A variational approach to frame-indifferent quasistatic viscoelasticity of rate type.

The three-dimensional dynamical model for nonlinear viscoelasticity of strain-rate type is investigated in a quasistatic setting under the assumption of higher-order regularity of the deformation, which in the literature is referred to as the case of non-simple materials. The existence of weak solutions is proven using a time-discretization technique while respecting the condition of dynamical frame indifference. Some observations on frame indifference for strain-rate-type stresses are made, and corrections are proposed for some related work in the literature. Finally, a counterexample is given to show that the assumed higher-order regularity is necessary in order to obtain the required compactness.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Nonlinear viscoelasticity of strain rate type: an overview.

There are some materials in nature that experience deformations that are not elastic. Viscoelastic materials are some of them. We come across many such materials in our daily lives through a number of interesting applications in engineering, material science and medicine. This article concerns itself with modelling of the nonlinear response of a class of viscoelastic solids. In particular, nonlinear viscoelasticity of strain rate type, which can be described by a constitutive relation for the stress function depending not only on the strain but also on the strain rate, is considered. This particular case is not only favourable from a mathematical analysis point of view but also due to experimental observations, knowledge of the strain rate sensitivity of viscoelastic properties is crucial for accurate predictions of the mechanical behaviour of solids in different areas of applications. First, a brief introduction of some basic terminology and preliminaries, including kinematics, material frame-indifference and thermodynamics, is given. Then, considering the governing equations with constitutive relationships between the stress and the strain for the modelling of nonlinear viscoelasticity of strain rate type, the most general model of interest is obtained. Then, the long-term behaviour of solutions is discussed. Finally, some applications of the model are presented.