Abstract: Shape memory materials are increasingly sought after to enable the design of unconventional structures, particularly in the aerospace community where deployable space structures and morphing aircraft have become important. These materials allow significant shape changes and spontaneously return to their original shape when triggered to do so. Shape memory polymer (SMP) is a well known class of these materials. A natural extension of shape memory polymer is elastic memory composite (EMC) which is, simply, SMP with fiber reinforcement. This material is very promising in that it inherits the very high strain capability and shape memory characteristics of the SMP and inherits high structural performance from the fiber reinforcement. When EMC is taken to elevated temperatures, where the shape-change and shape-recovery occurs, the SMP matrix behaves very elastomeric and linear. In this state, the fibers are slender columns on a compliant elastic foundation and buckle into highly uniform sinusoids when on the compression side of a bent EMC laminate. The buckling has a small characteristic wavelength (∼1mm) and is, therefore, termed microbuckling in the tradition of Walt Rosen. This is a fully elastic and recoverable deformation mode which allows fibers that normally fail at 1.5% compression strain to accommodate more than 10% composite compression strain. The seemingly obvious step of reinforcing shape memory polymer with fibers, therefore, results in a not-so-obvious synergistic relationship allowing a structurally efficient composite material, EMC, to endure very large deformations without degradation. The use of EMC is not yet wide-spread due, in part, to a lack of a fundamental understanding of its somewhat complex behavior in the elevated-temperature compliant-matrix state. However, the benefits of EMC would not be possible with a simple linear-elastic material and we must, therefore, increase our understanding. While the behavior of EMC in the rigid-matrix, straight-fiber state is understood with existing models; its highly nonlinear and somewhat complex behavior in the compliant-matrix post-microbuckled state cannot be understood with existing models. The following thesis is thus presented on the mechanics of post-microbuckled compliant-matrix composites.