Abstract
In this paper, systematic approaches to determine the polynomial stress functions for anisotropic plane problems are presented based on the Lekhnitskii’s theory of anisotropic elasticity. It is demonstrated that, for plane problems, there are at most four independent polynomials for arbitrary n-th order homogeneous polynomial stress functions: three independent polynomials for n equal to two and four for n greater than or equal to three. General expressions for such polynomial stress functions are derived in explicit forms. Unlike the isotropic case, the polynomials for anisotropic problems are functions of material constants, because the elastic constants cannot be eliminated in the governing equation for general anisotropic cases. The polynomials can be used as analytical trial functions to develop the new 8-node hybrid element (ATF-Q8) for anisotropic problems. This ATF-Q8 element demonstrated excellent performance in comparison with traditional numerical methods through several testing examples.
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Fu X.-R., Cen S., Li C.F., Chen X.-M.: Analytical trial function method for development of new 8-node plane element based on the variational principle containing Airy stress function. Eng. Comput. 27(4), 442–463 (2010)
Wang, M.-Z., Xu, B.-X., Zhao, Y.-T.: General representations of polynomial elastic fields. Accept by J. Appl. Mech-T. ASME (2011)
Cen S., Fu X., Zhou G., Zhou M., Li C.: Shape-free finite element method: the plane hybrid stress-function (HS-F) element method for anisotropic materials. Sci. China Phys. Mech. Astron. 54(4), 653–665 (2011)
Lekhnitskii S.G.: Anisotropic Plates. Gordon and Breach Science Publishers, New York (1968)
Vekua, I.N.: Generalized Analytic Functions, People’s Eduction Press (Chinese) (Translated from Russian) (1960)
Lekhnitskii S.G.: Theory of Elasticity of an Anisotropic Body. Mir Publishers, Moscow (1981)
Brown J.W., Churchill R.V.: Complex Variables and Applications. McGraw Hill Book Co., New York (2003)
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Zhao, YT., Wang, MZ., Chen, Y. et al. Polynomial stress functions of anisotropic plane problems and their applications in hybrid finite elements. Acta Mech 223, 493–503 (2012). https://doi.org/10.1007/s00707-011-0566-z
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DOI: https://doi.org/10.1007/s00707-011-0566-z