  The aim of this paper is to develop a general constitutive scheme within
continuum thermodynamics to describe the behavior of heat flow in deformable
media. Starting from a classical thermodynamic approach, the rate-type
constitutive equations are defined in the material (Lagrangian) description
where the standard time derivative satisfies the principle of objectivity. All
constitutive functions are required to depend on a common set of independent
variables and to be consistent with thermodynamics. The statement of the Second
Law is formulated in a general nonlocal form, where the entropy production rate
is prescribed by a non-negative constitutive function and the extra entropy
flux obeys a no-flow boundary condition. The thermodynamic response is then
developed based on Coleman-Noll procedure. In the local formulation, the free
energy potential and the rate of entropy production function are assumed to
depend on temperature, temperature gradient and heat-flux vector along with
their time derivatives. This approach results in rate-type constitutive
equations for the heat-flux vector that are intrinsically consistent with the
Second Law and easily amenable to analysis. A huge class of linear and
nonlinear models of the rate type are recovered (e.g., Cattaneo-Maxwell's,
Jeffreys-like, Green-Naghdi's, Quintanilla's and Burgers-like heat conductors).
In the (weakly) nonlocal formulation of the second law, both the entropy
production rate and an entropy extra-flux vector are assumed to depend on
temperature, temperature gradient and heat-flux vector along with their spatial
gradients and time derivatives. Within this (classical) thermodynamic framework
the nonlocal Guyer-Krumhansl model and some nonlinear generalizations devised
by Cimmelli and Sellitto are obtained
