539.3 Modeling of the stresses in thin composite cylindrical shells based on the asymptotic theory

Dimitrienko Y. I. (Bauman Moscow State Technical University), Gubareva E. A. (Bauman Moscow State Technical University), Pichugina A. Y. (Bauman Moscow State Technical University)


doi: 10.18698/2309-3684-2018-3-114132

The previously developed general asymptotic theory of thin multilayer shells is used for the case of cylindrical shells. The ratios are presented in explicit analytical form for all six components of the stress tensor in a thin multilayer elastic cylindrical shell, depending on the deformations, curvatures of the middle surface of the shell, as well as their derivatives along the longitudinal coordinates. The obtained formulas make it possible to calculate all the distributions of the components of the stress tensor over the thickness in a cylindrical shell after finding solutions to the two-dimensional problem of the theory of KirchhoffLyav shells. An example is given of the calculation of stresses in a cylindrical composite shell underaxisymmetric bending by pressure. To calculate stresses by these formulas, only a differentiation of displacements is required - a deflection and two displacements of the middle surface of the shell, for which an analytical solution is obtained.

[1] Lyav A. Matematicheskaya teoriya uprugosti [Mathematical theory of elasticity]. Moscow, ONTI Publ., 1935, 674 p.
[2] Timoshenko S.P., Vojnovskij-Kriger S. Plastinki i obolochki: per. s angl. [Of plates and shells]. Moscow, Nauka Publ., 1966, 635 p.
[3] Vasil'ev V.V. Mekhanika konstrukcij iz kompozicionnyh materialov [Mechanics of structures made of composite materials], Moscow, Mashinostroenie,Publ., 1988, 272 p.
[4] Grigolyuk E.I., Kulikov G.M. Mekhanika kompozitnikh materialov – Mechanics of Composite Materials, 1988, no. 4, pp. 693-704.
[5] Kohn R.V., Vogelius M. A new model of thin plates with rapidly varying thickness. Int. J. Solids and Struct, 1984, pp. 333–350.
[6] Gruttmann F., Wagner W. Shear correction factors in Timoshenko’s beam theory for arbitrary shaped cross–sections. Computational mechanics, v.27. 2001, pp.199-207.
[7] Ghugal Y.M., Shmipi R.P. A review of refined shear deformation theories for isotropic and anisotropic laminated beams. Journal of Reinforced Plastics and Composites, vol. 20, no. 3, 2001, pp. 255-272.
[8] Francesco T. Free vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method. Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 931–952.
[9] Zveryaev E.M., Makarov G.I. PMM — J. Appl. Math. Mech., 2008, vol. 72, iss. 2, pp. 308–321.
[10] Sheshenin S.V. Izv. RAN. MTT — Proc. of the Russ. Acad. Sci. Mech. Rigid Body, 2006, no. 6, pp. 71–79.
[11] Nazarov S.A., Svirs G.H., Sluckij A.S. Matematicheskij sbornik – Mathematical collection. 2011, vol. 202, no. 8, pp.41-80.
[12] Dimitrienko Yu.I. Vestnik MGTU im. N.E. Baumana. Seriya Estestvennye nauki — Herald of Bauman Moscow State Technical University, Natural Science Series, 2012, no. 3, pp. 86-100.
[13] Dimitrienko Yu,I., Yakovlev D.O. Mekhanika kompo-zitsionnykh materialov i konstruktsiy — Composite Mechanics and design, 2014, vol. 20, no. 2, pp. 260- 282.
[14] Dimitrienko Yu.I., Gubareva E.A., Yurin Yu.V. Matematicheskoe modelirovanie i chislennye metody — Mathematical Modeling and Computational Methods, 2014, no. 4, pp. 36–56.
[15] Dimitrienko Yu.I., Gubareva E.A., Sborschikov S.V. Matematicheskoe modelirovanie i chislennye metody — Mathematical Modeling and Computational Methods, 2014, no. 1, pp. 36–57.
[16] Dimitrienko Yu.I., Gubareva E.A., Shalygin I.S. Inzhenernyi zhurnal: nauka i innovatsii — Engineering Journal: Science and Innovation, 2015, iss. 5.
[17] Dimitrienko Yu.I., Gubareva E.A., Yurin Yu.V. Inzhenernyi zhurnal: nauka i innovatsii — Engineering Journal: Science and Innovation, 2016, iss. 12.
[18] Dimitrienko Yu.I., Gubareva E.A., Yakovlev D.O. Nauka i obrazovanie. Elektronnoe nauchno-telhnicheskoe izdanie — Science and Education. Electronic Scientific and Technical Joural, 2014, no. 10. doi: 10.7463/1014.0730105.
[19] Dimitrienko Yu.I., Dimitrienko I.D., Sborschikov S.V. Multiscale Hierarchical Modeling of Fiber Reinforced Composites by Asymptotic Homogenization Method. Applied Mathematical Sciences, Vol. 9, 2015, no. 145, 7211 7220
[20] Dimitrienko Yu.I., Dimitrienko I.D. Modeling of the thin composite laminates with general anisotropy under harmonic vibrations by the asymptotic homogenization method. Journal for Multiscale Computational Engineering. 2017 . № 15(3), pp. 219-237
[21] Dimitrienko Yu.I. Mekhanika sploshnoi sredy. V 4 tomakh [Continuum mechanics. In 4 vols.]. Vol. 4. Osnovy mekhaniki tverdogo tela [Fundamentals of solid mechanics]. Moscow, BMSTU Publ., 2013, 624 p.
[22] Dimitrienko Yu.I. Tenzornoe ischislenie [Tensor calculus]. Moscow, Vysshaya shkola Publ., 2001, 576 p.

Димитриенко Ю.И., Губарева Е.А., Пичугина А.Е. Моделирование напряжений в тонких композитных цилиндрических оболочках на основе асимптотической теории. Математическое моделирование и численные методы, 2018, № 3, с. 114–132.

Download article

Колличество скачиваний: 44