Tetrahedral Finite Element with Rotational Degrees of Freedom for Cosserat and Cauchy Continuum Problems
Publication: Journal of Engineering Mechanics
Volume 141, Issue 2
Abstract
This study formulates a simple tetrahedral finite element equipped with rotational degrees of freedom that can be used effectively to solve problems for both Cosserat and Cauchy continua. The formulation makes no assumption regarding the symmetry of the stress tensor, and such symmetry is achieved at convergence for the Cauchy problems. The numerical implementation of the new element is straightforward, and numerical tests demonstrate second-order accuracy in the case of both Cosserat and Cauchy elasticity.
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References
Allman, D. J. (1984). “A compatible triangular element including vertex rotations for plane elasticity analysis.” Comput. Struct., 19(1–2), 1–8.
Alshibli, K. A., Alsaleh, M. I., and Voyiadjis, G. Z. (2006). “Modelling strain localization in granular materials using micropolar theory: Numerical implementation and verification.” Int. J. Numer. Anal. Methods Geomech., 30(15), 1525–1544.
Baluch, M. H., Goldberg, J. E., and Koh, S. L. (1972). “Finite element approach to plane microelasticity.” J. Struct. Div., 98(9), 1957–1964.
Bauer, S., Schäfer, M., Grammenoudis, P., and Tsakmakis, C. (2010). “Three-dimensional finite elements for large deformation micropolar elasticity.” Comput. Methods Appl. Mech. Eng., 199(41–44), 2643–2654.
Chang, C. S., Wang, T. K., Sluys, L. J., and van Mier, J. G. M. (2002). “Fracture modeling using a micro-structural mechanics approach—I. Theory and formulation.” Eng. Fract. Mech., 69(17), 1941–1958.
Dassault Systèmes. (2010). ABAQUS 6.10 documentation: ABAQUS analysis user’s manual, 〈http://abaqusdoc.ucalgary.ca/books/usb/default.htm〉 (Mar. 31, 2014).
Eringen, A. C. (1999). “Theory of micropolar elasticity.” Microcontinuum field theories, Springer, New York, 101–248.
Gauthier, R. D., and Jahsman, W. E. (1976). “Bending of a curved bar of micropolar elastic material.” J. Appl. Mech., 43(3), 502–503.
Goldberg, J. E., Baluch, M. H., Korman, T., and Koh, S. L. (1974). “Finite element approach to bending of micropolar plates.” Int. J. Numer. Methods Eng., 8(2), 311–321.
Padovan, J. (1978). “Applications of 3-D finite element procedures to static and dynamic problems in micropolar elasticity.” Comput. Struct., 8(2), 231–236.
Pawlak, T. P., Yunus, S. M., and Cook, R. D. (1991). “Solid elements with rotational degrees of freedom: Part II—Tetrahedron elements.” Int. J. Numer. Methods Eng., 31(3), 593–610.
Pothier, A., and Rencis, J. J. (1994). “Three-dimensional finite element formulation for microelastic solids.” Comput. Struct., 51(1), 1–21.
Providas, E., and Kattis, M. A. (2002). “Finite element method in plane Cosserat elasticity.” Comput. Struct., 80(27–30), 2059–2069.
Riahi, A., and Curran, J. H. (2009). “Full 3D finite element Cosserat formulation with application in layered structures.” Appl. Math. Modell., 33(8), 3450–3464.
Sachio, N., Benedict, R., and Lakes, R. (1984). “Finite element method for orthotropic micropolar elasticity.” Int. J. Eng. Sci., 22(3), 319–330.
Sze, K. Y., and Pan, Y. S. (2000). “Hybrid stress tetrahedral elements with Allman’s rotational D.O.F.s.” Int. J. Numer. Methods Eng., 48(7), 1055–1070.
Tian, R., Matsubara, H., and Yagawa, G. (2006). “Advanced 4-node tetrahedrons.” Int. J. Numer. Methods Eng., 68(12), 1209–1231.
Tian, R., and Yagawa, G. (2005). “Generalized nodes and high-performance elements.” Int. J. Numer. Methods Eng., 64(15), 2039–2071.
Timoshenko, S., and Goodier, J. (1970). Theory of elasticity, McGraw Hill, New York.
Yunus, S. M., Pawlak, T. P., and Cook, R. D. (1991). “Solid elements with rotational degrees of freedom: Part 1—Hexahedron elements.” Int. J. Numer. Methods Eng., 31(3), 573–592.
Zienkiewicz, O., Taylor, R., and Zhu, J. (2005). The finite element method: Its basis and fundamentals, Elsevier Butterworth-Heinemann, Oxford, U.K.
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Published In
Journal of Engineering Mechanics
Volume 141 • Issue 2 • February 2015
Copyright
© 2014 American Society of Civil Engineers.
History
Received: Apr 6, 2014
Accepted: Aug 15, 2014
Published online: Sep 11, 2014
Published in print: Feb 1, 2015
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