In this paper, a central crack in a one-dimensional (1D) orthorhombic quasicrystal strip under anti-plane loadings has been studied. Fourier transform is applied to reduce the mixed boundary value problem of the crack to solving a system of simultaneous singular integral equations. Asymptotic expressions of the phonon and the phason fields near the crack tips have been obtained. The crack-tip singularities of the anti-plane crack have been investigated, and the intensity factors of the stresses in the phonon and the phason fields are derived explicitly. The stress intensity factors (SIFs) and the hoop stress intensity factors (HSIFs) have been determined to investigate the effect of the geometric size and the crack skewing phenomenon. This crack skewing phenomenon is different from the mode-III crack problems in isotropic elastic materials and transversely isotropic elastic materials where the crack propagates along the original crack plane.
ShechtmanDBlechIGratiasD, et al. Metallic phase with long-range orientational order and no translational symmetry. Phys Rev Lett1984; 53: 1951–1953.
2.
JanotC.Quasicrystals: a primer. Oxford: Clarendon Press; Oxford University Press, 1993.
3.
FanTY.Mathematical theory of elasticity of quasicrystals and its applications. Beijing, China: Science Press, 2016.
4.
DingDHYangWGHuCZ, et al. Generalized elasticity theory of quasicrystals. Phys Rev B1993; 48: 7003–7009.
5.
BakP.Phenomenological theory of icosahedral incommensurate (quasiperiodic) order in Mn-Al alloys. Phys Rev Lett1985; 54: 1517–1519.
6.
LevineDLubenskyTCOstlundS, et al. Elasticity and dislocations in pentagonal and icosahedral quasicrystals. Phys Rev Lett1985; 54: 1520–1523.
7.
ChenWQMaYLDingHJ.On three-dimensional elastic problems of one-dimensional hexagonal quasicrystal bodies. Mech Res Commun2004; 31: 633–641.
8.
WangX.The general solution of one-dimensional hexagonal quasicrystal. Mech Res Commun2006; 33: 576–580.
9.
GaoYRicoeurAZhangLL.Plane problems of cubic quasicrystal media with an elliptic hole or a crack. Phys Lett A2011; 375: 2775–2781.
10.
LiXFFanTYSunYE.A decagonal quasicrystal with a Griffith crack. Philos Mag A1999; 79: 1943–1952.
11.
ShiWC.Conservation integrals of any quasicrystal and application. Int J Fract2007; 144: 61–64.
12.
WangXPanE.Analytical solutions for some defect problems in 1D hexagonal and 2D octagonal quasicrystals. Pramana: J Phys2008; 70: 911–933.
13.
GuoJHYuJXingYM.Anti-plane analysis on a finite crack in a one-dimensional hexagonal quasicrystal strip. Mech Res Commun2013; 52: 40–45.
14.
SladekJSladekVAtluriSN.Path-independent integral in fracture mechanics of quasicrystals. Eng Fract Mech2015; 140: 61–71.
15.
DangHYLvSYFanCY, et al. Analysis of anti-plane interface cracks in one-dimensional hexagonal quasicrystal coating. Appl Math Model2020; 81: 641–652.
16.
HuKQGaoCFZhongZ, et al. Interaction of collinear interface cracks between dissimilar one-dimensional hexagonal piezoelectric quasicrystals. Z Angew Math Mech2021; 101: e202000360.
17.
ZhangLLZhangYMGaoY.General solutions of plane elasticity of one-dimensional orthorhombic quasicrystals with piezoelectric effect. Phys Lett A2014; 378: 2768–2776.
18.
YangLZGaoYPanE, et al. An exact closed-form solution for a multilayered one-dimensional orthorhombic quasicrystal plate. Acta Mech2015; 226: 3611–3621.
19.
SladekJSladekVPanE.Bending analysis of 1D orthorhombic quasicrystal plates. Int J Solids Struct2013; 50: 3975–3983.
20.
LiuG-TFanT-YGuoR-P.Governing equations and general solutions of plane elasticity of one-dimensional quasicrystals. Int J Solids Struct2004; 41: 3949–3959.
21.
SneddonIN.Fourier transforms. New York: McGraw-Hill, 1952.
22.
ErdoganF.Complex function technique. In: EringenAC (ed.) Continuum physics, vol. 2. New York: Academic Press, 1975, pp. 523–603.
23.
TheocarisPSLoakimidisNI.Numerical integration methods for the solution of singular integral equations. Q Appl Math1977; 35: 173–183.
24.
HuKQChenZT.Size effect on crack kinking in a piezoelectric strip under impact loading. Mech Mater2013; 61: 60–72.
25.
FabrikantVI.Computation of infinite integrals involving three Bessel functions by introduction of new formalism. Z Angew Math Mech2003; 83: 363–373.
26.
LiXFLeeKY.Three-dimensional electroelastic analysis of a piezoelectric material with a penny-shaped dielectric crack. J Appl Mech2004; 71: 866–878.
27.
HuKQChenZT.Boundary effect on crack kinking in a piezoelectric strip with a central crack. Theor Appl Fract Mech2016; 81: 11–24.
28.
FanTY.Mathematical theory and methods of mechanics of quasicrystalline materials. Engineering2013; 5: 407–448.
29.
ErdoganFGuptaGD.Stresses near a flat inclusion in bonded dissimilar materials. Int J Solids Struct1972; 8: 533–547.
30.
GradshteynISRyzhikIM.Tables of integrals, series and products. New York: Academic Press, 1965.
