Coupled systems mechanics

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Fundamental and plane wave solution in non-local bio-thermoelasticity diffusion theory

  • Kumar, Rajneesh (Department of Mathematics, Kurukshetra University) ;
  • Ghangas, Suniti (Department of Mathematics, M.D.S.D. Girls College) ;
  • Vashishth, Anil K. (Department of Mathematics, Kurukshetra University)
  • Received : 2020.10.28
  • Accepted : 2020.11.24
  • Published : 2021.02.25
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Abstract

This work is an attempt to design a dynamic model for a non local bio-thermoelastic medium with diffusion. The system of governing equations are formulated in terms of displacement vector field, chemical potential and the tissue temperature in the context of non local dual phase lag (NL DPL) theories of heat conduction and mass diffusion. Based on this considered model, we study the fundamental solution and propagation of plane harmonic waves in tissues. In order to analyze the behavior of the NL DPL model, we construct basic theorem in the terms of elementary function which determine the existence of three longitudinal and one transverse wave. The effects of various parameters on the characteristics of waves i.e., phase velocity and attenuation coefficients are elaborated by plotting various figures of physical quantities in the later part of the paper.

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References

  1. Bachher, M. and Sarkar, N. (2018), "Nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer", Wave. Random Complex Media, 29(4), 595-613 https://doi.org/10.1080/17455030.2018.1457230.
  2. Cattaneo, C. (1958), "A form of heat conduction equation which eliminates the paradox of instantaneous propagation", Comptes Rendus, 247, 431-433.
  3. Cheng, J. and Kar, A. (1997), "Mathematical model for laser densification of ceramic coating", J. Mater. Sci., 32, 6269-6278. https://doi.org/10.1023/A:1018693212407.
  4. Das, N., Sarkar, N. and Lahiri, A. (2019), "Reflection of plane waves from the stress-free isothermal and insulated boundaries of a nonlocal thermoelastic solid", Appl. Math. Model., 73, 526-544. https://doi.org/10.1016/j.apm.2019.04.028.
  5. Dong, Y., Cao, B.Y. and Guo, Z.Y. (2014), "Size dependent thermal conductivity of Si nanosystems based on phonon gas dynamics", Physica E, 56, 256. https://doi.org/10.1016/j.physe.2013.10.006.
  6. Eringen, A.C. (2002), Nonlocal Continuum Field Theories, Springer, New York.
  7. Guo, J.G. and Zhao,Y.P. (2005), "The size-dependent elastic properties of nanofilms with surface effects", J. Appl. Phys., 98, 074306. https://doi.org/10.1063/1.2071453.
  8. Gupta, M. and Mukhopadhyay, S. (2019), "A study on generalized thermoelasticity theory based on non-local heat conduction model with dual-phase-lag", J. Therm. Stress., 42(9), 1123-1135. https://doi.org/10.1080/01495739.2019.1614503.
  9. Guyer, R.A. and Krumhansl, J.A. (1966), "Solution of the linearized phonon Boltzmann equation", Phys. Rev., 148, 766. https://doi.org/10.1103/PhysRev.148.766
  10. Kansal, T. (2019), "Fundamental solution in the theory of thermoelastic diffusion materials with double porosity", J. Solid Mech., 11(2), 281-296.
  11. Kim, J.Y., Jang, K., Yang, S.J. and Baek, J.H. (2016), "Simulation study of the thermal and the thermoelastic effects induced by pulsed laser absorption in human skin", J. Korea. Phys. Soc., 68(8), 979-988. https://doi.org/10.3938/jkps.68.979.
  12. Kumar, R., Vashishth, A.K. and Ghangas, S. (2019a), "Nonlocal heat conduction approach in a bi-layer tissue during magnetic fluid hyperthermia with dual phase lag model", Bio-Med. Mater. Eng., 30, 387-402. https://doi.org/10.3233/BME-191061.
  13. Kumar, R., Vashishth, A.K. and Ghangas, S. (2019b), "Phase-lag effects in skin tissue during transient heating", Int. J. Appl. Mech. Eng., 24(3), 603-623. https://doi.org/10.2478/ijame-2019-0038.
  14. Kumar, R., Vashishth, A.K. and Ghangas, S. (2020), "Fundamental solution and study of plane waves in bio-thermoelastic medium with DPL", J. Solid Mech., 12(2), 278-296. https://doi.org/10.22034/JSM.2019.582000.1381.
  15. Li, X., Li, C., Xue, Z. and Tian, X. (2019), "Investigation of transient thermo-mechanical responses on the triple-layered skin tissue with temperature dependent blood perfusion rate", Int. J. Therm. Sci., 139, 339-349. https://doi.org/10.1016/j.ijthermalsci.2019.02.022.
  16. Li, X., Qin, Q.H. and Tian, X. (2020), "Thermo-viscoelastic analysis of biological tissue during hyperthermia treatment", Appl. Math. Model., 79, 881-895. https://doi.org/10.1016/j.apm.2019.11.007.
  17. Li, X., Xue, Z. and Tian, X. (2018), "A modified fractional order generalized bio-thermoelastic theory with temperature-dependent thermal material properties", Int. J. Therm. Sci., 132, 249-256. https://doi.org/10.1016/j.ijthermalsci.2018.06.007.
  18. Li, X., Zhong, Y., Smith, J. and Gu, C. (2017), "Non-fourier based thermal-mechanical tissue damage prediction for thermal ablation", Bioengineered, 8(1), 71-77. https://doi.org/10.1080/21655979.2016.1227609.
  19. Pennes, H.H. (1948), "Analysis of tissue and arterial blood temperature in the resting forearm", J. Appl. Physiol., 1, 93-122. https://doi.org/10.1152/jappl.1948.1.2.93
  20. Sarkar, N. (2020), "Thermoelastic responses of a finite rod due to nonlocal heat conduction", Acta Mech, 231, 947-955. https://doi.org/10.1007/s00707-019-02583-9.
  21. Sharma, S. and Kumar, 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.
  22. Sharma, S., Sharma, K. and Rani Bhargava, R. (2013), "Wave motion and representation of fundamental solution in electromicrostretch viscoelastic solids", Mater. Phys. Mech., 17, 93-110.
  23. Shen, W., Zhang, J. and Yang, F. (2005), "Modeling and numerical simulation of bioheat transfer and biomechanics in soft tissue", Math. Comput. Model., 41(11-12), 1251-1265. https://doi.org/10.1016/j.mcm.2004.09.006.
  24. Sherief, H.H., Hamza, F.A. and Saleh, H.A. (2004), "The theory of generalized thermoelastic diffusion", Int. J. Eng. Sci., 42, 591-608. https://doi.org/10.1016/j.ijengsci.2003.05.001.
  25. Sobolev, S. (1994), "Equations of transfer in non-local media", Int. J. Heat Mass Transf., 37, 2175. https://doi.org/10.1016/0017-9310(94)90319-0
  26. Svanadze, M. (2018a), Potential Method in Mathematical Theories of Multi-Porosity Media, Springer International Publishing.
  27. Svanadze, M. (2018b), "Fundamental solutions in the linear theory of thermoelasticity for solds with triple porosity", Math. Mech. Solid., 24(4), 919-938. https://doi.org/10.1177/1081286518761183.
  28. Tzou, D.Y. (1996), "A unified field approach for heat conduction from macro-to-microscales", ASME J. Heat Transf., 117, 8-16. https://doi.org/10.1115/1.2822329
  29. 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
  30. Tzou, D.Y. and Guo, Z.Y. (2010), "Nonlocal behavior in thermal lagging", Int. J. Therm. Sci., 49, 1133-1137. https://doi.org/10.1016/j.ijthermalsci.2010.01.022
  31. Vernotte, P. (1958), "Les paradoxes de la theorie continue de lequation de la chaleur", Comp. Rend, 246, 3154-3155.
  32. Xiong, C. and Guo, Y. (2017), "Electromagneto-thermoelastic diffusive plane waves in a half-space with variable material properties under fractional order thermoelastic diffusion", Int. J. Appl. Electromagnet. Mech., 53, 251-269. https://doi.org/10.3233/JAE-160038.
  33. Xu, F., Lu, T.J. and Seffen, K.A. (2008), "Biothermomechanical behavior of skin tissue", Acta Mech. Sin., 24(1), 1-23. https://doi.org/10.1007/s10409-007-0128-8.
  34. Xu, F., Seffen, K.A. and Lu, T.J. (2008), "Non-Fourier analysis of skin biothermomechanics", Int. J. Heat Mass Tran., 51(9-10), 2237-2259. https://doi.org/10.1016/j.ijheatmasstransfer.2007.10.024.
  35. 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, 123. https://doi.org/10.1016/j.ijengsci.2014.04.014.
  36. Yu, Y.J., Li, G.L. and Xue, Z.N. (2016), "The dilemma of hyperbolic heat conduction and its settlement by incorporating spatially nonlocal effect at nanoscale", Phys. Lett. A, 380, 255. https://doi.org/10.1016/j.physleta.2015.09.030.
  37. Yu, Y.J., Tian, X.G. and Lu, T.J. (2013), "Fractional order generalized electro-magneto-thermo-elasticity", Eur. J. Mech. A/Solid., 42, 188. https://doi.org/10.1016/j.euromechsol.2013.05.006.