Skip to main content
Log in

Deformation and Joule Heating of a Flexible Orthotropic Ring Plate of Variable Rigidity

  • Published:
International Applied Mechanics Aims and scope

The derivation and method of solution of the nonlinear equations of thermomagnetoelasticity of flexible orthotropic shells of revolution taking into account orthotropic conductivity and Joule heat are presented. The thermomagnetoelasticity of a flexible orthotropic ring plate of variable stiffness is analyzed using an axisymmetric problem statement and taking into account orthotropic conductivity and Joule heat.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. S. A. Ambartsumyan, G. E. Bagdasaryan, and M. V. Belubekyan, Magnetoelasticity of Thin Shells and Plates [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  2. R. E. Bellman and R. E. Kalaba, Quasilinearization and Nonlinear Boundary-Value Problems, Elsevier, New York (1965).

    MATH  Google Scholar 

  3. V. D. Budak, L. V. Mol’chenko, and A. V. Ovcharenko, Numerical-Analytical Solution of Boundary Value Problems of Magnetoelasticity of Flexible Shells [in Ukrainian], Ilion, Mykolaiv (2016).

    Google Scholar 

  4. V. D. Budak, L. V. Mol’chenko, and A. V. Ovcharenko, Nonlinear Magnetoelastic Shells [in Russian], Ilion, Nikolaev (2016).

    Google Scholar 

  5. S. K. Godunov, “Numerical solution of boundary-value problems for systems of linear ordinary differential equations,” Usp. Mat. Nauk, 16, No. 5, 171–174 (1963).

    MathSciNet  Google Scholar 

  6. Ya. M. Grigorenko and L. V. Mol’chenko, Fundamentals of the Theory of Plates and Shells with Elements of Magnetoelasticity [in Russian], VPTs Kyivs’kyi Universitet, Kyiv (2010).

  7. V. I. Dresvyannikov, “Nonstationary problems of the mechanics of elastoplastic conductive bodies subject to strong impulsive magnetic fields,” Prikl. Probl. Prochn. Plast., No. 19, 32–47 (1979).

    Google Scholar 

  8. L. D. Landau, and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford (1984).

    Google Scholar 

  9. J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Clarendon Press, Oxford (1957).

    MATH  Google Scholar 

  10. L. I. Sedov, A Course in Continuum Mechanics, Vol. 2, Wolters-Noordhoff, Groningen (1972).

    MATH  Google Scholar 

  11. Yu. I. Sirotin and M. P. Shaskol’skaya, Fundamentals of Crystal Physics [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  12. J. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York (1941).

    MATH  Google Scholar 

  13. I. E. Tamm, Fundamentals of the Theory of Electricity, Mir, Moscow (1979)

    Google Scholar 

  14. Y. H. Bian, “Analysis of nonlinear stresses and strains in a thin current-carrying elastic plate,” Int. Appl. Mech., 51, No. 1, 108–120 (2015).

    Article  MathSciNet  Google Scholar 

  15. Y. H. Bian and H. T. Zhao, “Analysis of thermal-Magnetic-Elastic stresses and strains in a thin current-carrying cylindrical shell,” Int. Appl. Mech., 52, No. 4, 437–448 (2016).

    Article  MathSciNet  Google Scholar 

  16. R. V. Dinzhos, V. P. Privalko and E. G. Privalko, “Relaxation in the cooling/heating cycles of polypropylene/organosilicananocomposites. I. Nonisothermal crystallization,” J. Macromolecular Sci. Phys., B44, 421–430 (2005).

    Article  Google Scholar 

  17. R. V. Dinzhos, V. P. Privalko and E. G. Privalko, “Enthalpy relaxation in the cooling/heating cycles of polypropylene/organosilicananocomposites. II. Melting behavior,” J. Macromol Sci. Phys., B44, 431–443 (2005).

    Google Scholar 

  18. R. Elhajjar, V. Saponara, and A. Muliana, Smart Composites. Mechanics and Design, CRC Press, New York (2013).

    Book  Google Scholar 

  19. K. Hutter, A. F. van de Ven, and A. Ursescu, Electromagnetic Field Matter Interactions in Thermoelastic Solids and Viscous Fluids, Springer, Berlin (2006).

    Google Scholar 

  20. L. V. Mol’chenko, L. N. Fedorchenko, and L. Ya. Vasil’eva, “Nonlinear theory of magnetoelasticity of shells of revolution with Joule heat taken into account,” Int. Appl. Mech., 54, No. 3, 306–314 (2018).

    Article  MathSciNet  Google Scholar 

  21. L. V. Mol’chenko and I. I. Loos, “Effect of conicity on axisymmetrical strain state of flexible orthotropic shell of revolution in magnetic field,” Int. Appl. Mech., 46, No. 11, 1261–1267 (2010).

    Article  Google Scholar 

  22. L. V. Mol’chenko and I. I. Loos, “Thermomagnetoelastic deformation of flexible isotropic shells of revolution subject to Joule heating,” Int. Appl. Mech., 55, No. 1, 68–78 (2019).

    Article  MathSciNet  Google Scholar 

  23. L. V. Mol’chenko and I. I. Loos, “Asymmetric deformation of sells of revolution of variable stiffness in a nostationary magnetic field,” Int. Appl. Mech., 55, No. 3, 311–320 (2019).

    Article  Google Scholar 

  24. L. V. Mol’chenko and I. I. Loos, “Thermomagnetoelastic deformation of a flexible orthotropic conical shell with electrical conductivity and Joule heat taken into account,” Int. Appl. Mech., 55, No. 5, 534–543 (2019).

    Article  MathSciNet  Google Scholar 

  25. L. V. Mol’chenko, I. I. Loos and V. N. Darmosyuk, “Thermomagnetoelastic deformation of flexible orthotropic shells of revolution of variable stiffness with Joule heat taken into account,” Int. Appl. Mech., 56, No. 4, 498–511 (2020).

    Article  MathSciNet  Google Scholar 

  26. F. C. Moon, Magneto-Solid Mechanics, Wiley, New York (1984).

    Google Scholar 

  27. N. M. Newmark, “A method of computation for structural dynamics,” J. Eng. Mech. Div. Proc. ASCE., 85, No. 7, 67–97 (1959).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to L. V. Mol’chenko.

Additional information

Translated from Prikladnaya Mekhanika, Vol. 57, No. 3, pp. 107–126, May–June 2021.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mol’chenko, L.V., Loos, I.I. Deformation and Joule Heating of a Flexible Orthotropic Ring Plate of Variable Rigidity. Int Appl Mech 57, 347–362 (2021). https://doi.org/10.1007/s10778-021-01086-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10778-021-01086-2

Keywords

Navigation