  The classical Stefan problem, concerning mere heat-transfer during
solid-liquid phase transition, is here enhanced towards mechanical effects. In
particular, the flow of the liquid phase as semi-compressible viscoelastic
fluid and creep and rupture of the solid phase considered in the Jeffreys
viscoelastic rheology exploiting the phase-field model. The Eulerian
description in stress/velocity or strain/velocity formulation is used with
convective and Zaremba-Jaumann corotational time derivatives. The L1-theory for
the heat equation is adopted for the Stefan problem in the relaxed modification
by allowing for superheating/supercooling effects during the solid-liquid phase
transition. A rigorous proof of existence of week solutions is provided for an
incomplete melting, exploiting a time-discretisation approximation.
