Surface elasticity effect on Plateau-Rayleigh instability in soft solids

TL;DR

The paper addresses how strain-dependent surface elasticity alters elasto-capillary Plateau-Rayleigh instabilities in soft cylinders by deriving a rigorously consistent 1d gradient model from a 3d bulk–surface elasticity framework and validating it with a Rayleigh-Ritz FE scheme. It reveals that surface elasticity and incompressibility can both delay or promote instability depending on the loading scenario, and that nonlinear post-bifurcation behavior such as strain softening and subcritical bifurcations can occur. The work provides a practical tool for calibrating surface parameters from measurable amplitude responses and offers a principled approach to designing solid-like materials with tailored surface effects. The combination of analytical reduction and robust 3d numerics advances understanding of elasto-capillary phenomena in solids and supports applications in soft robotics, bioengineering, and materials design.

Abstract

Soft solids exhibit instability and develop surface undulations due to surface effects, a phenomenon known as the elastic Plateau-Rayleigh (PR) instability, driven by the interplay of surface and bulk elasticity. Previous studies on the PR instability in solids mainly focused on the case of constant surface tension and ignored the effect of surface elasticity. It has been shown by experiments that the surface effects in solid-like materials depend both on the surface tension and surface elasticity, but little is known about the role of the latter in the elasto-capillary instabilities in soft solids. Here, we conduct an in-depth exploration of the effect of surface elasticity on the PR instability in an elastic cylinder by coupling theoretical and numerical methods. We derive an asymptotically consistent one-dimensional (1d) model to characterize the PR instability from three-dimensional (3d) nonlinear bulk-surface elasticity, and develop a new finite-element (FE) scheme for simulating 3d deformations of the bulk-surface system. The initiation and evolution of the PR instability are obtained analytically with the aid of the 1d model. The 1d results are further validated by the 3d FE simulations. By synthesizing the 1d analytic solutions and 3d numerical results, the effects of surface elasticity, surface compressibility, surface tension, axial force and geometrical size on the PR instability are thoroughly elucidated. Our results can be applied to calibrate surface parameters for solid-like materials and develop constitutive models for elastic surfaces.

Figures (10)

• Figure 1: A hyperelastic cylinder in (a) reference (undeformed) configuration and (b) current configuration. The cylinder is subjected to an axial force and surface stress.
• Figure 2: Homogeneous and inhomogeneous responses for the cases of fixed $\bar{\mu}_\text{s}=0$, $0.2$, $0.4$ respectively, $\bar{\nu}_\text{s}=0.2$ and $\bar{\gamma}=10$. (a) Homogeneous responses of the axial stress $\bar{N}$ varying with the stretch $\lambda$. (b) Dependence of the axial stress $\bar{N}$ on the stretch $\lambda_0=\lambda(0)$ at the necking starting point. (c) Necking amplitudes $a_\infty-a_0$ varying with the ratio of the axial stress to the Maxwell stress $\bar{N}/\bar{N}_\text{M}$. The gray dashed line in (a) represents the necking condition. The dotted parallel lines in (a) represent the corresponding Maxwell plateaus $\bar{N}_\text{M}$. The scattered hollow dots in (b) and (c) are the 3d FE results for $\bar{\mu}_\text{s}=0.2$. The solid points marked $1$-$5$ in (b) denote the different post-bifurcation stages. The deformations of the cylinder at these stages are presented in Fig. \ref{['fig:solution11b']}.
• Figure 3: Necking solutions for the radial stretch $a(L/A)$ at the five post-bifurcation stages marked in Fig. \ref{['fig:solution12a']}(b) for the three different surface-elasticity numbers: (a) $\bar{\mu}_\text{s}=0$, (b) $\bar{\mu}_\text{s}=0.2$ and (c) $\bar{\mu}_\text{s}=0.4$. The orange-colored section displays the morphology of the deformed cylinder at the final stage of bifurcation, which records the maximum necking amplitude.
• Figure 4: Homogeneous and inhomogeneous responses for the cases of fixed $\bar{\gamma}=10$, $\bar{\mu}_\text{s}=0.2$ and $\nu_\text{s}=0$, $0.4$, $0.8$ respectively. (a) Homogeneous responses for the axial stress $\bar{N}$ varying with the stretch $\lambda$. (b) Dependence of the axial stress $\bar{N}$ on the stretch $\lambda_0=\lambda(0)$ at the necking starting point. (c) Necking amplitudes $a_\infty-a_0$ varying with the ratio of the axial stress and the Maxwell stress $\bar{N}/\bar{N}_\text{M}$. The gray dashed line in (a) represents the necking condition. The dotted parallel lines in (a) represent the corresponding Maxwell plateaus $\bar{N}_\text{M}$.
• Figure 5: Homogeneous and inhomogeneous responses for the cases of fixed $\bar{N}=10$, $\bar{\mu}_\text{s}=0$, $0.2$, $0.4$ respectively, and $\nu_\text{s}=0.2$. (a) Homogeneous responses of the elaso-capillary number $\bar{\gamma}$ varying with the stretch $\lambda$. (b) Dependence of the $\bar{\gamma}$ on the stretch $\lambda_0=\lambda(0)$ at the bulging starting point. (c) Bulging amplitudes $a_0-a_\infty$ varying with ratio of the surface tension and its Maxwell level $\bar{\gamma}/\bar{\gamma}_\text{M}$. The gray dashed line in (a) represents the bulging condition. The dotted parallel lines in (a) represent the corresponding Maxwell plateaus $\bar{\gamma}_\text{M}$. The scattered hollow dots in (b) and (c) are the 3d FE results for $\bar{\mu}_\text{s}=0.2$. The solid points marked $1$-$5$ in (b) denote the different post-bifurcation stages. The deformations of the cylinder at these stages are presented in Fig. \ref{['fig:solution22b']}.