Nieliniowe zagadnienia ewolucyjne opisujące strukturalne przemiany fazowe w ciałach stałych

Zenon Kosowski

Abstract

In presented paper we examine a simple version of the Fried-Gurtin model for the isothermal phase transitions in solids. In this model one suppose that a mikroforce acts in response to the evolution of an order parameter, an internal scalar or vector variable, which characterises the phase of the solid. In the simplistic version, we assume small deformations and constant mass density. We also consider the scalar order parameter. Additionally, we postulate that the free energy is the sum of the elastic energy - the quadratic function of linearised strain tensor, the exchange energy ñrepresented by a double well potential, and energy accumulated in an interphase boundary ñthe quadratic function of the gradient of the order parameter. Balance laws for the linear momentum, the microforce and the second law of thermodynamics in the form of a dissipation inequality allow to infer equations of the model. The model leads to the system of coupled nonlinear partial di§erential equations. The hyperbolic subsystem for the displacement represents the linear momentum balance and the parabolic equation for the order parameter results from the microforce balance. The proof of the existence of global in time, weak solutions for three-dimensional problem with the initial and the boundary conditions is presented. The proof is based on the Feado-Galerkin approximation method. Besides, proofs of the existence and the uniqueness of a more regular solution in an arbitrary finite time interval are given. Such solution satisfis the hyperbolic subsystem in the weak sense and the parabolic equation in the classic sense. The proof of the existence is based on the Leray-Schauder fixed point theorem for continuous and compact map. The last part is devoted to the analysis of oneñdimensional problem. We present simplified proofs of the existence, the uniqueness, and regularity of the solution for this case. We demonstrate the existence of nontrivial stationary solutions, that is, solutions independent on time. Su¢ cient conditions for the existence of the solution with the phase transition are formulated. Besides, conditions for the stability of homogeneous stationary solutions are defined. We demonstrate some selected numerical simulations.
Diploma typeDoctor of Philosophy
Author Zenon Kosowski (FMIS)
Zenon Kosowski,,
- Faculty of Mathematics and Information Science
Title in PolishNieliniowe zagadnienia ewolucyjne opisujące strukturalne przemiany fazowe w ciałach stałych
Languagepl polski
Certifying UnitFaculty of Mathematics and Information Science (FMIS)
Disciplinemathematics / (mathematics domain) / (physical sciences)
Defense Date03-12-2010
End date16-12-2010
Supervisor Irena Pawłow (FMIS)
Irena Pawłow,,
- Faculty of Mathematics and Information Science

Internal reviewers Krzysztof Chełmiński (FMIS / DPDE)
Krzysztof Chełmiński,,
- Department of Partial Differential Equations
External reviewers Piotr Rybka
Piotr Rybka,,
-
Pages113
Keywords in Englishxxx
Abstract in EnglishIn presented paper we examine a simple version of the Fried-Gurtin model for the isothermal phase transitions in solids. In this model one suppose that a mikroforce acts in response to the evolution of an order parameter, an internal scalar or vector variable, which characterises the phase of the solid. In the simplistic version, we assume small deformations and constant mass density. We also consider the scalar order parameter. Additionally, we postulate that the free energy is the sum of the elastic energy - the quadratic function of linearised strain tensor, the exchange energy ñrepresented by a double well potential, and energy accumulated in an interphase boundary ñthe quadratic function of the gradient of the order parameter. Balance laws for the linear momentum, the microforce and the second law of thermodynamics in the form of a dissipation inequality allow to infer equations of the model. The model leads to the system of coupled nonlinear partial di§erential equations. The hyperbolic subsystem for the displacement represents the linear momentum balance and the parabolic equation for the order parameter results from the microforce balance. The proof of the existence of global in time, weak solutions for three-dimensional problem with the initial and the boundary conditions is presented. The proof is based on the Feado-Galerkin approximation method. Besides, proofs of the existence and the uniqueness of a more regular solution in an arbitrary finite time interval are given. Such solution satisfis the hyperbolic subsystem in the weak sense and the parabolic equation in the classic sense. The proof of the existence is based on the Leray-Schauder fixed point theorem for continuous and compact map. The last part is devoted to the analysis of oneñdimensional problem. We present simplified proofs of the existence, the uniqueness, and regularity of the solution for this case. We demonstrate the existence of nontrivial stationary solutions, that is, solutions independent on time. Su¢ cient conditions for the existence of the solution with the phase transition are formulated. Besides, conditions for the stability of homogeneous stationary solutions are defined. We demonstrate some selected numerical simulations.
Thesis file
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Citation count*5 (2020-09-26)

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