This work presents a generalized Kirchhoff-Love shell theory that can explicitly capture fiber-induced 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 classical Kirchhoff-Love shell theory to incorporate the in-plane bending resistance of fibers. It also extends existing second-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 tensor and the in-plane and out-of-plane moment tensors are then identified from the mechanical power balance. These tensors are all second order and symmetric in general. 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.

翻译：本文提出了一种广义Kirchhoff-Love壳理论，该理论能够显式捕捉纤维诱导的各向异性，不仅涵盖拉伸和面外弯曲，还包括面内弯曲。该框架特别适用于非均匀纤维材料，如纺织品、生物材料、复合材料及点阵结构。所述理论是经典Kirchhoff-Love壳理论的直接扩展，用于纳入纤维的面内弯曲抗力；同时，它将现有针对初始直纤维的二阶梯度Kirchhoff-Love壳理论推广至初始弯曲纤维。为描述多纤维族的附加运动学，本文提出了一种所谓的面内曲率张量——该张量为对称二阶张量。进而，根据机械功率平衡方程识别出有效应力张量、面内与面外弯矩张量，这些张量通常均为二阶对称张量。基于不同的机械功率平衡表达式，推导了超弹性材料的本构方程。同时给出了弱形式，这是基于无旋转有限元离散的计算壳公式所必需的。