Error message

  • Deprecated function: Unparenthesized `a ? b : c ? d : e` is deprecated. Use either `(a ? b : c) ? d : e` or `a ? b : (c ? d : e)` in include_once() (line 1439 of /home/science2016/public_html/includes/bootstrap.inc).
  • Deprecated function: Array and string offset access syntax with curly braces is deprecated in include_once() (line 3557 of /home/science2016/public_html/includes/bootstrap.inc).

Thermal stresses in a long cylinder under Gaussian-distributed heating in the framework of fractional thermoelasticity

Povstenko Yu.
AttachmentSize
PDF icon 2015_2_1_077_087.pdf290.51 KB

Keywords:

Abstract: 
An axisymmetric problem for Gaussian-distributed heating of a lateral surface of an infinite cylinder is solved in the framework of fractional thermoelasticity based on the time-fractional heat conduction equation with the Caputo derivative. The representation of stresses in terms of displacement potential and Love function is used to satisfy the boundary conditions on a surface of a cylinder. The results of numerical calculation are presented for different values of the order of fractional derivative and nondimensional time.
References: 
  1. Gurtin M. E., Pipkin A. C. A general theory of heat conduction with finite wave speeds. Arch. Rational Mech. Anal. 31, 113 (1968).
  2. Povstenko Y. Thermoelasticity which uses fractional heat conduction equation. Math. Meth. Phys.-Mech. Fields 51, 239 (2008). See also J. Math. Sci. 162, 296 (2009).
  3. Povstenko Y. Theory of thermoelasticity based on the space-time-fractional heat conduction equation. Phys. Scr. T. 136, 014017 (2009).
  4. Povstenko Y. Fractional Cattaneo-type equations and generalized thermoelasticity. J. Thermal Stresses. 34, 97 (2011).
  5. Povstenko Y. Theories of thermal stresses based on space-time-fractional telegraph equations. Comp. Math. Appl. 64, 3321 (2012).
  6. Cattaneo C. Sulla conduzione del calore. Atti Sem. Mat. Fis. Univ. Modena. 3, 83 (1948).
  7. Lord H. W., Shulman Y. A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids. 15, 299 (1967).
  8. Nigmatullin R. R. To the theoretical explanation of the “universal response”. Phys. Stat. Sol. (b). 123, 739 (1984).
  9. Green A. E., Naghdi P. M. Thermoelasticity without energy dissipation. J. Elast. 31, 189 (1993).
  10. Povstenko Y. Fractional heat conduction equation and associated thermal stresses. J. Thermal Stresses. 28, 83 (2005).
  11. Povstenko Y. Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder. Fract. Calc. Appl. Anal. 14, 418 (2011).
  12. Gorenflo R., Mainardi F. Fractional calculus: Integral and differential equations of fractional order. In: Carpinteri A., Mainardi F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 223–276. Springer (1997).
  13. Podlubny I. Fractional Differential Equations. Academic Press (1999).
  14. Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and Applications of Fractional Differential Equations. Elsevier (2006).
  15. Povstenko Y. Stresses exerted by a source of diffusion in a case of a non-parabolic diffusion equation. Int. J. Eng. Sci. 43, 977 (2005).
  16. 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 (2007).
  17. Povstenko Y. Fundamental solution to three-dimensional diffusion-wave equation and associated diffusive stresses. Chaos, Solitons and Fractals. 36, 961 (2008).
  18. Povstenko Y. Fundamental solutions to central symmetric problems for fractional heat conduction equation and associated thermal stresses. J. Thermal Stresses. 31, 127 (2008).
  19. Povstenko Y. Fractional heat conduction equation and associated thermal stresses in an infinite solid with spherical cavity. Quart. J. Mech. Appl. Math. 61, 523 (2008).
  20. Povstenko Y. Fractional radial heat conduction in an infinite medium with a cylindrical cavity and associated thermal stresses. Mech. Res. Commun. 37, 436 (2010).
  21. Povstenko Y. Dirichlet problem for time-fractional radial heat conduction in a sphere and associated thermal stresses. J. Thermal Stresses. 34, 51 (2011).
  22. Povstenko Y. Time-fractional radial heat conduction in a cylinder and associated thermal stresses. Arch. Appl. Mech. 82, 345 (2012).
  23. Povstenko Y. The Neumann boundary problem for axisymmetric fractional heat conduction in a solid with cylindrical hole and associated thermal stresses. Meccanica. 47, 23 (2012).
  24. Povstenko Y. Fractional Thermoelasticity. In: Hetnarski R. B. (ed.) Encyclopedia of Thermal Stresses. 4, 1778–1787. Springer (2014).
  25. Povstenko Y. Fractional Thermoelasticity. Springer (2015).
  26. Narahari Achar B. N., Hanneken J. W. Fractional radial diffusion in a cylinder. J. Mol. Liq. 114, 147 (2004).
  27. Povstenko Y. Fractional radial diffusion in a cylinder. J. Mol. Liq. 137, 46 (2008).
  28. Özdemir N., Karadeniz D. Fractional diffusion-wave problem in cylindrical coordinates. Phys. Lett. A. 372, 5968 (2008).
  29. Özdemir N., Agrawal O. P., Karadeniz D., Iskender B. B. Analysis of an axis-symmetric fractional diffusion-wave problem. J. Phys. A: Math. Theor. 42, 355208 (2009).
  30. Özdemir N., Karadeniz D., Iskender B. B. Fractional optimal control problems of a distributed systems in cylindrical coordinates. Phys. Lett. A. 373, 221 (2009).
  31. Povstenko Y. Fractional radial diffusion in an infinite medium with a cylindrical cavity. Quart. Appl. Math. 47, 113 (2009).
  32. Povstenko Y. Signaling problem for time-fractional diffusion-wave equation in a half-space in the case of angular symmetry. Nonlinear Dyn. 59, 593 (2010).
  33. Povstenko Y. Evolution of the initial box-signal for time-fractional diffusion-wave equation in a case of different spatial dimensions. Physica A. 381, 4696 (2010).
  34. Povstenko Y. Non-axisymmetric solutions to time-fractional heat conduction equation in a half-space in cylindrical coordinates. Math. Meth. Phys.-Mech. Fields. 54, 212 (2011).
  35. Povstenko Y. Solutions to time-fractional diffusion-wave equation in cylindrical coordinates. Adv. Difference Equat. 2011, 930297 (2011).
  36. Povstenko Y. Axisymmetric solutions to time fractional heat conduction equation in a half-space under Robin boundary conditions. Int. J. Diff. Equat. 2012, 154085 (2012).
  37. Povstenko Y. Axisymmetric solutions to fractional diffusion-wave equation in a cylinder under Robin boundary condition. Eur. Phys. J. Special Topics. 222, 1767 (2013).
  38. Sneddon I. N. The Use of Integral Transforms. McGraw-Hill (1972).
  39. Erdélyi A., Magnus W., Oberhettingerv F., Tricomi F. Higher Transcendental Functions. 3. McGraw-Hill (1955).
  40. Parkus H. Instationäre Wärmespannungen. Springer (1959).
  41. Nowacki W. Thermoelasticity. Pergamon Press; 2nd edition (1986).
  42. Gorenflo R., Loutchko J., Luchko Yu. Computation of the Mittag-Leffer function and its derivatives. Fract. Calc. Appl. Anal. 5, 491 (2002).
Bibliography: 
Math. Model. Comput. Vol.2, No.1, pp.77-87 (2015)