Geomechanics and Engineering

  • 제21권1호
  • /
  • Pages.85-93
  • /
  • 2020
  • /
  • 2005-307X(pISSN)
  • /
  • 2092-6219(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Fractional order thermoelastic wave assessment in a two-dimension medium with voids

  • Hobiny, Aatef D. (Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of Mathematics, King Abdulaziz University) ;
  • Abbas, Ibrahim A. (Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of Mathematics, King Abdulaziz University)
  • 투고 : 2019.11.10
  • 심사 : 2020.03.12
  • 발행 : 2020.04.10
⟨ 이전 논문 다음 논문 ⟩

초록

In this article, the generalized thermoelastic theory with fractional derivative is presented to estimate the variation of temperature, the components of stress, the components of displacement and the changes in volume fraction field in two-dimensional porous media. Easily, the exact solutions in the Laplace domain are obtained. By using Laplace and Fourier transformations with the eigenvalues method, the physical quantities are obtained analytically. The numerical results for all the physical quantities considered are implemented and presented graphically. The results display that the present model with the fractional derivative is reduced to the Lord and Shulman (LS) and the classical dynamical coupled (CT) theories when the fractional parameter is equivalent to one and the delay time is equal to zero and respectively.

키워드

과제정보

연구 과제 주관 기관 : King Abdulaziz University

This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No (DF-182-130-1441). The authors, therefore, gratefully acknowledge the DSR technical and financial support.

참고문헌

  1. Abbas, I.A. (2007), "Finite element analysis of the thermoelastic interactions in an unbounded body with a cavity", Forschung im Ingenieurwesen, 71(3-4), 215-222. https://doi.org/10.1007/s10010-007-0060-x.
  2. Abbas, I.A. (2009), "Generalized magneto-thermoelasticity in a nonhomogeneous isotropic hollow cylinder using the finite element method", Arch. Appl. Mech., 79(1), 41-50. https://doi.org/10.1007/s00419-008-0206-9.
  3. Abbas, I.A. (2014), "Nonlinear transient thermal stress analysis of thick-walled FGM cylinder with temperature-dependent material properties", Meccanica, 49(7), 1697-1708. https://doi.org/10.1007/s11012-014-9948-3
  4. Abbas, I.A. (2014), "The effects of relaxation times and a moving heat source on a two-temperature generalized thermoelastic thin slim strip", Can. J. Phys., 93(5), 585-590. https://doi.org/10.1139/cjp-2014-0387.
  5. Abbas, I.A. (2015), "The effects of relaxation times and a moving heat source on a two-temperature generalized thermoelastic thin slim strip", Can. J. Phys., 93(5), 585-590. https://doi.org/10.1139/cjp-2014-0387.
  6. Abbas, I.A. and Alzahrani, F.S. (2016), "Analytical solution of a two-dimensional thermoelastic problem subjected to laser pulse", Steel Compos. Struct., 21(4), 791-803. https://doi.org/10.12989/scs.2016.21.4.791.
  7. Abbas, I.A. and Kumar, R. (2016), "2D deformation in initially stressed thermoelastic half-space with voids", Steel Compos. Struct., 20(5), 1103-1117. https://doi.org/10.12989/scs.2016.20.5.1103.
  8. Abbas, I.A., Alzahrani, F.S. and Elaiw, A. (2019), "A DPL model of photothermal interaction in a semiconductor material", Waves Random Complex Media, 29(2), 328-343. https://doi.org/10.1080/17455030.2018.1433901.
  9. Abbas, I.A., El-Amin, M. and Salama, A. (2009), "Effect of thermal dispersion on free convection in a fluid saturated porous medium", Int. J. Heat Fluid Flow, 30(2), 229-236. https://doi.org/10.1016/j.ijheatfluidflow.2009.01.004.
  10. Abd-Elaziz, E.M., Marin, M. and Othman, M.I. (2019), "On the effect of Thomson and initial stress in a thermo-porous elastic solid under GN electromagnetic theory", Symmetry, 11(3), 413. https://doi.org/10.3390/sym11030413.
  11. Alzahrani, F.S. and Abbas, I.A. (2016), "The effect of magnetic field on a thermoelastic fiber-reinforced material under GN-III theory", Steel Compos. Struct., 22(2), 369-386. https://doi.org/10.12989/scs.2016.22.2.369.
  12. Biot, M.A. (1956), "Thermoelasticity and irreversible thermodynamics", J. Appl. Phys., 27(3), 240-253. https://doi.org/10.1063/1.1722351.
  13. Cattaneo, C. (1958), "A form of heat conduction equation which eliminates the paradox of instantaneous propagation", Compte Rendus, 247(4), 431-433.
  14. Das, N.C., Lahiri, A. and Giri, R.R. (1997), "Eigenvalue approach to generalized thermoelasticity", Indian J. Pure Appl. Math., 28(12), 1573-1594.
  15. Debnath, L. and Bhatta, D. (2014), Integral Transforms and Their Applications, Chapman and Hall, CRC.
  16. Deswal, S. and Kalkal, K.K. (2014), "Plane waves in a fractional order micropolar magneto-thermoelastic half-space", Wave Motion, 51(1), 100-113. https://doi.org/10.1016/j.wavemoti.2013.06.009.
  17. Ellahi, R., Sait, S.M., Shehzad, N. and Ayaz, Z. (2019), "A hybrid investigation on numerical and analytical solutions of electro-magnetohydrodynamics flow of nanofluid through porous media with entropy generation", Int. J. Numer. Meth. Heat Fluid Flow, 30(2), 834-854. https://doi.org/10.1108/HFF-06-2019-0506.
  18. Ezzat, M., El-Karamany, A. and El-Bary, A. (2016), "Modeling of memory-dependent derivative in generalized thermoelasticity", Eur. Phys. J. Plus, 131(10), 372. https://doi.org/10.1140/epjp/i2016-16372-3.
  19. Ezzat, M.A. (2011), "Theory of fractional order in generalized thermoelectric MHD", Appl. Math. Modell., 35(10), 4965-4978. https://doi.org/10.1016/j.apm.2011.04.004.
  20. Ezzat, M.A. and El-Bary, A.A. (2017), "A functionally graded magneto-thermoelastic half space with memory-dependent derivatives heat transfer", Steel Compos. Struct., 25(2), 177-186. https://doi.org/10.12989/scs.2017.25.2.177.
  21. Ezzat, M.A. and El-Karamany, A.S. (2011), "Fractional order theory of a perfect conducting thermoelastic medium", Can. J. Phys., 89(3), 311-318. https://doi.org/10.1139/P11-022.
  22. Ezzat, M.A. and El Karamany, A.S. (2011), "Theory of fractional order in electro-thermoelasticity", Eur. J. Mech. A/Solids, 30(4), 491-500. https://doi.org/10.1016/j.euromechsol.2011.02.004.
  23. Ezzat, M.A., AlSowayan, N.S., Al-Muhiameed, Z.I. and Ezzat, S.M. (2014), "Fractional modelling of Pennes' bioheat transfer equation", Heat Mass Transfer, 50(7), 907-914. https://doi.org/10.1007/s00231-014-1300-x.
  24. Hobiny, A.D. and Abbas, I.A. (2018), "Fractional order photo-thermo-elastic waves in a two-dimensional semiconductor plate", Eur. Phys. J. Plus, 133(6), 232. https://doi.org/10.1140/epjp/i2018-12054-6.
  25. Hussein, E.M. (2015), "Fractional order thermoelastic problem for an infinitely long solid circular cylinder", J. Therm. Stresses, 38(2), 133-145. https://doi.org/10.1080/01495739.2014.936253.
  26. Kakar, R. and Kakar, S. (2014), "Electro-magneto-thermoelastic surface waves in non-homogeneous orthotropic granular half space", Geomech. Eng., 7(1), 1-36. https://doi.org/10.12989/gae.2014.7.1.001.
  27. Karageorghis, A., Lesnic, D. and Marin, L. (2014), "A moving pseudo-boundary MFS for void detection in two-dimensional thermoelasticity", Int. J. Mech. Sci., 88, 276-288. https://doi.org/10.1016/j.ijmecsci.2014.05.015.
  28. Kaur, I. and Lata, P. (2020), "Stoneley wave propagation in transversely isotropic thermoelastic medium with two temperature and rotation", GEM-Int. J. Geomath., 11(1), 4. https://doi.org/10.1007/s13137-020-0140-8.
  29. Lata, P. and Kaur, H. (2019), "Deformation in transversely isotropic thermoelastic medium using new modified couple stress theory in frequency domain", Geomech. Eng., 19(5), 369-381. https://doi.org/10.12989/gae.2019.19.5.369.
  30. Lata, P. and Kaur, I. (2019), "Effect of time harmonic sources on transversely isotropic thermoelastic thin circular plate", Geomech. Eng., 19(1), 29-36. https://doi.org/10.12989/gae.2019.19.1.029.
  31. Lata, P. and Zakhmi, H. (2019), "Fractional order generalized thermoelastic study in orthotropic medium of type GN-III", Geomech. Eng., 19(4), 295-305. https://doi.org/10.12989/gae.2019.19.4.295.
  32. Lord, H.W. and Shulman, Y. (1967), "A generalized dynamical theory of thermoelasticity", J. Mech. Phys. Solids, 15(5), 299-309. https://doi.org/10.1016/0022-5096(67)90024-5.
  33. Marin, M. (2009), "On the minimum principle for dipolar materials with stretch", Nonlin. Anal. Real World Appl., 10(3), 1572-1578. https://doi.org/10.1016/j.nonrwa.2008.02.001.
  34. Marin, M. and Nicaise, S. (2016), "Existence and stability results for thermoelastic dipolar bodies with double porosity", Contin. Mech. Thermodyn., 28(6), 1645-1657. https://doi.org/10.1007/s00161-016-0503-4.
  35. Marin, M., Ellahi, R. and Chirila, A. (2017), "On solutions of Saint-Venant's problem for elastic dipolar bodies with voids", Carpathian J. Math., 33(2), 219-232. https://doi.org/10.37193/CJM.2017.02.09
  36. Marin, M., Vlase, S., Ellahi, R. and Bhatti, M. (2019), "On the partition of energies for the backward in time problem of thermoelastic materials with a dipolar structure", Symmetry, 11(7), 863. https://doi.org/10.3390/sym11070863.
  37. Mohamed, R., Abbas, I.A. and Abo-Dahab, S. (2009), "Finite element analysis of hydromagnetic flow and heat transfer of a heat generation fluid over a surface embedded in a non-Darcian porous medium in the presence of chemical reaction", Commun. Nonlin. Sci. Numer. Simul., 14(4), 1385-1395. https://doi.org/10.1016/j.cnsns.2008.04.006.
  38. Othman, M., Sarkar, N. and Atwa, S.Y. (2013), "Effect of fractional parameter on plane waves of generalized magneto-thermoelastic diffusion with reference temperature-dependent elastic medium", Comput. Math. Appl., 65(7), 1103-1118. https://doi.org/10.1016/j.camwa.2013.01.047.
  39. Othman, M.I. and Abd-Elaziz, E.M. (2019), "Influence of gravity and micro-temperatures on the thermoelastic porous medium under three theories", Int. J. Numer. Meth. Heat Fluid Flow, 29(9), 3242-3262. https://doi.org/10.1108/HFF-12-2018-0763.
  40. Othman, M.I. and Marin, M. (2017), "Effect of thermal loading due to laser pulse on thermoelastic porous medium under GN theory", Results Phys., 7, 3863-3872. https://doi.org/10.1016/j.rinp.2017.10.012.
  41. Sarkar, N. (2017), "Wave propagation in an initially stressed elastic half-space solids under time-fractional order two-temperature magneto-thermoelasticity", Eur. Phys. J. Plus, 132(4), 154. https://doi.org/10.1140/epjp/i2017-11426-8.
  42. Sarkar, N. and Lahiri, A. (2013), "The effect of fractional parameter on a perfect conducting elastic half-space in generalized magneto-thermoelasticity", Meccanica, 48(1), 231-245. https://doi.org/10.1007/s11012-012-9597-3.
  43. Sheikholeslami, M., Ellahi, R., Shafee, A. and Li, Z. (2019), "Numerical investigation for second law analysis of ferrofluid inside a porous semi annulus: An application of entropy generation and exergy loss", Int. J. Numer. Meth. Heat Fluid Flow, 29(3), 1079-1102. https://doi.org/10.1108/HFF-10-2018-0606.
  44. Sherief, H.H., El-Sayed, A.M.A. and Abd El-Latief, A.M. (2010), "Fractional order theory of thermoelasticity", Int. J. Solids Struct., 47(2), 269-275. https://doi.org/10.1016/j.ijsolstr.2009.09.034.
  45. Singh, B. (2007), "Wave propagation in a generalized thermoelastic material with voids", Appl. Math. Comput., 189(1), 698-709. https://doi.org/10.1016/j.amc.2006.11.123.
  46. Stehfest, H. (1970), "Algorithm 368: Numerical inversion of Laplace transforms [D5]", Commun. ACM, 13(1), 47-49. https://doi.org/10.1145/361953.361969.
  47. Sur, A. and Kanoria, M. (2014), "Fractional order generalized thermoelastic functionally graded solid with variable material properties", J. Solid Mech., 6(1), 54-69.
  48. Wang, Y., Liu, D. and Wang, Q. (2015), "Effect of fractional order parameter on thermoelastic behaviors in infinite elastic medium with a cylindrical cavity", Acta Mechanica Solida Sinica, 28(3), 285-293. https://doi.org/10.1016/S0894-9166(15)30015-X.
  49. Youssef, H.M. (2010), "Theory of fractional order generalized thermoelasticity", J. Heat Transfer, 132(6), 1-7. https://doi.org/10.1115/1.4000705.
  50. Youssef, H.M. (2012), "Two-dimensional thermal shock problem of fractional order generalized thermoelasticity", Acta Mechanica, 223(6), 1219-1231. https://doi.org/10.1007/s00707-012-0627-y.
  51. Youssef, H.M. and Al-Lehaibi, E.A. (2010), "Variational principle of fractional order generalized thermoelasticity", Appl. Math. Lett., 23(10), 1183-1187. https://doi.org/10.1016/j.aml.2010.05.008.

피인용 문헌

  1. A dual-phase-lag theory of thermal wave in a porothermoelastic nanoscale material by FEM vol.79, pp.1, 2020, https://doi.org/10.12989/sem.2021.79.1.001