Analysis of thin shells using anisotropic polyconvex energy densities

In this contribution the numerical framework for the simulation of anisotropic thin shells is presented on the basis of polyconvex strain energy functions. The nonlinear shell theory is based on the Reissner–Mindlin kinematic along with inextensible director vectors. For the variational framework we consider a three field variational functional taking into account independent displacements, enhanced strains and stress resultants, where the latter field is eliminated by the evaluation of some orthogonality conditions. The iterative enforcement of the zero normal stress condition at the integration points allows consideration of arbitrary three-dimensional constitutive equations. Due to the fact that we are interested in fiber-reinforced materials we consider an additive structure of the energy decoupled in an isotropic part for the matrix and a superposition of transversely isotropic parts for the fiber families. For the representation of the anisotropy we use the concept of structural tensors and formulate the strain energy function in terms of principal and mixed invariants of the right Cauchy–Green tensor and the structural tensor. In order to guarantee the existence of minimizers we focus on polyconvex strain energy functions. The anisotropy effect is documented in several representative examples.

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