Nonlinear and nonlocal models of heat conduction in continuum thermodynamics

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 a variant of the 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. Many 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). Owing to the (weakly) nonlocal formulation of the second law, weakly nonlocal models based on the heat-flux vector and its gradients are obtained within this (classical) thermodynamic framework. In particular, the nonlocal Guyer-Krumhansl model and some nonlinear generalizations devised by Cimmelli and Sellitto are obtained. Finally, we propose a new model where the heat flux dependence on temperature gradients is allowed up to second-order.