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Mathematical Modeling of Nonlinear Dynamics of Continuous Mechanical Structures with an Account of Internal and External Temperature Fields

  • Vadim A. Krysko
  • Jan AwrejcewiczEmail author
  • Maxim V. Zhigalov
  • Valeriy F. Kirichenko
  • Anton V. Krysko
Chapter
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 42)

Abstract

This chapter focuses on the construction of mathematical models of nonlinear dynamics of structural members in the form of plates and shallow shells, including internal and external temperature fields. The geometric nonlinearity is taken in the von Kármán form, and the physical nonlinearity is introduced based on the strain theory of plasticity, whereas the heat transfer processes are followed with the help of the Fourier principle. The variational formulation yields PDEs of different dimensions and different types (hyperbolic and hyperbolic–parabolic). Our considerations are based on the first-order kinematic Kirchhoff–Love model. The existence of a solution to the coupled problem of thermoelasticity of shells in the mixed form with a parabolic PDE governing heat transfer effects is rigorously proved. The economical (reasonably short computational time) algorithms devoted to the investigation of the coupled problems of the theory of shallow shells with the parabolic heat transfer equation based on the Faedo–Galerkin method in higher approximations and the finite difference method to second-order accuracy have been worked out. In order to solve the stationary problems of the theory of shells, we have extended and modified the classical relaxation method, exhibiting its effectiveness and high accuracy. A wide class of nonlinear vibrations of shells with various types of nonlinearity is studied.

References

  1. 1.
    Vlasov, V. Z. (1949). General theory of shells and its application to the engineering. NASA-TT F-99.Google Scholar
  2. 2.
    Novozhilov, V. V. (1964). Thin shell theory. Groningen: P. Noordhoff.CrossRefGoogle Scholar
  3. 3.
    Kauderer, H. (1958). Nichtlineare Mechanik. Berlin: Springer.CrossRefGoogle Scholar
  4. 4.
    Biot, M. A. (1956). Thermoelasticity and irreversible thermodynamics. Journal of Applied Physics, 27, 240–253.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kornishin, M. S. (1964). Nonlinear problems of theory of plates and shallow shells and methods of their solution. Moscow: Nauka.Google Scholar
  6. 6.
    Lions, J.-L. (1969). Some problems of solving non-linear boundary value problems. Paris: Dunod-Gauthier-Villars.Google Scholar
  7. 7.
    Kantorovich, L. V., & Akilov, G. P. (1982). Functional analysis. Oxford: Pergamon Press.CrossRefGoogle Scholar
  8. 8.
    Ladyzhenskaya, O. A. (1973). The boundary value problems of mathematical physics. Berlin: Springer.Google Scholar
  9. 9.
    Morozov, N. F. (1967). Investigation of nonlinear vibrations of thin plates with consideration of damping. Differential Equations, 3(4), 619–635 (in Russian).Google Scholar
  10. 10.
    Vaindiner, A. I. (1973). Some questions of approximation of functions of many variables and effective direct methods for solving problems of elasticity. Elasticity and Inelasticity, 3, 16–46 (in Russian).Google Scholar
  11. 11.
    Vorovich, I. I. (1957). On some direct methods in the non-linear theory of vibrations of curved shells. Izv. Akad. Nauk SSSR. Ser. Mat., 21, 747–784 (in Russian).Google Scholar
  12. 12.
    Mikhlin, S. G. (1970). Variational methods in mathematical physics. Oxford: Pergamon Press.zbMATHGoogle Scholar
  13. 13.
    Kirichenko, V. F., & Krysko, V. A. (1984). On the existence of solution of one nonlinear the problem of thermoelasticity. Differential Equations, XX(9), 1583–1588 (in Russian).Google Scholar
  14. 14.
    Petrovsky, I. G. (1992). Lectures of partial differential equations. New York: Dover.Google Scholar
  15. 15.
    Gurov, K. P. (1978). Phenomenological thermodynamics of irreversible processes. Moscow: Nauka (in Russian).Google Scholar
  16. 16.
    Ladyzhenskaya, O. A. (1969). The mathematical theory of viscous incompressible flow. New York: Gordon and Breach.zbMATHGoogle Scholar
  17. 17.
    Lions, J. L., & Magenes, E. (1961). Problemi ai limiti non omogenei, III. The Annali della Scuola Normale Superiore di Pisa, 15, 41103.MathSciNetGoogle Scholar
  18. 18.
    Volmir, A. S. (1972). The nonlinear dynamics of plates and shells. Moscow: Nauka (in Russian)Google Scholar
  19. 19.
    Holmes, P. J. (1979). A nonlinear oscillator with a strange attractor. Philosophical Transactions of the Royal Society A, 292, 419–425.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Shiau, A. S., Soong, T. T., & Roth, R. S. (1974). Dynamic buclung of conical shells with imperfections. AIAA Journal, 12(6), 24–30.zbMATHGoogle Scholar
  21. 21.
    Budiansky, B., Roth, R. S. (1962). Axisymmetric dynamic buckling of clamped shallow spherical shells (pp. 597–606). TN D-1510, NASA, Washington.Google Scholar
  22. 22.
    Danilovskaya, V. I. (1950). Thermal stresses in elastic half-space resulting from a sudden heating of its surface. Applied Mathematics and Mechanics, 14(3), 316–318 (in Russian).Google Scholar
  23. 23.
    Krysko, V. A. (1979). Dynamic buckling of shells, rectangular in plan, with finite displacements. Applied Mechanics, 15(11), 1059–1062.Google Scholar
  24. 24.
    Bolotin, V. V. (1956). Dynamic stability of elastic systems. Moscow: Gostehizdat (in Russian).Google Scholar
  25. 25.
    Kantor, B. Ya. (1971). Nonlinear problems in the theory of inhomogeneous shallow shells. Kiev: Naukova Dumka (in Russian).Google Scholar
  26. 26.
    Krysko, V. A. (1976). Nonlinear statics and dynamics of inhomogeneous membranes. Saratov: Publishing House Saratov University Press.Google Scholar
  27. 27.
    Harrik, I. Yu. (1955). On approximation of functions vanishing on the boundary of a region by functions of a special form. Mat. Sb. N.S., 37(79), 353384 (in Russian).Google Scholar
  28. 28.
    Volmir, A. S. (1967). Stability of deformable systems. Moscow: Nauka (in Russian).Google Scholar
  29. 29.
    Morozov, N. F. (1978). Selected two-dimensional problems of theory of elasticity. Leningrad: LGU (in Russian).Google Scholar
  30. 30.
    Kovalenko, A. D. (1970). Fundamentals of thermoelasticity. Kiev: Naukova Dumka (in Russian).Google Scholar
  31. 31.
    Podstrigatch, Ya. S., Koliano, Yu. M. (1976). Generalized thermomechanics. Kiev: Naukova Dumka (in Russian).Google Scholar
  32. 32.
    Awrejcewicz, J., Krysko, V. A., & Krysko, A. V. (2007). Thermo-dynamics of plates and shells. Berlin: Springer.zbMATHGoogle Scholar
  33. 33.
    Dennis, J. E., & Schnabel, R. B. (1983). Numerical methods for unconstrained optimization and nonlinear equations. Englewood Cliffs: Prentice-Hall.zbMATHGoogle Scholar
  34. 34.
    Awrejcewicz, J., Krysko, A. V., Zhigalov, M. V., & Krysko, V. A. (2017). Chaotic dynamic buckling of rectangular spherical shells under harmonic lateral load. Computers and Structures, 191, 80–99.CrossRefGoogle Scholar
  35. 35.
    Kantz, H. (1994). A robust method to estimate the maximal Lyapunov exponents of a time series. Physics Letters A, 185, 77–87.CrossRefGoogle Scholar
  36. 36.
    Rosenstein, M. T., Collins, J. J., & De Luca, C. J. F. (1993). Practical method for calculating largest Lyapunov exponents from small data sets. Physica D, 65, 117–134.MathSciNetCrossRefGoogle Scholar
  37. 37.
    Wolf, A., Swift, J. B., Swinney, H. L., & Vastano, J. A. (1985). Determining Lyapunov exponents from a time series. Physica D, 16, 285–317.MathSciNetCrossRefGoogle Scholar
  38. 38.
    Wessel, J. K., Kissell, J. R., Pantelakis, S. G., & Haidemenopoulos, G. N. (2004). The handbook of advanced materials: Enabling new design.  https://doi.org/10.1002/0471465186.Google Scholar
  39. 39.
    Newhouses, S., Ruelle, D., & Takens, F. (1978). Occurrence of strange Axiom A attractions near quasiperiodic flow on \(T^m\), \(m\ge 3\). Communications in Mathematical Physics, 64(1), 35–40.MathSciNetCrossRefGoogle Scholar
  40. 40.
    Rasskazov, A. O., Sokolov, I. I., & Shul’ga, N. A. (1986). Theory and calculation of layered orthotropic plates and shells. Kiev: Vishcha Shkola (in Russian).Google Scholar
  41. 41.
    Karchevskii, M. M. (1995). On the solvability of geometrically nonlinear problems of the theory of thin shells. News of Universities. Mathematics, 6(397), 30–36 (in Russian).Google Scholar
  42. 42.
    Grigolyuk, E. I., & Chulkov, P. P. (1973). Stability and vibration of three-layer shells. Moscow: Mashinostroyeniye.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Vadim A. Krysko
    • 1
  • Jan Awrejcewicz
    • 2
    Email author
  • Maxim V. Zhigalov
    • 1
  • Valeriy F. Kirichenko
    • 1
  • Anton V. Krysko
    • 3
  1. 1.Department of Mathematics and ModelingSaratov State Technical UniversitySaratovRussia
  2. 2.Department of Automation, Biomechanics and MechatronicsLodz University of TechnologyLodzPoland
  3. 3.Department of Applied Mathematics and Systems AnalysisSaratov State Technical UniversitySaratovRussia

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