**3.1 Basic equations, boundary and initial conditions**

There are over 50% icosahedral quasicrystals among observed the quasicrystals to date, this shows this kind of systems in the material presents the most importance. Within icosahedral quasicrystals, the icosahedral Al-Pd-Mn quasicrystals are concerned in particular by researchers, for which especially a rich set of experimental data for elastic constants accumulated so far, this is useful to the computational practice. So we focus on the elastohydrodynamics of icosahedral Al-Pd-Mn quasicrystals here. From the previous section we have known there are lack of measured data for phason elastic constants, the computation has to take some data which are obtained by Monte Carlo simulation, this makes some undetermined factors for computational results for decagonal quasicrystals. This shows the discussion on icosahedral quasicrystals is more necessary, and the formalism and numerical results are presented in the following.

If considering only the plane problem, especially for the crack problems, there are much of similarities with those discussed in the previous section. We present herein only the part that are different.

For the plane problem, i.e.,

$$\frac{\partial \langle \rangle}{\partial z} = 0 \tag{11}$$

Elasto-Hydrodynamics of Quasicrystals and Its Applications 439

1 231 23 <sup>2</sup> , , , , d , *RK K R c ccd d*

 

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

Consider an icosahedral quasicrystal specimen with a Griffith crack shown in Fig. 1, all parameters of geometry and loading are the same with those given in the previous, but in

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

0 00 0 00

 

 

( , ,) ( , ,) ( , ,) <sup>000</sup>

We now concentrate on investigating the phonon and phason fields in the icosahedral Al-

phonon elastic moduli, for phason ones 1 2 *K K* 72 MPa, 37 MPa and the constant relevant to diffusion coefficient of phason is

 μs/g . On the phonon-phason coupling constant, there is no measured result for icosahedral quasicrystals so far, we take

The problem is solved by the finite difference method, the principle, scheme and algorithm are illustrated as those in the previous section, and shall not be repeated here. The testing for the physical model, scheme, algorithm and computer program are similar to those given in Section 2. The numerical results for dynamic initiation of crack growth problem, the phonon and

> <sup>0</sup> ( ) lim ( ) ( ,0, ) *yy x a Kt x a x t*

the results are illustrated in Fig. 6, in which the comparison with those of crystals are shown, one can see the effects of phason and phonon-phason coupling are evident very much.

 

*<sup>y</sup> x z ttt*

( , ,) 0 ( , ,) 0 ( , ,) 0 ( , ,) 0 ( , ,) 0 ( , ,) 0

*xt yt zt xt yt zt*

*u xyt u xyt u xyt w xyt w xyt w xyt u xyt u xyt u xyt ttt*

0

and the normalized dynamics stress intensity factor (D.S.I.F.) 0 0 *K t K t ap* ( ) ( )/

000

5.1 g/cm and

for "decoupled quasicrystals" or crystals.

 74.2 GPa, 70.4 GPa 

*xy zy y xy zy wH H y at x L*

*p t HHH y H xL*

 

understood as a manmade damping coefficient as in the previous section.

<sup>3</sup> *d* do not represent wave speed, but are diffusive coefficients and parameter

the boundary conditions there are some different points, which are given as below

0, 0, 0, 0, 0, 0 on 0 for 0 0, 0, 0, 0, 0, 0 on for 0 ( ), 0, 0, 0, 0, 0 on for0

*u w H H x y H*

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

*x yx zx x yx zx xx yx zx xx yx zx yy xy zy yy xy zy yy xy zy yy xy zy*

 

Pd-Mn quasicrystal, in which we take <sup>3</sup>

*<sup>w</sup>* 1 / 4.8 10 m s/kg=4.8 10 cm

for quasicrystals, and *R* / 0

phason displacements are shown in Fig. 5.

The dynamic stress intensity factor *K t*( ) is defined by

19 3 10 3

1 2

(15)

*HHH x L y H*

*HHH y x at*

   

may be

(16)

(17)

of the

is used,

in which

0

The initial conditions are

 

*y*

*u*

**3.2 Some results** 

*R* / 0.01 

The linearized elasto-hydrodynamics of icosahedral quasicrystals have non-zero displacements , *u wz z* apart from ,, , *u uww <sup>x</sup> <sup>y</sup> <sup>x</sup> <sup>y</sup>* , so in the strain tensors

$$\boldsymbol{\omega}\_{ij} = \frac{1}{2} (\frac{\partial \boldsymbol{u}\_i}{\partial \boldsymbol{\alpha}\_j} + \frac{\partial \boldsymbol{u}\_j}{\partial \boldsymbol{\alpha}\_i}) \qquad \boldsymbol{w}\_{ij} = \frac{\partial \boldsymbol{w}\_i}{\partial \boldsymbol{\alpha}\_j}$$

it increases some non-zero components compared with those in two-dimensional quasicrystals. In connecting with this, in the stress tensors, the non-zero components increase too relatively to two-dimensional ones. With these reasons, the stress-strain relation presents different nature with that of decagonal quasicrystals though the generalized Hooke's law has the same form with that in one- and two-dimensional quasicrystals, i.e.,

$$
\sigma\_{ij} = \mathbb{C}\_{ijkl}\varepsilon\_{kl} + R\_{ijkl}w\_{kl} \qquad \mathsf{H}\_{ij} = R\_{klij}\varepsilon\_{kl} + K\_{ijkl}w\_{kl}
$$

In particular the elastic constants are quite different from those discussed in the previous sections, in which the phonon elastic constants can be expressed such as

$$\mathbf{C}\_{ijkl} = \lambda \delta\_{ij} \delta\_{kl} + \mu (\delta\_{ik} \delta\_{jl} + \delta\_{il} \delta\_{jk}) \tag{12}$$

and the phason elastic constant matrix [K] and phonon-phason coupling elastic one [R] are defined by the formulas of Fan's monograph [1], which are not listed here again. Substituting these non-zero stress components into the equations of motion

$$
\rho \frac{\partial^2 \boldsymbol{u}\_i}{\partial \mathbf{t}^2} = \frac{\partial \sigma\_{ij}}{\partial \mathbf{x}\_j} \text{, } \kappa \frac{\partial \mathbf{w}\_i}{\partial \mathbf{t}} = \frac{\partial H\_{ij}}{\partial \mathbf{x}\_j} \tag{13}
$$

and through the generalized Hooke's law and strain-displacement relation we obtain the final dynamic equations as follows

2 2 2 2 22 2 2 22 2 2 2 2 1 12 2 3 22 2 2 2 22 2 2 2 2 22 2 2 2 2 2 12 1 3 22 2 2 2 2 2 2 2 2 2 ( ) (2 ) ( ) (2 ) ( *y y xx x x x x yy y y y y x x z z uu u u w uw w c cc c c t x t x y yx y x y uu u u w uw w c cc c c t x t x y yx y x y u u c t xy t* 2 2 2 2 2 2 3 2 2 2 2 2 22 22 2 2 2 2 123 22 22 2 222 )( 2 ) ( )( )(2 ) *<sup>y</sup> x x z z <sup>z</sup> <sup>y</sup> <sup>x</sup> <sup>x</sup> xzz xxz ww ww w u c xy xy x y w u uuu u wd wd wd t x y x y x x y y x y* 

$$\begin{aligned} \frac{\partial w\_y}{\partial t} + \theta w\_y &= d\_1(\frac{\hat{\sigma}^2}{\hat{\alpha}\mathbf{x}^2} + \frac{\hat{\sigma}^2}{\hat{\alpha}\mathbf{y}^2})w\_y - d\_2 \frac{\hat{\sigma}^2 w\_z}{\hat{\alpha}\hat{\alpha}\mathbf{y}} + d\_3(\frac{\hat{\sigma}^2 u\_y}{\hat{\alpha}\mathbf{x}^2} + 2\frac{\hat{\sigma}^2 u\_x}{\hat{\alpha}\mathbf{x}\hat{\alpha}} - \frac{\hat{\sigma}^2 u\_y}{\hat{\alpha}\mathbf{y}^2} - 2\frac{\hat{\sigma}^2 u\_z}{\hat{\alpha}\mathbf{x}\hat{\alpha}}) \\ \frac{\partial w\_z}{\partial t} + \theta w\_z &= (d\_1 - d\_2)(\frac{\hat{\sigma}^2}{\hat{\alpha}\mathbf{x}^2} + \frac{\hat{\sigma}^2}{\hat{\alpha}\mathbf{y}^2})w\_z + d\_2(\frac{\hat{\sigma}^2 w\_x}{\hat{\alpha}\mathbf{x}^2} - \frac{\hat{\sigma}^2 w\_x}{\hat{\alpha}\mathbf{y}^2} - 2\frac{\hat{\sigma}^2 w\_y}{\hat{\alpha}\mathbf{x}\hat{\alpha}}) + d\_3(\frac{\hat{\sigma}^2}{\hat{\alpha}\mathbf{x}^2} + \frac{\hat{\sigma}^2}{\hat{\alpha}\mathbf{y}^2})u\_z \end{aligned} \tag{14}$$

in which

438 Hydrodynamics – Advanced Topics

The linearized elasto-hydrodynamics of icosahedral quasicrystals have non-zero

it increases some non-zero components compared with those in two-dimensional quasicrystals. In connecting with this, in the stress tensors, the non-zero components increase too relatively to two-dimensional ones. With these reasons, the stress-strain relation presents different nature with that of decagonal quasicrystals though the generalized Hooke's law has the same form with that in one- and two-dimensional

*ij C Rw H R Kw ijkl kl ijkl kl ij klij kl ijkl kl*

In particular the elastic constants are quite different from those discussed in the previous

( ) *Cijkl i*

and the phason elastic constant matrix [K] and phonon-phason coupling elastic one [R] are

, *ij <sup>i</sup>*

and through the generalized Hooke's law and strain-displacement relation we obtain the

 

*<sup>j</sup> kl ik jl il jk*

 

*j*

*w H t x*

( ) (2 )

( ) (2 )

2 2 2 2 2

*ww ww w*

*xy xy x y*

2 2 2 2 2 2 2

3 2 2 2 2 2 22 22 2 2 2 2 123 22 22 2 222

 

( )( )(2 )

*<sup>y</sup> <sup>x</sup> <sup>x</sup> xzz xxz*

2 2 2 2 2 22

( ) ( 2 2)

1 2 22 2 2 2 3 2 2

( )( ) ( 2 )( )

*w dd wd d u t x xy x y y x y*

*<sup>y</sup> z x <sup>x</sup> z z <sup>z</sup>*

*w u uuu u*

 

*t x y x y x x y y x y*

1 2 2 2 3 2 2

*y y <sup>y</sup> z xz y y*

*t x x y y x y x y x y*

*w u w uu u*

 

)( 2 )

*<sup>y</sup> x x z z <sup>z</sup>*

(12)

(13)

(14)

*j i i*

 

*j i j u w u w xx x*

<sup>1</sup> ( )

*ij ij*

displacements , *u wz z* apart from ,, , *u uww <sup>x</sup> <sup>y</sup> <sup>x</sup> <sup>y</sup>* , so in the strain tensors

2

sections, in which the phonon elastic constants can be expressed such as

2 2 *ij i j*

*u t x*

2 2 2 2 22 2 2 22 2 2 2 2 1 12 2 3 22 2 2 2 22 2 2 2 2 22 2 2 2 2 2 12 1 3 22 2

 

*c cc c c*

*c cc c c*

*u c*

*wd wd wd*

*wd wd d*

2

*w w w w*

*y y xx x x x x*

*uu u u w uw w*

*t x t x y yx y x y uu u u w uw w*

*t x t x y yx y x y*

*yy y y y y x x*

defined by the formulas of Fan's monograph [1], which are not listed here again. Substituting these non-zero stress components into the equations of motion

final dynamic equations as follows

2 2 2 2 2 2 2 2

 

*c t xy t*

*z z*

*u u*

(

quasicrystals, i.e.,

$$c\_1c\_1 = \sqrt{\frac{\lambda + 2\mu}{\rho}}, c\_2 = \sqrt{\frac{\mu}{\rho}}, c\_3 = \sqrt{\frac{R}{\rho}}, d\_1 = \frac{K\_1}{\kappa}, \text{ d}\_2 = \frac{K\_2}{\kappa}, d\_3 = \frac{R}{\kappa} \tag{15}$$

note that constants 1 2 *c c*, and <sup>3</sup> *c* have the meaning of elastic wave speeds, while 1 2 *d d*, and <sup>3</sup> *d* do not represent wave speed, but are diffusive coefficients and parameter may be understood as a manmade damping coefficient as in the previous section.

Consider an icosahedral quasicrystal specimen with a Griffith crack shown in Fig. 1, all parameters of geometry and loading are the same with those given in the previous, but in the boundary conditions there are some different points, which are given as below

0, 0, 0, 0, 0, 0 on 0 for 0 0, 0, 0, 0, 0, 0 on for 0 ( ), 0, 0, 0, 0, 0 on for0 0, 0, 0, 0, 0, 0 on 0 for 0 ( ) 0 *x yx zx x yx zx xx yx zx xx yx zx yy xy zy yy xy zy yy xy zy yy xy zy y u w H H x y H HHH x L y H p t HHH y H xL HHH y x at u* , 0, 0, 0, 0, 0 on 0 for ( ) *xy zy y xy zy wH H y at x L* (16)

The initial conditions are

$$\begin{aligned} \left.u\_x(\mathbf{x}, y, t)\right|\_{t=0} &= 0 & \left.u\_y(\mathbf{x}, y, t)\right|\_{t=0} &= 0 & \left.u\_z(\mathbf{x}, y, t)\right|\_{t=0} &= 0\\ \left.w\_x(\mathbf{x}, y, t)\right|\_{t=0} &= 0 & \left.w\_y(\mathbf{x}, y, t)\right|\_{t=0} &= 0 & \left.w\_z(\mathbf{x}, y, t)\right|\_{t=0} &= 0\\ \left.\frac{\partial u\_x(\mathbf{x}, y, t)}{\partial t}\right|\_{t=0} &= 0 & \left.\frac{\partial u\_y(\mathbf{x}, y, t)}{\partial t}\right|\_{t=0} &= 0 & \left.\frac{\partial u\_z(\mathbf{x}, y, t)}{\partial t}\right|\_{t=0} \end{aligned} \tag{17}$$

#### **3.2 Some results**

We now concentrate on investigating the phonon and phason fields in the icosahedral Al-Pd-Mn quasicrystal, in which we take <sup>3</sup> 5.1 g/cm and 74.2 GPa, 70.4 GPa of the phonon elastic moduli, for phason ones 1 2 *K K* 72 MPa, 37 MPa and the constant relevant to diffusion coefficient of phason is 19 3 10 3 *<sup>w</sup>* 1 / 4.8 10 m s/kg=4.8 10 cm μs/g . On the phonon-phason coupling constant, there is no measured result for icosahedral quasicrystals so far, we take *R* / 0.01 for quasicrystals, and *R* / 0 for "decoupled quasicrystals" or crystals. The problem is solved by the finite difference method, the principle, scheme and algorithm are illustrated as those in the previous section, and shall not be repeated here. The testing for the physical model, scheme, algorithm and computer program are similar to those given in Section 2. The numerical results for dynamic initiation of crack growth problem, the phonon and phason displacements are shown in Fig. 5.

The dynamic stress intensity factor *K t*( ) is defined by

$$K\_{\mathbf{I}}(t) = \lim\_{\mathbf{x} \to a\_0^+} \sqrt{\pi(\mathbf{x} - a\_0)} \sigma\_{yy}(\mathbf{x}, \mathbf{0}, t)$$

and the normalized dynamics stress intensity factor (D.S.I.F.) 0 0 *K t K t ap* ( ) ( )/ is used, the results are illustrated in Fig. 6, in which the comparison with those of crystals are shown, one can see the effects of phason and phonon-phason coupling are evident very much.

Elasto-Hydrodynamics of Quasicrystals and Its Applications 441

Fig. 7. Normalized stress intensity factor of propagating crack with constant crack speed

In Sections 1 through 3 a new model on dynamic response of quasicrystals based on argument of Lubensky et al is formulated. This model is regarded as an elastohydrodynamics model for the material, or as a collaborating model of wave propagation and diffusion. This model is more complex than pure wave propagation model for conventional crystals, the analytic solution is very difficult to obtain, except a few simple examples introduced in Fan's monograph [1]. Numerical procedure based on finite difference algorithm is developed. Computed results confirm the validity of wave propagation behaviour of phonon field, and behaviour of diffusion of phason field. The

The finite difference formalism is applied to analyze dynamic initiation of crack growth and crack fast propagation for two-dimensional decagonal Al-Ni-Co and three-dimensional icosahedral Al-Pd-Mn quasicrystals, the displacement and stress fields around the tip of stationary and propagating cracks are revealed, the stress present singularity with order 1/2 *r* , in which *r* denotes the distance measured from the crack tip. For the fast crack propagation, which is a nonlinear problem—moving boundary problem, one must provide additional condition for determining solution. For this purpose we give a criterion for checking crack propagation/crack arrest based on the critical stress criterion. Application of this additional condition for determining solution has helped us to achieve the numerical simulation of the moving boundary value problem and revealed crack length-time evolution. However, more important and difficult problems are left open for further study. Up to now the arguments on the physical meaning of phason variables based on hydrodynamics within different research groups have not been ended yet, see e.g. Coddens

[11], which may be solved by further experimental and theoretical investigations.

Details of this work can be given by Fan and co-workers [1], [10].

interaction between phonons and phasons are also recorded.

versus time.

**4. Conclusion and discussion** 

Fig. 5. Displacement components of quasicrystals versus time. (a)displacement component *ux* ; (b)displacement component *uy* ; (c)displacement component *wx* ;(d)displacement component *wy*

For the fast crack propagation problem the primary results are listed only the dynamic stress intensity factor versus time as below

Fig. 6. Normalized dynamic stress intensity factor of central crack specimen under impact loading versus time

For the fast crack propagation problem the primary results are listed only the dynamic stress

Fig. 6. Normalized dynamic stress intensity factor of central crack specimen under impact

Fig. 5. Displacement components of quasicrystals versus time. (a)displacement component *ux* ; (b)displacement component *uy* ; (c)displacement component *wx* ;(d)displacement component *wy*

intensity factor versus time as below

loading versus time

Fig. 7. Normalized stress intensity factor of propagating crack with constant crack speed versus time.

Details of this work can be given by Fan and co-workers [1], [10].
