Abstract
The purpose of this paper is to consider the interaction between many parallel dislocations and a wedge-shaped crack and their collective response to the external applied generalized stress in one-dimensional hexagonal piezoelectric quasicrystal, employing the complex variable function theory and the conformal transformation method; the problem for the crack is reduced to the solution of singular integral equations, which can be further reduced to solving Riemann–Hilbert boundary value problems. The analytical solutions of the generalized stress field are obtained. The dislocations are subjected to the phonon field line force, phason field line force, and line charge at the core. The positions of the dislocations are arbitrary, but the dislocation distribution is additive. The dislocation is not only subjected to the external stress and the internal stress generated by the crack, but also to the force exerted on it by other dislocations. The closed-form solutions are obtained for field intensity factors and the image force on a screw dislocation in the presence of a wedge-shaped crack and a collection of other dislocations. Numerical examples are provided to show the effects of wedge angle, dislocation position, dislocation distribution containing symmetric configurations and dislocation quantities on the field intensity factors, energy release rate, and image force acting on the dislocation. The principal new physical results obtained here are (1) the phonon stress, phason stress, and electric displacement singularity occur at the crack tip and dislocations cores, (2) the increasing number of dislocations always accelerates the crack propagation, (3) the effect of wedge angle on crack propagation is related to the distribution of dislocations, and (4) the results of the image force on the dislocation indicate that the dislocations can either be attracted or rejected and reach stable positions eventually.
Highlights
As a new structure of solid, quasicrystal (QC) was discovered by Shechtman et al [1] in 1984
3.1. e Electroelastic field of Parallel Screw Dislocations. e physical problem is shown in Figure 1. ere are n parallel screw dislocations located at zi(ri, θi)(i 1, 2, ..., n) with the Burgers vectors (0, 0, b(3i), b(⊥i), b(φi)) in the vicinity of a wedge-shaped crack in the 1D hexagonal QC piezoelectric material which is under the far-field uniform loading (σ∞ 32, H∞ 32, E∞ 2 ). e dislocations are subjected to the phonon field line force p, phason field line force q, and line charge v at the core. e wedge-shaped crack is infinitely long along the negative real axis, and α denotes the wedge angle
The field intensity factors, energy release rate, and the image force acting on the dislocation are expressed
Summary
As a new structure of solid, quasicrystal (QC) was discovered by Shechtman et al [1] in 1984. Chen et al [31] derived the closed solution of the electroelastic field for a screw dislocation situated near the tip two-bonded wedge-shaped dissimilar piezoelectric materials. Li and Zhao [33] coped with the problem of the interaction of a screw dislocation with the interface and wedge-shaped cracks in a one-dimensional hexagonal piezoelectric QC biomaterial using the conformal mapping method in conjunction with the image principle. Through the singular integral technique, we consider the antiplane problem of the interaction between many parallel screw dislocations and a wedge-shaped crack under remote uniform loadings, and the dislocations are subjected to the phonon field line force, phason field line force, and line charge at the core.
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