Evolution Equations for Non-Simple Viscoelastic Solids

Abstract

Objectivity and compatibility with thermodynamics of evolution equations are examined in connection with the modelling of viscoelastic solids. The purpose of the paper is to show that the evolution equation for the stress is eventually obtained by means of a tensorial internal variable within the framework of the reference configuration. The non-simple character is realized by gradients of the internal variable. The thermodynamic analysis is developed by investigating the entropy inequality in the reference configuration and allowing for a non-zero extra-entropy flux. It follows that the evolution for the Cauchy stress tensor involves the Oldroyd derivative, irrespective of the form of the non-local terms.

Thermodynamic restrictions and wave features of a non-linear Maxwell model

“…This investigation might be developed by following the lines of [9] by setting aside linearized approximations and evaluating the consequences of objective time derivatives in the evolution equation. Meanwhile an Eulerian setting might provide a scheme for viscous fluids, with a power-law of the stress, of the form D ¼ að1þb9T9 2 Þ n T, or more generally of the form D ¼ gðTÞ [17].…”

“…It is worth pointing out that a thermodynamic framework for rate-type constitutive equations is developed in [4], the thermodynamic requirement being an appropriate criterion of maximum rate of dissipation. In this paper, instead, the thermodynamic analysis is based on the Clausius-Duhem inequality as the requirement of the second law and the rate-type equation is framed within the scheme of materials with internal variables [7][8][9]. The second fold is to examine discontinuity waves and to determine the effects of constitutive functions on propagation properties.…”

“…Other fluid models involve rate-type constitutive equations such as Oldroyd-B fluid, Rivlin-Ericksen fluid and Burgers fluid possibly with non-linear terms [11][12][13]. The investigation of such models, within a thermodynamic setting based on the Clausius-Duhem inequality, requires a fully nonlinear scheme and a Lagrangian description as is done in [9].…”

Hyperbolic diffusion with Christov–Morro theory

“…Thus, we believe there is strong motivation to develop and analyse a hyperbolic theory of convective overturning of a solute in a layer of a porous material. To do this we employ recent work of Christov [4,5] who advocate the use of a Lie derivative and Morro [23,24] who develops a thermodynamically consistent theory which is compatible with Christov's derivative.…”

Nonlinear diffusion equations in fluid mixtures

“…for any constituent α. The compatibility of wave motion with the requirements of the entropy inequality has been investigated for a number of dissipative models as, e.g., those in [10,16,11]; see also [25]. Here waves are investigated in the particular case of inviscid and non-conducting constituents.…”