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Acta Mechanica

, Volume 230, Issue 10, pp 3723–3740 | Cite as

Fractional single-phase lag heat conduction and transient thermal fracture in cracked viscoelastic materials

  • Wenzhi Yang
  • Zengtao ChenEmail author
Original Paper
  • 57 Downloads

Abstract

In the present article, a thermo-viscoelastic model is developed to investigate fractional single-phase lag heat conduction and the associated transient thermal mechanical behavior of a cracked viscoelastic material under a thermal shock. To avoid the negative temperature distribution around cracks, which violates the second law of thermodynamics, the time-fractional single-phase lag heat conduction is introduced to analyze the transient temperature field around the cracks. The Fourier and Laplace transforms, coupled with the singular integral equations, are employed to solve the governing partial differential equations numerically. Both the results of temperature field and stress intensity factors (SIFs) show that the fractional single-phase lag heat conduction model is more accurate and reasonable compared to the conventional hyperbolic heat conduction. A significant difference in transient fracture behavior exists between viscoelastic and elastic materials. A sharp pulse of the SIFs at the early stage is observed and should be consider carefully to meet the requirement of increased application of viscoelastic composites under thermal loading.

Notes

Acknowledgements

Funding was provided by Natural Sciences and Engineering Research Council of Canada (2017–2022), China Scholarship Council (2016–2020).

References

  1. 1.
    Drury, J.L., Mooney, D.J.: Hydrogels for tissue engineering: scaffold design variables and applications. Biomaterials 24(24), 4337–4351 (2003)Google Scholar
  2. 2.
    Guo, M., Pitet, L.M., Wyss, H.M., Vos, M., Dankers, P.Y., Meijer, E.: Tough stimuli-responsive supramolecular hydrogels with hydrogen-bonding network junctions. J. Am. Chem. Soc. 136(19), 6969–6977 (2014)Google Scholar
  3. 3.
    Luo, F., Sun, T.L., Nakajima, T., Kurokawa, T., Zhao, Y., Sato, K., Ihsan, A.B., Li, X., Guo, H., Gong, J.P.: Oppositely charged polyelectrolytes form tough, self-healing, and rebuildable hydrogels. Adv. Mater. 27(17), 2722–2727 (2015)Google Scholar
  4. 4.
    Sun, J.-Y., Zhao, X., Illeperuma, W.R., Chaudhuri, O., Oh, K.H., Mooney, D.J., Vlassak, J.J., Suo, Z.: Highly stretchable and tough hydrogels. Nature 489(7414), 133 (2012)Google Scholar
  5. 5.
    Haag, S., Bernards, M.: Polyampholyte hydrogels in biomedical applications. Gels 3(4), 41 (2017)Google Scholar
  6. 6.
    Haraguchi, K.: Nanocomposite hydrogels. Curr. Opin. Solid State Mater. Sci. 11(3–4), 47–54 (2007)Google Scholar
  7. 7.
    Guedes, R.: Durability of polymer matrix composites: viscoelastic effect on static and fatigue loading. Compos. Sci. Technol. 67(11–12), 2574–2583 (2007)Google Scholar
  8. 8.
    Zhai, S., Zhang, P., Xian, Y., Zeng, J., Shi, B.: Effective thermal conductivity of polymer composites: theoretical models and simulation models. Int. J. Heat. Mass. Transf. 117, 358–374 (2018)Google Scholar
  9. 9.
    Chen, H., Ginzburg, V.V., Yang, J., Yang, Y., Liu, W., Huang, Y., Du, L., Chen, B.: Thermal conductivity of polymer-based composites: fundamentals and applications. Prog. Polym. Sci. 59, 41–85 (2016)Google Scholar
  10. 10.
    Ji, H., Sellan, D.P., Pettes, M.T., Kong, X., Ji, J., Shi, L., Ruoff, R.S.: Enhanced thermal conductivity of phase change materials with ultrathin-graphite foams for thermal energy storage. Energy Environ. Sci. 7(3), 1185–1192 (2014)Google Scholar
  11. 11.
    Li, X., Li, C., Xue, Z., Tian, X.: Analytical study of transient thermo-mechanical responses of dual-layer skin tissue with variable thermal material properties. Int. J. Therm. Sci. 124, 459–466 (2018)Google Scholar
  12. 12.
    Van Hees, J., Gybels, J.: C nociceptor activity in human nerve during painful and non painful skin stimulation. J. Neurol. Neurosurg. Psychiatry 44(7), 600–607 (1981)Google Scholar
  13. 13.
    Liu, Y.J., Xu, N.: Modeling of interface cracks in fiber-reinforced composites with the presence of interphases using the boundary element method. Mech. Mater. 32(12), 769–783 (2000)Google Scholar
  14. 14.
    Zhi-He, J., Naotake, 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)zbMATHGoogle Scholar
  15. 15.
    Erdogan, F., Wu, B.: The surface crack problem for a plate with functionally graded properties. J. Appl. Mech. 64(3), 449–456 (1997)zbMATHGoogle Scholar
  16. 16.
    Bao, G., Wang, L.: Multiple cracking in functionally graded ceramic/metal coatings. Int. J. Solids Struct. 32(19), 2853–2871 (1995)zbMATHGoogle Scholar
  17. 17.
    Wang, B., Mai, Y.: A cracked piezoelectric material strip under transient thermal loading. J. Appl. Mech. 69(4), 539–546 (2002)zbMATHGoogle Scholar
  18. 18.
    Ueda, S.: Thermally induced fracture of a piezoelectric laminate with a crack normal to interfaces. J. Therm. Stress. 26(4), 311–331 (2003)Google Scholar
  19. 19.
    Ueda, S.: Thermal stress intensity factors for a normal crack in a piezoelectric material strip. J. Therm. Stress. 29(12), 1107–1125 (2006)Google Scholar
  20. 20.
    Cattaneo, C.: A form of heat-conduction equations which eliminates the paradox of instantaneous propagation. C. R. 247, 431 (1958)zbMATHGoogle Scholar
  21. 21.
    Vernotte, P.: Some possible complications in the phenomena of thermal conduction. C. R. 252, 2190–2191 (1961)Google Scholar
  22. 22.
    Shaw, S., Mukhopadhyay, B.: A discontinuity analysis of generalized thermoelasticity theory with memory-dependent derivatives. Acta Mech. 228(7), 2675–2689 (2017)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Mondal, S., Pal, P., Kanoria, M.: Transient response in a thermoelastic half-space solid due to a laser pulse under three theories with memory-dependent derivative. Acta Mech. 230, 179–199 (2019)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Youssef, H.M.: Two-dimensional thermal shock problem of fractional order generalized thermoelasticity. Acta Mech. 223(6), 1219–1231 (2012)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Sur, A., Kanoria, M.: Fibre-reinforced magneto-thermoelastic rotating medium with fractional heat conduction. Procedia Eng. 127, 605–612 (2015)Google Scholar
  26. 26.
    Sur, A., Kanoria, M.: Modeling of memory-dependent derivative in a fibre-reinforced plate. Thin Wall Struct. 126, 85–93 (2018)Google Scholar
  27. 27.
    Mondal, S., Sur, A., Kanoria, M.: Transient response in a piezoelastic medium due to the influence of magnetic field with memory-dependent derivative. Acta Mech. 230, 2325–2338 (2019)MathSciNetGoogle Scholar
  28. 28.
    Sur, A., Pal, P., Mondal, S., Kanoria, M.: Finite element analysis in a fiber-reinforced cylinder due to memory-dependent heat transfer. Acta Mech. 230, 1607–1624 (2019)Google Scholar
  29. 29.
    Purkait, P., Sur, A., Kanoria, M.: Elasto-thermodiffusive response in a spherical shell subjected to memory-dependent heat transfer. Wave Random Complex Media 1–23 (2019).  https://doi.org/10.1080/17455030.2019.1599464
  30. 30.
    Li, W., Song, F., Li, J., Abdelmoula, R., Jiang, C.: Non-Fourier effect and inertia effect analysis of a strip with an induced crack under thermal shock loading. Eng. Fract. Mech. 162, 309–323 (2016)Google Scholar
  31. 31.
    Hu, K., Chen, Z.: 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)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Chang, D., Wang, B.: Transient thermal fracture and crack growth behavior in brittle media based on non-Fourier heat conduction. Eng. Fract. Mech. 94, 29–36 (2012)Google Scholar
  33. 33.
    Zhang, X., Chen, Z., Li, X.: Thermal shock fracture of an elastic half-space with a subsurface penny-shaped crack via fractional thermoelasticity. Acta Mech. 229(12), 4875–4893 (2018)MathSciNetGoogle Scholar
  34. 34.
    Zhang, X., Li, X.: 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)Google Scholar
  35. 35.
    Wang, B.: Transient thermal cracking associated with non-classical heat conduction in cylindrical coordinate system. Acta. Mech. Sin. 29(2), 211–218 (2013)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Zhang, X., Xie, Y., Li, X.: Transient thermoelastic response in a cracked strip of functionally graded materials via generalized fractional heat conduction. Appl. Math. Model. 70, 328–349 (2019)MathSciNetGoogle Scholar
  37. 37.
    Xue, Z., Chen, Z., Tian, X.: Thermoelastic analysis of a cracked strip under thermal impact based on memory-dependent heat conduction model. Eng. Fract. Mech. 200, 479–498 (2018)Google Scholar
  38. 38.
    Xue, Z., Chen, Z., Tian, X.: Transient thermal stress analysis for a circumferentially cracked hollow cylinder based on memory-dependent heat conduction model. Theor. Appl. Fract. Mech. 96, 123–133 (2018)Google Scholar
  39. 39.
    Kaminski, W.: Hyperbolic heat conduction equation for materials with a nonhomogeneous inner structure. J. Heat Transf. 112(3), 555–560 (1990)Google Scholar
  40. 40.
    Mitra, K., Kumar, S., Vedevarz, A., Moallemi, M.: Experimental evidence of hyperbolic heat conduction in processed meat. J. Heat Transf. 117(3), 568–573 (1995)Google Scholar
  41. 41.
    Braznikov, A., Karpychev, V., Luikova, A.: One engineering method of calculating heat conduction process. Inzhenerno Fizicheskij Zhurnal 28(4), 677–680 (1975)Google Scholar
  42. 42.
    Bai, C., Lavine, A.: On hyperbolic heat conduction and the second law of thermodynamics. J. Heat Transf. 117(2), 256–263 (1995)Google Scholar
  43. 43.
    Körner, C., Bergmann, H.: The physical defects of the hyperbolic heat conduction equation. Appl. Phys. A 67(4), 397–401 (1998)Google Scholar
  44. 44.
    Rubin, M.: Hyperbolic heat conduction and the second law. Int. J. Eng. Sci. 30(11), 1665–1676 (1992)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Zhang, W., Cai, X., Holm, S.: Time-fractional heat equations and negative absolute temperatures. Comput. Math. Appl. 67(1), 164–171 (2014)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Ezzat, M.A., El-Karamany, A.S.: Fractional thermoelectric viscoelastic materials. J. Appl. Polym. Sci. 124(3), 2187–2199 (2012)Google Scholar
  47. 47.
    Tarasov, V.E., Aifantis, E.C.: On fractional and fractal formulations of gradient linear and nonlinear elasticity. Acta Mech. 230, 2043–2070 (2019).  https://doi.org/10.1007/s00707-019-2373-x MathSciNetCrossRefGoogle Scholar
  48. 48.
    Cajić, M., Lazarević, M., Karličić, D., Sun, H., Liu, X.: Fractional-order model for the vibration of a nanobeam influenced by an axial magnetic field and attached nanoparticles. Acta Mech. 229, 4791–4815 (2018)MathSciNetGoogle Scholar
  49. 49.
    Atanackovic, T.M., Pilipovic, S.: On a constitutive equation of heat conduction with fractional derivatives of complex order. Acta Mech. 229, 1111–1121 (2018)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Ezzat, M., El-Karamany, A., El-Bary, A.: Generalized thermo-viscoelasticity with memory-dependent derivatives. Int. J. Mech. Sci. 89, 470–475 (2014)Google Scholar
  51. 51.
    Ezzat, M., El-Karamany, A., El-Bary, A.: Thermo-viscoelastic materials with fractional relaxation operators. Appl. Math. Model. 39(23–24), 7499–7512 (2015)MathSciNetGoogle Scholar
  52. 52.
    Ezzat, M.A., El-Bary, A.A.: On thermo-viscoelastic infinitely long hollow cylinder with variable thermal conductivity. Microsyst. Technol. 23, 3263–3270 (2017)Google Scholar
  53. 53.
    Sladek, J., Sladek, V., Zhang, C., Schanz, M.: Meshless local Petrov–Galerkin method for continuously nonhomogeneous linear viscoelastic solids. Comput. Mech. 37(3), 279–289 (2006)zbMATHGoogle Scholar
  54. 54.
    Cheng, Z., Meguid, S., Zhong, Z.: Thermo-mechanical behavior of a viscoelastic FGMs coating containing an interface crack. Int. J. Fract. 164(1), 15–29 (2010)zbMATHGoogle Scholar
  55. 55.
    Choi, H.J., Thangjitham, S.: Thermally-induced interlaminar crack-tip singularities in laminated anisotropic composites. Int. J. Fract. 60(4), 327–347 (1993)Google Scholar
  56. 56.
    Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids. Clarendon Press, Oxford (1959)zbMATHGoogle Scholar
  57. 57.
    Erdogan, F.: Interface cracking of FGM coatings under steady-state heat flow. Eng. Fract. Mech. 59, 361–380 (1998)Google Scholar
  58. 58.
    Zhou, Y., Li, X., Yu, D.: A partially insulated interface crack between a graded orthotropic coating and a homogeneous orthotropic substrate under heat flux supply. Int. J. Solids Struct. 47, 768–778 (2010)zbMATHGoogle Scholar
  59. 59.
    Christensen, R.M., Freund, L.: Theory of viscoelasticity. J. Appl. Mech. 38, 720 (1971)Google Scholar
  60. 60.
    Eringen, A.C.: Continuum Physics. Academic Press Inc, New York (1975). 632 pGoogle Scholar
  61. 61.
    Delale, F., Erdogan, F.: Effect of transverse shear and material orthotropy in a cracked spherical cap. Int. J. Solids Struct. 15(12), 907–926 (1979)zbMATHGoogle Scholar
  62. 62.
    Miller, M.K., Guy, J.: WT: numerical inversion of the Laplace transform by use of Jacobi polynomials. SIAM J. Numer. Anal. 3(4), 624–635 (1966)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Paulino, G., Jin, Z.-H.: Viscoelastic functionally graded materials subjected to antiplane shear fracture. J. Appl. Mech. 68(2), 284–293 (2001)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of AlbertaEdmontonCanada

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