Quasicrystals (Handbook of Metal Physics)

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Though some researchers believe that the hydrodynamics, i. In this case we give a general survey on the development of different theories and their applications below, which may be beneficial to readers who are interested in the dynamics of quasicrystals. This review is focused on the basic concepts and fundamental framework of the theory of elasto-dynamics for quasi-crystalline materials, including some 1D, 2D, and 3D quasicrystals.

A decomposition and superposition procedure has been suggested to simplify the elasto-dynamics problems of quasi-crystals. Simplification of the basic equations provides the possibility to solve boundary value or initial-boundary value problems of elasto-dynamics elasticity. The above may be seen as a development of the theory and methodology to those of classical dynamic elasticity.

Also, some elasto-dynamic problems for some 1Dand 2D quasi-crystals are studied, as are some examples of their theory.

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The law of the momentum conservation is assumed to hold for both the phonons and the phasons. For the elasto-dynamics, the viewpoint raised by Bak in linear and small deformation case the equations of motion are: The basic equations of anti-plane elasticity for one-dimensional hexagonal quasicrystals or three-dimensional icosahedral, and cubic quasicrystals take a similar form. Let us consider icosahedral quasicrystals first. The above formula can be regarded as anti-plane elasticity of one-dimensional quasicrystals and cubic quasicrystals as well. The solution of 7 can be determined by using the method for solving wave equations in mathematical physics.

Let us take the z -axis as periodic direction in point group 12 mm dodecagonal quasicrystals. Suppose that the field variables not depends on the coordinate z , then the elasticity problem of dodecagonal quasicrystals can be turned into the plane elasticity problem and anti-plane elasticity problem. Equation 18 provide a macro-description of vibration and wave propagation in two-dimensional quasicrystals with fold symmetry, which can be turned into the classical elastic wave equations. It is unavailable to obtain standard wave equations for elasto-dynamics of quasicrystals so far from the above four cases.

Let us consider a straight screw dislocation line parallel to the quasi-periodic axis which moves along a periodic x axis in the periodic plane. Suppose the dislocation moves with a constant velocity V. Henceforth we use x 1 , x 2 , t to denote the fixed coordinates and x , y to denote the moving coordinates.

Then the known wave Equation 7 can be ascribed to the Laplace equations, i. Thus s 2 can be regarded as the limit of the velocity from a moving dislocation. The above solution can be reduced to the solutions anti-plane elasticity cases of cubic or hexagonal quasicrystals. The material constants should be replaced similarly with Section 2. The fixed coordinates are x 1 , x 2 , t and the moving coordinates are x , y. It is easy to show that 45 satisfies the boundary conditions of the problem.

This is equal to the Yoffe solution [ 64 ], in which the stress intensity factor is also independent of the crack moving speed V. Fan and his co-workers finished many systematical studies on elasticity and elasto-dynamics of quasicrystals.

In the following sections, we will reveal the general solutions of the dynamic problems of the point group 12 mm two-dimensional dodecagonal quasicrystals and give an example using this approach. A moving Dugdale model of the point group 12 mm two-dimensional dodecagonal quasicrystals is solved in this paper. The Dugdale-type plastic zones ahead of the crack tip are also obtained for when a crack in a dodecagonal quasicrystal is moving or growing at a constant speed V in the direction O x 1. Suppose the phason field has no contribution on phonon field deformation, we will only discuss the phonon field, which can provide a macroscopic description for this kind dynamic problem.

In general, the plastic deformation will occur in the vicinity of the crack tip when it expands or propagates. So we extend the Dugdale-Barenblatt model for the moving crack problem.

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Obviously, this simplified model is linearized. So we can utilize the superposition principle to prescribe the above problem apart from a negligible constant term as follows: Because the stresses at the ends of the cohesive force zone, i. This value is corresponding to the result of the Dugdale model in nonlinear static fracture of isotropic body. It is obvious that the second equation of 57 is a linearized equation of hydrodynamics of quasicrystals of Lubensky et al. This operation is believed to be reconcile the contradiction. Consider a moving dislocation in anti-plane problem of three-dimensional icosahedral or cubic or one-dimensional hexagonal quasicrystal.

In this case the stress and strain relations are regarded as Equation 4. Suppose the screw dislocation has a speed V , which moves along direction O x. The problem 78 , 79 is solved by means of perturbation incorporating the varitional method. First, we determine the zeroth-order solution.

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The Equation in 90 is very complicated. It can be solved approximately by the variational method. It is pointed out that the solution is composed of three parts, namely, wave propagation, diffusion motion and their interaction. The part of the phonon field is governed by wave propagation, and the part of the phason field is governed by diffusion motion. The solution reveals that the phason field impedes dislocation. For illustration of its applications, we only present some solutions in this section. Because two-dimensional decagonal quasicrystals occupy an important position in all quasicrystals observed to date, herein we consider only the two-dimensional decagonal quasicrystals.

Referring to Figure 7 , suppose the periodic direction is the z axis and a Griffith crack penetrates along the z axis in the solids. If the elastic field is induced by the uniform tensile stress at the upper and lower surfaces of the specimen, it is obvious that all variables of the elastic field are independent of z , i. As shown in Figure 7 , the decagonal quasicrystal with a Griffith crack is a rectangular specimen with a central crack of length 2 a t subjected to a dynamic or static tensile stress at its ends; a t is the crack length which is a function of time.

Due to the symmetry of the specimen only the upper right quarter is considered. For dynamic initial growth of the crack, suppose that the crack is stable, i. Here we only consider dynamic initiation of crack growth, then fast propagation of the crack is omitted. The content can be seen in [ 23 , 62 , 63 ].

Due to symmetry of the specimen, it is sufficient to analyze the elastic field in the upper right quarter. The initial conditions are that the initial displacements and velocities of the phonon field and phason field are equal to zero, i. We shall solve the problem by using the finite difference method. To this end, the field variables and boundary-initial conditions will be expressed in terms of the displacements.

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This can be done through the constitutive Equation Due to the coupling effect of the phonon and phason, the normalized dynamics stress intensity factor between quasicrystals and classical crystals is quite different. This fact declares that the phonon wave propagation is dominant in coupling of the wave propagation-motion of diffusion, and there are oscillations in the figure of the stress intensity factor that are especially clear.

The oscillations declare that the reflection and diffraction between waves come from the crack surface and the specimen boundary surfaces. The oscillations may be greatly influenced by the specimen geometry and the material constants. Referring to Figure 9 , consider a centered crack specimen of icosahedral QCs of plane problem. Suppose a Griffith crack penetrates through the specimen of icosahedral QCs along the quasi-periodic direction, i.

Consider a deformation of the specimen induced by a uniform tensile stress at upper and lower surfaces of icosahedral QCs such that it is obvious that all variables of elastic field are independent of z , i. Then all the equations will be simple. Suppose the initial displacements and velocities of the phonon field and phason field are equal to zero, i. In order to obtain the numerical results, we now pay attention to investigating the effect of the phonon and phason fields in the icosahedral Al-Pd-Mn QCs.

The phonon-phason coupling constant has not been measured for icosahedral QCs so far. Below we graphically display the results. Firstly, we will discuss the dynamic stress intensity factor in the dynamic loadings for: Please see Table This difference in Figure 10 a can be summarized as: So it is predictable that the difference will become very remarkable when the loading sustains. Also if K 1 is taken as a fixed value, and K 2 takes different values, the dynamic stress intensity factors are hardly altered.

Thus it can be declared that K 1 occupies a central role in the deformation of QCs, and this effect cannot be ignored. In addition, we can infer that in three-dimensional icosahedral QCs, the two elastic constants of phason field, K 1 plays a leading role.

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The fast crack propagation also belongs to nonlinear problem in the deformation of QCs, in which the movement rule of crack is unknown in advance. In the field of fracture mechanics research, the researchers always make some simplifying assumptions to the movement rule of crack when they give the analysis for the propagation of the crack. We suppose that the crack extends along the x -axial direction and is related to y -axial with symmetry. Suppose the pull stress in the vicinity of crack tip is the parameter to control the fracture of QCs.

By applying stress criterion, the fast crack propagation of icosahedral QCs specimen with a center crack is discussed in this section.

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In the mathematical iteration, the pull stress perpendicular to the crack surface of a point in the vicinity of the crack tip is calculated at any time. The results can be reflected by Figure 11 , Figure 12 and Figure The results can be seen in these figures, which reveal: Referring to Figure 14 , consider an icosahedral QC with a centered crack, the length of which is 2 a. Without loss of generality, we only consider the upper right quarter of the cracked specimen. The initial conditions including the initial displacements and velocity ofthe phonon field and phason fieldare equal to zeroes, i.

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