Representation of stress and free energy for a viscoelastic body from a stressed reference

Abstract

Viscoelastic materials are often initially or residually stressed in the reference configuration. This stress and associated configuration may also evolve temporally without any external influence or loading. This paper develops a general framework representing the stress and Helmholtz potential from a stressed viscoelastic reference configuration. The Helmholtz function is assumed to depend on the initial stress, the initial stress rate (objective), the initial rate of deformation, and the right Cauchy–Green stretch tensor. The corresponding invariants appear not to have been have used earlier. New sets of spectral invariants are also determined and other possible sets of spectral invariants are identified. An appropriate constitutive relation and Helmholtz potential are obtained using a stress-free configuration associated with the stressed reference configuration. For this purpose, the initial strain tensors are determined in terms of initial stress and initial stress-rate by inverting a standard constitutive relation for a viscoelastic material. The model developed using this inverse approach is applied to the investigation of the stress-relaxation of a thick viscoelastic sphere from a stressed reference. Further, it is noted that measuring residual stress is quite challenging for viscoelastic materials. In the context of the present framework, a destructive, and a non-destructive approach to measure residual stress and residual stress-rate for viscoelastic materials is presented.