A thermodynamic approach to non-linear viscoelasticity (original) (raw)

Linear viscoelasticity and irreversible thermodynamics

Rheologica Acta, 1986

The underlying thermodynamic aspects of linear viscoelasticity are discussed. In particular, from the Extended Irreversible Thermodynamics theory we systematically derive the Maxwell model exhibiting its compatibility with thermodynamics and assessing its conditions of validity. We also calculate the equilibrium transverse velocity auto-correlation function and the frequency dependent shear viscosity. Nonlinear generalizations of our model are suggested and the possible role of extended thermodynamics in selecting constitutive equations is also discussed. ù

A Thermodynamic Theory of Solid Viscoelasticity. Part II:; Nonlinear Thermo-viscoelasticity

2002

This paper, second in the series of three papers, develops a general, nonlinear, non-isothermal, compressible theory for finite rubber viscoelasticity and specifies it in a form2 convenient for solving problems important to the rubber, tire, automobile, and air-space industries, among others. Based on the quasi-linear approach of non-equilibrium thermodynamics, a general nonlinear theory of differential type has been developed for arbitrary non-isothermal deformations of viscoelastic solids. In this theory, the constitutive equations were presented as the sum of a rubber elastic (equilibrium) and a liquid type viscoelastic (non-equilibrium) terms. These equations have then been simplified using several modeling and simplicity arguments.

Nonlinear Models of Thermo-Viscoelastic Materials

Materials, 2021

The paper develops a general scheme for viscoelastic materials, where the constitutive properties are described by means of measures of strain, stress, heat flux, and their time derivatives. The constitutive functions are required to be consistent with the second law of thermodynamics. Indeed, a new view is associated with the second law: the non-negative expression of the entropy production is set equal to a further constitutive function. The introduction of the entropy production as a constitutive function allows for a much wider range of models. Within this range, a scheme to obtain nonlinear models of thermo-viscoelastic materials subject to large deformations is established. Notably, the Kelvin–Voigt, Maxwell, Burgers, and Oldroyd-B viscoelastic models, along with the Maxwell–Cattaneo heat conduction, are obtained as special cases. The scheme allows also for modelling the visco-plastic materials, such as the Prandtl–Reuss work-hardening function and the Bingham–Norton fluid.

Irreversible thermodynamics and variational principles with applications to viscoelasticity

1962

A unified theory of the thermo-mechanical behavior of viscoelastic media is developed from studying the thermodynamics of irreversible processes, and includes discussions of the general equations of motion, crack propagation, variational principles, and approximate methods of stress analysis. The equations of motion in terms of generalized coordinates and forces are derived for systems in the neighborhood of a stable equilibrium state. They represent a modification of Biot's theory in that they contain explicit temperature dependence, and a thermodynamically consistent inclusion of the time-temperature superposition principle for treating media with temperature-dependent viscosity coefficients. The stress-strain-temperature and energy equations for viscoelastic solids follow immediately from the general equations and, along with equilibrium and strain-displacement relations, they form a complete set for the description of the thermomechanical behavior of media with temperature-d...

A Thermodynamic Theory Of Solid Viscoelasticity. Part 1: Linear Viscoelasticity

2002

The present series of three consecutive papers develops a general theory for linear and finite solid viscoelasticity. Because the most important object for nonlinear studies are rubber-like materials, the general approach is specified in a form convenient for solving problems important for many industries that involve rubber-like materials. General linear and nonlinear theories for non-isothermal deformations of viscoelastic solids are developed based on the quasi-linear approach of non-equilibrium thermodynamics. In this, the first paper of the series, we analyze non-isothermal linear viscoelasticity, which is applicable in a range of small strains not only to all synthetic polymers and bio-polymers but also to some non-polymeric materials. Although the linear case seems to be well developed, there still are some reasons to implement a thermodynamic derivation of constitutive equations for solid-like, non-isothermal, linear viscoelasticity. The most important is the thermodynamic m...

A Physicists' View on Constitutive Equations

Nonlinear hydrodynamic equations for viscoelastic media are discussed. We start from the recently derived fully hydrodynamic nonlinear description of permanent elasticity that utilizes the (Eulerian) strain tensor. The reversible quadratic nonlinearities in the strain tensor dynamics are of the 'lower convected' type, unambiguously. Replacing the (often neglected) strain diffusion by a relaxation of the strain as a minimal ingredient, a generalized hydrodynamic description of viscoelasticity is obtained. This can be used to get a nonlinear dynamic equation for the stress tensor (sometimes called constitutive equation) in terms of a power series in the variables. The form of this equation and in particular the form of the nonlinear convective term is not universal but depends on various material parameters. A comparison with existing phenomenological models is given. In particular we discuss how these ad-hoc models fit into the hydrodynamic description and where the various non-Newtonian contributions are coming from.

On rate-type viscoelasticity and the second law of thermodynamics

International Journal of Non-Linear Mechanics, 1984

Wededuceanenergyidentitywhich must besatisfied bythesmoothsolutionsofthesystem ofcquations~overningthcdynamicsofabodywithquasilincarra~e-typeconstitutivcequation. Wegive conditions when a unique energy function exists for rate-type viscoelaslicity. In the semilinear case we giveIheconditionswhenaunique,positiveandconvexenergyfunctionexistsandweobtainestimatesin energy [or the smooth solutions of jnitiai-boundary value problems. A viscoefastic approach to nonlinearelasticityisdiscussed.Finally,anexampleshowsthat thcsecondlawofthermodynamicsdoesnot imply stability.

Nonlinear Constitutive Laws in Viscoelasticity

Mathematics and Mechanics of Solids, 2007

The theory of viscoelasticity appears to play a central role in the description of materials which exhibit time dependent stress—strain behavior. Various materials like polymers, some soft biological tissues, and various foods have been already successfully modeled as nonlinear viscoelastic materials.The literature in these application areas is replete with different, seemingly unconnected nonlinear viscoelastic models. The aim of the present paper is to review the classical nonlinear viscoelastic models and provide a unifying framework using the continuum mechanics formalism.

On the thermodynamic consistency of Quasi-linear viscoelastic models for soft solids

Mechanics Research Communications, 2021

Originating in the field of biomechanics, Fung's model of quasi-linear viscoelasticity (QLV) is one of the most popular constitutive theories employed to compute the time-dependent relationship between stress and deformation in soft solids. It is one of the simplest models of nonlinear viscoelasticity, based on a time-domain integral formulation. In the present study, we consider the QLV model incorporating a single scalar relaxation function. We provide natural internal variables of state, as well as a consistent expression of the free energy to illustrate the thermodynamic consistency of this version of the QLV model. The thermodynamic formulation highlights striking similarities between QLV and the internal-variable models introduced by Holzapfel and Simo. Finally, the dissipative features of compressible QLV materials are illustrated in simple tension.

Stability of Non-Linear Constitutive Formulations for Viscoelastic Fluids

SpringerBriefs in Applied Sciences and Technology, 2014

Controversy about the frame indifference principle, the concept of non-local continuum field theories, local constitutive formulations, differential constitutive equations of linear viscoelasticity, Oldroyd, K-BKZ, FENE (Finitely Extensible Non-linear Elastic) class of constitutive equations, Smoluchowski and Fokker-Planck diffusion equations, constant stretch history flows, fading memory and nested integral representations of the stress, order fluids of the integral and differential type, constitutive formulations consistent with thermodynamics, maximization of the rate of dissipation in formulating thermodynamics compatible constitutive structures, Burgers equation which is finding a gradually widening niche in applications, minimum free energy and maximum recoverable work in the case of linearized viscoelastic constitutive structures, implicit constitutive theories, which define the stress field when the viscosity depends for instance on the constitutively undetermined pressure field, and which have found new focus in applications such as elastohydrodynamic lubrication are discussed and progress made is summarized. Canonical forms of Maxwell-like constitutive differential equations and single integral constitutive equations are presented and commented on together with the Hadamard and dissipative type of instabilities they may be subject to.