Computational Mesomechanics of

Composites Numerical analysis of the effect of microstructures of composites on their strength and damage resistance

LEON MISHNAEVSKY JR

Ris0 National Laboratory, Technical University of Denmark, Roskilde, Denmark

B I C E N T E N N I A L

3 I C E N T E N N I A L

John Wiley & Sons ; Ltd

Contents

About the Author xi Preface xiii Acknowledgements xvii

1 Composites 1 1.1 Classification and types of composites 1 1.2 Deformation, damage and fracture of composites: micromechanisms

and roles of phases 5 1.2.1 Particle and short fiber reinforced composites 5 1.2.2 Long fiber reinforced composites 8 1.2.3 Laminates 10

References 11

2 Mesoscale level in the mechanics of materials 13 2.1 On the definitions of scale levels: micro-and mesomechanics 13 2.2 Size effects 14

2.2.1 Brittle and quasi-brittle materials 14 2.2.2 Metals 15 2.2.3 Thinfilms 16

2.3 Biocomposites 17 2.3.1 Nacre 17 2.3.2 Sponge spicules 18 2.3.3 Bamboo 19 2.3.4 Teeth 20 2.3.5 Bones 21

2.4 On some concepts of the improvement of material properties 22 2.4.1 Gradient composite materials 23 2.4.2 The application of coatings 23 2.4.3 Layered metal matrix composites 23 2.4.4 Surface composites 24 2.4.5 Agglomerates of small scale inclusions and the 'double

dispersion' microstructures of steels 24 2.4.6 Inclusion networks 25

vi Contents

2.4.7 Interpenetrating phase composites (IPCs) 25 2.4.8 Hyperorganized structure control 26 2.4.9 Summary 27

2.5 Physical mesomechanics of materials 27 2.6 Topological and Statistical description of microstructures of composites 29 References 32

3 Damage and failure of materials: concepts and methods of modeling 37 3.1 Fracture mechanics: basic concepts 37

3.1.1 Griffim theory of brittle fracture 38 3.1.2 Stress field in the vicinity of a crack 39 3.1.3 Stress intensity factor and energy release rate 39 3.1.4 /-integral and other modeis of plastic effects 40

3.2 Statistical theories of strength 42 3.2.1 Worst flaw and weakest link theories 42 3.2.2 Random processes and stochastic equations 44 3.2.3 Fiber bündle modeis and chains of fiber bundles 46

3.3 Damage mechanics 48 3.3.1 Models of elastic solids with many cracks 48 3.3.2 Phenomenological analysis of damage evolution (continuum

damage mechanics) 49 3.3.3 Micromechanical modeis of void growth in ductile materials 50 3.3.4 Thermodynamic damage modeis 51 3.3.5 Nonlocal and gradient enhanced damage modeis 53

3.4 Numerical modeling of damage and fracture 55 References 60

4 Microstructure-strength relationships of composites: concepts and methods of analysis 65 4.1 Interaction between elements of microstructures: physical and

mechanical modeis 65 4.1.1 Theories of constrained plastic flow of ductile materials

reinforced by hard inclusions 66 4.1.2 Shear lag model and its applications 69

4.2 Multiscale modeling of materials and homogenization 71 4.2.1 Multiscale modeling 72 4.2.2 Homogenization 74

4.3 Analytical estimations and bounds of overall elastic properties of composites 75 4.3.1 Rule-of-mixture and classical Voigt and Reuss approximations 76 4.3.2 Hashin-Shtrikman bounds 78 4.3.3 Dilute distribution model 80 4.3.4 Effective field method and Mori-Tanaka model 80 4.3.5 Composite sphere and composite cylinder assemblage 81 4.3.6 Self-consistent modeis and other effective medium methods 82 4.3.7 Method of cells and transformation field analysis 84 4.3.8 Incorporation of detailed microstructural information:

three-point approximation 86

Contents v/7

4.3.9 Generalized continua: nonlocal and gradient-enhanced modeis 87 4.3.10 Nonlinear material behavior 88

4.4 Computational modeis of microstructures and strength of composites 89 4.4.1 Unit cell modeis of composites 90 4.4.2 How to incorporate real microstructures of materials into

numerical modeis 98 References 105

Computational experiments in the mechanics of materials: concepts and tools 115 5.1 Concept of computational experiments in the mechanics of materials 115 5.2 Input data for the simulations: determination of material properties 116

5.2.1 Nanoindentation 117 5.2.2 In-situ experiments using a scanning electron microscope 117 5.2.3 Inverse analysis 118

5.3 Program codes for the automatic generation of 3D microstructural modeis of materials 120 5.3.1 Program Meso3D for the automatic geometry-based

generation of 3D microstructural FE modeis of materials 120 5.3.2 Program Voxel2FEM for the automatic voxel-based

generation of 3D microstructural FE modeis of materials 123 References 127

Numerical mesomechanical experiments: analysis of the effect of microstructures of materials on the deformation and damage resistance 129 6.1 Finite dement modeis of composite microstructures 130 6.2 Material properties used in the simulations 130 6.3 Damage modeling in composites with the User Defined Fields 131

6.3.1 Damage mechanisms and failure conditions 131 6.3.2 Subroutine User Defined Field 132 6.3.3 Interface debonding 133

6.4 Stability and reproducibility of the simulations 134 6.5 Effect of the amount and volume content of particles on the

deformation and damage in the composite 137 6.6 Effect of particle clustering and the gradient distribution of particles 141 6.7 Effect of the variations of particle sizes on the damage evolution 145 6.8 Ranking of microstructures and the effect of gradient orientation 147 References 149

Graded particle reinforced composites: effect of the parameters of graded microstructures on the deformation and damage 151 7.1 Damage evolution in graded composites and the effect of the degree

of gradient 153 7.2 'Bilayer' model of a graded composite 159 7.3 Effect of the shape and orientation of elongated particles on the

strength and damage evolution: nongraded composites 161 7.4 Effect of the shape and orientation of elongated particles on the

strength and damage evolution: graded composites 165

Contents

7.5 Effect of Statistical variations of local strengths of reinforcing particles and the distribution of the particle sizes 168

7.6 Combined Reuss-Voigt model and its application to the estimation of stiffness of graded materials 172 7.6.1 Estimation of the stiffness of materials with arbitrarily

complex microstructures: combined Reuss-Voigt model 172 7.6.2 Effect of the degree of gradient on the elastic properties of

the composite 172 7.6.3 Inclined interface: effect of the orientation of interfaces on

the elastic properties 176 References 177

Particle clustering in composites: effect of clustering on the mechanical behavior and damage evolution 179 8.1 Finite element modeling of the effect of clustering of particles on the

damage evolution 179 8.1.1 Numerical modeis of clustered microstructures and

Statistical characterization of the microstructures 181 8.1.2 Simulation of damage and particle failure in clustered

microstructures 183 8.2 Analytical modeling of the effect of particle clustering on the

damage resistance 187 8.2.1 Cell array model of a composite 187 8.2.2 Probabilistic analysis of damage accumulation in a cell 191 8.2.3 Effect of particle clustering on the fracture toughness 193

References 195

Interpenetrating phase composites: numerical simulations of deformation and damage 197

9.1 Geometry-based and voxel array based 3D FE model generation: comparison 200

9.2 Gradient interpenetrating phase composites 202 9.3 Isotropie interpenetrating phase composites 207

9.3.1 Effect of the contiguity of interpenetrating phases on the strength of composites 207

9.3.2 Porous plasticity: open form porosity 210 References 212

Fiber reinforced composites: numerical analysis of damage initiation and growth 215 10.1 Modeling of strength and damage of fiber reinforced composites: a

brief overview 215 10.1.1 Shear lag based modeis and load redistribution Schemas 215 10.1.2 Fiber bündle model and its versions 219 10.1.3 Fracture mechanics based modeis and crack

bridging 220 10.1.4 Micromechanical modeis of damage and fracture 223

Contents ix

10.2 Mesomechanical simulations of damage initiation and evolution in fiber reinforced composites 225 10.2.1 Unit cell model and damage analysis 226 10.2.2 Numerical simulations: effect of matrix cracks on the fiber

fracture 228 10.2.3 Numerical simulations: interface damage initiation and its

interaction with matrix cracks and fiber fractures 229 References 235

11 Contact damage and wear of composite tool materials: micro-macro relationships 239 11.1 Micromechanical modeling of the contact wear of composites: a brief

overview 240 11.2 Mesomechanical simulations of wear of grinding wheels 243 11.3 Micro-macro dynamical transitions for the contact wear of

composites: 'black box modeling' approach 245 11.3.1 Model of the tool wear based on the 'black box modeling'

approach 245 11.3.2 Effect of the loading conditions on the tool wear 250 11.3.3 Experimental verification: approach and steady State

cutting regimes 252 11.4 Microscale scattering of the tool material properties and the

macroscopic efficiency of the tool 253 11.4.1 Statistical description of tool shapes 254 11.4.2 Effect of the scattering of the tool material properties on

the efficiency of the tool 257 11.4.3 Principle of the optimal tool design 259

References 261

12 Future fields: computational mesomechanics and nanomaterials 265 References 267

13 Conclusions 269

Index 271