Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andelfinger U (1991) Untersuchungen zur Zuverlässigkeit hybrid-gemischter Finiter Elemente für Flächentragwerke. PhD thesis, Institut für Baustatik, Universität Stuttgart
Andelfinger U, Ramm E (1993) EAS-elements for 2-dimensional, 3- dimensional, plate and shell structures and their equivalence to hr-elements. Int J Numer Meth Eng 36(8):1311–1337
Belytschko T, Bindemann LP (1986) Assumed strain stabilization of the 4-node quadrilateral with 1-point quadrature for nonlinear problems. Comput Meth Appl Mech Eng 54:279–301
Belytschko T, Ong JS-J, Liu WK, Kennedy JM (1984) Hourglass control in linear and nonlinear problems. Comput Meth Appl Mech Eng 43:251–276
Belytschko T, Ong JJS, Liu W, Kennedy J (1984) Hour glass control in linear and nonlinear problems. Comp Meth App Mech Eng (43): 251–276
Hughes TRJ (1987) The Finite Element Method. Prentice Hall, Englewood Cliffs, New Jersey
Jaquotte OP, Oden JT (1986) An accurate and efficient a posteriori control of hourglass instabilities in underintegrated linear and nonlinear elasticity. Comput Meth Appli Mech Eng 55:105–128
Korelc J, Wriggers P (1996) An efficient 3d enhanced strain element with taylor expansion of the shape functions. Comput Mech 19:30–40
Nadler B, Rubin MB (2003) A new 3-d finite element for nonlinear elastic-ity using the theory of a cosserat point. Int J Solids Struct 40:4585–4614
Pian THH (1964) Derivation of element stiffness matrices by assumed stress distributions. AIAA–J 2, 7:1333–1336
Pian THH, Sumihara K (1984) Rational approach for assumed stress finite elements. Int J Numer Meth Eng 20:1685–1695
Reese S, Wriggers P (2000) A stabilization technique to avoid hourglassing in finite elasticity. Int J Numer Meth Eng 48:79–110
Reese S, Wriggers P, Reddy BD (2000) A new locking-free brick element technique for large deformation problems in elasticity. Comput Struct 75:291–304
Simo JC, Armero F (1992) Geometrically nonlinear enhanced strain mixed methods and the method of incompatible modes. Int J Numer Meth Eng 33:1413–1449
Sussmann T, Bathe KJ (1987) A finite element formulation for nonlinear incompressible elastic and inelastic analysis. Comput Struct 26(1–2):357–409
Wriggers P, Korelc J (1996) On enhanced strain methods for small and finite deformations of solids. Comput Mech 18:413–428
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Loehnert, S., Boerner, E., Rubin, M. et al. Response of a nonlinear elastic general Cosserat brick element in simulations typically exhibiting locking and hourglassing. Comput Mech 36, 255–265 (2005). https://doi.org/10.1007/s00466-005-0662-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-005-0662-9