Department of Mathematics
Library of the School of Science

2023-06-13

2023

Zoi Stefania-Maria

Αθανασιάδης Χριστόδουλος, Ομότιμος Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ.
Κόττα-Αθανασιάδου Ευαγγελία (επιβλέπουσα), Αναπληρώτρια Καθηγήτρία, Τμήμα Μαθηματικών, ΕΚΠΑ.
Μπαρμπάτης Γεράσιμος, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ.
Σεβρόγλου Βασίλειος, Καθηγητής, Τμήμα Στατιστικής Και Aσφαλιστικής Eπιστήμης, Παν. Πειραιά.
Σμυρνέλης Παναγιώτης, Επίκουρος Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ.
Χαλικιάς Γιώργος, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ.
Χατζηνικολάου Μαρία, Καθηγήτρια, Σχολή Θετικών Επιστημών και Τεχνολογίας, ΕΑΠ.

Mathematical Theory of Thermoelasticity

English

Mathematical Theory of Thermoelasticity

In the present doctoral dissertation the mathematical theory of thermoelasticity and more precisely the scattering theory of thermoelastic waves is studied. Thermoelastic scattering problems for time-harmonic fields, for the cases of an impenetrable or a penetrable scattering object as well as for the case of a multi-layered scattering body with an impenetrable or a penetrable core, are formulated and studied. A four-dimensional form of the thermoelastic fields whose first three components correspond to the displacement fields and whose fourth components correspond to the temperature fields, is used to obtain the aforementioned scattering problem formulations in a unified form. Usage of the conditions on the scatterer's surface and usage of the conditions on the scatterer's layers surfaces respectively, lead to the derivation of alternative integral representations of the thermoelastic scattered field as well as to the construction of alternative expressions of the thermoelastic far-field patterns in which the scatterer's interior thermoelastic physical parameters and fields are incorporated. Uniqueness of solution for the thermoelastic scattering problems under study is proved via the usage of a Helmholtz's decomposition, Kupradze's thermoelastic radiation conditions and a Rellich's type Lemma. Existence of solution is also proved via the usage of single- and double- layer thermoelastic potentials and the Riesz-Fredholm theory. Next, scattering problems of time-harmonic elastic waves for a penetrable or a multi-layered thermoelastic scatterer with a penetrable core are formulated. Via the usage of the transmission conditions, alternative integral representations of the elastic scattered field as well as alternative expressions of the elastic far-field patters are constructed, containing the interior thermoelastic fields and the scatterer's physical parameters. Moreover, scattering relations corresponding to the reciprocity theorem, the general scattering theorem and the optical theorem are presented and proved. Finally, an inverse scattering problem of thermoelastic waves by a rigid thermoelastic scattering object that is apriori known to be of ellipsoidal shape and aims to the recovery of its unknown size and orientation is treated. A near-field method for solving the aforementioned inverse scattering problem via the usage of a finite number of measurements of the leading order-term of the low-frequency expansion of the thermoelastic scattered field is presented. Corresponding results for the geometrically degenerate case of the rigid thermoelastic sphere are obtained.

Science

scattering theory in thermoelasticity, thermoelastic scattering problems, time-harmonic fields, thermoelastic potentials, thermoelastic multi-layered scatterer, uniqueness and existence of solution of thermoelastic scattering problems, integral representations, far-field patterns, cross-sections, ellipsoidal scatterer, near-field data in thermoelastic scattering.

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https://pergamos.lib.uoa.gr/uoa/dl/object/3331257