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Cooperation with: P. Krejcí (Academy of Sciences of the Czech Republic, Prague), U. Stefanelli (Università di Pavia, Italy)
Supported by: DFG: ``HystereseOperatoren in PhasenfeldGleichungen'' (Hysteresis operators in phasefield equations)
Description:
To be able to deal with phase transitions, one has to take into account hysteretic phenomena that are modeled by hysteresis operators.
a)
The modeling of uniaxial nonlinear thermoviscoplastic
developments leads to the system
where u, , , and w are the unknowns displacement, absolute temperature, elastoplastic stress, and freezing index, respectively, , , C_{V}, and are positive constants, f, g are given functions, and , and are hysteresis operators. In [4], this system has been derived, its thermodynamic consistency has been proved, and the existence of a unique strong solution to an initialboundary value problem for this system has been shown. In [1], this existence result has been generalized to the case of a more general boundary condition for u and a weaker assumption for and . More precisely, the assumption that these operators are bounded has been replaced by the assumption that the clockwise admissible potential connected to these operators can be bounded from below by ,with some positive constant C.
A largetime asymptotic result for this system has been derived in [2]. An approach used therein to derive uniform estimates for the solutions to partial differential equations involving hysteresis operators has been further investigated in [3], leading to the notion of ``outward pointing hysteresis operators''. Moreover, in [3], generalizations of scalar PrandtlIshlinskii operators have been introduced and investigated with respect to their thermodynamic consistency.
b)
In [5], a control problem for an ordinary differential
system with hysteresis has been investigated. For controls
,one considers the system
(7) 
An approximation method and a conceptual algorithm are discussed as well.
References:



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