Ruffling a membrane

Soft biological tissue is often subjected to forces that affect the tissue’s geometry, and finite elasticity provides a robust theoretical framework for analyzing the mechanical behavior of those tissues. Although the theory can accommodate anisotropic, nonlinear, and inhomogeneous processes subjected to large stresses and strains, its complexity makes many problems intractable. For growing tissue, though, the slow addition of cells generates shape- or size-changing stresses that are small enough to model successfully (see Physics Today April 2007, page 20 ). So, too, are simple geometries for tissues in equilibrium, even after those tissues are subjected to large stresses. Two recent papers have looked at applying the theory to those cases in thin elastic disks. In one recent study, Julien Dervaux and Martine Ben Amar (both of École Normale Supérieure, Paris) looked at anisotropic growth rates: If growth was faster in the radial than in the circumferential direction, the disk became conelike, while a reversal of rates generated saddle shapes. A separate study by Jemal Guven (National Autonomous University of Mexico) along with Martin Müller (ENS) and Ben Amar looked at excessively large circumferences for a given radius. Using the fully nonlinear theory, the researchers found an infinity of quantized equilibrium states for an ever-increasing perimeter at fixed radius. The ripples around the edge grew in size and number—not unlike the flower petals shown here—eventually crowding together enough to touch, like the ruffled collar in a portrait by Rembrandt. For more on the elasticity of thin sheets, see the article in February 2007, page 33 . (J. Dervaux, M. Ben Amar, Phys. Rev. Lett. 101, 068101, 2008http://dx.doi.org/10.1103/Phys.Rev.Lett.101.068101 ; M. M. Müller, M. Ben Amar, J. Guven, Phys. Rev. Lett. 101, 156104, 2008.http://dx.doi.org/10.1103/Phys.Rev.Lett.101.156104 )