Interactions of a heat source moving over a visco-thermoelastic rod kept in a magnetic field in the Lord-Shulman model under a memory dependent derivative. (English) Zbl 1441.74102

Summary: The Fractional derivative is a widely accepted theory to describe physical phenomena and the processes with memory effects which is defined in the form of convolution having kernels as power functions. Due to the shortcomings of power-law distributions, some other forms of derivatives with few other kernel functions are proposed. This present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of the magnetic field and moving heat source in a thermoelastic rod in the context of the Lord-Shulman (LS) theory of generalized thermo-visco-elasticity. Both ends of the rod are fixed and are thermally insulated. Employing the Laplace transform as a tool, the problem has been transformed into the space-domain and solved analytically. Finally, solutions in the real-time domain are obtained on applying the numerical inversion of the Laplace transform, which has been carried out employing the Riemann sum approximation method. Numerical computations for stress, displacement, and temperature within the rod is carried out and have been demonstrated graphically. The results also demonstrate how the speed of the moving heat source influences the thermophysical quantities. It is observed that the temperature, thermally induced displacement and stress of the rod are found to decrease at large source speed. Also, significant differences in the thermophysical quantities are revealed due to the influence of the magnetic field and memory effect


74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74F05 Thermal effects in solid mechanics
74F15 Electromagnetic effects in solid mechanics
26A33 Fractional derivatives and integrals
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[1] Biot, MA, Thermoelasticity and irreversible thermodynamics, J. Appl. Phys., 27, 240-253 (1956) · Zbl 0071.41204
[2] Lord, HW; Shulman, Y., A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, 15, 299-309 (1967) · Zbl 0156.22702
[3] Sherief, HH, On uniqueness and stability in generalized thermoelasticity, Q. Appl. Math., 44, 773-778 (1987) · Zbl 0613.73010
[4] Ezzat, MA; El-Karamany, AS, On uniqueness and reciprocity theorems for generalized thermo-viscoelasticity with thermal relaxation, Can. J. Phys., 81, 823-833 (2003)
[5] Ezzat, MA; El-Karamany, AS, The uniqueness and reciprocity theorems for generalized thermoviscoelasticity for anisotropic media, J. Therm. Stresses, 25, 507-522 (2002)
[6] Ezzat, MA; El-Karamany, AS, Propagation of discontinuities in magneto-thermoelastic half-space, J. Therm. Stresses, 29, 331-358 (2006)
[7] El-Karamany, AS; Ezzat, MA, Discontinuities in generalized thermo-viscoelasticity under four theories, J. Therm. Stresses, 27, 1187-1212 (2004)
[8] El-Karamany, AS; Ezzat, MA, On the boundary integral formulation of thermo-viscoelasticity theory, Int. J. Eng. Sci., 40, 1943-1956 (2002) · Zbl 1211.74064
[9] El-Karamany, AS; Ezzat, MA, Boundary integral equation formulation for the generalized thermoviscoelasticity with two relaxation times, Appl. Math. Comput., 151, 347-362 (2004) · Zbl 1066.74523
[10] Al-Huniti, NS; Al-Nimr, MA; Naji, M., Dynamic response of a rod due to a moving heat source under the hyperbolic heat conduction model, J. Sound Vib., 242, 629-640 (2001)
[11] He, T.; Cao, L.; Li, S., Dynamic response of a piezoelectric rod with thermal relaxation, J. Sound Vib., 306, 897-907 (2007)
[12] Abbas, IA, Eigenvalue approach to fractional order generalized magneto-thermoelastic medium subjected to moving heat source, J. Magn. Magn. Mater., 377, 452-459 (2015)
[13] He, T.; Cao, L., A problem of generalized magneto-thermoelastic thin slim strip subjected to a moving heat source, Math. Comput. Modell., 49, 1710-1720 (2009) · Zbl 1165.74315
[14] Ilioushin, AA; Pobedria, BE, Fundamentals of the mathematical theory of thermal visco-elasticity (1970), Moscow: Nauka, Moscow
[15] Tanner, RI, Engineering Rheology (1988), Oxford: Oxford University Press, Oxford
[16] Ezzat, MA; Othman, MI; El-Karamany, AM, State space approach to two-dimensional generalized thermo-viscoelasticity with two relaxation times, Int. J. Eng. Sci., 40, 1251-1274 (2002) · Zbl 1211.74068
[17] Ezzat, MA; El-Karamany, AS, The uniqueness and reciprocity theorems for generalized thermo-viscoelasticity with two relaxation times, Int. J. Eng. Sci., 40, 1275-1284 (2002) · Zbl 1211.74067
[18] Othman, MIA, Generalized electromagneto-thermoviscoelastic in case of 2-D thermal shock problem in a finite conducting half-space with one relaxation time, Acta Mech., 169, 37-51 (2004) · Zbl 1063.74064
[19] Sarkar, N.; Bachher, M.; Lahiri, A., State-space approach to 3D generalized thermoviscoelasticity under Green-Nagdhi theory, NZJ Math., 46, 97-113 (2016) · Zbl 1398.74074
[20] Povstenko, YZ, Fractional radial heat conduction in an infinite medium with a cylindrical cavity and associated thermal stresses, Mech. Res. Commun., 37, 436-440 (2010) · Zbl 1272.74140
[21] El-Karamany, AS; Ezzat, MA, On fractional thermoelasticity, Math. Mech. Solid., 16, 334-346 (2011) · Zbl 1269.74055
[22] Ezzat, MA, Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer, Physica B: Condensed Matter, 405, 4188-4194 (2010)
[23] M. A. Ezzat and A. S. El Karamany, “Fractional order heat conduction law in magneto-thermoelasticity involving two temperatures,” Zeitschrift f¨ur angewandte Mathematik und Physik, 62, 937-952 (2011). · Zbl 1264.74049
[24] Ezzat, MA; El-Karamany, AS, Fractional thermoelectric viscoelastic materials, Appl. Polym. Sci., 124, 2187-2199 (2012)
[25] Bachher, M.; Sarkar, N.; Lahiri, A., Generalized thermoelastic ifinite medium with voids subjected to a instantaneous heat sources with fractional derivative heat transfer, Int. J. Mech. Sci., 89, 84-91 (2014)
[26] Bachher, M.; Sarkar, N.; Lahiri, A., Fractional order thermoelastic interactions in an infinite porous material due to distributed time-dependent heat sources, Meccanica, 50, 2167-2178 (2015) · Zbl 1325.74035
[27] Wang, J-L; Li, H-F, Surpassing the fractional derivative: Concept of the memory-dependent derivative, Comput. Math. Appl., 62, 1562-1567 (2011) · Zbl 1228.35267
[28] Yu, Y-J; Hu, W.; Tian, X-G, A novel generalized thermoelasticity model based on memory-dependent derivative, Int. J. Eng. Sci., 81, 123-134 (2014) · Zbl 1423.74253
[29] Ezzat, MA; El-Karamany, AS; El-Bary, AA, Generalized thermoelasticity with memory-dependent derivatives involving two temperatures, Mech. Adv. Mater. Struct., 23, 545-553 (2016)
[30] Ezzat, MA; El-Karamany, AS; El-Bary, AA, Generalized thermo-viscoelasticity with memory-dependent derivatives, Int. J. Mech. Sci., 89, 470-475 (2014)
[31] M. Ezzat, A. El-Karamany and A. El-Bary, “Modeling of memory-dependent derivatives in generalized thermoelasticity,” Eur. Phys. J. Plus, 131 (2016). · Zbl 1423.74026
[32] Al-Jamel, A.; Al-Jamal, MF; El-Karamany, A., A memory-dependent derivative model for damping in oscillatory systems, J. Vib. Control, 24, 2221-2229 (2018)
[33] Sarkar, N., A novel Pennes’ bioheat transfer equation with memory-dependent derivative, J. Math. Models Eng., 2, 151-158 (2016)
[34] Lotfy, K.; Sarkar, N., Memory-dependent derivatives for photothermal semiconducting medium in generalized thermoelasticity with two-temperature, Mech. Time-Depend. Mater., 21, 519-534 (2017)
[35] Shaw, S.; Mukhopadhyay, B., A discontinuity analysis of generalized thermoelasticity theory with memory-dependent derivatives, Acta Mech., 228, 2675-2689 (2017) · Zbl 1401.74078
[36] Kant, S.; Mukhopadhyay, S., An investigation on responses of thermoelastic interactions in a generalized thermoelasticity with memory-dependent derivatives inside a thick plate, Math. Mech. Solids, 24, 1081286518755562 (2019)
[37] Sur, A.; Pal, P.; Kanoria, M., Modeling of memory-dependent derivative in a fiber-reinforced plate under gravitational effect, J. Therm. Stresses, 41, 973-992 (2018)
[38] Sur, A.; Kanoria, M., Modeling of memory-dependent derivative in a fibre-reinforced plate, Thin-Walled Struct., 126, 85-93 (2018)
[39] Purkait, P.; Sur, A.; Kanoria, M., Thermoelastic interaction in a two dimensional infinite space due to memory dependent heat transfer, Int. J. Adv. Appl. Math. Mech., 5, 28-39 (2017) · Zbl 1460.74022
[40] Mondal, S.; Pal, P.; Kanoria, M., Transient response in a thermoelastic half-space solid due to a laser pulse under three theories with memory-dependent derivative, Acta Mech., 230, 179-199 (2019) · Zbl 1412.74020
[41] Sur, A.; Pal, P.; Mondal, S.; Kanoria, M., Finite element analysis in a fiber-reinforced cylinder due to memory-dependent heat transfer, Acta Mech., 230, 1607-1624 (2019) · Zbl 1429.74044
[42] Mondal, S.; Sur, A.; Kanoria, M., Transient response in a piezoelastic medium due to the influence of magnetic field with memorydependent derivative, Acta Mech., 230, 2325-2338 (2019) · Zbl 1428.74059
[43] Mondal, S.; Sarkar, N.; Sarkar, N., Waves in dual-phase-lag thermoelastic materials with voids based on Eringen’s nonlocal elasticity, J. Therm. Stresses, 42, 1035-1050 (2019)
[44] Mondal, S.; Sur, A.; Kanoria, M., A memory response in the vibration of a microscale beam induced by laser pulse, J. Therm. Stresses, 42, 1415-1431 (2019)
[45] Sarkar, N.; Mondal, S., Transient responses in a two-temperature thermoelastic infinite medium having cylindrical cavity due to moving heat source with memory-dependent derivative, Journal of Applied Mathematics and Mechanics/Zeitschrift f¨ur Angewandte Mathematik und Mechanik, 99, e201800343 (2019)
[46] S. Mondal, “Interactions due to a moving heat source in a thin slim rod under memory-dependent dual-phase lag magneto-thermovisco- elasticity,” Mech. Time-Depend. Mater. (2019).
[47] S. Mondal and M. Kanoria, “Thermoelastic solutions for thermal distributions moving over thin slim rod under memory-dependent three-phase lag magneto-thermoelasticity,” Mech. Based Des. Struct. Mach., 1-22 (2019).
[48] Tzou, D., Macro-to-Micro Heat Transfer (1996), Washington DC: Taylor & Francis, Washington DC
[49] Ezzat, MA; El-Karamany, AS, Fractional order theory of a perfect conducting thermoelastic medium, Can. J. Phys., 89, 311-318 (2011)
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