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Measure-valued solutions for models of ferroelectric materials

Published online by Cambridge University Press:  03 October 2014

Nataliya Kraynyukova  and
Sergiy Nesenenko
Nataliya Kraynyukova
Affiliation:
Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt, Germany, (kraynyukova@mathematik.tu-darmstadt.de; nesenenko@mathematik.tu-darmstadt.de)
Sergiy Nesenenko
Affiliation:
Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt, Germany, (kraynyukova@mathematik.tu-darmstadt.de; nesenenko@mathematik.tu-darmstadt.de)
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Abstract

In this work we study the solvability of the initial boundary-value problems that model the quasi-static nonlinear behaviour of ferroelectric materials. Similar to the metal plasticity, the energy functional of a ferroelectric material can be additively decomposed into reversible and remanent parts. The remanent part associated with the remanent state of the material is assumed to be a convex non-quadratic function f of internal variables. In this work we introduce the notion of the measure-valued solutions for the ferroelectric models, and show their existence in the rate-dependent case, assuming the coercivity of the function f. Regularizing the energy functional by a quadratic positive-definite term, which can be viewed as hardening, we show the existence of measure-valued solutions for the rate-independent and rate-dependent problems, avoiding the coercivity assumption on f.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 5 , October 2014 , pp. 935 - 963
Copyright
Copyright © Royal Society of Edinburgh 2014 

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