  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.
