  We employ a canonical variational framework for the predictive
characterization of structural instabilities that develop during the
diffusion-driven transient swelling of hydrogels under geometrical constraints.
The variational formulation of finite elasticity coupled with Fickian diffusion
has a two-field minimization structure, wherein the deformation map and the
fluid-volume flux are obtained as minimizers of a time-discrete potential
involving internal and external energetic contributions. Following spatial
discretization, the minimization principle is implemented using a conforming
Q$_1$RT$_0$ finite-element design, making use of the lowest-order
Raviart-Thomas-type interpolations for the fluid-volume flux. To analyze the
structural stability of a certain equilibrium state of the gel satisfying the
minimization principle, we apply the local stability criterion on the
incremental potential, which is based on the idea that a stable equilibrium
state has the lowest potential energy among all possible states within an
infinitesimal neighborhood. Using this criterion, it is understood that
bifurcation-type structural instabilities are activated when the coupled global
finite-element stiffness matrix loses its positive definiteness. This concept
is then applied to determine the onset and nature of wrinkling instabilities
occurring in a pair of representative film-substrate hydrogel systems. In
particular, we analyze the dependencies of the critical buckling load and mode
shape on the system geometry and material parameters.
