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Fractional Order Generalized Thermoelastic Problem in a Thick Circular Plate with Periodically Varying Heat Source

Year 2017, Volume: 20 Issue: 3, 132 - 138, 31.08.2017
https://doi.org/10.5541/eoguijt.336651

Abstract

This paper is concerned with fractional order thermoelastic response due
to a heat source whose magnitude varies periodically with time within the
context of generalized thermoelasticity with one relaxation time. Traction free
boundary conditions are considered and the thick circular plate is subjected to
a given axisymmetric temperature distribution. Integral transform technique is
used to derive the solution in the transformed domain. Laplace transforms are
inverted using a numerical scheme. Mathematical model is prepared for Copper
material and results for temperature, displacement and stress distributions are
computed and represented graphically.

References

  • [1] H. Lord, Y. Shulman, “A Generalized Dynamical theory of thermoelasticity,” J. Mechanics Physics Solids, 15, 299-307, 1967.
  • [2] S. H. Mallik, M. Kanoria, “A Two dimensional problem for a transversely isotropic generalized thermoelastic thick plate with spatially varying heat source,” European J. Mechanics A/Solids, 27, 607–621, 2008.
  • [3] N. M. El-Maghraby, “A two dimensional problem for a thick plate and heat sources in Generalized thermoelasticity,” J. Thermal Stresses, 28, 1227-1241, 2005.
  • [4] J. J. Tripathi, G. D. Kedar, K. C. Deshmukh, “Dynamic Problem of Generalized Thermoelasticity for a Semi-infinite Cylinder with Heat Sources,” J. Thermoelasticity , 2, 1-8, 2014.
  • [5] J. J. Tripathi, G. D. Kedar, K. C. Deshmukh, “Generalized thermoelastic diffusion problem in a thick circular plate with axisymmetric heat supply,” Acta Mech., 226, 2121-2134, 2015.
  • [6] J. J. Tripathi, G. D. Kedar, K. C. Deshmukh, “Two dimensional generalized thermoelastic diffusion in a half space under axisymmetric distributions,” Acta Mech., 226, 3263-3274, 2015.
  • [7] Y.Z. Povstenko, “Fractional heat conduction equation and associated thermal stress,” J. Thermal Stresses, 28, 83–102, 2005.
  • [8] Y.Z. Povstenko, “Fractional heat conduction equation and associated thermal stresses in an infinite solid with a spherical cavity,” Quart. J. Mech. Appl. Math., 61, 523–547, 2008.
  • [9] Y.Z. Povstenko, “Fractional radial diffusion in an infinite medium with cylindrical cavity,” Quart. Appl. Math., 67, 113–123, 2009.
  • [10] Y.Z. Povstenko, “Fractional radial heat conduction in an infinite medium with a cylindrical cavity and associated thermal stresses,” Mech. Res. Commun., doi: 10.1016/j.mechrescom. 2010.04.006.
  • [11] H. H. Sherief, A. El-Sayed, A. A. El-Latief, “Fractional order theory of thermoelasticity,” Int. J. Solids Struct. , 47, 269–275, 2010.
  • [12] M. A. Ezzat, A. S. El-Karamany, “Fractional order theory of a perfect conducting thermoelastic medium,” Can. J. Phys., 89, 311-318, 2011.
  • [13] M. A. Ezzat, A. S. El-Karamany, “Theory of Fractional order in electro-thermo-elasticity,” Eur. J. Mech. A/Solids, 30, 491-500, 2011.
  • [14] H. H. Sherief, A. El-Sayed, A. A. El-Latief, “Fractional order theory of thermoelasticity,” Int. J. Solids Struct., 47, 269–275, 2010.
  • [15] M. Baccher, “Deformations due to periodically varying heat sources in a reference temperature dependent thermoelastic porous material with a time free heat conduction law,” IRJET, 2, 145-152, 2015.
  • [16] M. Islam, M. Kanoria, “Short time analysis of magnetothermoelastic wave under fractional order heat conduction law,” J. Thermal Stresses, 38, 1219-1249, 2015.
  • [17] W. E. Raslan, “Application of Fractional order theory of thermoelasticity in a thick plate under axisymmetric temperature distribution,” J. Thermal Stresses, 38, 733-743, 2015.
  • [18] J.J. Tripathi, G.D. Kedar, K.C. Deshmukh, “Generalized thermoelastic diffusion in a thick circular plate including heat source,” Alexandria Engineering J., 55, 2241-2249, 2016.
  • [19] N, Sarkar, “Wave propagation in an initially stressed elastic half-space solids under time-fractional order two-temperature magneto-thermoelasticity,” Eur. Phys. J. Plus, 132, 154, 2017.
  • [20] J. J. Tripathi, G. D. Kedar, K. C. Deshmukh, “Dynamic problem offractional order thermoelasticity for a thick circular plate with finite wave speeds,” J. Thermal Stresses, 39, 220-230, 2016.
  • [21] S. D. Warbhe, J. J. Tripathi, K. C. Deshmukh, J. Verma, “Fractional Heat Conduction in a Thin Circular Plate With Constant Temperature Distribution and Associated Thermal Stresses,” J. Heat Transfer, 139, 044502, 2017.
  • [22] X. Chunbao, N. Yanbo, “Fractional-order generalized thermoelastic diffusion theory,” Applied Mathematics Mechanics, 38, 1091-1108, 2017.
  • [23] D. P. Gaver, “Observing Stochastic processes and approximate transform inversion,” Operations Res., 14, 444-459, 1966.
  • [24] H. Stehfast, “Algorithm 368, Numerical inversion of Laplace transforms,” Comm. Ass’n. Comp. Mach., 13, 47-49, 1970.
  • [25] H. Stehfast, “Remark on algorithm 368, Numerical inversion of Laplace transforms,” Comm. Ass’n. Comp., 3, 624, 1970.
  • [26] W. H. Press, B. P. Flannery, S. A. Teukolsky, W. A. Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, the art of scientific computing, 1986.
Year 2017, Volume: 20 Issue: 3, 132 - 138, 31.08.2017
https://doi.org/10.5541/eoguijt.336651

Abstract

References

  • [1] H. Lord, Y. Shulman, “A Generalized Dynamical theory of thermoelasticity,” J. Mechanics Physics Solids, 15, 299-307, 1967.
  • [2] S. H. Mallik, M. Kanoria, “A Two dimensional problem for a transversely isotropic generalized thermoelastic thick plate with spatially varying heat source,” European J. Mechanics A/Solids, 27, 607–621, 2008.
  • [3] N. M. El-Maghraby, “A two dimensional problem for a thick plate and heat sources in Generalized thermoelasticity,” J. Thermal Stresses, 28, 1227-1241, 2005.
  • [4] J. J. Tripathi, G. D. Kedar, K. C. Deshmukh, “Dynamic Problem of Generalized Thermoelasticity for a Semi-infinite Cylinder with Heat Sources,” J. Thermoelasticity , 2, 1-8, 2014.
  • [5] J. J. Tripathi, G. D. Kedar, K. C. Deshmukh, “Generalized thermoelastic diffusion problem in a thick circular plate with axisymmetric heat supply,” Acta Mech., 226, 2121-2134, 2015.
  • [6] J. J. Tripathi, G. D. Kedar, K. C. Deshmukh, “Two dimensional generalized thermoelastic diffusion in a half space under axisymmetric distributions,” Acta Mech., 226, 3263-3274, 2015.
  • [7] Y.Z. Povstenko, “Fractional heat conduction equation and associated thermal stress,” J. Thermal Stresses, 28, 83–102, 2005.
  • [8] Y.Z. Povstenko, “Fractional heat conduction equation and associated thermal stresses in an infinite solid with a spherical cavity,” Quart. J. Mech. Appl. Math., 61, 523–547, 2008.
  • [9] Y.Z. Povstenko, “Fractional radial diffusion in an infinite medium with cylindrical cavity,” Quart. Appl. Math., 67, 113–123, 2009.
  • [10] Y.Z. Povstenko, “Fractional radial heat conduction in an infinite medium with a cylindrical cavity and associated thermal stresses,” Mech. Res. Commun., doi: 10.1016/j.mechrescom. 2010.04.006.
  • [11] H. H. Sherief, A. El-Sayed, A. A. El-Latief, “Fractional order theory of thermoelasticity,” Int. J. Solids Struct. , 47, 269–275, 2010.
  • [12] M. A. Ezzat, A. S. El-Karamany, “Fractional order theory of a perfect conducting thermoelastic medium,” Can. J. Phys., 89, 311-318, 2011.
  • [13] M. A. Ezzat, A. S. El-Karamany, “Theory of Fractional order in electro-thermo-elasticity,” Eur. J. Mech. A/Solids, 30, 491-500, 2011.
  • [14] H. H. Sherief, A. El-Sayed, A. A. El-Latief, “Fractional order theory of thermoelasticity,” Int. J. Solids Struct., 47, 269–275, 2010.
  • [15] M. Baccher, “Deformations due to periodically varying heat sources in a reference temperature dependent thermoelastic porous material with a time free heat conduction law,” IRJET, 2, 145-152, 2015.
  • [16] M. Islam, M. Kanoria, “Short time analysis of magnetothermoelastic wave under fractional order heat conduction law,” J. Thermal Stresses, 38, 1219-1249, 2015.
  • [17] W. E. Raslan, “Application of Fractional order theory of thermoelasticity in a thick plate under axisymmetric temperature distribution,” J. Thermal Stresses, 38, 733-743, 2015.
  • [18] J.J. Tripathi, G.D. Kedar, K.C. Deshmukh, “Generalized thermoelastic diffusion in a thick circular plate including heat source,” Alexandria Engineering J., 55, 2241-2249, 2016.
  • [19] N, Sarkar, “Wave propagation in an initially stressed elastic half-space solids under time-fractional order two-temperature magneto-thermoelasticity,” Eur. Phys. J. Plus, 132, 154, 2017.
  • [20] J. J. Tripathi, G. D. Kedar, K. C. Deshmukh, “Dynamic problem offractional order thermoelasticity for a thick circular plate with finite wave speeds,” J. Thermal Stresses, 39, 220-230, 2016.
  • [21] S. D. Warbhe, J. J. Tripathi, K. C. Deshmukh, J. Verma, “Fractional Heat Conduction in a Thin Circular Plate With Constant Temperature Distribution and Associated Thermal Stresses,” J. Heat Transfer, 139, 044502, 2017.
  • [22] X. Chunbao, N. Yanbo, “Fractional-order generalized thermoelastic diffusion theory,” Applied Mathematics Mechanics, 38, 1091-1108, 2017.
  • [23] D. P. Gaver, “Observing Stochastic processes and approximate transform inversion,” Operations Res., 14, 444-459, 1966.
  • [24] H. Stehfast, “Algorithm 368, Numerical inversion of Laplace transforms,” Comm. Ass’n. Comp. Mach., 13, 47-49, 1970.
  • [25] H. Stehfast, “Remark on algorithm 368, Numerical inversion of Laplace transforms,” Comm. Ass’n. Comp., 3, 624, 1970.
  • [26] W. H. Press, B. P. Flannery, S. A. Teukolsky, W. A. Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, the art of scientific computing, 1986.
There are 26 citations in total.

Details

Subjects Engineering
Journal Section Regular Original Research Article
Authors

J.j. Tripathi

K.c. Deshmukh This is me

J. Verma This is me

Publication Date August 31, 2017
Published in Issue Year 2017 Volume: 20 Issue: 3

Cite

APA Tripathi, J., Deshmukh, K., & Verma, J. (2017). Fractional Order Generalized Thermoelastic Problem in a Thick Circular Plate with Periodically Varying Heat Source. International Journal of Thermodynamics, 20(3), 132-138. https://doi.org/10.5541/eoguijt.336651
AMA Tripathi J, Deshmukh K, Verma J. Fractional Order Generalized Thermoelastic Problem in a Thick Circular Plate with Periodically Varying Heat Source. International Journal of Thermodynamics. August 2017;20(3):132-138. doi:10.5541/eoguijt.336651
Chicago Tripathi, J.j., K.c. Deshmukh, and J. Verma. “Fractional Order Generalized Thermoelastic Problem in a Thick Circular Plate With Periodically Varying Heat Source”. International Journal of Thermodynamics 20, no. 3 (August 2017): 132-38. https://doi.org/10.5541/eoguijt.336651.
EndNote Tripathi J, Deshmukh K, Verma J (August 1, 2017) Fractional Order Generalized Thermoelastic Problem in a Thick Circular Plate with Periodically Varying Heat Source. International Journal of Thermodynamics 20 3 132–138.
IEEE J. Tripathi, K. Deshmukh, and J. Verma, “Fractional Order Generalized Thermoelastic Problem in a Thick Circular Plate with Periodically Varying Heat Source”, International Journal of Thermodynamics, vol. 20, no. 3, pp. 132–138, 2017, doi: 10.5541/eoguijt.336651.
ISNAD Tripathi, J.j. et al. “Fractional Order Generalized Thermoelastic Problem in a Thick Circular Plate With Periodically Varying Heat Source”. International Journal of Thermodynamics 20/3 (August 2017), 132-138. https://doi.org/10.5541/eoguijt.336651.
JAMA Tripathi J, Deshmukh K, Verma J. Fractional Order Generalized Thermoelastic Problem in a Thick Circular Plate with Periodically Varying Heat Source. International Journal of Thermodynamics. 2017;20:132–138.
MLA Tripathi, J.j. et al. “Fractional Order Generalized Thermoelastic Problem in a Thick Circular Plate With Periodically Varying Heat Source”. International Journal of Thermodynamics, vol. 20, no. 3, 2017, pp. 132-8, doi:10.5541/eoguijt.336651.
Vancouver Tripathi J, Deshmukh K, Verma J. Fractional Order Generalized Thermoelastic Problem in a Thick Circular Plate with Periodically Varying Heat Source. International Journal of Thermodynamics. 2017;20(3):132-8.