Viscoelasticity and accretive phase-change at finite strains
Andrea Chiesa, Ulisse Stefanelli
TL;DR
The paper addresses the coupled evolution of a two-phase viscoelastic solid undergoing irreversible accretive growth at finite strains, where the growing phase expands in its normal direction and the phases have distinct mechanical responses. It develops a variational framework with a phase indicator governed by a generalized eikonal equation for $\theta$, a diffused-interface energy $W_\varepsilon$, a viscosity dissipation $R_\varepsilon$, and a second-gradient regularization $H$, proving the existence of weak/viscosity solutions for both diffused-interface ($\varepsilon>0$) and sharp-interface ($\varepsilon=0$) formulations. An energy equality is established, accounting for viscous dissipation, external work, and the energetic cost of phase change, and the diffused-interface solutions are shown to converge to sharp-interface solutions as $\varepsilon\to0$. The results provide a mathematically rigorous foundation for growth-mechanics problems in contexts like tumor invasion and polymer gel swelling, enabling robust analysis of the fully coupled mechanics-growth system with finite strains.
Abstract
We investigate the evolution of a two-phase viscoelastic material at finite strains. The phase evolution is assumed to be irreversible: One phase accretes in time in its normal direction, at the expense of the other. Mechanical response depends on the phase. At the same time, growth is influenced by the mechanical state at the boundary of the accreting phase, making the model fully coupled. This setting is inspired by the early stage development of solid tumors, as well as by the swelling of polymer gels. We formulate the evolution problem by coupling the balance of momenta in weak form and the growth dynamics in the viscosity sense. Both a diffused- and a sharp-interface variant of the model are proved to admit solutions and the sharp-interface limit investigated.