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

TL;DR

This paper develops a comprehensive Kirchhoff-Love shell theory that explicitly models fiber-induced anisotropy, including in-plane bending, suitable for heterogeneous fibrous materials like textiles and composites.

Contribution

It extends classical shell theory to incorporate in-plane fiber bending and curved fibers, introducing an in-plane curvature tensor and deriving new constitutive equations.

Findings

The theory captures fiber-induced anisotropy in multiple deformation modes.

A symmetric in-plane curvature tensor is proposed for multiple fiber families.

The weak form facilitates computational implementation with finite element methods.

Abstract

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.