A bending theory of thermoelastic diffusion plates based on Green-Naghdi theory

TL;DR

This paper develops a bending plate theory for thermoelastic diffusion materials based on Green-Naghdi theory, incorporating wave propagation, energy dissipation, and well-posedness analysis for type II and III models.

Contribution

It introduces a new bending theory for thermoelastic diffusion plates that accounts for finite wave speeds and energy dissipation, with rigorous mathematical analysis of solutions.

Findings

Type II model allows wave propagation at finite speed without energy dissipation.

Type III model includes energy dissipation and faster decay of thermodynamic measures.

Proves well-posedness and asymptotic behavior of solutions using semigroup theory.

Abstract

This article is concerned with bending plate theory for thermoelastic diffusion materials under Green-Naghdi theory. First, we present the basic equations which characterize the bending of thin thermoelastic diffusion plates for type II and III models. The theory allows for the effect of transverse shear deformation without any shear correction factor, and permits the propagation of waves at a finite speed without energy dissipation for type II model and with energy dissipation for type III model. By the semigroup theory of linear operators, we prove the well-posedness of the both models and the asymptotic behavior of the solutions of type III model. For unbounded plate of type III model, we prove that a measure associated with the thermodynamic process decays faster than an exponential of a polynomial of second degree. Finally, we investigate the impossibility of the localization in time of solutions. The main idea to prove this result is to show the uniqueness of solutions for the backward in-time problem.