Growth of Acceleration Waves in Isotropic Elastic Materials

Abstract

The growth of acceleration waves propagating in an isotropic elastic material is examined. The underlying assumption made is that the material is at rest in its reference configuration. First, we establish that only longitudinal or transverse acceleration waves may propagate, and the wave speeds are determined. Then, we derive the differential equation governing the growth of the amplitude of longitudinal acceleration waves and obtain explicit solutions for plane, cylindrically or spherically expanding waves. These results indicate that the amplitude of acceleration waves may either decay to zero in infinite time or become infinite (shock formation) in finite time; the radius effect on the amplitudes of cylindrical or spherical acceleration waves are not of the form 1/r12 or 1/r, as predicted by lineary theory. On the other hand, the behavior of transverse waves is similar to that obtained by linear theory.