International Applied Mechanics

, Volume 54, Issue 1, pp 41–55 | Cite as

Generalized Method of Finite Integral Transforms in Static Problems for Anisotropic Prisms

Article
  • 2 Downloads

A new approach to solving three-dimensional elliptic linear boundary-value problems with nonseparable variables is developed using the ideas of methods of finite integral transforms. It consists in setting up a coupled system of three integral transforms with three pairs of independent variables of the domain, from which the transforms and kernels are determined. The approach is used to solve static problems for anisotropic prisms with elastic properties of low order of symmetry and arbitrary conditions on the faces. The approach is tested and the deformation of specific bodies of this class is analyzed.

Keywords

finite integral transform method new approach three-dimensional problems anisotropic prisms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. A. Grinberg, “New method for solving some boundary-value problems for equations of mathematical physics that allows separation of variables,” Izv. AN SSSR, Ser. Fiz., No. 10, 141–168 (1946).Google Scholar
  2. 2.
    V. V. Dykhta, Integral Transform Method in Wave Problems of Hydroacoustics [in Russian], Naukova Dumka, Kyiv (1981).MATHGoogle Scholar
  3. 3.
    N. S. Koshlyakov, E. B. Gliner, and M. M. Smirnov, Basic Differential Equations of Mathematical Physics [in Russian], Fizmatgiz, Moscow (1962).MATHGoogle Scholar
  4. 4.
    S. G. Lekhnitskii, Theory of Elasticity of Anisotropic Body [in Russian], Fiz.-Mat. Lit., Moscow (1977).MATHGoogle Scholar
  5. 5.
    A. V. Lykov, Theory of Heat Conduction [in Russian], Vysshaya Shkola, Moscow (1967).Google Scholar
  6. 6.
    Yu. E. Senitskii, Studying the Elastic Deformation of Structural Elements under Dynamic Loads using the Finite Transform Method [in Russian], Saratov Univ., Saratov (1985).Google Scholar
  7. 7.
    Yu. E. Senitskii, “Finite integral transform method. Its prospects in studying boundary-value problems of mechanics (review),” Vest. SamGTU, Ser. Mat., No. 22, 10–39 (2003).Google Scholar
  8. 8.
    Yu. E. Senitskii, “Finite integral transform method: generalization of the classical procedure of expansion into series of vector eigenfunctions,” Izv. Saratov. Univ., Ser. Mat.-Mekh.- Inform., No. 3(1), 61–89 (2011).Google Scholar
  9. 9.
    Ya. C. Uflyand, Integral Transforms in Elasticity Problems [in Russian], Nauka, Leningrad (1967).Google Scholar
  10. 10.
    V. K. Chibiryakov and A. M. Smolyar, “On one generalization of the finite integral transform method in the theory of thick plates,” in: Strength of Materials and Theory of Structures [in Russian], issue 42, Budivel’nik, Kyiv (1983), pp. 80–86.Google Scholar
  11. 11.
    E. I. Bespalova and A. B. Kitaygorodslii, “Advanced Kantorovich’s method for biharmonic problems,” J. Eng. Math., 46, 213–226 (2003).MathSciNetCrossRefGoogle Scholar
  12. 12.
    E. Bespalova and G. Urusova, “Solution of the Lame problem by the complete systems method,” Int. J. Comp. Meth. Eng. Sci. Mech., 14, No. 2, 159–167 (2013).MathSciNetCrossRefGoogle Scholar
  13. 13.
    E. I. Bespalova, “On the method of finite integral transforms in problems of statics of inhomogeneous plates,” Int. Appl. Mech., 50, No. 6, 651–663 (2014).ADSCrossRefMATHGoogle Scholar
  14. 14.
    A. M. Bidgoli, A. R. Daneshmehr, and R. Kolahchi, “Analytical bending solution of full clamped orthotropic rectangular plates resting on elastic foundations by the finite integral transform method,” J. Appl. Comp. Mech., 1, No. 2, 52–58 (2015).Google Scholar
  15. 15.
    N. Dernek, “On the solution of the e.p.d. equation using finite integral transformations,” Turkish J. Math., 21, 317–324 (1997).MathSciNetMATHGoogle Scholar
  16. 16.
    A. C. Eringen, “The finite Sturm–Liouwille transform,” Quart. J. Math., 2, No. 5, 120–131 (1954).ADSCrossRefMATHGoogle Scholar
  17. 17.
    A. C. Eringen, “Transform technique for boundary-value problems in fourth-order partial differential equations,” Quart. J. Math., 6, No. 24, 241–249 (1955).ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    V. I. Fabrikant, “Application of generalized images method to contact problems for a transversely isotropic elastic layer on a smooth half-space,” Archive Appl. Mech., 81, No. 7, 957–974 (2011).ADSCrossRefMATHGoogle Scholar
  19. 19.
    E. A. Gasimov, “Application of the finite integral transform method to solving a mixed problem with integrodifferential conditions for a nonclassical equation,” Diff. Eqs., 47, No. 3, 319–332 (2011).CrossRefGoogle Scholar
  20. 20.
    V. D. Kubenko, “A non-stationary problem for elastic half-plane under mixed boundary conditions,” Int. Appl. Mech., 52, No. 2, 105–118 (2016).ADSCrossRefMATHGoogle Scholar
  21. 21.
    V. D. Kubenko, “Nonstationary deformation of an elastic layer with mixed boundary conditions,” Int. Appl. Mech., 52, No. 6, 563–580 (2016).ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    H. Lamb, “On the propagation of tremors over the surface of an elastic solid,” Phil. Trans. Roy. Soc. of London, Ser. A, 203, 1–42 (1904).ADSCrossRefMATHGoogle Scholar
  23. 23.
    R. Li, Y. Zhong, B. Tian, and Y. Liu, “On the finite integral transform method for exact bending solutions of fully clamped orthotropic rectangular thin plates,” Appl. Math. Letters, 22, 1821–1827 (2009).MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    J. Ruan, X. Feng, G. Zhang, Y. Wang, and D. Fang, “Dynamic thermoelastic analysis of a slab using finite integral transformation method,” AIAA J., 48, No. 8, 1833–1839 (2010).ADSCrossRefGoogle Scholar
  25. 25.
    Yu. N. Shevchenko and V. G. Savchenko, “3-D problems of thermoviscoplasticity: Focus on Ukrainian studies,” Int. Appl. Mech., 52, No. 3, 217–271 (2016).ADSCrossRefMATHGoogle Scholar
  26. 26.
    S. Singh and P. K. Jain, “Finite integral transform method to solve asymmetric heat conduction in a multilayer annulus with time-dependent boundary conditions,” Nucl. Eng. Design, 241, No. 1, 144–154 (2011).CrossRefGoogle Scholar
  27. 27.
    I. N. Sneddon, Fourier Transforms, McGraw-Hill Book Company Inc., New York (1951).MATHGoogle Scholar
  28. 28.
    I. N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York (1972).MATHGoogle Scholar
  29. 29.
    I. N. Sneddon, Application of Integral Transforms in the Theory of Elasticity, McGraw-Hill, New York (1975).MATHGoogle Scholar
  30. 30.
    C. J. Tranter, Integral Transforms in Mathematical Physics, Wiley, New York (1951).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

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

Personalised recommendations