# Geometry and physics of wrinkling

## Overview

**Authors:** E.Cerda & L.Mahadevan

**Source:** Physical Review Letters, Vol.90, 7, (2003)

**Soft Matter key words:** wrinkling, Young's modulus, elastic sheet

## Abstract

In this publication, authors set out to develop a general theory of wrinkling, using elementary geometry and the physics of bending and stretching. As a result they produce scaling laws for the wrinkle wavelength <math>\lambda</math> and amplitude A. They proceed to test these scaling laws for various wrinkling circumstances: the wrinkling of a polyethylene sheet, the wrinkling of an apple, the wrinkling of human skin and even the wrinkling of polymerized vesicles used for drug delivery.

## Soft Matter Snippet

The derivation starts off by considering the stretching of a polyethylene sheet , clamped at the edges. Beyond a critical stretching strain the sheet wrinkles as depicted on figure 1. The functional for this process is:

<math>U = U_B + U_S -L</math>

Where <math>U_B</math> is the bending energy due to deformations on the y axis and is a function of the bending stiffness B. Accordingly, <math>U_S</math> is the stretching energy due to tension T(x) along the x direction. Variable L represents the condition of inextensibility that the sheet has to satisfy. Manipulating the equation leads the authors to an expression for U:

<math>U = B \kappa^2_n \Delta L + \pi^2 T \Delta /\kappa^2_n L</math>

Here <math>\Delta</math> is the imposed compressive transverse displacement and <math>\kappa_n</math> is the wave number. The wavelength and amplitude are obtained by minimizing <math>U</math>:

<math>\lambda = 2 \sqrt{\pi} (\frac{B}{T})^{1/4} L^{1/2} \sim (\frac{B}{K})^{1/4}</math>

<math>A = \frac{\sqrt{2}}{\pi} (\frac{\Delta}{W})^{1/2} \lambda \sim (\frac{\Delta}{W})^{1/2} \lambda</math>

Where the scaling law for the wavelength arises by a substitution of the tension-to-length ratio by the stiffness K of the elastic foundation: <math>K \sim \frac{T}{L^2}</math>

And now for the fun part! The authors test their model for skin wrinkling, a rather unusual soft matter application! They start off with the observation that the wavelength of human wrinkles is larger than both the elastic substrate thickness <math>H_s</math> on which the skin rests, as well as the thickness of the skin itself t. That is: <math>\lambda >> H_s >> t</math>. The wrinkle stiffness is of the order of:

<math>K \sim E_s \lambda^2/ H_s^3</math>

Consequently the wrinkle wavelength will scale as:

<math>\lambda = (tH_s)^{1/2} (E/E_s)^{1/6} \approx H_s</math>, by plugging in numbers.

A quick estimate for <math>H_s</math> yields: <math>2H_s \approx 5mm \Rightarrow \lambda \approx 2.5mm</math>

This is a good estimate!

A final note on the physics of skin wrinkling. Authors make a distinction between:

(a) Wrinkles due to excess skin, like the ones on elbows and knees. They refer to these as tension wrinkles and attribute them to the presence of prestress.

(b) Wrinkles where the skin is close to the bony skeleton and drapes it. These are compression wrinkles due to muscular stress in these areas.

(c) Combination wrinkles. On some unfortunate, but scientifically fascinating, sites both tension and compression occur. This leads for example to crow-feet patterns radiating from the eye.