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CivilComp Proceedings
ISSN 17593433 CCP: 75
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and Z. Bittnar
Paper 24
Displacement Finite Element Method for Couple Stress Theory E. Providas
Department of Civil Engineering, University of Thessaly, Volos, Greece E. Providas, "Displacement Finite Element Method for Couple Stress Theory", in B.H.V. Topping, Z. Bittnar, (Editors), "Proceedings of the Sixth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 24, 2002. doi:10.4203/ccp.75.24
Keywords: numerical methods, finite elements, boundaryvalue problems, gradient elasticity, couple stress, strain gradient theory.
Summary
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 energydensity 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 twodimensional 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 boundaryvalue problems in the twodimensional 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. References
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