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Thermoelasticity Based on Time-Fractional Heat Conduction Equation in Polar Coordinates

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Fractional Thermoelasticity

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 219))

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Abstract

The fundamental solutions to the first and second Cauchy problems and to the source problem are obtained for axisymmetric time-fractional heat conduction equation in an infinite plane in polar coordinates. Radial heat conduction in a cylinder and in an infinite solid with a cylindrical cavity is investigated. The Dirichlet boundary problems with the prescribed boundary value of temperature and the physical Neumann boundary problems with the prescribed boundary value of the heat flux are solved using the integral transform technique. The associated thermal stresses are studied. The numerical results are illustrated graphically. Figures show the characteristic features of temperature and stress distribution and represent the whole spectrum of order of time-derivative.

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References

  1. Abouelregal, A.E.: Generalized thermoelastic infinite transversely isotropic body with a cylindrical cavity due to moving heat source and harmonically varying heat. Meccanica 48, 1731–1745 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1972)

    Google Scholar 

  3. Allam, M.N., Elsibai, K.A., Abouelregal, A.E.: Thermal stresses in a harmonic field for an infinite body with a circular cylindrical hole without energy dissipation. J. Therm. Stress. 25, 57–68 (2002)

    Article  Google Scholar 

  4. Aouadi, M.: A generalized thermoelastic diffusion problem for an infinitely long solid cylinder. Int. J. Math. Math. Sci. 2006, 25976-1-15 (2006)

    Google Scholar 

  5. Bagri, A., Eslami, M.R.: Generalized coupled thermoelasticity of disks based on the Lord-Shulman model. J. Therm. Stress. 27, 691–704 (2004)

    Article  Google Scholar 

  6. Bagri, A., Eslami, M.R.: A unified generalized thermoelasticity formulation; application to thick functionally graded cylinders. J. Therm. Stress. 30, 911–930 (2007)

    Article  Google Scholar 

  7. Bagri, A., Eslami, M.R.: A unified generalized thermoelasticity; solution for cylinders and spheres. Int. J. Mech. Sci. 49, 1325–1335 (2007)

    Article  Google Scholar 

  8. Bagri, A., Eslami, M.R.: Generalized coupled thermoelasticity of functionally graded annular disk considering the Lord-Shulman theory. Compos. Struct. 83, 168–179 (2008)

    Article  Google Scholar 

  9. Bakhshi, M., Bagri, A., Eslami, M.R.: Coupled thermoelasticity of functionally graded disk. Mech. Adv. Mater. Struct. 13, 214–225 (2006)

    Article  Google Scholar 

  10. Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids, 2nd edn. Oxford University Press, Oxford (1959)

    Google Scholar 

  11. Chandrasekharaiah, D.S., Keshavan, H.R.: Axisymmetric thermoelastic interactions in an unbounded body with cylindrical cavity. Arch. Mech. 92, 61–76 (1992)

    MATH  Google Scholar 

  12. Chandrasekharaiah, D.S., Srinath, K.S.: Axisymmetric thermoelastic interactions without energy dissipation in an unbounded body with cylindrical cavity. J. Elast. 46, 19–31 (1997)

    Article  MATH  Google Scholar 

  13. El-Bary, A.A.: An infinite thermoelastic long annular cylinder with variable thermal conductivity. J. Appl. Sci. Res. 2, 341–345 (2006)

    Google Scholar 

  14. Erbay, S., Şuhubi, E.S.: Longitudinal wave propagation in a generalized thermoelastic cylinder. J. Therm. Stress. 9, 279–295 (1986)

    Article  Google Scholar 

  15. Furukawa, T., Noda, N., Ashida, F.: Generalized thermoelasticity for an infinite body with a circular cylindrical hole. JSME Int. J. Ser. I 33, 26–32 (1990)

    Google Scholar 

  16. Furukawa, T., Noda, N., Ashida, F.: Generalized thermoelasticity for an infinite solid cylinder. JSME Int. J. Ser. I 34, 281–286 (1991)

    Google Scholar 

  17. Galitsyn, A.S., Zhukovsky, A.N.: Integral Transforms and Special Functions in Heat Conduction Problems. Naukova Dumka, Kiev (1976) (in Russian)

    Google Scholar 

  18. Gorenflo, R., Loutchko, J., Luchko, Yu.: Computation of the Mittag-Leffler function and its derivatives. Fract. Calc. Appl. Anal. 5, 491–518 (2002)

    MATH  MathSciNet  Google Scholar 

  19. Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–208 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  20. He, T., Tian, X., Shen, Y.: A generalized electromagneto-thermoelastic problem for an infinitely long solid cylinder. Eur. J. Mech. A/Solids 24, 349–359 (2005)

    Article  MATH  Google Scholar 

  21. Ieşan, D.: Thermal stresses in inhomogeneous porous elastic cylinder. J. Therm. Stress. 30, 145–164 (2007)

    Article  Google Scholar 

  22. Kar, A., Kanoria, M.: Thermoelastic interaction with energy dissipation in an infinitely extended thin plate containing a circular hole. Far East J. Appl. Math. 24, 201–217 (2006)

    MATH  MathSciNet  Google Scholar 

  23. Kar, A., Kanoria, M.: Thermoelastic interaction with energy dissipation in a transversely isotropic thin circular disc. Eur. J. Mech. A/Solids 26, 969–981 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lamba, N.K., Khobragade, N.W.: Analysis of coupled thermal stresses in a axisymmetric hollow cylinder. Int. J. Latest Trend. Math. 1, 29–38 (2011)

    Google Scholar 

  25. Lenzi, E.K., da Silva, L.R., Silva, A.T., Evangelista, L.R., Lenzi, M.K.: Some results for a fractional diffusion equation with radial symmetry in a confined region. Phys. A 388, 806–810 (2009)

    Article  Google Scholar 

  26. Luikov, A.V.: Analytical Heat Diffusion Theory. Academic Press, New York (1968)

    Google Scholar 

  27. Misra, J.C., Chattopadhyay, N.C., Samanta, S.C.: Thermoviscoelastic waves in an infinite aelotropic body with a cylindrical cavity—a study under the review of generalized theory of thermoelasticity. Compos. Struct. 52, 705–717 (1994)

    Article  MATH  Google Scholar 

  28. Mukhopadhyay, S., Kumar, R.: Thermoelastic interactions on two-temperature generalized thermoelasticity in an infinite medium with a cylindrical cavity. J. Therm. Stress. 32, 341–360 (2009)

    Article  Google Scholar 

  29. Mukhopadhyay, S., Kumar, R.: Solution of a problem of generalized thermoelasticity of an annular cylinder with variable material properties by finite difference method. Comput. Methods Sci. Technol. 15, 169–176 (2009)

    Article  Google Scholar 

  30. Mukhopadhyay, S., Mukherjee, R.N.: Thermoelastic interaction in a transversally isotropic cylinder subjected to ramp type increase in boundary temperature and load. Indian J. Pure Appl. Math. 33, 635–646 (2002)

    MATH  Google Scholar 

  31. Narahari Achar, B.N., Hanneken, J.W.: Fractional radial diffusion in a cylinder. J. Mol. Liq. 114, 147–151 (2004)

    Google Scholar 

  32. Nigmatullin, R.R.: To the theoretical explanation of the “universal response”. Phys. Status Solidi (B) 123, 739–745 (1984)

    Article  Google Scholar 

  33. Nigmatullin, R.R.: The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Status Solidi (B) 133, 425–430 (1986)

    Article  Google Scholar 

  34. Noda, N., Furukawa, T., Ashida, F.: Generalized thermoelasticity in an infinite solid with a hole. J. Therm. Stress. 12, 385–402 (1989)

    Article  Google Scholar 

  35. Noda, N., Hetnarski, R.B., Tanigawa, Y.: Thermal Stresses, 2nd edn. Taylor and Francis, New York (2003)

    Google Scholar 

  36. Nowacki, W.: Thermoelasticity, 2nd edn. PWN-Polish Scientific Publishers, Warsaw and Pergamon Press, Oxford (1986)

    MATH  Google Scholar 

  37. Özdemir, N., Karadeniz, D.: Fractional diffusion-wave problem in cylindrical coordinates. Phys. Lett. A 372, 5968–5972 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  38. Özdemir, N., Karadeniz, D., Iskender, B.B.: Fractional optimal control problem of a distributed system in cylindrical coordinates. Phys. Lett. A 373, 221–226 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  39. Özdemir, N., Agrawal, O.P., Karadeniz, D., Iskender, B.B.: Fractional optimal control problem of an axis-symmetric diffusion-wave propagation. Phys. Scr. T 136, 014024-1-5 (2009)

    Google Scholar 

  40. Parkus, H.: Instationäre Wärmespannungen. Springer, Wien (1959)

    Book  MATH  Google Scholar 

  41. Povstenko, Y.: Fractional heat conduction equation and associated thermal stresses. J. Therm. Stress. 28, 83–102 (2005)

    Article  MathSciNet  Google Scholar 

  42. Povstenko, Y.: Stresses exerted by a source of diffusion in a case of a non-parabolic diffusion equation. Int. J. Eng. Sci. 43, 977–991 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  43. Povstenko, Y.: Two-dimensional axisymmentric stresses exerted by instantaneous pulses and sources of diffusion in an infinite space in a case of time-fractional diffusion equation. Int. J. Solids Struct. 44, 2324–2348 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  44. Povstenko, Y.: Fractional radial diffusion in a cylinder. J. Mol. Liq. 137, 46–50 (2008)

    Article  Google Scholar 

  45. Povstenko, Y.: Fractional radial diffusion in an infinite medium with a cylindrical cavity. Q. Appl. Math. 47, 113–123 (2009)

    MathSciNet  Google Scholar 

  46. Povstenko, Y.: Fractional radial heat conduction in an infinite medium with a cylindrical cavity and associated thermal stresses. Mech. Res. Commun. 37, 436–440 (2010)

    Article  MATH  Google Scholar 

  47. Povstenko, Y.: Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder. Fract. Calc. Appl. Anal. 14, 418–435 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  48. Povstenko, Y.: Time-fractional radial heat conduction in a cylinder and associated thermal stresses. Arch. Appl. Mech. 82, 345–362 (2012)

    Article  MATH  Google Scholar 

  49. Povstenko, Y.: The Neumann boundary problem for axisymmetric fractional heat conduction in a solid with cylindrical hole and associated thermal stresses. Meccanica 47, 23–29 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  50. Povstenko, Y.: Axisymmetric solutions to fractional diffusion-wave equation in a cylinder under Robin boundary condition. Eur. Phys. J. Spec. Top. 222, 1767–1777 (2013)

    Article  Google Scholar 

  51. Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and Series. Volume 2: Special Functions. Gordon and Breach, Amsterdam (1986)

    Google Scholar 

  52. Qi, H., Liu, J.: Time-fractional radial diffusion in hollow geometries. Meccanica 45, 577–583 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  53. Raslan, W.E.: Application of fractional order theory of thermoelasticity to a 1D problem for a cylindrical cavity. Arch. Mech. 66, 257–267 (2014)

    MATH  MathSciNet  Google Scholar 

  54. Shao, Z.S., Wang, T.J., Ang, K.K.: Transient thermo-mechanical analysis of functionally graded hollow circular cylinders. J. Therm. Stress. 30, 81–104 (2007)

    Article  Google Scholar 

  55. Sherief, H.H., Anwar, M.N.: A problem in generalized thermoelasticity for an infinitely long annular cylinder composed of two different materials. Acta Mech. 80, 137–149 (1989)

    Article  MATH  Google Scholar 

  56. Sherief, H.H., Elmisiery, A.E.M., Elhagary, M.A.: Generalized thermoelastic problem for an infinitely long hollow cylinder for short times. J. Therm. Stress. 27, 885–902 (2004)

    Article  Google Scholar 

  57. Sneddon, I.N.: The Use of Integral Transforms. McGraw-Hill, New York (1972)

    MATH  Google Scholar 

  58. Titchmarsh, E.C.: Eigenfunction Expansion Associated with Second-Order Differential Equations. Clarendon Press, Oxford (1946)

    Google Scholar 

  59. Wadhawan, M.C.: Thermoelastic response of a cylinder in the generalized dynamical theory of thermoelasticity. Pure Appl. Geophys. 102, 37–50 (1973)

    Article  Google Scholar 

  60. Youssef, H.M.: Generalized thermoelasticity of an infinite body with a cylindrical cavity and variable material properties. J. Therm. Stress. 28, 521–532 (2005)

    Article  Google Scholar 

  61. Youssef, H.M.: Problem of generalized thermoelastic infinite medium with cylindrical cavity subjected to a ramp-type heating and loading. Arch. Appl. Mech. 75, 553–565 (2006)

    Article  MATH  Google Scholar 

  62. Youssef, H.M.: Two-temperature generalized thermoelastic infinite medium with cylindrical cavity subjected to moving heat source. Arch. Appl. Mech. 80, 1213–1224 (2010)

    Article  MATH  Google Scholar 

  63. Youssef, H.M., Abbas, I.A.: Thermal shock problem of generalized thermoelasticity for an infinitely long annular cylinder with variable thermal conductivity. Comput. Methods Sci. Technol. 13, 95–100 (2007)

    Article  Google Scholar 

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  1. Institute of Mathematics and Computer Science, Jan Długosz University, Częstochowa, Poland

    Yuriy Povstenko

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Povstenko, Y. (2015). Thermoelasticity Based on Time-Fractional Heat Conduction Equation in Polar Coordinates. In: Fractional Thermoelasticity. Solid Mechanics and Its Applications, vol 219. Springer, Cham. https://doi.org/10.1007/978-3-319-15335-3_3

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  • DOI: https://doi.org/10.1007/978-3-319-15335-3_3

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  • Online ISBN: 978-3-319-15335-3

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