Dear Mr Reddy,

It depends on how vacuum is created.

Consider a plastic can with liquid subjected to lowered temperature.

Is that a case, within  your study?

Dear MS. Stromberg,

Let's say the air inside the membrane is pumped out to create a vacuum or equivalently, higher pressure is created around the membrane to crush it in. No liquids involved.

I want to know whether we can find minimizers of total potential energy (elastic strain energy + pressure work) in this case. I programmed the mathematics for outward inflation. When the inflating pressure is negative, no equilibrium is found.

Thanks very much for your time.

A membrane theory is not suitable in this case since a membrane cannot support compression. Therefore, what you want is an elastic shell or a cylindrical shell. The case of a cylindrical shell under pressure is known and leads to build-up of compressive stresses. Crushing will depend on inhomogeneities and is much harder to predict. Similarly the crushing of an elastic shell is a difficult problem. Recent work by Thompson  and Hutchinson on nonlinear shells under pressure is relevant for you.