Multiscale Modeling of Heterogenous Materials

Oana Cazacu is Professor in the Department of Mechanical and Aerospace Engineering at the University of Florida, REEF, Shalimar, USA.

• Foreword xiiiChapter 1. Accounting for Plastic Strain Heterogenities in Modeling Polycrystalline Plasticity: Microstructure-based Multi-laminate Approaches 1Patrick FRANCIOSI1.1. Introduction 11.2. Polycrystal morphology in terms of grain and sub-grain boundaries 21.2.1. Some evidence of piece-wise regularity for grain boundaries 21.2.2. Characteristics of plastic-strain due to sub-grain boundaries 31.3. Sub-boundaries and multi-laminate structure for heterogenous plasticity 51.3.1. Effective moduli tensor and Green operator of multi-laminate structures 61.3.2. Multi-laminate structures and piece-wise homogenous plasticity 101.4. Application to polycrystal plasticity within the affine approximation 101.4.1. Constitutive relations 101.4.2. Fundamental properties for multi-laminate modeling of plasticity 141.5. Conclusion 151.6. Bibliography 15Chapter 2. Discrete Dislocation Dynamics: Principles and Recent Applications 17Marc FIVEL2.1. Discrete Dislocation Dynamics as a link in multiscale modeling 172.2. Principle of Discrete Dislocation Dynamics 192.3. Example of scale transition: from DD to Continuum Mechanics 212.3.1. Introduction to a dislocation density model 212.3.1.1. Constitutive equations of a dislocation based model of crystal plasticity 222.3.1.2. Parameter identification 242.3.1.3. Application to copper simulations 252.3.1.4. Taking into account kinematic hardening 262.4. Example of DD analysis: simulations of crack initiation in fatigue 292.4.1. Case of single phase AISI 31GL 292.5. Conclusions 322.6. Bibliography 33 Chapter 3. Multiscale Modeling of Large Strain Phenomenain Polycrystalline Metals 37Kaan INAL and Raj. K. MISHRA3.1. Implementation of polycrystal plasticity in finite element analysis 393.2. Kinematics and constitutive framework 413.3. Forward Euler algorithm 443.4. Validation of the forward Euler algorithm 463.5. Time step issues in the forward Euler scheme 493.6. Comparisons of CPU times: the rate tangent versus the forward Euler methods 513.7. Conclusions 523.8. Acknowledgements 523.9. Bibliography 52Chapter 4. Earth Mantle Rheology Inferred from Homogenization Theories 55Olivier CASTELNAU, Ricardo LEBENSOHN, Pedro Ponte CASTAÑEDA and Donna BLACKMAN4.1. Introduction 554.2. Grain local behavior 574.3. Full-field reference solutions 594.4. Mean-field estimates 624.4.1. Basic features of mean-field theories 624.4.2. Results 644.5. Concluding observations 664.6. Bibliography 68Chapter 5. Modeling Plastic Anistropy and Strength Differential Effects in Metallic Materials 71Oana CAZACU and Frédéric BARLAT5.1. Introduction 715.2. Isotropic yield criteria 725.2.1. Pressure insensitive materials deforming by slip 725.2.2. Pressure insensitive materials deforming by twinning 735.2.3. Pressure insensitive materials with non-Schmid effects 765.2.4. Pressure sensitive materials 785.2.5. SD effect and plastic flow 805.3. Anisotropic yield criteria with SD effects 805.3.1. Cazacu and Barlat [CAZ 04] orthotropic yield criterion 805.3.2. Cazacu Plunkett Barlat yield criterion [CAZ 06] 825.4. Modeling anisotropic hardening due to texture evolution 835.5. Conclusions and future perspectives 865.6. Bibliography 87Chapter 6. Shear Bands in Steel Plates under Impact Loading 91George Z. VOYIADJIS and Amin H. ALMASRI6.1. Introduction 916.2. Viscoplasticity and constitutive modeling 926.3. Higher order gradient theory 976.4. Two-dimensional plate subjected to velocity boundary conditions 1026.5. Shear band in steel plate punch 1056.6. Conclusions 1086.7. Bibliography 109Chapter 7. Viscoplastic Modeling of Anisotropic Textured Metals 111Brian PLUNKETT and Oana CAZACU7.1. Introduction 1117.2. Anisotropic elastoviscoplastic model 1137.3. Application to zirconium. 1157.3.1. Quasi-static deformation of zirconium 1157.3.2. High strain-rate deformation of zirconium 1207.4. High strain-rate deformation of tantalum 1247.5. Conclusions1257.6. Bibliography 126Chapter 8. Non-linear Elastic Inhomogenous Materials: Uniform Strain Fields and Exact Relations 129Qi-Chang HE, B. BARY and Hung LE QUANG8.1. Introduction 1298.2. Locally uniform strain fields 1308.3. Exact relations for the effective elastic tangent moduli 1368.4. Cubic polycrystals 1398.5. Power-law fibrous composites 1448.6. Conclusion 1498.7. Bibliography 149Chapter 9. 3D Continuous and Discrete Modeling of Bifurcations in Geomaterials 153Florent PRUNIER, Félix DARVE, Luc SIBILLE and François NICOT9.1. Introduction 1539.2. 3D bifurcations exhibited by an incrementally non-linear constitutive relation 1559.2.1. Incrementally non-linear and piecewise linear relations 1559.2.2. 3D analysis of the second-order work with phenomenological constitutive models 1579.3. Discrete modeling of the failure mode related to second-order work criterion 1659.4. Conclusions 1739.5. Acknowledgements 1749.6. Bibliography 174Chapter 10. Non-linear Micro-cracked Geomaterials: Anisotropic Damage and Coupling with Plasticity 177Djimédo KONDO, Qizhi ZHU, Vincent MONCHIET and Jian-Fu SHAO10.1. Introduction 17710.2. Anisotropic elastic damage model with unilateral effects 17910.2.1. Homogenization of elastic micro-cracked media 17910.2.1.1. Micromechanics of media with random microstructure 17910.2.1.2. Application to micro-cracked media 18010.2.2. Micro-crack closure condition and damage evolution 18110.2.2.1. Micro-crack closure effects and unilateral damage 18110.2.2.2. Damage criterion and evolution law 18210.2.3. Non-local micromechanics-based damage model 18310.2.4. Application of the model 18410.2.4.1. Uniaxial tensile tests 18410.2.4.2. Predictions of the anisotropic damage model for William’s test 18510.2.4.3. Numerical analysis of Hassanzadeh’s direct tension test 18810.3. A new model for ductile micro-cracked materials 18810.3.1. Introductory observations 18810.3.2. Basic concepts and methodology of the limit analysis approach 19010.3.2.1. Representative volume element with oblate voids 19010.3.2.2. The Eshelby-like velocity field 19110.3.3. Determination of the macroscopic yield surface 19210.3.3.1. The question of the boundary conditions 19210.3.3.2. Principle of the determination of the yield function 19310.3.3.3. Closed form expression of the macroscopic yield function 19310.3.4. The particular case of penny-shaped cracks 19510.4. Conclusions 19710.5. Acknowledgement 19810.6. Appendix 19810.7. Bibliography 198Chapter 11. Bifurcation in Granular Materials: A Multiscale Approach 203François NICOT, Luc SIBILLE and Félix DARVE11.1. Introduction 20311.2. Microstructural origin of the vanishing of the second-order work 20511.2.1. The micro-directional model 20511.2.2. Microstructural expression of the macroscopic second-order work 20611.2.3. From micro to macro second-order work 20811.2.4. Micromechanical analysis of the vanishing of the second-order work 21011.3. Some remarks on the basic micro-macro relation for the second-order work 21211.4. Conclusion 21311.5. Bibliography 214Chapter 12. Direct Scale Transition Approach for Highly-filled Viscohyperelastic Particulate Composites: Computational Study 215Carole NADOT-MARTIN, Marion TOUBOUL, André DRAGON and Alain FANGET12.1. Morphological approach in the finite strain framework 21612.1.1. Geometric schematization 21612.1.2. Localization-homogenization problem 21712.1.2.1. Principal tools and stages 21712.1.2.2. Solving procedure 21912.2. Evaluation involving FEM/MA confrontations 22112.2.1. Material geometry, relative representations 22112.2.2. Loading paths, methodology of analysis 22312.2.3. MA estimates compared to FEM results for hyperelastic constituents 22512.2.4. Evaluation involving viscohyperelastic behavior of the matrix 22912.3. Conclusions and prospects 23212.4. Bibliography 234Chapter 13. A Modified Incremental Homogenization Approach for Non-linear Behaviors of Heterogenous Cohesive Geomaterials 237Ariane ABOU-CHAKRA GUÉRY, Fabrice CORMERY, Jian-Fu SHAO and Djimédo KONDO13.1. Introduction 23713.2. Experimental observations on the Callovo-Oxfordian argillite behavior 23813.2.1. Microstructure and mineralogical composition of the material 23813.2.2. Brief summary of the macroscopic behavior of the material 23913.3. Incremental formulation of the homogenized constitutive relation 24013.3.1. Introduction 24013.3.2. Limitations of Hill’s incremental method 24213.3.3. Modified Hill’s incremental method 24313.4. Modifying of the local constituents’ behaviors 24413.4.1. Elastoplastic behavior of the clay phase 24413.4.2. Elastic unilateral damage behavior of the calcite phase 24513.5. Implementation and numerical validation of the model 24713.5.1. Local integration of the micromechanical model 24713.5.2. Comparison with unit cell (finite element) calculation 24813.6. Calibration and experimental validations of the modified incremental micromechanical model 24813.7. Conclusions 24913.8. Acknowledgement 25113.9. Bibliography 251Chapter 14. Meso- to Macro-scale Probability Aspects for Size Effects and Heterogenous Materials Failure 253Jean-Baptiste COLLIAT, Martin HAUTEFEUILLE and Adnan IBRAHIMBEGOVIC14.1. Introduction 25314.2. Meso-scale deterministic model 25414.2.1. Structured meshes and kinematic enhancements 25514.2.2. Operator split solution for interface failure 25714.2.3. Comparison between structured and unstructured mesh approach 25814.3. Probability aspects of inelastic localized failure for heterogenous materials 25914.3.1. Meso-scale geometry description 26014.3.2. Stochastic integration 26114.4. Results of the probabilistic characterization of the two phase material 26314.4.1. Determination of SRVE size 26314.4.2. Numerical results and discussion 26414.5. Size effect modeling 26614.5.1. Random fields and the Karhunen-Loeve expansion 26714.5.2. Size effect and correlation length 26914.6. Conclusion 27114.7. Acknowledgments 27214.8. Bibliography 272Chapter 15. Damage and Permeability in Quasi-brittle Materials: from Diffuse to Localized Properties 277Gilles PIJAUDIER-CABOT, Frédéric DUFOUR and Marta CHOINSKA15.1. Introduction 27715.2. Mechanical problem – continuum damage modeling 27915.3. Permeability matching law 28115.3.1. Diffuse damage 28115.3.2. Localized damage – crack opening versus permeability 28115.3.3. Matching law 28315.4. Calculation of a crack opening in continuum damage calculations 28315.5. Structural simulations 28615.5.1. Mechanical problem – Brazilian splitting test 28715.5.2. Evolution of apparent permeability 28915.6. Conclusions 29115.7. Acknowledgement 29115.8. Bibliography 291Chapter 16. A Multiscale Modeling of Granular Materials with Surface Energy Forces 293Pierre-Yves HICHER and Ching S. CHANG16.1. Introduction 29316.2. Stress-strain model 29416.2.1. Inter-particle behavior 29616.2.1.1. Elastic part 29616.2.1.2. Plastic part 29616.2.1.3. Interlocking influence 29716.2.1.4. Elastoplastic force-displacement relationship 29816.2.2. Stress-strain relationship 29816.2.2.1. Micro-macro relationship 29816.2.2.2. Calculation scheme 30016.2.3. Summary of parameters 30116.3. Results of numerical simulation without surface energy forces consideration 30216.4. Granular material with surface energy forces: the example of lunar soil 30616.4.1. Van der Waals forces 30816.4.2. Triaxial tests with consideration of surface energy forces 31116.5. Summary and conclusion 31416.6. Bibliography 315Chapter 17. Length Scales in Mechanics of Granular Solids 319Farhang RADJAI17.1. Introduction 31917.2. Model description 32017.3. Force chains 32117.3.1. Probability density functions 32117.3.2. Bimodal character of stress transmission 32217.3.3. Spatial correlations 32417.4. Fluctuating particle displacements 32517.4.1. Uniform strain and fluctuations 32517.4.2. Scale-dependent pdfs 32617.4.3. Spatial correlations 32817.4.4. Granulence 32917.5. Friction mobilization 33017.5.1. Critical contacts 33017.5.2. Evolution of critical contacts 33017.5.3. Spatial correlations 33117.6. Conclusion 33217.7. Acknowledgements 33317.8. Bibliography 333List of Authors 337Index 341