On a qualitative effect arising from coupling in dynamic linear thermoelasticity
type: Item
instance of: scholarly article
title: On a qualitative effect arising from coupling in dynamic linear thermoelasticity
zbMATH Open document ID: 0601.73005
DOI: 10.1007/BF00251551
author: W. A. Day
published in: Archive for Rational Mechanics and Analysis
publication date: 1983-01-01T00:00:00Z
review text: Let \((u,T),\) \(u=u(x,t)=\) displacement, \(T=T(x,t)=\) temperature, \(x\in [0,1]\), \(t\in [0,\infty)\), denote a one-dimensional solution of linear dynamic coupled thermoelasticity with zero relaxation times for a homogeneous isotropic layer with clamped boundaries, nonhomogeneous boundary temperatures \(T(0,t)=f(t)\), \(T(1,t)=g(t)\) and arbitrary initial conditions. In the first paper [see the entry above (Zbl 0601.73004)] the author shows that if \(f(t)\) and \(g(t)\) are infinitely differentiable for \(t\in [0,\infty)\), and their derivatives of order \(n\geq 2\) are not too large in comparison to their derivatives of order 1 for large times, then for a sufficiently small thermoelastic coupling parameter \(\epsilon\) and for large times the following estimates hold: \(\max_{x\in [0,1]}T=(f+g)/2+| f-g| /2+o(1)\), \(\min_{x\in [0,1]}T=(f+g)/2- | f-g| /2+o(1)\). The proof is based on a large time series expansion of the solution into derivatives of \(f(t)\) and \(g(t)\) whose coefficients are generated from two sequences of polynomials defined by two coupled recurrence relations for \(x\in [0,1].\) In the second paper, a particular choice of the boundary temperatures: \(f(t)=g(t)=\exp(-st)\), \(s>0\), in the problem discussed in the first paper is made to obtain a closed form separable solution \((u,T)\) proportional to \(\exp(-st)\) and to prove that the solution has monotonic properties with respect to the thermoelastic coupling parameter \(\epsilon\). More precisely, it is proved that if the parameter s and an inertia parameter are suitably restricted, then the temperature T is an increasing function of \(\epsilon\) for each \(x\in (0,1)\) and \(t\in (-\infty,+\infty)\), while \(| u|\) is an increasing function of \(\epsilon\) for each x in the open intervals \((0,1/2)\), \((1/2,1)\) and \(t\in (-\infty,+\infty)\). \{Reviewer's remark: Note that the numbers (34) and (35) identifying corresponding formulas are missing.\}
Mathematics Subject Classification ID: 74F05
Mathematics Subject Classification ID: 74A15
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 3970161
zbMATH Keywords: separable solution
zbMATH Keywords: linear dynamic coupled thermoelasticity
zbMATH Keywords: zero relaxation times
zbMATH Keywords: homogeneous isotropic layer
zbMATH Keywords: clamped boundaries
zbMATH Keywords: nonhomogeneous boundary temperatures
zbMATH Keywords: closed form separable solution
zbMATH Keywords: monotonic properties
zbMATH Keywords: thermoelastic coupling parameter
MaRDI profile type: Publication
cites work: Q3273027
cites work: ASYMPTOTIC BEHAVIOUR IN DYNAMIC LINEAR THERMOELASTICITY: AN APPLICATION OF DIFFERENTIAL ITERATION
cites work: A property of the heat equation which extends to the thermoelastic equations
cites work: Further remarks on a property of the equations of dynamic thermoelasticity
cites work: Q3232201
full work available at URL: https://doi.org/10.1007/bf00251551
OpenAlex ID: W2018545437
description: scientific article; zbMATH DE number 3970161