Applied RVE Reconstruction and Homogenization of Heterogeneous Materials
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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
1
Literature Survey
1.1. Random heterogeneous material
Random heterogeneous material is a class of materials that is composed of different materials or states, such as a composite and a polycrystal. "Microscopic" length scale is much larger than the molecular scale but much smaller than the characteristic length of the macroscopic sample. The heterogeneous material can be assumed to be a continuum on the microscopic scale, and therefore its effective properties can be defined accordingly [TOR 02].
Statistical methods, using correlation functions, are among the most practical and powerful approaches to estimate properties of heterogeneous materials [TOR 02]. Properties of materials can be approximated by using different orders of the statistical correlation functions [TOR 97, TOR 02, PHA 03]. In multiphase materials, the first order correlation function represents volume fractions of different phases and does not describe any information about the distribution and morphology of phases [TOR 02].
If M-number of random points are inserted within a given microstructure and the number of points in phase i is counted as Mi, one-point probability function is then defined as the volume fraction through the following relation, as M (the total number) is increased to infinity:
[1.1]where Vi is the volume of phase i (Fi), Vtotal is the total volume and vi is the volume fraction of phase i. Clearly, for two phase's microstructure:
[1.2]1.2. Two-point probability functions
Now assign a vector starting at each of the random points in a heterogeneous microstructure. Depending on whether the beginning and the end of these vectors fall within phase-1 or phase-2, there will be four different probabilities: defined as Torquato and Haslach [TOR 02]:
[1.3]where Mij are the number of vectors with the beginning in phase-i (Øi) and the end in phase-j (Øj). Equation [1.3] defines a joint probability distribution function for the occurrence of events constructed by two-points (1 and 2) as the beginning and end of a vector when it is randomly inserted in a microstructure. The two-point function can be defined based on two other probability functions such that [TOR 02]:
[1.4]The first term on the right hand side is a conditional probability function. At very large distances, r8, the probability of occurrence of the beginning point does not affect the end point and the two-points become uncorrelated or statistically independent and the conditional probability function reduces to a one-point correlation function:
[1.5]The two-point function will then reduce to Torquato [TOR 02]:
[1.6]or
[1.7]For the case of a two-point function in a two-phase composite, we have symmetry for non-FGM microstructure [TOR 02]:
[1.8]For a three-phase composite, the indices (i, j) in the probability functions representation extend to three and as a result we have nine probabilities . Due to normality conditions the following equations are satisfied:
[1.9] [1.10] [1.11]Satisfying all three conditions for a three-phase composite (i, j? {1,2,3}) and knowing that the probability functions are symmetric results in the important conclusion that only three of the nine probabilities are independent variables. For instance, we can choose or (P11), or (P12), and or (P22) as the three probability parameters.
1.3. Two-point cluster functions
Two-point cluster function is the other microstructure descriptor of heterogeneous materials, which can reflect more precise information for heterogeneous materials [JIA 09]. The two-point cluster function (TPCCF) is the probability of finding both points (starting and ending point of vector ()) in the same cluster of one of the phase (i). This quantity is a useful signature of the microstructure as it reflects clustering information. Incorporation of such information in addition to the lower-order two-point cluster functions have led to the formulation of rigorous bounds on transport and mechanical properties of two-phase media [TOR 02, JIA 09].
1.4. Lineal-path function
The lineal-path function L(r) for n-phase heterogeneous materials gives the probability of finding a line segment of length r wholly in the target phase, when randomly thrown into the sample. The lineal-path function is an important statistical descriptor in determining the transport properties of heterogeneous materials and can be a function of interest in stereology.
1.5. Reconstruction
Experimental and numerical reconstruction of heterogeneous materials to obtain an accurate structure can be used to characterize and optimize heterogeneous materials. There are different experimental techniques such as X-ray tomography or focused ion beam/scanning electron microscopy (FIB/SEM), which are used to reconstruct three-dimensional microstructures. For numerical reconstruction, statistical information is extracted from the microstructure of the considered heterogeneous material and can be used to reconstruct three-dimensional microstructures [TOR 02, EDW 05, JIA 07, JIA 08, KET 11, REU 08, MER 00].
1.5.1. X-ray computed tomography (experimental)
X-ray computed tomography is a non-destructive technique that can be utilized to reconstruct micro-heterogeneous materials such as metal matrix composites. In this technique, an X-ray beam hits a rotating sample and two-dimensional projections are recorded using a detector on the other side of the sample (see Figure 1.1) [KET 11, MER 00].
Figure 1.1. Principle of X-ray tomography [MER 00]
In classical tomography (attenuation tomography), three-dimensional reconstruction is performed by combining the two-dimensional projections. This technique has some limitations, for example [KET 11]:
- - RESOLUTION limited to about 1000-2000 × the object cross-section diameter;
- - blurring of material boundaries;
- - weak attenuation contrasts for imaging;
- - complicated data acquisition and interpretation due to the image artifacts (beam hardening);
- - large data volumes and difficulty of visualization and analysis.
However, this technique also has several strengths, such as [KET 11]:
- - non-destructive 3D imaging;
- - easy sample preparation required;
- - extraction of sub-voxel level details.
1.5.2. X-ray computed tomography (applications to nanocomposites)
A composite specimen composed of 52% vol unidirectional glass fibers mixed into an epoxy matrix is analyzed and a 3D image is reconstructed using X-ray computed tomography. The internal microstructure of the specimen has been obtained using a high-resolution 3D X-ray imaging system (MicroXCT-400, Xradia). Figure 1.2(a) shows a sample of 2D projection generated by the X-rays passing through the specimen. A number of these X-ray projections have been acquired from the specimen in different angles (from -170° to +170° around the main axis of the specimen). Using a filtered back projection method, the 3D microstructure of the specimen has been reconstructed from these projection images. To eliminate noise and improve quality, a Gaussian smoothing filter has been applied to the raw data. The binary representation of the microstructure has been segmented from gray-scale data using a threshold filter. A 2D cross-section of the binary matrix is shown in Figure 1.2(b). Each voxel of the binary matrix (also known as label matrix) represents a cubic chunk of the material and a non-zero value is assigned to the each voxel corresponding to the phase occupying the location of the voxel. These operations have been performed in Matlab using the Image-Processing toolbox. Figure 1.2(c) also shows a volumetric rendering generated from the acquired data revealing the anisotropic arrangement of unidirectional glass fibers in the composite specimen.
1.5.3. FIB/SEM (experimental)
Focused ion beam (FIB) is a technique that can be used in materials science to modify and image the sample of interest.
An FIB setup is a scientific instrument that uses a focused beam of ions to image the sample. FIB is used to create very precise cut sections of a sample for imaging via SEM, STEM or TEM. FIB imaging can be applicable to image a sample directly. The contrast mechanism for FIB is different than for SEM or S/TEM. FIB can also be incorporated in SEM to investigated using either of the beams with both electron and ion beam columns. A dual beam FIB/SEM setup can be used for serial sectioning and 3D reconstruction of nanostructures [GIA 04]. In the next section, application of FIB/SEM for one sample of nanocomposite is explained.
Figure 1.2. a) Sample X-ray projection image used for reconstruction; b) 2D cross-section view of binary label matrix; c) 3D volume rendering of the arrangement of the glass fibers in the unidirectional composite specimen [BAN 14]
