Antiplane analysis for an elliptical inclusion in 1D hexagonal piezoelectric quasicrystal composites

An elliptical inclusion embedded in an infinite 1D hexagonal piezoelectric quasicrystal matrix is analysed in the framework of linear piezoelasticity of quasicrystals. Using the technique of conformal mapping, the closed-form solutions of the complex potentials, all the field quantities in the matrix and the inclusion are obtained under far-field antiplane mechanical loads of the phonon and phason fields and an inplane electrical load. Several special cases, such as a homogeneous material, a soft and permeable inclusion, an impermeable inclusion, a line inclusion, a rigid inclusion and a crack are reduced by the present results. Some numerical examples are provided to shows the variations of the stresses of the phonon and phason fields and the electric field with the shape of hole/inclusion, the dielectric permittivity and the distance from the hole/inclusion.     

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)     

Isotropic and anisotropic physical properties of quasicrystals

Since quasicrystals have positional and orientational long-range order, they are essentially anisotropic. However, the researches show that some physical properties of quasicrystals are isotropic. On the other hand, quasicrystals have additional phason degrees of freedom which can influence on their physical behaviours. To reveal the quasicrystal anisotropy, we investigate the quasicrystal elasticity and other physical properties, such as thermal expansion, piezoelectric and piezoresistance, for which one must consider the contributions of the phason field. The results indicate that: for the elastic properties, within linear phonon domain all quasicrystals are isotropic, and within nonlinear phonon domain the planar quasicrystals are still isotropic but the icosahedral quasicrystals are anisotropic. Moreover, the nonlinear elastic properties due to the coupling between phonons and phasons may reveal the anisotropic structure of QCs. For the other physical properties all quasicrystals behave like isotropic media except for piezoresistance properties of icosahedral quasicrystals due to the phason field.     

Elastic theory of icosahedral quasicrystals - application to straight dislocations

In quasicrystals, there are not only conventional, but also phason displacement fields and associated Burgers vectors. We have calculated approximate solutions for the elastic fields induced by two-, three- and fivefold straight screw- and edge-dislocations in infinite icosahedral quasicrystals by means of a generalized perturbation method. Starting from the solution for elastic isotropy in phonon and phason spaces, corrections of higher order reflect the two-, three- and fivefold symmetry of the elastic fields surrounding screw dislocations. The fields of special edge dislocations display characteristic symmetries also, which can be seen from the contributions of all orders. Received 21 February 2001 and Received in final form 27 June 2001     

Interaction between infinitely many dislocations and a semi-infinite crack in one-dimensional hexagonal quasicrystal

By means of analytic function theory, the problems of interaction between infinitely many parallel dislocations and a semi-infinite crack in one-dimensional hexagonal quasicrystal are studied. The analytic solutions of stress fields of the interaction between infinitely many parallel dislocations and a semi-infinite crack in one-dimensional hexagonal quasicrystal are obtained. They indicate that the stress concentration occurs at the dislocation source and the tip of the crack, and the value of the stress increases with the number of the dislocations increasing. These results are the development of interaction among the finitely many defects of quasicrystals, which possesses an important reference value for studying the interaction problems of infinitely many defects in fracture mechanics of quasicrystal.     

Lekhnitskii’s formalism of one-dimensional quasicrystals and its application

By generalizing the complex potential approach developed by Lekhnitskii, plane problems of one-dimensional quasicrystals are solved first by using an octet formalism for which there are four pairs of complex roots. The approach uses a representation of stresses and proceeds by integration of the expressions for deformations and application of the anisotropic constitutive law and the compatibility of displacements. To illustrate its utility, the generalized Lekhnitskii??s formalism is used to analyse the coupled phonon and phason fields in an infinite quasicrystal medium containing an elliptic rigid inclusion.     

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.     

Green's functions for infinite bi-material planes of cubic quasicrystals with imperfect interface

The problem of an infinite plane which is composed of two half-planes with different cubic quasicrystal materials subjected to line phonon and phason forces is investigated. By virtue of the general solution of cubic quasicrystals, a series of displacement functions is adopted to obtain Green's functions in the closed form. For the bonding along the bi-material interface three different models account for different coupling conditions.     

Experimental evaluation of phonon–phason coupling in icosahedral quasicrystals

By measuring phonon strain introduced in crystal approximants, the sign and magnitude of the phonon–phason coupling constant have been evaluated for icosahedral quasicrystals of Mg–Ga–Al–Zn and Al–Cu–Fe systems. The evaluated coupling constants are approximately ?0.04μ and 0.004μ (μ?=?shear modulus) for the former and the latter, respectively. They are in good agreement with the results of a previously reported theoretical calculation. Possible effects of phonon–phason coupling on the onset of phasonic elastic instability in icosahedral quasicrystals are discussed.     

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.     

PLANE ELASTICITY PROBLEM OF TWO-DIMENSIONAL OCTAGONAL QUASICRYSTALS AND CRACK PROBLEM

The plane elasticity theory of two-dimensional octagonal quasicrystals is developed in this paper. The plane elasticity problem of quasicrystals is reduced to a single higher-order partial differential equation by introducing a displacement function. As an example, the exact analytic solution of a Mode I Griffith crack in the material is obtained by using the Fourier transform and dual integral equations theory, then the displacement and stress fields, stress intensity factor and strain energy release rate can be calculated. The physical significance of the results relative to the phason and the difference between the mechanical behaviours of the crack problem in crystals and quasicrystals are figured out. These provide important information for studying the deformation and fracture of the new solid phase.     

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.     

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.     

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.     

Elasticity and dislocations in quasicrystals with 18-fold symmetry

Elasticity problems of quasicrystals with 18-fold rotational symmetry are studied. Constitutive equations and governing equations are obtained. For static elasticity problems, the displacement vectors in two phason fields are expressed in terms of two pairs of associated harmonic functions or two analytic functions. For dynamic problems, the displacement vectors can be represented in terms of an auxiliary function satisfying a fourth-order partial differential equation. A general solution of phasons is given by the solution of two diffusion equations. Phason elastic fields induced by a dislocation in a quasicrystal with 18-fold symmetry are determined and exhibit an inverse singularity.     

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.     

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.     

Crack and indentation problems for one-dimensional hexagonal quasicrystals

In this paper we develop a general method to solve elastic three-dimensional problems for one-dimensional hexagonal quasicrystals with point groups 6mm, 62_h2_h, m2_h and 6/mmm, including crack and indentation problems. Exact solutions are obtained by using Fourier series and Hankel transform methods. These results automatically reduce to those in the classical elasticity theory when the phason field is absent. Received 6 October 2000     

Phonon- and phason-dynamics in Al–Cu–Fe quasicrystals

The lattice dynamics of quasicrystals includes local phason jumps as well as phonons. Phason dynamics is important for the understanding of both the structure and atomic motion in quasicrystals, leading to short-ranged atomic motion not involving vacancies in addition to diffusion. We have studied the phason and phonon dynamics of icosahedral i-Al₆₂Cu_25.5Fe_12.5. Quasielastic Mössbauer spectroscopy (QMS) was used to probe the iron phason dynamics. Inelastic nuclear-resonant absorption (INA) of synchrotron radiation and inelastic neutron scattering (INS) were used to study the iron-partial as well as the total vibrational DOS (VDOS). We find from preliminary QMS studies that iron atoms jump on a time scale about two orders of magnitude slower than that found for copper. The EFG shows an abrupt change in slope at ca. 825 K which may be related to a transition from simple (isolated) to more complicated (co-operative) phason jumps. From INA we find that the iron-partial VDOS differs radically from that of the total (neutron-weighted) generalised VDOS measured by INS. Both these properties are related to the specific local environments of Fe and Cu in i-AlCuFe.