Phason Fields Research Articles

Quasi-Hamilton Principle of Quasi-Crystals Beam.

Quasi-crystals is not only a new structure of solids but also a new class of functional and structural materials. With the research and development of quasi-crystals, the mechanical properties of quasi-crystals get more and more attention. In the paper, quasi-Hamilton principle of quasi-crystals beam is established in non-conservative systems. And applying the quasi-Hamilton principle, all the equations of non-conservative quasi-crystals beam problem are obtained in the phonon field and the phason field respectively.

Dynamic Analysis of Cracked Octagonal Quasicrystals

We focus on the dynamic fracture problem of octagonal quasicrystals by applying a rectangular sample with a Griffith crack which is often used in classical elastic media based on the method of finite difference. This paper mainly is to investigate the variation of phonon, phason fields, and stress singularity around the crack tip including the stress intensity factor. In addition, the moving boundary due to the crack propagation has also been treated by introducing an additional condition for determining solution. The influence of wave propagation and diffusion in the dynamic process is also discussed in detail. Through comparing the results of octagonal quasicrystals with the results of crystal, this paper proclaims the influence of phonon‐phason coupling in dynamic fracture problem of octagonal quasicrystals which should not be neglected.

Generalized dynamics of moving dislocations in quasicrystals

A theoretical framework for dislocation dynamics in quasicrystals is provided according tothe continuum theory of dislocations. Firstly, we present the fundamental theory formoving dislocations in quasicrystals giving the dislocation density tensors and introducingthe dislocation current tensors for the phonon and phason fields, including the Bianchiidentities. Next, we give the equations of motion for the incompatible elastodynamics aswell as for the incompatible elasto-hydrodynamics of quasicrystals. We continue withthe derivation of the balance law of pseudomomentum thereby obtaining thegeneralized forms of the Eshelby stress tensor, the pseudomomentum vector, thedynamical Peach–Koehler force density and the Cherepanov force density forquasicrystals. The form of the dynamical Peach–Koehler force for a straight dislocationis obtained as well. Moreover, we deduce the balance law of energy that givesrise to the generalized forms of the field intensity vector and the elastic powerdensity of quasicrystals. The above balance laws are produced for both models.The differences between the two models and their consequences are revealed.The influences of the phason fields as well as of the dynamical terms are alsodiscussed.

The Interaction between a Dislocation and Circular Inhomogeneity in 1D Hexagonal Quasicrystals

The interaction between a dislocation and circular inhomogeneity in 1D hexagonal quasicrystals is investigated. By using the complex potential method, explicit solutions of complex potentials are obtained. The image force acting on the dislocation are also derived. The results show that the interface attracts the dislocation inside both the matrix and the inhomogeneity under most condition. The attraction increase with the increase of the elastic constant of phason field and the phonon-phason coupling elastic constant.

Stationary straight cracks in quasicrystals

Stationary straight cracks in quasicrystals in linear elastic setting are under scrutiny. The analysis is developed by using Stroh formalism which is modified to account for a totally degenerate eigenvalue problem: in fact, the fundamental matrix of the governing equations of motion admits a repeated eigenvalue corresponding to a single eigenvector. Cases of a semi-infinite rectilinear crack loaded along its margins and a crack of finite length under remote loading conditions are considered. Standard and phason stresses display square-root singularities at crack tip. The latter stresses represent peculiar microstructural inner actions occurring in quasicrystals and are determined by rearrangements assuring quasi-periodicity of the atomic tiling—modes described by a vector field, called phason field, collecting the local degrees of freedom exploited by the atoms within the material elements. Energy release rate increases with the coupling parameter between displacement and phason fields.

Collinear periodic cracks and/or rigid line inclusions ofantiplane sliding mode in one-dimensional hexagonal quasicrystal

For a one-dimensional (1D) hexagonal quasicrystal (QC), there is the periodic ( x 1 , x 2 ) -plane of atomic structures with the quasiperiodic direction x 3 -axis along which there exists a phason displacement. The macroscopically collinear periodic cracks and/or rigid line inclusions are placed on the periodic ( x 1 , x 2 ) -plane for finding out the influence of phason displacement on the related physical quantities. These two models are reduced to the Riemann–Hilbert problem of periodic analytic functions to obtain the closed-form solutions for the antiplane sliding mode. It is found that the phonon and phason stress intensity factors of cracks as well as the phonon and phason stress field intensity factors of rigid line inclusions are not related to the coupling of phonon and phason fields. These mean that there is not the influence of phason displacement on both the phonon stress intensity factor (usual stress intensity factor) of cracks and the phonon stress field intensity factor of rigid line inclusions. However, the energy release rates of periodic cracks and/or rigid line inclusions are obtained and affected not only by the periodicity of cracks and/or rigid line inclusions but also by the phason displacement.

The remarkable nature of one-dimensional quasicrystal

Abstract A material is spherically uniform when its generalized Hooke´s law at each material point referred to a spherical coordinate system is the same everywhere. We study a sphere of spherically uniform linear anisotropic one-dimensional quasicrystalline material subjected to radial phonon and phason strains on its surface. For a certain range of material parameters the phonon as well as the phason stress diverges at the center. The applied phason strain does not affect the phonon field but the phason field is disturbed by the phonon strain. “White hole” at the center may occur when the sphere is subjected to uniform tension while uniform pressure may cause a “black hole”. This behavior depends only on a dimensionless material parameter and the kind of applied strain but not on its magnitude.

COMPLEX VARIABLE METHOD FOR THE PLANE ELASTICITY OF QUASICRYSTALS WITH POINT GROUP 10

General complex variable method for solving plane elasticity problems of quasicrystals with point group 10 has been proposed. The stress and displacement components of phonon and phason fields are expressed by four arbitrary analytic functions. Explicit real-form displacement expressions for the dislocation problem of the quasicrystal is obtained through the use of this method.The interaction between two parallel dislocations is also discussed in detail. All the present results can be reduced to the exact solutions for the quasicrystals with point group 10 mm in the special case.

General solutions of equilibrium equations for 1D hexagonal quasicrystals

Three general solutions are presented for the coupled equilibrium equations in one-dimensional (1D) hexagonal quasicrystals (QCs). These solutions are expressed in terms of two displacement functions, which satisfy a quasi-harmonic equation and a sixth-order partial differential equation, respectively, as well as the theory of differential operator matrix developed by Wang (2006) [Wang, X., 2006. The general solution of one-dimensional hexagonal quasicrystal. Mech. Res. Commun. 33, 576–580]. However, it is difficult to obtain rigorous analytic solutions and not applicable in most cases, since a displacement function satisfies a higher-order equation. By utilizing a theorem, a decomposition and superposition procedure is taken to replace the sixth-order function with three quasi-harmonic functions, and the general solutions are simplified in terms of these quasi-harmonic functions. In consideration of different cases of three characteristic roots, each general solution possesses three cases, but all are in simple forms that are convenient to be used. As a special case, the general solutions for 1D hexagonal QCs can be degenerated into Elliott–Lodge (E–L) solutions of transversely isotropic elasticity when phonon–phason fields coupling effect is absent. Furthermore, it should be pointed out that the general solutions obtained here are complete in z-convex domains.

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.

Exact solutions of two semi-infinite collinear cracks in a strip of one dimensional hexagonal quasicrystal

Two semi-infinite collinear cracks in a strip of one dimensional hexagonal quasicrystal is considered. With the help of conformal transformation, exact solutions are constructed by means of complex variable method. The stress intensity factors for the phonon field and the phason field are obtained respectively, which are very useful in the field of fracture mechanics and engineering.

Boundary value problems of holomorphic vector functions in 1D QCs

By means of the generalized Stroh formalism, two-dimensional (2D) problems of one-dimensional (1D) quasicrystals (QCs) elasticity are turned into the boundary value problems of holomorphic vector functions in a given region. If the conformal mapping from an ellipse to a circle is known, a general method for solving the boundary value problems of holomorphic vector functions can be presented. To illustrate its utility, by using the necessary and sufficient condition of boundary value problems of holomorphic vector functions, we consider two basic 2D problems in 1D QCs, that is, an elliptic hole and a rigid line inclusion subjected to uniform loading at infinity. For the crack problem, the intensity factors of phonon and phason fields are determined, and the physical sense of the results relative to phason and the difference between mechanical behaviors of the crack problem in crystals and QCs are figured out. Moreover, the same procedure can be used to deal with the elastic problems for 2D and three-dimensional (3D) QCs.

Boundary Conditions for Plate Bending in One-dimensional Hexagonal Quasicrystals

For plate bending in one-dimensional (1D) hexagonal quasicrystals (QCs), the reciprocal theorem and the general solution for QCs media are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all order for plates of general edge geometry and loadings. Through generalizing the method developed by Gregory and Wan, a set of necessary conditions on the edge-data for the existence of a rapidly decaying solution is established. The prescribed data must satisfy these conditions in order that they should generate a decaying state. When a set of stress edge-data or mixed edge-data is imposed on the plate edge, these decaying state conditions for the case of axisymmetric deformation of 1D hexagonal QC plates are derived explicitly. They are then used for the correct formulation of boundary conditions for the plate theory solution (or the interior solution). Furthermore, in the absence of phonon–phason fields coupling effect, corresponding necessary conditions for the case of transversely isotropic elastic plates are derived subsequently, and their isotropic elastic counterparts are also obtained.

Stress potential function formulation and complex variable function method for solving theelasticity of quasicrystals of point group 10 and the exact solution for the notchproblem

The plane elasticity equations of two-dimensional quasicrystals of point group 10 arereduced to a single partial differential equation with eighth order by introducing a stresspotential function. Further, we develop the complex variable function method for classicalelasticity theory to that of the quasicrystals. The complex representations of stress anddisplacement components of phonon and phason fields in the quasicrystals are given. Withthe help of conformal transformation, an exact solution for the elliptic notch of thequasicrystals is presented. The solution of the Griffith crack problem as a special case ofthe results is also observed. This work shows that the stress potential and complex variablefunction methods are powerful for solving the complicated boundary value problems ofhigher order partial differential equations originating from quasicrystal elasticity.

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.

Conservation laws of a decagonal quasicrystal in elastodynamics

A special version of Noether's theorem for the sake of absolute invariance on invariant variational principles is used to obtain conservation laws of a three-dimensional solid periodically stacked in a two-dimensional quasiperiodic structure with decagonal symmetry. By applying Lie's infinitesimal criterion on the invariance to the Lagrangian density function under a continuous transformation group, it is found that there exist fourteen conservation laws (nine in physical space and five in material space). Conservation laws in physical space correspond to the known physical facts. Influence of the phason displacement field on the conservation laws in material space is presented. The infinitesimal symmetry-transformations from scale change and translation of coordinates as well as one coordinate rotation are verified to be true. Especially, although the angular momentum conservation laws do not contain the physical quantities in phason field, and the conservation of angular momentum of the phason displacement field itself does not hold, the physical quantities in phason field play an important role in the conservation law in material space derived from an infinitesimal coordinate rotation.

In situ high-resolution transmission electron microscopy observation of the phason-strain relaxation process in an Al-Cu-Co-Si decagonal quasicrystal

Transition process from a rational approximant state, which is regarded as containing a uniform phason strain in a quasicrystalline state, to the quasicrystalline state in the Al-Cu-Co-Si system has been observed by in situ high-temperature high-resolution transmission electron microscopy (HRTEM). The tiling pattern changing with lapse of time in the HRTEM image has been analyzed, and the spatial and temporal variations of the phason field have been deduced. The results show that two types of processes lead the phason-strain relaxation: one is the shrink of the area with the uniform strain by shifting the boundaries and the other a continuous strain-relaxation in the area outside of the boundary. Such processes arise from a combination of collective and successive phason flips like domino-toppling along different symmetry directions.

Eshelby’s problem of an inclusion of arbitrary shape in a decagonal quasicrystalline plane or half-plane

In this paper, a simple method based on analytical continuation and conformal mapping is presented to obtain an analytic solution for Eshelby’s problem of a two-dimensional inclusion of arbitrary shape in a decagonal quasicrystalline plane or half-plane. The inclusion in this study refers to a subdomain which tends to undergo uniform stress-free phonon and phason eigenstrains without the constraint of the surrounding matrix. Furthermore, the inclusion penetrates through the solid along the period direction. The present method leads to elementary expressions for the internal phonon and phason fields within the inclusion in an entire decagonal quasicrystalline plane. Several examples are discussed to illustrate the method.

CHARGE-DENSITY-WAVES IN THE PRESENCE OF IMPURITY POTENTIAL IN ONE-DIMENSIONAL SYSTEMS: A FIELD THEORY APPROACH

The effective Lagrangian model for charge-density waves interacting with an impurity potential in one-dimensional systems is considered in the dynamical phonon phase approach. Using the fermion–boson mapping we obtain the effective bosonized version of the model. The impurity potential breaks the linked electron–phonon symmetry, the phason field turns out to be in interaction with the electron–phonon condensate and the phonon field develops a non-vanishing vacuum expectation value. The effective fermionized version of the model corresponds to the chiral Gross–Neveu model with quartic self-interaction among a massless and a massive (electron) Fermi fields. The electron–phonon system exhibits a conserved topological charge which is independent of local variations of the phases of the phonon and electron fields. The equation of state of the associated statistical-mechanical system is obtained. For the dynamical phonon phase field the Kosterlitz–Thouless phase transition is suppressed.

Thermal phason fluctuation in an Al–Cu–Co decagonal quasicrystal

In situ high-temperature, high-resolution transmission electron microscopy (HRTEM) has been performed on an Al–Cu–Co decagonal quasicrystal, to investigate thermal phason fluctuations. The changing tiling pattern with elapsed time in the HRTEM image has been analyzed, and the spatial and temporal variations of the phason field have been deduced. In the temporal variation, abrupt and discrete changes have been observed in addition to smooth and continuous ones. The degrees of the spatial and temporal correlations of the phason fluctuation have been evaluated, both of which decrease continuously with increasing distance and time, respectively. The results have been discussed in light of the random tiling model of the quasicrystal.