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.

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.

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.

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.

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.

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.

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.

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.

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.

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 a 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.

Two kinds of contact problems in three-dimensional icosahedral quasicrystals

Two kinds of contact problems, i.e., the frictional contact problem and the adhesive contact problem, in three-dimensional （3D） icosahedral quasicrystals are dis- cussed by a complex variable function method. For the frictional contact problem, the contact stress exhibits power singularities at the edge of the contact zone. For the adhe- sive contact problem, the contact stress exhibits oscillatory singularities at the edge of the contact zone. The numerical examples show that for the two kinds of contact problems, the contact stress exhibits singularities, and reaches the maximum value at the edge of the contact zone. The phonon-phason coupling constant has a significant effect on the contact stress intensity, while has little impact on the contact stress distribution regu- lation. The results are consistent with those of the classical elastic materials when the phonon-phason coupling constant is 0. For the adhesive contact problem, the indentation force has positive correlation with the contact displacement, but the phonon-phason cou- pling constant impact is barely perceptible. The validity of the conclusions is verified.

Generalized 2D problem of icosahedral quasicrystals containing an elliptic hole

The generalized 2D problem of icosahedral quasicrystals containing an elliptic hole is considered by using the ex- tended Stroh formalism. The closed-form solutions for the displacements and stresses are obtained under general loading conditions. The solution of the Griffith crack problem as a special case of the results is also observed. The stress intensity factor and strain energy release rate are given. The effect of the phonon-phason coupling elastic constant on the mechanical behavior is also discussed.

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.

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.

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 modesⅠand 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.

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.

TIME-HARMONIC DYNAMIC GREEN'S FUNCTIONS FOR ONE-DIMENSIONAL HEXAGONAL QUASICRYSTALS

Quasicrystals have additional phason degrees of freedom not found in conventional crystals. In this paper, we present an exact solution for time-harmonic dynamic Green＇s function of one-dimensional hexagonal quasicrystals with the Laue classes 6/mh and 6/mhmm. Through the introduction of two new functions φ and ψ, the original problem is reduced to the determination of Green＇s functions for two independent Helmholtz equations. The explicit expressions of displacement and stress fields are presented and their asymptotic behaviors are discussed. The static Green＇s function can be obtained by letting the circular frequency approach zero.

Finite size specimens with cracks of icosahedral Al Pd Mn quasicrystals

Icosahedral quasicrystals are the most important and thermodynamically stable in all about 200 kinds of quasicrystals currently observed. Beyond the scope of classical elasticity, apart from a phonon displacement field, there is a phason displacement field in the elasticity of the quasicrystal, which induces an important effect on the mechanical properties of the material and makes an analytical solution difficult to obtain. In this paper, a finite element algorithm for the static elasticity of icosahedral quasicrystals is developed by transforming the elastic boundary value problem of the icosahedral quasicrystals into an equivalent variational problem. Analytical and numerical solutions for an icosahedral A1-Pd-Mn quasicrystal cuboid subjected to a uniaxial tension with different phonon-phason coupling parameters are given to verify the validity of the numerical approach. A comparison between the analytical and numerical solutions of the specimen demonstrates the accuracy and efficiency of the present algorithm. Finally, in order to reveal the fracture behavior of the icosahedral A1-Pd-Mn quasicrystal, a cracked specimen with a finite size of matter is investigated, both with and without phonon-phason coupling. Meanwhile, the geometry factors are calculated, including the stress intensity factor and the crack opening displacement for the finite-size specimen. Computational results reveal the importance of pbonon-phason coupling effect on the icosahedral A1-Pd-Mn quasicrystal. Furthermore, the finite element procedure can be used to solve more complicated boundary value problems.

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.

Anti-plane problem of nano-cracks emanating from a regular hexagonal nano-hole in one-dimensional hexagonal piezoelectric quasicrystals

By constructing a new conformal mapping function, we study the surface effects on six edge nano-cracks emanating from a regular hexagonal nano-hole in one-dimensional (1D) hexagonal piezoelectric quasicrystals under anti-plane shear. Based on the Gurtin–Murdoch surface/interface model and complex potential theory, the exact solutions of phonon field, phason field and electric field are obtained. The analytical solutions of the stress intensity factor of the phonon field, the stress intensity factor of the phason field, the electric displacement intensity factor and the energy release rate are given. The interaction effects of the nano-cracks and nano-hole on the stress intensity factor of the phonon field, the stress intensity factor of the phason field and the electric displacement intensity factor are discussed in numerical examples. It can be seen that the surface effect leads to the coupling of phonon field, phason field and electric field. With the decrease of cavity size, the influence of surface effect is more obvious.