On a nonlinear theory of plates and shells including consistent and inconsistent kinematics and the finite element method

Abstract

A nonlinear theory of plates and shells for thick and thin models accounting for consistent and inconsistent kinematical approximations is presented in general curvilinear tensorial form. The incremental shell governing equations for the finite element formulation of boundary value problems for finite strains and large deformations are derived employing the incremental virtual work principle defined in a Lagrangian description.Sample problems are investigated numerically employing a hyperelastic material model for which both general and specific forms are introduced. Finite element solutions of thick and thin plate and shell problems are obtained by employing lower and higher order displacement models and are compared with results deduced from application of the finite element method based on exact equations of continuum mechanics.It is found that the kinematical theories, which are based on one general formulation, adequately predict displacements and normal stresses. However, major discrepancies arise in the shear stress distributions at finite strains when exact and lower order theories are compared. Finally, it is noted that the consistent kinematical theory described by a cubic variation of in-plane and transverse displacements through the thickness coordinate yields the most accurate and meaningful solutions. Furthermore, it is shown that higher order kinematical theories become significant in the analysis of thick structures.