Steel and Composite Structures

  • 제45권4호
  • /
  • Pages.535-546
  • /
  • 2022
  • /
  • 1229-9367(pISSN)
  • /
  • 1598-6233(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Thermo-mechanical response of size-dependent piezoelectric materials in thermo-viscoelasticity theory

  • Ezzat, Magdy A. (Department of Mathematics, College of Science and Arts, Qassim University) ;
  • Al-Muhiameed, Zeid I.A. (Department of Mathematics, Faculty of Science, Qassim University)
  • 투고 : 2021.07.23
  • 심사 : 2022.11.14
  • 발행 : 2022.11.25
⟨ 이전 논문 다음 논문 ⟩

초록

The memory response of nonlocal systematical formulation size-dependent coupling of viscoelastic deformation and thermal fields for piezoelectric materials with dual-phase lag heat conduction law is constructed. The method of the matrix exponential, which constitutes the basis of the state-space approach of modern control theory, is applied to the non-dimensional equations. The resulting formulation together with the Laplace transform technique is applied to solve a problem of a semi-infinite piezoelectric rod subjected to a continuous heat flux with constant time rates. The inversion of the Laplace transforms is carried out using a numerical approach. Some comparisons of the impacts of nonlocal parameters and time-delay constants for various forms of kernel functions on thermal spreads and thermo-viscoelastic response are illustrated graphically.

키워드

참고문헌

  1. Abouelregal, A.E. and Marin, M. (2020a), "The size-dependent thermoelastic vibrations of nanobeams subjected to harmonic excitation and rectified sine wave heating", Mathematics, 8(7), 1128. https://doi.org/10.3390/math8071128.
  2. Abouelregal, A.E. and Marin, M. (2020b), "The response of nanobeams with temperature-dependent properties using statespace method via modified couple stress theory", Symmetry, 12(8), 1276. https://doi.org/10.3390/sym12081276
  3. Aldawody, D.A., Hendy, H.H. and Ezzat, M.A. (2019), "On dualphase-lag magneto-thermo-viscoelasticity theory with memory dependent derivative", Microsys. Tech., 25(8), 2915-2929. https://doi.org/10.1007/s00542-018-4194-6.
  4. Ashida, F. Andf Tauchert, T.R. (2004), "Piezothermoelastic response of a circular plate with thermal relaxation", J. Therm. Stress., 27(6), 513-536. https://doi.org/10.1080/01495730271405.
  5. Biot, M.A. (1956), "Thermoelasticity and irreversible thermodynamics", J. Appl. Phys., 27(3), 240-253. https://doi.org/10.1063/1.1722351.
  6. Caputo, M. (1967), "Linear models of dissipation whose Q is almost frequency independent II", Geophys. J. Int., 13(5) 529-539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x.
  7. Chandrasekharaiah, D.S. (1988), "A generalized linear thermoelasticity theory for piezoelectric media", Acta Mech., 71, 39-49. https://doi.org/10.1007/BF01173936.
  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. Diethelm, K. (2010), Analysis of Fractional Differential Equation: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer-Verlag, Berlin, Heideberg.
  10. El-Karamany, A.S. and Ezzat, M.A. (2002), "On the boundary integral formulation of thermo-viscoelasticity theory", Int. J. Eng. Sci., 40(17), 1943-1956. https://doi.org/10.1016/S0020-7225(02)00043-5.
  11. El-Karamany, A.S. and Ezzat, M.A. (2004), "Boundary integral equation formulation for the generalized thermoviscoelasticity with two relaxation times", Appl. Math. Compu., 151(2), 347-362. https://doi.org/10.1016/S0096-3003(03)00345-X.
  12. El-Karamany, A.S. and Ezzat, M.A. (2005), "Propagation of discontinuities in thermopiezoelectric rod", J. Therm. Stress., 28(10), 977-1030. https://doi.org/10.1080/01495730590964954.
  13. El-Karamany, A.S. and Ezzat, M.A. (2014), "On the dual-phaselag thermoelasticity theory", Meccanica, 49(1), 79-89. https://doi.org/10.1007/s11012-013-9774-z.
  14. El-Karamany, A.S. and Ezzat, M.A. (2016), "Thermoelastic diffusion with memory- dependent derivative", J. Therm. Stress., 39(9), 1035-1050. https://doi.org/10.1080/01495739.2016.1192847.
  15. Ezzat, M.A. and El-Bary, A.A. (2016), "Magneto-thermoelectric viscoelastic materials with memory-dependent derivative involving two-temperature", Int. J. Appl. Electromag.Mech., 50(4), 549 -567. https://doi.org/10.3233/JAE-150131.
  16. Ezzat, M.A. (1994), "State space approach to unsteady twodimensional free convection flow through a porous medium", Can. J. Phys., 72(5-6), 311-317. https://doi.org/10.1139/p94-045.
  17. Ezzat, M.A., El-Karamany, A.S., Samaan, A.A. (2004), "The dependence of the modulus of elasticity on reference temperature in generalized thermoelasticity with thermal relaxation", Appl. Math. Compu., 147(1), 169 -189. https://doi.org/10.1016/S0096-3003(02)00660-4.
  18. Ezzat, M.A. and El-Bary, A.A. (2011), "Combined heat and mass transfer for unsteady MHD flow of perfect conducting micropolar fluid with thermal relaxation", Energy Conver. Mang., 52(2), 934-945. https://doi.org/10.1016/j.enconman.2010.08.021.
  19. Ezzat, M.A., Othman, M.I. and Hemy, K.A. (1999), "A problem of a micropolar magnetohydrodynamic boundary-layer flow", Can. J. Phys., 77(10), 813-827. https://doi.org/10.1139/cjp-77-10-813.
  20. Ezzat, M.A. (2011), "Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer", Physica B, 406(1), 30-35. https://doi.org/10.1016/j.physb.2010.10.005.
  21. Ezzat, M. A. (2012), "State space approach to thermoelectric fluid with fractional order heat transfer", Heat Mass. Trans., 48(1), 71-82. https://doi.org/10.1007/s00231-011-0830-8.
  22. Ezzat, M.A., El-Karamany, A.S. and El-Bary, A.A. (2018), "Twotemperature theory in Green-Naghdi thermoelasticity with fractional phase-lag heat transfer", Micro. Sys. Technolo., 24(2), 951-961. https://doi.org/10.1007/s00542-017-3425-6.
  23. Ezzat, M.A. and El-Bary, A.A. (2017a), "Fractional magnetoThermoelastic materials with phase-lag Green-Naghdi theories", Steel Compos. Struct., 24(3), 297-307. https://doi.org/10.12989/scs.2017.24.3.297.
  24. Ezzat, M.A., El-Karamany, A.S. and El-Bary, A. (2016), "Electrothermoelasticity 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.
  25. Ezzat, M.A. and El-Bary, A.A. (2017b), "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.
  26. Ezzat, M.A., El-Karamany, A.S. and El-Bary, A.A. (2017c), "Thermoelectric viscoelastic materials with memory-dependent derivative", Smart Struc. Sys., 19(5), 539-551. https://doi.org/10.12989/sss.2017.19.5.539.
  27. Ezzat, M.A., El-Karamany, A.S. and El-Bary, A.A. (2017d), "On dual-phase-lag thermoelasticity theory with memory-dependent derivative", Mech. Adv. Mat. Struct., 24(11), 908-916. https://doi.org/10.1080/15376494.2016.1196793.
  28. Ezzat, M.A., El-Karamany, A.S. and Ezzat, S.M. (2012), "Twotemperature theory in magneto-thermoelasticity with fractional order dual-phase-lag heat transfer", Nuc. Eng. Des., 252, 267-277. https://doi.org/10.1016/j.nucengdes.2012.06.012.
  29. Galka, A., Telega, J.J. and Wojnar, R. (1992), "Homogenization and thermopiezoelectricity", Mech. Res. Commun.,19(4), 315-324. https://doi.org/10.1016/0093-6413(92)90050-K.
  30. Odegard, G.M., Gates, T.S., Nicholson, L.M. and Wise, K.E. (2002), "Equivalent-continuum modeling of nano-structured materials", Compos. Sci. Technol., 62(14), 1869-1880. https://doi.org/10.1016/S0266-3538(02)00113-6.
  31. Gross, B. (1953), Mathematical Structure of the Theories of Viscoelasticity", Hemann, Paris, France.
  32. He, J.H. (2001), "Derivation of generalized variational principles without using Lagrange multipliers. Part III: Applications to thermopiezoelectricity", Facta Univ., Ser. Mech. Autom. Cont. Robotics, 3(11), 5-12.
  33. Hellwege, K.H. (1979), Numerical Data and Functional Relationships in Science and Technology, 11, Springer, Berlin.
  34. Honig, G. and Hirdes, U. (1984), "A method for the numerical inversion of the Laplace transform", J. Comp. Appl. Math., 10(1), 113-132. https://doi.org/10.1016/0377-0427(84)90075-X.
  35. Landau, L.D. and Lifshitz, E.M. (1959), Electrodynamics of Continuous Media, Pergamon Press, Oxford, England.
  36. 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.
  37. Lata, P. (2018), "Effect of energy dissipation on plane waves in sandwiched layered thermoelastic medium", Steel Compos. Struct., Int. J., 27(4), 439-451. https://doi.org/10.12989/scs.2018.27.4.439.
  38. 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.
  39. Lata, P. and Singh, S. (2019), "Effect of nonlocal parameter on nonlocal thermoelastic soliddue to inclined load", Steel Compos. Struct., 33(1), 955-963. https://doi.org/10.12989/scs.2019.33.1.123.
  40. 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.
  41. Marin, M., Chirila, A., Oechsner, A. and Vlase, S. (2019a), "About finite energy solutions in thermoelasticity of micropolar bodies with voids", Bound. Value Probl., 89.
  42. Marin, M., Chirila, A. and Othman, M.I.A., (2019), "An extension of dafermos's results for bodies with a dipolar structure", Appl. Math. Compu., 361, 680-688.
  43. Mindlin, R.D. (1961), On the Equations of Motion of Piezoelectric Crystals, in Problems of Continuum Mechanics, SIAM, Philadelphia.
  44. Nowacki, W. (1978), "Some general theorems of thermopiezoelectricity", J. Therm. Stress., 1(2), 171-182. https://doi.org/10.1080/01495737808926940.
  45. Odegard, G.M., Gates, T.S., Nicholson, L.M. and Wise, K.E. (2002), "Equivalent-continuum modeling of nano-structured materials", Compos. Sci. Tech., 62(14), 1869-1880. https://doi.org/10.1016/S0266-3538(02)00113-6.
  46. Peddieson, J., Buchanan, G.R. and McNitt, R.P. (2003), "Application of nonlocal continuum models to nanotechnology", Int. J. Eng. Sci., 41(3-5), 305-312. https://doi.org/10.1016/S0020-7225(02)00210-0.
  47. Povstenko, Y.Z. (2015), Fractional thermoelasticity, Springer, New York, USA.
  48. Sabatier, J.A., Agrawal, O.P. and Machado, J.A.T. (2007), Advances in Fractional Calculus, Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands.
  49. Saravanos, D.A. and Heyliger, P.R. (1999), "Mechanics and computational models for laminated piezoelectric beams, plates and Shells", Appl. Mech. Rev., 52(10), 305-320. https://doi.org/10.1115/1.3098918.
  50. 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.
  51. 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.
  52. Sherief, H. and Abd El-Latief, A.M. (2013), "Effect of variable thermal conductivity on a half-space under the fractional order theory of thermoelasticity", Int. J. Mech. Sci., 74, 185-189. https://doi.org/10.1016/j.ijmecsci.2013.05.016.
  53. Sherief, H.H. and Hussein, E.M. (2018), "Contour integration solution for a thermoelastic problem of a spherical cavity", Appl. Math. Compu., 320, 557-571. https://doi.org/10.1016/j.amc.2017.10.024.
  54. Sunar, M. and Rao, S.S. (1999), "Recent advances in sensing and control of flexible structures via piezoelectric materials technology", Appl. Mech. Rev., 52(1), 1-16. https://doi.org/10.1115/1.3098923.
  55. Tauchert, T.R., Ashida, F., Noda, N., Adali, S. and Verijenko, V. (2000), "Developments in thermopiezoelasticity with relevance to smart composite structures", Compos. Struct., 48(1-3), 31-38. https://doi.org/10.1016/S0263-8223(99)00070-7
  56. Tzou, D.Y. (1996), Macro-to Micro-Scale Heat Transfer: The Lagging Behavior, CRC Press.
  57. Tzou, D.Y. and Guo, Z.Y. (2010), "Nonlocal behavior in thermal lagging", Int. J. Therm. Sci., 49(7), 1133-1137. https://doi.org/10.1016/j.ijthermalsci.2010.01.022.
  58. Wang, L. and Hu, H. (2005), "Flexural wave propagation in single-walled carbon nanotubes", Phys. Rev. B., 71, 195412. https://doi.org/10.1103/PhysRevB.71.195412.
  59. Wojnar, R., Bytner, S. and Galka, A. (1999), Effective Properties of Elastic Composites Subject to Thermal Fields, Thermal Stresses V, Lastran Corp., Rochester, N.Y.
  60. Yang, W. and Chen, Z. (2020), "Nonlocal dual-phase-lag heat conduction and the associated nonlocal thermal-viscoelastic analysis", Int. J. Heat Mass Trans. 156, 119752. https://doi.org/10.1016/j.ijheatmasstransfer.2020.119752.
  61. Yang, X.J. (2019), General Fractional Derivatives: Theory, Methods and Applications, Boca Raton, FL, USA: CRC Press.
  62. Yu, Y.J., Hu, W. and Tian, X.G. (2014), "A novel generalized thermoelasticity model based on memory-dependent derivative", Int. J. Eng. Sci., 8, 123-134. https://doi.org/10.1016/j.ijengsci.2014.04.014.
  63. Zhang, L., Bhatti, M.M., Michaelides, E., Marin, M. and Ellahi, R. (2022), "Hybrid nanofluid flow towards an elastic surface with tantalum and nickel nanoparticles, under the influence of an induced magnetic field", Eur. Phys. J. Special Topics, 231(3), 521-533. https://doi.org/10.1140/epjs/s11734-021-00409-1.
  64. Zhang, H., Xiaoyun, J. and Xiu, Y. (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.
  65. Zenkour, A.M. (2017), "Bending analysis of piezoelectric exponentially graded fiber-reinforced composite cylinders in hygrothermal environments", Int. J. Mech. Mat Des., 13, 515-529. https://doi.org/10.1007/s10999-016-9351-4.