Fundamental Solution and Study of Plane Waves in Bio-Thermoelastic Medium with DPL
الموضوعات :R Kumar 1 , A.K Vashishth 2 , S Ghangas 3
1 - Department of Mathematics, Kurukshetra University, Kurukshetra - 136119 Haryana, India
2 - Department of Mathematics, Kurukshetra University, Kurukshetra - 136119 Haryana, India
3 - Department of Mathematics, MDSD Girls College, Ambala City – 134002 Haryana, India
الکلمات المفتاحية: Bio-thermoelasticity, Wave propagation, Energy ratios, Fundamental solution, Dual-phase-lag,
ملخص المقالة :
The fundamental solution of the system of differential equations in bio-thermoelasticity with dual phase lag (DPL) in case of steady oscillations in terms of elementary function is constructed and basic property is established. The tissue is considered as an isotropic medium and the propagation of plane harmonic waves is studied. The Christoffel equations are obtained and modified with the thermal as well as bio thermoelastic coupling parameters. These equations explain the existence and propagation of three waves in the medium. Two of the waves are attenuating longitudinal waves and one is non-attenuating transverse wave. The thermal property has no effect on the transverse wave. The velocities and attenuating factors of longitudinal waves are computed for a numerical bioheat transfer model with phase lag. The variation with frequency, thermal parameters, blood perfusion parameter and phase lag parameter are presented graphically. Also the reflection of plane wave from a stress free isothermal boundary of isotropic bio-thermoelastic half space in the context of DPL theory of thermoelasticity is studied. The amplitude ratios of various reflected waves are obtained and these amplitude ratios are further used to obtain the energy ratios of various reflected waves. These energy ratios are function of the angle of incidence and bio-thermoelastic properties of the medium. The expressions of energy ratios have been computed numerically for a particular model to show the effect of Poisson ratio, blood perfusion rate and phase lag parameters.
[1] Hetnarski R.B., 1964, Solution of the coupled problem of thermoelasticity in form of a series of functions, Archives of Mechanics 16: 23-31.
[2] Hetnarski R.B., 1964, The fundamental solution of the coupled thermoelasticity problems for small times, Archives of Mechanics 16: 919-941.
[3] Svanadze M., 1996, The fundamental solution of the oscillation equations of the thermoelasticity theory of mixture of two elastic solids, Journal of Thermal Stresses 19(7): 633-648.
[4] Ciarletta M., Scalia A., Svanadze M., 2007, Fundamental solution in the theory of micropolar thermoelasticity for materials with voids, Journal of Thermal Stresses 30(3): 213-229.
[5] Svanadze M., 2005, Fundamental solution in the theory of consolidation with double porosity, Journal of the Mechanical Behavior of Materials 16(1-2): 123-130.
[6] Svanadze M., 2004, Fundamental solutions of the equations of the theory of thermoelasticity with microtemperatures, Journal of Thermal Stresses 27(2): 151-170.
[7] Scarpetta E., Svanadze M., Zampoli V., 2014, Fundamental solutions in the theory of thermoelasticity for solids with double porosity, Journal of Thermal Stresses 37(6): 727-74814.
[8] Svanadze M., 2016, Fundamental solutions in the theory of elasticity for triple porosity materials, Meccanica 51: 1825-1837.
[9] Svanadze M., 2018, Fundamental solutions in the linear theory of thermoelasticity for solids with triple porosity, Mathematics and Mechanics of Solids 24(4): 919-938.
[10] Sharma S., Sharma K., Rani Bhargava R., 2013, Wave motion and representation of fundamental solution in electro-microstretch viscoelastic solids, Materials Physics and Mechanics 17: 93-110.
[11] Sharma S., Sharma K., Rani Bhargava R., 2014, Plane waves and fundamental solution in an electro-microstretch elastic solids, Afrika Matematika 25(2): 983-997.
[12] Kumar R., Sharma K. D., Garg S. K., 2015, Fundamental solution in micropolar viscothermoelastic solids with void, International Journal of Applied Mechanics and Engineering 20(1): 109-125.
[13] Kumar R., Vohra R., Gorla M.G., 2016, Some considerations of fundamental solution in micropolar thermoelastic materials with double porosity, Archives of Mechanics 68(4): 263-284.
[14] Kumar R., Devi S., Sharma V., 2015, Plane wave and fundamental solution in a modified couple stress generalized thermoelastic with mass diffusion, Materials Physics and Mechanics 24: 72-85.
[15] Kumar R., Kansal T., 2011, Fundamental solution in the theory of thermo microstretch elastic diffusive solids, International Scholarly Research Network ISRN Applied Mathematics 2011: 764632.
[16] Kumar R., Kansal T., 2012, Fundamental solution in the theory of micropolar thermoelastic diffusion with voids, Computational and Applied Mathematics 31(1): 169-189.
[17] Kumar R., Kaur M., 2016, Plane waves and fundamental solutions in heat conducting micropolar fluid, Journal of Fluids 2016: 1453613.
[18] Xu F., Lu T., 2011, Introduction to Skin Biothermomechanics and Thermal Pain, Berlin (Heidelberg),Springer.
[19] Shen W.S., Zhang J., 2005, Modeling and numerical simulation of bioheat transfer and biomechanics in soft tissue, Mathematical and Computer Modelling 41: 1251-1265.
[20] Shen W.S., Zhang J., Yang F.Q., 2005, Skin thermal injury prediction with strain energy, International Journal of Nonlinear Sciences and Numerical Simulation 6: 317-328.
[21] Li X., Zhong Y., Jazar R., Subic A., 2014, Thermal-mechanical deformation modelling of soft tissues for thermal ablation, Bio-Medical Materials and Engineering 24: 2299-2310.
[22] Li X., Zhong Y., Subic A., Jazar R., Smith J., Gu C., 2016, Prediction of tissue thermal damage, Technology and Health Care 2: 5625-5629.
[23] Xu F., Seffen K., Lu T., 2008, Non-Fourier analysis of skin biothermomechanics, International Journal of Heat and Mass Transfer 51: 2237-2259.
[24] Li X., Zhong Y., Smith J., Gu C., 2017, Non-Fourier based thermal-mechanical tissue damage prediction for thermal ablation, Bioengineered 8(1): 71-77.
[25] Pennes H.H., 1948, Analysis of tissue and arterial blood temperatures in the resting human forearm, Journal of Applied Physiology 1: 93-122.
[26] Cattaneo C., 1958, A form of heat conduction equation which eliminates the paradox of instantaneous propagation, Journal of Applied Mathematics 247: 431-433.
[27] Vernotte P., 1958, Les paradoxes de la theorie continue de l’equation de la chaleur, Journal of Applied Mathematics 246: 3154-3155.
[28] Tzou D.Y., 1995, A unified field approach for heat conduction from macro-to-microscales, ASME Journal of Heat Transfer 117: 8-16.
[29] Shrama M.D., 2008, Wave propagation in thermoelastic saturated porous medium, Journal of Earth System Science 117(6): 951-958.
[30] Kupradze V.D., Gegelia T.G., Basheleishvili M.O., Buruchuladze T.V., 1979, Three Dimensional Problem of Mathematical Theory of Elasticity and Thermo-Elasticity, North-Holland Publishing, Company Amsterdam, New York, Oxford.
[31] Panji M., Kamalian M., Asgari Marnani J., Jafari M. K., 2013, Transient analysis of wave propagation problems by half-plane BEM, Geophysical Journal International 194: 1849-1865.
[32] Panji M., Kamalian M., Asgari Marnani J., Jafari M. K., 2014, Analysing seismic convex topographies by a half-plane time-domain BEM, Geophysical Journal International 197: 591-607.
[33] Panji M., Ansari B., 2017, Transient SH-wave scattering by the lined tunnels embedded in an elastic half-plane, Engineering Analysis with Boundary Elements 84: 220-230.
[34] Achenbach J.D., 1973, Wave Propagation in Elastic Solids, North-Holland Publishing, Amsterdam.
[35] Sharma K., Marin M., 2013, Effect of distinct conductive and thermodynaic temperatures on the reflection of plane waves in micropolar elastic half-space, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics 75(2): 121-132.
[36] Shrama K., 2012, Reflection of plane waves in thermodiffusive elastic half-space with voids, Multidiscipline Modeling in Materials and Structures 8(3): 269-296.
[37] Kumar R., Gupta V., 2013, Reflection and transmission of plane waves at the interface of an elastic half-space and a fractional order thermoelastic half-space, Archive of Applied Mechanics 83: 1109-1128.
[38] Saini R., 2015, Reflection/refraction at the interface of an elastic solid and a partially saturated porous solid containing liquid filled bound pores and a connected pore space saturated by two-phase fluid, Latin American Journal of Solids and Structures 12(10): 1870-1900.
[39] Kumar R., Vohra R., Gorla M.G., 2016, Reflection of plane waves in thermoelastic medium with double porosity, Multidiscipline Modeling in Materials and Structures 12(4): 748-778.
