorption gives very good agreement over most of the range of impact energies as shown in Figure 4.11. The deviation from the experimental results comes from a consistent over-prediction of the amount of 123 Chapter 4 Numerical Case Studies fibre breakage for a given impact energy which has been attributed to the coupling of the matrix/delamination damage and fibre breakage growth through the use of a single damage potential function for both types of damage. What is perhaps surprising is that the rate of energy absorption with increasing fibre breakage is constant (i.e. the curve is straight). In the previous paragraph we noted an under-prediction of the energy absorption predicted for a given delamination area (Figure 4.10). One would therefore expect to find the same trend in Figure 4.11 but we do not. In fact, this error in the predicted energy absorption is present in the numerical results but it has been masked by our assumption of a linear relationship between fibre breakage and delamination growth. The experimental trend in Figure 4.5 shows a bilinear variations which was ignored in favour of a simpler functional relationship for predicting fibre breakage from the damage measure available in the model (i.e. Equation 4.3). Figure 4.5 shows that above 2500 mm 2 of delamination, fibre breakage increases with delamination size at twice the rate it does below 2500 mm 2. The effect this has on the predicted fibre breakage is shown by the dashed line in Figure 4.11. The prediction is now consistent with the impactor energy loss in Figure 4.10. Figure 4.9, Figure 4.10, Figure 4.12, and Figure 4.15 all show deviations from the expected trends at an impact energy of approximately 35 J. Plots of the predicted damage distributions at 30 J (i.e. just before the change in behaviour), 33 J, and 35 J (Figure 4.19) clearly shows the development of a localization in the damage growth. The asymmetry of the damage growth pattern can be attributed to this localization phenomenon. It is interesting to note that the predictions of total delamination size do not seem to be influenced by this problem. Figure 4.8 shows no deviation from the experimental trend at energy levels above 20 J, nor does the total fibre breakage as a function of energy loss in Figure 4.11. The reason for this insensitivity is 124 Chapter 4 Numerical Case Studies unknown. Figure 4.20 shows fringe plots of average damage, (ft)i+ft) 2)/2, at each integration point through the thickness of the laminate. It is interesting to note that the size of the matrix/delamination damage zone (ft) > 0) does not change significantly through the thickness. However, the size of the most intensely damaged region (i.e. the zone where fibre failure is predicted) does. Moving from the impact face towards the back of the plate we observe a continuous decrease in the size of this zone (the black region corresponding to 0.8 < co< 1.0). By sublaminate 5, there is almost no indication of this level of damage. Moving from sublaminate 5 to sublaminate 6 we note an increase again. These observations are consistent with the detailed micrographical analyses by Delfosse et al. (1995) which show that fibre breakage decreases consistently as one moves away from the impact face. The amount of damage reaches a minimum at ply 21 (i.e. near the boundary between sublaminates 5 and 6 in the model) and then increases again significantly at the back-face. Finally, the predictions of projected delamination size are compared to C-scan images of delamination growth in Figure 4.21 and Figure 4.22. Both sets of images (numerical and C-scan) are to the same scale. The C-scans were taken across the width of the specimens so the whole plate is not visible. The box drawn around the numerical results highlights the location of the plate boundaries relative to the part of the plate modelled (i.e. a 127.0 mm by 76.2 mm simply supported plate). The boundary effect in the model are clearly visible at the higher incident energies (e.g. 46.2 J and 58.2 J in Figure 4.22). The low levels of damage predicted at the mesh boundaries are caused by local through-thickness shear strains, an effect of the boundary condition applied at the edge of the plate. The qualitative comparisons of the size of the 125 Chapter 4 Numerical Case Studies predicted and measured damage zones are extremely good over the whole range of impact energy levels. It is interesting to note that even the predicted localization of the damage growth along the length of the specimens is not unrealistic. Although, numerically this localization is confined to a region only one or two element wide, the C-scan images show the same preferred direction in the damage growth at impact energies of over 30 J. Time Histories of Force, Displacement, and Energy The force, energy, and displacement-time histories are presented after the damage growth prediction to highlight a fundamental difference in the approach used here. Unlike many other numerical models that have been proposed, the C O D A M model had as its foundation the experimental work on damage growth and stiffness loss. The physical interpretation of the predictions in terms of damage growth and the resulting stiffness loss were of primary interest and the results presented in the previous section show the predictions that can result from using an approach that is strongly tied to experiments. A more traditional method of assessing the accuracy of a given model has been the prediction of the impact force-time response for a given impact event. Here, our philosophy is that the accurate prediction of the force-time history is a by-product of a good damage model rather than a measure of one. The results of the prediction of the experimental force-time, energy-time, and force-displacement curves are summarized in Figure 4.23, Figure 4.24, and Figure 4.25 for the high mass (drop weight) experiments and Figure 4.26, Figure 4.27, and Figure 4.28 for the low mass (high velocity gas gun) tests. Consider first the high mass impact force-time histories shown in Figure 4.23. For impact energies ranging from 9.5 J to 58.2 J, the prediction of peak force and onset of damage, marked by the deviation of both numerical and experimental results from the smooth parabolic response observed in the undamaged elastic 9.5 J impact, are extremely good. 126 Chapter 4 Numerical Case Studies There is a slight over-prediction of the peak force which is observed to be related to a delay in the onset of damage, in all cases by approximately 0.4 ms. Within the region of the peak force, up to a time of 3 ms, the overall trends fit the experimental results accurately. However, after 3 ms, the region of the curve associated with the unloading of the plate and rebound of the projectile, we note that the response deviates slightly from the measured unloading. The over-prediction of the duration of the event in the 29.5 J test (a 0.2 ms difference) would go unnoticed if it were not for the more significant (up to 0.6 ms) over-predictions at higher energy levels. This type of response would be expected from a more compliant target. It should be noted that numerical results for incident energies over 30-35 J showed progressively increasing amounts of localization. A notable feature of a localized response, observed in the results discussed in Appendix A and in the next section for a brittle failure model, is an overly soft response. The low mass impact results (Figure 4.26) show similar trends. The predictions for low mass impacts are more noisy that the corresponding experimental results but this does not detract from the comparisons. Some of the smoothness of the experimental results are artificially induced to filter out the vibrations in the projectile. On the numerical side, the lack of damping in the numerical model (e.g. a rigid projectile and perfect boundary conditions) contributes to the oscillations observed. Again, for incident energies greater than the threshold at which localization was observed, there is a marked difference in the unloading segment of the curve. The high mass force-displacement results shown in Figure 4.27, and particularly in Figure 4.24, add insight into the slight over-prediction of the peak force. The numerical model is observed to be slightly stiffer than the actual plate. This is most easily seen in Figure 4.24 where the initial slope of the predicted force-displacement curves are steeper than the measured curves by almost 13%. The displacement at failure (or strain at failure), marked in the F E M prediction by the 127 Chapter 4 Numerical Case Studies sharp drop in the curve and in the experimental results by the onset of an oscillatory type response, is accurately predicted by the FEM model for all energy levels. Given the delay in the onset of damage observed in the force-displacement histories, a first conclusion might be that the threshold for the onset of matrix damage (i.e. F 1 ) is too high. However, these results show that is indeed not the case. It is more likely that the delay in the onset of damage and the over-prediction of the peak force are linked and result from a disparity in the material characterization data taken from the literature leading to an incorrect laminate stiffness. Note also that the overly soft unloading response is observed in the high mass curves although it is not readily apparent in the low mass gas gun results in Figure 4.27. A final aspect of the force-displacement results which is worth noting is that there is a slight under-prediction of the peak displacements at the lower energy levels of high mass impacts events. The predictions improve as the impact energy is increased with the experimental and predicted curves for the 58.2 I impact virtually overlapping up to the maximum displacement. The opposite trend is observed in the low mass impact predictions. At low energies, the peak displacements are in agreement. However, as the incident energy is increased, the maximum displacement is over-predicted by a continually increasing amount up to the 84.4 J impact where the difference is almost 2 mm on a 8 mm displacement. This is evidence of an inappropriately stiff initial modulus and an over-prediction of the softening. The response of a high mass (low velocity) impact with little or no damage will be dominated by the global bending stiffness of the plate. A stiffer plate will show lower displacements (Figure 4.24a and b). As the impact energy increases, more damage will be generated. An over-prediction of the stiffness reduction will result in a more compliant plate and, hence, higher displacements. In the case of the high mass impacts, this shows up as less and less of a difference between the measured and predicted 128 Chapter 4 Numerical Case Studies displacements (Figure 4.24c and d) until finally the opposing effects balance out (Figure 4.24e). Conversely, a low mass (high velocity) impact will be dominated by the local response. The global stiffness of the plate will contribute less to the peak displacements. As a result, an error in the initial plate stiffness will not be as large as it was in the low mass impacts (Figure 4.27a and b). Again, as the energy increases, damage begins to dominate the response through the stiffness reduction. At some stage the two effects balance out (Figure 4.27c) while at higher impact energies, the over-prediction of the stiffness loss results in more significant over-predictions of the peak displacement (Figure 4.27d and e). Finally, the comparisons of predicted and measured energy-time histories shown in Figure 4.25and Figure 4.28 are consistent with the observations made about the force-time and force-displacement results. Overall, the predictions follow the measured results. The major differences occur in the residual or absorbed energies with the numerical results generally under-predicting the amount of energy absorbed through damage. Experimental results for both low energy (=9 J) impacts show an energy absorption of approximately 2-3 J but no damage is associated with these tests. This small energy absorption is attributed to indentation and system losses and is a feature of all the experimental results. Numerically, the indentor is rigid, the contact is symmetric, the plate has no mechanism for indentation, and the boundary conditions are rigid from the point of view of energy absorption so that no such losses can be predicted. As the impact energy increases, these losses only account for part of the differences between measured and predicted residual energy. This under-prediction of the energy loss was also observed in the discussion of the damage results above. In general, the results of the high mass impacts show better correlation with the 33.4 J impact giving the best results by far. 129 Chapter 4 Numerical Case Studies Comparison with Other Models To assess the performance of the C O D A M model against current composite models in the LS-DYNA3D code, two other material models have been used to analyse the same problem. LS-DYNA3D MAT22 is a composite failure model based on the work of Chang and Chang (1987a) and is one of the most widely used composite models in L S - D Y N A 3 D (a number of applications of this model were identified in Section 2.5). Failure is predicted in a number of modes including matrix cracking and fibre splitting using modifications of the criteria proposed by Hashin (1980). The effect of the damage on the laminate response is modelled on a lamina-by-lamina level (an integration point per lamina) using the ply-discount method. Material properties required by the model were obtained from the literature and include the lamina elastic and strength properties summarized in Table 4.2. The second model is based on the C D M approach originally developed by Matzenmiller et al. (1995) which was implemented as a user material model in LS-DYNA3D during the preliminary stages of the current research (refer to Section 3.2 and Appendix A). The initial investigation of the application of C D M models to the prediction of damage in CFRP laminates published in Williams and Vaziri (1995 and 1998) used the same experimental data for comparisons. In this section, we add the predictions obtained using the C O D A M model and compare these new results with those obtained previously. The low mass, low energy impact results shown in Figure 4.29, Figure 4.30, and Figure 4.31 show that the results of the Chang and Chang (CC), M L T , and C O D A M models are comparable where little or no damage is observed. As shown in Figure 4.32 through Figure 4.37 for higher energy impacts (22.0 J and 33.41), the M L T and C O D A M models continue to provide good agreement with experiments particularly on peak force, event duration, and maximum 130 Chapter 4 Numerical Case Studies displacement. The M L T model better predicts the energy absorption in the 22.0 J event with the C O D A M showing less energy loss. The converse is true for the 33.4 J event. In both cases, however, the C C model shows a sudden drop in force at initiation of failure and generally over-predicts the stiffness loss. We also note an over-prediction of the energy absorption (Figure 4.33 and Figure 4.36) and of displacement (Figure 4.37). A l l are characteristic of a localization of the damage (refer to Appendix A and Figure A.11), a result of the catastrophic brittle failure predicted by the ply-discount scheme used in the CC model. At 56.5 J and 84.4 J the C O D A M model shows a significant improvement over the M L T model in the energy absorption and, in particular, the peak force with the M L T results over-predicting the peak forces by over 25% (Figure 4.38 and Figure 4.41). The C O D A M does over-predict both the event duration and the peak displacement for the 56.5 J event (Figure 4.38 and Figure 4.40) but the over-prediction of stiffness loss, of which these are both characteristic, has been attributed to the development of a localization of the damage growth. At 84.4 J the C O D A M model continue to give a much closer prediction of the peak force and, at this energy, predicts the event duration quite well (Figure 4.42). The M L T model, under-predicts the event duration and peak displacements, a sign of the under-prediction of the stiffness loss. At the same time, the results of the C O D A M model and CC model for both the 56.5 J and 84.4 J events have been becoming closer to each other, possibly due to the localization of damage growth in each. It is interesting to note that the experimental results seem to show more of the characteristics of this localized damage zone. The C-scans taken at progressively higher energy levels, shown previously in Figure 4.21 and Figure 4.22, indicated an elongation of the damage zone along the long axis of the plate. 131 Chapter 4 Numerical Case Studies Figure 4.44 through Figure 4.49 show the same series of curves (force and energy histories and force-displacement) for high mass impacts at 34.5 and 58.2 J. Again, the numerical predictions from the C O D A M model fit the experimental measurements very closely. The comparisons are particularly good at the higher energy level where the peak force, energy absorption, and peak plate displacement predictions are significantly better than either of the M L T or Chang and Chang results. 4.1.6 Parametric Studies A number of parametric studies have been carried out to assess the sensitivity of the model predictions to variations in the model parameters. The characterization of the matrix saturation (defined by co'm and R'E), the damage thresholds of the damage growth function (F 1 , F n , and Fm), the scale factor on y^ (S) in the damage potential function, and the mesh size were selected as the most critical parameters. Runs covering a range of impact energies from 5 J to 60 J, in intervals of 5 J, were performed for each parameter in the investigation. Based on the good agreement between the predicted results of the high and low mass impact and the shorter computation time of the gas gun simulations, only the low mass analyses were used during the parametric studies. The range selected for each parameter study is based on the range of values which one would realistically encounter when trying to characterize a material. Results of the studies, shown in Figure 4.50 through Figure 4.57, are presented in terms of the total damage growth and energy curves. While variations in these parameters will certainly influence the force-displacement histories, the identification of trends is most easily done on the basis of damage growth and energy loss. Trend lines, plotted for each series of results, are included to aid in the identification of overall trends in the results but should not be interpreted as 132 Chapter 4 Numerical Case Studies strict predictions of the damage and energy histories. Effect of Variations in the Characterization of Matrix Damage Saturation Figure 4.50 and Figure 4.51 present the results of parametric studies on the effects of variations in co'm and R'E, the values that characterize the material state at saturation of matrix damage. In Figure 4.50 two values of co'm (co'm =0.65 and co'm = 0.95) are compared to the baseline run (co'm = 0.8) and the experimental trends. Similarly, Figure 4.51 shows the results of three sets of numerical analyses using different residual stiffnesses at matrix damage saturation (R'E = 0.50, R'E=0.80, and the baseline value of R'E =0.65). In both cases, there are no observable differences between the baseline runs and the results of the parameter variations. The results for variations of R'E are significant because of the issues raised in Section 3.5.2 regarding the potential errors that might have been introduced by using simple mathematical techniques to characterize the stiffness loss due to matrix cracking and delamination. Figure 4.51 shows that the ply-discount prediction of stiffness loss used to derive the baseline value of R'E is sufficiently accurate to obtain good agreement. The model is shown to be insensitive to variations of R'E on the order of ±0.15, a range which is within the error normally attributed to a ply-discount prediction. Experimental results are still invaluable in identifying significant deviations from ply-discount theory resulting from stacking sequence effects, for example. Similarly, Figure 4.50 indicates that the simple assumptions used in Section 0 to characterize the state of damage at matrix saturation are sufficiently accurate. Similarly, the parametric study on co'm (Figure 4.50) provides evidence that the multidimensional response is insensitive to the damage level associated with matrix damage saturation. This is consistent with the discussion in Section 3.3.7 which showed the independence of the ID response with respect to co'm. 133 Chapter 4 Numerical Case Studies Effect of Variations in Damage Growth Thresholds Unlike the stiffness reduction and damage associated with matrix damage saturation, the damage thresholds can have a significant effect on the predicted response of the material. Figure 4.52, Figure 4.53, and Figure 4.54 show results of the parametric studies performed on F 1 , F n , and F m , respectively. The results shown in Figure 4.52 are for F 1 values of 0.005, 0.010 (the baseline), and 0.012. The value of 0.005 represents a more brittle matrix and 0.012 a tougher matrix (lower and higher strain to onset of matrix cracking, respectively). As might be expected, the threshold for matrix cracking has a significant effect on the total delamination area predicted (see Figure 4.52a and c). Decreasing the threshold significantly increases the matrix damage generated at a given impact energy. A drop in F 1 also decreases the predicted toughness as is evidenced by the increase in slope of the initial part of Figure 4.52c (note that the inverse of this slope is a measure of toughness, or energy absorbed per unit area of damage). Figure 4.52b also shows that the fibre breakage predicted is not sensitive to F 1 , nor is the overall energy loss as a function of impact energy. The smaller increase in F 1 from 0.010 to 0.012 results is a slight toughening of the response but the change is much less pronounced than that caused by the decrease from 0.010 to 0.005. Variations in F n have quite a different effect. As shown in Figure 4.53a and c, the matrix damage prediction is unaffected by F n . Figure 4.53b shows the shift in the fibre damage predicted. The rate of fibre breakage growth with increasing impact energy seems to remain the same except the initiation point shifts (as would be expected). The shift in fibre damage is also associated with a slight shift in the energy absorbed as a function of available energy. This shift is consistent with the results of the parametric study on F m (see Figure 4.54) which show the 134 Chapter 4 Numerical Case Studies high sensitivity of the predicted impactor energy loss to the fibre damage. Of all the parametric studies discussed up to this point, F m is the only parameter that seems to have a significant effect on the energy loss (Figure 4.54d). The variation in F r a is also observed to have a similar effect on matrix damage to that of the variation in F 1 with a much more brittle response predicted for low F m and a slightly tougher overall response for higher values of F m as shown in Figure 4.54c. Effect of Variations in the Characterization of the In-Plane Shear Strain Sensitivity of the Damage Potential Function The formulation of the damage potential function, F(- (i = 1,2) , has been highlighted as the least physical component of the model. Of the various parameters, the selection of the scaling factor on the in-plane shear strain (S) is the most difficult to argue from a physical standpoint. The value used for the baseline runs was selected on the basis of the results of a parametric study. Three values for S were selected, 10, 4, and 1. Decreasing values of S correspond to an increasing sensitivity of the damage growth to y xy • Figure 4.55 shows the results of the parametric study. A low value of S , e.g. 5 = 1, has a strong influence both matrix and fibre damage growth after initiation. The energy loss for a given impact energy also increases. However, the overall toughness of the material doesn't change (see Figure 4.55c). S is really a threshold for yxy a n ( ^ above a certain value, the influence of y ^ on the damage growth is negligible. From the results shown in Figure 4.55, this threshold is somewhere between 1 and 4. Mesh Size Sensitivity of the Damage Predictions The mesh sensitivity of damage models is well documented and a number of approaches have been used to address the problem. In the current investigation, the model development has 135 Chapter 4 Numerical Case Studies focused on the more fundamental aspects of the damage model, the experimental evidence for the proposed formulation and the physical implications of the assumptions made. Addressing the mesh sensitivity through the use of non-local damage theory, for example, has been left for future work. In an attempt to minimise the mesh size dependence of the damage mode, the analyses performed in this thesis have all used a uniform mesh with square elements. Nevertheless it is important to assess the sensitivity of the model to variations in the mesh size. A coarser 540 element (4.233 x 4.233 mm element size) and finer 2160 element (2.117 x2.117 mm element size) mesh were developed and the low mass, high velocity series of runs were repeated for each mesh. The results are compared to the experimental damage growth predictions as well as the results obtained from the baseline, 960 element mesh (3.175 x3.175 mm element size) in Figure 4.56. Decreasing the mesh size from 960 to 540 elements has no discernible effect on the results except for a minor decrease in the energy loss as a function of incident energy. The finer mesh gives marginally poorer predictions of total delamination damage as a function of impact energy (Figure 4.56a) and while shifting the curve, the results for energy loss as a function of impact energy show the same trends as for the coarser meshes. However, the finer 2160 element mesh does seem to give better predictions of fibre breakage as a function of impact energy and, in particular, delamination as a function of energy loss. In fact, up to an energy loss of 30-35 I, the numerical results for the 2160 element mesh follow the experimental curve very closely. Above this value, the deviation from the experimental trends is rapid and is once again associated with the development of a localization. Effect of the Damage Growth Law Assumption One of the major assumptions made throughout the course of this work has been that the matrix 136 Chapter 4 Numerical Case Studies damage continues to develop during the second phase of damage growth, after the initiation of fibre breakage. In Section 3.3.3 a number of other possibilities were discussed including the possibility of a trilinear damage growth relationship most of which are beyond the present capability of the model. However, it is of interest to see what happens if the opposite assumption is made. That is to assume that fibre damage initiates only after matrix cracking has saturated. Figure 4.57 shows the results. The same material characterization data has been used to generate both curves except that the point which defines the onset of the second phase of damage is associated with matrix damage saturation (i.e. ft)n = co'm and F n = E'm) for the case where matrix damage saturates before fibre breakage initiates. 4.1.7 IM6/937, a Brittle Matrix System Material Characterization The lamina elastic properties for the IM6/937 CFRP are listed in Table 4.4 as are the laminate properties derived using the P C - L A M I N A T E software (Radford, 1989). Based on the assumptions used to derive the characterization data for the T800H/3900-2 material in Section 4.1.5, only the constants associated with the material strengths (i.e. the damage thresholds F ^ F 1 1 , and F m ) are affected by the matrix system. The residual stiffness function will also change but only to the extent that F 1 , F n , and F m , and therefore ft)11, will change. The stiffness reduction rate (i.e. the slope of the normalized residual stiffness function) is set by the material lay-up and the characterization of the saturation of matrix/delamination damage, Co'm = 0.8 and R'E = R'G = 0.65 , remains the same for both material systems. As the FM6/937 material has a more brittle matrix, one would expect that the threshold for onset of matrix damage will be significantly lower than that of the T800H/3900-2. Similarly, F m , a 137 Chapter 4 Numerical Case Studies characteristic of the system (constituent materials and stacking sequence), will also be lower in the brittle matrix CFRP. However, F n , which is a fibre dominated property, should be less sensitive to the matrix system used. Initially, F n was left unchanged at 0.016 and F 1 was reduced to 0.005. An appropriate value for F m was obtained from results of Kongshavn and Poursartip for T300/593, another brittle matrix system. They found strains at rupture of approximately 3-3.5% . Fm was set to 0.032 accordingly. Results of using this material characterization in the CAI model showed less than half the matrix damage growth observed experimentally although, in comparison to the T800H/3900-2 results, the response was much closer to that of a brittle FRP (i.e. significantly more matrix damage growth). To allow the development of more damage, the matrix/delamination and fibre damage thresholds were reduced such that F 1 = 0.0 and F n = 0.012 ( F m was left unchanged at 0.032). The damage potential function was also modified but the formulation somewhat restricts the changes that can be made. Following the arguments made in deriving the scaling constants for these functions, K and L are related to the stacking sequence and it is difficult to argue making changes to their values. Similarly, the coefficients of the through-thickness shears are already set to 1, the minimum value. As a result, the only constant which can be varied is S , the constant applied to the in-plane shear strain. Based on the results of the parameter study in Section 4.1.6 which showed a strong sensitivity of the damage growth to a reduction in S , the value of 5 was reduced. Good agreement with the matrix damage predictions was found when a value of 5 = 1 was used indicating that matrix/delamination damage growth in a laminate with a brittle matrix system is more sensitive to in-plane shear than a tougher system. With co'm = 0.8 and R'E — R'G — 0.65 , fi>" = coE = 0.3 (Equation 4.12) and Gu = E11 = 0.869 138 Chapter 4 Numerical Case Studies (Equation 3.21). A summary of the C O D A M characterization data is given in Table 4.5. The damage growth and residual stiffness functions used for the LM6/937 are shown in Figure 4.58a and b, respectively, while the predicted ID stress-strain response is shown in Figure 4.59. Damage The damage growth and energy absorption predictions for IM6/937 CFRP obtained using the C O D A M model are summarized in Figure 4.60 through Figure 4.65. The predicted matrix damage growth (Figure 4.60) agrees fairly well with the measured delamination size over the full range of impact energies. The total fibre breakage is also in agreement with the experimental results for energies over 30 J as shown in Figure 4.61. The impact energy required to initiate a significant amount of fibre damage (observed experimentally at approximately 25 J) is under-predicted by 10 J. The close relationship between the predicted fibre and matrix damage observed in the T800/3900-2 results (Section 0) is seen again in Figure 4.62 for the IM6/937. Here the larger difference between the matrix and fibre damage thresholds result in more matrix damage growth before the onset of fibre damage than was predicted for the 3900-2 system. However, there is still a significant over-prediction of the amount of fibre damage for a given total delamination area. Note that in Figure 4.60 and Figure 4.61 the experimental trend lines are dashed for the results over 30 J. Delfosse and Poursartip observed that the matrix damage growth in this brittle matrix system is so large that the delamination growth approaches the boundaries. Therefore, less weight should be placed on results with an incident energy of 30 J or more because boundary conditions and edge effects begin to play a role in the damage growth. In Figure 4.63 and Figure 4.64, total delamination area and fibre breakage, respectively, are 139 Chapter 4 Numerical Case Studies shown as a function of the energy removed from the impactor (or energy absorbed by the plate). Although the numerical predictions are offset from the experimental trends, the energy absorption per unit area of damage (the inverse slopes of the trend lines) are in agreement for the matrix damage results. In contrast, the fibre breakage predicted for a given amount of energy absorption agrees well in magnitude over a range of impactor energy loss (from approximately 181 to 35 J) but the energy absorption per unit area of damage is roughly twice that observed experimentally (refer to Figure 4.64). Again, we observe the premature initiation of fibre damage. Experimentally, the energy absorption is attributed solely to matrix damage up to an energy loss of 15 J whereas the model predicts fibre breakage over the full range of energy absorption. Figure 4.65 shows the overall energy absorption as a function of incident energy. At low impact energies (i.e. where fibre breakage is not significant), impactor energy loss is under-predicted. Over approximately 18 J, the trend is good but the losses are over-predicted. In fact, above an impact energy of 30 J, the numerical predictions show that there is almost no energy recovered as residual projectile kinetic energy (i.e. Etoss ~ Eimpact) and all the impact energy is absorbed by the target. Figure 4.66 shows the results of the energy balance calculation discussed above in Section 0 for the T800/3900-2 system. For the IM6/937, the energy absorptions associated with matrix cracking/delamination and fibre breakage are 0.8 kJ/m 2 (Figure 4.63) and 150kJ/m 2 (Delfosse and Poursartip, 1997), respectively. The results are very good over the entire range of impact energy levels. However, note that the experimental elastic energies have been used rather than the predicted values (see the subsection entitled 'Predictions of Damage Growth' in Section 4.1.5). As noted in Figure 4.65, the predicted elastic energies recovered in the form of a residual projectile energy were extremely low at higher incident energies (>30 J) and the total 140 Chapter 4 Numerical Case Studies energy calculated based entirely on numerical results were almost 10 J lower than the incident energy. The numerical predictions of projected matrix/delamination damage are compared to the experimental C-scan measurements of projected delamination size in Figure 4.67 and Figure 4.68 for selected low and high mass impact events, respectively. The enormous effect that the lower toughness of the 937 matrix system has on the extent of delamination growth is clearly illustrated if one contrasts the results of these C-scan measurements with the C-scan images for the tougher 3900-2 based CFRP system shown previously in Figure 4.21 and Figure 4.22. The numerical predictions show low levels of matrix/delamination damage throughout the plate, even at low incident energy levels (e.g. 10 J in Figure 4.67). This is a direct result of using a matrix damage threshold of 0.0. However, the more intense region of matrix damage in the centre of the plate shows good agreement with the delamination size observed for the 10 J low mass impact (Figure 4.67) and the 11 J high mass impact (Figure 4.68). As the incident energy is increased, the model correctly predicts the significant growth of the delamination to the boundaries, however, a direct comparison of projected delamination size, possible with the T800/3900-2 results, is hampered by the interactions with the plate boundaries. Experimentally, the edge effects are clearly visible at incident energies over 30 J where the delamination zone has grown past the width of the backing plate opening. An accurate prediction of the boundary interactions and edge effects would require a more detailed prediction of the through-thickness stress state than is possible in a shell element. Here, a 3D brick formulation would be needed. Note the strong influence of the in-plane shear strain on the damage growth characterized by the 'x' shaped pattern of damage. When the damage zone grows to the boundaries, there is no 141 Chapter 4 Numerical Case Studies undamaged (or partially damaged) material to provide significant residual stiffness and the plate folds along the diagonals. This accounts for the high energy absorption observed in Figure 4.63, Figure 4.64, and Figure 4.65 at incident energy levels over 30 J. Experimentally, even though there is a significant amount of delamination and the plate is essentially split into a number of partially or fully debonded sublaminates, each of those loose bundles of plies has a residual stiffness. Although delamination area has increased significantly, fibre damage has not. Numerically, the inherent link between matrix/delamination damage growth and fibre breakage (and the associated stiffness loss) does not allow the same residual stiffness in a sublaminate (i.e. at an integration point). The stiffness reduction leads to the growth of more fibre damage and ultimately the complete loss of stiffness for elements within the damage zone. Force versus Displacement Curves The comparisons of predicted and measured force, energy, and displacement time histories are shown in Figure 4.69 through Figure 4.71 and Figure 4.72 through Figure 4.74 for the high mass and low mass impact events, respectively. Experimentally, we observe two characteristic changes in response on the force-time and force-displacement curves, one marking the onset of matrix damage, the other the onset of fibre damage. These are most easily observed in the high mass results (Figure 4.69, Figure 4.70). Figure 4.69a shows experimental results for a test in which no fibre breakage is observed. The change in behaviour that occurs approximately 1 ms into the impact event marks the onset of matrix damage and delamination. The stiffness reduction associated with this damage growth is clearly visible as a change in slope in the force-displacement curve shown in Figure 4.70a. Looking at the results of a higher energy impact in Figure 4.69c and Figure 4.70c, we again note the onset of matrix damage at 0.5 ms (a displacement of 1.4 mm and a force of 6000 N). 142 Chapter 4 Numerical Case Studies Furthermore, at 2.5 ms, we observe a sudden drop in force. This indicates the onset of fibre breakage and is the second characteristic point on the time histories of the brittle matrix system. Consider the high mass predictions in Figure 4.69 and Figure 4.70. The initial slopes of all three predicted responses compare well with the experiments indicating that the stiffness of the elastic plate has been correctly modelled in each case. Unlike the results for the T800/3900-2, we are not over-predicting the elastic moduli. At 11 J we observe a gradual change in the slope of the force-displacement curve (Figure 4.71a) although it is not as pronounced as the stiffness reduction observed experimentally. As a result, the model predicts a higher force, lower maximum plate displacement, and shorter event duration, all consistent with an under-prediction of the stiffness reduction associated with the damage (note that the area of matrix damage predicted was in agreement with the experimental results as shown in Figure 4.60). This is further supported by the plate energy-time graphs (Figure 4.71a) which show that the predicted energy loss is less than the measured energy absorption (refer to the initial part of the impactor energy loss versus impact energy curve in Figure 4.65 as well). At higher incident energy levels, 29.6 J and 33.7 J, where fibre damage is observed we see the characteristic sudden drop in the predicted force-time histories. This drop is larger than that observed experimentally and occurs earlier in the event. At these energy levels, the event duration and maximum displacement are also over-predicted and the energy absorption is over-predicted. The low mass predictions (Figure 4.72, Figure 4.73, and Figure 4.74) show similar trends although it is more difficult to identify the effects of matrix damage on the response. Here the over-prediction of the energy absorption is even larger than that observed in the high mass events (Figure 4.74). Overall, the predictions of the time histories are not as good as those made for the 143 Chapter 4 Numerical Case Studies T800/3900-2 system. 4.1.8 Case Study 1 Summary The results o f this case study show that the mode l developed i n this thesis is capable o f predict ing the force, displacement, and energy-time response o f a dynamica l ly loaded F R P composites. W h i l e other composi te damage and failure models have been used wi th some success to generate s imi la r t ime histories, the U B C - C O D A M 3 D s mode l provides a significant step forward i n the predict ion o f damage. Deta i l ed comparisons o f measured and predicted matr ix cracking/delaminat ion and fibre breakage areas for the impact events i n v o l v i n g the T800/3900-2 material system show that the mode l presented i n this thesis is capable o f producing a reasonable quantitative measure o f the damage growth. Compar i sons o f fringe plots o f the predicted matr ix damage zone compare favourably w i t h experimental C-scan images o f delaminat ion growth. The inferior performance o f the mode l i n predict ing the response o f the I M 6 / 9 3 7 panels can, i n large part, be traced to an over-predict ion o f the stiffness reduction associated w i t h a g iven size o f damage. A s the stiffness reduction attributed to matr ix damage is re lat ively smal l (Eu = 0.869 at the onset o f fibre damage), it is l i k e l y that the over-predict ion o f stiffness loss is associated wi th the predict ion o f fibre damage growth. T h i s is consistent w i t h the observed differences between the measured and predicted results for the T800/3900-2 system. T h e large matr ix/delaminat ion zone characteristic o f the brittle 937 matr ix s imp ly magnifies the effect o f the close relationship between matr ix and fibre damage growth predicted by the mode l . W h i l e the predictions presented i n Sect ion 4.1.7 for the I M 6 / 9 3 7 are not as good as those discussed i n Sect ion 0 for the T800/3900-2 , F igure 4.75 helps to put these results i n perspective. 144 Chapter 4 Numerical Case Studies Figure 4.75 shows the results of the TM6/937 predicted and measured projected delamination size plotted to scale with the T800/3900-2 predictions and measurements. The numerical model accurately captures the different behaviour associated with the tough and brittle matrix systems. Note that the only difference between the two models were the material characterizations. And therein lies the most significant conclusion which can be drawn from this case study. The predictions show that simple physical reasoning based almost entirely on widely available, published material characterization data, which in turn, is based on relatively simple standard test schemes, can be used to define the inputs required to characterize the UBC-CODAM3Ds model. 4.2 Case Study 2: Prediction of Force-Displacement Response and Damage Growth in an OCT Specimen. The second case study used to investigate the material response predicted by the UBC-CODAJVBDs model is that of a quasi-static analysis of crack growth in a prenotched CFRP panel. Of particular interest in this test, apart from the widely different loading scheme (i.e. low rate in-plane), is the origin of the experimental study upon which the model verification is based. The study used was that of Kongshavn and Poursartip (1997), based on the thesis work of Kongshavn (1996), who were interested in experimental characterization of progressive damage growth in FRP laminates. 4.2.1 Experimental Measurements Kongshavn and Poursartip (1997) carried out an experimental investigation with the goal of characterizing damage growth in notched composite laminates. Results presented include force-displacement curves, displacement field analyses, and damage growth characterizations for an Oversized Compact Tension (OCT) specimen which was designed by Kongshavn and 145 Chapter 4 Numerical Case Studies Poursartip (1997) in order to obtain stable growth of damage, a prerequisite for their detailed study of the softening behaviour of FRP laminates. The geometry of the OCT specimen is shown in Figure 4.76 while the dimensions of the two test specimens considered in the numerical study are shown in Table 4.6. The specimen is loaded quasi-statically in an Instron testing machine (see Figure 4.77). Lateral support for the specimen is provided by two teflon coated steel bars loosely clamped on either side of the specimen at the edge opposite to the notch. These supports effectively prevent through-thickness deformation and warping of the specimen under load. Kongshavn and Poursartip present a series of results for AS4/8552 sandwich panels with woven fabric outer plies and T300H/F593 [-45/90/45/0]2S quasi-isotropic CFRP panels. Of the tests reported on, two of the T300H/F593 panels (designated A3 and A4 as shown in Table 4.6) were chosen as the subject of the case study reported here. The major differences between the A3 and A4 specimens are the ratio of notch length to effective specimen width, known as the a/W ratio (see Figure 4.76). Before proceeding with an overview of the results it is helpful to reiterate the definitions for a few terms laid out by Kongshavn and Poursartip. First, the notch refers to the preformed through-thickness saw cut in the specimen. A crack is a zone of damage ahead of the notch which contains fibre damage, matrix cracking, and delamination which cause a through-thickness discontinuity in the material (i.e. a through crack, no load carrying capability across the damaged zone). In contrast, the process zone ahead of the crack contains some matrix and/or fibre damage, and/or delaminations but only in some of the plies (i.e. there is still a continuous load path across the damaged zone). Finally, the crack length is the extent of the through-thickness crack measured along the notch mid-plane. Note that this includes the effective length of the 146 Chapter 4 Numerical Case Studies notch (see Figure 4.76). Kongshavn and Poursartip employed a number of measurement techniques during the test series in order to characterize the growth of damage in the notched specimens. First, the force-displacement curve was measured by recording the applied load and displacement at the pinhole (hereafter referred to as the cross head displacement). Additional measurements of the displacement history were made using a clip gauge mounted on the back of the specimen, 10 mm from the notch tip (refer to Figure 4.76). A clip gauge was also mounted across the mouth of the crack to measure the Crack Mouth Opening Displacement (CMOD). The CMOD gauge was only used on the A4 specimen. The second type of measurements used were post-test non-destructive and destructive characterizations of damage. The techniques used included pulse-echo ultrasonic (PEUS) scans to map delaminations, deplies to measure fibre failure, and micrographical cross-sectional analyses to determine through-thickness fibre breakage, matrix cracking, and delamination. An example of the force-displacement response of the A3 specimen is shown in Figure 4.78a. The initial elastic response is followed by damage growth (indicated by the saw-tooth response), and ultimately crack growth with the corresponding sudden load drop. Figure 4.78b summarizes the results of the deplies and micrographical analyses and shows the extent of the damage zone which had developed at each of the points labelled in Figure 4.78a. It is interesting to note that up to state 3 in Figure 4.78, damage growth consists of the development of a wider and wider zone of damage but not extension of the notch in the form of a crack. From state 3 to state 4, the micrographical analysis confirms the sudden extension of the crack. This crack growth is stable. That is, the crack is arrested and another process zone develops ahead of the crack (i.e. state 5 in 147 Chapter 4 Numerical Case Studies Figure 4.78b). This is followed by further extension of the crack (state 6). The width of the damage zone is consistent along the length of the crack and was found to be 5 mm. A unique feature of these tests is the measurement of the displacement field on either side of the crack and growing zone of damage. A series of lines were inscribed on the A3 specimen (highlighted in Figure 4.77). Through digital photo analysis, Kongshavn and Poursartip were able to measure changes in the displacement field caused by damage growth. The results from the lines inscribed 20 mm from the notch mid-plane (line 4) on specimen A3 are shown in Figure 4.79. The curves correspond to the states labelled in Figure 4.78a and b. One first notices that during the damage growth phase, the line analysis shows an increase in the displacements only on the loading pin side of the notch tip. The experimental trend line rotates about a point that is close to the notch tip. After crack growth (i.e. from point B to D) the displacement field shifts rather than rotating. This shift is consistent with a crack opening displacement. From D to E where a process zone is developing in front of the new notch tip, the line analysis again shows a change in the slope but only a slight change in the zero displacement intercept. Once again, moving from E to F, which is characterized by further crack growth (e.g. the second load drop in Figure 4.78a), we note a shift in the curve. Through these results, Kongshavn and Poursartip were able to show that this displacement field analysis can be used as an indication of the development of damage and the extension of the crack. Further, the results indicate that the zero intercept of the experimental trend line gives a quantitative measure of the extent of damage growth (i.e. the location of the leading edge of the process zone). Finally, Kongshavn and Poursartip attempted to quantify the effect of the damage on the material properties. A number of small tensile specimens were cut from the damaged and undamaged 148 Chapter 4 Numerical Case Studies regions of the tested specimens. The results of this work provided a preliminary characteristic shape for the strain softening curve. These tests have been described in more detail at the end of Section 3.5.3. 4.2.2 Objective The objective of this second case study is relatively straightforward; to demonstrate the flexibility of the model by applying it to a completely different loading scheme and loading rate regime. The OCT specimen investigated by Kongshavn and Poursartip offers a unique subject as the impetus for the experimental study was the characterization of the strain softening behaviour in laminated FRPs. This behaviour forms the basis for the theoretical development of the model presented in this thesis. In fact, the study by Kongshavn and Poursartip has been used to guide the characterization of the materials in both case studies presented in this chapter. In effect, by modelling these tests we come full circle with the experimental evidence for progressive damage growth in laminated FRP composites. This is not the first attempt at modelling the OCT results. Work published by Williams et al. (1996) outlines a numerical study based on the ABAQUS Implicit F E M code. The results of their work demonstrated the need to incorporate a strain softening material response in order to obtain the overall material response observed experimentally. Furthermore the work justified the use of the preliminary strain softening curve obtained by Kongshavn and Poursartip as a valid material characterization. However, the material model used in the study was an adaptation of a simple orthotropic plasticity model with plastic softening instead of plastic hardening. The study described here represents a significant advance over the ABAQUS model. 149 Chapter 4 Numerical Case Studies 4.2.3 FEM Model The finite element model of the OCT specimen is shown in Figure 4.80. The deformable specimen is modelled using Belytschko-Tsai plane-stress quadrilateral shell elements. The material properties used to characterize the plate are discussed below in Section 4.2.4. A segment of the loading pin was also modelled using the same plane-stress elements. The loading pin was assumed to be rigid steel (refer to Table 4.1 for the material properties used). Only half of the OCT specimen was modelled with symmetry enforced by appropriate translational and rotational displacement boundary conditions placed along the notch mid-plane starting at the location of the notch tip (see Figure 4.80). An additional out-of-plane translational constraint was applied to all nodes in the model to mimic the experimental constraint on through-thickness bending and twisting. Thus, the model was restricted to purely in-plane deformation. The shape of the mesh was constrained by the requirement that elements in the damage growth zone must be of uniform size and shape (i.e. square). Outside of this zone, the mesh size sensitivity of the damage model is not an issue and it is possible to use mesh grading to optimize the mesh. The constraint on the mesh meant that it was not possible to locate nodes at points corresponding to the C M O D and clip gauge locations. For example, the C M O D gauge location on the A4 specimen falls halfway between two nodes (see the enlarged section of the mesh in Figure 4.80). The same is true of the clip gauge. As a result, the displacement histories of the two neighbouring nodes were averaged, e.g.: The displacement gradients are not large at the locations of the two gauges so the error induced by this averaging technique is small. 4 CMOD FEM ~ ~Z dCMOD + dcMOD ^ \ node 1 node 2 (4.13) 150 Chapter 4 Numerical Case Studies Note that the local notch tip geometry was not modelled explicitly. A detailed description of the sharp notch tip radius would require a very fine mesh, relative to the scale of the specimen. This has two adverse effects. The first is a significant reduction in the model time step associated with such small elements. The second, and perhaps most significant given the constitutive model being applied, is the mesh sensitivity of damage models which results from the finite strain energy absorption predicted by the strain softening curve. Results of the work by Williams et al. (1996) showed that the omission of a detailed description of the notch tip did not affect the overall results of the modelling work carried out as part of that investigation. In the experiments, the notch was designed in such a way as to give a less localized stress distribution, i.e. a large damage zone. This could have contributed to the good agreement found with the previous F E M work. Loading of the specimen was achieved by applying a monotonically increasing y-displacement to the loading pin rigid body. A number of preliminary runs were performed to ensure that the loading rate used was low enough to eliminate rate effects in the predicted plate stiffness. The value arrived at was 17.5 m/s giving an applied displacement of 1.75 mm (i.e. a relative cross head displacement of 3.5 mm) in 100 ms. Adequate constraint of the rigid body motion was achieved by constraining the displacement of the rigid body to a y-translation only. Rigid body rotation of the loading pin about the z-axis was allowed so as not to apply a shear load on the neighbouring plate elements. Displacement-time histories of the C M O D and clip gauges were gathered from the model (loading pin displacement is an input to the model). Kongshavn and Poursartip observed that the cross head displacements obtained from the experimental data acquisition system were not 151 Chapter 4 Numerical Case Studies accurate. Engels (1996) showed that the cross head displacement recorded and the actual pin hole displacement differed by a factor of approximately 2.1-2.2 in the elastic region of the response, a result of the compliance of the loading jig. As a result, the numerical results for cross head displacement used to generate the force-displacement curves are scaled by a factor of 2.1. The use of a simple scaling factor implies that the effect of the test frame compliance is linear over the entire loading range. While this is unlikely, the results are sufficiently accurate to allow a comparison of the numerical predictions and experimental measurements. The force-time history was obtained by tracking the total reaction force at the symmetry line. The displacement-time histories of the series of nodes which correspond to the location of two of the inscribed lines (line 1 and line 4, 5 mm and 20 mm from the notch mid-plane, respectively) were also collected from the model of the A3 specimen. The experimental line analysis of the A4 specimen was not available. 4.2.4 Material Characterization The two specimens modelled were both made of a T300H/F593 CFRP, a brittle matrix system similar to the EVI6/937. Elastic lamina properties for this system are shown in Table 4.7. Results of an earlier numerical study by Williams et al. (1996) indicated that the actual plate stiffness was slightly lower than the value predicted using the characterization data in Table 4.7. In order to correct this, the elastic moduli were scaled to fit the predicted elastic plate stiffness to the measured value. The resulting elastic properties are listed in Table 4.8 and have been adopted in the current study. Also shown in both Table 4.7 and Table 4.8 are the equivalent sublaminate elastic constants derived using the PC-LAMINATE package (Radford, 1989). As the F593 and 937 are both brittle matrices, the damage growth behaviours of both CFRPs 152 Chapter 4 Numerical Case Studies should be the same, to a first approximation. As a result, the HVI6/937 characterization data for the UBC-CODAM3Ds model, discussed in Section 4.1.7 above, was also used for the T300H/F593 (refer to Table 4.9). The only exception being the value of S , the scale factor on the in-plane shear strain in the damage potential functions (Equations 4.14 and 4.15). The results of the out-of-plane dynamic analyses on the IM6/937 material show the strong effect this parameter has on the damage growth which became dominated by the in-plane shear strain. In the OCT specimen, the nature of the loading (i.e. crack opening) results in a notch tip strain field which is prone to localization along an axis oriented at 45° to the notch mid-plane (i.e. in a shear dominated direction). As a result the value of S was increased to 2. As such, the hypothesis put forward during the characterization of the TM6/937 (i.e. that damage growth in brittle matrix systems is more strongly influenced by shear stain than it is in a tough matrix FRP) is maintained. S is still less than the value used to characterize the T800/3900-2 but the damage growth will be less sensitive to the shear localization than was observed in the IM6/937 results. The damage growth and residual stiffness functions used for the T30G7F593 are shown in Figure 4.81a and b respectively while the predicted ID stress-strain response is shown in Figure 4.82. 4.2.5 Model Predictions Force-Displacement Curves Figure 4.83a and b show the comparison of measured and predicted cross head and C M O D and clip gauge force-displacement curves, respectively, for the A4 specimen geometry. Note that the displacements shown in all the graphs presented in this section are the total displacements (e.g. the total displacement of one loading pin relative to the other). Therefore, the numerical 153 Chapter 4 Numerical Case Studies predictions plotted are twice the model outputs. Figure 4.83a indicates a certain amount of slackness in the experimental loading jig which causes the nonlinearity in the initial portion of the response but overall, the numerical model accurately predicts the stiffness of the OCT specimen in the elastic region. Unfortunately, problems with the experimental set-up resulted in the loss of a significant portion of the cross head displacement history. However, the C M O D and clip gauge results shown in Figure 4.83b capture the entire test. Here the offset between the measured and predicted clip gauge results is the result of slackness between the clip gauge and the specimen. The numerical model predicts the peak force to within 7%. Onset of damage growth is also predicted reasonably accurately with the experiments showing the first significant load drop at a C M O D displacement of 0.682 mm (labelled as point 1 in Figure 4.83b). Although the numerical model doesn't show the same load drop, there is a marked change in slope which occurs at roughly the same C M O D displacement. The use of a matrix damage threshold means that damage will grow throughout the loading, however, the stiffness reduction associated with this damage will be quite small. The change in the slope observed in the force-displacement curve (i.e. the change in the plate stiffness), is indicative of the initiation of a significant stiffness reduction (i.e. significant damage growth). At a C M O D displacement of 0.805 mm (point A in Figure 4.83b) the F E M model predicts the development of a large damage zone (i.e. crack growth) although the rate of damage growth seems to be lower than that observed in the experiment. This behaviour is consistent with the rapid growth of a damage zone ahead of the notch and corresponds to the load drop between points 2 and 3 on the experimental curve. Each segment of the predicted drop corresponds to an incremental growth of the damage zone by one element along the notch mid-plane. Experimentally, the response shows a more gradual development of a damage zone ahead of the notch between points 1 and 2, before the sudden 154 Chapter 4 Numerical Case Studies crack growth. In contrast, the numerical results show a slightly 'tougher' response after the predicted peak load. Other than a change in the slope of the force-displacement curve, there is no indication of this gradual damage growth in the model before the peak load is reached. Figure 4.84 shows the force-displacement curves for the A3 specimen geometry. Also shown are two other experimental results, specimens A l and A2. The specimen geometry used in these test was the same as that of the A3 specimen. The numerical prediction is extremely good. The peak force and displacement at the onset of crack growth are very close to those measured in specimen A l . Again we note the change in stiffness of the numerical result which occurs at roughly the same displacement as the first small load drop is observed in the A l specimen. It is particularly interesting to note the range of behaviours observed experimentally. The A l specimen shows a much more brittle response with a higher peak force than either of the other two specimens and a more sudden crack growth (i.e. no significant softening before the load drop). Specimen A3 shows a similar response to that of the A4 specimen discussed above while the response of the A2 specimen falls between that of A l and A3. These results show that the differences observed between the predicted and measured response of the A4 specimen may not be the result of an inconsistency in the model but rather an indication that the model simply captures one of the many force-displacement responses that are possible with the OCT test. There is not enough experimental data for one specimen geometry to make a formal statement about the repeatability of the force-displacement curves and so the pursuit of this hypothesis is left for future work. Damage Growth The experimental line analysis has been proven to be an effective tool for predicting the damage growth. Figure 4.85 shows the numerical predictions of the displacement fields at line 1 and line 4 (5 mm and 20 mm from the notch mid-plane, respectively) of the A3 specimen. The points on 155 Chapter 4 Numerical Case Studies the numerical force-displacement history which correspond to the six states shown are labelled A though F in Figure 4.84. Figure 4.86 shows the corresponding predictions of damage growth through fringe plots of average damage (((0, + OJ2)/2). Note the 'background' low level damage which is predicted throughout the plate. Again, this is a consequence of using a 0 strain as the threshold for matrix damage (refer to Section 4.1.7). The results for lines 1 and 4 (Figure 4.85a and b, respectively) show the same trends observed in the experimental results for line 4 (Figure 4.79), although the predicted changes in the displacement field are most easily seen in the results for line 1. The proximity of line 1 to the notch mid-plane makes it more sensitive to the changes in the displacement field which result from damage growth. Line 4 tends to show a smeared version of the same trends. From state A to B, there is no crack growth but there is the development of a diffuse damage zone ahead of the notch tip. This zone is three element wide (5.04 mm) and two element long (3.4 mm). The line analysis in Figure 4.85a shows a change in the slope of the displacement field between states A and B but little shift in the zero displacement intercept. The linear portion of the predicted trends for A and B, when extrapolated to the axis, shows that the zero displacement occurs roughly 4 mm (i.e. ~ 3.4 mm) ahead of the notch tip. From state B to C, there is a more significant magnification and growth of the damage zone to a distance of 8.4 mm (see Figure 4.86) consistent with a prediction of crack growth. The predicted displacement field at line 1, Figure 4.85a, shows the same shift in the curve between states B and C (not a rotation as was observed between A and B) that can be attributed to crack growth in the experimental results. The same is true from states C to D, D to E, and E to F. In all cases, an accurate prediction of the observed damage growth (Figure 4.86) can be made by extrapolating the linear portion of the displacement field back to a zero-displacement, the same technique that was verified 156 Chapter 4 Numerical Case Studies experimentally by Kongshavn and Poursartip. Having demonstrated that the numerical line analysis technique accurately models the same experimental analysis tool, it is of interest to see how close the numerical predictions are to the experimental measurements. Figure 4.87 again shows the predicted and measured force-displacement curves for the A3 specimen (previously shown in Figure 4.84). Here, selected experimental data points, which correspond to displacements at which the experimental line analysis was performed, are labelled as E X P 1, E X P 2, etc. For points E X P 1 and E X P 2, results from the F E M model at the corresponding cross head displacements have been chosen (FEM A ' and F E M B') . The difference in the response past the load drop complicates the selection of points based solely on cross head displacement. However, we are more interested in the predictions of displacement field at characteristic points (e.g. just before or just after the load drop). As a result, points F E M C and F E M D ' were chosen for comparison with E X P 3 and F E M E ' for comparison with E X P 4. The resulting plots of the measured and predicted line 4 displacement field are shown in Figure 4.88a and b. In all cases the agreement is extremely good. The overall trends are captured as are the magnitudes of the displacements indicating that the overall plate stiffness reduction is accurately modelled. The one area where the predictions deviate somewhat from the experiments is in the intercept with the zero-displacement axis. This seems to be a problem related to the distance from the damage growth region. Numerically, the sharp discontinuity in the displacement field caused by the damage zone appears to be smoothed out by the relatively undamaged material which lies between the line and the notch mid-plane. The same effect was noted in Figure 4.85. 157 Chapter 4 Numerical Case Studies As a final comparison, we return to the experimental damage growth prediction shown in Figure 4.78b and compare them with the numerical predictions shown in Figure 4.86. The process zone size is observed to be approximately 7 mm long. This is roughly the size of the damage zone just before crack growth (labelled as point 3 in Figure 4.78a and b). The numerical prediction from States B and C in Figure 4.86 show a damage zone that is between 3.4 and 8.4 mm. The initial state (B) is made up of partially damaged elements only (i.e. a process zone not a crack). After the load drop, the size of the experimental damage zone (crack and process zone) is 25 mm long. The numerical predictions for states D and E are 21.8 and 25.2 mm, respectively. The overall damage zone size (point 6 in Figure 4.78a and b) is 32 mm. From state F in Figure 4.86 it is predicted to be 33.6 mm. Finally, the width of the experimental damage is 5 mm. While the width of the predicted damage zone varies along its length, the initial process zone at states B and C are predicted to be between two to three elements in width or 3.36 to 5.04 mm. In all cases, the predicted damage zone sizes are remarkably similar to the measured values. 4.2.6 Case Study 2 Summary The second case study presented above provides a relatively simple application of the model developed in this thesis. Results show the predictive capability of the model in terms of overall damage zone size and material response for a problem involving quasi-static in-plane loading. More significant is the ability to quantitatively predict the overall trends of an experimental technique which was developed to characterize the very basis for the model, the strain softening or progressive damage growth characteristic of laminated FRPs. The one aspect of the response that the numerical model failed to predict correctly was the 158 Chapter 4 Numerical Case Studies progressive damage growth before extension of the crack. Variability in the experimental results does not allow a well formulated conclusion to be drawn and more experimental data and numerical analyses will be required. However, the predicted brittle behaviour is consistent with the results of the previous case study which raised questions about the close relationship which is predicted to exist between matrix/delamination damage and fibre breakage. An under-prediction of the stiffness loss resulting from matrix damage and an over-prediction of the fibre damage growth could lead to the brittle response which was observed in both the A3 and A4 specimens. 4.3 Summary In this chapter two case studies have been undertaken to investigate and verify the material response predicted by the UBC-CODAM3Ds model. The first rather extensive study demonstrates the ability of the model to quantitatively predict the damage growth in a normally impacted CFRP plate. The use of published experimental data to characterize the required input for the model was also described in some detail. The application of the model to a material system which exhibits a much different damage growth behaviour (i.e. a brittle matrix CFRP in contrast to a tough one) serves to further highlight the ease with which the model can be characterized based on physical interpretation of a material's response. This is in contrast to the parametric studies and user experience which are typically required by previous composite failure models. While results for the IM6/937 brittle matrix CFRP were not quite as good as those of the analyses of the T800/3900-2, they do show that the model is able to predict qualitative, and in most cases quantitative, trends in the experimental observations of damage growth. The results for the T800/3900-2 were extremely good. The quantitative comparisons of damage growth show 159 Chapter 4 Numerical Case Studies accurate predictions of both matrix/delamination and fibre damage over a wide range of impact energies. The predication of the time histories (force, displacement, and energy) match the experimental measurements, supporting our philosophy that the accurate predictions of the force-time history is a by-product of a good damage model rather than a measure of one. Comparisons with the results obtained from computational runs using an existing composite failure model and another C D M based damage model were also favourable. At lower incident energy levels, the UBC-CODAM3Ds model predictions were as good as those of the C D M model (the failure model performed poorly throughout the series of analyses) while at higher impact energies, the model developed in this thesis showed improved predictions over these other models. The second case study illustrates the ability of the model to predict damage growth in a wider range of loading conditions. The result of this study showed that the numerical model could be used effectively to perform a virtual experiment. The numerical results were obtained which mimicked the experimental measurement techniques including the force-displacement time histories and displacement field analyses. In all cases the numerical predictions compared favourably with the experimental measurements. The predictions of the extent of damage growth at characteristic points in the tests were also remarkably good. Both case studies highlight one aspect of the model which will need further development and refinement, that is the damage growth function. In both cases, there was evidence that the effect (i.e. amount of damage and/or magnitude of the associated stiffness reduction) of fibre breakage was over-predicted for a given amount of matrix/delamination damage. The result is an under-prediction of the energy absorption and, in the brittle matrix CFRPs, a response that is much too brittle. While there is confidence in the characterization of the damage thresholds, the damage 160 Chapter 4 Numerical Case Studies growth potential functions, and the relationship between fibre and matrix damage which is implied by those potential functions, are open to further investigation. It is clear that the original assumption that a single strain-based potential function is capable of describing not only the matrix/delamination damage growth but also fibre breakage is too simple. 161 Chapter 4 Numerical Case Studies Table 4.1 Elastic properties for steel. Parameter Value Units P 7810 kg/m3 E 208.5. GPa V 0.33 Table 4.2 Characterization data for the T800H/3900-2 plates. Elastic constants are from Ilcewicz (1992), the strength constants are from Straznicky et al. (1993), and the MLT constant, m, is from Williams and Vaziri (1995). Lamina Sub-laminate [45/90/-45/0] Parameter Value Parameter Value Units P 1543 P 1543 kg/m3 Ei 129.1 E x 48.37 GPa E 2 7.45 Ey 48.37 GPa Gl2 3.52 GXy 18.36 GPa V 1 2 0.33 V x y 0.320 tply 0.194 tsublam 0.775 mm x, 2089 N/A - MPa X c 1482 N/A - MPa Y, 79 N/A - MPa Y c 231 N/A - MPa Sc 133 N/A - MPa m 10 N/A -162 Chapter 4 Numerical Case Studies Table 4.3 UBC-CODAM3Ds damage model characterization for T800H/3900-2. Parameter Value Parameter Value Tensile Compressive Tensile Compressive 0.800 0.800 K x 1.000 1.200 R E 0.650 0.650 K y 1.333 1.400 R G 0.650 0.650 Lx 1.333 1.400 F 1 0.010 L y 1.000 1.200 F n 0.016 S x = S y 4.000 F m 0.040 T x = T y 1.000 ^ n coE 0.160 U x = U y 1.000 coG 0.160 E n 0.930 0.930 G n 0.930 0.930 163 Chapter 4 Numerical Case Studies Table 4.4 Characterization data for the IM6/937 plates. The elastic properties are from Razi (1998). Lamina Sub-laminate [45/0/-45/90] Parameter Value Parameter Value Units P 1543 P 1543 kg/m3 E, 152.4 E x 57.81 GPa E 2 9.24 Ey 57.81 GPa Gl2 4.83 GXy 21.98 GPa Vl2 0.33 v x y 0.315 tply 0.1958 tsublam 0.7833 mm Table 4.5 UBC-CODAM3Ds damage model characterization for IM6/937. Parameter Value Parameter Value Tensile Compressive Tensile Compressive 0.800 0.800 K X 1.000 1.200 R E 0.650 0.650 K Y 1.333 1.400 R G 0.650 0.650 L X 1.333 1.400 F 1 0.000 L y 1.000 1.200 F N 0.012 S X = S Y 1.000 F m 0.032 T X = T Y 1.000 coE 0.300 Ux=lJy 1.000 „ n coG 0.300 E n 0.869 0.869 G n 0.869 0.869 164 Chapter 4 Numerical Case Studies Table 4.6 Over/height compact tension (OCT) specimen dimensions. Letters refer to the schematic in Figure 4.76. All dimensions are in mm. (adapted from Kongshavn and Poursartip, 1997) Specimen ID a W aAV H B b c d e f A3 25.26 85.86 .29 200 3.35 12.7 19.05 7.8 N/A 5' A4 17.63 79.8 .22 207 3.35 22.24 19.05 9.5 28.72 2.58 ' 9 lines were inscribed above and 9 lines below the notch mid-plane ' 20 lines were inscribed above and 20 lines below the notch mid-plane Table 4.7 Characterization data for T300/F593. The elastic properties are from Razi (1995). Lamina Sub-laminate [-45/90/45/0] Parameter Value Parameter Value Units P 1543 P 1543 kg/m3 E, 115.2 E x 45.04 GPa E 2 8.83 Ey 45.04 GPa G12 4.55 G x y 17.15 GPa V l 2 0.34 VXy 0.313 tply 0.2032 tsublam ." 0.8092 mm 165 Chapter 4 Numerical Case Studies Table 4.8 Elastic properties for T300/F593 from Engels (1996). These values were used to fit the numerical predictions of the OCT specimen stiffness to experimental results from Kongshavn and Poursartip (1997). Lamina Sub-laminate [-45/90/45/0] Parameter Value Parameter Value Units P 1543 P 1543 kg/m3 Ei 99.65 E x 38.98 GPa E 2 7.65 Ey 38.98 GPa Gl2 3.94 GXy 14.85 GPa Vi2 0.34 VXy 0.313 tply 0.2032 tsublam 0.8092 mm Table 4.9 UBC-CODAM3Ds damage model characterization for T300H/F593. Parameter Value Parameter Value Tensile Compressive Tensile Compressive « r n 0.800 0.800 K x 1.000 1.200 R E 0.650 0.650 K y 1.333 1.400 R G 0.650 0.650 L X 1.333 1.400 F 1 0.000 L y 1.000 1.200 F n 0.012 S x = Sy 2.000 pin 0.032 TX=Ty 1.000 0.300 U X = U y 1.000 coG 0.300 E n 0.869 0.869 G n 0.869 0.869 166 Chapter 4 Numerical Case Studies Hardened Steel Hemispherical Indentor 0 25.4mm Rubber clamps • Composite Target Panel 101.6mm x 152.4mm Aluminum Test Frame Figure 4.1 Impact test specimen geometry. 167 Chapter 4 Numerical Case Studies Projected Figure 4.2 Post test C-scan measurements of delamination growth for (a) a 3 0 J high mass impact on a T 8 0 0 / 3 9 0 0 - 2 tough resin CFRP panel and (b) a 3 3 J low mass impact on an I M 6 / 9 3 7 brittle resin CFRP panel, (a) illustrates the method of calculating the projected area of delamination from the image and (b) highlights the characteristic spiral staircase pattern of delaminations between neighbouring plies. The highlighted pair of delaminations have occurred between a 0 ° and a - 4 5 ° lamina. Figure 4 . 3 Example of a deplied lamina showing the fibre breakage area measurement technique. The fibre breakage has been highlighted in black. Note that the length of the fibre breakage is determined perpendicular to the fibre direction and that multiple breaks of the same fibre are counted. 168 Chapter 4 Numerical Case Studies Projectile (4>25.4 mm, 60° arc) Simply supported Figure 4.4 Exploded view of the FEM model of the CAI impact specimen with the assembled mesh. The 960 element target mesh, corresponding to a 3.175x3.175 mm element size, is shown here. 169 Chapter 4 Numerical Case Studies ^ 1 2 0 1100 it I 60 + ts 3 20 o H 0 T800H/3900-2 / / / / / • 0.030 mm2/mm2 / + Open circles: quasi-static Grey circles: high mass 1 0 2000 4000 6000 Total delamination area (mm2) (a) JM6/937 / Open circles: quasi static ^ Grey squares: high mass Black squares: low mass / . / / / + + 10000 20000 30000 40000 Total delamination area (mm ) (b) Figure 4.5 Experimental measurements of fibre breakage area as a function of total delamination from quasi-static indentation and non-penetrating impacts of (a) T800/3900-2 and (b) IM6/937 CFRP coupons. 170 Chapter 4 Numerical Case Studies 1.0 0.8 0.2 o.o '- / ^ / / ; / / ' . I i 1 . i . i 1 . . i . 1 i i . i 0.00 0.01 0.02 0.03 0.04 Damage Growth Potential F(E) 0.05 1.0 II 0.8 • ffness 0.6 IStil 0.4 a s -Resic 0.2 :-0.0 0.0 0.2 -+- -+-0.4 0.6 Damage CO 0.8 1.0 (a) (b) Figure 4.6 T800H/3900-2 (a) damage growth function and (b) normalized residual stiffness function. 1000.0 0.00 0.01 0.02 0.03 Strain (mm/mm) 0.04 0.05 Figure 4.7 Predicted one-dimensional stress-strain response for T800H/3900-2. 171 Chapter 4 Numerical Case Studies Figure 4.8 Comparison of experimentally measured and predicted total delamination area as a function of incident energy for a [45/907-45/0]3S T800H/3900-2 CFRP plate. Figure 4.9 Comparison of experimentally measured and predicted total fibre breakage as a function of incident energy for a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 172 Chapter 4 Numerical Case Studies £ S ta v i. cs s o £ 3 o H Experiments Open circles: quasi-static Grey circles: high mass Numerical predictions Grey squares: high mass Black squares: low mass 10 20 30 Impactor energy loss (J) 40 50 Figure 4.10 Comparison of experimentally measured and predicted total delamination area as a function of impactor energy loss for a [45/90/-45/0]3S T800H/3900-2 CFRP plate. £ £ 4> OX) 03 « & -Q kl 3 o H 140 120 100 T800H/3900-2 With a bilinear relationship between fibre breakage and delamination size Numerical predictions Grey squares: high mass Black squares: low mass Experiments Open circles: quasi-static Grey circles: high mass 10 15 20 25 30 Impactor energy loss (J) 35 40 Figure 4.11 Comparison of experimentally measured and fibre breakage as a function of impactor energy loss for a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 173 Chapter 4 Numerical Case Studies Figure 4.12 Comparison of experimentally measured and predicted energy loss as a function of impact energy for a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 174 Chapter 4 Numerical Case Studies Figure 4.13 Prediction of total delamination area as a function of projectile energy loss from a single FEM run with an incident energy of 60 J (high and low mass events). Results shown are for a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 0 10 20 30 40 50 60 Impact energy (J) Figure 4.14 Prediction of total fibre breakage as a function of projectile energy loss from a single FEM run with an incident energy of 60 J (high and low mass events). Results shown are for a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 175 Chapter 4 Numerical Case Studies (b) Figure 4.15 Total system energy as a function of impact energy for a [45/90/-45/0]3S T800H/3900-2 C F R P plate. The numerical results of energy absorbed in matrix and fibre damage (Ematrix a i , d Efjorej respectively) are based on the F E M damage area predictions and experimental measurements of energy absorption for each type of damage. In (a) the residual kinetic energy has been used as a measure of elastic energy stored (Elastic) during the event while in (b) the elastic energy has been fit to the experimental data. 176 Chapter 4 Numerical Case Studies Figure 4.16 Comparison of experimentally measured and predicted projected delamination size as a function of impact energy for a [45/90/-45/0]3S T800H/3900-2 CFRP plate. Figure 4.17 Relationship between the experimentally measured ratio of the total delamination area to projected delamination sizes (Atotai/A p r 0j e c t e d) and impact energy. The ratios were obtained from micrographical analyses (total area) and C-scans (projected area) of the T800/3900-2 and IM6/937 impact results of Delfosse (1994). 177 Chapter 4 Numerical Case Studies 6000 Figure 4.18 Comparison of experimentally measured and predicted total fibre breakage as a function of delamination area for a [45/9G7-45/0]3S T800H73900-2 CFRP plate. 178 Chapter 4 Numerical Case Studies Figure 4.19 Predicted projected average matrix/delamination damage distributions for impact energies of (a) 30 J and (b) 33 J and (c) 35 J . Results shown are for low mass (high velocity) impacts on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 179 Chapter 4: Numerical Case Studies Figure 4.20 Predicted damage state for a high mass 46.2 J (v = 3.82 m/s, m — 6330 g) impact on a [45/90/-45/0]3s T800H/3900-2 CFRP plate showing the variation in the predicted averaged damage, (o^ + Ct)2)/2 through the thickness of the plate. 180 Chapter 4: Numerical Case Studies 9J 22 J 33 J 56 J 25.4 mm (co,+co2)/2 0.00 0.20 0.40 0.60 0.80 1.00 Figure 4.21 Comparison of predicted projected matrix/delamination damage and experimental C-scan images for low mass impact events on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. Results presented are for (a) 9.4 J (v = 7.74 m/s, m = 314g), (b) 22.0 J (v = 11.84 m/s, m = 314g), (c) 33.4 J (v =14.59 m/s, m = 314g), and (d) 56.4 J (v = 18.97 m/s, m = 314 g) impacts. 181 r Chapter 4: Numerical Case Studies Figure 4.22 Comparison of predicted projected matrix/delamination damage and experimental C-scan images for high mass impact events on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. Results presented are for (a) 9.5 J (v = 1.76 m/s, m = 6141 g), (d) 46.2 J (v = 3.82 m/s, m = 6330 g), (c) 58.2 J (v = 4.29 m/s, m = 6330 g) impacts. 182 Chapter 4 Numerical Case Studies Experimental (e) 58.2 J (v = 4.29 m/s, m = 6330 g) Figure 4.23 Comparison of predicted and measured force-time histories for high mass, low velocity (drop weight) events on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 183 Chapter 4 Numerical Case Studies Predicted Experimental 2.0 4.0 6.0 Displacement (mm) (a) 9.5 J (v = 1.76 m/s, m = 6141 g) 8.0 Experimental 2.0 4.0 6.0 Displacement (mm) (b) 29.5 J(v = 3.12 m/s, m= 6141 g) 8.0 Predicted Experimental 2.0 4.0 6.0 Displacement (mm) 8.0 2.0 4.0 6.0 Displacement (mm) 8.0 (c) 34.5 J (v =3.30 m/s, m = 6330 g) (d) 46.2 J (v = 3.82 m/s, m = 6330 g) 20000 Predicted 0.0 2.0 4.0 6.0 Displacement (mm) (e) 58.2 J (v = 4.29 m/s, m = 6330 g) 8.0 Figure 4.24 Comparison of predicted and measured force-displacement histories for high mass, low velocity (drop weight) events on a[45/90/-45/0]3S T800H/3900-2 CFRP plate. 184 Chapter 4 Numerical Case Studies 35 : Predicted 30 : 25 : h 2 0 ' j / Q : ' Experimental g 15 z 10 : 5 :_ 0 ; 1 , I . . • • 1 • • • i 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Time (ms) (c) 34.5 J (v = 3.30 m/s, m = 6330 g) Time (ms) (d) 46.2 J (v = 3.82 m/s, m = 6330 g) / Experimental / Predicted 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Time (ms) (e) 58.2 J (v = 4.29 m/s, m = 6330 g) Figure 4.25 Comparison of predicted and measured energy-time histories for high mass, low velocity (drop weight) events on a [4590/-45/0]3S T800H/3900-2 CFRP plate. 185 Chapter 4 Numerical Case Studies 20000 16000 Z 12000 + Experimental Predicted 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Time (ms) (a) 9.4 J (v = 7.74 m/s, m = 314 g) Predicted Experimental 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Time (ms) (b) 22.0 J (v = 11.84 m/s, m = 314 g) Experimenta] Experimental 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Time (ms) (c) 33.4 J (v = 14.59 m/s, m = 314 g) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Time (ms) (d) 56.4 J (v = 18.97 m/s, m = 314 g) Predicted Experimental 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Time (ms) (e) 84.4 J (v = 23.19 m/s, m = 314 g) Figure 4.26 Comparison of predicted and measured force-time histories for low mass, high velocity (gas gun) events on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 186 Chapter 4 Numerical Case Studies 20000 16000 Z 12000 4> | 8000 4000 0 Predicted / Experimental •if 0.0 2.0 4.0 6.0 Displacement (mm) (a) 9.4 J (v = 7.74 m/s, m = 314 g) 8.0 Predicted Experimental 2.0 4.0 6.0 Displacement (mm) 8.0 (b) 22.0 J (v = 11.84 m/s, m = 314 g) Predicted 2.0 4.0 6.0 Displacement (mm) 8.0 Experimental 2.0 4.0 6.0 Displacement (mm) 8.0 (c) 14.59 J (v = 14.59 m/s, m = 314 g) (d) 18.97 J (v = 18.97 m/s, m = 314 g) 20000 Predicted Experimental 2.0 4.0 6.0 8.0 Displacement (mm) (e) 84.4 J (v = 23.19 m/s, m = 314 g) 10.0 Figure 4.27 Comparison of predicted and measured force-displacement histories for low mass, high velocity (gas gun) events on a[4590/-45/0]3S T800H/3900-2 CFRP plate. 187 Chapter 4 Numerical Case Studies 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 Time (ms) Time (ms) (c) 14.59 J (v= 14.59 m/s, m = 314 g) (d) 18.97 J (v = 18.97 m/s, m = 314 g) 90 Time (ms) (e) 84.4 J (v = 23.19 m/s, m = 314 g) Figure 4.28 Comparison of predicted and measured energy-time histories for low mass, high velocity (gas gun) events on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 188 Chapter 4 Numerical Case Studies Figure 4.29 Comparison of predicted and measured force-time histories for a low mass 9.5 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 10 Time (ms) Figure 4.30 Comparison of predicted and measured energy-time histories for a low mass 9.5 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 189 Chapter 4 Numerical Case Studies Figure 4.31 Comparison of predicted and measured force-displacement histories for a low mass 9.5 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. Figure 4.32 Comparison of predicted and measured force-time histories for a low mass 22.0 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 190 Chapter 4 Numerical Case Studies Time (ms) Figure 4.33 Comparison of predicted and measured energy-time histories for a low mass 22.0 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. o fa 20000 16000 12000 + 8000 + 4000 C O D A M (heavy dashed line) M L T (solid line) Experimental (heavy solid line) Chang and Chang (dashed line) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Displacement (mm) 3.5 4.0 4.5 Figure 4.34 Comparison of predicted and measured force-displacement histories for a low mass 22.0 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 191 Chapter 4 Numerical Case Studies Figure 4.35 Comparison of predicted and measured force-time histories for a low mass 33.4 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. Figure 4.36 Comparison of predicted and measured energy-time histories for a low mass 33.4 J event on a [45/90/-45/0]3S T800H73900-2 CFRP plate. 192 Chapter 4 Numerical Case Studies Figure 4.37 Comparison of predicted and measured force-displacement histories for a low mass 33.4 J event on a [45/907-45/0]3S T800H/3900-2 CFRP plate. Figure 4.38 Comparison of predicted and measured force-time histories for a low mass 56.5 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 193 Chapter 4 Numerical Case Studies 0 K . i i . I i i i . I i i i i I . i i i I i i i i 1 i i i • I . i i i I i . i i I 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Time (ms) Figure 4.39 Comparison of predicted and measured energy-time histories for a low mass 56.5 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. Figure 4.40 Comparison of predicted and measured force-displacement histories for a low mass 56.5 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 194 Chapter 4 Numerical Case Studies 30000 25000 M L T (solid line) C O D A M (heavy dashed line) Experimental (heavy solid line) Chang and Chang (dashed line) 0.0 0.2 0.4 0.6 0.8 1.0 Time (ms) 1.2 1.4 1.6 Figure 4.41 Comparison of predicted and measured force-time histories for a low mass 84.4 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. Chang and Chang (dashed line) 0.0 0.2 0.4 0.6 0.8 1.0 Time (ms) 1.2 1.4 1.6 Figure 4.42 Comparison of predicted and measured energy-time histories for a low mass 84.4 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 195 Chapter 4 Numerical Case Studies 30000 25000 + 20000 M L T (solid line) v Experimental (heavy solid line) C O D A M (heavy dashed line) Chang and Chang (dashed line) 0.0 2.0 4.0 6.0 Displacement (mm) 8.0 10.0 12.0 Figure 4.43 Comparison of predicted and measured force-displacement histories for a low mass 84.4 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. o fa 20000 16000 12000 8000 4000 C O D A M (heavy dashed line) M L T (solid line) Chang and Chang (dashed line) 6.0 Figure 4.44 Comparison of predicted and measured force-time histories for a high mass 34.5 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 196 Chapter 4 Numerical Case Studies 40 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Time (ms) Figure 4.45 Comparison of predicted and measured energy-time histories for a high mass 34.5 J event on a [45/907-45/0]3S T800H/3900-2 CFRP plate. Figure 4.46 Comparison of predicted and measured force-displacement histories for a high mass 34.5 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 197 Chapter 4 Numerical Case Studies Figure 4.47 Comparison of predicted and measured force-time histories for a high mass 58.2 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. Figure 4.48 Comparison of predicted and measured energy-time histories for a high mass 58.2 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 198 Chapter 4 Numerical Case Studies 20000 1.0 Displacement (mm) Figure 4.49 Comparison of predicted and measured force-displacement histories for a high mass 58.2 J event on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 199 Chapter 4 Numerical Case Studies Figure 4.50 Effect of variations in the damage at saturation of matrix cracking (CO'm) on the predicted damage growth versus energy relationships for a normally impacted [45/90/-45/0]3S T800H/3900-2 CFRP plate. 200 Chapter 4 Numerical Case Studies Open triangles: R'm= 0.80 Black squares: R'm — . Open squares: R'm=0.50 = 0.65 (baseline) - 1500 + a IOOO o H 500 Experimental trend 10 20 30 40 Impact energy (J) 50 60 120 rf- 100 E E t 80 o H Open triangles: R'm = 0.80 Black squares: R'm = 0.65 (baseline) Open squares: R'm=0.50 10 20 30 40 Impact energy (J) 50 60 (a) (b) 3000 2000 4-to* / • X Experimental trend 7* Open triangles: R' m = 0.80 Black squares: R'm = 0.65 Oiaseline) Open squares: R'm= i i i 0.50 10 20 30 40 Impactor energy loss (J) 50 50 45 s 4 0 £ 35 ^ 30 Experimental trend Open triangles: R'm= 0.80 Black squares: R'm = 0.65 (baseline) Open squares: R'm=0.50 * 70 4-25 2  I 15 M io 10 20 30 40 Impact energy (J) 50 60 (C) (d) Figure 4.51 Effect of variations in the modulus at matrix damage saturation (Rm) on the predicted damage growth versus energy relationships for a normally impacted [45/90/-45/0]3S T800H/3900-2 CFRP plate. 201 Chapter 4 Numerical Case Studies 0 10 20 30 40 50 60 Impact energy (J) (a) 120 100 4-SS Open triangles: F =0.005 Black squares: F 1 = 0.010 (baseline) Open squares: F 1 =0.012 10 20 30 40 Impact energy (J) (b) 50 60 Experimental trend 10 20 30 40 Impactor energy loss (J) 50 50 45 40 + 35 30 25 20 --15 -10 5 0 Experimental trend Open triangles: F = 0.005 Black squares: F 1 = 0.010 (baseline) Open squares: F '= 0.012 (C) 0 10 20 30 40 Impact energy (J) (d) 50 60 Figure 4.52 Effect of variations in the damage potential at onset of matrix cracking (F1) on the predicted damage growth versus energy relationships for a normally impacted [45/90/-45/0]3S T800H/3900-2 CFRP plate. 202 Chapter 4 Numerical Case Studies 5000 ^ 4 5 0 0 § 4000 « 3000 c | 2500 | 2000 » 1500 a IOOO o H 500 0 • y \ Experimental trend // \ OpenTriangles: F ° = 0.015 Black squares: F n = 0.016 (baseline) Open Squares: F " = 0.020 10 20 30 40 Impactor energy loss (J) (C) 50 50 45 40 CA 35 o >, 30 0C B 25 <u u o 20 ttt a 15 S 10 5 0 Experimental trend OpenTriangles: F =0.015 Black squares: F n =0.016 (baseline) 10 20 30 40 Impact energy (J) (d) 50 60 Figure 4.53 Effect of variations in the damage potential at onset of fibre breakage (F ) on the predicted damage growth versus energy relationships for a normally impacted [45/90/-45/0]3S T800H/3900-2 CFRP plate. 203 Chapter 4 Numerical Case Studies 5000 .4500 4000 3500 3000 2500 2000 1500 1000 500 0 / * Experimental / trend ..4 F m =0.048 0 10 20 30 40 50 60 Impact energy (J) (a) 120 i # Experimental ^ 1 0 0 + F " 1 =0.020 / trend \ LU • / Fm =0.040 80 -|- \ » / (baseline) / I 60 € 40 H 20 F = 0.048 10 20 30 40 Impact energy (J) (b) 50 60 10 20 30 40 Impactor energy loss (J) 50 70 60 50 40 30 20 10 Experimental trend 10 20 30 40 Impact energy (J) 50 60 (C) (d) Figure 4.54 Effect of variations in the damage potential at rupture (F m ) on the predicted damage growth versus energy relationships for a normally impacted [45/90/-45/0]3S T800H/3900-2 CFRP plate. 204 Chapter 4 Numerical Case Studies 5000 4 - 4500 | | 4000 | ^ 3500 » 3000 e I 2500 « 1500 + T3 Black squares: S = 4 Open squares: S = 10 Experimental trend H 1 1— 20 30 40 Impact energy (J) (a) 60 120 100 80 60 40 20 0 Black squares: S = 4 Open squares: S = 10 Experimental trend 10 20 30 40 Impact energy (J) 50 60 (b) Open triangles: S = 1 Black squares: S = 4 Open squares: S = 10 10 20 30 40 Impactor energy loss (J) (c) 50 50 j 45 --s 40 --(fl C f l 35 --_© 30 --c 25 --u © 20 --a 15 -a 10 --5 --0 --Black squares: S = 4 Open squares: S = 10 10 20 30 40 Impact energy (J) (d) 50 60 Figure 4.55 Effect of variations in the damage potential function in-plane shear strain scale factor (S) on the predicted damage growth versus energy relationships for a normally impacted [45/90/-45/0]3S T800H/3900-2 CFRP plate. 205 Chapter 4 Numerical Case Studies 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0 - Black squares: 960 Elem (baseline) , ' Open Squares: 540 Elem f ° " 2160 Elem £C— i L—i 1— Experimental trend 10 20 30 40 Impact energy (J) (a) 50 60 120 100 DO a 80 + 60 4-40 4-H 20 Black squares: 960 Elem (baseline) Open Squares: 540 Elem 10 20 30 40 Impact energy (J) (b) 50 60 5000 Experimental trend 2160 Elem Black squares: 960 Elem (baseline) Open Squares: 540 Elem 10 20 30 40 Impactor energy loss (J) 50 50 45 S 4 0 | 35 4 30 | 25 0 2 0 1 15 M io 5 0 Experimental trend 2160 Elem 540 Elem 10 20 30 40 Impact energy (J) 50 60 (C) (d) Figure 4.56 Effect of variations in the mesh density on the predicted damage growth versus energy relationships for a normally impacted [45/90/-45/0]3S T800H/3900-2 CFRP plate. 206 Chapter 4 Numerical Case Studies 5000 . 4500 4000 3500 3000 2500 2000 1500 1000 500 0 Matrix damage saturates before v onset of fibre breakage / y/^ . / <4c / / V / V Fibre breakage and ^ matrix cracking in / Jf second phase of s/ damage growth Experimental / , trend 10 20 30 40 Impact energy (J) 50 60 (a) 120 100 it 80 + 60 -f 40 + H 20 Experimental trend Fibre breakage and matrix cracking in second phase of damage growth Matrix damage saturates before onset of fibre breakage 1 10 20 30 40 Impact energy (J) (b) 50 60 5000 , 4500 4000 3500 3000 2500 2000 1500 1000 500 0 Matrix damage 4 saturates before ' • y onset of fibre ^ breakage ' / y-mt-^ Experimental trend / // /*• //*\ / /A \ - Of \ Fibre breakage and matrix cracking in second phase of ...A 1 .1. damage growth 10 20 30 40 Impactor energy loss (J) (C) 50 50 45 s 4 0 I 3 5 I 3 0" I 2 5 -S3 20 - -g. 15 4-M 10 5 + 0 Matrix damage saturates before onset of fibre breakage Experimental trend Fibre breakage and matrix cracking in second phase of damage growth 0 10 20 30 40 Impact energy (J) (d) 50 60 Figure 4.57 Effect of the damage growth function assumptions on the damage growth and energy loss of a normally impacted [45/90/-45/0]3S T800H/3900-2 CFRP plate. Results shown represent the same material characterization but one assumes that fibre breakage initiates only after matrix cracking has saturated. The other prediction shown is based on the assumption of mixed fibre and matrix damage growth after onset of fibre breakage. 207 Chapter 4 Numerical Case Studies Damage Growth Potential F(e) Damage (0 (a) (b) Figure 4.58 IM6/937 (a) damage growth function and (b) normalized residual stiffness function. 1000.0 0.00 0.01 0.02 0.03 0.04 0.05 Strain (mm/mm) Figure 4.59 Predicted one-dimensional stress-strain response for IM6/937. 208 Chapter 4 Numerical Case Studies 35000 30000 25000 20000 es •S 15000 + E * 101 3 o IM6/937 Delamination approaching the boundaries . • Experiments Open circles: quasi-static Grey circles: high mass Black circles: low mass a g / • </\ / & <L aa r m / m \ Experimental trend • /fi " H A°. J i i Numerical predictions Grey squares: high mass Black squares: low mass 10 20 30 40 Impact energy (J) 50 60 Figure 4.60 Comparison of experimentally measured and predicted total delamination area as a function of incident energy for a [45/0/-45/90]3S IM6/937 CFRP plate. 120 6 E a M « es v u JQ V u .a c 3 o H 100 + 80 + 60 + 40 20 IM6V937 Experimental trend Numerical predictions Grey squares: high mass Black squares: low mass 0 j>« »»•»— df@8—8 0—o-0 10 20 D I • Experiments Open circles: quasi-static Grey circles: high mass Black circles: low mass 40 —I— 50 60 Impact energy (J) Figure 4.61 Comparison of experimentally measured and predicted total fibre breakage as a function of incident energy for a [45/0/-45/90]3S IM6/937 CFRP plate. 209 Chapter 4 Numerical Case Studies 120 IM6/937 100 + S B cu M cs *t es 0) t. -Q cu s-XI IS O H Numerical predictions Grey squares: high mass Black squares: low mass Experiments Open circles: quasi-static Grey circles: high mass Black circles: low mass 5000 10000 15000 20000 25000 30000 35000 Total delamination area (mm ) Figure 4.62 Comparison of experimentally measured and predicted total fibre breakage as a function of delamination area for a [45/0/-45/90]3S IM6/937 CFRP plate. 35000 30000 25000 IM6/937 < -Exp. only delamination Experiments Open circles: quasi-static Grey circles: high mass Black circles: low mass Exp. delamination and fibre breakage Numerical predictions Grey squares: high mass Black squares: low mass 10 15 20 Impactor energy loss (J) 25 30 Figure 4.63 Comparison of experimentally measured and predicted total delamination area as a function of impactor energy loss for a [45/0/-45/90]3S IM6/937 CFRP plate. 210 Chapter 4 Numerical Case Studies 100 90 ^ 80 E, 70 I 60 a Cj i-Xt O) u Xi !S 3 O H IM6/937 50 + 40 + 30 20 10 + Numerical predictions Grey squares: high mass Black squares: low mass Experimental trend 'g Experiments Open circles: quasi-static Grey circles: high mass Black circles: low mass -+-10 20 30 Impactor energy loss (J) —I— 40 50 Figure 4.64 Comparison of experimentally measured and fibre breakage as a function of impactor energy loss for a [45/0/-45/90]3S IM6/937 CFRP plate. 50 45 40 35 30 25 20 15 10 5 0 IM6/937 Numerical predictions Grey squares: high mass Black squares: low mass Experimental trend o Experiments Open circles: quasi-static Grey circles: high mass Black circles: low mass + 10 20 30 40 Impact energy (J) 50 60 Figure 4.65 Comparison of experimentally measured and predicted energy loss as a function of impact energy for a [45/0/-45/90]3S IM6/937 CFRP plate. 211 Chapter 4 Numerical Case Studies Impact energy (J) Figure 4.66 Total system energy as a function of impact energy for a [45/0/-45/90]3S IM6/937 CFRP plate. The numerical results of energy absorbed in matrix and fibre damage (Ematrix and Efjbre? respectively) are based on the FEM damage area predictions and experimental measurements of energy absorption by each mode of damage. The elastic energy stored (Elastic) is fit to the experimental trend. 212 Chapter 4: Numerical Case Studies Figure 4.67 Comparison of predicted projected damage and experimental C-scan images for low mass impact events on a [45/0/-45/90]3S IM6/937 CFRP plate. Results presented are for (a) 10.1 J (v = 8.01 m/s, m = 314 g), (b) 22.6 J (v = 12.00 m/s, m = 314 g), (c) 33.8 J (v = 14.67 m/s, m = 314 g), and (d) 43.6 J (v = 16.66 m/s, m = 314 g) impacts. 213 Chapter 4: Numerical Case Studies Figure 4.68 Comparison of predicted projected damage and experimental C-scan images for high mass impact events on a [45/0/-45/90]3S IM6/937 CFRP plate. Results presented are for (a) 11.0 J (v = 1.89 m/s, m = 6141 g), (b) 29.6 J (v = 3.10 m/s, m = 6141 g), (c) 33.7 J (v = 3.31 m/s, m = 6141 g), impacts. 214 Chapter 4 Numerical Case Studies 14000 Predicted Experimental 2.0 3.0 4.0 Time (ms) (a) 11.0 J(v= 1.89 m/s, m = 6141 g) 5.0 6.0 Experimental Predicted 0.0 1.0 2.0 3.0 4.0 Time (ms) (b) 29.6 J (v = 3.10 m/s, m = 6141 g) 5.0 6.0 14000 Predicted Experimental (c) 33.7 J (v = 3.31 m/s, m = 6141 g) Figure 4.69 Comparison of predicted and measured force-time histories for high mass, low velocity (drop weight impact) events on a [45/0/-45/90]3S IM6/937 CFRP plate. 215 Chapter 4 Numerical Case Studies 14000 Displacement (mm) (c) 33.7 J (v = 3.31 m/s, m = 6141 g) Figure 4.70 Comparison of predicted and measured force-displacement histories for high mass, low velocity (drop weight impact) events on a[45/0/-45/90]3S IM6/937 CFRP plate. 216 Chapter 4 Numerical Case Studies 12 i 10 • / / \ Experimental 8 • / \ \ / i 6 j / \ e M 4 ; / \ / / \ Predicted 2 • - IJ \ J 0 : J/ | | 1 1 I 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Time (ms) (a) 11.0 J(v= 1.89 m/s, m = 6141 g) Time (ms) (b) 29.6 J (v = 3.10 m/s, m = 6141 g) Time (ms) (c) 33.7 J (v = 3.31 m/s, m = 6141 g) Figure 4.71 Comparison of predicted and measured energy-time histories for high mass, low velocity (drop weight impact) events on a [45/0/-45/90]3S IM6/937 CFRP plate. 217 Chapter 4 Numerical Case Studies 16000 Time (ms) (e) 43.6 J (v = 16.66 m/s, m = 314 g) Figure 4.72 Comparison of predicted and measured force-time histories for low mass, high velocity (gas gun) events on a [45/0/-45/90]3S IM6/937 CFRP plate. 218 Chapter 4 Numerical Case Studies 16000 12000 4-8000 o fa 4000 Predicted / '•4 » Experimental 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Displacement (mm) (a) 10.1 J (v = 8.02 m/s, m = 314 g) 16000 12000 + z g" 8000 u o fa 4000 + 0 Experimental 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Displacement (mm) (b) 22.6 J (v = 12.00 m/s, m = 314 g) 16000 12000 8000 4000 Experimental Predicted 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Displacement (mm) (c) 29.6 J (v = 13.80 m/s, m = 314 g) 16000 12000 Experimental 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Displacement (mm) (d) 33.8 J (v = 14.67 m/s, m = 314 g) 16000 12000 + Experimental Predicted 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 Displacement (mm) (e) 43.6 J (v = 16.66 m/s, m = 314 g) Figure 4.73 Comparison of predicted and measured force-displacement histories for low mass, high velocity (gas gun) events on a[45/0/-45/90]3S IM6/937 CFRP plate. 219 Chapter 4 Numerical Case Studies 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Time (ms) (e) 43.6 (v = 16.66 m/s, m = 314 g) Figure 4.74 Comparison of predicted and measured energy-time histories for low mass, high velocity (gas gun) events on a [45/0/-45/90]3S IM6/937 CFRP plate. 220 Chapter 4 Numerical Case Studies 9000 Impact energy (J) Figure 4.75 Projected delamination size as a function of impact energy showing the difference in damage zone size between the brittle (IM6/937) and tough (T800H/3900-2) CFRP systems. Numerical predictions for each system are also shown. The FEM data shown combine the high and low mass results. 221 Chapter 4 Numerical Case Studies Lines inscribed on front surface at successive intervals of f mm from the notch mid-plane 2 mm T 4-<f>d < >+* •3 24.5 mm a W Notch mid-plane B Figure 4.76 Overheight compact tension (OCT) specimen geometry. Detailed dimensions for specimen A3 and A4 are listed in Table 4.6. (Adapted from Kongshavn, 1996) 222 Chapter 4 Numerical Case Studies Figure 4.77 OCT specimen in loading fixture showing (a) the location of the CMOD gauge and the out-of-plane displacement constaint and (b) the lines inscribed on the specimen surface for the displacement field analysis. The clip gauge is mounted behind the specimen and is therefore not visible. (Adapted from Kongshavn, 1996) 223 Chapter 4 Numerical Case Studies 7000 Cross head displacement (mm) (a) (b) Figure 4.78 OCT pinhole force-displacement curve and corresponding extent of damage growth for the A3 specimen. Note that no experimental measurements were available for the damage zone size at point 5. (Adapted from Kongshavn and Poursartip, 1997) 224 Chapter 4 Numerical Case Studies 1.6 1.2 S, •a 0.8 4-— "3. 0.6 0.4 0.2 0.0 -0.2 6 • 4 7 *v 3 ^ X . X , A ^ \ 2 \ \ . A A J •t \ 1 •» . X • \ n \ \ * * *v \ \ . \ e ^ A \ A N. N . A \ ^ \ \ A \ ^ \ * V . X. A \ \ • \ • • A \ A \ ^ B S . • \ . • \ V No 1 1 1 1 1 1 1 -30 -20 -10 0 10 20 Position in front of the notch tip (mm) 30 40 Figure 4.79 Experimental measurements of the displacement of the inscribed line 4 showing the measurements corresponding to the points labelled in Figure 4.78. Note that no experimental measurements were available for point 5. 225 Chapter 4 Numerical Case Studies Figure 4.80 Schematic of the OCT finite element model. The mesh shown is for the A3 specimen geometry although the A4 specimen mesh is very similar. The location of the CMOD gauge on the A4 specimen is highlighted. Note that it falls between two nodes. An average of the two neighbouring nodes was used during the post-processing to derive the predicted CMOD displacement. A similar scheme was used for the clip gauge which also falls between two nodes in both the A3 and A4 meshes. 226 Chapter 4 Numerical Case Studies 1.0 0.8 4-0.6 a 0.4 Q 0.2 + 0.0 7 / / y 0.00 0.01 0.02 0.03 0.04 Damage Growth Potential F(E) 0.05 1.0 T II 0.8 « -Tness 0.6 '--IStil 0.4 '--CQ 4j -'% 0.2 '--ft -0.0 0.0 0.2 0.4 0.6 Damage CO 0.8 1.0 (a) (b) Figure 4.81 T300/593 (a) damage growth function and (b) normalized residual stiffness function. 600.0 400.0 es ft. <u 200.0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 Strain (mm/mm) Figure 4.82 Predicted one-dimensional stress-strain response for T300/593. 227 Chapter 4 Numerical Case Studies 14000 j -12000 -Figure 4.83 Comparison of experimentally measured force-displacement curve with the corresponding UBC-CODAM3Ds prediction for (a) the cross head and (b) the clip and CMOD gauges of the A4 specimen. 228 Chapter 4 Numerical Case Studies 10000 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Cross head displacement (mm) Figure 4.84 Comparison of experimentally measured force-displacement history with the numerical prediction for the A3 specimen geometry. Note the range of responses measured from the different experimental specimens, all of which have the same geometry. The points labelled on the numerical prediction correspond to the line analysis displacement predictions shown below in Figure 4.85. 229 (b) Figure 4.85 Predicted line displacements for (a) line 1 and (b) line 4. Points correspond to the CMOD displacements labelled in Figure 4.84. 230 Chapter 4 Numerical Case Studies (co,+co2)/2 0.00 0.20 0.40 0.60 0.80 1.00 State A >• 33.6 State B State C State D State E mm State F 10 mm i 1 Figure 4.86 F E M predictions of damage growth corresponding to the line analyses shown in Figure 4.85. States correspond to the points highlighted on the force-displacement curve shown in Figure 4.84. 231 Chapter 4 Numerical Case Studies F E M C Numerical F E M E' 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Cross head displacement (mm) 3.5 4.0 Figure 4.87 Experimentally measured and predicted force-displacement history for the A3 specimen showing the reference points used to compare the measured and predicted displacements of the line 4 analysis (Figure 4.88 below). 232 Chapter 4 Numerical Case Studies 1.2 Position in front of the notch tip (mm) (b) Figure 4.88 Comparison of predicted line 4 displacements with experimental measurements. The labels of the experimental and FEM data used for comparison correspond to the points on the force- cross head displacement curve shown in Figure 4.87. 233 Chapter 5 Conclusions and Future Work 5. Conclusions and Future Work 5.1 Conclusions The main objective of this work was to develop a physically-based numerical model capable of predicting the damage growth in thin laminated FRP composite. The work was divided into two stages, the development and implementation of the damage model and the verification of the predicted response. Validation of the model was further subdivided into two case studies, the first involving the analysis of dynamically loaded CFRP plates and the second, the analysis of the in-plane quasi-static loading of a notched CFRP plate. The following highlights the accomplishments and conclusions that can be drawn from this work: 1. A plane stress damage model for laminated FRP plates has been developed The most significant features of the model are: • the physical basis for the predicted damage growth and residual stiffness functions • a sublaminate approach which allows ply stacking sequence effects to be incorporated implicitly in the model • the simple mathematical description of the experimental observations of damage growth and the effects of damage on the material response • the description of the damage growth behaviour has been formulated in a manner that is compatible with a continuum damage mechanics description of a composite sublaminate 2. Physically-based methodologies for characterizing the model have been presented A consequence of the physical basis for the proposed damage model is the ease with which it can be characterized: 234 Chapter 5 Conclusions and Future Work • input parameters for the model are based on published material characterization data, which in turn are based on relatively simple standard test schemes • a simple mathematical approach is used to derive many of the material constants required by the model • the strength of this approach has been demonstrated in the case studies used to validate the model • analyses based on material characterization data drawn directly from the literature showed excellent results without the need to calibrate the material inputs through parametric studies or previous experience • the application of the model to different material systems requires a very straightforward and intuitive modification of the material inputs 3. The model can be used to perform a virtual experiment Ultimately, the goal of any modelling work is to be able to perform an analysis that is not only able to predict the material response but can be used to present the results in a format that is comparable to an actual experiment: • both case studies demonstrated the use of the model as a tool to perform a virtual experiment • predictions of damage growth match experimental observations qualitatively and quantitatively • the measured global force and displacement-time histories were accurately predicted 5.2 Future Work While initial results from the model are encouraging, this thesis represents the first step in the ongoing development of a family of physically based C D M models applicable to a wide range of structural applications involving composite materials. The future development of the model 235 Chapter 5 Conclusions and Future Work would benefit from the incorporation of separate matrix/delamination and fibre damage growth functions. In this way, an improved description of the effects of the matrix and fibre systems on the material response would be possible. Furthermore, the damage potential functions could be reformulated to better capture the effects of the strain state on the initiation and growth of damage. This should include the effects of strain-rate on the damage growth. The bilinear residual stiffness and damage growth curves were able to capture the behaviour of the tough resin system investigated in Case Study 1. However, there are material responses that would be better described by a trilinear response. These include cross-ply laminates where matrix/delamination damage may saturate well before fibre damage and the more complex responses observed in quasi-isotropic lay-ups where fibre damage growth can be arrested through a variety of mechanisms. Published data were successfully used to characterize the materials investigated in the case studies. However, the lack of a complete set of characterization data for any one material meant that some of the parameters required were estimated from other similar materials. A complete experimental characterization of a material system should be considered as a precursor to a future application of the model. The model predictions were observed to be most sensitive to the damage growth function thresholds. The work of Kongshavn and Poursartip (1997), for example, proved to be particularly helpful in providing insight into these parameters. A continuation of their experimental investigation of strain softening would be very beneficial to the future development of the damage model. While the mesh sensitivity effects were only touched on briefly in this work, results of the application of the model do show a sharper localization of the damage than is observed 236 Chapter 5 Conclusions and Future Work experimentally. This should be addressed in the future development of the model through the implementation of a nonlocal theory. Finally, the next major step to be made in the model development is the extension of the model to include a more detailed description of the effects of the through-thickness damage on the material response. For efficient development, this work could be focused on a 2D axisymmetric model but the ultimate goal is the formulation of a full 3D brick formulation (UBC-CODAM3Db) currently underway at U B C . 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Zukas, J.A., Nicholas, T., Swift, H.F., et al., (1982), Impact Dynamics, John Wiley & Sons Inc., New York, N Y , USA. 248 Appendix A A CDM Model Applicable to Thin Laminated Structures A. A CDM Model Applicable to Thin Laminated Structures The continuum damage mechanics (CDM) model due to Matzenmiller et al. (1995) (originally published as Matzenmiller et al., 1991) was implemented in LS-DYNA3D to provide a means of assessing the effectiveness of C D M when applied to composite materials subjected to impact loading. While the theoretical development of the model has been published by Matzenmiller et al., few results obtained from this model have been found in the open literature (refer to Chapter 2). This is possibly due to the difficulty in obtaining the necessary material characterization data for the 25 material parameters required by the model, a problem which was highlighted in the original paper by Matzenmiller et al. (1995) and in subsequent applications of the model (see Agaram et al., 1997, for example). The discussion that follows outlines the model and is adapted from the original paper by Matzenmiller et al. (1995). Following the presentation of the theory, results of the application of the model to a normally impacted CFRP plate are presented. A.1 Outline of Model The following discussion outlines the LS-DYNA3D C D M model developed by Matzenmiller et al. (1995) (referred to in this discussion as the M L T model). The model had previously been implemented in LS-DYNA3D as a proprietary material model and was therefore inaccessible. To allow a detailed evaluation of the model and of the implementation of a C D M approach in a numerical code, the model described in Matzenmiller et al. (1995) was implemented as a vectorized User Material Model (UMAT46v) in LS-DYNA3D. The M L T model was developed to predict the damage growth and the effect of the resulting stiffness loss in a unidirectional lamina. The laminate response is then built up from each lamina using simple 249 Appendix A A C D M Model Applicable to Thin Laminated Structures laminate plate theory (i.e. no interaction between the laminae or plies in a laminate). The formulation of the model is based on a series of simplifying assumptions: 1. A unidirectional lamina can be represented as a homogenized continuum, irrespective of the damage state. 2. Linear elasticity applies if the damage state does not change (i.e. under conditions of unloading and reloading). 3. The nonlinear behaviour of the composite lamina is caused only by damage, and is not due to any nonlinear elastic or plastic deformation of the constituents. Of particular interest is the first assumption which leads directly to the type of damage assumed to exist in the lamina. Damage takes the form of disk-like cracks (i.e. independent of crack shape) oriented in the principal lamina directions (i.e. parallel and perpendicular to the fibre) as shown in Figure A. 1. A.l.l Selection of Damage Variables Two damage variables, cox and co2, are introduced to represent the relative size of the two damage areas (i.e. the area of the disk-like cracks) in the principal material directions. An additional damage parameter, cos, is introduced to account for the effect of damage on the shear response. This variable is less intuitive than ct)i and co2 but Matzenmiller et al. argue that the effect of damage on shear will be different than the effect on the longitudinal or transverse directions and hence an independent damage parameter is required. Note that the definition of the damage parameters, G), (i = 1,2, s), as representative of the damage area is not a strict assumption. No statement has been made regarding the inter-dependencies of the damage variables. It would be equally correct to consider a dependency on more fundamental variables which describe the attributes of the crack (e.g. crack geometry). 250 Appendix A A CDM Model Applicable to Thin Laminated Structures A. 1.2 Effective Stress-Strain Relationship In a uniaxial case, the damage is proposed to be analogous to the reduction in load bearing cross-sectional area: ^damaged (A.1) The equation for the effective stress, a, can therefore be written as: or *eff A-A -a = loss j _ ^ loss a = 1-6) (A.2) In a more general form for a multiaxial stress state, this simple linear relationship results in a transformation tensor between the damaged and equivalent undamaged states , [M] (defined previously in Section 3.1), of the form: [M] = 1 l - 0 ) i S Y M 0 1 l - 6 > 0 0 1 l-OJ. (A.3) From Equation 3.1: (AA) where {CT} = {cTj C T 2 T 1 2 } . In order to derive the constitutive equations, it is convenient to consider the compliance relationship: 251 Appendix A A CDM Model Applicable to Thin Laminated Structures {e} = [//°]{fj} where \H0' = _L_ E2 S Y M 0 0 U\2 (A.5) where , E2, V\2, and G®2 are the usual elasticity constants of the undamaged lamina, and {e} = {£, e 2 YnY • Combining Equations A.4 and A.5 we derive the compliance tensor for the damaged material, [H\. {e} = [H°}[M]{cj} = [H]{ a) (A.6) This, in turn, leads to the damaged or effective constitutive (material stiffness) tensor, [C], which is a function of the undamaged elastic constants and damage state defined by co: [C] = [HTl = ( l - f i > i ^ i ° S Y M Yk I - i - i 0 0 1 <l-fl>,X%. (A.7) A.1.3 Damage Thresholds The next step is to determine the damage thresholds, rj and r2, associated with each damage parameter (note that the shear damage threshold is assumed to be the same as the matrix dominated transverse damage threshold, r2). The concept of a threshold function, as implemented in the M L T model, is similar to the yield surface in plasticity theory. Within a certain region in stress space (or strain space), the state of damage in the material will not 252 Appendix A A CDM Model Applicable to Thin Laminated Structures change. This region, the elastic region, is bounded by a series of surfaces /,(o",0),r) associated with the different lamina failure modes (shown schematically in Figure A.2). The threshold functions then become monotonically increasing functions of cr and co which define the size of the elastic region as damage progresses. The prediction of failure is based on a maximum stress criterion with interactive terms assumed for the matrix dominated damage modes (transverse and shear damage) but not for the fibre dominated longitudinal damage. The equations used are of the form similar to Hashin's Criteria (Hashin, 1980), e.g.: Fibre direction Matrix direction a \Xc,t J 2 [ > 0 failed 1 \ (A.8) < 0 elastic 2 T12 2 + - 1 Y I s ) [ > 0 failed I < 0 elastic (A.9) where Xct, Yct, and S are the lamina strength parameters in the longitudinal and transverse directions and in shear, respectively. The surfaces which bound the elastic region can then be derived by substituting the effective stresses into the failure criteria. In symbolic notation: and (A. 10) / 2={cT } r [ F 2 ] { c r } - r 2 = 0 where [Fj] and [F 2] are tensorial representations of the failure criteria (e.g. Equations A.8 and A.9). Although it is useful to develop these equations in stress space, the implementation of this model 253 Appendix A A CDM Model Applicable to Thin Laminated Structures in a numerical code generally requires a strain-controlled approach. This has a secondary benefit if a strain softening response is to be incorporated. It is well-understood that stress-strain relationships which incorporate a descending branch are generally stable only under strain-controlled conditions. The corresponding relations for /j and f2 in strain space are: ft-{e}r[C][f,][C]{£}-^0 and (A. 11) 82={£}T[C][F2][C]{e}-r2=0 where [C] is the compliance tensor which is symmetric (i.e. [C] T = [C]). Therefore the threshold functions are defined as: r2 = { £ } r [ C ] [ F 1 ] [ C ] { £ } and (A. 12) = {e}T[C][F2][C]{e} The gradients of the loading functions, gi and g2, are indicators of the load state: dz . g = 0 and —-£>0 loading d£ dg . g = 0 and ——£ = 0 neutral loading (A. 13) d£ dg g = 0 and ——£<0 unloading d£ The solution algorithm for the M L T formulation is shown in Figure A.3. A.1.4 Damage Growth Law In their development, Matzenmiller et al. (1995) highlight the fact that to this point, no statement has been made about the characteristics of the damage variable. As such, the formulation is fairly general although it does present a rigorous mathematical statement of the problem. In the 254 Appendix A A CDM Model Applicable to Thin Laminated Structures examples presented in Matzenmiller et al. (1995), a damage variable of the form: co = l-e j _ me V X J (A.14) is proposed for the problem of uniaxial tension on a unidirectional laminate. The material stress versus strain curve predicted by this damage function is shown in Figure A.4. It was noted that this function, a Weibull distribution, can be derived from a statistical analysis of the probability of the failure of a bundle of fibres with initial defects. This damage function has been implemented in the LS-DYNA3D M L T composite damage model. The greatest difficulty in applying this damage growth law is obtaining the characterization data required for this model. The stress-strain relation for the example presented above, normalized to the failure or peak stress, is: _J_ CT me x, E°e — v -» J (A.15) Xt Xt While it may appear to be possible to obtain a value for m by fitting this function to a measured stress-strain curve, results of the application of the model (discussed below in Section A.A.2) have shown that the m parameter is not only a function of material but of loading rate and mesh size as well. A.2 Application and Results A.2.1 Material Driver Tests Following the successful addition of the M L T model to LS-DYNA3D as UMAT46v, a number of single element runs were carried out to investigate the predictions of the model. In particular, 255 Appendix A A CDM Model Applicable to Thin Laminated Structures the effects of the m parameters were investigated. Values for the elastic material properties and the lamina strength properties are generally available in the open literature or through experimental analysis. The exponent m, however, has less physical meaning. Figure A.5 shows the predicted response for a single element loaded in uniaxial tension. The material properties used in the analysis are for T800H/3900-2 and can be found in Table 4.2 of Chapter 4. Note that as m approaches infinity, the predicted response approaches that of the instantaneous failure models. As m decreases, not only does the strain softening behaviour become more gradual but the damage growth prior to the ultimate strength causes increasing non-linearity in the loading part of the curve. As a qualitative observation one can liken the response predicted using a high value for m to that of a more brittle material while a lower value of m represents a tougher material. However, the appropriate value for the exponent m may also be related to the mesh and loading conditions as will be discussed later in this section. Figure A.5 also highlights one of the weaknesses of the current model. The response is predicted by a single equation and, as a result, the loading and post-failure responses cannot be separated. This restricts the flexibility of the model. Consider, for example, a case where a linear elastic loading up to or near failure is required (i.e. a high value of m) while a gradual strain softening response is desired after the peak load is reached (i.e. a lower value of m). A.2.2 Comparison with Experimental Data Following the implementation of the model in LS-DYNA3D, a numerical study was undertaken to investigate the predictive capabilities of the model when applied to a practical problem. The problem considered was that of a composite plate subjected to a normal impact by a hemispherical steel indentor and is reported on in Williams and Vaziri (1995). The geometry of 256 Appendix A A CDM Model Applicable to Thin Laminated Structures the test specimen is identical to that used in the CODAJVDDs case study discussed in Section 4.1 as is the F E M mesh used (refer to Figure 4.1 and Figure 4.4 of Chapter 4). Experimental results for incident energy levels of approximately 9.5, 22 and 33 J were obtained for both drop-weight (high mass, low velocity) and gas-gun (low mass, high velocity) impact tests (Delfosse et al., 1995 and Delfosse and Poursartip, 1995 and 1997). The material used in the experimental study was a T800H/3900-2 CFRP with a laminate stacking sequence of [45/90/-45/0]3s. Lamina elastic and strength properties for the material were obtained from the literature and are presented in Table 4.2 of Chapter 4. Figure A.6 compares the unidirectional stress-strain response predicted by the M L T model for two values of m used in the study. The values used for m (2 and 10) were chosen to represent the extremes of the response predicted by the M L T model. Also shown in Figure A.6 is the response predicted by a more traditional instantaneous failure model due to Chang and Chang (LS-DYNA3D M A T 22) which was used to provide a comparison with the current predictive capability of LS-DYNA3D. As shown in the figure, the values of m represent two distinctly different responses, namely, a tough behaviour for m = 2 and a brittle behaviour for m = 10. The latter behaviour is very similar to the CC instantaneous unloading model up to the maximum stress level. Beyond this point, however, the M L T model with m = 10 exhibits a continuous strain softening response characteristic of quasi-brittle materials. The numerical simulations reported here use the three stress-strain models for comparative studies. Although the M L T model allows for different degrees of strain softening, m, in the longitudinal, transverse and in-plane shear loading modes, the same value of m was used for all three modes in the present analyses. Figure A.7 and Figure A.8 show a comparison between the measured contact force-time history 257 Appendix A A CDM Model Applicable to Thin Laminated Structures and those predicted using the C C and M L T models for the 9.4 J and 22.0 J, low mass (314 g) impact events, respectively. For the 9.4 J event, the numerical predictions virtually overlap and reflect the observed undamaged plate response. For the 22.0 J event, the predictions deviate noticeably at about 0.4 ms after impact. This is indicative of the onset of failure. The M L T model with m = 2 yields results that simulate an elastic response without damage. The C C model, on the other hand, predicts very sharp oscillations in the contact force after initial failure; a direct result of the abrupt unloading assumption in the model. Among the numerical results presented, the M L T model with m = 10 appears to yield a more realistic force-time profile. Figure A.9 shows a comparison of the force-time histories for the 33.4 J, low mass impact event (v 0 = 14.6 m/s, m = 314 g). Clearly the M L T model with m = 10 results in a better prediction of the peak forces and shape of the force-time curve. Figure A . 10 shows a similar improvement observed in the predicted force-displacement histories of the same low mass impact event. Figure A. 11 depicts the predicted state of fibre damage in the back-face at the conclusion of the impact event. Fibre failure is expected to be dominant on the back-face due to the global bending effects. For the M L T models, the fringe plots represent different degrees of damage marked by different values of co,. For the CC model, the solid patches represent zones that have completely failed while the remaining zones have remained entirely elastic. The ability to display the level of damage in a spatially continuous fashion is an attractive feature of the M L T model. To aid in the interpretation of the fringe plots, it is useful to note that the critical values of the damage parameter co, at the peak stress (cr = X,) are co, = 0.39 (for m =2) and co\ - 0.09 (for m = 10). It can be seen that for m = 2, the zones that have failed (i.e. co, > co\) are localized in the central region of the plate. In other words, the overall behaviour is close to 258 Appendix A A CDM Model Applicable to Thin Laminated Structures being elastic. The failed zones according to the CC model are spread over a long narrow strip, while the M L T model with m = 10 predicts a relatively smaller failure zone. Qualitatively, the latter prediction agrees well with the back-face fibre breakage zone detected using deply techniques previously reported in Delfosse et al. (1995). Figure A.12 illustrates the good agreement between the predicted and observed (through deply techniques) fibre breakage zone at the distal side of the laminate for the high velocity low mass impact at 33.4 J. Figure A.13, Figure A.14 and Figure A.15 show the corresponding plots for the 34.5 J, high mass impact event (v 0 = 3.3 m/s, m = 6330 g). Again, the predictions of the M L T model with m = 10 compare remarkably well with the experimental data. A.2.3 Mesh Sensitivity The mesh sensitivity of failure models is a well-known limitation of numerical prediction of damage in composites and is generally acknowledged in numerical work. One approach which has been used to overcome this effect is the implementation of a non-local formulation in which the stress at a point in the material is taken to be a function not only of the strain at that point but also the strain in the neighbouring material (as discussed in Belytschko et al., 1986, for example). However, the usual approach is to carry out a convergence study as a component of the numerical investigation. A brief study was conducted to investigate the relationship between m and the mesh size. Initial results show that as the element size decreases, results of the Chang and Chang material model diverge from the experimental results while a lower and lower value of m is required for each increase in mesh density to obtain agreement using the M L T model. This phenomenon reflects the increase in energy dissipation per unit volume as the element size decreases. While an 259 Appendix A A CDM Model Applicable to Thin Laminated Structures instantaneous failure model contains no mechanism to counter this effect, a strain softening model allows the post-failure energy absorption to be increased (or energy dissipation to decrease) to account for the change in element size. In the current form of the M L T model, changing the post-failure behaviour also changes the elastic behaviour, an undesirable characteristic of use of a single damage growth equation over the entire material response. A.2.4 Rate sensitivity The results discussed above in Section A.A.2.2 seem to show that the shape of the strain softening curve used in the analysis is independent of impact energy. However, more extensive application of the model indicates otherwise. Simulations performed to predict the force-time and force-displacement histories of specimens tested at 56.5 J (v 0 = 19.0 m/s, m = 314 g), 58.2 J (v 0 = 4.3m/s, m = 6330 g), and 84.4 J (v 0 = 23.2 m/s, m = 314 g) show that the use of an m value of 10 over-predicted the peak force and under-predicted the duration of the impact event (see Figure A. 16 and Figure A. 17 for example). In fact, as the impact velocity increases the predictions due to the Chang and Chang model improve relative to those of the M L T . Increasing m to 15 or 20 (i.e. more closely approximating a catastrophic failure of the lamina, see Figure A.5) improves the numerical predictions. These observations lead to the conclusion that the strain softening behaviour predicted by the model has to be a function of loading rate. This phenomenon has been discussed in the literature as well. Randies and Nemes (1992) and Nemes and Speciel (1996), for example, incorporated rate-dependent damage evolution in a continuum damage mechanics based model applied to high strain-rate deformation of graphite fibre reinforced composites. Of particular interest was their observation that the inclusion of rate-dependence resulted in a mesh insensitive model which converged to a finite amount of energy dissipation, a necessary feature of a dynamic strain softening model. Inclusion of rate effects is 260 Appendix A A CDM Model Applicable to Thin Laminated Structures equivalent to introducing an implicit length scale into the formulation. A.3 Summary and Conclusion The implementation of a C D M based composite damage model has demonstrated significant improvements in the prediction of damage growth and force-time history of non-penetrating and penetrating impact events on laminated structures when compared to the existing composite failure models. Results of the applications of these models have shown that the theoretical development of the model by Matzenmiller et al. (1995) does provide a useful framework for the further development of C D M based models for predicting damage growth in laminated structures. However, the lack of a standard testing procedure that can be used to characterize the strain softening behaviour severely limits the use of this model. There are a number of other issues which should also be addressed by a new damage model. These include: 1. The physical significance of the choice of damage parameter 2. Ease of material characterization 3. Stacking sequence or lay-up dependence of the damage growth in laminated structures 4. Rate dependence in non-carbon based systems 5. Mesh size dependence of the predicted damage growth 261 Appendix A A CDM Model Applicable to Thin Laminated Structures Transverse matrix failure (transverse loading and shear) Microcracks with normal n, 1 Damaged lamina with material coordinates e, and e2 Microcracks with normal n t 1: Longitudinal matrix and fibre failure (longitudinal loading dominated by fibre failure) Figure A.1 Idealized damage state in a unidirectional fibre-reinforced lamina. (Matzenmiller et al., 1995) Loading Figure A.2 Multi-surface loading criteria in the space of effective stress. (Matzenmiller et al., 1995) 262 Appendix A A CDM Model Applicable to Thin Laminated Structures Constitutive Law C=/(C°,co)h de co r e = e0 + de H g = / ( C , E ,r) g > 0 T C O = / ( E ) r=/(co,e) C=/(C°,a» &=/(C,e) Figure A.3 Solution algorithm for the MLT CDM model. 263 Appendix A A CDM Model Applicable to Thin Laminated Structures 2500 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 Strain (mm/mm) Figure A.5 Effect of variations in the exponent, m,-, on the longitudinal stress-strain behaviour predicted by the MLT model for a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 2500 0.000 0.020 0.040 0.060 0.080 0.100 0.120 Strain (mm/mm) Figure A.6 Comparison of the uniaxial stress-strain responses predicted by the MLT (m = 2 and m = 10) and Chang and Chang models. 264 Appendix A A CDM Model Applicable to Thin Laminated Structures Figure A.7 Comparison of predicted and measured force-time histories for a 9.4 J (v = 7.7 m/s, m = 314 g) impact on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. MLT (m=2) MLT (m=10) Experimental (heavy line) -1000. 0.00 0.20 0.40 0.60 Time (ms) 0.80 1.00 1.20 Figure A.8 Comparison of predicted and measured force-time histories for a 22.0 J (v = 11.8 m/s, m = 314 g) impact on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 265 Appendix A A CDM Model Applicable to Thin Laminated Structures Figure A.9 Comparison of predicted and measured force-time histories for a 33.4 J (v = 14.6 m/s, m = 314 g) impact on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. Figure A.10 Comparison of predicted and measured force-displacement histories for a 33.4 J (v = 14.6 m/s, m = 314 g) impact on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 266 Appendix A A C D M Model Applicable to Thin Laminated Structures (a) (b) Fibre Direction 45° 0) 1 0.000 0.065 0.130 0.195 0.260 0.325 0.390 (co") (C) (0 1 0.000 0.015 0.030 0.045 0.060 0.075 0.090 (to*) Figure A. l l Comparison of predicted back-face fibre damage for a 33.4 J (v = 14.6 m/s, m = 314 g) impact on a [45/90/-45/0]3S T800H/3900-2 CFRP plate showing results for (a) the Chang and Chang model and (b) the MLT model with m = 2 and (c) m = 10. 267 Appendix A A C D M Model Applicable to Thin Laminated Structures (a) (c) Figure A.12 Comparison of (a) and (b) experimentally observed and (c) numerically predicted back-face fibre damage for an impact energy of 33.4 J (v = 14.6 m/s, m = 314 g). The numerical prediction was made using the M L T model with m = 10. 268 Appendix A A CDM Model Applicable to Thin Laminated Structures 19000. T 0.00 1.00 2.00 3.00 4.00 5.00 6.00 Time (ms) Figure A.13 Comparison of predicted and measured force-time histories for a 34.5 J (v = 3.3 m/s, m = 6330 g) impact on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 0.00 1.00 2.00 3.00 4.00 5.00 6.00 Displacement (mm) Figure A.14 Comparison of predicted and measured force-displacement histories for a 34.5 J (v = 3.3 m/s, m = 6330 g) impact on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 269 Appendix A A C D M Model Applicable to Thin Laminated Structures Figure A.15 Comparison of predicted back-face fibre damage for a 34.5 J (v = 3.3 m/s, m = 6330 g) impact on a [45/90/-45/0]3S T800H/3900-2 CFRP plate showing results for (a) the Chang and Chang model and (b) the M L T model with m = 2 and (c) m = 10. 270 Appendix A A CDM Model Applicable to Thin Laminated Structures 19000. x 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Time (ms) Figure A.16 Comparison of predicted and measured force-time histories for a 58.2 J (v = 4.3 m/s, m = 6330 g) impact on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. 20000. x 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Displacement (mm) Figure A.17 Comparison of predicted and measured force-displacement histories for a 58.2 J (v = 4.3 m/s, m = 6330 g) impact on a [45/90/-45/0]3S T800H73900-2 CFRP plate. 271 Appendix B UBC-C0DAM3Ds Model Reference B. UBC-CODAM3Ds Model Reference B.l LS-DYNA3D UMAT47 Input Deck Description Below is a sample input deck for a material using UMAT47, the UBC-CODAM3Ds shell model. Note the keyword format used in the example presented here is supported by the ls936 and ls940 versions of LS-DYNA3D. $ T 8 0 0 / 3 9 0 0 - 2 T o u g h CFRP C o m p o s i t e $ Q u a s i - i s o t r o p i c Lay-up $ *MAT_USER_DEFINED_MATERIAL_MODELS $ m i d r o mt lmc n h v i o r t h o b u l k g $ i f i i i i e e 2 1 . 5 3 7 E - 0 3 47 48 40 1 35 36 $ i v e c t o r i e r o s i o n $ i f 0 0 $ a o p t maxc xp YP zp a l a2 a3 $ i f f f f f f f 0 .0 $ v l v2 v3 d l d2 d3 $ i f f f f f 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 $ 1 2 3 4 5 6 7 8 $ ex v x y gxy $ f f f f f f f f 4 840E+04 0 .320E+00 1 832E+04 0 000E+00 0 O0OE+0O 0 000E+00 0 000E+00 0 OOOE+00 $ 9 10 11 12 13 14 15 16 $ ey v y z g y z $ f f f f f f f f 4 840E+04 0 .340E+00 3 520E+03 0 000E+00 0 O0OE+0O 0 000E+00 0 000E+00 0 000E+00 $ 17 18 19 20 21 22 23 24 $ v x z g z x $ f f f f f f f f 0 000E+00 0 .340E+00 3 520E+03 0 000E+00 0 000E+00 0 OOOE+00 0 OOOE+00 0 OOOE+00 $ 25 26 27 28 29 30 31 32 •? $ f f f f f f f f 0 000E+00 0 .0OOE+0O 0 000E+00 0 000E+00 0 O0OE+00 0 000E+00 0 OOOE+00 0 OOOE+00 $ 33 34 35 36 37 38 39 40 $ bmod gmod $ f f f f f f f f 0 000E+00 0 .000E+00 4 481E+04 1 833E+04 0 000E+00 0 000E+00 0 000E+00 0 000E+00 $ 41 42 43 44 45 46 47 48 $ damdt $ f f f f f f f f 0 000E+00 0 .000E+00 0 000E+00 0 000E+00 0 O0OE+0O 0 010E+00 0 000E+00 0 OOOE+00 272 Appendix B UBC-C0DAM3Ds Model Reference For older versions of LS-DYNA3D using the fixed format input, the location of the required parameters (e.g. E\ as parameter 1 and G 2 3 as parameter 19) is identical but the format of the input decks is slightly different. For further details, refer to the appropriate LS-DYNA3D User Manual (e.g. Hallquist et al., 1994) under the section titled "Input for User Materials." Refer to Table B . l for a summary of the constants required by the User Defined Material Model definition cards. The values listed are valid for both Structured and Keyword LS-DYNA3D input deck formats. Table B.2 outlines the material constants that are entered in the U M A T cards, as shown above. The U M A T cards are used by the model to obtain the material elastic constants. The other material constants which define the damage growth curves, the residual stiffness functions, and the damage growth potential functions are read from the m a t i n . d a t file described below in Section B.B.2. Although there may seem to be a significant number of vacant positions available in the material input cards, these positions are being reserved for the constants required by the future development of a brick version of the model and the implementation of an erosion failure criterion. There is one other constant which is also read from the input cards. The value of Atdamout (Card 8, Columns 51-60) sets the output interval for the damout. d a t file which contains the models calculations of total matrix and fibre damage (refer to Section B.B.4 below for more details). B.2 UBC-CODAM3Ds Supplementary Data File Format The number of parameters required to define the bilinear functions for damage growth and residual stiffness requires the definition of a supplementary input data file. This data file must be named m a t i n . d a t and either the file or a link to a valid m a t i n . d a t file must be located in 273 Appendix B UBC-C0DAM3Ds Model Reference the directory in which LS-DYNA3D is being executed. The file format is as follows: c P r o p e r t i e s f o r T 8 0 0 / 3 9 0 0 - 2 Q u a s i CFRP c m o d u l u s d e g r a d a t i o n r u l e m o d u l i c r e x w t l r exwt2 rexwt3 r e x w c l rexwc2 rexwc3 1.000E+00 0.930E+00 0.000E+00 1.000E+00 0 .930E+00 0 .000E+00 c r e y w t l r eywt2 reywt3 r e y w c l reywc2 reywc3 1.000E+00 0.930E+00 O.OOOE+00 1.000E+00 0 .930E+00 O.OOOE+00 c r g x y w l rgxyw2 rgxyw3 1.000E+00 0.930E+00 O.OOOE+00 c damage g r o w t h f u n c t i o n c o n s t a n t s c f x w x l fxwx2 fxwx3 f y w y l fywy2 fywy3 0.010E+00 0.016E+00 0.055E+00 0.010E+00 0.016E+00 0 .055E+00 c w x l wx2 wx3 w y l wy2 wy3 0.000E+00 0.400E+00 1.000E+00 O.OOOE+00 0.400E+00 1.000E+00 c w s l ws2 ws3 0.000E+00 0.400E+00 1.000E+00 c damage p o t e n t i a l f u n c t i o n s c a l i n g c o n s t a n t s c k f x t k f x c l f x t l f x c s f x t f x u f x 1.000E+00 1.200E+00 1.333E+00 1.400E+00 4 .000E+00 1.000E+00 1.000E+00 c k f y t k f y c l f y t l f y c s f y t f y u f y 1.333E+00 1.400E+00 1.000E+00 1.200E+00 4 .000E+00 1.000E+00 1.000E+00 Note that the file format is fixed. As a result the comment lines (denoted by a c in the first column) must be included, even if they are left blank. A l l variables are read as single precision real values. Table B.3 contains an outline of the m a t i n . d a t file including the file format, the constants symbolic name as it appears in the theoretical development in Chapter 3, and the variable name used to identify the constant in the sample file above. B.3 History Variable Output In addition to the stresses, strains, energies and other standard output from the material model, L S - D Y N A allows U M A T developers to store other non-standard or material model specific parameters in so-called element history variables. These history variables are located in the D3 PLOTxx binary plot files starting at variable 1003 and are available for generating fringe plots and time history output in the same manner as the element stresses and strains, for example. The parameters available in the UBC-CODAM3Ds model are outlined in Table B.4 and include the 274 Appendix B UBC-C0DAM3Ds Model Reference element damage parameters (ft),), damage growth potentials (Ft), total strains (£ , ) , and normalized residual stiffness constants ( E x ) . Note that the variable identification numbering sequence shown in Table B.4 is not continuous. The gaps were left intentionally to allow the addition of the parameters required by the full 3D brick and 2D axisymmetric formulations. In this way, the variable naming and numbering schemes will be consistent between the models. B.4 Damage Summary File Part of the damage predictions come in the form of fringe and contour plots of the predicted integration point or sublaminate damage (history variables 1003, 1004, and 1005 in the LS-DYNA3D D 3 P L O T x x binary plot files, see Table B.4). The formulation of the model also allows a more direct comparison of predictions of matrix and fibre damage growth with experimental measurements through the integration point average matrix and fibre damages output as history variables 1024 and 1025. The method of calculating these values has been described in Section 3.6. The results of summing the integration point (i.e. sublaminate) fibre and matrix damage over the entire mesh are output in the damout. d a t file at an interval set by the parameter damdt variable in the material model input deck (refer to Table B.2). A sample output is shown below: 275 Appendix B UBC-C0DAM3Ds Model Reference Total damage as a fraction Time of damage at saturation Matrix Fibre Time M a t r i x M a t r i x F i b r e F i b r e % wx % wy % wx % wy 0.00000 0.000 0.000 0.000 0.000 0.10030 14.422 14.403 0.000 0.000 0.20024 14.422 14.403 0.000 0.000 0.30016 132.555 137.600 50.275 52.900 0.40004 132.571 138.681 50.275 53.102 0.50019 347.934 362.527 233.651 244.780 0.60012 385.116 399.025 270.858 281.673 0.70007 385.603 399.607 270.858 281.676 A l l four values (two matrix and two fibre, one for each material direction x and y) are calculated using the methods described in Section 3.6 and represent the total fraction of integration points which have reached matrix and fibre damage saturation. As an example, consider the output shown above. Assume that the model has been set up in such a way that there is one through-thickness integration points per sublaminate. Assume also that at saturation of matrix cracking and delamination, a sublaminate of the material being modelled equal in size to an element has been determined to contain 10 mm 2 of matrix damage. Then, at the last state shown in the output (a time of 0.7) the predicted total matrix damage area associated with cox and coy would be 3856 mm 2 and 3996 mm 2 respectively. For a quasi-isotropic lay-up, an average of these numbers would be a reasonable estimate of the total matrix damage (i.e. a matrix damage area of 3926 mm2). The same simple calculations can be used for total fibre damage area given that the area of fibre breakage at saturation is known. 276 Appendix B UBC-C0DAM3Ds Model Reference Table B.l LS-DYNA3D User Material Model definition constants for the CODAM3Ds composite damage model. Refer to the LS-DYNA3D User Manual (Hallquist et al., 1994) for additional details. Parameter L S - D Y N A 3 D Variable Name Value for U B C - C O D A M 3 s Material Number mt 47 Number of material constants to read lmc 48 Number of history variables nhv 25 Material bulk modulus position bulk 35 Material shear modulus position g 36 Vectorized model ivector 0 Element erosion flag ierosion 0 277 Appendix B U B C - C O D A M 3 D s Model Reference Table B.2 CODAM3Ds User Material Model input deck parameters. Refer to the LS-DYNA3D User Manual (Hallquist et al., 1994) for additional details. Card Colum n Paramete r Description Variable Name Card 3 1-10 Ex Young's modulus (in-plane x-dir.) ex 11-20 Poisson's ratio (in-plane) vxy 21-30 Gxy Shear modulus (in-plane) gxy Card 4 1-10 Ey Young's modulus (in-plane y-dir.) ey 11-20 vyz Poisson's ratio (through-thickness) vyz 21-30 Gyz Shear modulus (through-thickness) gyz Card 5 11-20 Vxz Poisson's ratio (through-thickness) vxz 21-30 Gzx Shear modulus (through-thickness) gzx Card 7 21-30 V ^mat Material bulk modulus N / A 31-40 Gmat Material shear modulus N / A Card 8 51-60 ^damout Output interval for the damout . d a t file damdt 'material parameter is required by the User Material interface but is not part of the constitutive model formulation. 278 Appendix B UBC-C0DAM3Ds Model Reference Table B.3 C O D M 3 D s User Material Model supplementary data file (matin.dat) input format reference. Card Column Parameter/Description Variable Name Equation/Figure Reference Card 1 1-80 Comment N/A N/A Card 2 1-80 Comment N/A N/A Card 3 1-80 Comment N/A N/A Card 4 1-10 Elx (tension) rexwtl 11-20 E^ (tension) rexwt2 Eqn. 3.24 21-30 E™ (tension) rexwt3 and 31-40 E\ (compression) rexwcl Fig. 3.7 41-50 E^ (compression) rexwc2 51-60 E^ (compression) rexwc3 Card 5 1-80 Comment N/A N/A Card 6 1-10 Ey (tension) reywtl 11-20 Ey1 (tension) reywt2 Eqn. 3.24 21-30 Ef (tension) reywt3 and 31-40 Ey (compression) reywcl Fig. 3.7 41-50 Ey (compression) reywc2 51-60 E^ (compression) reywc3 Card 7 1-80 Comment N/A N/A Card 8 1-10 G l y rgxywl Eqn. 3.24 11-20 Gxy rgxyw2 and 21-30 Gxy rgxyw3 Fig. 3.7 Continued on the following page 279 Appendix B UBC-CODAM3Ds Model Reference Table B.3 (Cont.) CODM3Ds User Material Model supplementary data file (matin.dat) input format reference. Card Column Parameter/Description Variable Name Equation/Figure Reference Card 9 1-80 Comment N/A N/A Card 10 1-80 Comment N/A N/A Card 11 1-10 Fl ± X fxwxl 11-20 F? fxwx2 Eqn. 3.14 21-30 Fm fxwx3 and 31-40 F1 y fywyl Fig. 3.5 41-50 Fn y fywy2 51-60 pin Fy fywy3 Card 12 1-80 Comment N/A N/A Card 13 1-10 wxl 11-20 wx2 Eqns. 3.14 & 3.24 21-30 cox wx3 and 31-40 co\ wyl Figs. 3.5 & 3.7 41-50 OJy wy2 51-60 coy wy3 Card 14 1-80 Comment N/A Card 15 31-40 wsl Eqns. 3.14 & 3.24 41-50 ws2 and 51-60 < ws3 Fig. 3.7 Continued on the following page 280 Appendix B UBC-C0DAM3Ds Model Reference Table B.3 (Cont.) C O D M 3 D s User Material Model supplementary data file (matin.dat) input format reference. Card Column Parameter/Description Variable Name Equation/Figure Reference Card 16 1-80 Comment N / A N / A Card 17 1-80 Comment N / A N / A Card 18 1-10 Kxt (tension) kfxt 11-20 Kxc (compression) kfxc 21-30 Lxt (tension) lfxt 31-40 Lxc (compression) lfxc Eqn. 3.18 41-50 sx sfx 51-60 tfx 61-70 ux ufx Card 19 1-80 Comment N / A N / A Card 20 1-10 Kyt (tension) kfyt 11-20 Kyc (compression) kfyc 21-30 Lyt (tension) lfyt 31-40 Lyc (compression) lfyc Eqn. 3.19 41-50 Sy sfy 51-60 Ty tfy 61-70 Uy ufy 281 Appendix B UBC-CODAM3Ds Model Reference Table B.4 C O D A M 3 D s User Material Model element history variable list. Variable ID in D3PLOT Binary Plot File Parameter Description 1003 cox Damage parameter (x- dir.) 1004 COy Damage parameter (y-dir.) 1006 (0s=^C02x+C0y-C0xC0y Damage parameter (off-axis) 1009 . Damage growth potential (x-dir.) 1010 Damage growth potential (y-dir.) 1012 Total in-plane strain (x- dir.) 1013 Sy Total in-plane strain (y- dir.) 1015 Total in-plane shear strain 1018 REX = Ex 1 Ex Residual stiffness (x- dir.) 1019 REy =EylEy Residual stiffness (y- dir.) 1021 RG^ =GxyIGx\ Residual stiffness (shear) 1024 <8=-2 < > v ® m x ®my , Average matrix damage as a fraction of matrix damage at saturation 1025 f 2 (afi , 03fy) Wft <°'jy) Average fibre damage as a fraction of matrix damage at saturation 282 

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