Quasistatic anti-plane motion in the simplest theory of nonlinear viscoelasticity

We consider a very simple model in the framework of differential viscoelastic materials which are isotropic and incompressible. In this model the Cauchy stress tensor is split in an elastic part and a dissipative part. The elastic part is derived from a strain-energy density function only of the first invariant of the Cauchy–Green strain tensor. The dissipative part is like the Navier–Stokes equations: linear in the stretching tensor with a constant viscosity parameter. For this model we provide some time and spatial estimates in the quasistatic approximations for the equations governing anti-plane shear motions. Several explicit examples for specific form of the strain energy are produced. Our results impose analytical restrictions on the mathematical properties of the strain energy to ensure a physical behavior in the creep and recovery experiments. Moreover, we show polynomial decay for the spatial behavior in the class of stress-hardening (or strain-stiffening) materials. For stress-softening materials a Phragmen–Lindelof alternative is proved.