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thermoplasticity; viscoelasticity; hysteresis; Prandtl-Ishlinskii operator; PDEs with hysteresis; thermodynamical consistency
In this paper, we develop a thermodynamically consistent description of the uniaxial behavior of thermovisco-elastoplastic materials for which the total stress $\sigma $ contains, in addition to elastic, viscous and thermic contributions, a plastic component $\sigma ^p$ of the form $\sigma ^p(x,t)={\mathcal P}[\varepsilon ,\theta (x,t)](x,t)$. Here $\varepsilon $ and $\theta $ are the fields of strain and absolute temperature, respectively, and $\lbrace {\mathcal P}[\cdot ,\theta ]\rbrace _{\theta > 0}$ denotes a family of (rate-independent) hysteresis operators of Prandtl-Ishlinskii type, parametrized by the absolute temperature. The system of momentum and energy balance equations governing the space-time evolution of the material forms a system of two highly nonlinearly coupled partial differential equations involving partial derivatives of hysteretic nonlinearities at different places. It is shown that an initial-boundary value problem for this system admits a unique global strong solution which depends continuously on the data.
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