Geomechanics and Engineering

  • 제23권4호
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  • Pages.393-403
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  • 2020
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  • 2005-307X(pISSN)
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  • 2092-6219(eISSN)
테크노프레스 (Techno-Press)
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DOI QR코드

DOI QR Code

Modeling of GN type III with MDD for a thermoelectric solid subjected to a moving heat source

  • Ezzat, Magdy A. (Department of Mathematics, College of Science and Arts, Al Qassim University)
  • 투고 : 2020.08.28
  • 심사 : 2020.11.11
  • 발행 : 2020.11.25
⟨ 이전 논문 다음 논문 ⟩

초록

We design the Green-Naghdi model type III (GN-III) with widespread thermoelasticity for a thermoelectric half space using a memory-dependent derivative rule (MDD). Laplace transformations and state-space techniques are used in order to find the general solution for any set of limit conditions. A basic question of heat shock charging half space and a traction-free surface was added to the formulation in the present situation of a traveling heat source with consistent heating speed and ramp-type heating. The Laplace reverse transformations are numerically recorded. There are called the impacts of several calculations of the figure of the value, heat source spead, MDD parameters, magnetic number and the parameters of the ramping period.

키워드

참고문헌

  1. Abd-Elaziz, E.M., Marin, M. and Othman, M.I.A. (2019), "On the effect of thomson and initial stress in a thermo-porous elastic solid under G-N electromagnetic theory", Symmetry, 11(3), 413. https://doi.org/10.3390/sym11030413.
  2. Biot, M.A. (1956), "Thermoelasticity and irreversible thermodynamics", J. Appl. Mech. Tech. Phys., 27(3), 240-253. https://doi.org/10.1063/1.1722351.
  3. Biswas, S. (2019a), "Modeling of memory-dependent derivatives in orthotropic medium with three-phase-lag model under the effect of magnetic field", Mech. Based Des. Struct. Mach., 47(3), 302-318. https://doi.org/10.1080/15397734.2018.1548968.
  4. Biswas, S. (2019b), "Modeling of memory-dependent derivatives with the state-space approach", Multidisc. Model. Mat Struct., 16(4), 657-677. https://doi.org/10.1108/MMMS-06-2019-0120.
  5. Biswas, S. and Mukhopadhyay, B. (2018), "Eigenfunction expansion method to analyze thermal shock behavior in magneto-thermoelastic orthotropic medium under three theories", J. Therm. Stresses, 41(3), 366-382. https://doi.org/10.1080/01495739.2017.1393780.
  6. Biswas, S., Mukhopadhyay, B. and Shaw, S. (2017), "Thermal shock response in magneto-thermoelastic orthotropic medium with three-phase-lag model", J. Electromag. Waves Appl., 31(9), 879-897. https://doi.org/10.1080/09205071.2017.1326851.
  7. Chandrasekharaiah, D.S. (1996) A uniqueness theorem in the theory of thermoelasticity without energy dissipation", J. Therm. Stresses, 19(3), 267-272. https://doi.org/10.1080/01495739608946173.
  8. Chandrasekharaiah, D.S. (1998), "Hyperbolic thermoelasticity: A review of recent literature", Appl. Mech. Rev., 51(12), 705-729. https://doi.org/10.1115/1.3098984.
  9. Chirita, S. and Ciarletta, M. (2010), "Reciprocal and variational principles in linear thermoelasticity without energy dissipation", Mech. Res. Commun., 37(3), 271-275. https://doi.org/10.1016/j.mechrescom.2010.03.001.
  10. Choudhuri, S.R. (2007), "On a thermoelastic three-phase-lag model", J. Therm. Stresses, 30(3), 231-238. https://doi.org/10.1080/01495730601130919.
  11. Ciarletta, M. (1999), "A theory of micropolar thermoelasticity without energy dissipation", J. Therm. Stresses, 22(6) 581-594. https://doi.org/10.1080/014957399280760.
  12. El-Karamany, A.S. and Ezzat, M.A. (2011), "On the two-temperature Green-Naghdi thermoelasticity theories", J. Therm. Stresses, 34(2), 1207-1226. https://doi.org/10.1080/01495739.2011.608313.
  13. El-Karamany, A.S. and Ezzat, M.A. (2016), "On the phase-lag Green-Naghdi thermoelasticity theories", Appl. Math. Model., 40(9-10), 5643-5659. https://doi.org/10.1016/j.apm.2016.01.010.
  14. Ezzat, M.A. (2008), "State space approach to solids and fluids", Can. J. Phys., 86(11), 1241-1250. https://doi.org/10.1139/P08-069.
  15. Ezzat, M.A. and El-Bary, A.A. (2016), "Modeling of fractional magneto-thermoelasticity for a perfect conducting materials", Smart Struct. Syst., 18(4), 707-731. https://doi.org/10.12989/sss.2016.18.4.707.
  16. 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.
  17. Ezzat, M.A. and Youssef, H.M. (2010), "Stokes' first problem for an electro-conducting micropolar fluid with thermoelectric properties", Can. J. Phys., 88(1) 35-48. https://doi.org/10.1139/P09-100.
  18. Ezzat, M.A., El-Karamany, A.S. and El-Bary, A. (2016), "Electro-thermoelasticity theory with memory-dependent derivative heat transfer", Int. J. Eng. Sci., 99(2), 22-38. https://doi.org/10.1016/j.ijengsci.2015.10.011.
  19. Ezzat, M.A., El-Karamany, A.S. and El-Bary, A. (2018), "Two-temperature theory in Green-Naghdi thermoelasticity with fractional phase-lag heat transfer", Microsyst. Technol., 24(2), 951-961. https://doi.org/10.1007/s00542-017-3425-6.
  20. Ezzat, M.A., El-Karamany, A.S. and El-Bary, A.A. (2015), "A novel magneto-thermoelasticity theory with memory-dependent derivative", J. Electromag. Waves Appl., 29(8), 1018-1031. https://doi.org/10.1080/09205071.2015.1027795.
  21. Ghazanfarian, J., Shomali, Z. and Abbassi, A. (2015), "Macro- to nano scale heat and mass transfer: The lagging behavior", Int. J. Thermophys., 36(7), 1416-1467. https://doi.org/10.1007/s10765-015-1913-4.
  22. Green, A. and Lindsay, K. (1972), "Thermoelasticity", J. Elasticity, 2(1), 1-7. https://doi.org/10.1007/BF00045689.
  23. Green, A.E. and Naghdi, P.M. (1991), "A re-examination of the basic postulates of thermomechanics", Proc. Royal Soc. London A, 432(1885), 171-194. https://doi.org/10.1098/rspa.1991.0012.
  24. Green, A.E. and Naghdi, P.M. (1992), "On undamped heat waves in an elastic solid", J. Therm. Stresses, 15(2), 253-264. https://doi.org/10.1080/01495739208946136.
  25. Green, A.E. and Naghdi, P.M. (1993), "Thermoelasticity without energy dissipation", J. Elasticity, 31(3), 189-208. https://doi.org/10.1007/BF00044969.
  26. Hetnarski, R.B. and Ignaczak, J. (2000), "Nonclassical dynamical thermoelasticity", Int. J. Solids Struct., 37(1), 215-224. https://doi.org/10.1016/S0020-7683(99)00089-X.
  27. Hiroshige, Y., Makoto, O. and Toshima, N. (2007), "Thermoelectric figure-of-merit of iodine-doped copolymer of phenylenevinylene with dialkoxyphenylenevinylene", Synth. Metals., 157(10-12), 467-474. https://doi.org/10.1016/j.synthmet.2007.05.003.
  28. Hohn, C., Galffy, M., Dascoulidou, A., Freimuth, A., Soltner, H. and Poppe, U. (1991), "Seebeck-effect in the mixed state of Y-Ba-Cu-O", Z. Phys. B., 85(2),161-168. https://doi.org/10.1007/BF01313216.
  29. Honig, G. and Hirdes, U. (1984), "A method for the numerical inversion of the Laplace transform", J. Comput. Appl. Math., 10(1), 113-132. https://doi.org/10.1016/0377-0427(84)90075-X.
  30. Kaliski, S. and Nowacki, W. (1963), "Combined elastic and electro-magnetic waves produced by thermal shock in the case of a medium of finite electric conductivity", Int. J. Eng. Sci., 1(2), 163-175. https://doi.org/10.1016/0020-7225(63)90031-4.
  31. Kothari, S. and Mukhopadhyay, S.A. (2011), "Problem on elastic half space under fractional order theory of thermoelasticity", J. Therm. Stresses, 34, 724-739. https://doi.org/10.1080/01495739.2010.550834.
  32. Kumar, R., Sharma, N. and Lata, P. (2016), "Effect of Hall current in a transversely isotropic magnetothermoelastic two temperature medium with rotation and with and without energy dissipation due to normal force", Struct. Eng. Mech., 57(1), 91-103. https://doi.org/10.12989/sem.2016.57.1.091.
  33. Lata, P. and Kaur, I. (2019), "Thermomechanical interactions in transversely isotropic magneto thermoelastic solid with two temperatures and without energy dissipation", Steel Compos. Struct., 32(6), 779-793. https://doi.org/10.12989/scs.2019.32.6.779.
  34. Lata, P. and Singh, S. (2019), "Effect of nonlocal parameter on nonlocal thermoelastic solitidue to inclined load", Steel Comps. Struct., 33(1), 955-963. https://doi.org/10.12989/scs.2019.33.1.123.
  35. Lata, P., Kumar, R. and Sharma, N. (2016), "Plane waves in an anisotropic thermoelastic", Steel Compos. Struct., 22(3), 567-587. https://doi.org/10.12989/scs.2016.22.3.567.
  36. Lord, H. 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.
  37. Lotfy, K. and Sarkar, N. (2017), "Memory-dependent derivatives for photothermal semiconducting medium in generalized thermoelasticity with two-temperature", Mech. Time-Dep. Mater., 21(4), 519-534. https://doi.org/10.1007/s11043-017-9340-5.
  38. Mahan, G., Sales, B. and Sharp, J. (1997), "Thermoelectric materials: New approaches to an old problem", Phys. Today, 50(3), 42-47. https://doi.org/10.1063/1.881752.
  39. Marin, M. (1995), "On existence and uniqueness in thermoelasticity of micropolar bodies", Comptes Rendus de l'Academie des Sciences Paris, 321(12), 375-480.
  40. Marin, M. (1996), "Some basic theorems in elastostatics of micropolar materials with voids", J. Comput. Appl. Math., 70(1), 115-126. https://doi.org/10.1016/0377-0427(95)00137-9.
  41. Marin, M. (2009), "On the minimum principle for dipolar materials with stretch", Nonlin. Anal., 10(3), 1572-1578. https://doi.org/10.1016/j.nonrwa.2008.02.001.
  42. Marin, M. (2010), "A partition of energy in thermoelasticity of microstretch bodies", Nonlin. Anal., 11(4), 2436-2447. https://doi.org/10.1016/j.nonrwa.2009.07.014.
  43. Marin, M. and Lupu, M. (1998), "On harmonic vibrations in thermoelasticity of micropolar bodies", J. Vib. Control, 4(5), 507-518. https://doi.org/10.1177/107754639800400501.
  44. Marin, M. and Stan, G. (2013), "Weak solutions in elasticity of dipolar bodies with stretch", Car. J. Math., 29(1), 33-40. https://doi.org/10.37193/CJM.2013.01.12
  45. Mukhopadhyay, S. and Kumar, R. (2009), "Thermoelastic interactions on two-temperature generalized thermoelasticity in an infinite medium with a cylindrical cavity", J. Therm. Stresses, 32(4), 341-360. https://doi.org/10.1080/01495730802637183.
  46. Othman, M.I., Ezzat, M.A., Zaki, S.A. and El-Karamany, A.S. (2002), "Generalized thermo-viscoelastic plane waves with two relaxation times", Int. J. Eng. Sci., 40(12), 1329-1347. https://doi.org/10.1016/S0020-7225(02)00023-X.
  47. Povstenko, Y.Z. (2009), "Thermoelasticity that uses fractional heat conduction equation", J. Math. Sci., 162(2), 296-305. https://doi.org/10.1007/s10958-009-9636-3.
  48. Rowe, D.M. (1995), Handbook of Thermoelectrics, CRC Press.
  49. Sharma, K. (2010), "Boundary value problem in generalized thermodiffusive elastic medium", J. Solid Mech., 2(4), 348-362.
  50. Sharma, S. and Sharma, K. (2014), "Influence of heat sources and relaxation time on temperature distribution in tissues", Int. J. Appl. Mech. Eng., 19(2), 427-433. https://doi.org/10.2478/ijame-2014-0029.
  51. Sharma, S., Sharma, K. and Bhargava, R. (2016), "Effect of viscosity on wave propagation in anisotropic thermoelastic Green-Naghdi theory type-II and type-III", Mater. Phys. Mech., 16(2), 144-158.
  52. Shaw, S. (2019), "Theory of generalized thermoelasticity with memory-dependent derivatives", J. Eng. Mech., 145(3), 04019003. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001569.
  53. Shercliff, J.A. (1979), "Thermoelectric magnetohydrodynamics", J. Fluid Mech., 191(3), 231-251. https://doi.org/10.1017/S0022112079000136.
  54. Sherief, 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.
  55. Sherief, H.H. (1986), "Fundamental solution of generalized thermoelastic problem for short times", J. Therm. Stress., 9(2), 151-164. https://doi.org/10.1080/01495738608961894.
  56. Sherief, H.H. and Raslan, W.E. (2016), "Thermoelastic interactions without energy dissipation in an unbounded body with a cylindrical cavity", J. Therm. Stresses, 39(3), 326-332. https://doi.org/10.1080/01495739.2015.1125651.
  57. Sur, A. and Kanoria, M. (2019), "Memory response on thermal wave propagation in an elastic solid with voids", Mech. Based Des. Struct. Mach., 48(3), 326-347. https://doi.org/10.1080/15397734.2019.1652647.
  58. Tiwari, R. and Mukhopadhyay, S. (2018), "Analysis of wave propagation in the presence of a continuous line heat source under heat transfer with memory dependent derivatives", Math. Mech. Solids, 23(5), 820-834. https://doi.org/10.1177/1081286517692020.
  59. Tritt, T.M. (2000), Semiconductors and Semimetals, in Recent Trends in Thermoelectric Materials Research, Academic Press, San Diego, California, U.S.A.
  60. Tschoegl, N.W. (1997), "Time dependence in material properties: An overview", Mech. Time-Depend. Mat., 1(1), 3-31. https://doi.org/10.1023/A:1009748023394.
  61. Wang, J.L. and Li, H.F. (2011), "Surpassing the fractional derivative: Concept of the memory-dependent derivative", Comput. Math. Appl., 62(3), 1562-1567. https://doi.org/10.1016/j.camwa.2011.04.028.
  62. Xue, Z.N., Chen, Z.T. and Tian, X.G. (2018), "Thermoelastic analysis of a cracked strip under thermal impact based on memory-dependent heat conduction model", Eng. Fract. Mech., 200, 479-498. https://doi.org/10.1016/j.engfracmech.2018.08.018.
  63. Yu, B., Jiang, X. and Xu, H. (2015), "A novel compact numerical method for solving the two dimensional non-linear fractional reaction-subdiffusion equation", Num. Algor., 68(4), 923-950. https://doi.org/10.1007/s11075-014-9877-1.
  64. Yu, Y.J. and Deng, Z.C. (2020), "New insights on microscale transient thermoelastic responses for metals with electron-lattice coupling mechanism", Eur. J. Mech. A/Solids, 80, 103887. https://doi.org/10.1016/j.euromechsol.2019.103887.
  65. Yu, Y.J., Hu, W. and Tian, X.G. (2014), "A novel generalized thermoelasticity model based on memory-dependent derivative", Int. J. Eng. Sci., 81(3-4), 123-134. https://doi.org/10.1016/j.ijengsci.2014.04.014.
  66. Yu, Y.J., Tian, X.G. and Tian, J.L. (2013), "Fractional order generalized electro-magneto-thermo-elasticity", Eur. J. Mech. A/Solids, 42,188-202. https://doi.org/10.1016/j.euromechsol.2013.05.006.
  67. Zhang, H., Jiang, X. and Yang, X. (2018), "A time-space spectral method for the time-space fractional Fokker-Planck equation and its inverse problem", Appl. Math. Comput., 320, 302-318. https://doi.org/10.1016/j.amc.2017.09.040.