Localized necking of a dielectric membrane

We first revisit the problem of localized bulging of an inflated rubber tube, and show that the associated bifurcation condition is equivalent to the vanishing of the determinant of the Hessian of the strain-energy density function when the latter is viewed as a function of the internal volume v and the axial stretch λz. This led us to conjecture and verify that when a dielectric plane membrane is subjected to the combined action of an electric field and in-plane mechanical stretching, localized necking will occur when the determinant of the Hessian of the free-energy function becomes zero. One situation in which this bifurcation condition is satisfied is when the nominal electric field as a function of the nominal electric displacement reaches a maximum under a dead load. This corrects the widespread mis-conception that when the nominal electric field reaches a maximum, a so-called pull-in instability would occur whereby the membrane thins down uniformly, leading to wrinkle formation or dielectric failure. We highlight the fact that the above bifurcation condition can be satisfied, and hence localized necking may occur, even if the nominal electric field does not have a maximum at all. This happens, for instance, when a rectangular membrane is uni-axially stretched and then has its four edges fixed before it is subjected to an electric field.