A general theory for anisotropic Kirchhoff-Love shells with embedded fibers and in-plane bending

In this work we present a generalized Kirchhoff-Love shell theory that can capture anisotropy not only in stretching and out-of-plane bending, but also in in-plane bending. This setup is particularly suitable for heterogeneous and fibrous materials such as textiles, biomaterials, composites and pantographic structures. The presented theory is a direct extension of existing Kirchhoff-Love shell theory to incorporate the in-plane bending resistance of fibers. It also extends existing high gradient Kirchhoff-Love shell theory for initially straight fibers to initially curved fibers. To describe the additional kinematics of multiple fiber families, a so-called in-plane curvature tensor - which is symmetric and of second order - is proposed. The effective stress, in-plane and out-of-plane moment tensors are then identified from the mechanical power balance. These tensors are all second order and symmetric for general materials. The constitutive equations for hyperelastic materials are derived from different expressions of the mechanical power balance. The weak form is also presented as it is required for computational shell formulations based on rotation-free finite element discretizations.