A generalized Cosserat point element (CPE) for isotropic nonlinear elastic materials including irregular 3-D brick and thin structures

Abstract

A generalized form for the strain energy of inhomogeneous deformations is developed for a 3-D brick Cosserat Point Element (CPE) which includes full coupling of bending and torsional modes of deformation. The constitutive coefficients, which depend on the reference geometry of the element, are determined by solving eighteen bending problems and six torsion problems on special elements that are parallelepipeds with two right angles. The resulting constitutive coefficients ensure that the strain energy for inhomogeneous deformations remains a positive definite function of the inhomogeneous strain measures for all reference element shapes. A number of example problems are considered which show that the generalized CPE produces results as accurate as enhanced strain and incompatible elements for thin structures and is free of hourglass instabilities typically predicted by these enhanced elements in regions experiencing combined high compression with bending. Recently, Nadler and Rubin [2003] developed a 3-D eight noded brick element based on the theory of a Cosserat point [Rubin 1995; 2000]. This Cosserat point element (CPE) can be used to formulate the numerical solution of dynamic problems for nonlinear hyperelastic materials. The kinematics of the CPE are characterized by eight element director vectors and the kinetics propose eight balance laws of director momentum to determine the dynamic response of the element. The locations of the nodes in the current deformed configuration are characterized by eight nodal director vectors and the element directors are related to the nodal directors by standard tri-linear shape functions. Moreover, the CPE theory considers the element as a structure and introduces a strain energy function which characterizes the response of the structure. Also, the nodal forces are related to derivatives of the strain energy function through algebraic relations in a similar manner to the relationship of the stress to derivatives of the strain energy function in the full three-dimensional theory of hyperelastic materials. The CPE can easily be implemented into standard finite element programs. Specifically, the nodal positions (nodal director vectors) are used as input variables to determine the nodal forces and tangent stiffness as output variables. Here, the element assembly and solution procedures in the computer program FEAP [Taylor 2005] were used to satisfy the equilibrium equations by updating the present nodal director vectors. In the standard finite element procedures for hyperelastic materials the response of the element is determined by integrals over the element region which assume that the kinematic approximation is valid pointwise. This is in contrast with the CPE which needs no integration over the element region. In