Effect of Two Temperatures and Thermal Phase-lags in a Thick Plate due to a Ring Load with Axisymmetric Heat Supply
Kumar Rajneesh, Sharma Nidhi, Lata Parveen
1 Department of Mathematics, Kurukshetra University
Kurukshetra, Haryana, India
E-mail: rajneesh_kuk@rediffmail.com2 Department of Mathematics, MM University
Mullana, Ambala, Haryana, India
E-mail: nidhi_kuk26@rediffmail.com3 Department of Basic and Applied Sciences, Punjabi University
Patiala, Punjab, India
∗E-mail: parveenlata@pbi.ac.in
Received:
Received: 31 Janurary 2016; revised: 22 September 2016; accepted: 26 September 2016; published online: 30 September 2016
DOI: 10.12921/cmst.2016.0000005
Abstract:
The present investigation concerns thermomechanical interactions in a homogeneous isotropic thick plate in the light of the two-temperature thermoelasticity theory with dual phase lag due to a ring load. The upper and lower ends of the thick plate are traction free and subjected to an axisymmetric heat supply. The solution is obtained by using Laplace and Hankel transform techniques. The analytical expressions of displacement components, stresses, conductive temperature, temperature change and cubic dilatation are computed in a transformed domain. The numerical inversion technique has been applied to obtain the results in the physical domain. Numerically simulated results are depicted graphically. The effect of thermal phase-lags and two temperatures are shown on the various components. Some particular cases of the result are also deduced from the present investigation.
Key words:
Hankel transform, isotropic, Laplace transform, ring load, two phase lags, two-temperature
References:
[1] C. Catteno, A form of heat conduction equation which elim-
inates the paradox of instantaneous propagation, Compute
Rendus 247, 431-433 (1958).
[2] P. Vernotte, Les paradox de la theorie continue de l’equation
de la chaleur, Compute Rendus 246, 3145-3155 (1958).
[3] D.Y. Tzou, Macro to microscale Heat Transfer; The lagging
behaviour, Taylor and Francis, Washington DC, USA 1996.
[4] R. Quintanilla, R. Racke, A note on stability in dual-phase-
lag heat conduction, International Journal of Heat and Mass
Transfer 9(7-8), 1209-1213 (2006).
[5] R. Kumar, and S. Mukhopdhyay, Effects of thermal relaxation
times on plane wave propagation under two temperature ther-
moelasticity, International Journal Of Engineering Sciences
48(2), 128-139 (2010).
[6] Y. Chou, R.J. Yang, Two dimensional dual-phase-lag ther-
mal behaviour in single-/multi-layer structures using CESE
method, International Journal of Heat and Mass Transfer 52,
239-249 (2009).
[7] J. Zhou, Y. Zang, J.K. Chen, An axisymmetric dual-phase-lag
bioheat model for laser heating of living tissues, International
Journal of Thermal Sciences 48, 1477-1485 (2009).
[8] R. Kumar, V. Chawla, I.A. Abbas, Effect of viscosity on wave
propagation in anisotropic thermoelastic medium with three-
phase-lag model, Theoret. Appl. Mech 39(4), 313-341 (2012).
[9] X.H. Ying, J.X. Yun, Time fractional dual-phase-lag heat
conduction equation, Chin. Phys. B 24(3), 034401 (2015).
[10] I.A. Abdallah, Dual phase lag Heat Conduction and Ther-
moelastic properties of a semi infinite medium Induced by
Ultrashort Pulsed layer, Progress in Physics 3, 60-63 (2009).
[11] A.S. Rukolaine, Unphysical effects of the dual-phase-lag
model of heat conduction, International Journal of Heat and
Mass Transfer 78, 58-63 (2014).
[12] J.J Tripathi, G.D. Kedar, K.C. Deshmukh, Generalized ther-
moelastic diffusion problem in a thick circular plate with ax-
isymmetric heat supply, Acta Mechanica 226(7), 2121-2134
(2015).
[13] P.J. Chen, M.E. Gurtin, On a theory of heat conduction involv-
ing two parameters, Zeitschrift für angewandte Mathematik
9
[14] P.J. Chen, M.E. Gurtin, W.O. Williams, A note on simple
heat conduction, Journal of Applied Mathematics and Physics
(ZAMP) 19, 969-70 (1968).
[15] P.J. Chen, M.E. Gurtin, W.O. Williams, On the thermodynam-
ics of non simple elastic materials with two temperatures,
ZAMP 20, 107-112 (1969).
[16] B.A. Boley, I.S. Tolins, Transient coupled thermoelastic
boundary value problem in the half space, Journal of Applied
Mechanics 29, 637-646 (1962).
[17] W.E. Warren, P.J. Chen, Wave propagation in the two temper-
ature theory of thermoelasticity, Journal of Acta Mechanica
16, 21-33 (1973).
[18] Youssef, H.M. Theory of two – temperature thermoelasticity
without energy dissipation, Journal of Thermal Stresses,34,
138-146. (2011)
[19] I.A. Abbas, R. Kumar, L.S. Reen, Response of thermal sources
in transversely isotropic thermoelastic materials without en-
ergy dissipation and with two temperatures, Canadian Journal
of Physics 92(11), 1305-11 (2014).
[20] R. Kumar, S. Mukhopadhyay, Analysis of the Effects of Phase-
lags on propagation of harmonic Plane Waves in Thermoelas-
tic Media, CMST 16(1), 19-28 (2010).
[21] H.M. Youssef, Theory of two temperature generalized ther-
moelasticity, IMA Journal of Applied Mathematics 71(3), 383-
390 (2006).
[22] R. Kumar, K.D. Sharma, S.K. Garg, Effect of two temperature
on reflection coefficient in micropolar thermoelastic media
with and without energy dissipation, Advances in Acoustics
and Vibrations (2014).
[23] S. Kaushal, R. Kumar, A. Miglani, Wave propagation in tem-
perature rate dependent thermoelasticity with two tempera-
tures, Mathematical Sciences 5, 125-146 (2011).
[24] S. Kaushal, N. Sharma, R. Kumar, Propagation of waves in
generalized thermoelastic continua with two temperature, In-
ternational Journal of Applied Mechanics and Engineering 15,
1111-1127 (2010).
[25] M.A. Ezzat, E.S. Awad, Constitutive relations, uniqueness of
solution and thermal shock application in the linear theory of
micropolar generalized thermoelasticity involving two temper-
atures, Journal of Thermal Stresses 33(3), 225-250 (2010).
[26] K. Sharma, M. Marin, Effect of distinct conductive and ther-
modynamic temperatures on the reflection of plane waves in
micropolar elastic half-space, U.P.B.Sci.Bull Series 75(2),
121-132 (2013).
[27] S.Y. Atwa, A. Jahangir, Two temperature effects on plane
waves in generalized Thermo-Microstretch Elastic Solid, In-
ternational Journal of Thermophysics 35,175-193 (2014).
[28] S. Mondal, S.H. Malik, S. Kanoria, Fractional order two tem-
perature dual phase lag thermoelasticity with variable ther-
mal conductivity, International Scholarly Research Notices, 646049 (2014).
[29] S.M. Said, Deformation of a rotating two temperature gener-
alized magneto thermoelastic medium with initial heat source
due to hydrostatic initial stress, Meccanica 50(8), 2077-2091
(2015).
[30] S.M. Said, Two-temperature generalized magneto-
thermoelastic medium for dual-phase-lag model under
the effect of gravity field and hydrostatic initial stress,
Multidiscipline Modeling in Materials and Structures 12(2),
362-383 (2016).
[31] K. Lofty, Elastic wave motions for a photothermal medium of
a dual phase lag model with an internal heat source and grav-
itational field, Canadian Journal of Physicsl 94(4), 400-409
(2016).
[32] D.P. Gaver, Observing stochastic processes and approximate
transform inversion, Operations Res. 14, 444-459 (1966).
[33] H. Stehfast, Algorithm 368, Numerical inversion of Laplace
Transforms, Comm. Ass’n. Mach, 13, pp.47-49 (1970).
[34] H. Stehfast; Remark on Algorithm 368, Numerical inversion
of Laplace Transforms, Comm. Ass’n. Mach, 3, 624 (1970).
[35] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.A. Vatterling,
Numerical recipes, Cambridge University Press, Cambridge,
the art of scientific computing 1986.
[36] R.S. Dhaliwal, A. Singh, Dynamic coupled thermoelasticity,
Hindustance Publisher Corp., New Delhi India, 1980.
The present investigation concerns thermomechanical interactions in a homogeneous isotropic thick plate in the light of the two-temperature thermoelasticity theory with dual phase lag due to a ring load. The upper and lower ends of the thick plate are traction free and subjected to an axisymmetric heat supply. The solution is obtained by using Laplace and Hankel transform techniques. The analytical expressions of displacement components, stresses, conductive temperature, temperature change and cubic dilatation are computed in a transformed domain. The numerical inversion technique has been applied to obtain the results in the physical domain. Numerically simulated results are depicted graphically. The effect of thermal phase-lags and two temperatures are shown on the various components. Some particular cases of the result are also deduced from the present investigation.
Key words:
Hankel transform, isotropic, Laplace transform, ring load, two phase lags, two-temperature
References:
[1] C. Catteno, A form of heat conduction equation which elim-
inates the paradox of instantaneous propagation, Compute
Rendus 247, 431-433 (1958).
[2] P. Vernotte, Les paradox de la theorie continue de l’equation
de la chaleur, Compute Rendus 246, 3145-3155 (1958).
[3] D.Y. Tzou, Macro to microscale Heat Transfer; The lagging
behaviour, Taylor and Francis, Washington DC, USA 1996.
[4] R. Quintanilla, R. Racke, A note on stability in dual-phase-
lag heat conduction, International Journal of Heat and Mass
Transfer 9(7-8), 1209-1213 (2006).
[5] R. Kumar, and S. Mukhopdhyay, Effects of thermal relaxation
times on plane wave propagation under two temperature ther-
moelasticity, International Journal Of Engineering Sciences
48(2), 128-139 (2010).
[6] Y. Chou, R.J. Yang, Two dimensional dual-phase-lag ther-
mal behaviour in single-/multi-layer structures using CESE
method, International Journal of Heat and Mass Transfer 52,
239-249 (2009).
[7] J. Zhou, Y. Zang, J.K. Chen, An axisymmetric dual-phase-lag
bioheat model for laser heating of living tissues, International
Journal of Thermal Sciences 48, 1477-1485 (2009).
[8] R. Kumar, V. Chawla, I.A. Abbas, Effect of viscosity on wave
propagation in anisotropic thermoelastic medium with three-
phase-lag model, Theoret. Appl. Mech 39(4), 313-341 (2012).
[9] X.H. Ying, J.X. Yun, Time fractional dual-phase-lag heat
conduction equation, Chin. Phys. B 24(3), 034401 (2015).
[10] I.A. Abdallah, Dual phase lag Heat Conduction and Ther-
moelastic properties of a semi infinite medium Induced by
Ultrashort Pulsed layer, Progress in Physics 3, 60-63 (2009).
[11] A.S. Rukolaine, Unphysical effects of the dual-phase-lag
model of heat conduction, International Journal of Heat and
Mass Transfer 78, 58-63 (2014).
[12] J.J Tripathi, G.D. Kedar, K.C. Deshmukh, Generalized ther-
moelastic diffusion problem in a thick circular plate with ax-
isymmetric heat supply, Acta Mechanica 226(7), 2121-2134
(2015).
[13] P.J. Chen, M.E. Gurtin, On a theory of heat conduction involv-
ing two parameters, Zeitschrift für angewandte Mathematik
9
[14] P.J. Chen, M.E. Gurtin, W.O. Williams, A note on simple
heat conduction, Journal of Applied Mathematics and Physics
(ZAMP) 19, 969-70 (1968).
[15] P.J. Chen, M.E. Gurtin, W.O. Williams, On the thermodynam-
ics of non simple elastic materials with two temperatures,
ZAMP 20, 107-112 (1969).
[16] B.A. Boley, I.S. Tolins, Transient coupled thermoelastic
boundary value problem in the half space, Journal of Applied
Mechanics 29, 637-646 (1962).
[17] W.E. Warren, P.J. Chen, Wave propagation in the two temper-
ature theory of thermoelasticity, Journal of Acta Mechanica
16, 21-33 (1973).
[18] Youssef, H.M. Theory of two – temperature thermoelasticity
without energy dissipation, Journal of Thermal Stresses,34,
138-146. (2011)
[19] I.A. Abbas, R. Kumar, L.S. Reen, Response of thermal sources
in transversely isotropic thermoelastic materials without en-
ergy dissipation and with two temperatures, Canadian Journal
of Physics 92(11), 1305-11 (2014).
[20] R. Kumar, S. Mukhopadhyay, Analysis of the Effects of Phase-
lags on propagation of harmonic Plane Waves in Thermoelas-
tic Media, CMST 16(1), 19-28 (2010).
[21] H.M. Youssef, Theory of two temperature generalized ther-
moelasticity, IMA Journal of Applied Mathematics 71(3), 383-
390 (2006).
[22] R. Kumar, K.D. Sharma, S.K. Garg, Effect of two temperature
on reflection coefficient in micropolar thermoelastic media
with and without energy dissipation, Advances in Acoustics
and Vibrations (2014).
[23] S. Kaushal, R. Kumar, A. Miglani, Wave propagation in tem-
perature rate dependent thermoelasticity with two tempera-
tures, Mathematical Sciences 5, 125-146 (2011).
[24] S. Kaushal, N. Sharma, R. Kumar, Propagation of waves in
generalized thermoelastic continua with two temperature, In-
ternational Journal of Applied Mechanics and Engineering 15,
1111-1127 (2010).
[25] M.A. Ezzat, E.S. Awad, Constitutive relations, uniqueness of
solution and thermal shock application in the linear theory of
micropolar generalized thermoelasticity involving two temper-
atures, Journal of Thermal Stresses 33(3), 225-250 (2010).
[26] K. Sharma, M. Marin, Effect of distinct conductive and ther-
modynamic temperatures on the reflection of plane waves in
micropolar elastic half-space, U.P.B.Sci.Bull Series 75(2),
121-132 (2013).
[27] S.Y. Atwa, A. Jahangir, Two temperature effects on plane
waves in generalized Thermo-Microstretch Elastic Solid, In-
ternational Journal of Thermophysics 35,175-193 (2014).
[28] S. Mondal, S.H. Malik, S. Kanoria, Fractional order two tem-
perature dual phase lag thermoelasticity with variable ther-
mal conductivity, International Scholarly Research Notices, 646049 (2014).
[29] S.M. Said, Deformation of a rotating two temperature gener-
alized magneto thermoelastic medium with initial heat source
due to hydrostatic initial stress, Meccanica 50(8), 2077-2091
(2015).
[30] S.M. Said, Two-temperature generalized magneto-
thermoelastic medium for dual-phase-lag model under
the effect of gravity field and hydrostatic initial stress,
Multidiscipline Modeling in Materials and Structures 12(2),
362-383 (2016).
[31] K. Lofty, Elastic wave motions for a photothermal medium of
a dual phase lag model with an internal heat source and grav-
itational field, Canadian Journal of Physicsl 94(4), 400-409
(2016).
[32] D.P. Gaver, Observing stochastic processes and approximate
transform inversion, Operations Res. 14, 444-459 (1966).
[33] H. Stehfast, Algorithm 368, Numerical inversion of Laplace
Transforms, Comm. Ass’n. Mach, 13, pp.47-49 (1970).
[34] H. Stehfast; Remark on Algorithm 368, Numerical inversion
of Laplace Transforms, Comm. Ass’n. Mach, 3, 624 (1970).
[35] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.A. Vatterling,
Numerical recipes, Cambridge University Press, Cambridge,
the art of scientific computing 1986.
[36] R.S. Dhaliwal, A. Singh, Dynamic coupled thermoelasticity,
Hindustance Publisher Corp., New Delhi India, 1980.
