A holed membrane at finite equibiaxial stretch

TL;DR

This paper analyzes the deformation and stress distribution in a stretched thin neo-Hookean membrane with a central hole, providing exact and approximate solutions across different stretch regimes and revealing critical stress ratios and energy differences.

Contribution

It introduces a simplified differential equation and closed-form solutions for the membrane with a hole, covering finite, intermediate, and large deformations, with validation against numerical results.

Findings

Stress ratio at the hole boundary approaches 4 at large stretches with a small hole.

Difference in stored strain energy becomes significant only at finite stretches.

Membrane flaw tearing likely occurs only at finite stretch levels.

Abstract

The deformation and stress distribution in a stretched thin neo-Hookean circular membrane with a hole at its center are analyzed within the framework of finite deformation elasticity. Initially, we derive a simple form for the differential governing equation to the problem. This enables us to introduce a closed-form solution in the limit of infinite stretch. Subsequently, we propose approximate solutions for intermediate and large deformations. These approximations approach the exact solutions in the limits of small and infinite stretches. The transition stretch at which the membrane behavior switches from the intermediate to the large deformation approximation is determined too. Comparison of our solution and approximations to corresponding numerical results reveal a neat agreement for any stretch and ratio between the hole to the membrane radii. In the limit of large stretches and a small hole, the ratio of the hoop stress at the hole boundary to the nominal stress is 4, which is twice the corresponding ratio in the small deformation limit. Comparison of the strain energy stored in the membrane to the one in a membrane without a hole reveals that only at finite stretches the difference between these energies becomes meaningful. This implies that it is likely that a flaw in a membrane will tear out only at a finite level of stretches.