Vibrations of Inhomogeneous Viscothermoelastic Nonlocal Hollow Sphere under the effect of Three-Phase-Lag Model
محورهای موضوعی : Mechanics of Solids
S.R Sharma
1
,
M.K Sharma
2
,
D.K Sharma
3
1 - Chitkara University School of Engineering and Technology, Chitkara University, Himachal Pradesh, 174103, India
2 - Department of Mathematics, Maharaja Agrasen University, Baddi Solan, 174103, India
3 - Department of Mathematics, Maharaja Agrasen University, Baddi Solan, 174103, India
کلید واژه: Three-phase-lag model, Nonlocal elasticity, Functionally graded material, vibrations, Frequency shift, Dual–phase–lag model,
چکیده مقاله :
Herein, the free vibrations of inhomogeneous nonlocal viscothermoelastic sphere with three-phase-lag model of generalized thermoelasticity have been addressed. The governing equations and constitutive relations with three-phase-lag model have been solved by using non-dimensional quantities. The simple power law has been presumed to take the material in radial direction. The series solution has been established to derive the solution analytically. The relations of frequency equations for the continuation of viable modes are developed in dense form. The analytical results have been authenticated by the reduction of nonlocal and three–phase–lag parameters. To investigate the quality of vibrations, frequency equations are determined by applying the numerical iteration method. MATLAB software tools have been used for numerical computations and simulations to present the results graphically subject to natural frequencies, frequency shift, and thermoelastic damping. The numerical results clearly show that the variation of vibrations is slightly larger in case of nonlocal elastic sphere in contrast to elastic sphere.
[1] Nowacki W., 1975, Dynamic Problems of Thermoelasticity, Noordhof, Leyden, The Netherlands.
[2] Lord H.W., Shulman Y., 1967, Generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15: 299-309.
[3] Green A.E., Lindsay K.A., 1972, Thermoelasticity, Journal of Elasticity 2: 1-7.
[4] Green A.E., Naghdi P.M., 1993, On thermoelasticity without Energy Dissipation, Journal of Elasticity 31: 189-208.
[5] Chandrasekharaiah D.S., 1998, Hyperbolic thermoelasticity: A review of recent literature, Applied Mechanics Reviews 51: 705-729.
[6] Tzou D.Y., 1995, A unified field approach for heat conduction from macro to micro-scales, ASME Journal of Heat Transfer 117: 8-16.
[7] Choudhuri S.R., 2007, On a thermoelastic three-phase-lag model, Journal of Thermal Stresses 30: 231-238.
[8] Biswas S., Mukhopadhyay B., 2018, Rayleigh surface wave propagation in transversely isotropic medium with three-phase-lag model, Journal of Solid Mechanics 10(1): 175-185.
[9] Quintanilla R., 2009, A well-posed problem for the three-dual-phase-lag heat conduction, The Journal of Thermal Stresses 32: 1270-1278.
[10] Sharma D.K., Bachher M., Sarkar N., 2020, Effect of phase-lags on the transient waves in an axisymmetric functionally graded viscothermoelastic spherical cavity in radial direction, International Journal of Dynamics and Control DOI: 10.1007/s40435-020-00659-2.
[11] Eringen A.C., 2002, Nonlocal Continuum Field Theories, Springer Verlag, New York.
[12] Ghadiri M., Shafiei, N., Hossein Alavi S., 2017, Vibration analysis of a rotating nanoplate using nonlocal elasticity theory, Journal of Solid Mechanics 9(2): 319-337.
[13] Li C., Tian X., He T., 2020, Nonlocal thermo-viscoelasticity and its application in size-dependent responses of bi-layered composite viscoelastic nano-plate under non uniform temperature for vibration control, Mechanics of Advanced Materials and Structures Doi:10.1080/15376494.2019.1709674.
[14] Zarei M., Ghalami-Choobar M., Rahimi, G.H., Faghani, G.R., 2018, Free vibration analysis of non-uniform circular nanoplate, Journal of Solid Mechanics 10(2): 400-415.
[15] Najafizadeh M.M., Raki M., Yousefi P., 2018, Vibration analysis of FG nanoplate based on third-order shear deformation theory (TSDT) and nonlocal elasticity, Journal of Solid Mechanics 10(3): 464-475.
[16] Bachher M., Sarkar N., 2019, Nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer, Waves in Random and Complex Media 29: 595-613.
[17] Mondal S., Sarkar N., Sarkar N., 2019, Waves in dual-phase-lag thermoelastic materials with voids based on Eringen’s nonlocal elasticity, The Journal of Thermal Stresses 42: 1035-1050.
[18] Asbaghian Namin S.F., Pilafkan R., 2018, Influences of small-scale effect and boundary conditions on the free vibration of nano-plates: A molecular dynamics simulation, Journal of Solid Mechanics 10(3): 489-501.
[19] Sarkar N., De S., Sarkar N., 2019, Waves in nonlocal thermoelastic solids of type II, The Journal of Thermal Stresses 42: 1153-1170.
[20] Othman M.I.A., Ezzat M.A., Zaki S.A., El-Karamany A.S., 2002, Generalized thermo-viscoelastic plane waves with two relaxation times, International Journal of Engineering Science 40: 1329-1347.
[21] Othman M.I.A., Hasona W.M., Mansour N.T., 2015, The effect of Magnetic field on generalized thermoelastic medium with two temperature under three–phase–lag Model, Multidiscipline Modeling in Materials and Structures 11(4): 544-557.
[22] Soltani P., Bahramian R., Saberian, J., 2015, Nonlinear vibration analysis of the fluid-filled single walled carbon nanotube with the shell model based on the nonlocal elacticity theory, Journal of Solid Mechanics 7(1):58-70.
[23] Marin M., Craciun E-M., Pop N., 2016, Consideration on mixed initial-boundary value problems for micropolar porous bodies, Dynamic System and Applications 25: 175-196.
[24] Lamb H., 1881, On the vibrations of an elastic sphere, Proceedings of the London Mathematical Society 13: 189-212.
[25] Sato Y., Usami T., 1962, Basic study on the oscillation of homogeneous elastic sphere; Part I, Frequency of the free oscillations, Geophysics Magazine 31: 15-24.
[26] Sato Y., Usami T., 1962, Basic study on the oscillation of a homogeneous elastic sphere; Part II, Distribution of displacement, Geophysics Magazine 31: 25-47.
[27] Hsu M.H., 2007, Vibration Analysis of Annular Plates, Tamkang Journal of Science and Engineering 10(3): 193-199.
[28] Keles I., Tutuncu N., 2011, Exact analysis of axisymmetric dynamic response of functionally graded Cylinders (or disks) and Spheres, Journal of Applied Mechanics 78: 061014.
[29] Sharma J.N., Sharma D. K., Dhaliwal S.S., 2012, Three-dimensional free vibration analysis of a viscothermoelastic hollow sphere, Open Journal of Acoustics 2: 12-24.
[30] Sharma J.N., Sharma D. K., Dhaliwal S.S., 2013, Free vibration analysis of a rigidly fixed viscothermoelastic hollow sphere, Indian Journal of Pure and Applied Mathematics 44: 559-586.
[31] Sharma D.K., Sharma J.N., Dhaliwal S.S., Walia V., 2014, Vibration analysis of axisymmetric functionally graded viscothermoelastic spheres, Acta Mechanica Sinica 30: 100-111.
[32] Nejad M.Z., Rastgoo A., Hadi A., 2014, Effect of exponentially-varying properties on displacements and stresses in pressurized functionally graded thick spherical shells with using iterative technique, Journal of Solid Mechanics 6(4):366-377.
[33] Sharma D.K., 2016, Free vibrations of homogenous isotropic viscothermoelastic spherical curved plates, Journal of Applied Science and Engineering 19(2): 135-148.
[34] Biswas S., Mukhopadhyay B., 2019, Three-dimensional vibration analysis in transversely isotropic cylinder with matrix frobenius method, The Journal of Thermal Stresses 42(10):1207-1228.
[35] Biswas S., 2019, Eigenvalue approach to a magneto-thermoelastic problem in transversely isotropic hollow cylinder: comparison of three theories, Waves in Random and Complex Media Doi:10.1080/17455030.2019.1588484.
[36] Sharma D. K., Sharma S. R., Walia V., 2018, Analysis of axisymmetric functionally graded forced vibrations due to heat sources in viscothermoelastic hollow sphere using series solution, AIP Conference Proceedings 1975: 030010.
[37] Sharma D.K., Mittal H., Sharma S.R., 2019, Forced vibration analysis in axisymmetric functionally graded viscothermoelastic hollow cylinder under dynamic pressure, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences.
[38] Manthena V.R., Lamba N.K., Kedar G.D., 2018, Mathematical modeling of thermoelastic state of a thick hollow cylinder with nonhomogeneous material properties, Journal of Solid Mechanics 10(1): 142-156.
[39] Manthena V.R., Lamba N.K., Kedar G.D., 2018, Estimation of thermoelastic state of a thermally sensitive functionally graded thick hollow cylinder: A mathematical model, Journal of Solid Mechanics 10(4): 766-778.
[40] Sharma D.K., Mittal H., 2019, Analysis of free vibrations of axisymmetric functionally graded generalized viscothermoelastic cylinder using series solution, Journal of Vibration Engineering & Technologies DOI: 10.1007/s42417-019-00178-1.
[41] Riaz A., Ellahi, R., Bhatti M.M., Marin M., 2019, Study of heat and mass transfer in the Eyring–Powell model of fluid propagating peristaltically through a rectangular compliant channel, Heat Transfer Research 50(16): 1539-1560.
[42] Bhatti M.M., Ellahi R., Zeeshan A., Marin M., Ijaz N., 2019, Numerical study of heat transfer and Hall current impact on peristaltic propulsion of particle-fluid suspension with compliant wall properties, Modern Physics Letters B 33(35): 1950439.
[43] Sharma D.K., Thakur P.C., Sarkar N., Bachher, M., 2020, Vibrations of a nonlocal thermoelastic cylinder with void, Acta Mechanica 231: 2931-2945.
[44] Sharma D.K., Thakur D., Walia V., Sarkar N., 2020, Free vibration analysis of a nonlocal thermoelastic hollow cylinder with diffusion, The Journal of Thermal Stresses 43(8): 981-997.
[45] Biswas S., 2020, Fundamental solution of steady oscillations equations in nonlocal thermoelastic medium with voids, The Journal of Thermal Stresses 43(3): 284-304.
[46] Pramanik A.S., Biswas S., 2020, Surface waves in nonlocal thermoelastic medium with state space approach, The Journal of Thermal Stresses 43(6): 667-686.
[47] Noda N., Jin Z. H., 1993, Thermal stress intensity factor for a crack in a strip of functionally graded material, International Journal of Solids and Structures 30: 1039-1056.
[48] Tomantschger K.W. 2002, Series solutions of coupled differential equations with one regular singular point, Journal of Computational and Applied Mathematics 140: 773-783.
[49] Cullen C.G., 1972, Matrices and Linear Transformation, Addison–Wesley Pub., Reading Massachusetts.
[50] Neuringer J.L., 1978, The Fröbenius method for complex roots of the indicial equation, International Journal of Mathematical Education in Science and Technology 9: 71-77.
