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

Abstract

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 L1$$ {L}^1 $$‐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.