International Applied Mechanics

, Volume 55, Issue 5, pp 487–494 | Cite as

Three-Dimensional Analysis of the Stress–Strain State of Inhomogeneous Hollow Cylinders Using Various Approaches

  • A. Ya. GrigorenkoEmail author
  • S. N. Yaremchenko

The stress–strain state of an inhomogeneous hollow cylinder with clamped edges is studied using the three-dimensional elasticity theory. The problem is solved with the spline-approximation and finite-element methods. To reduce the system of partial differential equations to a system of ordinary high-order differential equations, two-dimensional splines are applied. The one-dimensional problem is solved with the method of discrete orthgonalization. The results obtained with the spline-approximation and finite-element methods for an open inhomogeneous cylinder with a radially varying elastic modulus are compared.


stress–strain state three-dimensional elasticity theory hollow cylinder finite length spline collocation finite-element method 


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  1. 1.
    O. C. Zienkiewicz, The Finite-Element Method in Engineering Science, McGraw-Hill, New York (1971).Google Scholar
  2. 2.
    A. Bahri, M. Salehi, and M. Akhlaghi, “Three-dimensional static and dynamic analysis of the functionally gradient cylinder bonded to the laminated plate under general loading,” Mech. Adv. Mater. Struct., 23, No. 12, 1437–1453 (2016).CrossRefGoogle Scholar
  3. 3.
    V. Birman and L.W. Byrd, “Modeling and analysis of functionally graded materials and structures,” Appl. Mech. Rev., 60, 195–215 (2009).ADSCrossRefGoogle Scholar
  4. 4.
    E. Ghafoori and M. Asghari, “Three-dimensional elasticity analysis of functionally graded rotating cylinders with variable thickness profile,” Proc. Inst. Mech. Eng., Part C; J. Mech. Eng. Sci., 226, No. 3, 585–594 (2012).Google Scholar
  5. 5.
    A. Ya. Grigorenko, A. S. Bergulev, and S. N. Yaremchenko, “Numerical solution of bending problems for rectangular plates,” Int. Appl. Mech., 49, No. 1, 81–94 (2013).ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Ya. Grigorenko, W. H. Muller, R. Wille, and S. N. Yaremchenko, “Numerical solution of the problem on the stress-strain state in hollow cylinders using spline approximations,” J. Math, Sci., 180, No. 2, 135–145 (2012).CrossRefGoogle Scholar
  7. 7.
    A. Ya. Grigorenko, N. P. Yaremchenko, and S. N. Yaremchenko, “Analysis of the axisymmetric stress–strain state of a continuously inhomogeneous hollow sphere,” Int. Appl. Mech., 54, No. 5, 577–583 (2018).ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Ya. Grigorenko and S. N. Yaremchenko, “Analysis of the stress–strain state of inhomogeneous hollow cylinders,” Int. Appl. Mech., 52, No. 4, 342–349 (2016).ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Ya. M. Grigorenko and N. N. Kryukov, “Use of spline approximation to study displacement and stress fields in cylinders with different boundary conditions on the ends,” Int. Appl. Mech., 33, No. 12, 958–965 (1997).ADSCrossRefGoogle Scholar
  10. 10.
    Ya. M. Grigorenko and L. S. Rozhok, “Effect of change in the curvature parameters on the stress state of concave corrugated hollow cylinders,” Int. Appl. Mech., 54, No. 3, 266–273 (2018).ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Ya. M. Grigorenko and L. S. Rozhok, “Layered inhomogeneous hollow cylinders with concave corrugations under internal pressure,” Int. Appl. Mech., 54, No. 5, 531–538 (2018).ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    V. G. Karnaukhov, V. I. Kozlov, A. V. Zavgorodnii, and I. N. Umrykhin, “Forced resonant vibrations and self-heating of solids of revolution made of a viscoelastic piezoelectric material,” Int. Appl. Mech., 51, No. 6, 614–622 (2015).ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    N. N. Kryukov, “Solution of problems of the stressed state of thick-walled orthotropic cylindrical shells with the aid of spline functions,” Int. Appl. Mech., 29, No. 7, 541–547 (1993).ADSCrossRefGoogle Scholar
  14. 14.
    Y. Miyamoto, W. A. Kaysser, B. H. Rabin, A. Kawasaki, and R. G. Ford, Functionally Graded Materials. Design, Processing and Applications, Kluwer Academic Publishers, Boston (1999).Google Scholar
  15. 15.
    S. Suresh and A. Mortensen, Fundamentals of Functionally Graded Materials, Maney, London (1998).Google Scholar
  16. 16.
    B. Woodward and M. Kashtalyan, “Three-dimensional elasticity solution for bending of transversely isotropic functionally graded plates,” European J. Mech. A/Solids, 30, No. 5, 715–718 (2011).ADSMathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.S. P. Timoshenko Institute of MechanicsNational Academy of Sciences of UkraineKyivUkraine

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