Decay of waves in strain gradient porous elasticity with Moore–Gibson–Thompson dissipation

Abstract

We study a one-dimensional problem arising in strain gradient porous-elasticity. Three different Moore–Gibson–Thompson dissipation mechanisms are considered: viscosity and hyperviscosity on the displacements, and weak viscoporosity. The existence and uniqueness of solutions are proved. The energy decay is also shown, being polynomial for the two first situations, unless a particular choice of the constitutive parameters is made in the hyperviscosity case. Finally, for the weak viscoporosity, only the slow decay can be expected. This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)’.

A Moore‐Gibson‐Thompson heat conduction equation for non centrosymmetric rigid solids

“…In this work, we want to focus on the theory of heat conduction called Moore-Gibson-Thompson. It is worth recalling that this theory has received much attention in the last four years [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] since many authors have investigated the qualitative and quantitative properties of the solutions to this equation. If we consider the type III heat conduction theory proposed by Green-Naghdi [6], we can see that it is based on the constitutive equation in the case of centrosymmetric materials:…”

“…In this work, we want to focus on the theory of heat conduction called Moore‐Gibson‐Thompson. It is worth recalling that this theory has received much attention in the last four years [10–24] since many authors have investigated the qualitative and quantitative properties of the solutions to this equation. If we consider the type III heat conduction theory proposed by Green‐Naghdi [6], we can see that it is based on the constitutive equation in the case of centrosymmetric materials: $$$$\begin{equation*} q_i=k_{ij} \alpha _{,j}+ \kappa _{ij}^* \theta _{,j}, \end{equation*}$$$$where $$q_i$$ is the heat flux vector, θ is the temperature and α is the thermal displacement.…”