Statistical correlation functions are a well-known class of statistical descriptors that can be used to describe the morphology and the microstructure-properties relationship. A comprehensive study has been performed for the use of these correlation functions for the reconstruction and homogenization in nano-composite materials. Correlation functions are measured from different techniques such as microscopy (SEM or TEM), small angle X-ray scattering (SAXS) and can be generated through Monte Carlo simulations.  In this book, different experimental techniques such as SAXS and image processing are presented, which are used to measure two-point correlation function correlation for multi-phase polymer composites.

Higher order correlation functions must be calculated or measured to increase the precision of the statistical continuum approach. To achieve this aim, a new approximation methodology is utilized to obtain N-point correlation functions for multiphase heterogeneous materials. The two-point functions measured by different techniques have been exploited to reconstruct the microstructure of heterogeneous media.

Statistical continuum theory is used to predict the effective thermal conductivity and elastic modulus of polymer composites. N-point probability functions as statistical descriptors of inclusions have been exploited to solve strong contrast homogenization for effective thermal conductivity and elastic modulus properties of heterogeneous materials.  Finally, reconstructed microstructure is used to calculate effective properties and damage modeling of heterogeneous materials.

Yves Rémond is Distinguished Professor (Exceptional Class) at the University of Strasbourg in France.

Saïd Ahzi is a Research Director of the Materials Science and Engineering group at Qatar Environment and Energy Research Institute (QEERI) and Professor at the College of Science & Engineering, Hamad Bin Khalifa University, Qatar Foundation, Qatar.

Majid Baniassadi is Assistant Professor at the School of Mechanical Engineering, University of Tehran, Iran.

Hamid Garmestani is Professor of Materials Science and Engineering at Georgia Institute of Technology, USA and a Fellow of the American Society of Materials (ASM International).

Preface ix

Introduction xiii

Chapter 1. Literature Survey 1

1.1. Random heterogeneous material 1

1.2. Two-point probability functions 2

1.3. Two-point cluster functions 4

1.4. Lineal-path function 4

1.5. Reconstruction 4

1.5.1. X-ray computed tomography (experimental) 4

1.5.2. X-ray computed tomography (applications to nanocomposites) 6

1.5.3. FIB/SEM (experimental) 6

1.5.4. Reconstruction using statistical descriptor (numerical) 10

1.6. Homogenization methods for effective properties 11

1.7. Assumption of statistical continuum mechanics 12

1.8. Representative volume element 13

Chapter 2. Calculation of Two-Point Correlation Functions 15

2.1. Introduction 15

2.2. Monte Carlo calculation of TPCF 17

2.3. Two-point correlation functions of eigen microstructure 19

2.4. Calculation of two-point correlation functions using SAXS or SANS data 21

2.4.1. Case study for structural characterization using SAXS data 24

2.5. Necessary conditions for two-point correlation functions 28

2.6. Approximation of two-point correlation functions 30

2.6.1. Examination of the necessary conditions for the proposed estimation 34

2.6.2. Case study for the approximation of a TPCF 39

2.7. Conclusion 42

Chapter 3. Approximate Solution for N-Point Correlation Functions for Heterogeneous Materials 43

3.1. Introduction 43

3.2. Approximation of three-point correlation functions 45

3.2.1. Decomposition of higher order statistics 45

3.2.2. Decomposition of two-point correlation functions 46

3.2.3. Decomposition of three-point correlation functions 47

3.3. Approximation of four-point correlation functions 51

3.4. Approximation of N-point correlation functions 56

3.5. Results 60

3.5.1. Computational verification 60

3.5.2. Experimental validation 62

3.6. Conclusions 66

Chapter 4. Reconstruction of Heterogeneous Materials Using Two-Point Correlation Functions 67

4.1. Introduction 67

4.2. Monte Carlo reconstruction methodology 69

4.2.1. 3D cell generation 72

4.2.2. Cell distribution 75

4.2.3. Cell growth 77

4.2.4. Optimization of the statistical correlation functions 79

4.2.5. Percolation 79

4.2.6. Three-phase solid oxide fuel cell anode microstructure 81

4.2.7. Reconstruction of multiphase heterogeneous materials 82

4.3. Reconstruction procedure using the simulated annealing (SA) algorithm 86

4.4. Phase recovery algorithm 91

4.5. 3D reconstruction of non-eigen microstructure using correlation functions 96

4.5.1. Microstructure reconstruction using Monte Carlo methodology 96

4.5.2. Sample production 97

4.5.3. Monte Carlo calculation of a two-point correlation function 98

4.5.4. Microstructure optimization 99

4.5.5. Results and discussion 99

4.6. Conclusion 101

Chapter 5. Homogenization of Mechanical and Thermal Behavior of Nanocomposites Using Statistical Correlation Functions: Application to Nanoclay-based Polymer Nanocomposites 103

5.1. Introduction 103

5.2. Modified strong-contrast approach for anisotropic stiffness tensor of multiphase heterogeneous materials 104

5.3. Strong-contrast approach to effective thermal conductivity of multiphase heterogeneous materials 112

5.4. Simulation and experimental verification 117

5.4.1. Computer-generated model 118

5.4.2. Thermal conductivity 120

5.4.3. Mechanical model 122

5.4.4. Experimental part 125

5.5. Results and discussion 127

5.5.1. Thermal conductivity 127

5.5.2. Thermo-mechanical properties 128

5.6. Conclusion 130

Chapter 6. Homogenization of Reconstructed RVE 133

6.1. Introduction 133

6.2. Finite element homogenization of the reconstructed RVEs 134

6.2.1. Reconstruction of FIB-SEM RVEs 134

6.2.2. Finite element analysis of RVEs 138

6.3. Finite element homogenization of the statistical reconstructed RVEs 141

6.3.1. FEM analysis of reconstruction RVE using statistical correlation functions 141

6.3.2. Finite element analysis of RVEs 143

6.4. FEM analysis of debonding-induced damage model for polymer composites 149

6.4.1. Representative volume element (RVE) 150

6.4.2. Cohesive zone model 152

6.4.3. Material behavior and FE simulation 157

6.4.4. The effect of the GNP’s volume fraction and aspect ratio in perfectly bonded nanocomposite  158

6.4.5. Comparing the effect of the GNP’s volume fraction and aspect ratio in perfectly bonded and cohesively bonded nanocomposites 160

6.4.6. The effect of the GNP’s aspect ratio and volume fraction in weakly bonded nanocomposite 163

6.5. Conclusion and future work 166

Appendices 169

Appendix A 171

Appendix B 175

Bibliography 179

Index 185