Transversely Isotropic Magneto-Visco Thermoelastic Medium with Vacuum and without Energy Dissipation

Document Type: Research Paper

Authors

1 Department of Mathematics, Kurukshetra University, Kurukshetra, Haryana , India

2 Research Scholar ,IKG Panjab Technical University, Kapurthala ,Punjab, India

3 Department of Mathematics, DAVIET, Jalandhar ,Punjab, India

Abstract

In the present investigation the disturbances in a homogeneous transversely isotropic magneto-Visco thermoelastic rotating medium with two temperature due to thermomechanical sources has been addressed. The thermoelasticity theories developed by Green-Naghdi (Type II and Type III) both with and without energy dissipation has been applied to the thermomechanical sources. The Laplace and Fourier transform techniques have been applied to solve the present problem. As an application, the bounding surface is subjected to concentrated and distributed sources (mechanical and thermal sources). The analytical expressions of displacement, stress components, temperature change and induced magnetic field are obtained in the transformed domain. Numerical inversion techniques have been applied to obtain the results in the physical domain. Numerical simulated results are depicted graphically to show the effect of viscosity on the resulting quantities. Some special cases of interest are also deduced from the present investigation.                         

Keywords


[1] Arani A.G., Salari M., Khademizadeh H., Arefmanesh A., 2009, Magneto thermoelastic transient response of a functionally graded thick hollow sphere subjected to magnetic and thermoelastic fields, Archieve of Applied Mechanics 79: 481.
[2] Atwa S.Y., Jahangir A., 2014, Two temperature effects on plane waves in generalized thermo micro stretch elastic solid, International Journal of Thermophysics 35: 175-193.
[3] AI-Basyouni K.S., Mahmoud S.E., Alzahrani E.O., 2014, Effect of rotation, magnetic field and a periodic loading on radial vibrations thermo-viscoelastic non-homogeneous media, Boundary Value Problems 2014: 166.
[4] Boley B.A., Tolins I.S., 1962, Transient coupled thermoelastic boundary value problem in the half space, Journal of Applied Mechanics 29: 637-646.
[5] Borrelli A., Patria M.C., 1991, General analysis of discontinuity waves in thermoviscoelastic solid of integral type, International Journal of Non-Linear Mechanics 26: 141.
[6] Chandrasekharaiah D. S., 1998, Hyperbolic thermoelasticity: A review of recent literature, Applied Mechanics Reviews 51: 705-729.
[7] Chen P.J., Gurtin M.E., 1968, On a theory of heat conduction involving two parameters, Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 19: 614-627.
[8] Chen P.J., Gurtin M.E., Williams W.O., 1968, A note on simple heat conduction, Journal of Applied Mathematics and Physics 19: 969-970.
[9] Chen P.J., Gurtin M.E., Williams W.O., 1969, On the thermodynamics of non-simple elastic materials with two temperatures, Journal of Applied Mathematics and Physics 20: 107-112.
[10] Corr D.T., Starr M.J.,Vanderky Jr.R., Best T.M., 2001, A nonlinear generalized Maxwell fluid Model, Journal of Applied Mechanics 68: 787-790.
[11] Das P., Kanoria M., 2014, Study of finite thermal waves in a magneto thermoelastic rotating medium, Journal of Thermal Stresses 37(4): 405-428.
[12] Dhaliwal R.S., Singh A., 1980, Dynamic Coupled Thermoelasticity, Hindustan Publisher Corp, New Delhi, India.
[13] Ezzat M.A., Awad E.S., 2010, Constitutive relations, uniqueness of solution and thermal shock application in the linear theory of micropolar generalized thermoelasticity involving two temperatures, Journal of Thermal Stresses 33(3): 225-250.
[14] Ezzat M.A., 1997, State approach to generalized magneto- thermoelasticity with two relaxation times in a medium of perfect conductivity, International Journal of Engineering Science 35: 741-752.
[15] Freudenthal A.M., 1954, Effect of rheological behavior on thermal stresses, Journal of Applied Physics 25: 1110-1117.
[16] Green A.E., Naghdi P.M., 1991, A re-examination of the basic postulates of thermomechanics, Proceedings of Royal Society of London A 432: 171-194.
[17] Green A.E., Naghdi P.M.,1992, On undamped heat waves in an elastic solid, Journal of Thermal Stresses 15: 253-264.
[18] Green A.E., Naghdi P.M., 1993, Thermoelasticity without energy dissipation, Journal of Elasticity 31: 189-208.
[19] Hilton H.H., 2014, Coupled longitudinal 1–d thermal and viscoelastic waves in media with temperature dependent material properties, Engineering Mechanics 21(4): 219-238.
[20] Honig G., Hirdes U., 1984, A method for the inversion of Laplace transform, Journal of Computational and Applied Mathematics 10: 113-132.
[21] Iesan D., Scalia A., 1989, Some theorems in the theory of thermo viscoelasticity, Journal of Thermal stresses 12: 225-239.
[22] Kaliski S., 1963, Absorption of magneto-viscoelastic surface waves in a real conductor magnetic field, Proceedings of Vibration Problems 4: 319-329.
[23] Kaushal S., Kumar R., Miglani A., 2011, Wave propagation in temperature rate dependent thermoelasticity with two temperatures, Mathematical Sciences 5: 125-146.
[24] Kaushal S., Sharma N., Kumar R., 2010, Propagation of waves in generalized thermoelastic continua with two temperature, International Journal of Applied Mechanics and Engineering 15: 1111-1127.
[25] Khademizadeh H., Arani A.G., Salari M., 2008, Stress analysis of magneto thermoelastic and induction magnetic field in FGM hallow sphere, Journal of Simulation and Analysis of Novel Technologies in Mechanical Engineering 1 (1): 49.
[26] Kumar R., Chawla V., Abbas I.A., 2012, Effect of viscosity on wave propagation in anisotropic thermoelastic medium with three-phase-lag model, Journal of Theoretical and Applied Mechanics 39(4): 313-341.
[27] Kumar R., Devi S., 2010, Magneto thermoelastic (Type-II and III) half-space in contact with vacuum, Applied Mathematical Sciences 69(4): 3413- 3424.
[28] Kumar R., Kansal T., 2010, Effect of rotation on Rayleigh lamb waves in an isotropic generalized thermoelastic diffusive plate, Journal of Applied Mechanics and Technical Physics 51(5): 751-761.
[29] Kumar R., Mukhopdhyay S., 2010, Effects of thermal relaxation times on plane wave propagation under two temperature thermoelasticity, International Journal of Engineering Sciences 48(2): 128-139.
[30] Kumar R., Rupender., 2009, Effect of rotation in magneto-micro polar thermoelastic medium due to mechanical and thermal sources, Chaos Solitons and Fractals 41: 1619-1633.
[31] Kumar R., Sharma K.D., Garg S.K., 2014, Effect of two temperature on reflection coefficient in micro polar thermoelastic media with and without energy dissipation, Advances in Acoustics and Vibrations 2014: 846721.
[32] Lofty K., Hassan W., 2013, Effect of rotation for two temperature generalized thermoelasticity of two dimensional unde thermal shock problem, Mathematical Problems in Engineering 2013: 297274.
[33] Mahmoud S.R., 2013, An analytical solution for effect of magnetic field and initial stress on an infinite generalized thermoelastic rotating non homogeneous diffusion medium, Abstract and Applied Analysis 2013: 284646.
[34] Othman M.I.A., Zidan M.E.M., Hilai M.I.M., 2013, Effect of rotation on thermoelastic material with voids and temperature dependent properties of type-III, Journal of Thermoelasticity 1(4):1-11.
[35] Pal P.C., 2000, A note on the torsional body forces in a viscoelastic half-space, Indian Journal of Pure and Applied Mathematics 31(2): 207-210.
[36] Press W.H., Teukolshy S.A., Vellerling W.T., Flannery B.P., 1986, Numerical Recipes in FORTRAN, Cambridge University Press, Cambridge.
[37] Quintanilla R., 2002, Thermoelasticity without energy dissipation of materials with microstructure, Journal of Applied Mathematical Modeling 26: 1125-1137.
[38] Sarkar N., Lahiri A., 2012, Temperature rate dependent generalized thermoelasticity with modified Ohm's law, International Journal of Computational Materials Science and Engineering 1(4): 1250031.
[39] Sharma N., Kumar R., Lata P., 2015, Disturbance due to inclined load in transversely isotropic thermoelastic medium with two temperatures and without energy dissipation, Material Physics and Mechanics 22: 107-117.
[40] Sharma K., Bhargava R.R., 2014, Propagation of thermoelastic plane waves at an imperfect boundary of thermal conducting viscous liquid/generalized thermoelastic solid, Afrika Matematika 25: 81-102.
[41] Sharma K., Marin M., 2013, Effect of distinct conductive and thermodynamic temperatures on the reflection of plane waves in micro polar elastic half-space, UPB Scientific Bulletin 75(2): 121-132.
[42] Sharma S., Sharma K., Bhargava R.R., 2013, Effect of viscosity on wave propagation in anisotropic thermoelastic with Green- Naghdi theory Type-II and Type-III, Materials Physics and Mechanics 16: 144-158.
[43] Singh B., Bala K., 2012, Propagation of waves in a two- temperature rotating thermoelastic solid half- space without energy dissipation, Applied Mathematics 3(12):1903.
[44] Slaughter W.S., 2002, The Linearized Theory of Elasticity, Birkhausar.
[45] Voigh W., 1887,Theoritishestudien under der Elastizitatsner – haltnirse der Kristalle, Abhandlungen der Akademie der Wissenschaften in Göttingen 34(3): 51.
[46] Warren W.E., Chen P.J., 1973, Wave propagation in the two temperature theory of thermoelasticity, Journal of Acta Mechanica 16: 21-33.
[47] Yadav R., Kalkal K.K., Deswal S., 2015, Two-temperature generalized thermo viscoelasticity with fractional order strain subjected to moving heat source: state space approach, Journal of Mathematics 2015: 487513.
[48] Youssef H.M., 2006, Theory of two temperature generalized thermoelasticity, IMA Journal of Applied Mathematics 71(3): 383-390.
[49] Youssef H.M., 2011, Theory of two - temperature thermoelasticity without energy dissipation, Journal of Thermal Stresses 34: 138-146.