Mathematical Models of Higher Orders pp 21-131 | Cite as

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

## 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.Vlasov, V. Z. (1949). General theory of shells and its application to the engineering. NASA-TT F-99.Google Scholar
- 2.Novozhilov, V. V. (1964).
*Thin shell theory*. Groningen: P. Noordhoff.CrossRefGoogle Scholar - 3.Kauderer, H. (1958).
*Nichtlineare Mechanik*. Berlin: Springer.CrossRefGoogle Scholar - 4.Biot, M. A. (1956). Thermoelasticity and irreversible thermodynamics.
*Journal of Applied Physics*,*27*, 240–253.MathSciNetCrossRefGoogle Scholar - 5.Kornishin, M. S. (1964).
*Nonlinear problems of theory of plates and shallow shells and methods of their solution*. Moscow: Nauka.Google Scholar - 6.Lions, J.-L. (1969).
*Some problems of solving non-linear boundary value problems*. Paris: Dunod-Gauthier-Villars.Google Scholar - 7.Kantorovich, L. V., & Akilov, G. P. (1982).
*Functional analysis*. Oxford: Pergamon Press.CrossRefGoogle Scholar - 8.Ladyzhenskaya, O. A. (1973).
*The boundary value problems of mathematical physics*. Berlin: Springer.Google Scholar - 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.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.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.Mikhlin, S. G. (1970).
*Variational methods in mathematical physics*. Oxford: Pergamon Press.zbMATHGoogle Scholar - 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.Petrovsky, I. G. (1992).
*Lectures of partial differential equations*. New York: Dover.Google Scholar - 15.Gurov, K. P. (1978).
*Phenomenological thermodynamics of irreversible processes*. Moscow: Nauka (in Russian).Google Scholar - 16.Ladyzhenskaya, O. A. (1969).
*The mathematical theory of viscous incompressible flow*. New York: Gordon and Breach.zbMATHGoogle Scholar - 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.Volmir, A. S. (1972).
*The nonlinear dynamics of plates and shells*. Moscow: Nauka (in Russian)Google Scholar - 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.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.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.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.Krysko, V. A. (1979). Dynamic buckling of shells, rectangular in plan, with finite displacements.
*Applied Mechanics*,*15*(11), 1059–1062.Google Scholar - 24.Bolotin, V. V. (1956).
*Dynamic stability of elastic systems*. Moscow: Gostehizdat (in Russian).Google Scholar - 25.Kantor, B. Ya. (1971). Nonlinear problems in the theory of inhomogeneous shallow shells. Kiev: Naukova Dumka (in Russian).Google Scholar
- 26.Krysko, V. A. (1976).
*Nonlinear statics and dynamics of inhomogeneous membranes*. Saratov: Publishing House Saratov University Press.Google Scholar - 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.Volmir, A. S. (1967).
*Stability of deformable systems*. Moscow: Nauka (in Russian).Google Scholar - 29.Morozov, N. F. (1978).
*Selected two-dimensional problems of theory of elasticity*. Leningrad: LGU (in Russian).Google Scholar - 30.Kovalenko, A. D. (1970).
*Fundamentals of thermoelasticity*. Kiev: Naukova Dumka (in Russian).Google Scholar - 31.Podstrigatch, Ya. S., Koliano, Yu. M. (1976).
*Generalized thermomechanics*. Kiev: Naukova Dumka (in Russian).Google Scholar - 32.Awrejcewicz, J., Krysko, V. A., & Krysko, A. V. (2007).
*Thermo-dynamics of plates and shells*. Berlin: Springer.zbMATHGoogle Scholar - 33.Dennis, J. E., & Schnabel, R. B. (1983).
*Numerical methods for unconstrained optimization and nonlinear equations*. Englewood Cliffs: Prentice-Hall.zbMATHGoogle Scholar - 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.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.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.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.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.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.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.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.Grigolyuk, E. I., & Chulkov, P. P. (1973).
*Stability and vibration of three-layer shells*. Moscow: Mashinostroyeniye.Google Scholar