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

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 nonlocalities. In both cases, the restrictions placed by an entropy inequality are solved, and an explicit solution for the constitutive equations is provided.