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

Global Regular Solutions to a Kelvin-Voigt Type Thermoviscoelastic System

A classical 3-D thermoviscoelastic system of Kelvin-Voigt type is considered. The existence and uniqueness of a global regular solution is proved without small data assumption. The existence proof is based on the successive approximation method. The crucial part constitute a priori estimates on an arbitrary finite time interval, which are derived with the help of the theory of anisotropic Sobolev spaces with a mixed norm.Comment: 52 page

Asymptotic Behavior of the Solutions to a Landau-Ginzburg System with Viscosity for Martensitic Phase Transitions in Shape Memory Alloys

In this paper, we investigate the system of partial differential equations governing the dynamics of martensitic phase transitions in shape memory alloys under the presence of a (possibly small) viscous stress. The corresponding free energy is assumed in Landau-Ginzburg form and nonconvex as function of the order parameter. Results concerning the asymptotic behavior of the solution as time tends to infinity are proved, and the compactness of the orbit is shown

Dynamical System Methods in Fluid Dynamics

The workshop was organized around the infusion of new techniques from dynamical systems, geometric methods, multiscale analysis, scientific computation, and control theory into traditional methods in fluid mechanics. It was well attended with about 45 participants with broad geographic representation from all continents. There was an excellent blend of senior researchers, students, postdocs and junior faculty

Some gradient theories in linear visco-elastodynamics towards dispersion and attenuation of waves in relation to large-strain models

Various spatial-gradient extensions of standard viscoelastic rheologies of the Kelvin-Voigt, Maxwell's, and Jeffreys' types are analyzed in linear one-dimensional situations as far as the propagation of waves and their dispersion and attenuation. These gradient extensions are then presented in the large-strain variants where they are sometimes used rather for purely analytical reasons either in the Lagrangian or the Eulerian formulations without realizing this wave-propagation context

Nonlinear Anisotropic Viscoelasticity

In this paper we revisit the mathematical foundations of nonlinear viscoelasticity. We study the underlying geometry of viscoelastic deformations, and in particular, the intermediate configuration. Starting from the multiplicative decomposition of deformation gradient into elastic and viscous parts \mathbf{F}=\Fe\Fv\,, we point out that \Fv can be either a material tensor (\Fe is a two-point tensor) or a two-point tensor (\Fe is a spatial tensor). We show that based on physical grounds the second choice is unacceptable. It is assumed that the free energy density is the sum of an equilibrium and a non-equilibrium part. The symmetry transformations and their action on the total, elastic, and viscous deformation gradients are carefully discussed. Following a two-potential approach the governing equations of nonlinear viscoelasticity are derived using the Lagrange-d'Alembert principle. We discuss the constitutive and kinetic equations for compressible and incompressible isotropic, transversely isotropic, orthotropic, and monoclinic viscoelastic solids. We finally semi-analytically study creep and relaxation in three examples of universal deformations