Thermodynamically consistent gradient elasticity with an internal variable

TL;DR

This paper develops a thermodynamically consistent gradient elasticity theory incorporating internal variables, ensuring compatibility with thermodynamics and accounting for memory effects and dissipation in elastic materials.

Contribution

It introduces a new gradient elasticity framework using nonequilibrium thermodynamics with internal variables, ensuring thermodynamic consistency and including gradient effects in stress analysis.

Findings

Gradient terms contribute to stress even without dissipation.

The theory incorporates memory effects and dissipation in small strain elasticity.

Constitutive relations are derived to be thermodynamically compatible.

Abstract

The role of thermodynamics in continuum mechanics and the derivation of the proper constitutive relations is a discussed subject of Rational Mechanics. The classical literature did not use the accumulated knowledge of thermostatics and was very critical with the heuristic methods of irreversible thermodynamics. In this paper, a small strain gradient elasticity theory is constructed with memory effects and dissipation. The method is nonequilibrium thermodynamics with internal variables; therefore, the constitutive relations are compatible with thermodynamics by construction. Thermostatic Gibbs relation is introduced for elastic bodies with a single tensorial internal variable. The thermodynamic potentials are first-order weakly nonlocal, and the entropy production is calculated. Then the constitutive functions and the evolution equation of the internal variable is constructed. The second law analysis has shown a contribution of gradient terms to the stress, also without dissipation.