Closed-form solutions of an elliptical crack subjected to coupled phonon–phason loadings in two-dimensional hexagonal quasicrystal media

Abstract

An elliptical crack subjected to coupled phonon–phason loadings in a three-dimensional body of two-dimensional hexagonal quasicrystals is analytically investigated. Owing to the existence of the crack, the phonon and phason displacements are discontinuous along the crack face. The phonon and phason displacement discontinuities serve as the unknown variables in the generalized potential function method which are used to derive the boundary integral equations. These boundary integral equations governing Mode I, II, and III crack problems in two-dimensional hexagonal quasicrystals are expressed in integral differential form and hypersingular integral form, respectively. Closed-form exact solutions to the elliptical crack problems are first derived for two-dimensional hexagonal quasicrystals. The corresponding fracture parameters, including displacement discontinuities along the crack face and stress intensity factors, are presented considering all three crack cases of Modes I, II, and III. Analytical solutions for a penny-shaped crack, as a special case of the elliptical problem, are given. The obtained analytical solutions are graphically presented and numerically verified by the extended displacement discontinuities boundary element method.

A Phase Field Approach to Two-Dimensional Quasicrystals with Mixed Mode Cracks

“…These solutions could have applications in 3D contact and crack problems in QCs. Li et al [ 13 , 14 ] derived solutions for elliptical crack and planar crack problems in 2D hexagonal QCs. Li and Shi [ 15 ] employed the method of potential function theory to solve plane defect problems originating from two-dimensional decagonal QCs.…”

Three-dimensional thermo-electro-elastic field in one-dimensional hexagonal piezoelectric quasi-crystal weakened by an elliptical crack

“…Many classical methods in elasticity theory were extended to analyze crack problems in QCs, such as the integral transformation method [6, 7], Stroh formalism [8, 9], the potential theory method [10–13], the displacement discontinuity method [14–18], the dislocation layer method [19, 20], and the weight function method [21]. By using these methods, some classical crack problems, including those for line cracks [8, 14], half-infinite cracks [12, 13], penny-shaped cracks [10–13], elliptical cracks [22], and interface cracks [9, 15, 16, 23], were considered, and the corresponding analytical or semi-analytical solutions were obtained.…”

Analysis solution method for 3D planar crack problems of two-dimensional hexagonal quasicrystals with thermal effects

“…With the help of these general solutions in terms of quasi-harmonic functions [37,39] conjugated with the generalized method of potential theory, some 3D exact analyzes of planar crack in 2D hexagonal QCs were conducted, such as the cases of Model I crack [40] and symmetry temperature loadings [41]. Without considering thermal effects, Li et al [42] took the phonon and phason displacement discontinuities as the unknown variables of generalized potential function method and first derived closed-form exact solutions to the elliptical crack problems for 2D hexagonal QCs. Zhao et al [43] extended boundary integral equation method to investigate 3D planar crack problem for 2D hexagonal QCs.…”

“…Eqs (41). and(42), it is found that Mode II and III field intensity factors are very brief in structure, and the phonon and phason loadings are decoupled in Mode II and III field intensity factors. However, when inserting Eq.…”