WIAS Preprint No. 1616, (2011)

Global existence result for thermoviscoelastic problems with hysteresis



Authors

  • Paoli, Laetitia
  • Petrov, Adrien

2010 Mathematics Subject Classification

  • 35A01 35Q80 74C05

Keywords

  • Existence result, generalized standard materials, heat equation, enthalpy transformation, maximal monotone operators, doubly nonlinear equations, plasticity, shape-memory alloys

Abstract

We consider viscoelastic solids undergoing thermal expansion and exhibiting hysteresis effects due to plasticity or phase transformations. Within the framework of generalized standard solids, the problem is described in a 3D setting by the momentum equilibrium equation, the flow rule describing the dependence of the stress on the strain history, and the heat transfer equation. Under appropriate regularity assumptions on the data, a local existence result for this thermodynamically consistent system is established, by combining existence results for ordinary differential equations in Banach spaces with a fixed-point argument. Then global estimates are obtained by using both the classical energy estimate and more specific techniques for the heat equation introduced by Boccardo and Gallouet. Finally a global existence result is derived.

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