Dynamic crushing of cellular materials: Continuum-based wave models for the transitional and shock modes

As shown in the extensive studies of the dynamic responses of cellular materials, when the impact velocity is high, ‘shock’ waves can be generated. Because of the nature of the cellular structure, behind the ‘shock (or compaction) front’, there is a region of thickness approximately one single-cell-layer, across which the deformation can vary enormously, with strains of the order of ∼0.8, say. This is due to the extensive and progressive crushing of the cells. The compressed part of the cellular material is crushed and densified as the material crosses the front. Depending on the details of the cellular geometry, this locally large deformation can be very intricate to model, however, a first order ‘shock’ model can be defined, which permits a useful understanding of the phenomenology of the dynamic deformation of cellular materials, particularly metal foams. However, when the impact velocity is not very high, there exists a different type of front behind which the strain, though plastic, does not reach the densification strain. Based on one-dimensional continuum-based stress wave theory with a ‘rigid unloading’ assumption, in this paper a theoretical framework is established to explore the corresponding inherent mechanisms as a simple extension of the original ‘shock’ theory. Two models, namely the Shock-Mode model and the Transitional-Mode model, are introduced. The distributions of stress, strain and velocity in the foam rod are derived. The theoretical results show that for a Shock Mode, behind the front the initial strain remains constant and the initial stress varies proportionally with the square of the impact velocity, but for a Transition Mode, the initial strain and stress behind the front reduce linearly with reducing impact velocity. The critical impact velocities for modes transition are predicted. Two dimensionless parameters, namely the shock-enhancement parameter and the stress-hardening parameter, are defined and the features of the theoretical predictions are presented. Compared to the experimental results, the responses at the ends of foam rod are well predicted by the present models and also by the R-P-P-L model. However, deformation mechanisms uncovered by the present modes and the R-P-P-L model are very different when the impact velocity is not very high. The present simple, wave-based models extend the understanding of metallic foams to loading over a wider range of impact velocities than the previous models. In particular, the sub-sh