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A polymer network can imbibe water, forming an aggregate called hydrogel, andundergo large and inhomogeneous deformation with external mechanicalconstraint. Due to the large deformation, nonlinearity plays a crucial role,which also causes the mathematical difficulty for obtaining analyticalsolutions. Based on an existing model for equilibrium states of a swollenhydrogel with a core-shell structure, this paper seeks analytical solutions ofthe deformations by perturbation methods for three cases, i.e. free-swelling,nearly free-swelling and general inhomogeneous swelling. Particularly for thegeneral inhomogeneous swelling, we introduce an extended method of matchedasymptotics to construct the analytical solution of the governing nonlinearsecond-order variable-coefficient differential equation. The analyticalsolution captures the boundary layer behavior of the deformation. Also,analytical formulas for the radial and hoop stretches and stresses are obtainedat the two boundary surfaces of the shell, making the influence of theparameters explicit. An interesting finding is that the deformation ischaracterized by a single material parameter (called the hydrogel deformationconstant), although the free-energy function for the hydrogel contains twomaterial parameters. Comparisons with numerical solutions are also made andgood agreements are found.

Analytical Solutions for the Equilibrium states of a Swollen Hydrogel Shell and an Extended Method of Matched Asymptotics