Numerical determination of the effective transport properties of heterogeneous materials by digital image analysis
Date
2022
Authors
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Journal ISSN
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Publisher
University of Delaware
Abstract
Multiscale heterogeneous materials, such as composites, display complex microstructure morphologies. They are produced by mixing two or more material phases, of widely differing properties, resulting in the formation of superior materials which meet specific requirements. The successful application of these materials, hinges on the development of a comprehensive computational scheme which links microstructure geometry to performance. This entails predicting their effective properties, based on the knowledge of the microgemetry, spatial distribution, and property of their constituents. Many effective medium models have been developed long ago towards the estimation of the effective transport properties of two-phase multiscale media. However, these models were limited to the analysis of idealized isotropic microstructures, whereby the full details of their morphology were assumed to be unknown. Additionally, these early models were only applicable to microstructures having low volume fractions of the constituent phases, and did not consider the effect of microstructure geometry. ☐ Thanks to advances in microstructure imaging methods, such as scanning electron microscopy and X-ray microtomography, real microstructures of multiscale, heterogeneous materials, including composites, can be obtained with astonishing accuracy, at the sub-micrometer scale. This promoted the use of microstructural digital images, and the development of numerical methods for the analysis of heterogeneous materials. For example, in finite element (FE)-based homogenization, digital image analysis has been used to extract the microgeometric properties of the constituent phases of materials. Here each pixel (for 2D microstructures) or voxel (for 3D microstructures) of the image is represented as a finite element, allowing a FE mesh of the material to be generated, upon which the homogenization problem is solved. But, it is crucial to note that when large amounts of microstructure data are obtained from microstructural images, creating a FE model from that data may be quite difficult due to the large number of elements needed to be generated in the process. This highlights a serious flaw with FE image-based homogenization method. The Fast Fourier Transform (FFT) approach is an established alternative to FE homogenization. Here, numerical computations are performed on equispaced grids, and the material phases associated with each grid are analogous to the pixels or voxels of the microstructural digital images, thus eliminating the need for meshing, and improving computational efficiency, which represents the core advantage of the method. The main disadvantage, however, is the poor convergence rate and accuracy of results for microstructures having large contrasts in the properties of constituent phases. ☐ To address the shortcomings of FE and FFT homogenization approaches, this thesis presents the concept of homogenization by boundary integral equations (BIEs), using information obtained from real microstructural images as input data. These input data consist of microstructural topological features such as radii, spatial distribution, axes, and angles of orientation of inclusions in multiphase materials. They are retrieved from microstructural digital images using Computer Vision (CV)-based algorithms, and are used for the reconstruction of piecewise smooth or smooth curve or surface approximations of the inclusion boundaries, upon which the proposed BIE is solved to determine effective properties. In comparison to FE and FFT methods, the proposed BIE model has several major advantages: first, the dimensionality of the problem is significantly reduced because the computations are carried out on the boundaries of the material phases; second, the effective properties are obtained as analytical functions in terms of representations that separate the dependence on phase microgeometric distribution from the dependence on the phase contrast parameter; lastly, the BIE method maintains a very high degree of accuracy over the whole range of values that may be attained for the phase contrast parameter and the volume fractions of the inclusions. ☐ We validated the accuracy of our CV-based algorithms for the reconstruction of inclusion boundaries, derived from microstructural images in 2D and 3D spaces. Specifically, we quantitatively compared results of the topological features of a wide range of inclusion geometries, computed by the CV-based algorithms, to their actual (“true”) values. We obtained relative errors less than 0.5% at sufficiently high microstructural image resolutions. ☐ The accuracy of the BIE model for the analysis of 2D and 3D microstructures was also validated (using the microstructural topological data obtained from the CV-based algorithms) by comparing results for the homogenized tensors to their high precision values. For heat conduction problems, the results for the effective conductivity tensors showed good agreement with their high precision values, over the full range of inclusion conductivity, i.e., from zero (voids) to infinity (perfectly conducting inclusions). The BIE model was also extended to the analysis of two-phase dielectric media, for estimating effective permittivity tensors. Its accuracy and reliability were verified by comparing computed results to the classical Bergman-Milton bounds for effective permittivity tensors. Our BIE model gave reliable results for two-phase microstructures, and showed good agreement with the classical Bergman-Milton bounds. We also showed that inclusion geometry (at fixed volume fractions) has a profound impact on the effective dielectric loss constant, as well as the loss tangent, for two-phase isotropic microstructures. In particular, comparing 4 cases of isotropic microstructures, having inclusions of circular, square, annular and star shapes, we found that the microstructure having the annular-shaped inclusion yielded the largest effective dielectric loss constant, as well as the largest effective loss tangent. Such microstructures would be of interest in microwave applications as they dissipate the largest amount of heat per unit volume. ☐ Applications of the BIE model in the field of Plasmonics, for the design of spectrally selective absorbers used in photovoltaics and thermoelectric applications, were also considered. We showed that certain metal-dielectric composites can be characterized by resonant electrostatic conditions, which are completely microstructure dependent, and at which their absorptive capacity (for electromagnetic waves of a particular frequency/spectrum) peaks.
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Keywords
Composite materials, Data analysis, Effective properties, Homogenization, Image processing, Microstructure reconstruction, Numerical determination, Digital analysis