Keywords: numerical methods, finite elements, couple stress elasticity, strain gradient theory, plates.

The linear couple stress theory of elasticity is proposed by Mindlin and Tiersten [1], while the two-dimensional version of the theory is treated separately by Mindlin [2]. The couple stress theory is an extension of the classical theory where the material can transmit couple stress as well as the usual force. Its main difference from the micropolar theory [3] is that material rotations are not independent but they are defined in terms of displacement components. There is now enough evidence [4] that certain problems involving high stress concentrations and significant dependence on the size of a structural element and the dominant micro-structural length scale of the material can be modeled more efficiently by the couple stress theory or the micropolar theory.

The application of the displacement based finite element method to micropolar theory is presented by the authors in [5], while some preliminary results regarding the formulation of displacement finite element models for the couple stress elasticity are given in [6]. It is noted that in the potential energy functional for couple stress theory, second order derivatives of displacement appear and therefore the interpolating displacement fields should be at least continuous. This requirement of high continuity has led most of researchers to devise alternative mixed (or hybrid) finite elements requiring only continuity ([7,8,9,10]). Mixed finite element models, however, are not trouble free. One of their main drawbacks is the large number of degrees of freedom which they possess and the fact that they may work well on certain problems and perform very poorly on other problems of the same category. Moreover, in commercial finite element programs for practical structural analysis, displacement based finite elements are almost exclusively used.

This paper deals with the formulation of the basic 3-node,18-degree of freedom triangular element of couple stress elasticity. Cubic interpolation polynomials are employed for the approximation of the displacement fields within one element and all stiffness matrices and load vectors are derived explicitly. The performance of the present element is assessed and a comparison is made with an alternative displacement based 3-node, 9-degree of freedom triangle [6].

The present finite element model seems to produce results of acceptable accuracy but inferior to results obtained by the simpler and less expensive 3-node, 9-dof triangular element, which is proven to be a very good performer.

1

R. D. Mindlin and H.F. Tiersten, "Effects of couple stress in linear elasticity", Archs ration. Mech. Anal., 11, 415-448, 1962.

2

R.D. Mindlin, "Influence of couple-stresses on stress concentrations", Exp. Mech.,1, 1-7, 1963. doi:10.1007/BF02327219

3

A.C. Eringen, "Linear theory of Micropolar Elasticity", J. Math. Mech., 15 (6), 909-923, 1966.

4

N.A. Fleck and J.W. Hutchinson, "Strain gradient plasticity", Adv. Appl. Mech., 33, 295-361, 1997. doi:10.1016/S0065-2156(08)70388-0

5

E. Providas, M.A. Kattis, "Finite element method in plane Cosserat elasticity", Computers & Structures, 80, 2059-2069, 2002. doi:10.1016/S0045-7949(02)00262-6

6

E. Providas, "Displacement finite element method for couple stress theory", 6th International Conference on Computational Structures Technology, Prague, 4-6 Sept., 2002. doi:10.4203/ccp.75.24

7

L.R. Hermann, "Mixed finite elements for couple-stress analysis", in "Hybrid and mixed finite element methods", S.N.Alturi, R.H. Gallagher and O.C. Zienkiewicz, (Editors), John Wiley & Sons Ltd., 1-17, 1983.

8

R.D. Wood, "Finite element analysis of plane couple-stress problems using first order stress functions", Int. J. Numer. Meth. Engng., 26, 489-509, 1988. doi:10.1002/nme.1620260214

9

Z.C. Xia and J.W. Hutchinson, "Crack tip fields in strain gradient plasticity", J. Mech. Phys. Solids, 44, 1621-1648, 1996. doi:10.1016/0022-5096(96)00035-X

10

J.Y. Shu, W.E. King and N. A. Fleck, "Finite elements for materials with strain gradient effects", Int. J. Numer. Meth. Engng., 44, 373-391, 1999. doi:10.1002/(SICI)1097-0207(19990130)44:3<373::AID-NME508>3.0.CO;2-7

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