A second gradient theory of thermoviscoelasticity

Abstract

This paper is concerned with a rate-type theory of thermoviscoelasticity in which the second gradient of the displacement and the second temperature gradient are added to the classical set of independent constitutive variables. Viscoelasticity and related phenomena are of great importance in the study of biological materials. An adequate modeling of rubber-like materials and of biological soft tissues requires the use of the theory of viscoelasticity. Introduction of the concept of thermal displacement and the theory of multipolar continua allows us to show that Green-Naghdi thermomechanics can be used to derive a second gradient theory. The basic equations of the theory are established and the boundary conditions associated to nonsimple materials are investigated. The stress tensor and hyperstress tensor are shown to depend on the first and second temperature gradients. For rigid heat conductors we find that the temperature satisfies a fourth order equation. The boundary-initial-value problems are formulated. A uniqueness result in the dynamic theory of thermoviscoelastic materials is presented. We establish an existence result and prove the analyticity of the solutions. As a consequence, the exponential decay of the solutions and their impossibility of localization are obtained.

Keywords:

• A fourth-order heat equation
• boundary conditions
• dissipation inequality
• existence and uniqueness
• stress tensors dependent on temperature gradients
• thermoviscoelasticity

Mathematics Subject Classification 2010::

• 35L56
• 74A15
• 74A20
• 74A30
• 74D05
• 74H25
• 74H45

Disclosure statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Funding

This paper is part of the project “Qualitative and numerical analyses of problems in Thermomechanics” [Ref. PID2023-146359NB-I00], which is currently under evaluation in the Spanish Ministry of Science, Innovation and Universities.