×

Dynamic problem of fractional thermoelasticity in bounded cylindrical domain with relaxation time. (English) Zbl 1448.74029

Maity, Damodar (ed.) et al., Advances in fluid mechanics and solid mechanics. Proceedings of the 63rd congress of the Indian Society of Theoretical and Applied Mechanics (ISTAM), Bangalore, India, December 20–23, 2018. Singapore: Springer. Lect. Notes Mech. Engin., 139-154 (2020).
Summary: A fractional heat conduction model of a solid heat conductor is designed in the bounded cylindrical domain. The solid heat conductor under consideration is assumed to be in the form of a thick circular plate. The boundaries of the thick circular plate are traction free and subjected to externally applied axisymmetric heat source. Governing heat conduction equation of this model has been designed in the context of time fractional derivative with one-relaxation time. The solution of fractional heat conduction equation in association with Caputo time fractional derivative has been found by transforming the original boundary value problem into eigenvalue problem through the integral transforms. The inversion of Laplace transforms in terms of infinite series approximations has been achieved numerically using Gaver-Stehfest algorithm. The convergence of infinite series solutions has been discussed. Illustratively, the numerical scheme has been employed to partially distributed heat flux, and thermal behavior of a heat conductor has been discussed numerically and studied graphically. Results obtained are compared with coupled thermoelasticity, fractional thermoelasticity, and generalized thermoelasticity.
For the entire collection see [Zbl 1445.74001].

MSC:

74F05 Thermal effects in solid mechanics
74H10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics
74K20 Plates
80A19 Diffusive and convective heat and mass transfer, heat flow
26A33 Fractional derivatives and integrals

Software:

Algorithm 368
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Povstenko YZ (2005) Fractional heat conduction equation and associated thermal stress. J Therm Stress 28:83-102. https://doi.org/10.1080/014957390523741 · doi:102&publication_year=2005&doi=10.1080/014957390523741
[2] Povstenko YZ (2009) Thermoelasticity that uses fractional heat conduction equation. J Math Sci 162:296-305. https://doi.org/10.1007/s10958-009-9636-3 · doi:10.1007/s10958-009-9636-3
[3] Caputo M (1967) Linear models of dissipation whose Q is almost frequency independent-II. Geophy J Int. 13:529-539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x · doi:10.1111/j.1365-246X.1967.tb02303.x
[4] Sherief HH, El-Sayed A, El-Latief A (2010) Fractional order theory of thermoelasticity. Int J Solid Struct 47:269-275. https://doi.org/10.1016/j.ijsolstr.2009.09.034 · Zbl 1183.74051 · doi:10&doi=10.1016/j.ijsolstr.2009.09.034
[5] Hussein EM (2015) Fractional order thermoelastic problem for an infinitely long circular cylinder. J Therm Stress 38:133-145. https://doi.org/10.1080/01495739.2014.936253 · doi:10.1080/01495739.2014.936253
[6] Tripathi JJ, Kedar GD, Deshmukh KC (2016) Dynamic problem of fractional order thermoelasticity for a thick circular plate with finite wave speeds. J Therm Stress 39:220-230. https://doi.org/10.1080/01495739.2015.1124646 · doi:10.1080/01495739.2015.1124646
[7] Sherief HH, Raslan WE (2019) Fundamental solution for a line source of heat in the fractional order theory of thermoelasticity using the new Caputo definition. J Therm Stress 42(1):18-28. https://doi.org/10.1080/01495739.2018.1525330 · doi:10.1080/01495739.2018.1525330
[8] Sherief, H. H., Hamza, F. A., El-latief Abd, A. M.: Wave propagation study for axisymmetric 2D problems of a generalized thermo-visco-elastic half space . J. Therm. Stresses. https://doi.org/10.1080/01495739.2019.1587326 · doi:10.1080/01495739.2019.1587326
[9] Cotterell B, Parkes EW (1960) Thermal buckling Of circular plates. Ministry of Aviation, Aeronautical Research Council, Reports and memoranda No. \(3245^*\), (1960). http://naca.central.cranfield.ac.uk/reports/arc/rm/3245.pdf
[10] Gaver DP (1966) Observing stochastic processes and approximate transform inversion. Oper Res 14:444-459. https://doi.org/10.1287/opre.14.3.444 · doi:10.1287/opre.14.3.444
[11] Stehfest H (1970) Algorithm 368, numerical inversion of Laplace transforms. Asson Comput Mach 13:47-49. https://doi.org/10.1145/361953.361969 · doi:10.1145/361953.361969
[12] Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, New-York · Zbl 1092.45003 · doi:10.1016/S0304-0208(06)80001-0
[13] Pan I, Das S (2013) Intelligent fractional order systems and control: an introduction. Springer, New-York · Zbl 1250.93004 · doi:10.1007/978-3-642-31549-7_9
[14] Miller K, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, Inc, New York · Zbl 0789.26002
[15] Özisik MN (1968) Boundary value problems of heat conduction. International Textbook Company, Scranton, Pennsylvania
[16] Kuznetsov A (2013) On the convergence of the Gaver-Stehfest algorithm. J Numer Anal 51:2984-2998. https://doi.org/10.1137/13091974X · Zbl 1461.65258 · doi:10.1137/13091974X
[17] Lord HW, Shulman Y (1967) A generalized dynamical theory of thermoelasticity. J Mech 15(5):299- · Zbl 0156.22702 · doi:10.1016/0022-5096(67)90024-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.