Direct numerical simulation and modelling techniques for membrane wrinkling
Cirak, F.¹; Mosler, J.^2
1University of Cambridge; ²Ruhr Univesity Bochum

There are a growing number of space missions in different planning stages that use large membrane structures. Because a membrane has a very small bending stiffness, wrinkles with various amplitudes and frequencies may occur easily under compression. In this work, we investigate two alternative, complementary techniques for the analysis of elastic and inelastic membranes at finite strains.

In the first approach, the membrane structure is modelled as a thin shell and discretised with the subdivision finite elements [1, 2]. Using the Kirchhoff-Love energy functional, the stable equilibrium configurations of the shell can be determined with the principle of minimum potential energy. In this finite element approach, the shell is discretised with smooth shape functions that have square-integrable curvatures. Subdivision surfaces deliver, in a particularly natural and efficient way, the requisite smooth shape functions on general unstructured meshes [3]. This approach yields a discrete algebraic problem, which can be solved iteratively.

In the second approach, the bending stiffness of the membrane is neglected and the wrinkles are not explicitly resolved. The membrane is analysed with a variational model similar to that proposed by Pipkin [3]. Pipkin's analytical approach allows to reformulate the classical tension field theory as a variational problem and is ideally suited for finite elements. For isotropic and anisotropic membranes, Pipkin proved, under few conditions, that the quasi-convexification of the membrane energy defines a relaxed energy functional whose derivatives yield the membrane stresses. Although Pipkin's concept is ideally suited for computations, a fully variational numerical framework had not been developed until recently [4].

The efficiency and performance of the resulting algorithms will be demonstrated by means of several numerical examples, such as torsion of elastic and elasto-plastic circular membranes and the inflation of airbags.

REFERENCES: [1] Cirak, F. and Ortiz, M. Fully C-1-conforming subdivision elements for finite deformation thin-shell analysis. Int. J. Numer. Meth. Eng. 51, 813-833, 2001.
[2] Cirak, F., Cisternas, J.E. , Cuitino, A.M., et al. Oscillatory thermomechanical instability of an ultrathin catalyst. Science 300, 1932-1936, 2003.
[3] Pipkin, A.C. Relaxed energy densities for large deformations of membranes. IMA J. of Appl. Math. 52, 297–308, 1994.
[4] Mosler, J. A novel variational algorithmic formulation for wrinkling at finite strains based on energy minimization: application to mesh adaption. Comp. Meth. Appl. Mech. Eng, in press, 2007.