Three-Dimensional Non-Linear Shell Theory for Flexible Multibody Dynamics

Abstract

In flexible multibody systems, many components are approximated as shells. Classical shell theories, such as Kirchhoff or Reissner-Mindlin shell theory, form the basis of the analytical development for shell dynamics. While such approach is capable of capturing the kinetic energy of the system accurately, it cannot represent the strain energy adequately. For instance, it is well known from three-dimensional elasticity theory that the normal material line will warp under load for laminated composite shells, leading to three-dimensional deformations that generate complex stress states. To overcome this problem, a novel three-dimensional shell theory is proposed in this paper. Kinematically, the problem is decomposed into an arbitrarily large rigid-normal-material-line motion and a warping field. The sectional strains associated with the rigid-normal-material-line motion and the warping field are assumed to remain small. As a consequence of this kinematic decomposition, the governing equations of the problem fall into two distinct categories: the global equations describing geometrically exact shells and the local equations describing local deformations. The governing equations for geometrically exact shells are nonlinear, two-dimensional equations, whereas the local equations are linear, one dimensional, provide the detailed distribution of three-dimensional stress and strain fields. Based on a set of approximated solutions, the local equations is reduced to the corresponding global equations. In the reduction process, a 9 × 9 sectional stiffness matrix can be found, which takes into account the warping effects due to material heterogeneity. In the recovery process, three-dimensional stress and strain fields at any point in the shell can be recovered from the two-dimensional shell solution. The proposed method proposed is valid for anisotropic shells with arbitrarily complex through-the-thickness lay-up configuration.