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We employ a canonical variational framework for the predictivecharacterization of structural instabilities that develop during thediffusion-driven transient swelling of hydrogels under geometrical constraints.The variational formulation of finite elasticity coupled with Fickian diffusionhas a two-field minimization structure, wherein the deformation map and thefluid-volume flux are obtained as minimizers of a time-discrete potentialinvolving internal and external energetic contributions. Following spatialdiscretization, the minimization principle is implemented using a conformingQ$_1$RT$_0$ finite-element design, making use of the lowest-orderRaviart-Thomas-type interpolations for the fluid-volume flux. To analyze thestructural stability of a certain equilibrium state of the gel satisfying theminimization principle, we apply the local stability criterion on theincremental potential, which is based on the idea that a stable equilibriumstate has the lowest potential energy among all possible states within aninfinitesimal neighborhood. Using this criterion, it is understood thatbifurcation-type structural instabilities are activated when the coupled globalfinite-element stiffness matrix loses its positive definiteness. This conceptis then applied to determine the onset and nature of wrinkling instabilitiesoccurring in a pair of representative film-substrate hydrogel systems. Inparticular, we analyze the dependencies of the critical buckling load and modeshape on the system geometry and material parameters.

Transient stability analysis of composite hydrogel structures based on a minimization-type variational formulation