Continua with non-local constitutive laws: exploitation of entropy inequality

TL;DR

This paper develops a framework for non-local constitutive laws in continuum mechanics using entropy inequalities, providing explicit solutions for complex fluid models with gradient-dependent behaviors.

Contribution

It introduces an extended Liu procedure to ensure entropy consistency in non-local constitutive models, including fluids with internal variables and Korteweg fluids.

Findings

Explicit solutions for non-local constitutive equations

Entropy inequality constraints are effectively solved

Framework applicable to complex fluid models

Abstract

In this paper, we consider a system of balance laws sufficiently general to contain the equations describing the thermomechanics of a one-dimensional continuum; this system involves some constitutive functions depending on the elements of the so called state space assumed to contain the spatial gradients of some of the unknown fields. The compatibility of the constitutive equations with an entropy-like principle is considered via an extended Liu procedure by using as constraints both the balance equations and some of their gradient extensions. This procedure is then applied to the equations of a fluid whose description involves an internal variable and first order non-local constitutive relations, and to a Korteweg fluid with second order non-localities. In both cases, the restrictions placed by an entropy inequality are solved, and an explicit solution for the constitutive equations is provided.