**1. Introduction**

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Quasicrystal as a new structure of solids as well as a new material, has been studied over twenty five years. The elasticity and defects play a central role in field of mechanical behaviour of the material, see e.g. Fan [1]. Different from crystals and conventional engineering materials, quasicrystals have two different displacement fields: phonon field <sup>123</sup> *uu u u* (,,) and phason field 123 *ww w w* (,,) , which is a new degree of freedom to condensed matter physics as well as continuum mechanics, this leads to two strain tensors such as

$$\boldsymbol{\omega}\_{i\dot{j}} = \frac{1}{2} (\frac{\partial \boldsymbol{u}\_i}{\partial \boldsymbol{\alpha}\_j} + \frac{\partial \boldsymbol{u}\_j}{\partial \boldsymbol{\alpha}\_i}) \; \boldsymbol{w}\_{i\dot{j}} = \frac{\partial \boldsymbol{w}\_i}{\partial \boldsymbol{\alpha}\_j} \tag{1}$$

We call the first of equation (1) as phonon strain tensor, the second as phason strain tensor, respectively. The corresponding stress tensor is *ij* and *Hij* .

The constitutive law is the so-called generalized Hooke's law as follows

$$\begin{aligned} \sigma\_{ij} &= \mathbf{C}\_{ijkl}\boldsymbol{\varepsilon}\_{kl} + \mathbf{R}\_{ijkl}\boldsymbol{\pi}\_{kl} \\ \mathbf{H}\_{ij} &= \mathbf{K}\_{ijkl}\mathbf{w}\_{kl} + \mathbf{R}\_{klij}\boldsymbol{\varepsilon}\_{kl} \end{aligned} \tag{2}$$

in which *Cijkl* denotes the phonon elastic tensor, *Kijkl* the phason one, and *Rijkl* the phononphason coupling one, respectively. It is evident that the appearance of the new degree freedom yields a great challenge to the continuum mechanics.

In the dynamic process of quasicrystals problem presents further complexity. According to the point of view of Lubensky et al. [2,3], phonon represents wave propagation, while phason represents diffusion in the dynamic process. Following the argument of Lubensky et al., Rochal and Lorman [4] and Fan [1,5] put forward the equations of motion of quasicrystals as follows

$$
\rho \frac{\partial^2 u\_i}{\partial t^2} = \frac{\partial \sigma\_{ij}}{\partial x\_j} \tag{3}
$$

Elasto-Hydrodynamics of Quasicrystals and Its Applications 431

, 2

Note that constants 1 2 *c c*, and <sup>3</sup> *c* have the meaning of elastic wave speeds, while 1 *d* and <sup>2</sup> *d* do

A decagonal quasicrystal with a crack is shown in Fig.1. It is a rectangular specimen with a central crack of length 2() *a t* subjected to a dynamic or static tensile stress at its edges ED and FC, in which *a t*( )represents the crack length being a function of time, and for dynamic initiation of crack growth, the crack is stable, so 0 *at a* ( ) constant , for fast crack propagation, *a t*( ) varies with time. At first we consider dynamic initiation of crack growth, then study crack fast propagation. Due to the symmetry of the specimen only the upper

Referring to the upper right part and considering a fix grips case, the following boundary

0, 0, 0, 0 on 0 for 0 ( )

*pt H H y H x L*

*H H xL yH*

*H H y x at*

(7)

0, 0, 0, 0 on 0 for ( )

*u wH y at x L*

0, 0, 0, 0 on 0 for 0 0, 0, 0, 0 on for 0 ( ), 0, 0, 0 on for 0

*u wH x yH*

 

1 2 31 <sup>2</sup> , ,, *LM M R K c c cd*

<sup>1</sup> *d* and <sup>2</sup>

not represent wave speed, and <sup>2</sup>

right quarter is considered.

Fig. 1. The specimen with a central crack

*x yx x yx xx yx xx yx yy xy yy xy yy xy yy xy y xy y xy*

 

 

 

conditions should be satisfied:

1

<sup>2</sup> *d* are diffusive coefficients in physical meaning.

*<sup>R</sup> <sup>d</sup>* 

where

$$\kappa \frac{\partial w\_i}{\partial t} = \frac{\partial H\_{ij}}{\partial \mathbf{x}\_j} \tag{4}$$

Equation (3) is the equation of motion of conventional elastodynamics, and equation (4) is the linearized equation of hydrodynamics of Lubensky et al., so equations (3), (4) are elastohydrodynamic equations of quasicrystals.

The equations (1)-(4) are the basis of dynamic analysis of quasicrystalline material.
