On shock waves in materials with memory

Abstract

Shock wave propagation in materials with memory is studied through the method of discontinuities of Achenbach and Reddy [3]. We consider linear ageing and nonageing viscoelastic materials with creep-relaxation kernels containing a weak (integrable) singularity and presented by fractional models. It is shown that for this class of materials, the existence of discontinuities of any order is not possible and that the solutions behind the wavefront are infinitely smooth. This conclusion is confirmed through the integral transforms method. The obtained results are also valid for phenomenological nonlinear viscoelastic theories. The problem of the general nonlinear case is discussed. Relationships for shock wave propagation in classic viscoelastic materials (spring--pot models) follow from the presented ones as a limit case for the singularity parameter tending to zero. The established results lead to numerous applications, for example in materials science, civil and seismic engineering.