On a qualitative effect arising from coupling in dynamic linear thermoelasticity
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.\}
scientific article; zbMATH DE number 3970161