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

TL;DR

This paper extends the classical Stefan problem to include mechanical effects, viscoelastic fluids, and fracture, providing a rigorous existence proof for weak solutions in a complex thermomechanical context.

Contribution

It introduces a comprehensive thermomechanical model with fracture and fluid flow, using advanced mathematical tools to prove the existence of weak solutions.

Findings

Existence of weak solutions for the extended Stefan problem.

Incorporation of viscoelastic fluid and fracture mechanics.

Handling of superheating and supercooling effects in phase transition.

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 $L^1$-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, employinga time-discretisation approximation.