Three-dimensional non-linear shell theory for flexible multibody dynamics

In flexible multibody systems, many components are approximated as shells. Classical theories, such as Kirchhoff and Reissner–Mindlin shell theories, are based on a priori kinematic assumptions. While such approach captures the kinetic energy of the system accurately, it cannot represent the strain energy adequately. Indeed, three-dimensional elasticity theory indicates that the normal material line will warp under load, leading to three-dimensional deformations and complex stress states. To overcome this problem, a novel three-dimensional shell theory is proposed in this paper. Kinematically, the problem is decomposed into a large rigid-normal-material-line motion and a warping field. The strains associated with the rigid-normal-material-line motion and the warping field are assumed to remain small. Consequently, the governing equations of the problem fall into two categories: the global equations describing geometrically exact shells and the local equations describing local deformations. The geometrically exact shell equations are nonlinear, two-dimensional equations, whereas the linear, local equations provide the detailed distribution of three-dimensional stress and strain fields. A shell stiffness matrix is found that reflects the effects of warping due to material heterogeneity and curvature. Three-dimensional stress and strain fields are recovered from the two-dimensional shell solution. The proposed approach is valid for anisotropic shells with arbitrary through-the-thickness lay-up configuration undergoing large motion but small strain.