Some Remarks on Materials with Memory: Heat Conduction and Viscoelasticity

Abstract

Materials with memory are here considered. The introduction of the dependence on time not only via the present, but also, via the past time represents a way, alternative to the introduction of possible non linearities, when the physical problem under investigation cannot be suitably described by any linear model. Specifically, the two different models of a rigid heat conductor, on one side, and of a viscoelastic body, on the other one, are analyzed. In them both, to evaluate the quantities of physical interest a key role is played by the past history of the material and, accordingly, the behaviour of such materials is characterized by suitable constitutive equations where Volterra type kernels appear. Specifically, in the heat conduction problem, the heat flux is related to the history of the temperature-gradient while, in isothermal viscoelasticity, the stress tensor is related to the strain history. Then, the notion of equivalence is considered to single out and associate together all those different thermal histories, or, in turn, strain histories, which produce the same work. The corresponding explicit expressions of the minimum free energy are compared.

Integro-differential models for percutaneous drug absorption

“…Moreover, there is an overestimation of the travelling waves velocity. The drawback observed in equation (3) is also present in the Fisher equation [4][5][6][7][8][9][10] and in the classical heat equation [11][12][13]. In order to overcome these unphysical properties, in the context of the Fisher equation, the flux defined by Fick's law is replaced by a flux with a memory term [4][5][6].…”

Singular kernel problems in materials with memory

“…Indeed, throughout the whole article, the two different models are compared in connection to the singular kernel problems under investigation. In particular, analogies between the two models of materials with memory both under the physical as well as under the functional spaces point of view are shown in [66], here the comparison is extended to singular kernel problems which were not considered previously. In particular, the analogy remains valid when the regularity requirement of the kernel is relaxed to consider also the case when, in turn, the heat flux relaxation function and the relaxation modulus are unbounded at the origin provided they are integrable (L 1 ).…”

“…Notably, as pointed out in [66], the analogous functional spaces, wherein the solutions of the evolution problems are looked for, are obtained in the cases of the two different models. Indeed, throughout the whole article, the two different models are compared in connection to the singular kernel problems under investigation.…”

Existence, uniqueness and exponential decay: An evolution problem in heat conduction with memory

“…Notably, there are interesting analogies, as far as the function spaces which contain physically meaningful states of the system are concerned, between the case of a rigid heat conductor with memory and an isothermal viscoelastic body, according to results in [4].…”

“…where k c (ω) denotes the Fourier cosine transform, see for instance [1], of k(|τ −s|), which, by definition, is an even function of its real nonnegative argument, 4 and where…”