Nonlinear shell problem formulation accounting for through-the-thickness stretching and its finite element implementation

We first review some recent and current research works attributing to a very significant progress on shell problem theoretical foundation and numerical implementation attained over a period of the last several years. We then discuss theoretical formulations of shell model accounting for the trough-the-thickness stretching, which allows for large deformations and direct use of 3d constitutive equations. Three different possibilities for implementing this model within the framework of the finite element method are examined, one leading to 7 nodal parameters and the remaining two to 6 nodal parameters. Comparisons are performed of the 7- parameter shell model with no simplification of kinematic terms and 7-parameter shell model which exploits usual simplifications of Green-Lagrange strain measures. Comparisons are also presented of two different ways of implementing the incompatible mode method for reducing the number of shell model parameters to 6. One implementation uses additive decomposition of the strains and the other additive decomposition of the deformation gradient. Several numerical examples are given to illustrate performance of the shell elements developed herein.