Skip to main content
Log in

Reflection Phenomenon of Thermoelastic Wave in a Micropolar Semiconducting Porous Medium

  • Published:
Mechanics of Solids Aims and scope Submit manuscript

Abstract

In this research, the nonlocal thermo-elasticity theory in the context of fractional order time derivative three phase lag model is employed to explore the reflection of waves in a micro-polar semiconducting medium with voids. Two dispersion relations for speed-frequency are computed, corresponding to longitudinal and transverse waves. The dilatational waves are affected by voids, thermal, and nonlocal factors, whereas transverse waves are mainly influenced by nonlocality and micro-polarity. Analytically, reflection coefficients and their related energy ratios are computed. For a specific material, Silicon, the findings are analyzed visually. In addition to semiconductor nanoparticle systems, the research might be beneficial for semiconductor nanostructure, geology, and seismology.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.

Similar content being viewed by others

REFERENCES

  1. A. C. Eringen and E. S. Suhubi, “Non-linear theory of micro-elastic solids,” Int. J. Eng. Sci. 2 (2), 189–203 (1964). https://doi.org/10.1016/0020-7225(64)90004-7

    Article  MATH  Google Scholar 

  2. A. C. Eringen, “Linear theory of micropolar elasticity,” J. Math. Mech. 15 (6), 909–923. (1966). http://www.jstor.org/stable/24901442909-923.

    MathSciNet  MATH  Google Scholar 

  3. W. Nowacki, “Couple-stresses in the theory of thermoelasticity,” in Irreversible Aspects of Continuum Mechanics and Transfer of Physical Characteristics in Moving Fluids (Springer, Vienna, 1968), pp. 259–278. https://doi.org/10.1016/0020-7225(69)90030-5

    Book  Google Scholar 

  4. A. C. Eringen, Foundations of Micropolar Thermoelasticity (Springer, Berlin, 1970). https://doi.org/10.1007/978-3-7091-2904-3

    Book  MATH  Google Scholar 

  5. A. C. Eringen, Microcontinuum Field Theories. I. Foundations and Solids (Springer Science & Business Media, 2012). https://doi.org/10.1007/978-1-4612-0555-5

    Book  MATH  Google Scholar 

  6. A. C. Eringen, Nonlocal Continuum Field Theories (Springer, Berlin, 2002). https://doi.org/10.1007/b97697

    Book  MATH  Google Scholar 

  7. B. S. Altan, “Uniqueness in the linear theory of nonlocal elasticity,” Bull. Tech. Univ. Istanb. 37, 373–385 (1984).

    MathSciNet  MATH  Google Scholar 

  8. S. Chirita, “On some boundary value problems of nonlocal elasticity,” An. Şti. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.) 22 (2), 231–237 (1976).

  9. B. Craciun, “On nonlocal thermoelsticity,” Ann. St. Univ., Ovidus Constanta 5 (1), 29–36 (1996).

    Google Scholar 

  10. D. G. B. Edelen and N. Laws. “On the thermodynamics of systems with nonlocality,” Arch. Rational Mech. Anal. 43 (1), 24–35 (1971). https://doi.org/10.1007/BF00251543

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. D. G. B. Edelen, A. E. Green, and N. Laws, “Nonlocal continuum mechanics,” Arch. Rational Mech. Anal. 43 (1), 36–44 (1971). https://doi.org/10.1007/BF00251544

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. A. C. Eringen and D. G. B. Edelen, “On nonlocal elasticity,” Int. J. Eng. Sci. 10 (3), 233–248 (1972). https://doi.org/10.1016/0020-7225(72)90039-0

    Article  MathSciNet  MATH  Google Scholar 

  13. A. C. Eringen, “Nonlocal polar elastic continua,” Int. J. Eng. Sci. 10 (1), 1–16 (1972). https://doi.org/10.1016/0020-7225(72)90070-5

    Article  MathSciNet  MATH  Google Scholar 

  14. A. C. Eringen, “Nonlocal continuum theory of liquid crystals,” Mol. Cryst. Liq. Cryst., 75 (1), 321–343 (1981). https://doi.org/10.1080/00268948108073623

    Article  Google Scholar 

  15. A. C. Eringen, “Memory-dependent nonlocal electromagnetic elastic solids and superconductivity,” J. Math. Phys. 32 (3), 787–796 (1991). https://doi.org/10.1063/1.529372

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Ieşan Dorin, “A theory of thermoelastic materials with voids,” Acta Mech. 60 (1), 67–89 (1986). https://doi.org/10.1007/bf0130294

    Article  Google Scholar 

  17. S. Dilbag, G. Kaur, and S. K. Tomar, “Waves in nonlocal elastic solid with voids,” J. Elasticity 128 (1), 85–114 (2017). https://doi.org/10.1007/s10659-016-9618-x

    Article  MathSciNet  MATH  Google Scholar 

  18. P. S. Casas and R. Quintanilla, “Exponential decay in one-dimensional porous-thermo-elasticity,” Mech. Res. Comm. 32 (6), 652–658 (2005). https://doi.org/10.1016/j.mechrescom.2005.02.015

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Magaña and R. Quintanilla, “On the time decay of solutions in one-dimensional theories of porous materials,” Int. J. Solids Struct. 43 (11–12), 3414–3427 (2006). https://doi.org/10.1016/j.ijsolstr.2005.06.077

  20. A. Magaña and R. Quintanilla, “On the time decay of solutions in porous-elasticity with quasi-static microvoids,” J. Math. Anal. Appl. 331 (1), 617–630 (2007). https://doi.org/10.1016/j.jmaa.2006.08.086

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Soufyane, et al. “General decay of solutions of a linear one-dimensional porous-thermoelasticity system with a boundary control of memory type,” Nonlin. Anal.: Theory, Meth. Appl. 72 (11), 3903–3910 (2010). https://doi.org/10.1016/j.na.2010.01.004

    Article  MATH  Google Scholar 

  22. M. Bachher, N. Sarkar, and A. Lahiri. “Generalized thermoelastic infinite medium with voids subjected to a instantaneous heat sources with fractional derivative heat transfer,” Int. J. Mech. Sci. 89, 84–91 (2014). https://doi.org/10.1016/j.ijmecsci.2014.08.029

    Article  Google Scholar 

  23. M. Bachher, N. Sarkar, and A. Lahiri. “Fractional order thermoelastic interactions in an infinite porous material due to distributed time-dependent heat sources,” Meccanica 50 (8), 2167–2178 (2015). doi:https://doi.org/10.1007/s11012-015-0152-x

    Article  MathSciNet  MATH  Google Scholar 

  24. Mitali Bachher and Nantu Sarkar, “Nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer,” Waves Rand. Compl. Media 29 (4), 595–613 (2019). doi:https://doi.org/10.1080/17455030.2018.1457230

    Article  ADS  MathSciNet  Google Scholar 

  25. Siddhartha Biswas and Nantu Sarkar, “Fundamental solution of the steady oscillations equations in porous thermoelastic medium with dual-phase-lag model,” Mech. Mater. 126, 140–147 (2018). doi:https://doi.org/10.1016/j.mechmat.2018.08.008

    Article  Google Scholar 

  26. Nantu Sarkar and S. K. Tomar, “Plane waves in nonlocal thermoelastic solid with voids,” J. Therm. Stress. 42 (5), 580–606 (2019). https://doi.org/10.1080/014895739.2018.1554395

    Article  Google Scholar 

  27. Yu. Z. Povstenko, “Fractional heat conduction equation and associated thermal stress,” J. Therm. Stress. 28 (1), 83–102 (2004). https://doi.org/10.1080/014957390523741

    Article  MathSciNet  Google Scholar 

  28. M. Caputo, “Linear models of dissipation whose Q is almost frequency independent-II,” Geophys. J. Int. 13 (5), 529–539 (1967). https://doi.org/10.1111/j.1365-246x.1967.tb02303.x

    Article  ADS  Google Scholar 

  29. Hany H. Sherief, A. M. A. El-Sayed, and AM Abd El-Latief, “Fractional order theory of thermoelasticity,” Int. J. Solids Struct. 47 (2), 269–275 (2010). https://doi.org/10.1016/j.ijsolstr.2009.09.034

    Article  Google Scholar 

  30. Hamdy M. Youssef, “Theory of fractional order generalized thermoelasticity,” J. Heat Transf. 132 (6) (2010). https://doi.org/10.1115/1.4000705

  31. Magdy A. Ezzat, Ahmed S. El-Karamany, and Shereen M. Ezzat, “Two-temperature theory in magneto-thermoelasticity with fractional order dual-phase-lag heat transfer,” Nucl. Eng. Des. 252, 267–277 (2012). https://doi.org/10.1016/j.nucengdes.2012.06.012

    Article  Google Scholar 

  32. Magdy A. Ezzat, Ahmed S. El Karamany, and Mohsen A. Fayik, “Fractional order theory in thermoelastic solid with three-phase lag heat transfer,” Arch. Appl. Mech. 82 (4), 557–572 (2012). https://doi.org/10.1007/s00419-011-0572-6

    Article  ADS  MATH  Google Scholar 

  33. F. Hamza, M. Abdou, and A. M. Abd El-Latief, “Generalized fractional thermoelasticity associated with two relaxation times,” J. Therm. Stress. 37 (9), 1080–1098 (2014). https://doi.org/10.1080/01495739.2014.936196

    Article  Google Scholar 

  34. F. Hamza, A. M. Abd El-Latief, and M. Abdou, “1D applications on fractional generalized thermoelasticity associated with two relaxation times,” Mech. Advanc. Mater. Struct. 23 (6), 689–703 (2016). https://doi.org/10.1080/15376494.2015.1029158

    Article  Google Scholar 

  35. A. Ibrahim and S. Abbas, “Generalized thermo elastic interaction in functional graded material with Fractional order three-phase lag heat transfer,” J. Central South Uni. 22 (5), 1606–1613 (2015). https://doi.org/10.1007/s11771-015-2677-5

    Article  Google Scholar 

  36. Y. Q. Song, J. T. Bai, and Z. Y. Ren, “Study on the reflection of photothermal waves in a semiconducting medium under generalized thermoelastic theory,” Acta Mech. 223 (7), 1545–1557 (2012). https://doi.org/10.1007/s00707-012-0677-1

    Article  MathSciNet  MATH  Google Scholar 

  37. Feixiang Tang and Yaqin Song, “Wave reflection in semiconductor nanostructures,” Theor. Appl. Mech. Lett. 8 (3), 160–163 (2018). https://doi.org/10.1016/j.taml.2018.03.003

    Article  Google Scholar 

  38. Ibrahim A. Abbas, Faris S. Alzahrani, and Ahmed Elaiw, “A DPL model of photothermal interaction in a semiconductor material,” Waves Rand. Complex Media 29 (2), 328–343 (2019). https://doi.org/10.1080/17455030.2018.1433901

    Article  Google Scholar 

  39. J. Adnan, Ali Hashmat, and Aftab Khan, “Reflection phenomena of waves in a semiconductor nanostructure elasticity medium,” Waves Rand. Complex Media 31 (6), 1818–1834 (2021). doi..https://doi.org/10.1080/17455030.2019.1705425

    Article  MATH  Google Scholar 

  40. Ali Hashmat, Adnan Jahangir, and Aftab Khan, “Reflection of waves in a rotating semiconductor nanostructure medium through torsion-free boundary condition,” Indian J. Phys. 94 (12), 2051–2059 (2020). https://doi.org/10.1007/s12648-019-01652

    Article  ADS  Google Scholar 

  41. Ali Hashmat, Aftab Khan, and Adnan Jahangir, “Transmission phenomenon at the interface between isotropic and semiconductor nanostructure based on nonlocal theory,” Waves Rand. Complex Media 1–21 (2021). https://doi.org/10.1080/17455030.2021.1919339

  42. Ali Hashmat, Adnan Jahangir, and Aftab Khan, “Reflection of plane wave at free boundary of micro-polar nonlocal semiconductor medium,” J. Thermal Stress. 44 (11), 1307–1323 (2021). https://doi.org/10.1080/01495739.2021.1973632

    Article  Google Scholar 

  43. Mondal Sudip, Nihar Sarkar, and Nantu Sarkar, “Waves in dual-phase-lag thermoelastic materials with voids based on Eringen’s nonlocal elasticity,” J. Thermal Stress. 42 (8), 1035–1050 (2019). https://doi.org/10.1080/01495739.2019.1591249

    Article  Google Scholar 

  44. Stephen C. Cowin and Jace W. Nunziato, “Linear elastic materials with voids,” J. Elasticity 13 (2), 125–147 (1983). https://doi.org/10.1007/BF00041230

    Article  MATH  Google Scholar 

  45. M. Bachher, N. Sarkar, and A. Lahiri, “Generalized thermoelastic infinite medium with voids subjected to a instantaneous heat sources with fractional derivative heat transfer,” Int. J. Mech. Sci. 89, 84–91 (2014). https://doi.org/10.1016/j.ijmecsci.2014.08.029

    Article  Google Scholar 

  46. Noël Challamel, et al. “A nonlocal Fourier’s law and its application to the heat conduction of one-dimensional and two-dimensional thermal lattices,” Comptes Rendus Mécanique, 344 (6), 388-401 (2016). https://doi.org/10.1016/j.crme.2016.01.001

    Article  ADS  Google Scholar 

  47. J. D. Achenbach, Wave Propagation in Elastic Solids (North-Holland Publ. Comp., Amsterdam, 1973).

    MATH  Google Scholar 

  48. X. Zeng, Y. Chen and J. D. Lee, “Determining material constants in nonlocal micromorphic theory through phonon dispersion relations,” Int. J. Eng. Sci. 44, 1334–1345 (2006). https://doi.org/10.1016/j.ijengsci.2006.08.00248

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hashmat Ali.

Ethics declarations

The authors declare that they have no conflicts of interest.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ali, H., Jahangir, A. & Azhar, E. Reflection Phenomenon of Thermoelastic Wave in a Micropolar Semiconducting Porous Medium. Mech. Solids 57, 856–869 (2022). https://doi.org/10.3103/S0025654422040021

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0025654422040021

Keywords:

Navigation