# Osmotic collapse of a void in an elastomer: breathing, buckling and creasing

[Entry by Jianyu Li, AP225, Fall, 2010]

## Abstract

This paper reports how to define and differentiate three kinds of deformation mechanism during the osmotic collapse of a void in elastomer, which includes breathing, buckling and creasing. The difference in humidity between the unsaturated air and water-filled elastomer results in tension of the liquid inside. If the tension is low, the void will shrink, but maintain the shape, otherwise two kinds of instability will occur. What is the critical conditions of the events? It is the main point of this work, which shows the wall thickness dominates the process: an elastomer with thin walls prefers to buckling, by contrast, thick wall to creasing.

## Tension

As mentioned above, the humidity gradient produces the tension in the liquid water inside the void. To derive the tension, the paper first describes the chemical potential of water in three conditions. 1. For water in the gas: <math>\mu=kTlog(p/p_0)</math>. 2. For water in a dilute solution: <math>\mu=-\Omega ckT</math>. 3. For water subject to stress: <math>\mu=-\Omega(\sigma+p_0)</math>. Combine those equation, the tension in the liquid water is given <math>\sigma=\frac{kT}{\Omega}log(\frac{p_0}{p})=ckT</math>.

## Summary of Instability==Breathing=

The stress <math>\sigma</math> depends on the ratio of the void radius before and after deformation, for the cylindrical void, it is given by <math>\frac{\sigma}{G}=-1/2+1/2(a/A)^{-2}-log(a/A)</math>; for a spherical void, the relaltion is <math>\frac{\sigma}{G}=-5/2+2(a/A)^{-1}+1/2(a/A)^{-4}</math>. Their curve is displayed in Fig. 4.

### Buckling

When the deformation deviates from breathing by a field of strain with infinitesimal amplitude and finite space, buckling occurs. Linear perburbation analysis is applied to analyze the mechanism, briefly speaking, a field of infinitesimal strain is added by a linear superposition of a set of sinusoidal eigenfields whose amplitude decays exponentially. Biot's analysis gave the critical conditoin for buckling is <math>\lambda_3/\lambda_1=3.383</math>, as used by the author, to estimate the critical tension condition in the cases of cylindarical and spherical voids.

### Creasing

Different from buckling, creasing is triggered by a field of strain finite in amplitude, but infinitesimal in space. The critical condition is <math>\lambda_3/lambda_1=2.4</math>. <math>\sigma_{cr}=1.12G</math> for cylindrical void; <math>\sigma_{cr}=1.75G</math> for spherical void. Dimensional analysis shows that the free energy per unit thickness takes the form <math>\Delta \Pi=G L^2 f(\frac{\sigma}{G})</math>, where <math>f(\frac{\sigma}{G})</math> function is obtained by ABAQUS. File:6.jpg