Hexagonal Quasicrystals Research Articles

Interaction between infinitely many dislocations and a semi-infinite crack in one-dimensional hexagonal quasicrystal**Project supported by the National Natural Science Foundation of China (Grant Nos. 11462020, 11262017, and 11262012) and the Key Project of Inner Mongolia Normal University, China (Grant No. 2014ZD03).

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.

Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects. Part II: Numerical solutions

• 2D hexagonal quasicrystals planar crack with thermal effects analyzed numerically. • Green's functions are derived for uniform triangular and rectangular elements. • The EDD-BEM is proposed for planar crack analysis in infinite 2D hexagonal QC body. • Elliptical crack with different loadings are investigated and presented graphically. The extended displacement discontinuity (EDD) boundary element method is developed to analyze an arbitrarily shaped planar crack in two-dimensional (2D) hexagonal quasicrystals (QCs) with thermal effects. The EDDs include the phonon and phason displacement discontinuities and the temperature discontinuity on the crack face. Green's functions for uniformly distributed EDDs over triangular and rectangular elements for 2D hexagonal QCs are derived. Employing the proposed EDD boundary element method, a rectangular crack is analyzed to verify the Green's functions by discretizing the crack with rectangular and triangular elements. Furthermore, the elliptical crack problem for 2D hexagonal QCs is investigated. Normal, tangential, and thermal loads are applied on the crack face, and the numerical results are presented graphically.

Analysis of cracks in one-dimensional hexagonal quasicrystals with the heat effect

The extended displacement discontinuity (EDD) method is proposed to analyze cracks in the periodical plane of one-dimensional (1D) hexagonal quasicrystals with the heat effect. Based on the operator theory and the Fourier transform, the fundamental solutions for EDDs are derived, where the EDDs include phonon and phason displacement discontinuities and the temperature discontinuity. The EDD boundary integral equation method is used to analyze the singularities of the near-crack tip fields, and the extended stress intensity factor (ESIF) expressions are obtained in terms of the EDDs across the crack faces. The EDD boundary element method is proposed to calculate the ESIFs of cracks in 1D hexagonal quasicrystals. COMSOL software is used to validate the developed method. The influences of applied mechanical and heat loads on cracks in a finite plate are investigated.

Fracture analysis of one-dimensional hexagonal quasicrystals: Researches of a finite dimension rectangular plate by boundary collocation method

As an important supplement and development to crystallography, the applications about quasicrystal materials have played a core role in many fields, such as manufacturing and the space industry. Due to the sensitivity of quasicrystals to defects, the research on the fracture problem of quasicrystals has attracted a great deal of attention. We present a boundary collocation method to research fracture problems for a finite dimension rectangular one-dimensional hexagonal quasicrystal plate. Because mode I and mode II problems for onedimensional hexagonal quasicrystals are like that for the classical elastic materials, only the anti-plane problem is discussed in this paper. The correctness of the present numerical method is verified through a comparison of the present results and the existing results. And then, the size effects on stress field, stress intensity factor and energy release rate are discussed in detail. The obtained results can provide valuable references for the fracture behavior of quasicrystals.

The interaction between a screw dislocation and a wedge-shaped crack in one-dimensional hexagonal piezoelectric quasicrystals**Project supported by the National Natural Science Foundation of China (Grant Nos. 11262017, 11262012, and 11462020), the Natural Science Foundation of Inner Mongolia Autonomous Region, China (Grant No. 2015MS0129), the Programme of Higher-level Talents of Inner Mongolia Normal University (Grant No. RCPY-2-2012-K-035), and the Key Project

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.

The fracture behavior of two asymmetrical limited permeable cracks emanating from an elliptical hole in one-dimensional hexagonal quasicrystals with piezoelectric effect

By modeling that the surfaces of the cracks and hole are in limited permeable boundary conditions, the anti-plane problem of an elliptical hole with two asymmetrical cracks in one-dimensional (1D) hexagonal quasicrystals (QCs) with piezoelectric effect is investigated by adopting the technique of conformal mapping and the Stroh-type formulism. The analytic solutions of the field intensity factors and energy release rate are presented. With the variation of the hole-size and the crack length, the proposed model can be reduced to some classic crack models. In the absence of the electric load or the phason field, the present results match with the classical results of 1D hexagonal QCs or piezoelectric materials provided in the open literature. Numerical examples are then conducted to reveal the effects of the hole-size, the crack length, the coupling coefficient and applied mechanical/electric loads on the field intensity factors and the energy release rate.

Eshelbian mechanics of novel materials: Quasicrystals

In this work, the so-called Eshelbian or configurational mechanics of quasicrystals is presented. Quasicrystals are considered as a prototype of novel materials. Material balance laws for quasicrystalline materials with dislocations are derived in the framework of generalized incompatible elasticity theory of quasicrystals. Translations, scaling transformations as well as rotations are examined; the latter presents particular interest due to the quasicrystalline structure. This derivation provides important quantities of the Eshelbian mechanics, as the Eshelby stress tensor, the scaling flux vector, the angular momentum tensor, the configurational forces (Peach–Koehler force, Cherepanov force, inhomogeneity force or Eshelby force), the configurational work, and the configurational vector moments for dislocations in quasicrystals. The corresponding [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-integrals for dislocation loops and straight dislocations in quasicrystals are derived and discussed. Moreover, the explicit formulas of the [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-integrals for parallel screw dislocations in one-dimensional hexagonal quasicrystals are obtained. Through this derivation, the physical interpretation of the [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-integrals for dislocations in quasicrystals is revealed and their connection to the Peach–Koehler force, the interaction energy and the rotational vector moment (torque) of dislocations in quasicrystals is established.

Fundamental solutions and analysis of three-dimensional cracks in one-dimensional hexagonal piezoelectric quasicrystals

In this study, three-dimensional cracks in one-dimensional hexagonal piezoelectric quasicrystals are investigated. Based on the general solutions and Hankel transform, the fundamental solutions for unit point extended displacement discontinuities (including the phonon and phason displacement discontinuities and the electric potential discontinuity) are derived. The extended displacement discontinuity boundary integral equations for a planar crack of arbitrary shape in the periodic plane of one-dimensional hexagonal piezoelectric quasicrystals are established in terms of the extended displacement discontinuities. Using the boundary integral equation method, the singularities of near-crack border fields are obtained and the stress and electric displacement intensity factors are derived.

Crack tip plasticity of a thermally loaded penny-shaped crack in an infinite space of 1D QC

The present work is concerned with a penny-shaped Dugdale crack embedded in an infinite space of one-dimensional (1D) hexagonal quasicrystals and subjected to two identical axisymmetric temperature loadings on the upper and lower crack surfaces. Applying Dugdale hypothesis to thermo-elastic results, the extent of the plastic zone at the crack tip is determined. The normal stress outside the plastic zone and crack surface displacement are derived in terms of special functions. For a uniform loading case, the corresponding results are presented by simplifying the preceding results. Numerical calculations are carried out to show the influence of some parameters.

Crack tip plasticity of a half-infinite Dugdale crack embedded in an infinite space of one-dimensional hexagonal quasicrystal

The present paper is devoted to determining the crack tip plasticity of a half-infinite Dugdale crack embedded in an infinite space of one-dimensional hexagonal quasicrystal. A pair of equal but opposite line loadings is assumed to be exerted on the upper and lower crack lips. By applying the Dugdale hypothesis together with the elastic results for a half-infinite crack, the extent of the plastic zone in the crack front is estimated. The normal stress outside the enlarged crack and crack surface displacements are explicitly presented, via the principle of superposition. The validity of the present solutions is discussed analytically by examining the overall equilibrium of the half-space.

Analytic solutions of problem about a circular hole with a straight crack in one-dimensional hexagonal quasicrystals with piezoelectric effects

The anti-plane shear problem about a circular hole with a straight crack in one-dimensional (1D) hexagonal quasicrystals (QCs) with piezoelectric effects is investigated by means of the complex variable function method and using the technique of conformal mapping. The analytic solutions of the stress intensity factors (SIFs) and the electric displacement intensity factors (EDIFs) with electrically impermeable and permeable conditions were obtained, and these solutions have an important theoretical significance for the engineering application of QCs materials. When the circle radius tends to zero, the present results can be reduced to the cases of the Griffith crack. In the absence of the phason field, the obtainable results in this paper agrees well with the results for piezoelectric materials. Numerical analysis is then conducted to discuss the influences of geometric parameters and applied mechanical/electric loads on the field intensity factors and energy release rate.

An antiplane shear crack moving in one-dimensional hexagonal quasicrystals

An investigation is presented of an antiplane shear strip crack subjected to representative general non-constant crack-face loading conditions which is moving uniformly in one-dimensional hexagonal quasicrystals, using an extended technique of continuous dislocation layers. Readily-calculable explicit closed-form representations are determined and discussed for the components of the phonon and phason stress fields created throughout the media. Illustrative numerical results are presented graphically. As a special case, the corresponding expressions are deduced explicitly for a non-uniformly loaded stationary crack.

Interaction among <i>n</i> Parallel Dislocations in One-Dimensional Hexagonal Quasicrystals

By means of analytic function theory, the problems of interaction amongparallel dislocations in one-dimensional hexagonal quasicrystals are investigated. The interaction force of parallel dislocations in the material is obtained in forms of complex variable function firstly, which is the versions of well-known Peach-Koehler formula in one-dimensional hexagonal quasicrystals on parallel dislocations. These results are development of the corresponding parts of quasicrystals. Meanwhile, in this paper, we firstly give the equivalent action point of parallel dislocations in one-dimensional hexagonal quasicrystals, which be of important reference value to researching the interaction problems of many dislocations in fracture mechanics of quasicrystals.

General solutions of plane problem in one-dimensional quasicrystal piezoelectric materials and its application on fracture mechanics

Based on the fundamental equations of piezoelasticity of quasicrystals (QCs), with the symmetry operations of point groups, the plane piezoelasticity theory of onedimensional (1D) QCs with all point groups is investigated systematically. The governing equations of the piezoelasticity problem for 1D QCs including monoclinic QCs, orthorhombic QCs, tetragonal QCs, and hexagonal QCs are deduced rigorously. The general solutions of the piezoelasticity problem for these QCs are derived by the operator method and the complex variable function method. As an application, an antiplane crack problem is further considered by the semi-inverse method, and the closed-form solutions of the phonon, phason, and electric fields near the crack tip are obtained. The path-independent integral derived from the conservation integral equals the energy release rate.

Fundamental solutions in a half space of two-dimensional hexagonal quasicrystal and their applications

Fundamental phonon-phason field in a half-infinite space of two-dimensional hexagonal quasicrystal is derived, on the basis of general solutions in terms of quasi-harmonic functions, by virtue of the trial-and-error technique. Extended Boussinesq and Cerruti problems are studied. Appropriate potential functions are assumed and corresponding fundamental solutions are explicitly derived in terms of elementary functions. The boundary integral equations governing the contact and crack problems are constructed from the present fundament solutions. The obtained analytical solutions can serve as guidelines for future indentation tests via scanning probe microscopy and atomic force microscopy methods.

The Antiplane Shearing Problem of Circular Holes with 4k Periodic Cracks in One-Dimensional Hexagonal Quasicrystals

Using the method of complex analysis and by constructing conformal mapping, the study investigated the antiplane shearing problem of circular holes with 4k periodic cracks in one-dimensional hexagonal quasicrystals. Wherefrom we simulated the problem of antiplane shearing in circular holes of cross-cracks, symmetrical four-cracks, as well as symmetrical eight-cracks, and provided an analytic solution to the crack tip stress intensity factor (SIF).

Electric-Elastic Field Induced by a Straight Dislocation in One-Dimensional Quasicrystals

By using the generalized Stroh formalism, the electric-elastic eld induced by a straight dislocation parallel to a periodic axis of a one-dimensional quasicrystal is obtained. The derivation is concise and the solution is in an exact closed form. As an illustration, the electric-elastic elds around a straight dislocation in a one-dimensional hexagonal quasicrystal are studied. Besides the interesting numerical results presented, the generalized Stroh formalism can be applied to more complicated dislocation problems in quasicrystals.

Row of shear cracks moving in one-dimensional hexagonal quasicrystalline materials

Representations for the stress fields created around an infinite row of collinear, antiplane shear cracks moving within one-dimensional hexagonal quasicrystals, and the resulting stress intensity factors and the J-integral, are determined in closed-form and discussed, using an extended method of dislocation layers. The solutions for a finite quasicrystalline plate containing a single moving crack and a plate with a moving edge crack are also provided by this analysis.

Indentation on two-dimensional hexagonal quasicrystals

This paper is concerned with contact problem for a half-space of two-dimensional hexagonal quasi-crystal punched by three common indenters (cylindrical flat-ended, conical and spherical punches). Based on the Green’s functions for the half-space subjected to an external phonon source exerted on the surface, the superposition principle is applied to constructing the boundary integral equation. Relations between the indentation force and the penetration depth, and the indentation stiffness constants are explicitly obtained for these three indenters. Complete and exact fields in the half-space are given in terms of elementary functions. The present theoretical solutions can not only serve as benchmark for computational contact mechanics, but also apply to guiding future indentation experiments.

The Anti-Plane Shear Problem of Two Symmetric Cracks Originating from an Elliptical Hole in 1D Hexagonal Piezoelectric QCs

Using the complex variable function method and the technique of conformal mapping, the fracture mechanics of two symmetric collinear cracks originating from an elliptical hole in a one-dimensional (1D) hexagonal piezoelectric quasicrystals (QCs) are investigated under anti-plane shear loading and electric loading. The crack is assumed to be either electrical impermeable or permeable. The exact solutions in closed-form of the stress intensity factors (SIFs) of the phonon field and the phason field, and the electric displacement intensity factors (EDIFs) are obtained. In the limiting cases, the new results such as Griffith crack, a circular hole with equal two edge cracks and cross crack can be obtained from the present solutions. In the absence of the phason field, the obtainable results in this paper match with the classical results.