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Transient thermoelastic response in a cracked strip of functionally graded materials via generalized fractional heat conduction. (English) Zbl 07186570
Summary: This work is devoted to analyzing a thermal shock problem of an elastic strip made of functionally graded materials containing a crack parallel to the free surface based on a generalized fractional heat conduction theory. The embedded crack is assumed to be insulated. The Fourier transform and the Laplace transform are employed to solve a mixed initial-boundary value problem associated with a time-fractional partial differential equation. Temperature and thermal stresses in the Laplace transform domain are evaluated by solving a system of singular integral equations. Numerical results of the thermoelastic fields in the time domain are given by applying a numerical inversion of the Laplace transform. The temperature jump between the upper and lower crack faces and the thermal stress intensity factors at the crack tips are illustrated graphically, and phase lags of heat flux, fractional orders, and gradient index play different roles in controlling heat transfer process. A comparison of the temperature jump and thermal stress intensity factors between the non-Fourier model and the classical Fourier model is made. Numerical results show that wave-like behavior and memory effects are two significant features of the fractional Cattaneo heat conduction, which does not occur for the classical Fourier heat conduction.

74F05 Thermal effects in solid mechanics
74R10 Brittle fracture
74E05 Inhomogeneity in solid mechanics
26A33 Fractional derivatives and integrals
80A19 Diffusive and convective heat and mass transfer, heat flow
Full Text: DOI
[1] Ootao, Y.; Tanigawa, Y., Transient thermoelastic analysis for a functionally graded hollow cylinder, J. Therm. Stresses, 29, 11, 1031-1046 (2006)
[2] Peng, X. L.; Li, X. F., Transient response of temperature and thermal stresses in a functionally graded hollow cylinder, J. Therm. Stresses, 33, 5, 485-500 (2010)
[3] Zhou, Y. T.; Lee, K. Y.; Yu, D. H., Transient heat conduction in a functionally graded strip in contact with well stirred fluid with an outside heat source, Int. J. Heat Mass Transf., 54, 25-26, 5438-5443 (2011) · Zbl 1231.80036
[4] Noda, N.; Jin, Z. H., Thermal stress intensity factors for a crack in a strip of a functionally gradient material, Int. J. Solids Struct., 30, 8, 1039-1056 (1993) · Zbl 0767.73055
[5] Jin, Z. H.; Noda, N., Transient thermal stress intensity factors for a crack in a semi-infinite plate of a functionally gradient material, Int. J. Solids Struct., 31, 2, 203-218 (1994) · Zbl 0799.73055
[6] Erdogan, F.; Wu, B. H., Crack problems in FGM layers under thermal stresses, J. Therm. Stresses, 19, 3, 237-265 (1996)
[7] Chiu, T. C.; Tsai, S. W.; Chue, C. H., Heat conduction in a functionally graded medium with an arbitrarily oriented crack, Int. J. Heat Mass Transf., 67, 514-522 (2013)
[8] Zhang, Y.; Guo, L.; Guo, F.; Zhong, S., Fracture analysis of a nonhomogeneous coating/substrate system with an interface under thermal shock, Acta Mech., 225, 9, 2485-2500 (2014) · Zbl 1302.74153
[9] Lord, H. W.; Shulman, Y., A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, 15, 5, 299-309 (1967) · Zbl 0156.22702
[10] Tzou, D. Y., A unified field approach for heat conduction from macro- to micro-scales, J. Heat Transf., 117, 1, 8 (1995)
[11] Sherief, H. H.; El-Maghraby, N. M., An internal penny-shaped crack in an infinite thermoelastic solid, J. Therm. Stresses, 26, 4, 333-352 (2003)
[12] Mallik, S. H.; Kanoria, M., A unified generalized thermoelasticity formulation: application to penny-shaped crack analysis, J. Therm. Stresses, 32, 9, 943-965 (2009)
[13] Hu, K. Q.; Chen, Z. T., Thermoelastic analysis of a partially insulated crack in a strip under thermal impact loading using the hyperbolic heat conduction theory, Int. J. Eng. Sci., 51, 144-160 (2012) · Zbl 1423.74237
[14] Rahideh, H.; Malekzadeh, P.; Haghighi, M. R.G., Heat conduction analysis of multi-layered FGMs considering the finite heat wave speed, Energy Convers. Manage., 55, 14-19 (2012)
[15] Babaei, M. H.; Chen, Z., Transient hyperbolic heat conduction in a functionally graded hollow cylinder, J. Thermophys. Heat Transf., 24, 2, 325-330 (2010)
[16] Fu, J. W.; Akbarzadeh, A.; Chen, Z. T.; Qian, L.; Pasini, D., Non-Fourier heat conduction in a sandwich panel with a cracked foam core, Int. J. Therm. Sci., 102, 263-273 (2016)
[17] Keles, I.; Conker, C., Transient hyperbolic heat conduction in thick-walled FGM cylinders and spheres with exponentially-varying properties, Eur. J. Mech. A Solids, 30, 3, 449-455 (2011) · Zbl 1278.80002
[18] Daneshjou, K.; Bakhtiari, M.; Parsania, H.; Fakoor, M., Non-Fourier heat conduction analysis of infinite 2D orthotropic FG hollow cylinders subjected to time-dependent heat source, Appl. Therm. Eng., 98, 582-590 (2016)
[19] Akbarzadeh, A.; Cui, Y.; Chen, Z., Thermal wave: from nonlocal continuum to molecular dynamics, RSC Adv., 7, 22, 13623-13636 (2017)
[20] Podlubny, I., Fractional Differential Equations (1998), Academic press: Academic press New York · Zbl 0922.45001
[21] Ortigueira, M. D., Fractional Calculus for Scientists and Engineers, Springer Science & Business Media, 84 (2011) · Zbl 1251.26005
[22] Metzler, R.; Klafter, J., The random Walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1, 1-77 (2000) · Zbl 0984.82032
[23] Kimmich, R., Strange kinetics, porous media, and NMR, Chem. Phys., 284, 1-2, 253-285 (2002)
[24] Povstenko, Y. Z., Fractional heat conduction equation and associated thermal stress, J. Therm. Stresses, 28, 1, 83-102 (2004)
[25] Sherief, H. H.; El-Sayed, A. M.A.; El-Latief, A. M.A., Fractional order theory of thermoelasticity, Int. J. Solids Struct., 47, 269-275 (2010) · Zbl 1183.74051
[26] Youssef, H. M., Theory of fractional order generalized thermoelasticity, J. Heat Transf., 132, 6, 061301 (2010)
[27] Ezzat, M. A.; Karamany, A. S.E., Theory of fractional order in electro-thermoelasticity, Eur. J. Mech. A. Solids, 30, 4, 491-500 (2011) · Zbl 1278.74036
[28] Ezzat, M. A., Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer, Physica B, 406, 1, 30-35 (2011)
[29] Qi, H.; Jiang, X., Solutions of the space-time fractional Cattaneo diffusion equation, Physica A, 390, 11, 1876-1883 (2011) · Zbl 1225.35253
[30] Jiang, X.; Qi, H., Thermal wave model of bioheat transfer with modified Riemann-Liouville fractional derivative, J. Phys. A Math. Theor., 45, 48, 485101 (2012) · Zbl 1339.80006
[31] Ghazizadeh, H. R.; Azimi, A.; Maerefat, M., An inverse problem to estimate relaxation parameter and order of fractionality in fractional single-phase-lag heat equation, Int. J. Heat Mass Transf., 55, 7-8, 2095-2101 (2012)
[32] Qi, H.; Guo, X., Transient fractional heat conduction with generalized Cattaneo model, Int. J. Heat Mass Transf., 76, 535-539 (2014)
[33] Liu, L.; Zheng, L.; Liu, F.; Zhang, X., An improved heat conduction model with Riesz fractional Cattaneo-Christov flux, Int. J. Heat Mass Transf., 103, 1191-1197 (2016)
[34] Zhang, X. Y.; Li, X. F., Thermal shock fracture of a cracked thermoelastic plate based on time-fractional heat conduction, Eng. Fract. Mech., 171, 22-34 (2017)
[35] Zhang, X. Y.; Li, X. F., Transient thermal stress intensity factors for a circumferential crack in a hollow cylinder based on generalized fractional heat conduction, Int. J. Therm. Sci., 121, 336-347 (2017)
[36] Liu, L.; Zheng, L.; Liu, F., Research on macroscopic and microscopic heat transfer mechanisms based on non-Fourier constitutive model, Int. J. Heat Mass Transf., 127, 165-172 (2018)
[37] Jumarie, G., Derivation and solutions of some fractional black-scholes equations in coarse-grained space and time. application to Merton’s optimal portfolio, Comput. Math. Appl., 59, 3, 1142-1164 (2010) · Zbl 1189.91230
[38] Theocaris, P.; Ioakimidis, N., Numerical integration methods for the solution of singular integral equations, Q. Appl. Math., 35, 1, 173-183 (1977) · Zbl 0353.45016
[39] Stehfest, H., Numerical inversion of Laplace transforms, Commun. ACM, 13, 1, 47-49 (1970)
[40] Li, X. F., Dynamic analysis of a cracked magnetoelectroelastic medium under antiplane mechanical and inplane electric and magnetic impacts, Int. J. Solids Struct., 42, 11, 3185-3205 (2005) · Zbl 1142.74014
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