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Phase-field models with hysteresis in one-dimensional thermoviscoplasticity. (English) Zbl 1034.34053
Here, a combined one-dimensional model is introduced for thermoviscoplastic behavior under solid-solid phase transformations that incorporates the occurrence of hysteresis effects in both the strain-stress law and the phase transition described by the evolution of a phase-field (which is usually closely related to an order parameter of the phase transition). Hysteresis is accounted for using the mathematical theory of hysteresis operators developed in the past thirty years. The model extends recent work of the first two authors on phase-field models with hysteresis [J. Math. Anal. Appl. 252, 198–219 (2000; Zbl 0983.74048), Nonlinear Anal., Theory Methods Appl. 39A, 569–586 (2000; Zbl 0941.35123)] to the case when mechanical effects can no longer be ignored or even prevail. It leads to a strongly nonlinear coupled system of partial differential equations in which hysteresis nonlinearities occur at several places, even under time and space derivatives. We show the thermodynamic consistency of the model, and we prove its well-posedness.

MSC:
34C55 Hysteresis for ordinary differential equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
47J40 Equations with nonlinear hysteresis operators
74K05 Strings
74N30 Problems involving hysteresis in solids
80A22 Stefan problems, phase changes, etc.
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