Response of curved surfaces to point indentation : implications on the behavior of small-scale structures

Published online on June 3, 2008
Proc. Natl. Acad. Sci. USA, 10.1073/pnas.0707364105




APPLIED PHYSICAL SCIENCES

Localized and extended deformations of elastic shells

Ashkan Vaziri and L. Mahadevan*

School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138

Edited by John W. Hutchinson, Harvard University, Cambridge, MA, and approved March 11, 2008 (received for review August 7, 2007)

Abstract

The dried raisin, the crushed soda can, and the collapsed bicycle inner tube exemplify the nonlinear mechanical response of naturally curved elastic surfaces with different intrinsic curvatures to a variety of different external loads.

To understand the formation and evolution of these features in a minimal setting, we consider a simple assay: the response of curved surfaces to point indentation.

We find that for surfaces with zero or positive Gauss curvature, a common feature of the response is the appearance of faceted structures that are organized in intricate localized patterns, with hysteretic transitions between multiple metastable states.

In contrast, for surfaces with negative Gauss curvature the surface deforms nonlocally along characteristic lines that extend through the entire system.

These different responses may be understood quantitatively by using numerical simulations and classified qualitatively by using simple geometric ideas.
Our ideas have implications for the behavior of small-scale structures.


nonlinear mechanics | pattern formation | physical geometry