A comparison between the recently developed Cosserat brick element (see [9]) and other standard elements known from the literature is presented in this paper. The Cosserat brick element uses a director vector formulation based on the theory of a Cosserat point. The strain energy for a hyperelastic element is split additively into parts for homogeneous and inhomogeneous deformations respectively. The kinetic response due to inhomogeneous deformations uses constitutive constants that are determined by analytical solutions of a rectangular parallelepiped to the deformation modes of bending, torsion and hourglassing. Standard tests are performed which typically exhibit hourglassing or locking for many other finite elements. These tests include problems for beam and plate bending, shell structures and nearly incompressible materials, as well as for buckling under high pressure loads. For all these critical tests the Cosserat brick element exhibits robustness and reliability. Moreover, it does not depend on user-tuned stabilization parameters. Thus, it shows promise of being a truly user-friendly element for problems in nonlinear elasticity.

SUMMARYThis paper describes an improvement of the Cosserat point element formulation for initially distorted, non-rectangular shaped elements in 2D. The original finite element formulation for 3D large deformations shows excellent behaviour for sensitive geometries, large deformations, coarse meshes, bending dominated and stability problems without showing undesired effects such as locking or hourglassing, as long as the initial element shape resembles that of a rectangular parallelepiped. In the following, an extension of this element formulation for 2D plane strain is presented which has the same good properties also for the case of non-rectangular initial element shapes. Results of numerical tests are presented, that clearly show the advantages of the improved Cosserat point element compared to the standard displacement elements and the original version of the Cosserat point element.

A three-dimensional 8-node brick continuum finite element formulation for incompressible finite elasticity is presented. The core idea is to introduce a substructure consisting of eight sub-elements inside each finite element, further referred to as macro-element. For each of the subelements, the deformation is averaged. The weak form for each sub-element is based on the Hu-Washizu principle. The response of each sub-element is assembled and projected onto the eight external nodes of the macro-element. The introduction of deformable sub-elements in case of incompressible elasticity has two major advantages. Firstly, it is possible to suppress locking by evaluating the volumetric part of the response only in the macro-element instead of in each of the sub-elements. Secondly, no integration is necessary due to the use of averaged deformations on the sub-element level. The idea originates from the Cosserat point element developed in Nadler and Rubin (Int J Solids Struct 40:4585-4614, 2003). A consistent transition between the Cosserat point macro-element and a displacement macro-element formulation using a kinematical description from the enhanced strain element formulation (Flanagan, Belytschko in Int J Numer Methods Eng 17: 1981) or (Belytschko et al. in Comput Methods Appl Mech Eng 43:251-276, 1984) and the principle of Hu-Washizu is presented. The performance is examined by means of numerical examples.

The theory of Cosserat points is the basis of a 3D finite element formulation for large deformations in structural mechanics, that recently was presented by [1]. First investigations [2] have revealed, that this formulation is free of showing undesired locking or hourglassing‐phenomena. It additionally shows excellent behaviour for any type of incompressible material, for large deformations and sensitive structures such as plates or shells. The formulation initially was restricted to a Neo‐Hookean material. This work will present the extension to a general elastic Ogden material and the verification of the chosen model. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The theory of Cosserat points is the basis of a 3D finite element formulation allowing for large deformations in structural mechanics, that recently was presented by [1]. First attempts have revealed, that this formulation is free of showing undesired locking or hourglassing‐phenomena. It additionally shows excellent behaviour for any type of incompressible material, for large deformations and sensitive structures such as plates or shells. Within the theory of Cosserat points, the position vectors X and x, are described through director vectorsDi and di by use of trilinear shape functions Ni for an 8‐node brick element.The special choice of shape functions Ni allows for director vectors with which the deformation can be split into a homogeneous and an inhomogeneous part. This split enables the use of stiffnesses that correspond to different deformation modes. Analytical solutions to the inhomogeneous deformation modes are incorporated in the formulation and avoid the undesired phenomena. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)