A coupled Legendre-Laguerre polynomial method with analytical integration for the Rayleigh waves in a quasicrystal layered half-space with an imperfect interface

The Laguerre polynomial method has been successfully used to investigate the dynamic responses of a half-space.However,it fails to obtain the correct stress at the interfaces in a layered half-space,especially when there are significant differences in material properties.Therefore,a coupled Legendre-Laguerre polynomial method with analytical integration is proposed.The Rayleigh waves in a one-dimensional(1D)hexagonal quasicrystal(QC)layered half-space with an imperfect interface are investigated.The correctness is validated by comparison with available results.Its computation efficiency is analyzed.The dispersion curves of the phase velocity,displacement distributions,and stress distributions are illustrated.The effects of the phonon-phason coupling and imperfect interface coefficients on the wave characteristics are investigated.Some novel findings reveal that the proposed method is highly efficient for addressing the Rayleigh waves in a QC layered half-space.It can save over 99%of the computation time.This method can be expanded to investigate waves in various layered half-spaces,including earth-layered media and surface acoustic wave(SAW)devices.

Precipitation of multi-type nano-quasicrystals in a Mg-Zn-Y alloy

Mg-6Zn-1Y(at.%)ribbons with strengthening precipitates of multi-type nanoquasicrystals were prepared by melt-spinning followed by aging treatments.Microstructural evolution of the rapidly solidified ribbons during isothermal aging was comprehensively studied using various electron microscopy techniques.Two new kinds of decagonal quasicrystals were formed in aged ribbons,in addition to precipitation of nanometer icosahedral quasicrystals.Atomic-resolution observations reveal that both decagonal quasicrystals can be modeled by quasiperiodic tiling with decagonal clusters of 2.5 nm in diameter,but overlap of neighboring clusters in both decagonal quasicrystals is different from the Gummelt model observed in other quasicrystals.A shell composed of complex Laves Mg-Zn domains was formed surrounding each decagonal quasicrystal precipitate upon prolonged aging.In addition,all kinds of nanoprecipitates exhibit excellent structure and size stability at 573 K.Our findings may have implications for not only fundamental studies about quasicrystals,but also microstructural manipulation of high-performance Mg alloys.

Love wave propagation in one-dimensional piezoelectric quasicrystal multilayered nanoplates with surface effects

The exact solutions for the propagation of Love waves in one-dimensional(1D)hexagonal piezoelectric quasicrystal(PQC)nanoplates with surface effects are derived.An electro-elastic model is developed to investigate the anti-plane strain problem of Love wave propagation.By introducing three shape functions,the wave equations and electric balance equations are decoupled into three uncorrelated problems.Satisfying the boundary conditions of the top surface on the covering layer,the interlayer interface,and the matrix,a dispersive equation with the influence of multi-physical field coupling is provided.A surface PQC model is developed to investigate the surface effects on the propagation behaviors of Love waves in quasicrystal(QC)multilayered structures with nanoscale thicknesses.A novel dispersion relation for the PQC structure is derived in an explicit closed form according to the non-classical mechanical and electric boundary conditions.Numerical examples are given to reveal the effects of the boundary conditions,stacking sequence,characteristic scale,and phason fluctuation characteristics on the dispersion curves of Love waves propagating in PQC nanoplates with surface effects.

Hamiltonian system for the inhomogeneous plane elasticity of dodecagonal quasicrystal plates and its analytical solutions

A Hamiltonian system is derived for the plane elasticity problem of two-dimensional dodecagonal quasicrystals by introducing the simple state function. By using symplectic elasticity approach, the analytic solutions of the phonon and phason displacements are obtained further for the quasicrystal plates. In addition, the effectiveness of the approach is verified by comparison with the data of the finite integral transformation method.

Study on the effective elastic performance of composites containing decagonal symmetric two-dimensional quasicrystal coatings

On account of the Mori-Tanaka approach,the effective elastic performance of composites containing decagonal symmetric two-dimensional(2D)quasicrystal(QC)coatings is studied.Explicit expressions for the effective elastic constants of rare-earth QC reinforced magnesium-based composites are provided.Detailed discussion is presented on the effects of the volume fraction of the inclusions,the aspect ratio of the inclusions,the coating thickness,and the coating material parameters on the effective elastic constants of the composites.The results indicate that considering the coating increases the effective elastic constants of the composites to some extent.

On Discrete Hopf Fibrations, Grand Unification Groups, the Barnes-Wall, Leech Lattices, and Quasicrystals

A discrete Hopf fibration of S15 over S8 with S7 (unit octonions) as fibers leads to a 16D Polytope P16 with 4320 vertices obtained from the convex hull of the 16D Barnes-Wall lattice Λ16. It is argued (conjectured) how a subsequent 2-1 mapping (projection) of P16 onto a 8D-hyperplane might furnish the 2160 vertices of the uniform 241 polytope in 8-dimensions, and such that one can capture the chain sequence of polytopes 241,231,221,211in D=8,7,6,5dimensions, leading, respectively, to the sequence of Coxeter groups E8,E7,E6,SO(10)which are putative GUT group candidates. An embedding of the E8⊕E8and E8⊕E8⊕E8lattice into the Barnes-Wall Λ16 and Leech Λ24 lattices, respectively, is explicitly shown. From the 16D lattice E8⊕E8one can generate two separate families of Elser-Sloane 4D quasicrystals (QC’s) with H4 (icosahedral) symmetry via the “cut-and-project” method from 8D to 4D in each separate E8 lattice. Therefore, one obtains in this fashion the Cartesian product of two Elser-Sloane QC’s Q×Qspanning an 8D space. Similarly, from the 24D lattice E8⊕E8⊕E8one can generate the Cartesian product of three Elser-Sloane 4D quasicrystals (QC’s) Q×Q×Qwith H4 symmetry and spanning a 12D space.

Green's functions of two-dimensional piezoelectric quasicrystal in half-space and bimaterials

In this paper,we obtain Green’s functions of two-dimensional(2D)piezoelectric quasicrystal(PQC)in half-space and bimaterials.Based on the elastic theory of QCs,the Stroh formalism is used to derive the general solutions of displacements and stresses.Then,we obtain the analytical solutions of half-space and bimaterial Green’s functions.Besides,the interfacial Green’s function for bimaterials is also obtained in the analytical form.Before numerical studies,a comparative study is carried out to validate the present solutions.Typical numerical examples are performed to investigate the effects of multi-physics loadings such as the line force,the line dislocation,the line charge,and the phason line force.As a result,the coupling effect among the phonon field,the phason field,and the electric field is prominent,and the butterfly-shaped contours are characteristic in 2D PQCs.In addition,the changes of material parameters cause variations in physical quantities to a certain degree.

Thermal-induced interfacial behavior of a thin one-dimensional hexagonal quasicrystal film

In this paper,we investigate the interfacial behavior of a thin one-dimensional(1D)hexagonal quasicrystal(QC)film bonded on an elastic substrate subjected to a mismatch strain due to thermal variation.The contact interface is assumed to be nonslipping,with both perfectly bonded and debonded boundary conditions.The Fourier transform technique is adopted to establish the integral equations in terms of interfacial shear stress,which are solved as a linear algebraic system by approximating the unknown phonon interfacial shear stress via the series expansion of the Chebyshev polynomials.The expressions are explicitly obtained for the phonon interfacial shear stress,internal normal stress,and stress intensity factors(SIFs).Finally,based on numerical calculations,we briefly discuss the effects of the material mismatch,the geometry of the QC film,and the debonded length and location on stresses and SIFs.

A Dugdale-Barenblatt model for elliptical orifice problem with asymmetric cracks in one-dimensional orthorhombic quasicrystals

By means of Muskhelishvili’s method and the technique of generalized conformal mapping,the physical plane problems are transformed into regular mathematical problems in quasicrystals(QCs).The analytical solution to an elliptical orifice problem with asymmetric cracks in one-dimensional(1D)orthorhombic QCs is obtained.By using the Dugdale-Barenblatt model,the plastic simulation at the crack tip of the elliptical orifice with asymmetric cracks in 1D orthorhombic QCs is performed.Finally,the size of the atomic cohesive force zone is determined precisely,and the size of the atomic cohesive force zone around the crack tip of an elliptical orifice with a single crack or two symmetric cracks is obtained.

Analytic solution of quasicrystal microsphere considering the thermoelectric effect and surface effect in the elastic matrix

The incorporation of the quasicrystalline phase into the metal matrix offers a wide range of potential applications in particle-reinforced metal-matrix composites.The analytic solution of the piezoelectric quasicrystal(QC)microsphere considering the thermoelectric effect and surface effect contained in the elastic matrix is presented in this study.The governing equations for the QC microsphere in the matrix subject to the external electric loading are derived based on the nonlocal elastic theory,electro-elastic interface theory,and eigenvalue method.A comparison between the existing results and the finite-element simulation validates the present approach.Numerical examples reveal the effects of temperature variation,nonlocal parameters,surface properties,elastic coefficients,and phason coefficients on the phonon,phason,and electric fields.The results indicate that the QC microsphere enhances the mechanical properties of the matrix.The results are useful for the design and understanding of the characterization of QCs in micro-structures.

An Initial Research on the Geometric Theory of Quasicrystal Structures

The geometric theory of quasicrystal structure is an important subject in quasicrystal research. The authors deduce the quasicrystal plane geometric lattices from the stereograms of quasicrystal space geometric lattice , and put them together to form the geometric lattices of quasicrystal structures . The general characteristics of quasicrystal geometric lattices , the relation between structural models and geometric lattices , and the relation formula (k=0 , 2 , 4 , 6 , 8, 10,12) of the symmetric axis between quasicrystal and crystal are discussed based on the quasicrystal space geometric lattices. This is of significant in quasicrystal research .

Elastic Field of a Straight Dislocation in One Dimensional Hexagonal Quasicrystals *L

Aim To study dislocation elasticity theory in quasicrystals. Methods A dislocation was separated into pure edge part and pure screw part and their superposition was used to find the elastic field. Results and Conclusion The elastic solution of a straight dislocation parallel to the quasiperiodic axis in 1D hexagonal quasicrystals was obtained and the generalized Peach Koehler force on a dislocation in quasicrystals was given.

Effect of applied pressure on microstructure and mechanical properties of Mg-Zn-Y quasicrystal-reinforced AZ91D magnesium matrix composites prepared by squeeze casting

The Mg-Zn-Y quasicrystal-reinforced AZ91 D magnesium matrix composites were prepared by squeeze casting process. The effects of applied pressure on microstructure and mechanical properties of the composites were investigated. The results show that squeeze casting process is an effective method to refine the grain. The composites are mainly composed of α-Mg, β-Mg17Al12 and Mg3Zn6Y icosahedral quasicrystal phase（I-phase）. With the increase of applied pressure, the contents of β-Mg17Al12 phase and Mg3Zn6 Y quasicrystal particles increase, further matrix grain refinement occurs and coarse dendritic α-Mg transforms into equiaxed grain structure. The composite exhibits the maximum ultimate tensile strength and elongation of 194.3 MPa and 9.2% respectively when the applied pressure is 100 MPa, and a lot of dimples appear on the tensile fractography. Strengthening mechanisms of quasicrystal-reinforced AZ91 D magnesium matrix composites are chiefly fine-grain strengthening and quasicrystal particles strengthening.

Anti-Plane Elasticity Problem and Mode Ⅲ Crack Problem of Cubic Quasicrystal

The fracture theory of cubic quasicrystal was developed. The exact analytic solution of a Mode Ⅲ Griffith crack in the material was obtained by using the Fourier transform and dual integral equations theory, and so the displacement and stress fields, the stress intensity factor and strain energy release rate were determined. The results show that the stress intensity factor is independent of material constants, and the strain energy release rate is dependent on all material constants. These provide important information for studying the deformation and fracture of the new solid material.

Contact Problem in Decagonal Two-Dimensional Quasicrystal

As a new structure of solid matter quasicrystal brings profound new ideas to the traditional condensed matter physics, its elastic equations are more complicated than that of traditional crystal. A contact problem of decagonal two? dimensional quasicrystal material under the action of a rigid flat die is solved satisfactorily by introducing displacement function and using Fourier analysis and dual integral equations theory, and the analytical expressions of stress and displacement fields of the contact problem are achieved. The results show that if the contact displacement is a constant in the contact zone, the vertical contact stress has order -1/2 singularity on the edge of contact zone, which provides the important mechanics parameter for contact deformation of the quasicrystal.

Influence of heat treatment on electrochemical properties of Ti_(1.4)V_(0.6)Ni alloy electrode containing icosahedral quasicrystalline phase

The structures and electrochemical properties of the Ti1.4V0.6Ni ribbon before and after heat treatment are investigated systematically. The structure of the sample is characterized by X-ray powder diffraction analysis. Electrochemical properties including the discharge capacity, the cyclic stability and the high-rate discharge ability are tested. X-ray powder diffraction analysis shows that after heat treatment at 590 °C for 30 min, all samples mainly consist of the icosahedral quasicrystal phase (I-phase), Ti2Ni phase (FCC), V-based solid solution phase (BCC) and C14 Laves phase (hexagonal). Electrochemical measurements show that the maximum discharge capacity of the alloy electrode after heat treatment is 330.9 mA?h/g under the conditions that the discharge current density is 30 mA/g and the temperature is 30 °C. The result indicates that the cyclic stability and the high-rate discharge ability are all improved. In addition, the electrochemical kinetics of the alloy electrode is also studied by electrochemical impedance spectroscopy (EIS) and hydrogen diffusion coefficient (D).

Several Fundamental Theorems and Conservative Integrals in Quasicrystal Elasticity Theory

Aim To extend several fundamental theorems of conventional elasticity theory to quasicrystalelasticity theory. Methods The basic governing equations of quasicrystal elasticity theory and Gauss's theorem were applied in the derivation. Results and Conclusion The principle of virtual work, Betti's reciprocal theorem and the uniqueness theorem of quasicrystal elasticity theory are proud, and some conservative integrals in quasicrystal elasticty theory are obtained.

Mathematical Theory and Methods of Mechanics of Quasicrystalline Materials

The review is devoted to introduce the recent development of the study in mathematical theory and methods of mechanics of quasicrystals, respectively. The mechanics of quasicrystalline materials includes elasticity, plasticity, defects, dynamics, fracture etc. In the article some relevant measured data are collected for some important quasicrystal systems, which are necessary for understanding physics and applications of the materials. It is very interesting that the mathe-matical theory and solving methods of the mechanics of quasicrystals have developed rapidly in recent years, which is strongly supported by the experiments and applications. The theoretical development strongly enhances the understanding in-depth the physics including mechanics of the materials. The mathematical theory and computational methods provide a basis to the applications of quasicrystals as functional and structural materials in practice as well. More recently the quasicrystals in soft matter are observed, which challenge the study of based on the quasicrystals of binary and ternary alloys and greatly enlarge the scope of the materials and have aroused a great deal attention of researchers, an introduction about this new phase and its mathematical theory is also given in the review.

Icosahedral quasicrystalline phase in an as-cast Mg-Zn-Er alloy

The microstructure of an as-cast Mg-Zn-Er alloy was investigated through scanning electron microscopy （SEM） and transmission electron microscopy （TEM） equipped with energy dispersive spectroscopy （EDS）. The results indicate that two different second phases, one with eutectoid-lamellar morphology and the other with granular shape, distribute in the α-Mg matrix. The coexistence of the face-centered icosahedral quasicrystalline phase （I-phase） and W-phase with the face-centered cubic structure is found in the as-cast alloy. The coexistence of I-phase and W-phase in the Mg-Zn-Er alloy is because the W-phase is the primary phase and the I-phase forms by peritectic reaction during solidification.

Anti-plane Analysis of a Circular Hole withThree Unequal Cracks in One-dimensionalHexagonal Piezoelectric Quasicrystals

This paper employes variable function method and the technique of conformal mappingto discuss the anti-plane problem of a circular hole with three unequal cracksin a one-dimensional （1D） hexagonal piezoelectric quasicrystal. Based on the piezoelectricityfundamental equations of quasicrystal materials and the symmetry of1D hexagonal quasicrystal and its linear piezoelectricity effect, 1D hexagonal quasicrystalcontrol equations of anti-plane problem are derived. Applying Cauchyintegral formula, the analytical expressions for the crack tip filed intensity factorsare presented with the assumption that the crack are electrical impermeable andelectrical permeable. With the variation of the hole-size and the crack length, someof the new model of crack are obtained. In the absence of the electric load, theresults match with the classical ones. The numerical results indicate the effects ofgeometric parameters on the field intensity factors. It is verified that the horizontalcrack length and the circle radius can easily promote crack growth. Researchon such issues will provide reliable theoretical value for the engineering materialspreparation and application.