Acceleration waves in von Kármán plate theory

Abstract

In the present work we study acceleration waves in plate-like bodies using the balance laws for the time-dependent von Karman equations presented in [P. A. Djondjorov and V. M. Vassilev, Conservation Laws and Group-Invariant Solutions of the von Karman Equations, Int. J. Nonlinear Mech. 31 (1), pp. 73-87, (1986)]. Two of these balance laws correspond to the time-dependent von Karman equations themselves. They are used to define acceleration waves in thin isotropic elastic plates as discontinuity solutions with finite jumps of second derivatives of the displacement field at a certain curve - the wave front. These two balance laws lead through Hadamard's lemma and Kotchin's theorem to a set of jump conditions on the wave front. Similarly, the other balance laws lead to additional and independent jump conditions on the curve of discontinuity. Several examples are given to illustrate the effect of involving jump conditions of that kind in analysis of acceleration waves in plates. The examples are composed on the basis of three families of group-invariant solutions to the time-dependent von Karman equations. They present three kinds of waveforms pertaining to the class of the so-called relatively undistorded progressive waves. We analyze the propagation of this special kind of acceleration waves into a known state of plate motion.