Transient stability analysis of composite hydrogel structures based on a minimization-type variational formulation

TL;DR

This paper develops a variational framework to predict and analyze transient structural instabilities, such as wrinkling, in hydrogels undergoing diffusion-driven swelling, using finite element methods and stability criteria.

Contribution

It introduces a coupled variational formulation and finite-element implementation for analyzing hydrogel stability and buckling, advancing predictive capabilities for complex swelling behaviors.

Findings

Identified conditions for bifurcation and instability onset.

Determined how geometry and material properties influence buckling.

Validated the approach with hydrogel film-substrate systems.

Abstract

We employ a canonical variational framework for the predictive characterization of structural instabilities that develop during the diffusion-driven transient swelling of hydrogels under geometrical constraints. The variational formulation of finite elasticity coupled with Fickian diffusion has a two-field minimization structure, wherein the deformation map and the fluid-volume flux are obtained as minimizers of a time-discrete potential involving internal and external energetic contributions. Following spatial discretization, the minimization principle is implemented using a conforming Q$_1$RT$_0$ finite-element design, making use of the lowest-order Raviart-Thomas-type interpolations for the fluid-volume flux. To analyze the structural stability of a certain equilibrium state of the gel satisfying the minimization principle, we apply the local stability criterion on the incremental potential, which is based on the idea that a stable equilibrium state has the lowest potential energy among all possible states within an infinitesimal neighborhood. Using this criterion, it is understood that bifurcation-type structural instabilities are activated when the coupled global finite-element stiffness matrix loses its positive definiteness. This concept is then applied to determine the onset and nature of wrinkling instabilities occurring in a pair of representative film-substrate hydrogel systems. In particular, we analyze the dependencies of the critical buckling load and mode shape on the system geometry and material parameters.