Fractional Thermoelasticity Model of a 2D Problem of Mode-I Crack in a Fibre-Reinforced Thermal Environment

Document Type: Research Paper


1 Department of Mathematics, Faculty of Science, King Abdulaziz University P.O. Box 80203, Jeddah 21589, Saudi Arabia

2 Department of Mathematics, Faculty of Science, Kafrelsheikh University Kafrelsheikh 33516, Egypt

3 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt


A model of fractional-order of thermoelasticity is applied to study a 2D problem of mode-I crack in a fibre-reinforced thermal environment. The crack is under prescribed distributions of heat and pressure. The normal mode analysis is applied to deduce exact formulae for displacements, stresses, and temperature. Variations of field quantities with the axial direction are illustrated graphically. The results regarding the presence and absence of fiber reinforcement and fractional parameters are compared. Some particular cases are also investigated via the generalized thermoelastic theory. The presented results can be applied to design different fibre-reinforced isotropic thermoelastic elements subjected to the thermal load in order to meet special technical requirements.


Main Subjects

[1] Nowacki, W., Dynamic Problems of Thermoelasticity, Noordho., Leyden, The Netherlands (1975).

[2] Nowacki, W., Thermoelasticity, 2nd edition, Pergamon Press, Oxford (1986).

[3] Lord, H.W. and Shulman, Y., A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15 (1967) 299–309.

[4] Green, A.E. and Lindsay, K.A., Thermoelasticity, J. Elasticity 2 (1972) 1–7.

[5] Green, A.E. and Naghdi, P.M., Thermoelasticity without energy dissipation, J. Elasticity 31 (1993) 189–209.

[6] Ignaczak, J. and Ostoja-Starzewski, M., Thermoelasticity with Finite Wave Speeds, Oxford University Press, New York, p. 413, 2010.

[7] Belfield, A.J., Rogers, T.G. and Spencer, A.J.M., Stress in elastic plates rein-forced by fibre lying in concentric circles, J. Mech. Phys. Solids 31 (1983) 25–54.

[8] Verma, P.D.S. and Rana, O.H., Rotation of a circular cylindrical tube reinforced by fibers lying along helices, Mech. Mat. 2 (1983) 353–359.

[9] Sengupta, P.R. and Nath, S., Surface waves in fibre-reinforced anisotropic elastic media, Sadhana 26 (2001) 363–370.

[10] Hashin, Z. and Rosen, W.B., The elastic moduli of fibre-reinforced materials, J. Appl. Mech. 31 (1964) 223–232.

[11] Singh, B. and Singh, S.J., Reflection of planes waves at the free surface of a fibrereinforced elastic half-space, Sadhana 29 (2004) 249–257.

[12] Singh, B., Wave propagation in an incompressible transversely isotropic fibre-reinforced elastic media, Arch. Appl. Mech. 77 (2007) 253–258.

[13] Singh, B., Effect of anisotropy on reflection coefficients of plane waves in fibre-reinforced thermoelastic solid, Int. J. Mech. Solids 2 (2007) 39–49.

[14] Kumar, R. and Gupta, R., Dynamic deformation in fibre-reinforced anisotropic generalized thermoelastic solid under acoustic fluid layer, Multidispline Modeling Mater. Struct. 5(3) (2009) 283–288.

[15] Abbas, I.A. and Abd-Alla, A.N., Effect of initial stress on a fiber-reinforced anosotropic thermoelastic thick plate, Int. J. Thermophys. 32(5) (2011) 1098–1110.

[16] Ailawalia, P. and Budhiraja, S., Fibre-reinforced generalized thermoelastic medium under hydrostatic initial stress, Engineering 3 (2011) 622–631.

[17] Prabhakar, S., Melnik, R.V.N., Neittaanmki, P., and Tiihonen, T., Coupled magneto-thermo-electromechanical effects and electronic properties of quantum dots, J. Comput. Theor. Nanosci. 10 (2013) 534–547.

[18] El-Naggar, A.M., Kishka, Z., Abd-Alla, A.M., Abbas, I.A., Abo-Dahab, S.M., and Elsagheer, M., On the initial stress, magnetic field, voids and rotation effects on plane waves in generalized thermoelasticity, J. Comput. Theor. Nanosci. 10 (2013) 1408–1417.

[19] Abd-Alla, A.M., Abo-Dahab, S.M., and Al-Thamali, T.A., Love waves in a non-homogeneous orthotropic magneto-elastic layer under initial stress overlying a semi-infinite medium, J. Comput. Theor. Nanosci. 10 (2013) 10–18.

[20] Podlubny, I., Fractional Differential Equations, Academic Press, New York, 1999.

[21] Othman, M.I.A., Sarkar, N., Atwa, S.Y., Effect of fractional parameter on plane waves of generalized magneto-thermoelastic diffusion with reference temperature-dependent elastic medium, Comput. Math. Appl. 65 (2013) 1103–1118.

[22] Povstenko, Y.Z., Fractional heat conduction equation and associated thermal stress, J. Therm Stresses 28 (2005) 83–102.

[23] Youssef, H., Theory of fractional order generalized thermoelasticity, J. Heat Trans 132 (2010) 1–7.

[24] Sherief, H.H., El-Sayed, A.M.A., Abd El-Latief, A.M., Fractional order theory of thermoelasticity, Int. J. Solids Struct. 47 (2010) 269–275.

[25] Ezzat, M.A., El Karamany, A.S., Fractional order heat conduction law in magneto-thermoelasticity involving two temperatures, Z. Angew. Math. Phys. 62 (2011) 937–952.

[26] Abouelregal, A.E., Fractional order generalized thermo-piezoelectric semi-infinite medium with temperature-dependent properties subjected to a ramp-type heating, J. Thermal Stresses 34(11) (2011) 1139–1155.

[27] Zenkour, A.M., Abouelregal, A.E., 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. 65(1) (2014) 149–164.

[28] Caputo, M., Linear model of dissipation whose Q is almost frequency independent–II, Geophys J. R. Astron Soc. 13 (1967) 529–539.

[29] Abbas, I.A., Zenkour, A.M., Two-temperature generalized thermoelastic interaction in an infinite fiber-reinforced anisotropic plate containing a circular cavity with two relaxation times, J. Comput. Theor. Nanosci. 11 (2014) 1–7.

[30] Kimmich, R., Strange kinetics, porous media, and NMR, J. Chem. Phys. 284 (2002) 243–285.

[31] Cheng, J.C. and Zhang, S.Y., Normal mode expansion method for lasergenerated ultrasonic lamb waves in orthotropic thin plates, Appl. Phys. B 70 (2000) 57–63.

[32] Allam, M.N., Elsibai, K.A. and Abouelregal, A.E., Electromagneto-thermoelastic problem in a thick plate using Green and Naghdi theory, Int. J. Eng. Sci. 47 (2009) 680–690.

[33] Abouelregal A.E., Zenkour A.M., Effect of fractional thermoelasticity on a twodimensional problem of a mode I crack in a rotating fibre-reinforced thermoelastic medium, Chinese Physics B 22 (2013) 108102.