Multiscale Modeling of Complex Materials (eBook)

Preface 6
Contents 8
Atomistic-Continuum Couplings for Dynamic Fracture 9
1 Introduction 9
2 Molecular Dynamics 11
2.1 Governing equations in the atomistic domain 11
2.2 Equilibrium of the atomistic domain 12
3 Continuum Model 13
4 Coupling Scheme 14
4.1 Coupling functions 14
4.2 Discretized problem 16
4.3 Time integration scheme 18
4.4 Energy transfer between the atomistics and continuum domains 20
5 Mechanical quantities in the atomistic domain 25
6 Examples: Crack propagation under velocity loading 27
7 Concluding Remarks 31
Bibliography 33
On the method of virtual power in the mechanics of non-classical continua 37
1 Classical mechanics and classical continua 37
1.1 The traditional approach 38
1.2 Sketch of the proofs 39
1.3 The indifference of power 44
1.4 The method of virtual power 45
1.5 The variational approach 46
1.6 Bounded Cauchy fluxes 47
2 Non-classical continua 50
2.1 Continua with microstructure 50
2.2 Continua with scalar microstructure 53
2.3 Continua with vectorial microstructure 54
2.4 Continua with tensorial microstructure 57
2.5 Second-gradient continua 60
Bibliography 62
Adaptive Concurrent Multi-level Modeling of Multi-scale Ductile Failure in Heterogeneous Metallic Materials 67
1 Introduction 67
2 Levels in the Concurrent Multi-Scale Modeling Framework 71
2.1 Computational Subdomain Level-0 (?l0) 72
2.2 Computational Sub-domain Level-1 ?l1 79
2.3 Computational Sub-domain Level-2 (?l2) 82
2.4 Computational Sub-domain Level-tr (?tr) 82
3 Mesh Adaptivity and Level Change Criteria 82
3.1 Mesh Refinement for level-0 Elements 83
3.2 Criteria for Switching from Level-0 to Level-1 Elements 83
3.3 Criteria for Switching from Level-1 to Level-2 Elements 84
4 Coupling Multiple Levels in the Concurrent Setting 85
4.1 Weak Form for the Concurrent Multi-scale Model 85
4.2 Finite Element Discretization for Multi-scale Analysis 87
4.3 Iterative Solution of the Coupled Multi-Scale System 88
5 Numerical Studies and Validations 90
5.1 Effect of Interface ?int between Macro and Micro Sub-domains 90
5.2 Calibration and Validation of the Level-1 to Level-2 Criteria 93
5.3 Validation of the Multi-level Model Against Micromechanical Analysis 96
6 Ductile Failure of a Cast Aluminum Tension Bar 99
6.1 Parameters in the HCPD Model for Level-0 Elements 102
6.2 Initial Model and Level Changes in a Tension Test 103
6.3 Simulation Through Complete Micro-cracking 104
7 Summary and Conclusions 108
8 Acknowledgments 109
Bibliography 110
Fractals and Randomness in Mechanics of Materials 115
1 Mechanics of Random Media 116
1.1 Basic Concepts 116
The RVE postulate 119
Hill condition - mechanical versus energy definitions 120
Hierarchies of mesoscale bounds 122
Examples of hierarchies of mesoscale bounds 125
Physically nonlinear elastic microstructures 125
Power-law materials 126
Finite elasticity of random composites 127
Elastic-plastic microstructures 131
Viscoelastic microstructures 135
Thermoelastic microstructures 136
Linear case 136
Non-linear case 141
Comparison of scaling trends 143
Scaling and stochastic evolution in damage phenomena 145
1.2 Fractals in Mechanics of Materials 149
Morphogenesis of fractals at elasto-plastic transitions 149
Background 149
Model 3d material 149
Scaling function at the elastic-plastic transition 151
Fractals and avalanches at elastic-plastic-brittle transitions in disordered media 153
Homogenization of fractal media 154
1.3 Closing Comments 154
Acknowledgement 154
References 155
Modelling of damage and fracture processes of ceramic matrix composites under mechanical loading 159
1. Introduction – variety of different composite materials 159
2. Strategy in the Multiscale Modelling of CMC’s 162
3. Experimental Prerequisites for the Multiscale Model 164
4. Multiscale model for the two-phase ceramic composites 167
5. Numerical examples 178
6. Conclusions 181
Acknowledgement 182
7. References 182
Multiscale Modeling of Damage in Composite Materials 187
1. Introduction 187
2. Damage in composite materials 188
3. Damage mechanics 193
4. Synergistic Damage Mechanics (SDM) 212
5. Concluding Remarks 215
Acknowledgement 216
References 216
Molecular Approaches for Multifield Continua: origins and current developments 218
1 Introduction. Multiscale approaches: a short review 219
2 The Nineteenth century molecular models with a glance at modern discrete–continuum theories 226
2.1 Cauchy model for elasticity 227
2.2 Voigt’s model with particle rotations 231
2.3 Poincaré’s refined molecular model for elasticity 235
2.4 Discrete–continuum theories: new perspectives 238
3 Multifield continua, basics synoptic 240
4 A molecular–multifield model for composites 249
4.1 Lattice model of a material with fibres and flaws 250
4.2 Micro–macro transition 253
4.3 Continuum with rigid and affine structure 256
5 Case study: a one–dimensional microcracked bar 262
6 Final remarks 267
Acknowledgements 268
Bibliography 268