Derivation of the Variants of the Burgers Model Using a Thermodynamic Approach and Appealing to the Concept of Evolving Natural Configurations

Abstract

Viscoelastic rate-type fluid models involving the stress and frame-indifferent time derivatives of second order, like those in Burgers’ model, are used to describe the complicated response of fluid like materials that are endowed with a complex microstructure that allows them to possess two different relaxation mechanisms as well as other non-Newtonian characteristics. Such models are used in geomechanics, biomechanics, chemical engineering and material sciences. We show how to develop such rate-type fluid models that include the classical Burgers’ model as well as variants of Burgers’ model, using a thermodynamic approach based on constitutive assumptions for two scalar quantities (namely, how the material stores energy and how the energy is dissipated) and appealing to the concept of natural configuration associated with the placement of the body that evolves as the body deforms.

A Simple Construction of a Thermodynamically Consistent Mathematical Model for Non-Isothermal Flows of Dilute Compressible Polymeric Fluids

“…This system of governing equations is essentially the same as that reported by Öttinger and Grmela [18], albeit the model has been derived in a different manner. In fact our approach is similar to that used in the derivation of macroscopic viscoelastic rate-type models, see especially Rajagopal and Srinivasa [37], Málek et al [38], Hron et al [17], Málek et al [39], Málek et al [40], and Dostalík et al [41]. For a coupling between microscopic and macroscopic models, see also Souček et al [42].…”

Nonlinear Models of Thermo-Viscoelastic Materials

“…For definiteness, we now show that a class of models for heat conduction, of the Maxwell-Cattaneo type [28], follows from (35). Let ψ depend on q R via ξ = |q R | n , n ≥ 2.…”

“…While sometimes the model is motivated by rheological analogs [34] and framed within a scheme with intermediate reference configurations [35], here, the model is investigated within the thermodynamic approach developed so far.…”

Nonlocal extension of causal thermodynamics of the isotropic cosmic fluid

“…The analog of the Hooke's term, which in classical viscoelasticity is of the form π = Eǫ, does not appear in the equation (7), however, it would be interesting to study the analog of the classical model, associated with the constitutive equation of the type τ π+µπ = ν ǫ+Eǫ, which is extensively used in the transient and steady-state rheologies, as well as in the theory of viscoelastic media and materials (see, e.g., [18]). This equation is associated with the Burgers model (see, e.g., [19,20]), for which the constitutive equation is of the second order in time derivatives for the stress π…”