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Cosserat Spectrum of an Axisymmetric Elasticity Problem for a Finite-Length Solid Cylinder

Published online by Cambridge University Press:  02 July 2018

Yu. V. Tokovyy
Yu. V. Tokovyy*
Affiliation:
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of UkraineLviv, Ukraine
*
*Corresponding author (tokovyy@gmail.com)
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Abstract

An algorithm for the computation and analysis of the Cosserat spectrum for an axisymmetric elasticity boundary-value problem in a finite-length solid cylinder with boundary conditions in terms of stresses is proposed. By making use of the cross-wise superposition method, the spectral problem is reduced to systems of linear algebraic equations. A solution method for the mentioned systems is presented and the asymptotic behavior of the Cosserat eigenvalues is established. On this basis, the key features of the Cosserat spectrum for the mentioned problem are analyzed with special attention given to the effect of the cylinder aspect ratio.

Keywords

Cosserat spectrumAxisymmetric elasticity problemFinite-length solid cylinder
Type
Research Article
Information
Journal of Mechanics , Volume 35 , Issue 3 , June 2019 , pp. 343 - 349
Copyright
© The Society of Theoretical and Applied Mechanics 2018 

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References

REFERENCES

Cosserat, E. et Cosserat, F., “Sur les Ėquations de la Théorie de l′Ėlasticité,” Comptes Rendus de l'Académie des Sciences, 126, pp. 1089–1091 (1898).Google Scholar
Mikhlin, S. G., “The Spectrum of a Family of Operators in the Theory of Elasticity,” Russian Mathematical Surveys, 28, pp. 45–88 (1973).Google Scholar
Kozhevnikov, A., “A History of the Cosserat Spectrum,” in Rossmann, J., Takac, P., Wildenhain, G. (eds.), Operator Theory: Advances and Applications, The Maz'ya Anniversary Collection, 109, pp. 223–234 (1999).Google Scholar
Soykan, Y. and Göcen, M., “A Short History of the Cosserat Spectrum Problem,” Karaelmas Science and Engineering Journal, 1, pp. 55–58 (2011).Google Scholar
Sadd, M. H., Elasticity, Theory, Applications, and Numerics, 3rd ed., Academic Press, Oxford, 2014.Google Scholar
Valeev, V. E., “The Cosserat Spectrum of the Boundary-Value Problem of the Elasticity Theory,” Vestnik St. Petersburg University, Mathematics, 33, pp. 10–15 (2000).Google Scholar
Knops, R. J. and Payne, L. E., Uniqueness Theorems in Linear Elasticity, Springer, New York (1971).Google Scholar
Cosserat, E. et Cosserat, F., “Sur la Déformation Infiniment Petite d'un Corps Ėlastique Soumis à des Forces Données,” Comptes Rendus de l'Académie des Sciences, 133, pp. 271–273 (1901).Google Scholar
Cosserat, E. et Cosserat, F., “Sur un Point Critique Particulier de la Solution des Ėquations de l'Ėlasticité, dans le cas où les Efforts sur la Frontière sont Donnés,” Comptes Rendus de l'Académie des Sciences, 133, pp. 382–384 (1901).Google Scholar
Schermann, D. J., “Sur la Distribution des Nombres Caractéristiques d'Equations Intégrales du Problème Plan de la Théorie d'Elasticité,” Publications de l'Institut Séismologique de l'Académie des Sciences de URSS, 82, pp. 1–24 (1938).Google Scholar
Trefftz, E., “Mathematische Elastizitätstheorie,” in: von Grammel, R., Henning, F. et al. (eds.) Handbuch der Physik, Vol. 6: „ Mechanik der Elastischen Körper, Springer, Berlin (1928).Google Scholar
Mazya, V. G. and Mikhlin, S. G., “On the Cosserat Spectrum of the Equations of the Theory of Elasticity,” Vestnik Leningradskogo Univiversiteta Mathematika, 3, pp. 58–63 (1967) [in Russian].Google Scholar
Mikhlin, S. G., “On Cosserat Functions,” in: Problemy Matemematicheskogo Analiza. Kraevye Zadachi i Integralnye Uravneniya, Izdatelstvo Leningradskogo Universiteta, pp. 56–59 (1966) [in Russian].Google Scholar
Algazin, S. D., “Cosserat Spectrum of the First Boundary-Value Problem of Elasticity,” Journal of Applied Mechanics and Technical Physics, 54, pp. 287–294 (2013).Google Scholar
Liu, W. and Markenscoff, X., “The Cosserat Spectrum for Cylindrical Geometries (Part 1: Discrete Subspace),” International Journal of Solids and Structures, 37, pp. 1165–1176 (2000).Google Scholar
Liu, W. and Markenscoff, X., “The Cosserat Spectrum for Cylindrical Geometries (Part 2: ũ(-1) Subspace and Applications),” International Journal of Solids and Structures, pp. 1177–1190 (2000).Google Scholar
Grinchenko, V. T., “The Biharmonic Problem and Progress in the Development of Analytical Methods for the Solution of Boundary-Value Problems,” Journal of Engineering Mathematics, 46, pp. 281–297 (2003).Google Scholar
Zernov, V., Pichugin, A. V. and Kaplunov, J., “Eigenvalue of a Semi-Infinite Elastic Strip,” Proceedings of the Royal Society of London A, 462, pp. 1255–1270 (2006).Google Scholar
Meleshko, V. V., “Selected Topics in the History of the Two-Dimensional Biharmonic Problem,” Applied Mechanics Reviews, 56, pp. 33–85 (2003).Google Scholar
Meleshko, V. V., Tokovyy, Yu. V. and Barber, J. R., “Axially Symmetric Temperature Stresses in an Elastic Isotropic Cylinder of Finite Length,” Journal of Mathematical Sciences, 176, pp. 646–669 (2011).Google Scholar
Meleshko, V. V. and Tokovyy, Yu. V., “Equilibrium of an Elastic Finite Cylinder under Axisymmetric Discontinuous Normal Loadings,” Journal of Engineering Mathematics, 78, pp. 143–166 (2013).Google Scholar
Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, 4th ed. Cambridge University Press, Cambridge (1927).Google Scholar