A degenerating PDE system for phase transitions and damage

Abstract

In this paper, we analyze a PDE system arising in the modeling of phase transition and damage phenomena in thermoviscoelastic materials. The resulting evolution equations in the unknowns ϑ (absolute temperature), u (displacement), and χ (phase/damage parameter) are strongly nonlinearly coupled. Moreover, the momentum equation for u contains χ-dependent elliptic operators, which degenerate at the pure phases (corresponding to the values χ = 0 and χ = 1), making the whole system degenerate. That is why, we have to resort to a suitable weak solvability notion for the analysis of the problem: it consists of the weak formulations of the heat and momentum equation, and, for the phase/damage parameter χ, of a generalization of the principle of virtual powers, partially mutuated from the theory of rate-independent damage processes. To prove an existence result for this weak formulation, an approximating problem is introduced, where the elliptic degeneracy of the displacement equation is ruled out: in the framework of damage models, this corresponds to allowing for partial damage only. For such an approximate system, global-in-time existence and well-posedness results are established in various cases. Then, the passage to the limit to the degenerate system is performed via suitable variational techniques.