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L. Jentsch; D. Natroshvili : Thermoelastic Oscillations of Anisotropic Bodies (Sommerfeld 96 - Workshop)

L. Jentsch; D. Natroshvili : Thermoelastic Oscillations of Anisotropic Bodies (Sommerfeld 96 - Workshop)

Author(s) :
L. Jentsch; D. Natroshvili
Title :
Thermoelastic Oscillations of Anisotropic Bodies (Sommerfeld 96 - Workshop)
Electronic source :
[gzipped dvi-file] 29 kB
[gzipped ps-file] 68 kB
Preprint series
Technische Universität Chemnitz-Zwickau, Fakultät für Mathematik (Germany). Preprint 96-13, 1996
Published in :
Erhard Meister (Ed.): Modern Mathematical Methods in Diffraction Theory and its Applications in Engineering, Proc. of the Sommerfeld '96 Workshop, Freudenstadt, September 30 - October 4, 1996; Methoden und Verfahren der mathematischen Physik, Band 42. Frankfurt am Main: Peter Lang. p. 68-87,
Appeared in :
Erhard Meister (Ed.): Modern Mathematical Methods in Diffraction Theory and its Applications in Engineering, Proc. of the Sommerfeld '96 Workshop, Freudenstadt, September 30 - October 4, 1996; Methoden und Verfahren der mathematischen Physik, Band 42. Frankfurt am Main: Peter Lang. p. 68-87
Mathematics Subject Classification :
73D30 [ Linear vibrations of solids ]
73B40 [ Anisotropic materials ]
Abstract :
Three-dimensional basic problems of statics, pseudo-oscillations, general dynamics and steady state oscillations of the thermoelasticity of isotropic bodies have been completely investigated by many authors. In particular, exterior steady state oscillation problems have been studied on the basis of Sommerfeld-Kupradze radiation conditions in the thermoelasticity, and the uniqueness theorems were proved with the help of the well-known Rellich's lemma, since the components of the displacement vector and the temperature in the isotropic case can be represented as a sum of metaharmonic functions .

Unfortunately, the methods of investigation of thermoelastic steady state oscillation problems developed for the isotropic case are not applicable in the case of general anisotropy. This is stipulated by a very complicated form of the corresponding characteristic equation which plays a significant role in the study of far field behaviour of solutions to the oscillation equa- tions.

We note that the basic and crack type boundary value problems (BVPs) for the pseudo-oscillation equations of the thermoelasticity theory in the anisotropic case are considered in [3,14].

To the best of the authors' knowledge the problems of thermoelastic steady oscillations for anisotropic bodies have not been treated in the scientific literature.

In the present paper we will consider a wide class of basic and mixed type BVPs for the equations of thermoelastic steady state oscillations. We will formulate thermoelastic radiation conditions for an anisotropic medium (the generalized Sommerfeld-Kupradze type radiation conditions) and prove the uniqueness theorems in corresponding spaces. To derive these conditions we have essentially applied results of Vainberg.

Further, using the potential method and the theory of pseudodifferential equations on manifolds we will prove existence theorems in various functional spaces and establish the smoothness properties of solutions.

Keywords :
thermoelasticity, oscillations, anisotropic bodies, potential methods
Language :
english
Publication time :
10/1996