A novel variational algorithmic formulation for wrinkling at finite strains based on energy minimization: Application to mesh adaption

This paper is concerned with an efficient novel algorithmic formulation for wrinkling at finite strains. In contrast to previously published numerical implementations, the advocated method is fully variational. More precisely, the parameters describing wrinkles or slacks, together with the unknown deformation mapping, are computed jointly by minimizing the potential energy of the considered mechanical system. Furthermore, the wrinkling criteria are naturally included within the presented variational framework. The proposed method allows to employ three-dimensional constitutive models without any additional modification, i.e., a projection in plane stress space is not required. Analogously to the wrinkling parameters, the non-vanishing out-of-plane component of the strain tensor results conveniently from relaxing the respective Helmholtz energy of the membrane. The proposed framework is very general and does not rely on any assumption regarding the symmetry of the material, i.e., arbitrary anisotropic hyperelastic models can be consistently taken into account. The advantages associated with such a variational method are manifold. For instance, it opens up the possibility of applying standard optimization algorithms to the numerical implementation. This is especially important for highly non-linear or singular problems such as wrinkling. On the other hand, minimization principles provide a suitable basis for a posteriori error estimation and thus, for adaptive finite element formulations. As a prototype, a variational error indicator leading to an efficient h-adaption for wrinkling is briefly discussed. The performance of the wrinkling approach is demonstrated by selected finite element analyses.