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