Fundamental solution for extended dislocation in one‐dimensional piezoelectric quasicrystal and application to fracture analysis

Fan, CuiYing; Chen, Shuai; Xu, Guoyan; Zhang, Qiaoyun

doi:10.1002/zamm.201800232

Cited by 7 publications

(4 citation statements)

References 32 publications

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“…For a 1D hexagonal piezoelectric quasicrystal, the anti-plane phonon field displacement u z and phason field displacement w z are coupled with the electric fields E x and E y in the plane and are irrelevant to the vertical co-ordinate z, i.e., u z = u z (x, y), w = w(x, y), E x = E x (x, y), and E y = E y (x, y). The basic equation is as follows [8,28,30].…”

Section: Basic Equationsmentioning

confidence: 99%

“…Li and Liu [29] employed the Stroh formula to analyze the electroelasticity of icosahedral quasicrystals with straight dislocations. Fan et al [30] deduced a basic solution for extended dislocations in 1D hexagonal piezoelectric quasicrystals. Lou et al [31] studied a thin elastic inclusion in infinite 1D hexagonal quasicrystals using a hypersingular integral equation.…”

Section: Introductionmentioning

confidence: 99%

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Analytical Solution of the Interference between Elliptical Inclusion and Screw Dislocation in One-Dimensional Hexagonal Piezoelectric Quasicrystal

Zhang,

Li,

Ding

2023

Crystals

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This study examines the interference problem between screw dislocation and elliptical inclusion in one-dimensional hexagonal piezoelectric quasicrystals. The general solutions are obtained using the complex variable function method and the conformal transformation technique. When the screw dislocation is located outside or inside the elliptical inclusion, the perturbation method and Laurent series expansion are employed to derive explicit analytical expressions for the complex potentials in the elliptical inclusion and the matrix, respectively. Considering four types of far-field force and electric loading conditions, analytical solutions for various specific cases are obtained by using matrix operations. Expressions for the phonon field stress, phason field stress, and electric displacement are given for special cases, including the absence of a dislocation, the presence of an elliptical hole, and the interference between a screw dislocation and circular inclusion, as well as the case of a circular hole. The design and analysis of quasicrystal inclusion structures can benefit from the results of this work.

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Section: Basic Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analytical Solution of the Interference between Elliptical Inclusion and Screw Dislocation in One-Dimensional Hexagonal Piezoelectric Quasicrystal

Zhang,

Li,

Ding

2023

Crystals

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“…For 2D deformations in the x 1 − x 2 plane, the phonon and phason displacements depend only on the x 1 and x 2 coordinates. Based on the Stroh formalism for a 1D hexagonal piezoelectric QC [24], the phonon and phason displacements u = ( u 1 u 2 u 3 w 2 ) T and the phonon and phason stress function vector Φ can be expressed as…”

Section: Basic Equations and Stroh Formalismmentioning

confidence: 99%

Fundamental solutions and analysis of an interfacial crack in a one-dimensional hexagonal quasicrystal bi-material

Fan

Chen

Zhang

et al. 2020

Mathematics and Mechanics of Solids

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Using the Stroh formalism, Green’s functions are obtained for phonon and phason dislocations and opening displacements on the interface of a one-dimensional hexagonal quasicrystal bi-material. The integro-differential equations governing the interfacial crack are then established, and the singularities of the phonon and phason displacements at the crack tip on the interface are analyzed. To eliminate the oscillating singularities, we represent the delta function in terms of the Gaussian distribution function in the Green’s functions and the integro-differential equations, which helps reduce these equations to the standard integral equations. Finally, a boundary element numerical approach is also proposed to solve the integral equation for the crack opening displacements, the asymptotic expressions of the extended intensity factors, and the energy release rate in terms of the crack opening displacements near the crack tip. In numerical examples, the effect of the Gaussian parameter on the numerical results is discussed, COMSOL software is used to validate the analytical solution, and the influence of the different phonon and phason loadings on the interfacial crack behaviors is further investigated.

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“…Due to these interesting properties, QCs can be used as sensors, actuators, coatings, thermoelectric converters and so on [5]. As such, the investigation on QCs' functions and behaviours has become an essential issue in the field of condensed matter physics, and many topics have been investigated, including defect problems [6][7][8][9][10][11], contact issues [12,13] and static as well as vibration analyses of layered QC structure [14][15][16]. In particular, Zhang et al [17] investigated a spheroidal inclusion embedded within an infinite matrix of a one-dimensional (1D) piezoelectric QC and obtained the explicit expressions.…”

Section: Introductionmentioning

confidence: 99%

Axisymmetric bending analysis of functionally graded one-dimensional hexagonal piezoelectric quasi-crystal circular plate

Qin

et al. 2020

Proc. R. Soc. A.

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Within a framework of the state space method, an axisymmetric solution for functionally graded one-dimensional hexagonal piezoelectric quasi-crystal circular plate is presented in this paper. Applying the finite Hankel transform onto the state space vector, an ordinary differential equation with constant coefficients is obtained for the circular plate provided that the free boundary terms are zero and an exponential function distribution of material properties is assumed. The ordinary differential equation is then used to obtain the stress, displacement and electric components in the physical domain of the elastic simply supported circular plate through the use of the propagator matrix method and the inverse Hankel transform. The numerical studies are carried out to show the validity of the present solution and reveal the influence of material inhomogeneity on the axisymmetric bending of the circular plate with different layers and loadings, which provides guidance for the design and manufacture of functionally graded one-dimensional hexagonal piezoelectric quasi-crystal circular plate.

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Fundamental solution for extended dislocation in one‐dimensional piezoelectric quasicrystal and application to fracture analysis

Cited by 7 publications

References 32 publications

Analytical Solution of the Interference between Elliptical Inclusion and Screw Dislocation in One-Dimensional Hexagonal Piezoelectric Quasicrystal

Analytical Solution of the Interference between Elliptical Inclusion and Screw Dislocation in One-Dimensional Hexagonal Piezoelectric Quasicrystal

Fundamental solutions and analysis of an interfacial crack in a one-dimensional hexagonal quasicrystal bi-material

Axisymmetric bending analysis of functionally graded one-dimensional hexagonal piezoelectric quasi-crystal circular plate

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