  Subjected to compressive stresses, soft polymers with stiffness gradients can
display various buckling patterns. These compressive stresses can have
different origins, like mechanical forces, temperature changes, or, for
hydrogel materials, osmotic swelling. Here, we focus on the influence of the
transient nature of osmotic swelling on the initiation of buckling in confined
layered hydrogel structures. A constitutive model for transient hydrogel
swelling is outlined and implemented as a user-subroutine for the commercial
finite element software Abaqus. The finite element procedure is benchmarked
against linear perturbation analysis results for equilibrium swelling showing
excellent correspondence. Based on the finite element results we conclude that
the initiation of buckling in a two-layered hydrogel structure is highly
affected by transient swelling effects, with instability emerging at lower
swelling ratios and later in time with a lower diffusion coefficient. In
addition, for hard-on-soft systems the wavelength of the buckling pattern is
found to decrease as the diffusivity of the material is reduced for gels with a
relatively low stiffness gradient between the substrate and the upper film.
This study highlights the difference between equilibrium and transient swelling
when it comes to the onset of instability in hydrogels, which is believed to be
of importance as a fundamental aspect of swelling as well as providing input to
guiding principles in the design of specific hydrogel systems.
