Construction of a class of integral models for heat flow in materials with memory

On the thermodynamics of viscoelastic materials of single-integral type

“…'cf., e.g., [9] and the references cited therein. 2Cf., e.g., [1] and the references cited therein. 3 In this regard we are indebted to Coleman [2] for his general thermodynamical theory of materials with fading memory.…”

“…If we define a new variable co(t) through u> = 6~le -r], (1)(2)(3)(4)(5)(6)(7) then, by (1.3) and (1.4), co is given by a constitutive equation of single-integral form:…”

A note on Onsager’s relations

“…The well-known modification by Cattaneo [2] of Fourier's law of heat conduction is included as a special case in (1.1), but Fourier's law itself is not included. Thermodynamics of rigid heat conductors with the heat flux dependent on the summed histories of the gradient of temperature was examined by Gurtin and Pipkin [9], Coleman, Fabrizio and Owen [5], and Brandon and Hrusa [1]. I also refer to these papers for further reference.…”

“…In this paper I extend first some of the results of [1,9] to deformable bodies, and then I proceed to consider the approximations of the constitutive equations (1.1) appropriate for slow processes with small gradients of temperature. I shall show that the approximate constitutive equations are (1) the approximate constitutive equations obey the Clausius-Duhem inequality with a positive production of entropy given by a quadratic form in the nonequilibrium parameters F, g; (2) the approximate constitutive equations for the energy, entropy, and Helmholtz free energy obey the principle of local equilibrium (i.e., are independent of F, g); (3) the matrix of kinetic coefficients occurring in the approximate constitutive equations for the stress and the heat flux vector obeys Onsager's type of symmetry.…”

“…The particular form given above is first given by Brandon and Hrusa [1] for one-dimensional rigid heat conductors. Formally, relations (4.10)-(4.13) can be obtained from the abstract results of Gurtin and Hrusa [8] by identifying their paths with triplets (F, 6, G), where G is the indefinite integral of g; the values of their single-integral law with the values of the triplets (o, -r], -£/0), with the integrand written in the form m(F(t), F(t -5), 6(t), G(0 -G(t -s), s), and with their thermodynamic potential identified with y .…”