Behavior of functionally graded semiconducting rod with internal heat source under a thermal shock

Document Type : Research Paper


1 Department of Mathematics, University Institute of Science Chandigarh University, Gharaun-Mohali Punjab, India.

2 Department of Mathematics and Computer Science, Transilvania University of Brasov, 500036 Brasov, Romania.

3 Academy of Romanian Scientists, Ilfov Street, 3, 050045 Bucharest, Romania.


The research article investigates the behavior of a functionally graded

semiconducting rod with internal heat source of length l under the thermal shock. A sudden heat source is applied to the left boundary of

the finite rod. The equations of motion are solved analytically and the analytical expressions of displacement, carrier density, temperature

distribution and stresses are obtained. The numerical values of these expressions are calculated and presented graphically to show

the effect of non-homogeneity parameter on the components.

The variations of the parameters are shown for different theories of thermoelasticity namely modiffed Green-Lindsay(MGL) theory,

Green-Lindsay(GL) theory, Lord-Shulman(LS) theory and Coupled(CT) theory.


Main Subjects

[1]          M. A. Biot, Thermoelasticity and irreversible thermodynamics, Journal of applied physics, Vol. 27, No. 3, pp. 240-253, 1956.
[2]          H. W. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids, Vol. 15, No. 5, pp. 299-309, 1967.
[3]          A. E. Green, K. Lindsay, Thermoelasticity, Journal of elasticity, Vol. 2, No. 1, pp. 1-7, 1972.
[4]          A. Green, P. Naghdi, Thermoelasticity without energy dissipation, Journal of elasticity, Vol. 31, No. 3, pp. 189-208, 1993.
[5]          J. I. Richard B. Hetnarski, GENERALIZED THERMOELASTICITY, Journal of Thermal Stresses, Vol. 22, No. 4-5, pp. 451-476, 1999/06/01, 1999.
[6]          J. Ignaczak, M. Ostoja-Starzewski, 2009, Thermoelasticity with finite wave speeds, OUP Oxford,
[7]          I. Abbas, A. Hobiny, M. Marin, Photo-thermal interactions in a semi-conductor material with cylindrical cavities and variable thermal conductivity, Journal of Taibah University for Science, Vol. 14, No. 1, pp. 1369-1376, 2020.
[8]          M. Marin, A. Hobiny, I. Abbas, The Effects of Fractional Time Derivatives in Porothermoelastic Materials Using Finite Element Method, Mathematics, Vol. 9, pp. 1606, 07/07, 2021.
[9]          M. Marin, A. Seadawy, S. Vlase, A. Chirila, On mixed problem in thermoelasticity of type III for Cosserat media, Journal of Taibah University for Science, Vol. 16, No. 1, pp. 1264-1274, 2022.
[10]        L. Codarcea-Munteanu, M. Marin, S. Vlase, The study of vibrations in the context of porous micropolar media thermoelasticity and the absence of energy dissipation, 1402.
[11]        M. Luminita Scutaru, S. Vlase, M. Marin, Symmetrical Mechanical System Properties-Based Forced Vibration Analysis, 1402.
[12]        S. Vlase, M. Marin, A. Elkhalfi, P. Ailawalia, Mathematical model for dynamic analysis of internal combustion engines, Journal of Computational Applied Mechanics, Vol. 54, No. 4, pp. 607-622, 2023.
[13]        M. Koizumi, FGM activities in Japan, Composites Part B: Engineering, Vol. 28, No. 1, pp. 1-4, 1997/01/01/, 1997.
[14]        T.-K. Nguyen, K. Sab, G. Bonnet, First-order shear deformation plate models for functionally graded materials, Composite Structures - COMPOS STRUCT, Vol. 83, pp. 25-36, 03/01, 2008.
[15]        Y. Khalfi, M. S. A. Houari, A. Tounsi, A refined and simple shear deformation theory for thermal buckling of solar functionally graded plates on elastic foundation, International Journal of Computational Methods, Vol. 11, pp. 1350077, 10/01, 2014.
[16]        J. Reddy, C. Chin, Thermomechanical analysis of functionally graded cylinders and plates, Journal of thermal Stresses, Vol. 21, No. 6, pp. 593-626, 1998.
[17]        B. Sankar, J. Tzeng, Thermal Stresses in Functionally Graded Beams, Aiaa Journal - AIAA J, Vol. 40, pp. 1228-1232, 06/01, 2002.
[18]        S. Mallik, M. Kanoria, Generalized thermoelastic functionally graded solid with a periodically varying heat source, International Journal of Solids and Structures, Vol. 44, pp. 7633-7645, 11/01, 2007.
[19]        I. Abbas, A. Zenkour, LS model on electro–magneto–thermoelastic response of an infinite functionally graded cylinder, Composite Structures, Vol. 96, pp. 89–96, 02/01, 2013.
[20]        A. Gunghas, R. Kumar, S. Deswal, K. Kalkal, Influence of Rotation and Magnetic Fields on a Functionally Graded Thermoelastic Solid Subjected to a Mechanical Load, Journal of Mathematics, Vol. 2019, 06/10, 2019.
[21]        K. Kalkal, A. Gunghas, S. Deswal, Two-dimensional magneto-thermoelastic interactions in a micropolar functionally graded solid, Mechanics Based Design of Structures and Machines, Vol. 48, pp. 1-22, 08/20, 2019.
[22]        M. Barak, P. Dhankhar, Effect of inclined load on a functionally graded fiber-reinforced thermoelastic medium with temperature-dependent properties, Acta Mechanica, Vol. 233, No. 9, pp. 3645-3662, 2022.
[23]        S. K. Sheokand, K. K. Kalkal, S. Deswal, Thermoelastic interactions in a functionally graded material with gravity and rotation under dual-phase-lag heat conduction, Mechanics Based Design of Structures and Machines, Vol. 51, No. 6, pp. 3026-3045, 2023.
[24]        J. Gordon, R. Leite, R. Moore, S. Porto, J. Whinnery, Long‐transient effects in lasers with inserted liquid samples, Journal of Applied Physics, Vol. 36, No. 1, pp. 3-8, 1965.
[25]        D. Kliger, 2012, Ultrasensitive laser spectroscopy, Elsevier,
[26]        A. C. Tam, Applications of photoacoustic sensing techniques, Reviews of Modern Physics, Vol. 58, No. 2, pp. 381-431, 04/01/, 1986.
[27]        D. M. Todorović, P. M. Nikolić, A. I. Bojičić, Photoacoustic frequency transmission technique: Electronic deformation mechanism in semiconductors, Journal of Applied Physics, Vol. 85, No. 11, pp. 7716-7726, 1999.
[28]        Y. Song, D. Todorovic, B. Cretin, P. Vairac, Study on the generalized thermoelastic vibration of the optically excited semiconducting microcantilevers, International Journal of Solids and Structures, Vol. 47, pp. 1871-1875, 07/01, 2010.
[29]        Y. Song, J. Bai, Z. Ren, Reflection of Plane Waves in a Semiconducting Medium under Photothermal Theory, International Journal of Thermophysics, Vol. 33, 07/01, 2012.
[30]        A. Mandelis, M. Nestoros, C. Christofides, Thermoelectronic-wave coupling in laser photothermal theory of semiconductors at elevated temperatures, Optical Engineering, Vol. 36, No. 2, pp. 459-468, 1997.
[31]        D. M. Todorovic, Plasma, thermal, and elastic waves in semiconductors, Review of Scientific Instruments, Vol. 74, pp. 582-585, 02/01, 2003.
[32]        M. Othman, R. Tantawi, E. Eraki, Effect of initial stress on a semiconductor material with temperature dependent properties under DPL model, Microsystem Technologies, Vol. 23, 12/01, 2017.
[33]        K. Lotfy, The elastic wave motions for a Photothermal medium of a dual-phase-lag model with an internal heat source and gravitational field, Canadian Journal of Physics, Vol. 94, 02/09, 2016.
[34]        M. I. Othman, M. Fekry, M. Marin, Plane waves in generalized magneto-thermo-viscoelastic medium with voids under the effect of initial stress and laser pulse heating, Struct. Eng. Mech, Vol. 73, No. 6, pp. 621-629, 2020.
[35]        M. Othman, R. Tantawi, E. Eraki, Effect of the gravity on the photothermal waves in a semiconducting medium with an internal heat source and one relaxation time, Waves in Random and Complex Media, Vol. 27, pp. 1-21, 03/27, 2017.
[36]        K. Lotfy, N. Sarkar, Memory-dependent derivatives for photothermal semiconducting medium in generalized thermoelasticity with two-temperature, Mechanics of Time-Dependent Materials, Vol. 21, 11/01, 2017.
[37]        K. Lotfy, Photothermal waves for two temperature with a semiconducting medium under using a dual-phase-lag model and hydrostatic initial stress, Waves in Random and Complex media, Vol. 27, No. 3, pp. 482-501, 2017.
[38]        A. Hobiny, I. Abbas, Fractional Order GN Model on Photo-Thermal Interaction in a Semiconductor Plane, Silicon, Vol. 12, 08/01, 2020.
[39]        K. Lotfy, A. El-Bary, W. Hassan, M. Ahmed, Hall current influence of microtemperature magneto-elastic semiconductor material, Superlattices and Microstructures, Vol. 139, pp. 106428, 2020.
[40]        P. Ailawalia, A. Kumar, Ramp Type Heating in a Semiconductor Medium under Photothermal Theory, Silicon, Vol. 12, No. 2, pp. 347-356, 2020/02/01, 2020.
[41]        K. Lotfy, A novel model of magneto photothermal diffusion (MPD) on polymer nano-composite semiconductor with initial stress, Waves in Random and Complex Media, Vol. 31, pp. 1-18, 01/17, 2019.
[42]        A. K. Khamis, K. Lotfy, A. El-Bary, Effect of variable thermal conductivity of semiconductor elastic medium during photothermal excitation subjected to thermal ramp type, Waves in Random and Complex Media, Vol. 32, No. 1, pp. 78-90, 2022.
[43]        A. K. Khamis, A. El-Bary, K. Lotfy, Electromagnetic Hall current and variable thermal conductivity influence for microtemperature photothermal excitation process of semiconductor material, Waves in Random and Complex Media, Vol. 32, No. 1, pp. 406-423, 2022.
[44]        K. Lotfy, Microtemperature photothermal excitation of semiconductor material influenced by electromagnetic Hall current and variable thermal conductivity, Waves in Random and Complex Media, pp. 1-18, 2023.
Volume 55, Issue 1
January 2024
Pages 51-61
  • Receive Date: 01 January 2024
  • Revise Date: 25 January 2024
  • Accept Date: 25 January 2024