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.

1

A.C. Eringen, "Theory of Micropolar Elasticity", Chapter 7 of Fracture, Vol II, Liebowitz H. ed., Academic Press, 1968.

2

A.C. Eringen, "Microcontinuum Field Theories", Vol I, Foundations and Solids, Springer, 1999.

3

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.

4

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.

5

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

6

S. Nakamura, R.S. Lakes, "Finite element analysis of Saint-Venant end effects in micropolar elastic solids", Engineering Computations, 12, 571-587, 1995.

7

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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