Keywords: energy minimization, wrinkling, rotation-free triangular shell element, membranes experiments.

Over recent years, new structural concepts for large spacecraft applications involving thin film surfaces have been designed. Compared to traditional spacecraft structures, these "gossamer structures" could provide many advantages such as reduced mass and package volume. However, the materials used in gossamer structures (as very thin KaptonR film) cannot support compressive stresses because of their small bending rigidity. The result of compressive stress is that buckling occurs and wrinkles are formed. The existence of wrinkled regions may affect the performance and reliability of the flexible gossamer structure (as in an antenna or a reflector). Thus, the prediction of wrinkle patterns in a membrane surface is one of the many current technological interests in aerospace industry.

This paper is concerned with an efficient algorithmic formulation for the wrinkling at finite strains of very thin structures made of hyperelastic material. This problem is highly non linear since it deals with finite displacements and rotations as well as finite strains. Furthermore, the material nonlinear behavior is described by the use of a hyperelastic constitutive relationship.

The classical approach to investigate the wrinkles in membrane structures is to perform a bifurcation analysis using membrane elements or shell elements. Two methods are also possible. The first one is to use thin shell elements in conjunction with geometric imperfections embedded into the initial mesh to perform a post buckling analysis. The second one involves the detection of critical points based on the singularity of the tangent stiffness matrix, then the switching on a bifurcated branch. This specific treatment of the buckling in the classical finite element method usually leads to heavy computational times.

In this work, the problem of wrinkling is solved by directly minimizing the total potential energy of the structure. The numerical solution is carried out by means of an iterative method like the conjugate gradient or the Newton one. Although the proposed approach is theoretically equivalent to the traditional finite element method, it proves to be an attractive alternative which is particularly efficient for thin wrinkled structures. The main interest is that no specific buckling analysis is carried out during the calculation.

All the numerical computations are performed using two different elements. The two shell elements are formed by the superimposition of the membrane element and a plate bending element. The membrane stiffness is evaluated using the metric tensor in the current and reference configuration. The bending stiffness is obtained using the discrete Kirchhoff theory plate formulation for the first shell element. For the second element, a rotation-free triangular shell element formulation is used. The bending behavior is approximated from the displacement of a patch of four triangular shell elements. First, the proposed finite element procedure is validated through two well-known problems: the wrinkling prediction of rectangular shell-membrane under transverse in-plane displacement and the wrinkling prediction in a square membrane under corner loads. Then comparison is done with an original biaxial experiment performed using cruciform membrane specimens. An experimental biaxial device dedicated to the study of membranes was used. The device enables membranes to be stretched along two uncoupled axes. The wrinkled shape of the membrane was acquired by using a whole field measure based on the fringe analysis method. This optical method instantly gives the whole three-dimensional shape of the membrane. The aim of this experiment was to observe the influence of the membrane thickness on the wrinkle pattern and the ability of the above mentioned finite element procedure to reproduce them.

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