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Multiple collinear Griffith cracks in a one-dimensional hexagonal quasicrystalline layer

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Abstract

The fracture behavior of multiple collinear Griffith cracks in a one-dimensional quasicrystalline layer is investigated in this paper. Using the Fourier transform technique, the mixed boundary value problem is reduced to a system of Cauchy singular integral equations. The integral equations are further reduced to a system of algebraic equations. The stress intensity factors of the phonon and phason fields and the energy release rate are determined. Numerical results reveal the effects of geometric size, the distance of the cracks and loading parameter ratio of the phonon and phason fields on crack propagation and growth. The results seem useful for the design of quasicrystalline materials, structures and devices of high performance.

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References

  1. Shechtman, D., Blech, I., Gratias, D., Cahn, J.W.: Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953 (1984)

    Article  Google Scholar 

  2. Dubois, J.M.: Useful quasicrystals. World Scientific, Singapore (2005)

    Book  Google Scholar 

  3. Bak, P.: Symmetry, statibility and elastic properties of icosahedron incommensurate crystals. Phys. Rev. B 32, 5764–5772 (1985)

    Article  MathSciNet  Google Scholar 

  4. Levine, D., Lubensky, T.C., Ostlund, S., Ramaswamy, S., Steinhardt, P.J.: Elasticity and dislocations in pentagonal and icosahedral quasicrystals. Phys. Rev. Lett. 54, 1520–1523 (1985)

    Article  Google Scholar 

  5. Hu, C.Z., Wang, R.H., Ding, D.H.: Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Rep. Prog. Phys. 63, 1–39 (2000)

    Article  MathSciNet  Google Scholar 

  6. Fan, T.Y., Mai, Y.W.: Elasticity theory, fracture mechanics and some relevant thermal properties of quasicrystal materials. Appl. Mech. Rev. 57, 325–344 (2004)

    Article  Google Scholar 

  7. Fan, T.Y.: The mathematical elasticity of quasicrystals and its applications. Springer, Heidelberg (2011)

    Book  Google Scholar 

  8. Hu, C.Z., Yang, W.Z., Wang, R.H.: Symmetry and physical properties of quasicrystals. Adv. Phys. 17, 345–376 (1997)

    Google Scholar 

  9. Li, X.F., Fan, T.Y., Sun, Y.F.: A decagonal quasicrystal with a Griffith crack. Philos. Mag. A 79, 1943–1952 (1999)

    Article  Google Scholar 

  10. Zhou, W.M., Fan, T.Y.: Plane elasticity problem of two-dimensional octagonal quasicrystals and crack problem. Chin. Phys. 10, 743–747 (2001)

    Article  Google Scholar 

  11. Zhu, A.Y., Fan, T.Y.: Elastic analysis of a mode II crack in an icosahedral quasicrystal. Chin. Phys. 16, 1111–1118 (2007)

    Article  Google Scholar 

  12. Gao, Y., Ricoeur, A., Zhang, L.: Plane problems of cubic quasicrystal media with an elliptic hole or a crack. Phys. Lett. A 375, 2775–2781 (2011)

    Article  Google Scholar 

  13. Li, L.H., Fan, T.Y.: Exact solutions of two semi-infinite collinear cracks in a strip of one dimensional hexagonal quasicrystal. Appl. Math. Comput. 196, 1–5 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Guo, Y.C., Fan, T.Y.: Mode-II Griffith crack in decagonal quasicrystals. Appl. Math. Mech. 22, 1311–1317 (2001)

    Article  MATH  Google Scholar 

  15. Guo, J.H., Liu, G.T.: Exact analytic solutions for an elliptic hole with asymmetric collinear cracks in a one-dimensional hexagonal quasicrystal. Chin. Phys. B 17, 2610–2610 (2008)

    Article  Google Scholar 

  16. Guo, J.H., Lu, Z.X.: Exact solution of four cracks originating from an elliptical hole in one-dimensional hexagonal quasicrystals. Appl. Math. Comput. 217, 9397–9403 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Guo, J.H., Yu, J., Si, R.: A semi-inverse method of a Griffith crack in one dimensional hexagonal quasicrystals. Appl. Math. Comput. 219, 7445–7449 (2013)

    MathSciNet  MATH  Google Scholar 

  18. Guo, J.H., Yu, J., Xing, Y.M.: Anti-plane analysis on a finite crack in a one-dimensional hexagonal quasicrystal strip. Mech. Res. Comm. 52, 40–45 (2013)

    Article  Google Scholar 

  19. Radi, E., Mariano, P.M.: Stationary straight cracks in quasicrystals. Int. J. Fract. 166, 105–120 (2010)

    Article  MathSciNet  Google Scholar 

  20. Radi, E., Mariano, P.M.: Dynamic steady-state crack propagation in quasicrystals. Math. Meth. Appl. Sci. 34, 1–23 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Radi, E., Mariano, P.M.: Steady-state crack propagation of dislocations in quasicrystals. Proc. Roy. Soc. A 467, 3490–3508 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tupholme, G.E.: Row of shear cracks moving in one-dimensional hexagonal quasicrystalline materials. Eng. Fract. Mech. 134, 451–458 (2015)

    Article  Google Scholar 

  23. Ding, D.H., Yang, W., Hu, C., Wang, R.: Generalized elasticity theory of quasicrystals. Phys. Rev. B 48, 7003–7010 (1993)

    Article  Google Scholar 

  24. Li, X.Y., Li, P.D.: Three-dimensional thermo-elastic general solutions of one-dimensional hexagonal quasi-crystal and fundamental solutions. Phys. Lett. A 376, 2004–2009 (2012)

    Article  MATH  Google Scholar 

  25. Erdogan, F., Gupta, G.D.: On the numerical solution of singular integral equations. Q. Appl. Math. 29, 525–539 (1972)

    MathSciNet  MATH  Google Scholar 

  26. Yang, L.Z., Gao, Y., Pan, E., Waksmanski, N.: An exact closed-form solution for a multilayered one-dimensional orthorhombic quasicrystal plate. Acta Mech. 226, 3611–3621 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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  1. College of Civil Engineering, Hebei University of Engineering, Handan, 056038, Hebei, People’s Republic of China

    Y. S. Li

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  1. Y. S. Li
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Li, Y.S. Multiple collinear Griffith cracks in a one-dimensional hexagonal quasicrystalline layer. Acta Mech 227, 3671–3686 (2016). https://doi.org/10.1007/s00707-016-1688-0

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  • DOI: https://doi.org/10.1007/s00707-016-1688-0

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