The Stefan problem in a thermomechanical context with fracture and fluid flow

The classical Stefan problem, concerning mere heat-transfer during solid-liquid phase transition, is here enhanced towards mechanical effects. The Eulerian description at large displacements is used with convective and Zaremba-Jaumann corotational time derivatives, linearized by using the additive Green-Naghdi's decomposition in (objective) rates. In particular, the liquid phase is a viscoelastic fluid while creep and rupture of the solid phase is considered in the Jeffreys viscoelastic rheology exploiting the phase-field model and a concept of slightly (so-called semi) compressible materials. The L-1-theory for the heat equation is adopted for the Stefan problem relaxed by allowing for kinetic superheating/supercooling effects during the solid-liquid phase transition. A rigorous proof of existence of weak solutions is provided for an incomplete melting, employing a time discretization approximation.

Automated summary

This article extends the classical Stefan problem by considering mechanical effects during solid-liquid phase transition. It uses the Eulerian description with convective and Zaremba-Jaumann corotational time derivatives, linearized through the additive Green-Naghdi's decomposition in objective rates. The liquid phase is modeled as a viscoelastic fluid, while the solid phase incorporates creep and rupture via the Jeffreys viscoelastic rheology and the phase-field model for slightly compressible materials. The L-1 theory is applied to the heat equation in the Stefan problem, allowing for kinetic superheating/supercooling effects. A rigorous proof of weak solution existence is provided for cases of incomplete melting using a time discretization approximation.