Stress Distributions in Partly Wrinkled Anisotropic Membranes

The search for the equilibrium configurations of initially flat elastic membranes subject to com-patible systems of loads constitutes a fundamental problem in biomedical engineering and, at the same time, an attractive object of research in aircraft, spacecraft and civil engineering. The main difficulties encountered in solving these problems, stem from the well-known feature of partly wrinkled membranes of exhibiting different mechanical properties within the various regions (taut, wrinkled and inactive, respectively) that at equilibrium compose the deformed surface. No doubt, the localisation of these regions constitutes the main unknown of the prob-lem, so, when dealing with cases where symmetries are absent, the search for solutions always presents a high degree of complexity with respect to the cases documented in literature. In order to simplify the problem, buckling and wrinkling phenomena, which reveals the inability of the membrane in sustaining compressive stresses, are usually considered as physical rather then geometrical nonlinearities. On the other hand, there’s no denying that most fabrics used in flexible structures are anisotropic themselves, therefore, determining the equilibrium configura-tions of anisotropic membranes results in an even more complicated problem to handle. Apart from the pioneering solutions based on “Tension Field Theory”, a number of interesting problems [1] related to isotropic elastic membranes were recently solved based on the ideas by Pipkin [2] of using a “relaxed strain energy” function for the membrane. This approach presents the considerable advantage that no a priori knowledge is needed about the lines separating the different region types (taut, wrinkled or inactive) resulting at equilibrium; moreover, it is easily implementable and flexible enough. To solve problems of plane infinitesimal wrinkly elasticity where the membrane interacts with beam elements, a simpler model was proposed in [3]. The existence and uniqueness of a relaxed energy function for anisotropic elastic membranes was recently established by Epstein in the finite elasticity context [4]. However, in order to ap-ply the Pipkin’ approach, a number of issues need to be still analyzed. In particular, the concepts of natural width or natural contraction used to point-wise specify the equilibrium state of an isotropic membrane (taut, wrinkled, or slack) on the basis of the knowledge of the displacement field, need to be accurately revisited since now principal stress and strains are no longer coaxial.