On a System of Nonlinear PDEs with Temperature-Dependent Hysteresis in One-Dimensional Thermoplasticity

In this paper, we develop a thermodynamically consistent description of the uniaxial behaviour of thermoelastoplastic materials that are characterized by a constitutive law of the form s x, t.sPw« , u x,t.x x,t., where « , s , u denote the fields of strain, stress, and absolute temperature, respectively, and where Pw?, ux4 denotes a family of rate-independent. hysteresis operators of u)0 Prandtl]Ishlinskii type, parametrized by the absolute temperature. The system of state equations governing the space-time evolution of the material are derived. It turns out that the resulting system of two nonlinearly coupled partial differential equations involves partial derivatives of hysteretic nonlinearities at different places. It is shown that an initial-boundary value problem for this system admits a global weak solution. The paper can be regarded as a first step towards a thermodynamic theory of rate-independent hysteresis operators depending on temperature.