Supersonic wave propagation in heterogeneous solids and the evolution of energy distribution

Abstract

Dynamic homogenization of heterogeneous materials has received a lot of attention recently, due to the fact that it can avoid expensive direct simulations with high space-time resolution. In this paper, the applicability of effective homogenous media on some peculiar dynamic phenomena is discussed. Longitudinal wave velocity has been believed to be the upper limit of mechanical signal speeds in a homogenous elastic solid according to the classical elasticity theory. However, for heterogeneous materials, we demonstrate numerically and theoretically that if the wave impedances of the constituent phases are not equal, there must be the supersonic phenomenon such that some wave may propagate faster than the longitudinal wave in the effective homogenous medium. Especially, if two materials with the same longitudinal wave velocity are used to compose a heterogeneous medium, the effective longitudinal wave speed will almost definitely be lower than its local counterpart. We also investigate the evolution of energy distribution over time when the wave propagates through heterogeneous media with various gradients, which indicates, interestingly, that the smoother the change in wave impedances of the heterogeneous medium, the more energy in “before-wave” region and the less energy in “after-wave” region. The corresponding dispersion curves and group speed curves also show dramatic change for heterogeneous media with different smoothness, which can partially explain but not quantitively predict these phenomena. We attempt to develop an effective medium model to reproduce the energy evolution but fail. All these findings pose a big challenge to the proper establishment of dynamic homogenization models.