The quasi-static and dynamic compressive behavior of open-cell foams, textile cores, and pyramidal truss cores were investigated using a combination of experimental apparatus. Quasi-static tests were performed using a miniature loading stage and a Kolsky bar apparatus was used for intermediate deformation rates. For high deformation rates, a gas gun was employed. Optical observations of the sample deformation were performed in real time by means of high-speed photography. The deformation modes were investigated in detail from acquired images and digital image correlation. For the open cell foams, comparison between deformation fields under quasi-static and Kolsky bar loading revealed a moderate micro-inertia effect, where the inertia associated to the bending and buckling of ligaments delayed strain localization. Gas gun experiments performed on the same samples revealed a totally different deformation mode. A crashing shock wave was generated at the impact surface and propagated through the specimen. In these experiments both forward and reverse impact tests were performed to interrogate the state of stress in front and behind the shock wave front. Through these experiments, it was confirmed that the generation and propagation of shock waves within foam materials greatly enhance their energy absorption. For the case of textile cores, the mechanical response was found to be similar to the open cell foam materials. No significant difference in load-deformation histories and failure modes were observed between quasi-static and intermediate deformation rates. As in the case of open cell foams, at high deformation rates, the failure mode was governed by the development of a crushing shock wave. For the truss cores, significant deformation rate effects on peak stress and energy absorption were identified. Inertia effects appeared to dominate the core response because of two effects: i) the propagation of a plastic wave along the truss members, and ii) buckling induced lateral motion. In this article we provide a quantification of load-deformation response and associated failure modes across the sample as captured by high speed photography and image correlation.