Keywords: Cosserat point element, hyperelasicity, quadratic, tetrahedral element.

It is well known that generating a mesh for a general three-dimensional region is easier using tetrahedral elements than using brick elements. However, homogeneously deformable four node tetrahedral elements exhibit a stiff response to bending relative to eight node brick elements, which admit inhomogeneous deformations. Quadratic tetrahedral elements based on quadratic shape functions with full integration eliminate stiffness to bending modes but are known to exhibit inaccurate response for nearly incompressible materials. Mixed methods can be used for nearly incompressible materials, but they exhibit soft response to bending. Also, it is known that full integration elements exhibit poor response for contact problems with large deformations. In particular, the default quadratic tetrahedral element in the commercial code ABAQUS is an undocumented patented modified element. This modified element is indeed more robust than the full integration element but it is inaccurate in bending.

The objective of this work is to develop a quadratic ten node tetrahedral Cosserat point element (CPE) for nonlinear isotropic hyperelastic materials. In the standard Bubnov-Galerkin approach the kinematic approximation is assumed to be valid pointwise in the element and the constitutive equation is evaluated at the Gauss points of integration to determine element stiffnesses. In contrast, in the CPE approach [1,2] the kinematic approximation is only used to connect nodal director vectors (i.e. nodal positions) to element director vectors. Specifically, hyperelastic constitutive equations of the CPE are developed by treating the element as a structure with a strain energy function that models the response to all possible modes of deformation. The coefficients in the strain energy function for inhomogeneous deformation modes are determined by matching small deformation solutions for bending problems. Examples show that the resulting CPE exhibits accurate, robust response with a smooth transition from compressible to nearly incompressible material behavior. This new CPE can be used for thin structures and three-dimensional bodies. It also outperforms the modified element of ABAQUS both in accuracy and robustness, even for contact problems with large deformations.

1

B. Nadler, M.B. Rubin, "A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point", Int. J. Solids Struct., 40, 4585-4614, 2003. doi:10.1016/S0020-7683(03)00210-5

2

M. Jabareen, M.B. Rubin, "A generalized Cosserat point element (CPE) for isotropic nonlinear elastic materials including irregular 3-D brick and thin structures", J. of Mech. of Materials and Struct., 3, 1465-1498, 2008. doi:10.2140/jomms.2008.3.1465

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