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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.

Additive manufacturing allows for the manufacture of complicated geometric features which are impractical or impossible to manufacture by other means. One of these types of additive manufacturing exclusive geometric features is the collection of intricate and periodic strut-like geometries referred to by the term of lattice structures. Lattice structures are desired for their ability to provide stiffness through the large number of supporting members while employing void space within the geometry as a means of reducing part material volume. Strut thicknesses of every lattice in a part are generally not well optimized in order to maximize part stiffness, and often every lattice unit cell is identical throughout the part. This thesis presents a lattice density optimization methodology based on the widely used SIMP topology optimization which is able to find the optimal strut thickness profile for maximizing part stiffness. The material property interpolation scheme used in SIMP optimization is replaced by a new interpolation scheme which more accurately captures the material properties of the prescribed lattice structure at arbitrary density. The resulting lattice structure maintains the same weight reduction as compared to a SIMP lattice replacement scheme while achieving a lower compliance of the part to applied forces.

Lightweight materials such as Aluminum are prevalent in aerospace and automotive vehicles, but the use of lighter Magnesium alloys will significantly increase fuel efficiency and cut emissions. Magnesium alloys present a wide array of unsolved scientific challenges, such as the deformation response of the slip and twin systems and the influence of dislocation interactions and twinning on tensile and fatigue behavior. In this thesis, a parallel three-dimensional(3D) crystal plasticity finite element open-source code was developed based on the deal.II finite element framework as part of PRISMS-Plasticity. Rate-independent crystal plasticity was implemented by developing a nonlinear algorithm which enables all the slip systems to lie on or inside the yield surface, and a consistent tangent modulus ensures convergence for small loading increments. A twin activation mechanism was incorporated into the framework based on a quadrature point sensitive scheme. Furthermore, by bounding the L2-norm of the plastic-slip, load-step adaptivity is enabled. The code demonstrates parallel performance and scaling on large-scale problems running on hundreds of processors. Using experimental microstructure images as input, the code has been used to compute, validate and investigate response of crystalline aggregates to mechanical loading; this leads to insights on slip and twin activity. Boundary value problems were set up to compare the displacement and strain fields obtained by Scanning Electron Microscope - surface Digital Image Correlation (DIC) experiments for Magnesium alloy WE-43 T5 and T6 tempers with the crystal plasticity finite element simulations. The results indicate a strong correlation between experiments and crystal plasticity finite element simulations. For further insight into the material behavior and to interpret the surface observations better, it is important to know the subsurface effects on the surface behavior. 3D reconstruction of microstructures is growing to be a major topic of interest in the field of modeling and simulation for comparison with experiments. An inverse Voronoi problem approach is used to construct an approximate Voronoi representation of the surface microstructure by generating a convexified representation of the microstructure. The output is combined with random sections of Electron backscatter diffraction observations to build a 3D microstructure. Comparisons are made with surface DIC measurements for random samples of 3D microstructures and they indicate the effect of the underlying microstructure on the surface plastic strain. These developed methods will serve as powerful tools in an Integrated Computational Materials Engineering framework towards accelerating alloy development and in better understanding the mechanical behavior of materials.

This thesis outlines the development of computational modeling tools used to predict the elastic properties of composite lamina from representative volume elements (RVE) using numerical methods. The homogenization approach involves the use of Gauss’s Theorem to simply the average volumetric strain integral into a surface integral containing which is defined by surface displacements and their direction. Simulations of RVEs under specific loading conditions (longitudinal tension or shear and transverse tension or shear) are then performed in the software package ABAQUS to obtain the surface displacements. It was found that obtaining quality meshes and applying periodic boundary conditions for each RVE was very user and time-intensive, thus, modeling tools were developed to automate the modeling process. The homogenized composite lamina properties are obtained using C++ code to generate models, batch scripts to run successive simulations in ABAQUS, and C++ code to extract surface displacements from the output data and calculate the final properties. The model generation code contains many user-controllable features such as constituent material properties, fiber volume fraction, mesh density, and type of applied boundary conditions. The modeling tools are then applied in a sensitivity study modeling a unit cell to identify which constituent properties have the largest impact on out-of-plane lamina properties. The framework is then extended to model lamina RVEs consisting of two different fiber materials. The location and percent composition of the two different fiber materials are varied to analyze the effect on out-of-plane lamina properties and identify optimal material combination.

The representative volume element (RVE) of asphalt concrete with applications to low temperatures is investigated based on statistical analysis and modeling of an extensive set of experiments. The experimental part consisted of three-point bending creep tests performed on beams of different sizes: 6.25 × 12.5 × 100 mm (1x), 12.5 × 25 × 200 mm (2x), and 18.75 × 37.5 × 300 mm (3x). The creep stiffness of ten asphalt mixtures was determined at three low pavement service temperature levels. The experimental results showed that a representative creep stiffness of asphalt concrete can be obtained from testing at least three replicates of the thin (1x) mixture beams, even though the mixtures contained aggregate sizes that are larger than the smallest dimension of the beam. In the theoretical part, asphalt concrete specimens of different sizes were analyzed based on digital image analysis, micromechanics, and finite element modeling. The volumetric fractions and particle size distributions of the different specimen sizes were estimated from their binary images after digital processing. The volumetric fraction and the average size distribution of aggregates for the 3x, 2x, and 1x specimens were found to be very similar and based on these mixture properties, RVE sizes were proposed. Detailed information on the internal structure of asphalt concrete was investigated by estimating the spatial correlation functions of specimens of different sizes. No large differences were observed between the correlation functions of the specimens of different sizes. Micromechanical models were used to investigate the influence of the specimen size on the relaxation modulus of asphalt concrete. The empirical model predicted fairly well the relaxation modulus. However, the higher order models were poor predictors of this parameter due to the high stiffness contrast of the phases. Finally, the results from two-dimensional finite element simulations of tension tests were used to suggest minimum sizes for the RVE of asphalt concrete for low temperature applications.