Effects of Viscosity on a Thick Circular Plate in Thermoelastic Diffusion Medium

Document Type: Research Paper

Authors

1 Department of Mathematics, Kurukshetra University, Kurukshetra, India

2 Department of Mathematics & Statistics, Himachal Pradesh University, Shimla, India

10.22034/jsm.2019.667247

Abstract

The problem treated here is to determinethe viscosity effect on stresses, temperature change and chemical potential in a circular plate. The mathematical formulation is applied to two theories of thermoelastic diffusion developed by Sherief et al. [27] with one relaxation time and Kumar and Kansal [9]with two relaxation times. Laplace and Hankel transform techniques are used to obtain the expression for the displacement components, stresses, temperature change and chemical potential. The resulting quantities are computed numerically and depicted graphically by using numerical inversion technique for a particular model. Effect of viscosity is shown in the normal stress, tangential stress, temperature change and chemical potential. Some particular cases of interest are also deduced. Viscoelastic materials play an important role in many branches of engineering, technology and, in recent years, biomechanics. Viscoelastic materials, such as amorphous polymers, semicrystalline polymers, and biopolymers, can be modelled in order to determine their stress or strain interactions as well as their temporal dependencies.                     

Keywords

[1]Boit M., 1956, Thermoelasticity and irreversible thermodynamics, Journal of Applied Physics of Solids 27: 240-253.
[2] Daliwal R.S., Singh A., 1980, Dynamical Coupled Thermoelasticity, Hindustan Publishers, Delhi.
[3] Duhamel J., 1837, Une memoire sur les phenomenes thermo-mechanique, Journal de l'École Polytechnique 15: 1-31.
[4] El-Maghraby N.M., Abdel-Halim A.A., 2010, A generalized thermoelsticity problem for a half space with heat sources under axisymmetric distributions, Australian Journal of Basic and Applied Sciences 4: 3803-3814.
[5] Ezzat M.A., Othman M.I., El-Karamany A.S., 2002, State space approach to generalized thermo-viscoelasticity with two relaxation times, International Journal of Engineering Science 40: 283-302.
[6] Ezzat M.A., El-Karamany A.S., Samaan A.A., 2001, State-space formulation to generalized thermoviscoelasticity with thermal relaxation, Journal of Thermal Stresses 24: 823-846.
[7] Ezzat M.A., 2006, The relaxation effects of the volume properties of electrically conducting viscoelastic material, Materials Science and Engineering B 130: 11-23.
[8] Green A.E., Lindsay K.A., 1972, Thermoelasticity, Journal of Elasticity 2: 1-7.
[9] Kumar R., Kansal T., 2008, Propagation of lamb waves in transversely isotropic thermoelastic diffusion plate, The International Journal of Solids and Structures 45: 5890-5913.
[10] Kumar R., Chawla V., Abbas I.A., 2012, Effect of viscosity on wave propagation in anisotropic thermoelastic medium with three-phase-lag model, Theoretical and Applied Mechanics 39(4): 313-341.
[11] 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.
[12] Kumar R., Sharma K.D., Garg S.K., 2015, Reflection of plane waves in transversely isotropic micropolar viscothermoelastic solid, Materials Physics and Mechanics 22: 1-14.
[13] Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of Mechanics and Physics of Solids 15: 299-309.
[14] Marin M., 2008, Weak solutions in elasticity of dipolar porous materials, Mathematical Problems in Engineering 2008: 158908.
[15] Marin M., 2010, Lagrange identity method for microstretch thermoelastic materials, Journal of Mathematical Analysis and Applications 363: 275-286.
[16] Marin M., Stan G., 2013, Weak solutions in elasticity of dipolar bodies with stretch, Carpathian Journal of Mathematics 29(1): 33-40.
[17] Mukhopadhyay S., Kumar R., 2010, Analysis of phase-lag effects on wave propagation in a thick plate under axisymmetric temperature distribution, Acta Mechanica 210: 331-344.
[18] Neumann F., 1955, Vorlesungen Uber die Theorie der Elasticitat, Breslau, Meyer.
[19] Nowacki W., 1974, Dynamical problems of thermo diffusion in solids I, Bulletin of the Polish Academy of Sciences Technical Sciences 22: 55-64.
[20] Nowacki W., 1974, Dynamical problems of thermo diffusion in solids II, Bulletin of the Polish Academy of Sciences Technical Sciences 22: 129-135.
[21] Nowacki W., 1974, Dynamical problems of thermo diffusion in solids III, Bulletin of the Polish Academy of Sciences Technical Sciences 22: 257-266.
[22] Nowacki W., 1976, Dynamical problems of thermo diffusion in solids, Engineering Fracture Mechanics 8: 261-266.
[23] 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.
[24] Podstrigach I.S., 1961, Differential equations of the problem of thermodiffusion in isotropic deformed solid bodies, Dop Akad Nauk Ukr SSR 1961: 169-172.
[25] Sharma K., Kumar P., 2013, Propagation of plane waves and fundamental solution in thermoviscoelastic medium with voids, Journal of Thermal Stresses 36: 94-111.
[26] Sharma D.K., Parkash I., Dhaliwal S.S., Walia V., Chandel S., 2017, Effect of magnetic field on transient wave in viscothermoelastic half space, International Journal of Computational and Applied Mathematics 12(2): 343-364.
[27] Sherief H.H., Saleh H., Hamza F., 2004, The theory of generalized thermoelastic diffusion, International Journal of Engineering Sciences 42: 591-608.
[28] Sherief H.H., Saleh H., 2005, A half-space problem in the theory of generalized thermoelastic diffusion, International Journal of Solids and Structures 42: 4484-4493.
[29] Svanadze M.M., 2016, Plane waves and problems of steady vibrations in the theory of viscoelasticity for Kelvin-Voigt materials with double porosity, Archive of Mechanics 68(6): 441-458.
[30] Tripathi J.J., Kedar G.D., Deshmukh K.C., 2014, Dynamic problem of generalized thermoelasicity for a semi-infinite cylinder with heat sources, Journal of Thermoelasticity 2: 1-8.
[31] Tripathi J.J., Kedar G.D., Deshmukh K.C., 2015, Generalized thermoelastic diffusion problem in a thick circular plate with axisymmetric heat supply, Acta Mechanica 226(7): 2121-2134.