Heterogeneous materials may become structurally unstable under an applied stress. In this thesis, the effects of two dissimilar stresses are studied. First, the methods of statistical mechanics are used to analyze the effects of mechanical stress on the strength of heterogeneous materials. A phenomenological multi-scale model is presented that analyzes inelastic deformation in a model natural composite, nacre. A kinetic Monte Carlo technique is developed to study the mechanical response of the biopolymer. The results of this model are used to generate a cellular automata model of the composite material. Under certain conditions, the sizes of plastic events in this model follow an apparent power-law distribution. The dynamics are found to be similar to earthquakes, where the slip sizes exhibit a scale-free distribution.

Second, a continuum theory is generated to understand how a model microstructure responds to thermal stresses. At elevated temperatures, a structure may spontaneously change shape in order to minimize its overall surface energy. To this end, the stability of a hollow-core dislocation to pearling and coarsening is considered using a linear stability analysis. There is a competition between elastic energy and the anisotropic surface energy. It is shown that sufficiently small hollow-core dislocations are stable with respect to both forms of structural instability, suggesting a route to stabilize nanometer-scale wires.