Analysis of Reflection Phenomena in a Fiber Reinforced Piezo-Thermoelastic Half Space with Diffusion and Two-Temperature
محورهای موضوعی : Engineering
K Jain
1
(
Department of Mathematics, Guru Jambheshwar University of Science and Technology, Hisar-125001, Haryana, India
)
S Kumar
2
(
Department of Basic and Applied Sciences, BPS Mahila Vishwavidyalaya, Khanpur Kalan, Sonepat-131305, Haryana, India
)
S Deswal
3
(
Department of Mathematics, Guru Jambheshwar University of Science and Technology, Hisar-125001, Haryana, India
)
کلید واژه: Diffusion, Piezo-thermoelasticity, Reflection phenomena, Generalized thermoelasticity, Two-temperature, Fiber reinforcement,
چکیده مقاله :
Present work is concerned with the analysis of transient wave phenomena in a piezo-thermoelastic medium with diffusion, fiber reinforcement and two-temperature, when an elastic wave is made incident obliquely at the traction free plane boundary of the considered medium. The formulation is applied under the purview of generalized theory of thermoelasticity with one relaxation time. The problem is solved analytically and it is found that there exists four coupled quasi waves: qp (quasi-p ), qMD (quasi mass diffusion), qT (quasi thermal) and qSV (quasi-SV ) waves propagating with different speeds in a two-dimensional model of the solid. The amplitude ratios, phase velocities and energy ratios for the reflected waves are derived and the numerical computations have been carried out with the help of MATLAB programming. Effect of presence of diffusion is analyzed theoretically, numerically and graphically. The number of reflected waves reduce to three in the absence of diffusion as qMD wave will disappear in that case which is physically admissible. Influence of piezoelectric effect, two temperature and anisotropy is discussed on different characteristics of reflected waves such as phase velocity and reflection coefficients. It has been verified that there is no dissipation of energy at the boundary surface during reflection. Thus, the energy conservation law holds at the surface. Finally, all the reflection coefficients are represented graphically through 3D plots to estimate and highlight the effects of frequency and angle of incidence.
[1] Biot M.A., 1956, Thermoelasticity and irreversible thermodynamics, Journal of Applied Physics 27: 240-253.
[2] Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15: 299-309.
[3] Ignaczak J., Ostoja-Starzewski M., 2010, Thermoelasticity with Finite Wave Speeds, Mathematical Monographs, Oxford University Press, Oxford.
[4] Lata P., 2018, Reflection and refraction of plane waves in layered nonlocal elastic and anisotropic thermo-elastic medium , Structural Engineering and Mechanics 66: 113-124.
[5] Lata P., 2018, Effect of energy dissipation on plane waves in sandwiched layered thermoelastic medium, Steel and Composite Structures 27: 439-451.
[6] Lata P., Kaur I., 2019, Plane wave propagation in transversely isotropic magneto-thermoelastic rotating medium with fractional order generalized heat transfer, Structural Monitoring and Maintenance-An International Journal 6: 191-218.
[7] Lata P., Kaur I., 2019, Thermomechanical interactions due to time harmonic sources in a transversely isotropic magneto thermoelastic solids with rotation, International Journal of Microstructure and Materials Properties 14: 549-577.
[8] Mindlin R.D., 1961, On the Equations of Motion of Piezoelectric Crystals, Problems of Continuum Mechanics, N.I. Muskelishvili’s Birthday 70, SIAM, Philadelphia 1961: 282-290.
[9] Nowacki W., 1978, Some general theorems of thermo-piezoelectricity, Journal of Thermal Stresses 1: 171-182.
[10] Nowacki W., 1979, Foundations of Linear Piezoelectricity, Parkus H., (Ed.), Electromagnetic Interactions in Elastic Solids, Springer, Wien, Chapter 1.
[11] Nowacki W., 1983, Mathematical Models of Phenomenological Piezoelectricity, New problems in Mechanics of Continua, University of Waterloo Press, Ontario 1983: 29-49.
[12] Chandrasekharaiah D.S., 1984, A temperature rate dependent theory of piezoelectricity, Journal of Thermal Stresses 7: 293-306.
[13] Chandrasekharaiah D.S., 1988, A generalized linear thermoelasticity theory of piezoelectric media, Acta Mechanica 71: 39-49.
[14] Sharma J.N., Walia V., Gupta S.K., 2008, Reflection of piezothermoelastic waves from the charge and stress free boundary of a transversely isotropic half space, International Journal of Engineering Science 46: 131-146.
[15] Kuang Z.B., Yuan X.G., 2011, Reflection and transmission of waves in pyroelectric and piezoelectric materials, Journal of Sound and Vibration 330: 1111-1120.
[16] Yuan X., Jiang Q., 2016, Reflection of plane waves from rotating pyroelectric half-space under initial stresses, Zeitschrift für Angewandte Mathematik und Mechanik 97: 365-374.
[17] Vashishth A.K., Sukhija H., 2015, Reflection and transmission of plane waves from fluid- piezothermo- elastic solid interface, Applied Mathematics and Mechanics 36: 11-36.
[18] Othman M.I.A., Ahmed E.A.A., 2019, Exact analytical solution of a homogeneous anisotropic piezo- thermoelasic half-space of a hexagonal type under different fields with three theories, Microsystem Technologies 25: 1423-1435.
[19] Pipkin A.C., 1973, Finite Deformations of Ideal Fiber-Reinforced Composites, Sendeckyj G.P., (Ed.), Composites Materials, Academic Press, New York 2: 251-308.
[20] Rogers T.G., 1975, Anisotropic Elastic and Plastic Materials, Thoft Christensen P., (Ed.), Continuum Mechanics Aspects of Geodynamics and Rock Fracture Mechanics, Reidel, Dordrecht 1975: 177-200.
[21] Belfield A.J., Rogers T.G., Spencer A.J.M., 1983, Stress in elastic plates reinforced by fiber lying in concentric circles, Journal of the Mechanics and Physics of Solids 31: 25-54.
[22] Singh S.S., Tomar S.K., 2007, Shear waves at a corrugated interface between two dissimilar fiber reinforced elastic half spaces, Journal of Mechanics of Materials and Structures 2: 167-188.
[23] Abbas I.A., Abd-Alla A.N., Othman M.I.A., 2011, Generalized magnetothermoelasticity in a fiber-reinforced anisotropic half-space, International Journal of Thermophysics 32: 1071-1085.
[24] Said S.M., Othman M.I.A., 2016, Effects of gravitational and hydrostatic initial stress on a two-temperature fiber-reinforced thermoelastic medium for three-phase-lag, Journal of Solid Mechanics 8: 806-822.
[25] Deswal S., Sheokand S.K., Kalkal K.K., 2019, Reflection at the free surface of fiber-reinforced thermoelastic rotating medium with two-temperature and phase-lag, Applied Mathematical Modelling 65: 106-119.
[26] Lata P., Kaur I., 2020, Thermomechanical interactions in transversely isotropic magneto-thermoelastic medium with fractional order generalized heat transfer and hall current, Arab Journal of Basic and Applied science 27: 13-26.
[27] Chen P.J., Gurtin M.E., 1968, On a theory of heat conduction involving two temperatures, Zeitschrift für Angewandte Mathematik und Physik 19: 614-627.
[28] Chen P.J., Gurtin M.E., Williams W.O., 1968, A note on non-simple heat conduction, Zeitschrift für Angewandte Mathematik und Physik 19: 969-970.
[29] Chen P.J., Gurtin M.E., Williams W.O., 1969, On the thermodynamics of non-simple elastic material with two temperatures, Zeitschrift fur angewandte Mathematik und Physik 20: 107-112.
[30] Youssef H.M., 2006, Theory of two-temperature generalized thermoelasticity, IMA Journal of Applied Mathematics 71: 383-390.
[31] Youssef H.M., Bassiouny E., 2008, Two-temperature generalized thermopiezoelasticity for one dimensional problems—State space approach, Computational Methods in Science and Technology 14: 55-64.
[32] Islam M., Kanoria M., 2014, One-dimensional problem of a fractional order two-temperature generalized thermo-piezoelasticity, Mathematics and Mechanics of Solids 19: 672-693.
[33] Kaur I., Lata P., 2019, Effect of hall current on propagation of plane wave in transversely isotropic thermoelastic medium with two temperature and fractional order heat transfer, SN Applied Sciences 1: 900.
[34] Lata P., Kaur I., 2019, Thermomechanical interactions in transversely isotropic magneto thermoelastic solid with two temperatures and without energy dissipation, Steel & Composite Structures 32: 779-793.
[35] Nowacki W., 1974, Dynamical problems of thermodiffusion in solids-I, Bulletin of the Polish Academy of Sciences, Technical Sciences 22: 55-64.
[36] Nowacki W., 1974, Dynamical problems of thermodiffusion in solids-II, Bulletin of the Polish Academy of Sciences, Technical Sciences 22: 129-135.
[37] Nowacki W., 1974, Dynamical problems of thermodiffusion in solids-III, Bulletin of the Polish Academy of Sciences, Technical Sciences 22: 257-266.
[38] Sherief H.H., Hamza F.A., Saleh H.A., 2004, The theory of generalized thermoelastic diffusion, International Journal of Engineering Science 42: 591-608.
[39] Kuang Z.B., 2010, Variational principles for generalized thermodiffusion theory in pyroelectricity, Acta Mechanica 214: 275-289.
[40] Kumar R., Chawla V., 2012, General steady-state solution and Green’s function in orthotropic piezothermoelastic diffusion medium, Archives of Mechanics 64: 555-579.
[41] Kumar R., Sharma P., 2016, Analysis of plane waves in anisotropic piezothermoelastic diffusive medium, Multidiscipline Modeling in Materials and Structures 12: 93-109.
[42] Kaur I., Lata P., 2019, Rayleigh wave popagation in transversely isotropic magneto thermoelastic
medium with three-phase=lag heat transfer and diffusion, International Journal of Mechanical and Materials Engineering 14: 12.
[43] Kansal T., 2019, Fundamental solution in the theory of thermoelastic diffusion materials with double porosity, Journal of Solid Mechanics 11: 281-296.
[44] Bijarnia R., Singh B., 2004, Propagation of plane waves in an isotropic two-temperature thermoelastic solid half-space with diffusion, Annals of Solid and Structural Mechanics 6: 37-45.
