Formation of creases on the surfaces of elastomers and gels

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Entry by Jianyu Li, AP225, Fall, 2010

Introduction

 When an elastomer or a gel is bent, we might observe some creases created on the surface. Biot`s linear perturbation analysis predicts the inhomogeneous deformation in a rubber occurs upon a critical strain, <math>\epsilon_biot=0.46</math>. However, experimental observations indicate the real critical strain is <math>\epsilon_exp=0.35</math>. Obviously, the theoretical value for creasing does not agree with experiment results. 
 Someone thinks there are two modes of instability occur at different critical strains. One mode is described by Biot`s linear perturbation analysis, thus restricted his search within the deformation with infinitesimal strains. The other one is the creasing which initially is small, and grows under large strains. 
 In order to explain the phenomenon clearly, this paper uses the neo-Hookean model to calculate the change in the free-energy density through the scaling theory. The theoretical results match with the experimental observation perfectly. 

The Critical Strain

An undeformed block of elastomer is as the reference state, on the surface, there are three markers: A', O, A. Under compression, the elastomer might either deform homogeneously [Fig.2(b)], or form a crease [Fig.2(c)], as illustrated in Figure 2.
Fig.2. Schematic a semi-infinite block of elastomer under compression
 Treat the deformation under the plane-strain conditions. The free energy density is <math>W=\frac{\mu}{2}*(\lambda_1^2+\lambda_2^2+\lambda_3^3-3)</math>, where <math>\mu</math> is the shear modulus, <math>\lambda</math> is the principal stretches. The elastomer is assumed to be incompressible, <math>\lambda_1*\lambda_2*\lambda_3=1</math>. The segment OA rotates to the vertical position, could be solved as a part of the boundary-value problem. 
 Based on dimensional analysis, the elastic energy is <math>U=\mu*L^2*f(\epsilon)</math>,<math>f(\epsilon)</math> is a dimensionless function. Figure 3 is the plot of <math>f</math> and <math>\epsilon</math>. 

The difference in elastic energy between the creased state and the homogeneous state

 It is found that at the value of strain,about 0.35, <math>f</math> decreases to zero. That means when <math>\epsilon>\epsilon_c</math>, the creased state has a lower energy than the homogeneous state, thus the crease will definitely occur. This result agrees with the experimental value quite well.

The Plane-strain Problem

The authors further discuss an equivalent plane-strain problem <math>(\lambda_2=1)</math>. As illustrated in Figure 4, an intermediate state is added as another reference state.
A generalized plane-strain problem and its equivalent plane-strain problem.

<math>\lambda_1=\lambda_1'/\sqrt{\lambda_2},\lambda_3=\lambda_3'/\sqrt{\lambda_2},\lambda_2=\lambda_2</math>

 So the free-energy density <math>W=\frac{\mu}{2*\lambda_2}*[(\lambda_1')^2+(\lambda_3')^2]+\frac{\mu}{2}*(\lambda_2^2-3)</math>
 The critical strain <math>\epsilon_c=0.35</math> equals to <math>\lambda_1=0.65</math>. Correspondingly, <math>\lambda_3/\lambda_1=2.4</math>, which is valid for any arbitrary value of <math>\lambda_2</math>. For Biot's theory, <math>\lambda_3/\lambda_1=3.4</math>

A Swelling Gel

 Consider a gel, <math>\eta</math> is the ratio of the thickness of the swollen gel and that of the initial gel. With the same procedure, <math>\eta_c=2.4, \eta_biot=3.4</math>, comparable to experimental observations 2.0-3.7. Note that this analysis does not account for the migration of solvent in a gel. Further work is needed.