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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 85
Edited by: B.H.V. Topping
Paper 24

Towards Hybrid Equilibrium Elements for Microcontinua

E.A.W. Maunder and C.W. Smith

School of Engineering, Computer Science and Mathematics, University of Exeter, United Kingdom

Full Bibliographic Reference for this paper
E.A.W. Maunder, C.W. Smith, "Towards Hybrid Equilibrium Elements for Microcontinua", in B.H.V. Topping, (Editor), "Proceedings of the Fifteenth UK Conference of the Association of Computational Mechanics in Engineering", Civil-Comp Press, Stirlingshire, UK, Paper 24, 2007. doi:10.4203/ccp.85.24
Keywords: cellular solids, chiral, honeycomb, hybrid element, micropolar, Cosserat.

Recent developments of man-made cellular materials for lightweight structures for aerospace and marine applications have lead to increasing interest in modelling their behaviour using microcontinuum [1,2,3] theories, such as Cosserat theory. This approach allows their complex and unusual microstructures, which give rise to non-traditional behaviours such as a negative Poisson's ratio, to be taken into account [4]. New elements based on such theories can be implemented into finite element code [5,6] to facilitate modelling of larger scale structures. Applications are not however limited to new materials since large civil engineering structures such as tall buildings which contain a repetitive form of substructure have been effectively modelled in this way using a micropolar theory [2].

This paper investigates the extension of hybrid equilibrium concepts, as developed for the classical plane stress form of linear elasticity, to the micropolar theory which exploits couple-stresses and non-symmetric stress tensors. We begin by first considering micropolar theory in the context of triangular primitive elements [7], which could be assembled into macro-elements of a more general shape. The aim with macro-elements is to effectively remove spurious kinematic modes and thereby create stable forms of hybrid elements.

It is shown that a primitive hybrid element based on linear fields of stress and quadratic fields of couple-stress is hyperstatic and free of spurious kinematic modes. The direction of further investigation is outlined, with the longer term aim of including chiral characteristics of the microstructure.

A.C. Eringen, "Theory of Micropolar Elasticity", Chapter 7 of Fracture, Vol II, Liebowitz H. ed., Academic Press, 1968.
A.C. Eringen, "Microcontinuum Field Theories", Vol I, Foundations and Solids, Springer, 1999.
R.S. Lakes, "Elastic freedoms in cellular solids and composite materials", Mathematics of Multiscale Materials, K. Golden, G. Grimmert, R. James, G. Milton, P. Sen (Editors), IMA, 99, Springer, 129-153, 1998.
F. Scarpa, S. Blain, T. Lew, D. Perrott, M. Ruzzene, J.R. Yates, "Analytical and experimental analysis on the elastic buckling of hexagonal chiral cell honeycombs", Accepted in Composite Material Engineering A.
R.D. Wood, "Finite element analysis of plane couple-stress problems using first order stress functions", Int. J. Num. Meth. In Eng. 26, 489-509, 1988. doi:10.1002/nme.1620260214
S. Nakamura, R.S. Lakes, "Finite element analysis of Saint-Venant end effects in micropolar elastic solids", Engineering Computations, 12, 571-587, 1995.
E.A.W. Maunder, J.P. Moitinho de Almeida, "A triangular hybrid equilibrium plate element of general degree", Int. J. Num. Meth. In Eng. 63, 315-350, 2005. doi:10.1002/nme.1271

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