Dynamic response of a bounded, three-dimensional, viscoelastic body

Abstract

The isothermal response of a bounded, three-dimensional viscoelastic body, subject to arbitrary body forces, initial conditions, and boundary conditions, is considered within the linear theory of viscoelasticity. The viscoelastic properties of the body are assumed to be isotropic and homogeneous. The boundary surface areas over which the boundary conditions are specified are assumed to remain constant with time. The response is first found formally in terms of a causal Green’s function. It is then shown that when Poisson’s ratio is constant, the causal Green’s function can be expanded in a series of spatial eigenfunctions of an associated elastic eigenvalue problem. The resulting solution for the general problem is an eigenfunction series with Laplace transformed time-dependent coefficients. The solution is inverted for two special cases, when the body forces and boundary conditions are separable in space and time, and when the relaxation function in simple shear has the form of a series of decaying exponentials. Subject Classification: [43]40.20; [43]35.50.