WIAS Preprint No. 219, (1996)

On a System of Nonlinear PDE's with Temperature-Dependent Hysteresis in One-Dimensional Thermoplasticity


  • Krejčí, Pavel
    ORCID: 0000-0002-7579-6002
  • Sprekels, Jürgen

2010 Mathematics Subject Classification

  • 35G25 73B30 73E60 73B05


  • Thermoplasticity, hysteresis, Prandtl-Ishlinskii operator, weak solutions, PDE's with hysteresis




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 σ(x,t)= 𝒫[εθ(x,t)](x,t), where ε, σ, θ denote the fields of strain, stress and absolute temperature, respectively, and where {𝒫[·, θ]}θ>0 denotes a family of(rate-independent) hysteresis operators of 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.

Appeared in

  • J. Math. Anal. Appl. 209 (1997) pp. 25-46

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