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A doubly nonlinear problem associated with a mathematical model for piezoelectric material behavior
Author(s) -
Kraynyukova N.,
Alber H.D.
Publication year - 2012
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201000129
Subject(s) - monotone polygon , lipschitz continuity , mathematics , nonlinear system , monotonic function , piezoelectricity , function (biology) , subderivative , mathematical analysis , operator (biology) , regularization (linguistics) , pure mathematics , regular polygon , convex optimization , physics , computer science , geometry , biochemistry , chemistry , quantum mechanics , repressor , evolutionary biology , artificial intelligence , gene , acoustics , transcription factor , biology
We consider a mathematical model, which describes piezoelectric material behavior. This model is similar to models of plasticity theory. However, piezoelectric models describe coupled mechanical and electrical material behavior. Therefore they contain additional nonlinearities in the piezoelectric tensor and in the enthalpy function, which is non quadratic. These nonlinearities cause difficulties in the proof of existence theorems. Under the assumption that the piezoelectric tensor is constant (i.e. independent of P), we show how the system of model equations can be reduced to a doubly nonlinear evolution equation of the form z t ∈ G(‐Mz‐Φ (z) + f), which contains a composition of two monotone operators. The monotone mapping G is a subdifferential of the indicator function of some convex set while the second monotone mapping Φ is the Nemyckii operator of a monotone function. We prove existence of strong solutions, if Φ is replaced by a regularization. If in addition Φ is Lipschitz continuous we can show that the solution is unique.