The Stefan problem for complete melting of finitely strained solids into viscoelastic fluids

The compressible fluid-solid interaction (FSI) with a thermomechanical phase transition is formulated at large strains within the Eulerian frame. For the deviatoric part, the Jeffreys (also called anti-Zener) rheology with an additional viscosity is adopted. The core philosophy governing the mechanical solid-liquid transition is that the viscous (or viscoplastic) response is temperature-dependent and may fully degenerate to a viscoelastic fluid during thawing, so that there is no elastic response on the shear distortion. This behavior enables the free flow of the fluid, its subsequent freezing into a new configuration, and potential re-melting back into a fluid, allowing such cycles to repeat indefinitely. The classical Stefan problem, associated with the latent heat of the first-order (thawing-freezing) phase transition, is augmented by incorporating kinetic overheating and undercooling. The analysis by a time discretization with an appropriate truncation is applied to a higher-gradient modification of the original formulation, utilizing the concept of multipolar nonsimple continua.