CIMMS Lunchtime Series: Indentation, Puncturing and Cavitation of Thin Sheets

Dr. David J. Steigmann, Department of Mechanical Engineering, University of California at Berkeley

We use membrane theory to analyze the puncturing of a thin solid circular isotropic elastic sheet by a rigid axisymmetric indenter. A solution is obtained in which a hole is formed at the center of the sheet with an interior annulus in frictionless contact with the cylindrical surface. The contacting part is in a state of pure hoop stress with the corresponding hoop stretch exhibiting a strong singularity at the origin. Conditions are given ensuring that the solution has finite total energy and it is shown to be energetically favored over unpunctured states for transverse indenter displacements exceeding a finite critical value.

The material model used has also been shown to predict cavitation, the spontaneous formation of a traction-free hole following a sequence of uniform plane deformations of a disc of material. This is energetically preferred over uniform deformations when the material dilation reaches a critical value. This value plays the role of a material parameter. The same model is shown to accommodate puncturing.

Conventional indentation is also predicted. This is the classical unilateral obstacle problem in which the membrane maintains contact with the rigid body and the deformation is continuous everywhere. Using the minimum-energy principle as a selection criterion, we show that for a monotone sequence of indenter displacements, deformations of the sheet starting from a stress-free state begin with a conventional indentation phase, followed by puncturing and eventual cavitation. This furnishes a theoretical description of a ubiquitous failure mode which has thus far received virtually no attention in the literature.