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

TL;DR

This work formulates a canonical variational framework for diffusion-driven transient swelling in hydrogels under geometric constraints, using a two-field minimization over the deformation map ${\boldsymbol{\varphi}}$ and fluid-volume flux ${\mathbb{H}}$. A time-discrete incremental potential $\Pi^\tau$ is constructed and discretized with conforming Q1RT0 elements to achieve a symmetric, positive-definite stiffness for stable states, enabling a local stability analysis via the smallest eigenvalue $\lambda_{\min}$ of the global stiffness. Structural instabilities, such as wrinkling, are predicted by bifurcation when $\lambda_{\min}$ crosses zero, and the onset and mode shapes are explored in representative film-substrate hydrogel bilayers and tubes. The constitutive model combines a neo-Hookean mechanical term with Flory–Rehner-type chemistry and a diffusion dissipation potential, with a preswollen reference state to avoid dry-state singularities; the results reveal how film thickness, modulus ratio, substrate thickness, and geometry govern critical growth and wrinkle patterns, aligning with analytical and experimental trends in comparable hyperelastic 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.

Figures (14)

• Figure 1: Primary fields of the coupled problem and the boundary conditions. Within the minimization-based variational framework, finite elasticity coupled with diffusion is modeled using two primary fields, namely the deformation ${\boldsymbol{\varphi}}$ and the fluid-volume flux $\mathbb{H}$. The total boundary $\partial{\mathcal{B}}_0$ is split into $\partial{\mathcal{B}}_0^{\,{\boldsymbol{\varphi}}}$ and $\partial{\mathcal{B}}_0^{\,{\boldsymbol{\mathnormal t}}}$ for the mechanical fields and $\partial{\mathcal{B}}_0^{\,h}$ and $\partial{\mathcal{B}}_0^{\,\mu}$ for the chemical fields such that $\partial{\mathcal{B}}_0^{\,{\boldsymbol{\varphi}}}\cap\partial{\mathcal{B}}_0^{\,{\boldsymbol{\mathnormal t}}}=\partial{\mathcal{B}}_0^{\,h}\cap\partial{\mathcal{B}}_0^{\,\mu}=\varnothing$. The vector ${\boldsymbol{\mathnormal n}}_0({\boldsymbol{\mathnormal X}})$ represents the unit outward normal to the boundary $\partial{\mathcal{B}}_0$ of the continuum in the reference configuration. Conditions for the various fields on their respective boundaries are shown.
• Figure 2: Geometric description of the hydrogel bilayer and the applied load profile. a) Fluid diffuses into the bilayer from the top of the film and is allowed to accumulate inside by making the remaining edges of the bilayer impermeable. Suitable mechanical boundary conditions are applied so that the bilayer is allowed to swell only along the direction normal to the film surface. The dimensions of the substrate are taken to be $H=0.5\,$mm and $L=2\,$mm for all the analyses. b) Chemical potential on the film surface is increased linearly from its initial value $\mu_0$ to that of the diffusing fluid $\bar{\mu}=0$ in $1\,$s and then held constant. The resulting gradient in chemical potential along the bilayer thickness drives the diffusion process.
• Figure 3: Influence of film thickness on the critical buckling characteristics. a) Critical growth $g_c$ and b) wrinkle count $N_c$ as functions of the film thickness $w$ for a bilayer having modulus ratio $\gamma^f/\gamma^s=8$, preswelling factor $J_0^{f,s}=1.01$ and interaction parameter $\chi^{f,s}=0.1$. The colored curves $N=\left\lbrace1,1.5,2 \right\rbrace$ represent the extensions of the respective isomodes. Intersection of two successive isomodes indicates a shift in buckling mode due to the existence of an alternate buckled state at a lower critical load.
• Figure 4: Effect of shear-modulus ratio on the critical buckling characteristics. Variations in a) critical growth $g_c$ and b) wrinkle count $N_c$ with respect to the shear-modulus ratio $\gamma^f/\gamma^s$ of a hydrogel bilayer with film thickness $w=0.01\,$mm. The material parameters $J_0^{f,s}=1.01$ and $\chi^{f,s}=0.1$ are used. $g_c$ is observed to be a decreasing function of $\gamma^f/\gamma^s$, whereas $N_c$ shows an increasing-decreasing trend.
• Figure 5: Critical buckling modes of the bilayer for selected film thicknesses and shear-modulus ratios. For a given modulus ratio, fewer wrinkles are seen with increasing film thickness as described in Fig. \ref{['fig:bi_film']}b. Furthermore, the number of wrinkles can be seen to decrease with increasing modulus ratio for a fixed bilayer geometry as illustrated in Fig. \ref{['fig:bi-modrat']}b.