Internal heat source in a temperature dependent thermoelastic half space with microtemperatures

Document Type: Research Paper

Authors

1 a Department of Mathematics and Humanities, Maharishi Markandeshwar University, Sadopur-Ambala, Haryana, India

2 Department of Applied Sciences, D. A.V. Institute of Engineering and Technology, Jalandhar, Punjab, India

Abstract

A two dimensional deformation due to internal heat source in a thermoelastic solid with microtemperatures under the dependence of modulus of elasticity and thermal conductivity on reference temperature has been studied. A mechanical force of constant magnitude is applied at the free surface of thermoelastic half space. The normal modes technique has been applied to obtain the exact expressions for the components of normal displacement, microtemperature, normal force stress, temperature distribution, heat flux moment tensor and tangential couple stress for thermoelastic solid with microtemperatures. The effect of internal heat source, thermal conductivity and microrotation on the derived components have been derived analytically. The graphical results are shown in the presence and absence of thermal conductivity and microrotation to show the appreciable effect of rotation and temperature on the quantities. The problem may also be extended to show the effect of different types of mechanical and thermal sources applied in the medium.

Keywords

Main Subjects


[1]           A. C. Eringen, E. Suhubi, Nonlinear theory of simple micro-elastic solids—I, International Journal of Engineering Science, Vol. 2, No. 2, pp. 189-203, 1964.

[2]           E. Suhubl, A. C. Eringen, Nonlinear theory of micro-elastic solids—II, International Journal of Engineering Science, Vol. 2, No. 4, pp. 389-404, 1964.

[3]           A. Eringen, Linear theory of micropolar elasticity", ONR Technical Report No. 29, School of Aeronautics, Aeronautics and Engineering Science, Purdue University, West Lafayette, IN, Purdue University, 1965.

[4]           A. C. Eringen, A unified theory of thermomechanical materials, International Journal of Engineering Science, Vol. 4, No. 2, pp. 179-202, 1966.

[5]           A. C. Eringen, Linear theory of micropolar elasticity, Journal of Mathematics and Mechanics, pp. 909-923, 1966.

[6]           R. A. Grot, Thermodynamics of a continuum with microstructure, International Journal of Engineering Science, Vol. 7, No. 8, pp. 801-814, 1969.

[7]           P. Říha, On the microcontinuum model of heat conduction in materials with inner structure, International Journal of Engineering Science, Vol. 14, No. 6, pp. 529-535, 1976.

[8]           D. I. Quintanilla, R, On a theory of thermoelasticity with microtemperatures, Journal of Thermal Stresses, Vol. 23, No. 3, pp. 199-215, 2000.

[9]           D. Ieşan, On a theory of micromorphic elastic solids with microtemperatures, Journal of Thermal Stresses, Vol. 24, No. 8, pp. 737-752, 2001.

[10]         M. Ezzat, M. Othman, A. El-Karamany, The dependence of the modulus of elasticity on the reference temperature in generalized thermoelasticity, Journal of thermal stresses, Vol. 24, No. 12, pp. 1159-1176, 2001.

[11]         P. S. Casas, R. Quintanilla, Exponential stability in thermoelasticity with microtemperatures, International Journal of Engineering Science, Vol. 43, No. 1-2, pp. 33-47, 2005.

[12]         A. Scalia, M. Svanadze, On the representations of solutions of the theory of thermoelasticity with microtemperatures, Journal of Thermal Stresses, Vol. 29, No. 9, pp. 849-863, 2006.

[13]         D. Ieşan, Thermoelasticity of bodies with microstructure and microtemperatures, International Journal of Solids and Structures, Vol. 44, No. 25-26, pp. 8648-8662, 2007.

[14]         M. Aouadi, Some theorems in the isotropic theory of microstretch thermoelasticity with microtemperatures, Journal of Thermal Stresses, Vol. 31, No. 7, pp. 649-662, 2008.

[15]         D. Ie? an, R. Quintanilla, On thermoelastic bodies with inner structure and microtemperatures, Journal of Mathematical Analysis and Applications, Vol. 354, No. 1, pp. 12-23, 2009.

[16]         A. Scalia, M. Svanadze, R. Tracinà, Basic theorems in the equilibrium theory of thermoelasticity with microtemperatures, Journal of Thermal Stresses, Vol. 33, No. 8, pp. 721-753, 2010.

[17]         R. Quintanilla, On growth and continuous dependence in thermoelasticity with microtemperatures, Journal of Thermal Stresses, Vol. 34, No. 9, pp. 911-922, 2011.

[18]         H. Steeb, J. Singh, S. K. Tomar, Time harmonic waves in thermoelastic material with microtemperatures, Mechanics Research Communications, Vol. 48, pp. 8-18, 2013.

[19]         S. Chiriţă, M. Ciarletta, C. D’Apice, On the theory of thermoelasticity with microtemperatures, Journal of Mathematical Analysis and Applications, Vol. 397, No. 1, pp. 349-361, 2013.

[20]         R. Kumar, M. Kaur, Reflection and refraction of plane waves at the interface of an elastic solid and microstretch thermoelastic solid with microtemperatures, Archive of Applied Mechanics, Vol. 84, No. 4, pp. 571-590, 2014.

[21]         N. Noda, Thermal stresses in materials with temperature-dependent properties, Applied Mechanics Reviews, Vol. 44, No. 9, pp. 383-397, 1991.

[22]         H. M. Youssef, Dependence of modulus of elasticity and thermal conductivity on reference temperature in generalized thermoelasticity for an infinite material with a spherical cavity, Applied Mathematics and Mechanics, Vol. 26, No. 4, pp. 470-475, 2005.

[23]         M. I. Othman, K. Lotfy, Two-dimensional problem of generalized magneto-thermoelasticity with temperature dependent elastic moduli for different theories, Multidiscipline Modeling in Materials and Structures, Vol. 5, No. 3, pp. 235-242, 2009.

[24]         M. A. Biot, 1964, Mechanics of incremental deformations,

[25]         R. Kumar, S. Devi, Thermomechanical intereactions in porous generalized thermoelastic material permeated with heat sources, Multidiscipline Modeling in Materials and Structures, Vol. 4, No. 3, pp. 237-254, 2008.

[26]         K. Lotfy, Transient disturbance in a half-space under generalized magneto-thermoelasticity with a stable internal heat source under three theories, Multidiscipline Modeling in Materials and Structures, Vol. 7, No. 1, pp. 73-90, 2011.

[27]         K. Lotfy, Transient thermo-elastic disturbances in a visco-elastic semi-space due to moving internal heat source, International Journal of Structural integrity, Vol. 2, No. 3, pp. 264-280, 2011.

[28]         M. Othman, State space approach to the generalized thermoelastic problem with temperature-dependent elastic moduli and internal heat sources, Journal of applied mechanics and technical physics, Vol. 52, No. 4, pp. 644, 2011.

[29]         R. Kumar, S. Devi, Deformation in porous thermoelastic material with temperature dependent properties, Applied Mathematics & Information Sciences, Vol. 5, No. 1, pp. 132-147, 2011.

[30]         P. Ailawalia, S. Budhiraja, Internal Heat Source in Temperature Rate Dependent Thermoelastic Medium with Hydrostatic Initial Stress, Mechanics and Mechanical Engineering, Vol. 20, No. 3, pp. 263-277, 2016.

[31]         M. I. Othman, M. E. Zidan, M. I. Hilal, Effect of gravitational field and temperature dependent properties on two-temperature thermoelastic medium with voids under GN theory, Computers, Materials & Continua, Vol. 40, No. 3, pp. 179-201, 2014.

[32]         R. Kumar, M. Kaur, S. Rajvanshi, Plane wave propagation in microstretch thermoelastic medium with microtemperatures, Journal of Vibration and Control, Vol. 21, No. 16, pp. 3403-3416, 2015.

[33]         P. Ailawalia, S. K. Sachdeva, D. Pathania, Two dimensional deformation in microstretch thermoelastic half space with microtemperatures and internal heat source, Cogent Mathematics, Vol. 2, No. 1, pp. 1086293, 2015.

[34]         P. Ailawalia, S. K. Sachdeva, D. S. Pathania, Plane strain problem in a rotating microstretch thermoelastic solid with microtemperatures, Theoretical and Applied Mechanics, Vol. 44, No. 1, pp. 51-82, 2017.

[35]         A. C. Eringen, Plane waves in nonlocal micropolar elasticity, International Journal of Engineering Science, Vol. 22, No. 8-10, pp. 1113-1121, 1984.

[36]         R. Dhaliwal, A. Singh, Dynamic coupled thermoelasticity Hindustan Publ, Corp., New Delhi, 1980.