Plane waves in simple elastic solids and discontinuous dependence of solution on boundary conditions

Abstract

Using stress as the dependent variable instead of the deformation gradient, plane waves of finite amplitude in simple elastic solids are studied. For isotropic materials there are two plane polarized simple waves as well as shock waves and one circularly polarized simple wave which can also be regarded as a shock wave. With the aid of the stress paths for simple waves and shock waves in the stress space introduced here, one can see clearly what combination of simple waves and/or shock waves is needed to satisfy the initial and boundary conditions. We use second order isotropic hyperelastic materials to illustrate the ideas. In one example we show that the solution requires as many as four simple waves. In another we show that depending on the boundary condition there are more than eight possible solutions to the problem. We also present an example in which the solution does not depend continuously on the boundary condition. This implies that in experiments if the applied load at the boundary is not properly controlled, any slight deviation in the applied load would result in a finite different response in the material.