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 ixIntroduction xiiiChapter 1 Literature Survey 11.1 Random heterogeneous material 11.2 Two-point probability functions 21.3 Two-point cluster functions 41.4 Lineal-path function 41.5 Reconstruction 41.5.1 X-ray computed tomography (experimental) 41.5.2 X-ray computed tomography (applications to nanocomposites) 61.5.3 FIB/SEM (experimental) 61.5.4 Reconstruction using statistical descriptor (numerical) 101.6 Homogenization methods for effective properties 111.7 Assumption of statistical continuum mechanics 121.8 Representative volume element 13Chapter 2 Calculation of Two-Point Correlation Functions 152.1 Introduction 152.2 Monte Carlo calculation of TPCF 172.3 Two-point correlation functions of eigen microstructure 192.4 Calculation of two-point correlation functions using SAXS or SANS data 212.4.1 Case study for structural characterization using SAXS data 242.5 Necessary conditions for two-point correlation functions 282.6 Approximation of two-point correlation functions 302.6.1 Examination of the necessary conditions for the proposed estimation 342.6.2 Case study for the approximation of a TPCF 392.7 Conclusion 42Chapter 3 Approximate Solution for N-Point Correlation Functions for Heterogeneous Materials 433.1 Introduction 433.2 Approximation of three-point correlation functions 453.2.1 Decomposition of higher order statistics 453.2.2 Decomposition of two-point correlation functions 463.2.3 Decomposition of three-point correlation functions 473.3 Approximation of four-point correlation functions 513.4 Approximation of N-point correlation functions 563.5 Results 603.5.1 Computational verification 603.5.2 Experimental validation 623.6 Conclusions 66Chapter 4 Reconstruction of Heterogeneous Materials Using Two-Point Correlation Functions 674.1 Introduction 674.2 Monte Carlo reconstruction methodology 694.2.1 3D cell generation 724.2.2 Cell distribution 754.2.3 Cell growth 774.2.4 Optimization of the statistical correlation functions 794.2.5 Percolation 794.2.6 Three-phase solid oxide fuel cell anode microstructure 814.2.7 Reconstruction of multiphase heterogeneous materials 824.3 Reconstruction procedure using the simulated annealing (SA) algorithm 864.4 Phase recovery algorithm 914.5 3D reconstruction of non-eigen microstructure using correlation functions 964.5.1 Microstructure reconstruction using Monte Carlo methodology 964.5.2 Sample production 974.5.3 Monte Carlo calculation of a two-point correlation function 984.5.4 Microstructure optimization 994.5.5 Results and discussion 994.6 Conclusion 101Chapter 5 Homogenization of Mechanical and Thermal Behavior of Nanocomposites Using Statistical Correlation Functions: Application to Nanoclay-based Polymer Nanocomposites 1035.1 Introduction 1035.2 Modified strong-contrast approach for anisotropic stiffness tensor of multiphase heterogeneous materials 1045.3 Strong-contrast approach to effective thermal conductivity of multiphase heterogeneous materials 1125.4 Simulation and experimental verification 1175.4.1 Computer-generated model 1185.4.2 Thermal conductivity 1205.4.3 Mechanical model 1225.4.4 Experimental part 1255.5 Results and discussion 1275.5.1 Thermal conductivity 1275.5.2 Thermo-mechanical properties 1285.6 Conclusion 130Chapter 6 Homogenization of Reconstructed RVE 1336.1 Introduction 1336.2 Finite element homogenization of the reconstructed RVEs 1346.2.1 Reconstruction of FIB-SEM RVEs 1346.2.2 Finite element analysis of RVEs 1386.3 Finite element homogenization of the statistical reconstructed RVEs 1416.3.1 FEM analysis of reconstruction RVE using statistical correlation functions 1416.3.2 Finite element analysis of RVEs 1436.4 FEM analysis of debonding-induced damage model for polymer composites 1496.4.1 Representative volume element (RVE) 1506.4.2 Cohesive zone model 1526.4.3 Material behavior and FE simulation 1576.4.4 The effect of the GNP’s volume fraction and aspect ratio in perfectly bonded nanocomposite 1586.4.5 Comparing the effect of the GNP’s volume fraction and aspect ratio in perfectly bonded and cohesively bonded nanocomposites 1606.4.6 The effect of the GNP’s aspect ratio and volume fraction in weakly bonded nanocomposite 1636.5 Conclusion and future work 166Appendices 169Appendix A 171Appendix B 175Bibliography 179Index 185

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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 ixIntroduction xiiiChapter 1 Literature Survey 11.1 Random heterogeneous material 11.2 Two-point probability functions 21.3 Two-point cluster functions 41.4 Lineal-path function 41.5 Reconstruction 41.5.1 X-ray computed tomography (experimental) 41.5.2 X-ray computed tomography (applications to nanocomposites) 61.5.3 FIB/SEM (experimental) 61.5.4 Reconstruction using statistical descriptor (numerical) 101.6 Homogenization methods for effective properties 111.7 Assumption of statistical continuum mechanics 121.8 Representative volume element 13Chapter 2 Calculation of Two-Point Correlation Functions 152.1 Introduction 152.2 Monte Carlo calculation of TPCF 172.3 Two-point correlation functions of eigen microstructure 192.4 Calculation of two-point correlation functions using SAXS or SANS data 212.4.1 Case study for structural characterization using SAXS data 242.5 Necessary conditions for two-point correlation functions 282.6 Approximation of two-point correlation functions 302.6.1 Examination of the necessary conditions for the proposed estimation 342.6.2 Case study for the approximation of a TPCF 392.7 Conclusion 42Chapter 3 Approximate Solution for N-Point Correlation Functions for Heterogeneous Materials 433.1 Introduction 433.2 Approximation of three-point correlation functions 453.2.1 Decomposition of higher order statistics 453.2.2 Decomposition of two-point correlation functions 463.2.3 Decomposition of three-point correlation functions 473.3 Approximation of four-point correlation functions 513.4 Approximation of N-point correlation functions 563.5 Results 603.5.1 Computational verification 603.5.2 Experimental validation 623.6 Conclusions 66Chapter 4 Reconstruction of Heterogeneous Materials Using Two-Point Correlation Functions 674.1 Introduction 674.2 Monte Carlo reconstruction methodology 694.2.1 3D cell generation 724.2.2 Cell distribution 754.2.3 Cell growth 774.2.4 Optimization of the statistical correlation functions 794.2.5 Percolation 794.2.6 Three-phase solid oxide fuel cell anode microstructure 814.2.7 Reconstruction of multiphase heterogeneous materials 824.3 Reconstruction procedure using the simulated annealing (SA) algorithm 864.4 Phase recovery algorithm 914.5 3D reconstruction of non-eigen microstructure using correlation functions 964.5.1 Microstructure reconstruction using Monte Carlo methodology 964.5.2 Sample production 974.5.3 Monte Carlo calculation of a two-point correlation function 984.5.4 Microstructure optimization 994.5.5 Results and discussion 994.6 Conclusion 101Chapter 5 Homogenization of Mechanical and Thermal Behavior of Nanocomposites Using Statistical Correlation Functions: Application to Nanoclay-based Polymer Nanocomposites 1035.1 Introduction 1035.2 Modified strong-contrast approach for anisotropic stiffness tensor of multiphase heterogeneous materials 1045.3 Strong-contrast approach to effective thermal conductivity of multiphase heterogeneous materials 1125.4 Simulation and experimental verification 1175.4.1 Computer-generated model 1185.4.2 Thermal conductivity 1205.4.3 Mechanical model 1225.4.4 Experimental part 1255.5 Results and discussion 1275.5.1 Thermal conductivity 1275.5.2 Thermo-mechanical properties 1285.6 Conclusion 130Chapter 6 Homogenization of Reconstructed RVE 1336.1 Introduction 1336.2 Finite element homogenization of the reconstructed RVEs 1346.2.1 Reconstruction of FIB-SEM RVEs 1346.2.2 Finite element analysis of RVEs 1386.3 Finite element homogenization of the statistical reconstructed RVEs 1416.3.1 FEM analysis of reconstruction RVE using statistical correlation functions 1416.3.2 Finite element analysis of RVEs 1436.4 FEM analysis of debonding-induced damage model for polymer composites 1496.4.1 Representative volume element (RVE) 1506.4.2 Cohesive zone model 1526.4.3 Material behavior and FE simulation 1576.4.4 The effect of the GNP’s volume fraction and aspect ratio in perfectly bonded nanocomposite 1586.4.5 Comparing the effect of the GNP’s volume fraction and aspect ratio in perfectly bonded and cohesively bonded nanocomposites 1606.4.6 The effect of the GNP’s aspect ratio and volume fraction in weakly bonded nanocomposite 1636.5 Conclusion and future work 166Appendices 169Appendix A 171Appendix B 175Bibliography 179Index 185

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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 ixIntroduction xiiiChapter 1 Literature Survey 11.1 Random heterogeneous material 11.2 Two-point probability functions 21.3 Two-point cluster functions 41.4 Lineal-path function 41.5 Reconstruction 41.5.1 X-ray computed tomography (experimental) 41.5.2 X-ray computed tomography (applications to nanocomposites) 61.5.3 FIB/SEM (experimental) 61.5.4 Reconstruction using statistical descriptor (numerical) 101.6 Homogenization methods for effective properties 111.7 Assumption of statistical continuum mechanics 121.8 Representative volume element 13Chapter 2 Calculation of Two-Point Correlation Functions 152.1 Introduction 152.2 Monte Carlo calculation of TPCF 172.3 Two-point correlation functions of eigen microstructure 192.4 Calculation of two-point correlation functions using SAXS or SANS data 212.4.1 Case study for structural characterization using SAXS data 242.5 Necessary conditions for two-point correlation functions 282.6 Approximation of two-point correlation functions 302.6.1 Examination of the necessary conditions for the proposed estimation 342.6.2 Case study for the approximation of a TPCF 392.7 Conclusion 42Chapter 3 Approximate Solution for N-Point Correlation Functions for Heterogeneous Materials 433.1 Introduction 433.2 Approximation of three-point correlation functions 453.2.1 Decomposition of higher order statistics 453.2.2 Decomposition of two-point correlation functions 463.2.3 Decomposition of three-point correlation functions 473.3 Approximation of four-point correlation functions 513.4 Approximation of N-point correlation functions 563.5 Results 603.5.1 Computational verification 603.5.2 Experimental validation 623.6 Conclusions 66Chapter 4 Reconstruction of Heterogeneous Materials Using Two-Point Correlation Functions 674.1 Introduction 674.2 Monte Carlo reconstruction methodology 694.2.1 3D cell generation 724.2.2 Cell distribution 754.2.3 Cell growth 774.2.4 Optimization of the statistical correlation functions 794.2.5 Percolation 794.2.6 Three-phase solid oxide fuel cell anode microstructure 814.2.7 Reconstruction of multiphase heterogeneous materials 824.3 Reconstruction procedure using the simulated annealing (SA) algorithm 864.4 Phase recovery algorithm 914.5 3D reconstruction of non-eigen microstructure using correlation functions 964.5.1 Microstructure reconstruction using Monte Carlo methodology 964.5.2 Sample production 974.5.3 Monte Carlo calculation of a two-point correlation function 984.5.4 Microstructure optimization 994.5.5 Results and discussion 994.6 Conclusion 101Chapter 5 Homogenization of Mechanical and Thermal Behavior of Nanocomposites Using Statistical Correlation Functions: Application to Nanoclay-based Polymer Nanocomposites 1035.1 Introduction 1035.2 Modified strong-contrast approach for anisotropic stiffness tensor of multiphase heterogeneous materials 1045.3 Strong-contrast approach to effective thermal conductivity of multiphase heterogeneous materials 1125.4 Simulation and experimental verification 1175.4.1 Computer-generated model 1185.4.2 Thermal conductivity 1205.4.3 Mechanical model 1225.4.4 Experimental part 1255.5 Results and discussion 1275.5.1 Thermal conductivity 1275.5.2 Thermo-mechanical properties 1285.6 Conclusion 130Chapter 6 Homogenization of Reconstructed RVE 1336.1 Introduction 1336.2 Finite element homogenization of the reconstructed RVEs 1346.2.1 Reconstruction of FIB-SEM RVEs 1346.2.2 Finite element analysis of RVEs 1386.3 Finite element homogenization of the statistical reconstructed RVEs 1416.3.1 FEM analysis of reconstruction RVE using statistical correlation functions 1416.3.2 Finite element analysis of RVEs 1436.4 FEM analysis of debonding-induced damage model for polymer composites 1496.4.1 Representative volume element (RVE) 1506.4.2 Cohesive zone model 1526.4.3 Material behavior and FE simulation 1576.4.4 The effect of the GNP’s volume fraction and aspect ratio in perfectly bonded nanocomposite 1586.4.5 Comparing the effect of the GNP’s volume fraction and aspect ratio in perfectly bonded and cohesively bonded nanocomposites 1606.4.6 The effect of the GNP’s aspect ratio and volume fraction in weakly bonded nanocomposite 1636.5 Conclusion and future work 166Appendices 169Appendix A 171Appendix B 175Bibliography 179Index 185

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