Geometry and Physics of Wrinkling

The wrinkling of thin elastic sheets occurs over a range of length scales, from the fine scale patterns in substrates on which cells crawl to the coarse wrinkles seen in clothes. Motivated by the wrinkling of a stretched elastic sheet, we deduce a general theory of wrinkling, valid far from the onset of the instability, using elementary geometry and the physics of bending and stretching. Our main result is a set of simple scaling laws; the wavelength of the wrinkles $\lambda \sim K^{-1/4}$, where $K$ is the stiffness due to an “elastic substrate” effect with a multitude of origins, and the amplitude of the wrinkle $A\sim \lambda$. These could form the basis of a highly sensitive quantitative wrinkling assay for the mechanical characterization of thin solid membranes.

Figure 1

Wrinkles in a polyethylene sheet of length $L\approx 25\mathrm{c}\mathrm{m}$, width $W\approx 10\mathrm{c}\mathrm{m}$, and thickness $t\approx 0.01\mathrm{c}\mathrm{m}$ under a uniaxial tensile strain $\gamma \approx 0.10$. (Figure courtesy of K. Ravi-Chandar)

Figure 2

Wrinkling of skin. (a) Wrinkles induced in the skin of an apple ($\approx 5\mathrm{c}\mathrm{m}$) by the shrinking of the flesh. Observe that the wrinkles are orthogonal to the free boundary where the drying first starts. (b) Compression wrinkles induced on the back of one’s hand by bunching up the $\mathrm{\text{skin}}+\mathrm{\text{substrate}}$. The wavelength in such a situation is predicted to scale as the thickness of the layer, consistent with observations.

Figure 3

The basis for wrinkling assays of thin solid films. (a) Wrinkles on a thin elastic substrate induced by the forces exerted by a cell (figure courtesy of K. Burton, reprinted from 19 with permission by the American Society of Cell Biology). Typical wavelengths are in the range of $\mu \mathrm{m}$, and lengths are in the range of $10\mu \mathrm{m}$. (b) Wrinkles on a vesicle ($\approx 10\mu \mathrm{m}$ that is solid in its plane; observe that the wrinkles appear at $45°$ to the direction to flow-induced shear, corresponding to the direction of maximum compression (figure courtesy of H. Rehage, reprinted from 20 with permission of Elsevier Science).