Exponential stability in type III thermoelasticity with microtemperatures

Abstract

In this paper we consider the type III thermoelastic theory with microtemperatures. We study the time decay of the solutions and we prove that, under suitable conditions for the constitutive tensors, the solutions decay exponentially. This fact is in somehow shocking because it diers from the behavior of the solutions in the classical model of thermoelasticity with microtemperatures.

Thermoelasticity of Moore–Gibson–Thompson type with history dependence in the temperature

“…Type III theory attracted a lot of interest, witnessed by a flurry of publications appeared on the argument (e.g. [5,15,24,26,27,30,32,33,34,35,36,42,43,45,49]). In the nontrivial case when both k and k * are positive, and we substitute q i into (1.2), we obtain a generalization of the Fourier classical heat equation, namely,…”

Exponential decay in one‐dimensional Type II/III thermoelasticity with two porosities

“…It is important to analyze in detail polynomial q 4 (n), which appears in the denominator: 12 ). Notice that q 4 (n) has no real roots because its coefficients are all strictly positive.…”

“…There is in the literature a huge quantity of publications analyzing this FIGURE 1 A fairy cake and a piece of travertine theory. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] It has been applying to geological materials such as rocks or soils as well as to manufactured materials such as ceramics or pressed powders.…”

Analysis of a Poro-Thermo-Viscoelastic Model of Type III

“…It is based on the work by Green and Nagdhi [23], where the thermal displacement was included as a new independent variable into the model (even of the elastic displacement), and it is usually called type III thermal law. Since then, many authors worked on this theory (see, e.g., [4,[24][25][26][27][28][29]). Here, our aim is to extend the results provided in [30], where a general thermoelastic law was studied from both mathematical (existence and uniqueness) and numerical (stability, a priori error estimates) points of view.…”

“…where u 0 , ϕ 0 and ψ 0 represent initial conditions for the displacement, the volume fraction and the thermal displacement. Therefore, the thermomechanical problem of a centrosymmetric, isotropic and homogeneous thermo-viscoelastic body within the type III thermal theory is written as follows (see [26,27,32]).…”