Thermopiezoelectricity without energy dissipation

Abstract

In this paper, we use the Green-Naghdi theory of thermomechanics of continua to derive a linear theory of thermopiezoelectricity of a body with inner structure. This theory permits propagation of thermal waves at finite speed. We establish a uniqueness result and a continuous dependence of the solutions upon initial data and body supplies. Some applications (the problem of a concentrated heat source, the problem of an impulsive body force, the deformation of a thick-walled spherical shell) are presented.

Asymptotic Partition of Energy in Micromorphic Thermopiezoelectricity

“…Throughout this article we assume that 0 , I, , a are positive and the quadratic forms 2 1 ij e ij = ii jj + + 2 + e ii e jj + 2 + ii e jj + ij ji + 4 + ij e ji + 2 + 2 + e ij e ji + + ij ij (8) 2 2 ijk = 2 1 iij jkk + 2 2 iij kjk + 3 iij kkj + 4 ijj ikk + 2 5 ijj kik + 6 iji kjk + 7 ijk ijk + 8 ijk jki + ijk kij + 9 ijk ikj + 10 ijk jik + 11 ijk kji (9) are positive definite in terms of ij , e ij and ijk , respectively. This means that there exist the positive constants m , m , M , M such that…”

“…We recall that the linear theory of microstretch piezoelectricity was studied in [5,6], while in [7] some important theorems have been proven for microstretch thermopiezoelectricity. Moreover, the basic equations which govern the bending of microstretch piezoelectric plates have been treated in [8], while a linear theory of microstretch thermopiezoelectricity without energy dissipation has been presented in [9].…”

On the backward in time problem for the thermoelasticity with two temperatures

“…Several thermoelastic theories mainly based on the heat conduction theory of Cattaneo and Maxwell [29] (see Hetnarski and Ignaczack [14,15], the books by Ignaczack and Ostoja-Starzewski [21], Straughan [46] and the references cited therein) have been proposed since the decade of the 1960's. Recently, Green and Naghdi [11,12,13] postulated other theories that are currently the subject of deep research (see Iesan [18,19]; Iesan and Quintanilla [20]; Lazzari and Nibbi [22]; Leseduarte et al [23,24]; Liu and Quintanilla [27]; Liu and Lin [26], Messaoudi and Soufyane [30]; Puri and Jordan [32]; Qin et al [33]; Quintanilla [34,35,36,38]; Quintanilla and Racke [41]; Quintanilla and Straughan [43,44]; Yang and Wang [49], among others). These theories are based on well accepted thermomechanical axioms and therefore they are consistent from the mathematical standpoint.…”

Hölder stability in Type III thermoelastodynamics

“…Type III is the more general theory and it contains the other two as limiting cases. We can recall several papers (see Iesan [8,9]; Iesan and Quintanilla [10]; Lazzari and Nibbi [13]; Leseduarte et al [14,15]; Liu and Quintanilla [16,17]; Liu and Lin [18]; Messaoudi and Soufyane [19]; Puri and Jordan [20]; Qin et al [21]; Quintanilla [22][23][24][25][26][27]; Quintanilla and Racke [28]; Quintanilla and Straughan [29][30][31]; Yang and Wang [34], among others), where existence, uniqueness, continuous dependence, spatial and time behavior have been studied.…”