Abstract
The plane problems of an elliptic hole and a crack in three-dimensional quasicrystals subject to far-field loadings are studied. The generalized Stroh formalism is adopted here, and the explicit solutions for the coupled fields are obtained in the closed form. When the elliptic hole reduces to a crack, the analytical expressions for both the entire fields and the asymptotic fields near the crack tip are determined. The crack theory of quasicrystals, including the determination of the field intensity factors, crack opening displacements, crack tip energy release rates and so on, is a prerequisite. Applying Betti’s theorem of reciprocity, the weight functions for a quasicrystal body with a crack are derived. The weight functions provide a means of calculating the intensity factors for the crack when both phonon and phason point forces are imposed at arbitrary locations.
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Athanasiou N.S., Politis C., Spirlet J.C., Baskoutas S., Kapaklis V.: The significance of valence electron concentration on the formation mechanism of some ternary aluminum-based quasicrystals. Int. J. Mod. Phys. B 16(31), 4665–4683 (2002)
Barnett D.M., Lothe J.: Line force loadings on anisotropic half-spaces and wedges. Phys. Norv. 8, 13–22 (1975)
Bueckner H.F.: Novel principle for the computation of stress intensity factors. Z. Angew. Math. Mech. 50, 529–546 (1970)
Dai M.X., Urban K.: Twins in icosahedral Al-Cu-Fe. Philos. Mag. Lett. 67(2), 67–71 (1993)
Ding D.H., Yang W.G., Hu C.Z., Wang R.H.: Generalized elasticity theory of quasicrystals. Phys. Rev. B 48(10), 7003–7010 (1993)
Ebert P., Feuerbacher M., Tamura N., Wollgarten M., Urban K.: Evidence for a cluster-based structure of AlPdMn single quasicrystals. Phys. Rev. Lett. 77(18), 3827–3830 (1996)
Fan T.Y., Mai Y.W.: Elasticity theory, fracture mechanics, and some relevant thermal properties of quasi-crystalline materials. Appl. Mech. Rev. 57, 325–343 (2004)
Gao Y., Ricoeur A.: Three-dimensional analysis of a spheroidal inclusion in a two-dimensional quasicrystal body. Philos. Mag. 92(34), 4334–4353 (2012)
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)
Gao Y., Zhao B.S.: General solutions of three-dimensional problems for two-dimensional quasicrystals. Appl. Math. Model. 33(8), 3382–3391 (2009)
Gao Y., Zhao Y.T., Zhao B.S.: Boundary value problems of holomorphic vector functions in 1D QCs. Phys. B 394(1), 56–61 (2007)
Hu C.Z., Wang R.H., Ding D.H.: Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Rep. Prog. Phys. 63(1), 1–39 (2000)
Hu C.Z., Wang R.H., Yang W.G., Ding D.H.: Point groups and elastic properties of two-dimensional quasicrystals. Acta Crystallogr. Sect. A 52, 251–256 (1996)
Hwu C.: Thermal stresses in an anisotropic plate disturbed by an insulated elliptic hole or crack. ASME J. Appl. Mech. 57(4), 916–922 (1990)
Letoublon, A., De Boissieu,M., Boudard,M., Mancini, L.,Gastaldi, J., Hennion, B., Caudron, R., Bellissent, R.: Phason elastic constants of the icosahedral Al-Pd-Mn phase derived from diffuse scattering measurements.Philos. Mag. Lett. 81(4), 273–283 (2001)
Levine D., Lubensky T.C., Ostlund S., Ramaswamy S., Steinhardt P.J., Toner J.: Elasticity and dislocations in pentagonal and icosahedral quasicrystals. Phys. Rev. Lett. 54(14), 1520–1523 (1985)
Levine D., Steinhardt P.J.: Quasi-crystals: a new class of ordered structure. Phys. Rev. Lett. 53(26), 2477–2480 (1984)
Levine D., Steinhardt P.J.: Quasicrystals. 1. Definition and structure. Phys. Rev. B 34(2), 596–616 (1986)
Li X.F., Fan T.Y.: New method for solving elasticity problems of some planar quasicrystals and solutions. Chin. Phys. Lett. 15, 278–280 (1998)
Li X.F., Fan T.Y., Sun Y.F.: A decagonal quasicrystal with a Griffith crack. Philos. Mag. A 79, 1942–1953 (1999)
Lubensky T.C., Ramaswamy S., Joner J.: Hydrodynamics of icosahedral quasicrystals. Phys. Rev. B 32(11), 7444–7452 (1985)
Ma L.F., Chen Y.H.: Weight functions for interface cracks in dissimilar anisotropic piezoelectric materials. Int. J. Fract. 110(3), 263–279 (2001)
McMeeking R., Ricoeur A.: The weight function for cracks in piezoelectrics. Int. J. Solids Struct. 40(22), 6143–6162 (2003)
Muskhelishvili N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen (1953)
Pak Y.E.: Crack extension force in a piezoelectric material. ASME J. Appl. Mech. 57, 647–653 (1990)
Park J.Y., Ogletree D.F., Salmeron M., Ribeiro R.A., Canfield P.C., Jenks C.J., Thiel P.A.: High frictional anisotropy of periodic and aperiodic directions on a quasicrystal surface. Science 309(5739), 1354–1356 (2005)
Park J.Y., Sacha G.M., Enachescu M., Ogletree D.F., Ribeiro R.A., Canfield P.C., Jenks C.J., Thiel P.A., Saenz J.J., Salmeron M.: Sensing dipole fields at atomic steps with combined scanning tunneling and force microscopy. Phys. Rev. Lett. 95(13), 136802 (2005)
Peng Y.Z., Fan T.Y.: Perturbation method solving elastic problems of icosahedral quasicrystals containing a circular crack. Chin. Phys. 9, 764–766 (2000)
Rice J.R.: Some remarks on elastic crack-tip stress fields. Int. J. Solids Struct. 8, 751–758 (1972)
Rice, J.R.: Weight function theory for three-dimensional elastic crack analysis. In: Wei, R.P., Gangloff, R.P. (eds.) Fracture Mechanics: Perspectives and Directions (Twentieth Symposium). American Society for Testing and Materials, Philadelphia, pp. 29–57 (1989)
Shechtman D., Blech I., Gratias D., Cahn J.W.: Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53(20), 1951–1953 (1984)
Socolar J.E.S.: Simple octagonal and dodecagonal quasicrystals. Phys. Rev. B 39(15), 10519–10551 (1989)
Stadnik Z.: Physical Properties of Quasicrystals, vol. 126. Springer, Berlin (1999)
Stroh A.N.: Dislocations and cracks in anisotropic elasticity. Philos. Mag. 3, 625–646 (1958)
Stroh A.N.: Steady state problems in anisotropic elasticity. J. Math. Phys. 41, 77–103 (1962)
Suo Z., Kuo C.M., Barnett D.M., Willis J.R.: Fracture mechanics for piezoelectric ceramics. J. Mech. Phys. Solids 40(4), 739–765 (1992)
Tanaka K., Mitarai Y., Koiwa M.: Elastic constants of Al-based icosahedral quasicrystals. Philos. Mag. A 73(6), 1715–1723 (1996)
Ting T.C.T.: Anisotropic Elasticity: Theory and Applications. Oxford University Press, Oxford (1996)
Ting T.C.T.: Recent developments in anisotropic elasticity. Int. J. Solids Struct. 37(1–2), 401–409 (2000)
Wollgarten M., Beyss M., Urban K., Liebertz H., Koster U.: Direct evidence for plastic deformation of quasicrystals by means of a dislocation mechanism. Phys. Rev. Lett. 71(4), 549–552 (1993)
Zhou W.M., Fan T.Y.: Plane elasticity problem of two-dimensional octagonal quasicrystals and crack problem. Chin. Phys. 10, 743–747 (2001)
Zhu A.Y., Fan T.Y.: Elastic analysis of a mode II crack in an icosahedral quasicrystal. Chin. Phys. 16(4), 1111–1118 (2007)
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Gao, Y., Ricoeur, A., Zhang, LL. et al. Crack solutions and weight functions for plane problems in three-dimensional quasicrystals. Arch Appl Mech 84, 1103–1115 (2014). https://doi.org/10.1007/s00419-014-0868-4
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DOI: https://doi.org/10.1007/s00419-014-0868-4