Abstract
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.
Highlights
Real materials exhibit a variety of inelastic phenomena
A way of overcoming this problem is to consider a physically relevant regularization, which can be done by adding capillarity (e.g. [31] or [32]) or viscosity effects into the equation. We focus on the latter method, in which the stress includes a viscosity term proportional to the strain rate yxt, a general form of which can be written as follows: ρR ytt = σx
Given a dynamical system starting from an initial state, it is difficult to predict how the system will evolve as time increases
Summary
Real materials exhibit a variety of inelastic phenomena. Viscoelasticity, plasticity and fracture are just a few to mention. Any observable quantity must be independent of the particular orthogonal basis in which it is computed Rather than stating it in its most general form, we would like to adopt the version for elastic materials, which says (cf [19]): The principle of frame-indifference: Constitutive functions are invariant under rigid motions and time shifts. To express this principle as a mathematical statement, first we note that a change of observer (or equivalently the orthogonal basis in which the observable quantity is computed) can be seen as application of rigid-body motions on the current configuration
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