Analysis of Constitutive Assumptions for the Strain Energy of a Generalized Elastic Membrane in a Nonlinear Contact Problem

Abstract

For very thin shell-like structures it is common to ignore bending effects and model the structure using simple membrane theory. However, since the thickness of the membrane is not modeled explicitly in simple membrane theory it is not possible to use the three-dimensional strain energy function directly. Approximations must be introduced like the assumptions of: no thickness changes, generalized plane stress or incompressibility. In contrast, the theory of a Cosserat generalized membrane uses the three-dimensional strain energy function directly, it includes both thickness changes and shear deformation and it allows contact conditions to be formulated on the interface of the membrane with another body instead of on the middle surface of the membrane. A specific nonlinear contact problem is used to study these effects and comparison is made with solutions of a hierarchy of theories which include different levels of deformation through the thickness of the membrane and different formulations of the contact conditions. The results indicate that within the context of a simple membrane the assumption of generalized plane stress is best for this problem and that a generalized contact condition extends the range validity of the simple membrane solution to thicker membranes.