  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.
