Skip to main content
SpringerLink
Log in

Thermal shock fracture of an elastic half-space with a subsurface penny-shaped crack via fractional thermoelasticity

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

This paper analyzes a thermal shock problem of an elastic half-space with a penny-shaped crack near the surface based on a fractional thermoelasticity theory. The embedded crack is assumed to be insulated. The Hankel transform and Laplace transform are employed to solve an initial-boundary value problem associated with a fractional partial differential equation. Explicit expressions of temperature and thermal stresses induced by the penny-shaped crack are obtained by solving a system of singular integral equations. Numerical results of the thermoelastic fields in the time domain are given by applying the numerical inverse Laplace transform. The temperature jump between the upper and lower crack surfaces and the thermal stress intensity factors at the crack front are illustrated graphically for various relaxation times and fractional orders, as well as the distance between the crack plane and the half-space surface. A comparison of the temperature, thermal stresses, and their intensity factors is made when adopting the fractional heat conduction model and the classical Fourier heat conduction model. Numerical results show that the temperature overshooting phenomenon may occur for the fractional heat conduction model, whereas it does not occur for the classical Fourier heat conduction model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Olesiak, Z., Sneddon, I.N.: The distribution of thermal stress in an infinite elastic solid containing a penny-shaped crack. Arch. Ration. Mech. Anal. 4, 238–254 (1959)

    Article  MathSciNet  Google Scholar 

  2. Florence, A.L., Goodier, J.N.: The linear thermoelastic problem of uniform heat flow disturbed by a penny-shaped insulated crack. Int. J. Eng. Sci. 1, 533–540 (1963)

    Article  MathSciNet  Google Scholar 

  3. Kaczynski, A., Matysiak, S.J.: On the three-dimensional problem of an interface crack under uniform heat flow in a bimaterial periodically-layered space. Int. J. Fract. 123, 127–138 (2003)

    Article  Google Scholar 

  4. Chen, W.Q., Ding, H.J., Ling, D.S.: Thermoelastic field of a transversely isotropic elastic medium containing a penny-shaped crack: exact fundamental solution. Int. J. Solids Struct. 41, 69–83 (2004)

    Article  Google Scholar 

  5. Rekik, M., Neifar, M., El-Borgi, S.: An axisymmetric problem of a partially insulated crack embedded in a graded layer bonded to a homogeneous half-space under thermal loading. J. Therm. Stress. 34, 201–227 (2011)

    Article  Google Scholar 

  6. Li, X.F., Lee, K.Y.: Effect of heat conduction of penny-shaped crack interior on thermal stress intensity factors. Int. J. Heat Mass Transf. 91, 127–134 (2015)

    Article  Google Scholar 

  7. Li, X.Y., Li, P.D., Kang, G.Z., Chen, W.Q., Muller, R.: Steady-state thermo-elastic field in an infinite medium weakened by a penny-shaped crack: complete and exact solutions. Int. J. Solids Struct. 84, 167–182 (2016)

    Article  Google Scholar 

  8. Li, P.D., Li, X.Y., Kang, G.Z.: Axisymmetric thermo-elastic field in an infinite space containing a penny-shaped crack under a pair of symmetric uniform heat fluxes and its applications. Int. J. Mech. Sci. 115, 634–644 (2016)

    Article  Google Scholar 

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

    Article  Google Scholar 

  10. Tzou, D.Y.: A unified field approach for heat conduction from macro- to micro-scales. J. Heat Transf. 117, 8–16 (1995)

    Article  Google Scholar 

  11. Sherief, H.H., El-Maghraby, N.M.: An internal penny-shaped crack in an infinite thermoelastic solid. J. Therm. Stress. 26, 333–352 (2003)

    Article  Google Scholar 

  12. Mallik, S.H., Kanoria, M.: A unified generalized thermoelasticity formulation: application to penny-shaped crack analysis. J. Therm. Stress. 32, 943–965 (2009)

    Article  Google Scholar 

  13. Chen, Z.T., Hu, K.Q.: Thermo-elastic analysis of a cracked half-plane under a thermal shock impact using the hyperbolic heat conduction theory. J. Therm. Stress. 35, 342–362 (2012)

    Article  Google Scholar 

  14. 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)

    Article  MathSciNet  Google Scholar 

  15. Fu, J.W., Akbarzadeh, A.H., Chen, Z.T., Qian, L.F., Pasini, D.: Non-Fourier heat conduction in a sandwich panel with a cracked foam core. Int. J. Therm. Sci. 102, 263–273 (2016)

    Article  Google Scholar 

  16. Guo, S.L., Wang, B.L.: Thermal shock fracture of a cylinder with a penny-shaped crack based on hyperbolic heat conduction. Int. J. Heat Mass Transf. 91, 235–245 (2015)

    Article  Google Scholar 

  17. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Academic press, Cambridge (1998)

    MATH  Google Scholar 

  18. Ortigueira, M.D.: Fractional Calculus for Scientists and Engineers, vol. 84. Springer, Berlin (2011)

    MATH  Google Scholar 

  19. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  MathSciNet  Google Scholar 

  20. Povstenko, Y.Z.: Fractional heat conduction equation and associated thermal stress. J. Therm. Stress. 28, 83–102 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  22. Youssef, H.M.: Theory of fractional order generalized thermoelasticity. J. Heat Transf. 132, 061301 (2010)

    Article  Google Scholar 

  23. Ezzat, M.A.: Thermoelectric MHD with modified Fourier’s law. Int. J. Therm. Sci. 50, 449–455 (2011)

    Article  Google Scholar 

  24. Ezzat, M.A.: Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer. Phys. B 406, 30–35 (2011)

    Article  Google Scholar 

  25. Akbarzadeh, A.H., Cui, Y., Chen, Z.T.: Thermal wave: from nonlocal continuum to molecular dynamics. RSC Adv. 7, 13623–13636 (2017)

    Article  Google Scholar 

  26. Youssef, H.M.: Two-dimensional thermal shock problem of fractional order generalized thermoelasticity. Acta Mech. 223, 1219–1231 (2012)

    Article  MathSciNet  Google Scholar 

  27. Sur, A., Kanoria, M.: Fractional order two-temperature thermoelasticity with finite wave speed. Acta Mech. 223, 2685–2701 (2012)

    Article  MathSciNet  Google Scholar 

  28. Blasiak, S.: Time-fractional heat transfer equations in modeling of the non-contacting face seals. Int. J. Heat Mass Transf. 100, 79–88 (2016)

    Article  Google Scholar 

  29. Yu, Y.J., Tian, X.G., Lu, T.J.: On fractional order generalized thermoelasticity with micromodeling. Acta Mech. 224, 2911–2927 (2013)

    Article  MathSciNet  Google Scholar 

  30. Xu, H.Y., Jiang, X.Y.: Time fractional dual-phase-lag heat conduction equation. Chin. Phys. B 24, 034401 (2015)

    Article  Google Scholar 

  31. 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)

    Article  Google Scholar 

  32. 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)

    Article  Google Scholar 

  33. Erdogan, F., Gupta, G.D.: On the numerical solution of singular integral equations. Q. Appl. Math. 29, 525–534 (1972)

    Article  MathSciNet  Google Scholar 

  34. Arin, K., Erdogan, F.: Penny-shaped crack in an elastic layer bonded to dissimilar half spaces. Int. J. Eng. Sci. 9, 213–232 (1971)

    Article  Google Scholar 

  35. Miller, M.K., Guy Jr., W.: Numerical inversion of the Laplace transform by use of Jacobi polynomials. SIAM J. Numer. Anal. 3, 624–635 (1966)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Civil Engineering, Central South University, Changsha, 410075, People’s Republic of China

    Xue-Yang Zhang & Xian-Fang Li

  2. Department of Mechanical Engineering, University of Alberta, Edmonton, AB, T6G 2G8, Canada

    Xue-Yang Zhang & Zeng-Tao Chen

Authors
  1. Xue-Yang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zeng-Tao Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xian-Fang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xian-Fang Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, XY., Chen, ZT. & Li, XF. Thermal shock fracture of an elastic half-space with a subsurface penny-shaped crack via fractional thermoelasticity. Acta Mech 229, 4875–4893 (2018). https://doi.org/10.1007/s00707-018-2252-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-018-2252-x

Search

Navigation