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In this thesis we are interested in the propagation of low-frequency linear elastic waves through composite materials. We use a variety of dynamic homogenization techniques to find the effective elastic properties of some composites. We consider composites with isotropic phases for ease of exposition but the theory could easily be extended to anisotropic inclusions or host.We use a Representative Volume Element approach with the Method of Asymptotic Homogenization to model a random fibre-reinforced composite. The fibres are all aligned in the same direction and are taken to be of infinite extent, so the composite is essentially two-dimensional. For a random composite we have considered the anti-plane case for SH wave propagation and the in-plane case for P and SV elastic wave propagation, extending the previous published work of Parnell and Abrahams (2006), (2008a), in which a periodic fibre-reinforced composite was studied. We also show, for a simple example, that it is possible to extend the Representative Volume Element method to a three-dimensional particulate composite.In this thesis an Integral Equation Method for homogenization is also studied, with application to periodic fibre-reinforced composites. We have extended the work of Parnell and Abrahams (2008b), which considered SH wave propagation only, to the case of P and SV wave propagation.

Ultra-high temperature ceramics (UHTCs) are a class of materials that maintain their structural integrity at high temperatures, e.g. 2000 °C. They have been limited in their aerospace applications because of their relatively high density and the difficulty involved in forming them into complex shapes, like leading edges and inlets. Recent advanced processing techniques have made significant headway in addressing these challenges, where the introduction of multiscale porosity has resulted in lightweight UHTCs dubbed multiscale porous UHTCs. The effect of multiscale porosity on material properties must be characterized to enable design, but doing so experimentally can be costly, especially when attempting to replicate hypersonic flight conditions for relevant testing of selected candidate samples. As such, this dissertation seeks to computationally characterize the thermomechanical properties of multiscale porous UHTCs, specifically titanium diboride, and validate those results against experimental results so as to build confidence in the model. An implicit quasi-static variant of the Material Point Method (MPM) is developed, whose capabilities include intrinsic treatment of large deformations and contact which are needed to capture the complex material behavior of the as-simulated porous UHTC microstructures. It is found that the MPM can successfully obtain the elastic thermomechanical properties of multiscale porous UHTCs over a wide range of temperatures. Furthermore, characterizations of post-elastic behavior are found to be qualitatively consistent with data obtained from uniaxial compression experiments and Brazilian disk experiments.

Heterogeneity in materials leads to increased complication of physics, at the level or scale where heterogeneity exists, making analysis and material behavior predictions more difficult. As a result of the intricate descriptions or understanding of the physics required to determine the behavior of a heterogenous material or system, practitioners frequently prefer the use of effective properties or responses. This body of work is dedicated to performing computational analyses on the thermal behavior of heterogeneous materials and evaluating the corresponding analyses using formal verification approaches. Custom finite element method code was used for solving the heat equation in a variety of two-dimensional heterogeneous systems. Formal verification methods—including the method of manufactured solutions and the grid convergence index—were used to assess code performance and solution veracity. Functional correlations were drawn between properties of heterogeneity and effective thermal responses.

The effective properties of cellular materials are dependent both on the fixed material properties of the constituent material and the geometry of the cellular structure. As a result, the effective properties can be altered through geometric modifications without changes to the constituent material. Cellular materials present new opportunities in mechanical design due to the ability to produce new materials with customized properties which can improve the performance of existing designs. However, the task of designing the geometry to achieve desired properties presents additional challenges to the design process. To increase the viability of customizable cellular materials in new product design, new methods are needed to help designers develop new materials efficiently and effectively. Two such design methods are presented in this thesis; for the design of honeycomb structures to achieve two effective shear properties simultaneously, and for the design of a compliant skin structure to achieve desired shape morphing behavior. The design methods presented here reflect an effort to develop systematic and automatable processes for the design of new cellular materials for two separate applications, using two separate approaches to the design problem. This thesis discusses the development of both cellular structure design methods and the respective design algorithms created to implement the methods automatically. Numerical analysis is used to test the effectiveness of the methods for their respective design applications and design examples are provided for each method.

The dissertation is concerned with the study of the mechanical time-dependent behavior of viscoelastic composite materials and structures. The analysis of such materials should not only take into account their complex structure, but also the time-varying properties of one or more constituents. The first part of the dissertation is concerned with the calculation of local time-varying fields in the viscoelastic fiber-reinforced composites and porous media. A two-dimensional model that represents a section perpendicular to the axes of the fibers is employed. The analysis adopts the correspondence principle based on the Laplace transform, and the problem is treated with the use of a direct boundary integral approach. The unknown boundary parameters are approximated by the truncated Fourier series. The procedures of the analytical (for the case of porous viscoelastic media) and numerical (for the case of fiber-reinforced composites) inversion of the Laplace transform are used to obtain time-varying fields anywhere in the matrix or inside the inclusions. The developed approach possesses several advantages in regard to computational efficiency and accuracy if compared with conventional methods based on the use of collocation and discretization techniques. The second part of the dissertation is concerned with the evaluation of the effective transverse properties of the viscoelastic fiber-reinforced composites. The developed approach relies on the knowledge of local stress fields and adopts the equivalent inhomogeneity technique modified for the case of viscoelastic composite's matrix. The solution is obtained in the Laplace domain, and the Laplace inversion is required to arrive at the time-varying effective properties. The developed approach directly takes into account the interactions between the inhomogeneities. The final part of the dissertation is dealing with the problem of thermal stress evolution in viscoelastic composite structures. These stresses are due to mismatch between the coefficients of thermal expansion of the composite's constituents. The proposed approach employs the Volterra correspondence principle and relies on the ability to obtain the analytical solution for the corresponding elastic problem. The approach adopts the discrete (matrix) representation of Volterra type operators. Particular attention is devoted to the analysis of thermal stress evolution in viscoelastic asphalt binders at low temperatures.

We discuss the effective macroscopic properties of inhomogeneous materials, such as composite or polycrystalline media, ferromagnetic domain structures and inhomogeneous superconductors. In particular, we use Bergman's spectral theory to analyze the effects of disorder on a periodic composite system such as a colloidal crystal; we further extend that spectral theory to describe polycrystalline media and use that extension to analyze the third-order nonlinear response of a polycrystalline material. For the problem of inhomogeneous materials in an external magnetic field, we demonstrate that in certain 3-constituent composites the Hall effect in the metallic component can lead to either saturating or non-saturating effective magnetoresistance, depending on the constituent volume fractions; we identify and study the percolation problem discriminating between those two cases. We also demonstrate that metallic ferromagnets can exhibit negative magnetoresistance originating from the changes in the magnetic domain structure; we suggest quantitative models describing this behavior for some domain structure geometries. For superconducting media, we demonstrate that inhomogeneities in the magnitude of the order parameter lead to an additional absorption at finite frequencies, and develop sum rules for this absorption for a number of possible models. We then discuss how the fast suppression of the superconductivity in Zn-doped YBa 2 Cu 3 O 7-δ can be described by a simple percolation model based on the local suppression of the order parameter in the regions surrounding Zn atoms. We also discuss how those various changes in the magnitude of the order parameter affect the nonlinear properties such as the intermodulation current. Finally, in the last chapter we consider a problem that is somewhat complimentary to the other problems discussed in this thesis, and analyze the origin of inhomogeneities in the Mg 1-x Al x B 2 at certain Al concentrations. We use the results of ab initio simulations to develop a model for the energetics of various ionic arrangements and demonstrate that the free energy profile based on this model predicts two regions of phase-separation instability, consistent with experiment.