Modeling of Viscoelastic Shear: A Nonlinear Stick-Slip Formulation

We present a class of nonlinear dynamic viscoelastic models for materials subjected to shear stress. The model equations are based on a continuum variation of a reptation model in which chemically cross-linked (CC) systems of molecules act as constraint boxes per unit volume for physically constrained (PC) systems of molecules. Results from validating the model with dynamic shear experiments are given and a stability analysis for the corresponding linearized systems is discussed. In this paper we derive for the rst time a continuum model combined with molecular based internal dynamics for shear deformations in viscoelastic materials. The resulting partial dieren tial equation model is coupled to ordinary dieren tial equations for internal strains via a nonlinear stick-slip molecular theory. An initial step toward validating the models with carefully designed experiments is discussed. Various molecular and phenomenological models have been proposed to model rubber deformations. One early and inuen tial paper on viscoelasticity is that of Pipkin and Rogers (20). Their approach (like many subsequent contributions to the literature) is phenomenological, and is based on incorporating the strain/stress his- tory in the relaxation model in a mathematically sound way leading to an appropriate relaxation kernel. The construction incorporates step strain data obtained from ex- periments. The approach presented in this paper is a combination of micro and macro-scale considerations. The relationship between the constrained (PC) and con- straining (CC) molecules embodies an attempt to construct relaxation kernels in a direct way. The relaxation kernel incorporates the strain/stress and strain rate his- tory and the stress/strain relationship is a nonlinear integral expression as is the case in (20).