Large Elastic Deformation of an Inflatable Membrane of Revolution

Abstract

A general theory of large deformation of an anisotropic, elastic material has been applied to the problem of an inflatable, transversely isotropic membrane of revolution. Two states, the unstrained, and the strained states, are investigated. In the unstrained state, the thickness of the membrane is assumed to be uniform everywhere, but not in the strained state. In each state, the membrane is a smooth surface of revolution and is subjected to a uniform internal gas pressure. A set of static equilibrium equations is derived in terms of a strain energy density function and its derivatives, through the use of the calculus of variations. In order to carry out a numerical evaluation from the equilibrium equations, an explicit form of the energy density function has to be obtained. This function can be expressed in terms of two principal stretches. A finite-differe nce method is used to reduce the differential equations into the form of difference equations. The inflation of the membrane is determined. The results are compared with the isotropic case. N recent years, the use of inflated structures has become attractive to aerospace engineers and to military equipment and communications satellite designers. Most of these structures are thin-walled shells filled with pressurized gas. The advantages of light weight and small folded volume make it possible for such structures to be carried conveniently to their destinations for performing their designed functions. In space exploration, such a structure can be ejected from a booster outside the atmosphere of the Earth to form a space station or a space laboratory. In re-entry operation, the space capsule as it descends down to the ocean surface can be supported by an inflated circular raft for a period of time while waiting to be picked up by a helicopter. Similar ideas are also applicable for pilots ejected from airplanes. An inflatable tent can be considered as another example of such structures. In many cases, such as the one being attempted in this investigation, a large elastic deformation theory must be used. Many important researches in the area of large deformation of elastic membranes are summarized and recorded in the second edition of the book by Green and Adkin.1 Most of the studies are based upon a well-known strain energy function which was suggested by Mooney.2 Ericksen and Rivlin3 extended the earlier studies from isotropic to transversely isotropic materials. They showed that the strain energy density function for such a material can be expressed in terms of five scalar invariants. The strain energy density function for anisotropic materials has been studied by Smith and Rivlin.4 The authors discussed the restrictions imposed by symmetry on the form of this function, for elastic materials belonging to various crystal classes. The function was assumed to be a polynomial in gijt the deformation gradient tensor. This polynomial must be form-invariant under a group of transformations depending on the symmetry of materials. The problem of determination of the limitation imposed on the strain