Weak solutions to a model for phase separation coupled with finite-strain viscoelasticity subject to external distortion

We study the coupling of a viscoelastic deformation governed by a Kelvin–Voigt model at equilibrium, based on the concept of second-grade nonsimple materials, with an inelastic deformation due to volumetric swelling, described via a phase-field variable subject to a Cahn–Hilliard model expressed in a Lagrangian frame. Such models can be used to describe the time evolution of hydrogels in terms of phase separation within a deformable substrate. The equations are mainly coupled via a multiplicative decomposition of the deformation gradient into both contributions and via a Korteweg term in the Eulerian frame. To treat time-dependent Dirichlet conditions for the deformation, an auxiliary variable with fixed boundary values is introduced, which results in another multiplicative structure. Imposing suitable growth conditions on the elastic and viscous potentials, we construct weak solutions to this quasistatic model as the limit of time-discrete solutions to incremental minimization problems. The limit passage is possible due to additional regularity induced by the viscous stress and the elastic hyperstress.