Minimum free energy in the frequency domain for a heat conductor with memory (Q612881)

This article considers the heat conduction problem within the framework of materials with memory. It investigates the problem of finding an explicit form for the minimum free energy related to a particular state of a linear rigid heat conductor based on general constitutive equations with memory effects for the heat flux satisfying a general notion of the dynamical system. These constitutive equations for the internal energy and the heat flux are expressed as linear functionals of the histories of temperature and its gradient, respectively, together with the present value of the latter quantity. Using the standard method for determining the minimum free energy associated with a given state of a material, two integral equations of the second kind are obtained. The thermodynamic properties, derived for the kernels related to the expressions for the internal energy and the heat flux, together with some theorems on factorization allow solving the two integral equations of second kind in the frequency domain. An explicit expression for the minimum free energy is thus derived. Another equivalent expression is also obtained for the minimum free energy and is used to derive explicit formulae for the case of a discrete spectrum model material response.