Second-order elastic solutions for spherical gels subjected to spherically symmetric dilatation

The problem of a spherically symmetric dilatation in a spherical gel, important for understanding the mechanical behavior of hydrogels and biogels (e.g., drug-delivery vehicles, mineralized organic gels), is solved analytically using second-order elasticity theory. The solutions are developed via a perturbation method up to the cubic terms of the strains. Using elastic constants for a gel derived from an earlier paper (Wu and Kirchner, 2010), the stresses, displacements and energy due to a dilatation in an unconstrained sphere and in a spherical shell bonded to a rigid core are calculated. The results have the advantage of being in analytical closed-form, and are consistent with previous numerical simulations. The nonlinear part of the stress, displacement and energy is shown to be significantly larger than the linear part, and they may have different signs.