Modeling of viscoelasticity through relations between strain, stress, and their derivatives

What is it about?

The paper is devoted to the modeling of nonlinear viscoelastic materials. The constitutive equations are considered in differential form via relations between strain, stress, and their derivatives in the Lagrangian description. The thermodynamic consistency is established by using the Clausius--Duhem inequality through a procedure that involves two uncommon features. Firstly, the entropy production is regarded as a positive-valued constitutive function per se. This view implies that the inequality is in fact an equation. Secondly, this statement of the second law is investigated by using an algebraic representation formula, thus arriving at quite general results for rate terms that are usually overlooked in thermodynamic analyses. Starting from strain-rate or stress-rate equations, the corresponding finite equations are derived. It then emerges that a greater generality of the constitutive equations of the classical models, such as those of Boltzmann and Maxwell, are obtained as special cases.

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Why is it important?

The modeling of viscoelastic solids is often developed through memory functionals for the stress in terms of the strain history. Yet, memory functionals make it more involved than any account of nonlinearity and affect compatibility with thermodynamics. That is why, alternatively, this paper provides a general account of thermodynamically consistent modeling of viscoelasticity through rate equations as relations between strain, stress, and their derivatives. Two features, characteristic of this paper, are unusual in the literature. Firstly, the entropy production rate is regarded as provided by a constitutive function to be determined or chosen. Secondly, the use of a representation formula enables vector (or tensor) unknowns to comprise an arbitrary term whose choice leads to qualitatively new constitutive equations.

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