Keywords: numerical methods, finite elements, boundary-value problems, gradient elasticity, couple stress, strain gradient theory.

The classical theory of elasticity, in spite its great success in modeling quite satisfactorily most engineering structure problems, fails to give adequate answers in certain applications involving high stress concentrations and significant dependence on the ratio of a dimension of the structural element to a characteristic material length parameter. In these cases the use of generalized theories seem to be more appropriate. A generalization of the classical theory has been developed by Cosserat brothers in which each material point can rotate independently of translation and the material can transmit couple stress as well as the usual force. Mindlin has proposed extended linear theories of elasticity in which the potential energy-density is a function of the gradients of the strain in addition to the strain. A special case of these extended theories, and the simplest possible type of generalization of the classical theory, is the couple stress theory that includes the effects of couple stress in addition to the usual force, while the couple stresses are related to strain gradients via the shear modulus and the characteristic material length. Its main difference from the Cosserat theory is that material rotations are not independent but they are defined in terms of displacements. A particularly comprehensive study of the linear couple stress theory is presented by Mindlin and Tiersten [1], while the two-dimensional version of the theory is treated separately by Mindlin [2].

The equations of linear couple stress theory of elasticity are considerably more involved than of the classical theory. Despite its complexity several problems have been solved analytically. The extension of the displacement finite element method to couple stress problems, however, is not straightforward as it is the case for the Cosserat theory [3]. In the potential energy functional, 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 finite elements requiring only continuity ([4,6,5,7,8]).

The present paper is concerned with the extension of the traditional displacement finite element method to the solution of boundary-value problems in the two-dimensional linear couple stress theory of elasticity. Three triangular finite elements of the displacement field are discussed. Of particular interest is a triangular finite element with merely a total of nine degrees of freedom obtained through a modified principle of potential energy with relaxed continuity requirements. Regarding the fact that there are not any lower order displacement finite elements for couple stress elasticity, this element is important for both the theoretical point of view as well as the practical use. All finite element equations are derived analytically and no numerical integration or other numerical tricks are used. The patch test is satisfied, while numerical results of an application of the finite element model to a boundary value problem with known solution indicate good performance.

1

R. D. Mindlin and H.F. Tiersten, "Effects of couple stress in linear elasticity", Archs ration. Mech. Anal., 11, 415-448, 1962. doi:10.1007/BF00253946

2

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

3

E. Providas and M.A. Kattis, "Finite element method in plane Cosserat elasticity", in "Computational Techniques for Materials, Composites and Composite Structures", B.H.V. Topping (Editor), Civil-Comp Press, 47-56, 2000. doi:10.4203/ccp.67.1.6

4

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.

5

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

6

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

7

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

8

E. Amanatidou and N. Aravas, quot;Mixed finite element formulations of strain-gradient elasticity problems", Comput. Methods. Appl. Mech. Engrg., 191, 1723-1751, 2002. doi:10.1016/S0045-7825(01)00353-X

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