  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.
