New computational approaches for wrinkled and slack membranes

The static response of thin, wrinkled membranes is studied using both a tension field approximation based on plane stress conditions and a 3D nonlinear elasticity formulation, discretized through 8-noded Cosserat point elements. While the tension field approach only obtains the wrinkled/slack regions and at best a measure of the extent of wrinkliness, the 3D elasticity solution provides, in principle, the deformed shape of a wrinkled/slack membrane. However, since membranes barely resist compression, the discretized and linearized system equations via both the approaches are ill-conditioned and solutions could thus be sensitive to discretizations errors as well as other sources of noises/imperfections. We propose a regularized, pseudo-dynamical recursion scheme that provides a sequence of updates, which are almost insensitive to the regularizing term as well as the time step size used for integrating the pseudo-dynamical form. This is borne out through several numerical examples wherein the relative performance of the proposed recursion scheme vis-à-vis a regularized Newton strategy is compared. The pseudo-time marching strategy, when implemented using 3D Cosserat point elements, also provides a computationally cheaper, numerically accurate and simpler alternative to that using geometrically exact shell theories for computing large deformations of membranes in the presence of wrinkles.