Complex variable method for plane elasticity of icosahedral quasicrystals and elliptic notch problem

The complex variable method for the plane elasticity theory of icosahedral quasi-crystals is developed. Based on the general solution obtained previously, complex representations of stress and displacement components of phonon and phason fields in the quasicrystals are given. With the help of conformal transformation, an analytic solution for the elliptic notch problem of the material is presented. The solution of the Griffith crack problem can be observed as a special case of the results. The stress intensity factor and energy release rate of the crack are also obtained.     

Elliptic hole in octagonal quasicrystals

The stress potential function theory for plane elasticity of octagonal quasicrystals is developed. By introducing stress functions, a large number of basic equations involving elasticity of octagonal quasicrystals are reduced to a single partial differential equation. Furthermore, we develop the complex variable function method (Lekhnitskii method) for anisotropic elasticity theory to that for quasicrystals. With the help of conformal transformation, an exact solution for the elliptic hole of quasicrystals is presented. The solution of the Griffith crack problem, as a special case of the results, is obtained. As a consequence, the phonon stress intensity factor is derived analytically.     

Elliptic holes in octagonal quasicrystals

The stress potential function theory for the plane elasticity of octagonal quasicrystals is developed. By introducing stress functions, a large number of basic equations involving the elasticity of octagonal quasicrystals are reduced to a single partial differential equation. Furthermore, we develop the complex variable function method (Lekhnitskii method) for anisotropic elasticity theory to that for quasicrystals. With the help of conformal transformation, an exact solution for the elliptic hole of quasicrystals is presented. The solution of the Griffith crack problem, as a special case of the results, is obtained. As a consequence, the phonon stress intensity factor is derived analytically.     

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.     

Fundamentals in generalized elasticity and dislocation theory of quasicrystals: Green tensor,dislocation key-formulas and dislocation loops

The present work provides fundamental quantities in generalized elasticity and dislocation theory of quasicrystals. In a clear and straightforward manner, the three-dimensional Green tensor of generalized elasticity theory and the extended displacement vector for an arbitrary extended force are derived. Next, in the framework of dislocation theory of quasicrystals, the solutions of the field equations for the extended displacement vector and the extended elastic distortion tensor are given; that is, the generalized Burgers equation for arbitrary sources and the generalized Mura–Willis formula, respectively. Moreover, important quantities of the theory of dislocations as the Eshelby stress tensor, Peach–Koehler force, stress function tensor and the interaction energy are derived for general dislocations. The application to dislocation loops gives rise to the generalized Burgers equation, where the displacement vector can be written as a sum of a line integral plus a purely geometric part. Finally, using the Green tensor, all other dislocation key-formulas for loops, known from the theory of anisotropic elasticity, like the Peach–Koehler stress formula, Mura–Willis equation, Volterra equation, stress function tensor and the interaction energy are derived for quasicrystals.     

GENERALIZATION OF ESHELBY'S METHOD TO THE ANISOTROPIC ELASTICITY THEORY OF DISLOCATIONS IN QUASICRYSTALS

The general expressions of the elastic fields induced by straight dislocations in quasicrystals have been given according to Eshelby's method which was used to treat the anisotropic elasticity of dislocations in crystals. As an example, the elastic displacement vector, the stress tensor and the elastic energy density of a screw dislocation line lying on the quasiperiodic plane of decagonal quasicrystals are calculated.     

GENERALIZATION OF ESHELBY''S METHOD TO THE ANISOTROPIC ELASTICITY THEORY OF DISLOCATIONS IN QUASICRYSTALS

The general expressions of the elastic fields induced by straight dislocations in quasicrystals have been given according to Eshelby's method which was used to treat the anisotropic elasticity of dislocations in crystals. As an example, the elastic displacement vector, the stress tensor and the elastic energy density of a screw dislocation line lying on the quasiperiodic plane of decagonal quasicrystals are calculated.     

Non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane

The non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane was investigated using the generalized Almansi's theorem and the Schmidt method. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the jumps in displacements across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of crack length, the distance between two collinear cracks and the lattice parameter on the stress field, the electric displacement field and the magnetic flux field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress, electric displacement or magnetic flux singularities are present at the crack tips in a magnetoelectroelastic composite material plane. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion.     

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.     

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     

A Dugdale–Barenblatt model for a strip with a semi-infinite crack embedded in decagonal quasicrystals

The present study is to determine the solution of a strip with a semi-infinite crack embedded in decagonal quasicrystals, which transforms a physically and mathematically daunting problem. Then cohesive forces are incorporated into a plastic strip in the elastic body for nonlinear deformation. By superposing the two linear elastic fields, one is evaluated with internal loadings and the other with cohesive forces, the problem is treated in Dugdale-Barenblatt manner. A simple but yet rigorous version of the complex analysis theory is employed here, which involves conformal mapping technique. The analytical approach leads to the establishment of a few equations, which allows the exact calculation of the size of cohesive force zone and the most important physical quantity in crack theory: stress intensity factor. The analytical results of the present study may be used as the basis of fracture theory of decagonal quasicrystals.     

Plastic analysis of the crack problem in two-dimensional decagonal Al-Ni-Co quasicrystalline materials of point group 10,10

The fundamental plastic nature of the quasicrystalline materials remains an open problem due to its essential complicacy. By developing the proposed generalized cohesive force model, the plastic deformation of crack in point group 10, 10 decagonal quasicrystals is analysed strictly and systematically. The crack tip opening displacement （CTOD） and the size of the plastic zone around the crack tip are determined exactly. The quantity of the crack tip opening displacement can be used as a parameter of nonlinear fracture mechanics of quasicrystalline material. In addition, the present work may provide a way for the plastic analysis of quasicrystals.     

Icosahedral quasicrystals solids with an elliptic hole under uniform heat flow

The complex variable method for solving the two-dimensional thermal stress problem of icosahedral quasicrystals is stated. The closed-form solutions for icosahedral quasicrystals containing an elliptical hole subjected to a remote uniform heat flow are obtained. When the hole degenerates into a crack, the explicit solutions for the stress intensity factors is presented.     

On dynamic plane elasticity problems of 2D quasicrystals

In this Letter, the dynamic plane elasticity problems of 2D quasicrystals is considered. By use of the Fourier transform and matrix transformations the system is reduced to uncoupled ordinary differential equations. Fourier images of Green's functions for dynamic plane elasticity problems of 2D dodecagonal, pentagonal and decagonal quasicrystals are obtained explicitly by the suggested method.     

Diffraction of elastic waves by a sub-surface crack (anti-plane motion)

A rigorous theory of the diffraction of SH-waves by a stress-free crack embedded in a semi-infinite elastic medium is presented. The incident time-harmonic SH-wave is taken to be either a uniform plane wave or a cylindrical wave originating from a surface line-source. The resulting boundary-value problem for the unknown jump in the particle displacement across the crack is solved by employing an integral equation approach. The unknown quantity is expanded in a complete sequence of Chebyshev polynomials. By writing the Green function as a Fourier integral, an infinite system of linear, algebraic equations for the expansion coefficients is obtained. Numerical results are presented for the particle displacement at the surface of the half-space, the far field radiation characteristic, the scattering cross-section of the crack and the dynamic stress intensity factor at the crack tips, for a range of geometrical parameters.     

Basic solution of four parallel non-symmetric permeable Mode-III cracks in a piezoelectric/piezomagnetic composite plane

The interaction of four parallel non-symmetric permeable cracks in a piezoelectric/piezomagnetic composite plane subjected to anti-plane shear stress loading was studied by the Schmidt method. The problem was formulated through a Fourier transform into four pairs of dual integral equations, in which unknown variables are jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the relationships among the electric displacement, magnetic flux and stress fields near the crack tips were obtained. The results show that the stress, the electric displacement and the magnetic flux intensity factors at the crack tips depend on the lengths and spacing of cracks. It was also revealed that the crack shielding effect is present in piezoelectric/piezomagnetic composites.     

Diffuse scattering from dodecagonal quasicrystals

General formulae for thermal diffuse scattering from quasicrystals are applied to the case of dodecagonal quasicrystals from corresponding elasticity theory. Contours of constant diffuse scattering intensity are illustrated. Unlike ordinary crystals, shapes of isointensity contours are much more complicated and vary even among the collinear Bragg spots. Diffuse scattering patterns in the plane perpendicular to a given zone axis are associated with corresponding specific elastic constants. Information about elastic constants can be extracted from quantitative analysis of diffuse scattering patterns. Received 7 December 1998 and Received in final form 12 March 1999     

Analytic solutions to a finite width strip with a single edge crack of two-dimensional quasicrystals

In this paper, we investigate the well-known problem of a finite width strip with a single edge crack, which is useful in basic engineering and material science. By extending the configuration to a two-dimensional decagonal quasicrystal, we obtain the analytic solutions of modes I and II using the transcendental function conformal mapping technique. Our calculation results provide an accurate estimate of the stress intensity factors K_I and K_II, which can be expressed in a quite simple form and are essential in the fracture theory of quasicrystals. Meanwhile, we suggest a generalized cohesive force model for the configuration to a two-dimensional decagonal quasicrystal. The results may provide theoretical guidance for the fracture theory of two-dimensional decagonal quasicrystals.     

准晶的弹性,塑性与位错

本文较系统地综述了适合准晶的线弹性理论，准品位错的基本理论和实验鉴定，准晶位错的弹性理论，以及准晶高温塑性形变的微观机制。     

Elastic fields around a nanosized elliptic hole in decagonal quasicrystals

Based on the variational principle, a continuum theory of surface elasticity and new boundary conditions for qua- sicrystals is proposed. The effect of the residual surface stress on a decagonal quasicrystal that is weakened by a nanoscale elliptical hole is considered. The explicit expressions for the hoop stress along the edge of the hole are obtained using the Stroh formalism. The results show that the residual surface stress and the shape of the hole have a significant effect on the elastic state around the hole.