Advertisement

Plane Analysis for an Inclusion in 1D Hexagonal Quasicrystal Using the Hypersingular Integral Equation Method

  • Fei Lou
  • Ting Cao
  • Taiyan QinEmail author
  • Chunhui Xu
Original Article
  • 14 Downloads

Abstract

A model of a thin elastic inclusion embedded in an infinite 1D hexagonal quasicrystal is discussed. The atomic arrangements of the matrix and the inclusion are both periodic along the \(x_{1}\)-direction and quasiperiodic along the \(x_{2}\)-direction in the \(ox_{1}x_{2}\)-coordinate system. Using the hypersingular integral equation method, the inclusion problem is reduced to solving a set of hypersingular integral equations. Based on the exact analytical solution of the singular phonon and phason stresses near the inclusion front, a numerical method of the hypersingular integral equation is proposed using the finite-part integral method. Finally, the numerical solutions for the phonon and phason stress intensity factors of some examples are given.

Keywords

Elastic inclusion 1D hexagonal quasicrystal Hypersingular integral equations 

Notes

Acknowledgements

The authors would like to express their special thanks to the National Natural Science Foundation of China (Project No. 11172320 and No. 11272341).

References

  1. 1.
    Shechtman D, Blech I, Gratias D, et al. Metallic phase with long-range orientational order and no translational symmetry. Phys Rev Lett. 1984;53:1951–3.CrossRefGoogle Scholar
  2. 2.
    Levine D, Joseph SP. Quasicrystals: a new class of ordered structures. Phys Rev Lett. 1984;53:2477–80.CrossRefGoogle Scholar
  3. 3.
    Yang W, Ding DH, Wang RH, et al. Thermodynamics of equilibrium properties of quasicrystals. Z Phys B Condens Matter. 1997;100:447–54.CrossRefGoogle Scholar
  4. 4.
    Biggs BD, Li Y, Poon SJ. Electronic properties of icosahedral, approximant, and amorphous phases of an Al–Cu–Fe alloy. Phys Rev B. 1991;43:8747–50.CrossRefGoogle Scholar
  5. 5.
    Pierce FS, Guo Q, Poon SJ. Enhanced insulatorlike electron transport behavior of thermally tuned quasicrystalline states of Al–Pd–Re alloys. Phys Rev Lett. 1994;73:2220.CrossRefGoogle Scholar
  6. 6.
    Bak P. Phenomenological theory of icosahedral incommensurate (quasiperiodic) order in Mn–Al alloys. Phys Rev Lett. 1985;54:1517.CrossRefGoogle Scholar
  7. 7.
    Levine D, Lubensky TC, Ostlund S, et al. Elasticity and dislocations in pentagonal and icosahedral quasicrystals. Phys Rev Lett. 1985;54:1520.CrossRefGoogle Scholar
  8. 8.
    Peng Y, Fan T. Elastic theory of 1D-quasiperiodic stacking of 2D crystals. J Phys Condens Matter. 2000;12:9381–7.CrossRefGoogle Scholar
  9. 9.
    De P, Pelcovits RA. Disclinations in pentagonal quasicrystals. Phys Rev B. 1987;36:9304–7.CrossRefGoogle Scholar
  10. 10.
    Ding DH, Wang RH, Yang WG, et al. General expressions for the elastic displacement fields induced by dislocations in quasicrystals. J Phys Condens Matter. 1995;7:5423.CrossRefGoogle Scholar
  11. 11.
    Fan TY, Guo LH. The final governing equation and fundamental solution of plane elasticity of icosahedral quasicrystals. Phys Lett A. 2005;341:235–9.CrossRefGoogle Scholar
  12. 12.
    Li LH, Fan TY. Final governing equation of plane elasticity of icosahedral quasicrystals and general solution based on stress potential function. Chin Phys Lett. 2006;23:2519–21.CrossRefGoogle Scholar
  13. 13.
    Eshelby JD. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond. 1957;241:376–96.MathSciNetzbMATHGoogle Scholar
  14. 14.
    Muskhelishvili NI. Some basic problems of the mathematical theory of elasticity. Math Gaz. 1953;48:351.Google Scholar
  15. 15.
    Wang ZY, Zhang HT, Chou YT. Characteristics of the elastic field of a rigid line inhomogeneity. J Appl Mech. 1985;52:729–829.Google Scholar
  16. 16.
    Atkinson C. Some ribbon-like inclusion problems. Int J Eng Sci. 1973;11:243–66.CrossRefGoogle Scholar
  17. 17.
    Qin TY, Tang RJ. Finite-part integral and boundary element method to solve inclusion problems. Acta Mater Compos Sin. 1996;13:65–70.Google Scholar
  18. 18.
    Tao FM, Zhang MH, Tang RJ. The interaction problem between the elastic line inclusions. Appl Math Mech. 2002;23:371–9.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Chen TY. The rotation of a rigid ellipsoidal inclusion embedded in an anisotropic piezoelectric medium. Int J Solids Struct. 1993;30:1983–95.CrossRefGoogle Scholar
  20. 20.
    Deng W, Meguid SA. Analysis of conducting rigid inclusion at the interface of two dissimilar piezoelectric materials. J Appl Mech. 1998;65:76–84.CrossRefGoogle Scholar
  21. 21.
    Pan E. Eshelby problem of polygonal inclusions in anisotropic piezoelectric full- and half-planes. J Mech Phys Solids. 2004;52:567–89.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ru CQ. Eshelby inclusion of arbitrary shape in an anisotropic plane or half-plane. Acta Mech. 2003;160:219–34.CrossRefGoogle Scholar
  23. 23.
    Li G, Wang BL, Han JC. Exact solution for elliptical inclusion in magnetoelectroelastic materials. Int J Solids Struct. 2010;47:419–26.CrossRefGoogle Scholar
  24. 24.
    Wang YZ. Interfacial line inclusion between two dissimilar thermo-electro-magneto-elastic solids. Acta Mech. 2015;226:2861–72.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Guo XP, Chen JF, Yu HL, et al. A study on the microstructure and tribological behavior of cold-sprayed metal matrix composites reinforced by particulate quasicrystal. Surf Coat Technol. 2015;268:94–8.CrossRefGoogle Scholar
  26. 26.
    Sakly A, Kenzari S, Bonina D, et al. A novel quasicrystal-resin composite for stereolithography. Mater Design (1980–2015). 2014;56:280–5.CrossRefGoogle Scholar
  27. 27.
    Wang X. Eshelby’s problem of an inclusion of arbitrary shape in a decagonal quasicrystalline plane or half-plane. Int J Eng Sci. 2004;42:1911–30.MathSciNetCrossRefGoogle Scholar
  28. 28.
    Shi W. Collinear periodic cracks and/or rigid line inclusions ofantiplane sliding mode in one-dimensional hexagonal quasicrystal. Appl Math Comput. 2009;215:1062–7.MathSciNetzbMATHGoogle Scholar
  29. 29.
    Wang X, Schiavone P. Decagonal quasicrystalline elliptical inclusions under thermomechanical loading. Acta Mech Solida Sin. 2014;27:518–30.CrossRefGoogle Scholar
  30. 30.
    Guo JH, Zhang ZY, Xing YM. Antiplane analysis for an elliptical inclusion in 1D hexagonal piezoelectric quasicrystal composites. Philos Mag. 2016;96:349–69.CrossRefGoogle Scholar
  31. 31.
    Guo JH, Yu J, Si RL. A semi-inverse method of a Griffith crack in one-dimensional hexagonal quasicrystals. Appl Math Comput. 2013;219:7445–9.MathSciNetzbMATHGoogle Scholar
  32. 32.
    Guo JH, Pan E. Three-phase cylinder model of one-dimensional hexagonal piezoelectric quasi-crystal composites. J Appl Mech Trans ASME. 2016;83:081007.CrossRefGoogle Scholar
  33. 33.
    Wang Y, Guo JH. Effective electroelastic constants for three-phase confocal elliptical cylinder model in piezoelectric quasicrystal composites. Appl Math Mech Engl Ed. 2018;39:797–812.MathSciNetCrossRefGoogle Scholar
  34. 34.
    Ioakimidis NI. Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity. Acta Mech. 1982;45:31–47.MathSciNetCrossRefGoogle Scholar
  35. 35.
    Kaya AC, Erdogan F. On the solution of integral equations with strongly singular kernels. Q Appl Math. 1987;45:105–22.MathSciNetCrossRefGoogle Scholar
  36. 36.
    Ding DH, Yang W, Hu C, et al. Generalized elasticity theory of quasicrystals. Phys Rev B Condens Matter. 1993;48:7003–10.CrossRefGoogle Scholar
  37. 37.
    Ding HJ, Wang GQ, Chen WQ. Fundamental solutions for plane problem of piezoelectric materials. Sci China Ser E Technol Sci. 1997;40:331–6.CrossRefGoogle Scholar
  38. 38.
    Erdogan F. Mixed boundary value problems in mechanics of materials. Multiscale Funct Graded Mater. 2008;973:772–83.Google Scholar
  39. 39.
    Fan CY, Yuan YP, Pan YB, et al. Analysis of cracks in one-dimensional hexagonal quasicrystals with the heat effect. Int J Solids Struct. 2017;120:146–56.CrossRefGoogle Scholar
  40. 40.
    Lee JS, Jiang LZ. Exact electroelastic analysts of piezoelectric laminae via state space approach. Int J Solids Struct. 1996;33:977–90.CrossRefGoogle Scholar
  41. 41.
    Li XY, Li PD, Kang GZ. Crack tip plasticity of a thermally loaded penny-shaped crack in an infinite space of 1D QC. Acta Mech Solida Sin. 2015;28:471–83.CrossRefGoogle Scholar
  42. 42.
    Zhang LL, Wu D, Xu WS, et al. Green’s functions of one-dimensional quasicrystal bi-material with piezoelectric effect. Phys Lett A. 2016;380:3222–8.MathSciNetCrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics 2019

Authors and Affiliations

  1. 1.College of ScienceChina Agricultural UniversityBeijingChina
  2. 2.College of EngineeringChina Agricultural UniversityBeijingChina

Personalised recommendations