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CivilComp Proceedings
ISSN 17593433 CCP: 79
PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and C.A. Mota Soares
Paper 135
An Assessment of Displacement Finite Elements for Couple Stress Elasticity E. Providas+ and M.A. Kattis*
+Department of Exact Sciences, Higher Technological Education Institute of Larissa, Greece
E. Providas, M.A. Kattis, "An Assessment of Displacement Finite Elements for Couple Stress Elasticity", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Seventh International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 135, 2004. doi:10.4203/ccp.79.135
Keywords: numerical methods, finite elements, couple stress elasticity, strain gradient theory, plates.
Summary
The linear couple stress theory of elasticity is proposed by Mindlin and
Tiersten [1], while the twodimensional 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 microstructural 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 3node,18degree 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 3node, 9degree 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 3node, 9dof triangular element, which is proven to be a very good performer. References
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