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

TL;DR

This paper develops analytical solutions for the equilibrium deformation states of a swollen hydrogel shell using perturbation methods and an extended matched asymptotics approach, capturing boundary layer effects and parameter influences.

Contribution

It introduces an extended method of matched asymptotics for nonlinear swelling equations, providing explicit analytical formulas for deformation and stresses in hydrogel shells.

Findings

Analytical solutions match well with numerical results.

Deformation characterized by a single material parameter.

Boundary layer behavior accurately captured by the method.

Abstract

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.