Transient thermal stress intensity factors for an edge crack in a thin elastic plate via fractional-order framework

محورهای موضوعی : Applied Mechanics

Apeksha Sanjayrao Balwir ¹ , Dilip Kamdi ² , Vinod Varghese ³

1 - Gondwana University, Gadchiroli
2 - Rashtrapita Mahatma Gandhi Art's, Comm & Sci College, Saoli, Gadchiroli, India
3 - Department of Mathematics, Sushilabai Bharti Science College, Arni, Yavatmal, India

تاریخ دریافت : 1402/08/07 تاریخ پذیرش : 1403/06/23 تاریخ انتشار : 1403/07/03

کلید واژه: fractional calculus, non-Fourier heat conduction, integral transform approach, fractional calculus, integral transform,

چکیده مقاله :

In this study, the analysis focuses on a transient thermoelastic problem in an isotropic homogeneous elastic plate that is exposed to heat loading within the framework of the fractional-order theory. The sectional heat supply is applied on both the front edge and the farthest edge of the rectangular plate. The integral transformation was considered as a means to solve the main governing equations. The Mittag-Leffler function is utilized to express the analytical solution for temperature change, displacement, and stress response. The investigation also encompasses the study of thermoelastic behaviors in a plate featuring a central crack. The stress intensity factors at the fracture tip are determined numerically using the weight function method in this proposed solution. The findings are depicted using numerical computations, considering the material as a media, and visually represented in graphical form. The findings of this investigation indicate that the utilization of the theory of generalized thermoelasticity with fractional order heat transfer provides a more precise depiction of the behavior shown by the constituent particles comprising an elastic body, in comparison to the theory of generalized thermoelasticity with integer order.

چکیده انگلیسی:

In this study, the analysis focuses on a transient thermoelastic problem in an isotropic homogeneous elastic plate that is exposed to heat loading within the framework of the fractional-order theory. The sectional heat supply is applied on both the front edge and the farthest edge of the rectangular plate. The integral transformation was considered as a means to solve the main governing equations. The Mittag-Leffler function is utilized to express the analytical solution for temperature change, displacement, and stress response. The investigation also encompasses the study of thermoelastic behaviors in a plate featuring a central crack. The stress intensity factors at the fracture tip are determined numerically using the weight function method in this proposed solution. The findings are depicted using numerical computations, considering the material as a media, and visually represented in graphical form. The findings of this investigation indicate that the utilization of the theory of generalized thermoelasticity with fractional order heat transfer provides a more precise depiction of the behavior shown by the constituent particles comprising an elastic body, in comparison to the theory of generalized thermoelasticity with integer order.

منابع و مأخذ:

[1] Y.Z. Povstenko, Fractional heat conduction equation and associated thermal stress, J. Therm. Stresses, Vol. 28, No. 1, pp. 83-102, 2004.
[2] Y.Z. Povstenko, Two-dimensional axisymmetric stresses exerted by instantaneous pulses and sources of diffusion in an infinite space in a case of time-fractional diffusion equation, Int J Solids Struct, Vol. 44, No. 7–8, pp. 2324-2348, 2007.
[3] H.M. Youssef and E.A. Al-Lehaibi, Variational principle of fractional order generalized thermoelasticity, Appl. Math. Lett., Vol. 23, No. 10, pp. 1183-1187, 2010.
[4] H.H. Sherief, A.M.A. El-Sayed, A.M. Abd El-Latief, Fractional order theory of thermoelasticity, Int J Solids Struct, Vol. 47, No. 2, pp. 269-275, 2010.
[5] M.A. Ezzat and A.S. El-Karamany, Two-temperature theory in generalized magneto-thermoelasticity with two relaxation times, Meccanica, Vol. 46, pp. 785–794, 2011.
[6] H.M. Youssef and E.A. Al-Lehaibi, Fractional order generalized thermoelastic infinite medium with cylindrical cavity subjected to harmonically varying heat, Sci. Res., Vol. 3, No. 1, pp. 32-37, 2011.
[7] Y. Povstenko, Time-fractional radial heat conduction in a cylinder and associated thermal stresses, Arch. Appl. Mech., Vol. 82, pp. 345–362, 2012.
[8] A. Sur and M. Kanoria, Fractional order two-temperature thermoelasticity with finite wave speed, Acta Mech., Vol. 223, pp. 2685-2701, 2012.
[9] Y. Wang, X. Zhang and X. Song, A generalized theory of thermoelasticity based on thermomass and its uniqueness theorem, Acta Mech., Vol. 225, pp. 797–808, 2014.
[10] A.M. Zenkour and A.E. Abouelregal, State-space approach for an infinite medium with a spherical cavity based upon two-temperature generalized thermoelasticity theory and fractional heat conduction, Z. Angew. Math. Phys., Vol. 65, pp. 149–164, 2014.
[11] R. Yadav, K.K. Kalkal and S. Deswal, Two-temperature generalized thermoviscoelasticity with fractional order strain subjected to moving heat source: State space approach, J. Math., 2015, Article ID 487513, 13 pages, 2015.
[12] N.D. Gupta and N.C. Das, Eigenvalue approach to fractional order generalized thermoelasticity with line heat source in an infinite medium, J. Therm. Stresses, Vol. 39, pp. 977-990, 2016.
[13] S.S. Sheoran and P. Kundu, Fractional order generalized thermoelasticity theories: A review, Int. J. Adv. Appl. Math. and Mech., Vol. 3, No. 4, 76–81, 2016.
[14] I.A. Abbas, Fractional order generalized thermoelasticity in an unbounded medium with cylindrical cavity, J. Eng. Mech., Vol. 142, pp. 04016033-1-5, 2016.
[15] M. Bachher and N. Sarkar, Fractional order magneto-thermoelasticity in a rotating media with one relaxation time, Mathematical Models in Engineering, Vol. 2, No. 1, pp. 56-68, 2016.
[16] Y. Povstenko, D. Avci, E. İskender and Ö. Necati, Control of thermal stresses in axissymmetric problems of fractional thermoelasticity for an infinite cylindrical domain, Therm. Sci., Vol. 21, No. 1A, pp. 19-28, 2017.
[17] C. Xiong and Y. Niu, Fractional-order generalized thermoelastic diffusion theory, Appl. Math. Mech. -Engl. Ed., Vol. 38, pp. 1091–1108, 2017.
[18] A. Chirilă and M. Marin, The theory of generalized thermoelasticity with fractional order strain for dipolar materials with double porosity, J Mater. Sci., Vol. 53, pp. 3470–3482. 2018.
[19] I.A. Abbas, A Study on Fractional Order Theory in Thermoelastic Half-Space under Thermal Loading, Phys Mesomech, Vol. 21, pp. 150–156, 2018.
[20] P. Lata, Fractional order thermoelastic thick circular plate with two temperatures in frequency domain, Appl. Appl. Math., Vol. 13, pp. 1216 – 1229, 2018.
[21] S. Mondal, A. Sur, M. Kanoria, Magneto-thermoelastic interaction in a reinforced medium with cylindrical cavity in the context of Caputo–Fabrizio heat transport law, Acta Mech., pp. 1-18, 2019.
[22] G. Mittal and V.S. Kulkarni, Two temperature fractional order thermoelasticity theory in a spherical domain, J. Therm. Stresses, Vol. 42, No. 9, pp. 1136-1152, 2019.
[23] A. Sur and M. Kanoria, Fractional order two-temperature thermoelasticity with finite wave speed, Acta Mech., Vol. 223, pp. 2685-2701, 2012. DOI: 10.1007/s00707-012-0736-7
[24] H. R. Ghazizadeh, M. Maerefat, and A. Azimi, Explicit and implicit finite difference schemes for fractional Cattaneo equation, J. Comput. Phys., Vol. 229, No. 19, pp. 7042–7057, 2010. DOI: 10.1016/j.jcp.2010.05.039.
[25] T. T. Lam, A unified solution of several heat conduction models, Int. J. Heat Mass Transf., Vol. 56, No. 1–2, pp. 653–666, 2013. DOI: 10.1016/j.ijheatmasstransfer.2012.08.055.
[26] C. Cattaneo, Sur uneForme de I’equation de la Chaleur Eliminant le Paradoxed’une Propagation Instantanee’, ComptesRendus de l’Académie des Sciences, Vol. 247, pp. 431-433, 1958.
[27] M. N. Özisik, Heat Conduction, John Wiley & Sons, New York, 1993.
[28] H.H. Sherief, A. El-Sayed and A. El-Latief, Fractional order theory of thermoelasticity, Int J Solids Struct, Vol. 47, No. 2, pp. 269-275, 2010.
[29] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.
[30] S. Liang, W. Ranchao, and L. Chen, Laplace transform of fractional order differential equation, Electron. J. Differ. Equ., Vol. 139, pp. 1–15, 2015. URL: http://ejde.math.txstate.edu
[31] X. Y. Zhang, and X. F. Li, Thermal shock fracture of a cracked thermoelastic plate based on time–fractional heat conduction, Engng Fract Mech, Vol. 171, pp. 22-34, 2017. DOI: 10.1016/j.engfracmech.2016.11.033
[32] H. Tada, P. C. Paris, G. R. Irwin, The stress analysis of cracks handbook, 3rd edn, ASME, New York, 2000.
[33] D. P. Gaver, Observing stochastic processes and approximate transform inversion, Oper. Res., Vol. 14, No. 3, pp. 444–459, 1966. DOI:10.1287/opre.14.3.444.
[34] H. Stehfest, Algorithm 368, Numerical inversion of Laplace transforms, Comm. Assn. Comp. Mach., Vol. 13, No. 1, pp. 47–49, 1970. DOI:10.1145/361953.361969.
[35] H. Stehfest, Remark on algorithm 368: Numerical inversion of Laplace transforms, Commun. Assn. Comput. Mach., Vol. 13, No. 10, pp. 624, 1970. DOI:10.1145/355598.362787.
[36] A. Kuznetsov, On the convergence of the Gaver–Stehfest algorithm, SIAM J. Num. Anal., Vol. 51, No. 6, pp. 2984–2998, 2013. DOI:10.1137/13091974X.