Growth and decay of one-dimensional acceleration waves in non-linear materials with anelastic response

Abstract

The growth behavior of acceleration waves which propagate in non-linear materials which exhibit anelastic response, in the sense of Eckart, is examined. The differential equation which governs the growth behavior of the wave is a generalized Riccati equation; this sets the present work off from previous studies of wave propagation in non-linear materials where, invariably, the governing equation is of Bernoulli type. It is noted that the Riccati equation derived reduces to one of Bernoulli type if the motion which defines the preferred time-dependent reference configuration is a homogeneous motion. Under the assumption that a continuous particular integral of the governing Riccati equation exists, the equation is reduced to one of Bernoulli type which is then readily integrated so as to yield the general solution. As part of our analysis, we apply the Quasilinearization method of Bellman & Kalaba and obtain an analytic expression which bounds solutions of the Riccati equation arising here; from the various upper bounds for the amplitude of the wave that are obtained we derive certain results concerning the asymptotic behavior of the amplitudes of waves that are either initially compressive or initially expansive.