A Cosserat point element (CPE) for nearly planar problems (including thickness changes) in nonlinear elasticity

This paper is concerned with nonlinear elastic deformations of a body that is a right-cylinder bounded by two major non-planar surfaces that are located symmetrically relative to the body’s planar middle surface. Deformations which maintain this symmetry are called nearly planar since they allow non-uniform thickness changes. Here, a four node quadrilateral Cosserat point element (CPE) is developed for these nearly planar problems. This planar CPE has three degrees of freedom at each node: two associated with the standard in-plane nodal displacements and one associated nodal thickness changes. The planar CPE can model problems where the thickness is specified (including plane strain) and where the normal component of the nodal traction vanishes (modified plane stress). In particular, thickness changes are modeled and the nonlinear three-dimensional constitutive equation is used without plane stress type approximations. Example problems are considered which compare the response of the planar CPE with other element formulations in the computer codes ABAQUS, ADINA, ANSYS and FEAP.