On the interaction between dislocations and cracks in one-dimensional hexagonal quasi-crystals

The solution of elasticity problems involving the interaction between dislocations and cracks plays a fundamental role in many practical and theoretical applications. Although elasticity problems involving dislocation or cracks in quasi-crystals have been investigated in many papers, the analysis is limited to a single defect. This paper investigates the interaction of defects in one-dimensional hexagonal quasi-crystals using the complex variable function method. The interaction force between two parallel dislocations is presented and the analytic solutions of elastic fields of interaction between a dislocation and a crack are obtained. A version of the well-known Peach－Koehler formula in one-dimensional hexagonal quasi-crystals is given.     

Analytical solutions for some defect problems in 1D hexagonal and 2D octagonal quasicrystals

We study some typical defect problems in one-dimensional (1D) hexagonal and two-dimensional (2D) octagonal quasicrystals. The first part of this investigation addresses in detail a uniformly moving screw dislocation in a 1D hexagonal piezoelectric quasicrystal with point group 6mm. A general solution is derived in terms of two functions φ ₁, φ ₂, which satisfy wave equations, and another harmonic function φ ₃. Elementary expressions for the phonon and phason displacements, strains, stresses, electric potential, electric fields and electric displacements induced by the moving screw dislocation are then arrived at by employing the obtained general solution. The derived solution is verified by comparison with existing solutions. Also obtained in this part of the investigation is the total energy of the moving screw dislocation. The second part of this investigation is devoted to the study of the interaction of a straight dislocation with a semi-infinite crack in an octagonal quasicrystal. Here the crack penetrates through the solid along the period direction and the dislocation line is parallel to the period direction. We first derive a general solution in terms of four analytic functions for plane strain problem in octagonal quasicrystals by means of differential operator theory and the complex variable method. All the phonon and phason displacements and stresses can be expressed in terms of the four analytic functions. Then we derive the exact solution for a straight dislocation near a semi-infinite crack in an octagonal quasicrystal, and also present the phonon and phason stress intensity factors induced by the straight dislocation and remote loads.      

一维六方准晶中螺形位错与楔形裂纹的相互作用

采用保角映射方法和扰动技巧,研究了一维六方准晶中螺形位错和半无限楔形裂纹的相互作用. 讨论了位错的位置和楔形角对作用在位错上的力的影响,得到了应力强度因子和作用在 位错上的力的解析解.此外,还详细地讨论了位错对裂纹的影响.当楔形角参数λ=1/2时, 半无限楔形裂纹退化成半无限尖裂纹,相应的结果 可以作为特殊情况而直接得到.     

Complex variable function method for the plane elasticity and the dislocation problem of quasicrystals with point group 10 mm

General complex variable function method for solving plane elasticity and the dislocation problems of quasicrystals with point group 10 mm has been proposed. All the fields variables are expressed by four arbitrary analytic functions. Analytical displacement expressions for the dislocation problem of the quasicrystal is obtained. The interaction between two parallel dislocations is also discussed in detail.     

The interaction between a screw dislocation and a wedge-shaped crack in one-dimensional hexagonal piezoelectric quasicrystals

Based on the fundamental equations of piezoelasticity of quasicrystal material,we investigated the interaction between a screw dislocation and a wedge-shaped crack in the piezoelectricity of one-dimensional hexagonal quasicrystals.Explicit analytical solutions are obtained for stress and electric displacement intensity factors of the crack,as well as the force on dislocation.The derivation is based on the conformal mapping method and the perturbation technique.The influences of the wedge angle and dislocation location on the image force are also discussed.The results obtained in this paper can be fully reduced to some special cases already available or deriving new ones.     

磁电弹性体中多条位错与Griffith裂纹

根据本征方程，研究磁电弹性体中若干平行螺型位错与Griffith裂纹的相互作用.结合Muskhelishvili方法和算子理论，得到磁电弹性体中由位错和裂纹所诱导的应力场、电场和磁场的解析解.数值算例表明：在裂纹的端点及位错点上仍然存在应力的奇异性，离位错点越远处广义力越小，结论与已有的结果相符，证明了结论的正确性.当位错点与裂纹端点距离越近时，裂纹与位错间的应力场越小，并逐渐趋近于零.     

一维六方准晶非周期平面内的平面应变理论

针对一维六方准晶中一种新平面弹性问题, 即非周期平面内的平面弹性问题, 通过引入应力势函数, 建立了一维六方准晶中非周期平面内的平面应变理论. 作为应用, 求解了一维六方准晶中垂直于准周期方向的椭圆孔口问题, 得到了其弹性应力场的解析解. 在极限情形下, 可给出裂纹问题的解. 关键词： 一维六方准晶 椭圆孔口 广义解析函数     

Exact analytic solutions for an elliptic hole with asymmetric collinear cracks in a one-dimensional hexagonal quasi-crystal

Using the complex variable function method and the technique of conformal mapping, the anti-plane shear problem of an elliptic hole with asymmetric colfinear cracks in a one-dimensional hexagonal quasi-crystal is solved, and the exact analytic solutions of the stress intensity factors （SIFs） for mode Ⅲ problem are obtained. Under the limiting conditions, the present results reduce to the Griffith crack and many new results obtained as well, such as the circular hole with asymmetric collinear cracks, the elliptic hole with a straight crack, the mode T crack, the cross crack and so on. As far as the phonon field is concerned, these results, which play an important role in many practical and theoretical applications, are shown to be in good agreement with the classical results.     

Exact solution of two collinear cracks normal to the boundaries of a 1D layered hexagonal piezoelectric quasicrystal

The electroelastic problem related to two collinear cracks of equal length and normal to the boundaries of a one-dimensional hexagonal piezoelectric quasicrystal layer is analysed. By using the finite Fourier transform, a mixed boundary value problem is solved when antiplane mechanical loading and inplane electric loading are applied. The problem is reduce to triple series equations, which are then transformed to a singular integral equation. For uniform remote loading, an exact solution is obtained in closed form, and explicit expressions for the electroelastic field are determined. The intensity factors of the electroelastic field and the energy release rate at the inner and outer crack tips are given and presented graphically.     

Elastic analysis of a mode Ⅱ crack in an icosahedral quasicrystal

Based on the displacement potential functions, the elastic analysis of a mode II crack in an icosahedral quasicrystal is performed by using the Fourier transform and dual integral equation theory. By the solution, the analytic expressions for the displacement field and stress field are obtained. The asymptotic behaviours of the phonon and phason stress fields around the crack tip indicate that the stresses near the crack tip exhibit a square root singularity. The most important physical quantities of fracture theory, crack stress intensity factor and energy release rate, are evaluated in an explicit version.     

In-situ transmission electron microscopy observations and molecular dynamics simulations of dislocation-defect interactions in ion-irradiated copper

An in-situ transmission electron microscopy straining technique has been used to investigate the dynamics of dislocation-defect interactions in ion-irradiated copper and the subsequent formation of defect-free channels. Defect removal frequently required interaction with multiple dislocations, although screw dislocations were more efficient at annihilating defects than edge dislocations were. The defect pinning strength was determined from the dislocation curvature prior to breakaway and exhibited values ranging from 15 to 175 MPa. Pre-existing dislocations percolated through the defect field but did not show long-range motion, indicating that they are not responsible for creating the defect-free channels and have a limited contribution to the total plasticity. Defect-free channels were associated with the movement of many dislocations, which originated from grain boundaries or regions of high stress concentration such as at a crack tip. These experimental results are compared with atomistic simulations of the interaction of partial dislocations with defects in copper and a dispersed-barrier-hardening crystal plasticity model to correlate the observations to bulk mechanical properties.     

Dynamic Effects in a Defective System of Crystal

Works devoted to studying the interaction between dislocations and structural defects in a crystal lattice in the dynamic region of velocities are considered. Analytic expressions for the force of dynamic drag and the dynamic yield stress for a wide range of problems of dislocation dynamics are obtained using the theory of dynamic interaction between structural defects.     

Energy characteristics of edge dislocations in quasicrystals

A model is developed for describing phason defects in quasicrystals in the form of dilation filaments. This model is used to calculate the energy of edge dislocations in quasicrystals including the interaction of this type of dislocation with its “intrinsic” phason defects and with the equilibrium phason defects present in a quasicrystal. It is shown that the contribution of “intrinsic” phason defects to the total energy of an edge dislocation in a quasicrystal is substantial. Fiz. Tverd. Tela (St. Petersburg) 39, 2003–2007 (November 1997)     

The partition function and state equation of the point group 12mm two-dimensional dodecagonal quasicrystal

This study extends the analysis of the vibrational spectrum of the one-dimensional hexagonal quasicrystal to the two-dimensional quasicrystal with the 12mm point group. Several thermodynamic functions and the state equation of the material are shown. Received 18 July 2002 Published online 27 January 2003 RID="a" ID="a"e-mail: litongtzb@263.net     

Dynamic behaviour of the icosahedral Al——Pd——Mn quasicrystal with a Griffith crack

The dynamic response of an icosahedral Al--Pd--Mn quasicrystal with a Griffith crack to impact loading is investigated in this paper. The elastohydrodynamic model for the wave propagation and diffusion together with their interaction is adopted. Numerical results of stress, displacement and dynamic stress intensity factors are obtained by using the finite difference method. The effects of wave propagation, diffusion and phonon--phason coupling on the quasicrystal in the dynamic process are discussed in detail, where the phason dynamics is explored particularly.     

Confinement scaling laws for the conventional reversed-field pinch

Mechanical behavior of the Si(111)/Si(3)N4(0001) interface is studied using million atom molecular dynamics simulations. At a critical value of applied strain parallel to the interface, a crack forms on the silicon nitride surface and moves toward the interface. The crack does not propagate into the silicon substrate; instead, dislocations are emitted when the crack reaches the interface. The dislocation loop propagates in the (1; 1;1) plane of the silicon substrate with a speed of 500 (+/-100) m/s. Time evolution of the dislocation emission and nature of defects is studied.     

Elastic interaction between two parallel dislocations in non-parallel slip planes

The elastic interaction between two parallel dislocations which can glide in non-parallel slip planes is studied under the simplifying assumption that the dislocation glide velocity is proportional to stress. The motion of the two dislocations is represented by a motion of one reference point in a configuration plane. It is concluded that the contribution of the long-range elastic interaction between individual dislocations from different slip systems to work hardening is negligible, compared to the contribution from the formed attractive junctions. Especially, two parallel edge dislocations with mutually perpendicular Burgers vectors can co-exist in minimum energy positions, however, they can be separated by an arbitrarily small external stress.     

Elastic interaction of a dislocation array with a phase boundary

The equilibrium configuration of an array of dislocations in parallel equidistant slip planes under an external shear stress near a welded boundary between two isotropic half-spaces having different elastic constants is computed. For large external stress, the dislocations are arranged into an arc concave when seen from the boundary. It is concluded that such an arc is formed at the tip of a twin or of a martensitic plate near a phase boundary. The tensile stress across the boundary due to an edge dislocation array is discussed in connection with the formation of an interfacial crack.     

AXISYMMETRIC ELASTICITY PROBLEM OF CUBIC QUASICRYSTAL

A method for analyzing the elasticity problem of cubic quasicrystal is developed. The axisymmetric elasticity problem of cubic quasicrystal is reduced to a single higher-order partial differential equation by introducing a displacement function. As an example, the solutions of elastic field of cubic quasicrystal with a penny-shaped crack are obtained, and the stress intensity factor and strain energy release rate are determined.     

Shielding effect of a nano-circular inclusion acting on semi-infinite wedge cracks

The model of a screw dislocation near a semi-infinite wedge crack tip inside a nano-circular inclusion is proposed to investigate the shielding effect of nano inclusions acting on cracks. Utilizing the complex function method, the closed-form solutions of the stress fields in the matrix and the inclusion region are derived. The stress intensity factor, the image force, as well as the critical loads for dislocation emission are discussed in detail. The results show that the nano inclusion not only enhances the shielding effect exerted by the dislocation, but also provides a shielding effect itself. Moreover, dislocations may be trapped in the nano inclusion even if the matrix is softer than the inclusion. This helps the dislocation shield crack, and reduces the dislocation density within the matrix.