Advertisement

On the dynamic bending of layered plates

  • Ya. F. Kayuk
Article

The dynamic bending of layered metallic plates is studied. Their layers are reinforced with wires, and there are interfacial layers between the contact surfaces. A general approach to solving the corresponding dynamic problems is proposed. It is the principle of virtual work. A system of ordinary differential constitutive equations is obtained and used to determine the amplitude coefficients of the series of coordinate functions that approximates the deflections. The constitutive equations are reduced to Volterra equations of the second kind. Algorithms are developed to calculate the coefficients of the constitutive equations depending on the coordinate functions and on the reinforcement pattern. Possible methods to set up coordinate functions and solve integral equations are indicated

Keywords

layered metallic plate reinforcement principle of virtual work constitutive equations Volterra equations 

References

  1. 1.
    S. A. Ambartsumyan, General Theory of Anisotropic Shells [in Russian], Nauka, Moscow (1974).Google Scholar
  2. 2.
    Ya. M. Grigorenko and A. T. Vasilenko, Static Problems for Inhomogeneous Anisotropic Shells [in Russian], Nauka, Moscow (1992).Google Scholar
  3. 3.
    A. N. Guz (ed.), T. Kabelka, S. Markus, et al., Dynamics and Stability of Laminated Composite Materials [in Russian], Naukova Dumka, Kyiv (1991).Google Scholar
  4. 4.
    M. Kravchuk, Method of Moments Applied to Solve Linear Differential and Integral Equations [in Ukrainian], Vseukr. Akad. Nauk, Kyiv (1932).Google Scholar
  5. 5.
    Ya. F. Kayuk and A. O. Cherkas, “Dynamic deformation of cylindrical shells made of inhomogeneous materials,” Visn. Donetsk. Univ., Ser. A, Pryr. Nauky, No. 2, 131–136 (2002).Google Scholar
  6. 6.
    Ya. F. Kayuk and A. O. Cherkas, “Dynamic deformation of layered beams with interfacial layers,” Nauk. Visti NTUU “KPI,” No. 2, 52–58 (2002).Google Scholar
  7. 7.
    Ya. F. Kayuk, “Contribution of M. V. Ostrogradskii and M. P. Kravchuk to the development of direct methods for solving problems of mathematical physics,” in: Proc. Ukrainian Math. Congr. on Numerical Mathematics and Mathematical Problems in Mechanics [in Ukrainian], Inst. Mat. NAN Ukrainy, Kyiv (2002), pp. 48–64.Google Scholar
  8. 8.
    K. G. Kreider (ed.), Metallic Matrix Composites, Vol. 4 of the eight-volume series L. J. Broutman and R. H. Krock, Composite Materials, Academic Press, New York (1974), pp. 420–421.Google Scholar
  9. 9.
    B. L. Pelekh and F. N. Fleishman, “Influence of thin interfacial layers on the macroscopic characteristics of composite materials,” Izv. AN SSSR, Mekh. Tverd. Tela, No. 2, 121–124 (1984).Google Scholar
  10. 10.
    V. G. Piskunov and A. O. Rasskazov, “Evolution of the theory of laminated plates and shells,” Int. Appl. Mech., 38, No. 2, 135–166 (2002).CrossRefGoogle Scholar
  11. 11.
    V. G. Piskunov, Yu. M. Fedorenko, and I. M. Didychenko, “Dynamics of inelastic laminated composite shells,” Mech. Comp. Mater., 31, No. 1, 56–62 (1995).CrossRefGoogle Scholar
  12. 12.
    A. O. Rasskazov, “Vibration theory of multilayer orthotropic shells,” Int. Appl. Mech., 13, No. 8, 759–764 (1977).MATHMathSciNetGoogle Scholar
  13. 13.
    A. O. Rasskazov, I. I. Sokolovskaya, and N. A. Shul’ga, “Comparative analysis of several shear models in problems of equilibrium and vibrations for multilayer plates,” Int. Appl. Mech., 19, No. 7, 633–638 (1983).MATHGoogle Scholar
  14. 14.
    V. L. Rvachev and L. V. Kurpa, R-Functions in Plate Problems [in Russian], Naukova Dumka, Kyiv (1987).Google Scholar
  15. 15.
    G. B. Sinyaraev, N. A. Vatolin, B. G. Trusov, and G. K. Moiseev, Computers in the Theory of Dynamic Calculations of Metallurgy Processes [in Russian], Nauka, Moscow (1982).Google Scholar
  16. 16.
    M. Kh. Shorshorov, L. M. Ustinov, and L. E. Gukasyan, “Relationship between the strength of the fiber–matrix interface and the tensile strength of aluminum–boron composite,” Fiz. Khim. Obrab. Mater., No. 3, 132–137 (1979).Google Scholar
  17. 17.
    M. E. Babeshko and Yu. N. Shevchenko, “Elastoplastic stress–strain state of flexible layered shells made of isotropic and transversely isotropic materials with different moduli and subjected to axisymmetric loading,” Int. Appl. Mech., 43, No. 11, 1208–1217 (2007).CrossRefGoogle Scholar
  18. 18.
    G. L. Gorinin and Yu. V. Nemirovskii, “Transverse vibrations of laminated beams in three-dimensional formulation,” Int. Appl. Mech., 41, No. 6, 631–645 (2005).CrossRefGoogle Scholar
  19. 19.
    Ya. M. Grigorenko, N. N. Kryukov, and N. S. Yakovenko, “Using spline function to solve boundary-value problems for laminated orthotropic trapezoidal plates of variable thickness,” Int. Appl. Mech., 41, No. 4, 413–420 (2005).CrossRefGoogle Scholar
  20. 20.
    Ya. F. Kayuk, “Dynamic problem for layered shells of revolution with interfacial phenomena taken into account,” Int. Appl. Mech., 44, No. 7, 775–787 (2008).CrossRefMathSciNetGoogle Scholar
  21. 21.
    Ya. F. Kayuk and M. K. Shekera, “On one dynamic problem for structurally inhomogeneous beam,” Int. Appl. Mech., 43, No. 11, 1256–1263 (2007).CrossRefGoogle Scholar
  22. 22.
    I. F. Kirichok and M. V. Karnaukhov, “Single-frequency vibrations and vibrational heating of a piezoelectric circular sandwich plate under monoharmonic electromechanical loading,” Int. Appl. Mech., 44, No. 1, 65–72 (2008).CrossRefGoogle Scholar
  23. 23.
    L. V. Kurpa and G. N. Timchenko, “Studying the free vibrations of multilayer plates with a complex planform,” Int. Appl. Mech., 42, No. 1, 103–109 (2006).CrossRefGoogle Scholar
  24. 24.
    N. P. Semenyuk and V. M. Trach, “Stability of axially compressed cylindrical shells made of reinforced materials with specific fiber orientation within each layer,” Int. Appl. Mech., 42, No. 3, 318–324 (2006).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.S. P. Timoshenko Institute of MechanicsNational Academy of Sciences of UkraineKyivUkraine03057

Personalised recommendations