WAVES IN NONLINEAR-ELASTIC “RATE-TYPE” VISCOUS MATERIALS

Abstract

Provided the effects of nonlinearity and dissipation are small but definitely not negligible, a wide class of wave phenomena can be investigated by means of the quasilinear parabolic equation known as Burgers' equation. The analysis of Burgers' equation in its form corresponding to the one-dimensional propagation of a longitudinal plane wave in homogeneous isotropic solids constitutes the main conent of this paper. The closed system of equations forming the basis for the derivation of Burgers' equation and representing nonlinear-elastic “rate-type” viscous media is considered. The importance of the geometric nonlinearity caused by the strain-displacement relation relative to that of the physical nonlinearity due to the stress-strain relation is discussed. Quantitative results illustrating how the parameters of the input (maximum amplitude and frequency) influence the distortion of an initially sinusoidal pulse are given for aluminium. A method for the evaluation of the parameter characterizing the physical (or material) nonlinearity is presented. It involves the approximation of a material' s experimental stress-strain curve by the quadratic stress-strain relation.The results show that:(1) Consideration of nonlinearity brings qualitatively new effects into the study of wave phenomena.(E. g., the distortion of the wave profile, which leads to the formation of a “weak shock”.)(2) Various materials show a notably nonlinear behaviour, as expressed, in particular, in terms of the shapes of their experimental stress-strain curves.(3) The effect of both the physical and the geometric nonlinearities together must be taken into account in the study of nonlinear wave phenomena.