Time Domain Modeling of Linear Viscoelasticity Using Anelastic Displacement Fields

Abstract

A time domain model of linear viscoelasticity is developed based on a decomposition of the total displacement field into two parts: one elastic, the other anelastic. The anelastic displacement field is used to describe that part of the strain that is not instantaneously proportional to stress. General coupled constitutive equations for (1) the total and (2) the anelastic stresses are developed in terms of the total and anelastic strains, and specialized to the case of isotropic materials. A key feature of the model is the absence of explicit time dependence in the constitutive equations. Apparent time-dependent behavior is described instead by differential equations that govern (1) the motion of mass particles and (2) the relaxation of the anelastic displacement field. These coupled governing equations are developed in a parallel fashion, involving the divergence of appropriate stress tensors. Boundary conditions are also treated: the anelastic displacement field is effectively an internal field, as it is driven exclusively through coupling to the total displacement, and cannot be directly affected by applied loads. In order to illustrate the use of the method, model parameters for a commonly-used high damping polymer are developed from available complex modulus data.