**2. The elasto-hydrodynamics of two-dimensional decagonal quasicrystals and application to dynamic fracture**

## **2.1 Statement of formulation and sample problem**

Among over 200 quasicrystals observed to date, there are over 70 two-dimensional decagonal quasicrystals, so this kind of solid phases play an important role in the material. For simplicity, here only point group 10mm two-dimensional decagonal quasicrystals will be considered. We denote the periodic direction as the *z* axis and the quasiperiodic plane as the *x y* plane. Assume that a Griffith crack in the solid along the periodic direction, i.e., the *z* axis. It is obvious that elastic field induced by a uniform tensile stress at upper and lower surfaces of the specimen is independent of *z* , so ( )/ 0 *z* . In this case, the stressstrain relations are reduced to

$$\begin{aligned} \sigma\_{xx} &= L(\varepsilon\_{xx} + \varepsilon\_{yy}) + 2M\varepsilon\_{xx} + R(w\_{xx} + w\_{yy}) \\ \sigma\_{yy} &= L(\varepsilon\_{xx} + \varepsilon\_{yy}) + 2M\varepsilon\_{yy} - R(w\_{xx} + w\_{yy}) \\ \sigma\_{xy} &= \sigma\_{yx} = 2M\varepsilon\_{xy} + R(w\_{yx} - w\_{xy}) \\ H\_{xx} &= K\_1 w\_{xx} + K\_2 w\_{yy} + R(\varepsilon\_{xx} - \varepsilon\_{yy}) \\ H\_{yy} &= K\_1 w\_{yy} + K\_2 w\_{xx} + R(\varepsilon\_{xx} - \varepsilon\_{yy}) \\ H\_{xy} &= K\_1 w\_{xy} - K\_2 w\_{yx} - 2R\varepsilon\_{xy} \\ H\_{yx} &= K\_1 w\_{yx} - K\_2 w\_{xy} + 2R\varepsilon\_{xy} \end{aligned} \tag{5}$$

where 12 11 12 *LC M C C* , ( )/2 are the phonon elastic constants, *K*<sup>1</sup> and *K*<sup>2</sup> are the phason elastic constants, *R* phonon-phason coupling elastic constant, respectively.

Substituting equations (5) into equations (3), (4) we obtain the equations of motion of decagonal quasicrystals as following:

$$\begin{aligned} \frac{\partial^2 u\_x}{\partial t^2} &= c\_1^2 \frac{\partial^2 u\_x}{\partial x^2} + (c\_1^2 - c\_2^2) \frac{\partial^2 u\_y}{\partial x \partial y} + c\_2^2 \frac{\partial^2 u\_x}{\partial y^2} + c\_3^2 (\frac{\partial^2 w\_x}{\partial x^2} + 2 \frac{\partial^2 w\_y}{\partial x \partial y} - \frac{\hat{\sigma}^2 w\_x}{\partial y^2}) \\ \frac{\partial^2 u\_y}{\partial t^2} &= c\_2^2 \frac{\partial^2 u\_y}{\partial x} + (c\_1^2 - c\_2^2) \frac{\partial^2 u\_x}{\partial x \partial y} + c\_1^2 \frac{\partial^2 u\_y}{\partial y^2} + c\_3^2 (\frac{\partial^2 w\_y}{\partial x^2} - 2 \frac{\partial^2 w\_x}{\partial x \partial y} - \frac{\hat{\sigma}^2 w\_y}{\partial y^2}) \\ \frac{\partial w\_x}{\partial t} &= d\_1^2 (\frac{\partial^2 w\_x}{\partial x^2} + \frac{\partial^2 w\_x}{\partial y^2}) + d\_2^2 (\frac{\partial^2 u\_x}{\partial x^2} - 2 \frac{\partial^2 u\_y}{\partial x \partial y} - \frac{\hat{\sigma}^2 u\_x}{\partial y^2}) \\ \frac{\partial w\_y}{\partial t} &= d\_1^2 (\frac{\partial^2 w\_y}{\partial x^2} + \frac{\partial^2 w\_y}{\partial y^2}) + d\_2^2 (\frac{\partial^2 u\_y}{\partial x^2} + 2 \frac{\partial^2 u\_x}{\partial x \partial y} - \frac{\hat{\sigma}^2 u\_y}{\partial y^2}) \end{aligned} \tag{6}$$

where

430 Hydrodynamics – Advanced Topics

*w H t x*

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 elasto-

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

 

elastic constants, *R* phonon-phason coupling elastic constant, respectively.

2 22 2 2

*c cc c c*

2 22 2 2

*c cc c c*

2 22 2

22 2 2 2

( )( 2 )

1 2 22 2 2

*y yy y y x*

*w ww u u <sup>u</sup> d d t x xy x y y*

( )( 2

 

*y x xx x*

*w ww u u d d t xy x*

 

2 2 1 2 22 2

2 2

*H Kw Kw R H Kw Kw R H Kw Kw R H Kw Kw R*

 

 

*xy yx xy yx xy xx xx yy xx yy yy yy xx xx yy xy xy yx xy yx yx xy xy*

 

 

**2. The elasto-hydrodynamics of two-dimensional decagonal quasicrystals** 

Among over 200 quasicrystals observed to date, there are over 70 two-dimensional decagonal quasicrystals, so this kind of solid phases play an important role in the material. For simplicity, here only point group 10mm two-dimensional decagonal quasicrystals will be considered. We denote the periodic direction as the *z* axis and the quasiperiodic plane as the *x y* plane. Assume that a Griffith crack in the solid along the periodic direction, i.e., the *z* axis. It is obvious that elastic field induced by a uniform tensile stress at upper and lower surfaces of the specimen is independent of *z* , so ( )/ 0 *z* . In this case, the stress-

> ( )2 ( ) ( )2 ( ) 2( )

*xx xx yy xx xx yy yy xx yy yy xx yy*

 

*L MR w w L MR w w M Rw w*

 

 

> 2 2

where 12 11 12 *LC M C C* , ( )/2 are the phonon elastic constants, *K*<sup>1</sup> and *K*<sup>2</sup> are the phason

Substituting equations (5) into equations (3), (4) we obtain the equations of motion of

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

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

*u u u w uw w*

*y y x x xx x*

2 2 2 12 1 3 22 2

*t x x y yx y x y*

*y y yy y x x*

2 2 2 2 22 2

 

( ) ( ) (5)

(6)

 

 

( ) (2 )

( ) (2 )

*x y y*

2 2

)

*x*

*u*

hydrodynamic equations of quasicrystals.

**and application to dynamic fracture** 

strain relations are reduced to

decagonal quasicrystals as following:

**2.1 Statement of formulation and sample problem** 

*ij i j*

(4)

$$c\_1 = \sqrt{\frac{L + 2M}{\rho}}, c\_2 = \sqrt{\frac{M}{\rho}}, c\_3 = \sqrt{\frac{R}{\rho}}, d\_1 = \sqrt{\frac{K\_1}{\kappa}} \quad d\_2 = \sqrt{\frac{R}{\kappa}}$$

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 not represent wave speed, and <sup>2</sup> <sup>1</sup> *d* and <sup>2</sup> <sup>2</sup> *d* are diffusive coefficients in physical meaning.

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 right quarter is considered.

Fig. 1. The specimen with a central crack

Referring to the upper right part and considering a fix grips case, the following boundary conditions should be satisfied:

$$\begin{aligned} \mu\_x &= 0, \sigma\_{yx} = 0, \sigma\_{x} = 0, H\_{yx} = 0 & \text{on } x = 0 \quad \text{for } 0 \le y \le H\\ \sigma\_{xx} &= 0, \sigma\_{yx} = 0, H\_{xx} = 0, H\_{yx} = 0 & \text{on } x = L \text{ for } 0 \le y \le H\\ \sigma\_{yy} &= p(t), \sigma\_{xy} = 0, H\_{yy} = 0, H\_{xy} = 0 & \text{on } y = H \text{ for } 0 \le x \le L\\ \sigma\_{yy} &= 0, \sigma\_{xy} = 0, H\_{yy} = 0, H\_{xy} = 0 & \text{on } y = 0 \text{ for } 0 \le x \le a(t)\\ \mu\_y &= 0, \sigma\_{xy} = 0, w\_y = 0, H\_{xy} = 0 & \text{on } y = 0 \text{ for } a(t) \le x \le L \end{aligned} \tag{7}$$

Elasto-Hydrodynamics of Quasicrystals and Its Applications 433

Fig. 2. (a) Displacement component of phonon field *ux* versus time

Fig. 2. (b) Displacement component of phonon field *uy* versus time

in which 0 *p*() () *t p f t* is a dynamic load if *f* ( )*t* varies with time, otherwise it is a static load (i.e., if *f*( )*t const* ), and 0 *p const* with the stress dimension. . The initial conditions are

$$\begin{aligned} \left.u\_x(\mathbf{x}, \mathbf{y}, t)\right|\_{t=0} &= 0 & \left.u\_y(\mathbf{x}, \mathbf{y}, t)\right|\_{t=0} &= 0\\ \left.w\_x(\mathbf{x}, \mathbf{y}, t)\right|\_{t=0} &= 0 & \left.w\_y(\mathbf{x}, \mathbf{y}, t)\right|\_{t=0} &= 0\\ \left.\frac{\partial u\_x(\mathbf{x}, \mathbf{y}, t)}{\partial t}\right|\_{t=0} &= 0 & \left.\frac{\partial u\_y(\mathbf{x}, \mathbf{y}, t)}{\partial t}\right|\_{t=0} &= 0 \end{aligned} \tag{8}$$

For implementation of finite difference all field variables in governing equations (6) and boundary-initial conditions (7), (8) must be expressed by displacements and their derivatives. This can be done through the constitutive equations (2). The detail of the finite difference scheme is omitted here but can be referred to Fan [1].

For the related parameters in this section, the experimentally determined mass density for decagonal Al-Ni-Co quasicrystal 3 -3 4.186 10 g mm is used and phonon elastic moduli are <sup>12</sup> <sup>2</sup> <sup>12</sup> <sup>2</sup> <sup>11</sup> <sup>12</sup> *C C* 2.3433 10 dyn/ cm , 0.5741 10 d yn/cm <sup>10</sup> <sup>2</sup> (10 dyn/cm GPa) which are obtained by resonant ultrasound spectroscopy, refer to Chernikov et al [6], we have also chosen phason elastic constants <sup>12</sup> <sup>2</sup> <sup>1</sup> *K* 1.22 10 dyn/cm and <sup>12</sup> <sup>2</sup> <sup>2</sup> *K* 0.24 10 dyn/ cm <sup>10</sup> <sup>2</sup> (10 dyn/cm GPa) estimated by Monto-Carlo simulation given by Jeong and Steinhardt [7] and 19 3 10 3 1 / 4.8 10 m s/kg=4.8 10 cm μs/g  *w* which measured by de Boussieu and collected by Walz in his master thesis [8].The coupling constant *R* has been measured for some special cases recently, see Chapter 6 and Chapter 9 of monograph written by Fan [1] respectively. In computation we take *R M*/ 0.01 for coupling case corresponding to quasicrystals, and *R M*/ 0 for decoupled case which corresponds to crystals.

#### **2.2 Examination on the physical model**

In order to verify the correctness of the suggested model and the numerical simulation, we first explore the specimen without a crack. We know that there are the fundamental solutions characterizing time variation natures based on wave propagation of phonon field and on motion of diffusion of phason, respectively according to mathematical physics

$$\begin{cases} u - e^{i\alpha(t-x/c)} \\ w - \frac{1}{\sqrt{t-t\_0}} e^{-(x-x\_0)^2/\Gamma\_w(t-t\_0)} \end{cases} \tag{9}$$

where is a frequency and *c* a speed of the wave, *t* the time and 0*t* a special value of *t* , *x* the distance, 0 *x* a special value of *x* , and *<sup>w</sup>* the kinetic coefficient of phason defined previously. Comparison results are shown in Fig.2 (a-c), in which the solid line represents the numerical solution of quasicrystals and the dotted line represents fundamental solution given by formulas (9). From Fig. 2(a) and (b) we can see that both displacement components of phonon field are in excellent agreement to the fundament solutions of mathematical physics. However, there are some differences because the phonon field is influenced by phason field

in which 0 *p*() () *t p f t* is a dynamic load if *f* ( )*t* varies with time, otherwise it is a static load

0 0 0 0

 

(8)

<sup>2</sup> *K* 0.24 10 dyn/ cm

 

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

*xt yt xt yt*

*u xyt u xyt w xyt w xyt u xyt u xyt t t*

*<sup>y</sup> <sup>x</sup> t t*

( , ,) ( , ,) 0 0

For implementation of finite difference all field variables in governing equations (6) and boundary-initial conditions (7), (8) must be expressed by displacements and their derivatives. This can be done through the constitutive equations (2). The detail of the finite

For the related parameters in this section, the experimentally determined mass density for

<sup>10</sup> <sup>2</sup> (10 dyn/cm GPa) estimated by Monto-Carlo simulation given by Jeong and Steinhardt

and collected by Walz in his master thesis [8].The coupling constant *R* has been measured for some special cases recently, see Chapter 6 and Chapter 9 of monograph written by Fan [1] respectively. In computation we take *R M*/ 0.01 for coupling case corresponding to

In order to verify the correctness of the suggested model and the numerical simulation, we first explore the specimen without a crack. We know that there are the fundamental solutions characterizing time variation natures based on wave propagation of phonon field and on motion of diffusion of phason, respectively according to mathematical physics

( /)

*i txc*

*w e t t*

*u e*

 

0

distance, 0 *x* a special value of *x* , and *<sup>w</sup>* the kinetic coefficient of phason defined previously. Comparison results are shown in Fig.2 (a-c), in which the solid line represents the numerical solution of quasicrystals and the dotted line represents fundamental solution given by formulas (9). From Fig. 2(a) and (b) we can see that both displacement components of phonon field are in excellent agreement to the fundament solutions of mathematical physics. However, there are some differences because the phonon field is influenced by phason field

2 0 0

is a frequency and *c* a speed of the wave, *t* the time and 0*t* a special value of *t* , *x* the

1 *<sup>w</sup>*

( )/ ( )

*xx tt*

quasicrystals, and *R M*/ 0 for decoupled case which corresponds to crystals.

<sup>11</sup> <sup>12</sup> *C C* 2.3433 10 dyn/ cm , 0.5741 10 d yn/cm <sup>10</sup> <sup>2</sup> (10 dyn/cm GPa) which are obtained by resonant ultrasound spectroscopy, refer to Chernikov et al [6], we have also

which measured by de Boussieu

4.186 10 g mm is used and phonon elastic moduli

<sup>1</sup> *K* 1.22 10 dyn/cm and <sup>12</sup> <sup>2</sup>

(9)

0 0

(i.e., if *f*( )*t const* ), and 0 *p const* with the stress dimension. .

difference scheme is omitted here but can be referred to Fan [1].

decagonal Al-Ni-Co quasicrystal 3 -3

are <sup>12</sup> <sup>2</sup> <sup>12</sup> <sup>2</sup>

chosen phason elastic constants <sup>12</sup> <sup>2</sup>

[7] and 19 3 10 3 1 / 4.8 10 m s/kg=4.8 10 cm μs/g *-*

The initial conditions are

*w*

where  **2.2 Examination on the physical model** 

Fig. 2. (a) Displacement component of phonon field *ux* versus time

Fig. 2. (b) Displacement component of phonon field *uy* versus time

Elasto-Hydrodynamics of Quasicrystals and Its Applications 435

are the same with those given in classical references for conventional fracture dynamics, discussed in Fan's monograph [1] in detail. At first the comparison to the classical exact analytic solution is carried out, in this case we put 0 *w w x y* (i.e., 1 2 *KKR* 0 ) for the numerical solution. The comparison has been done with the key physical quantity—

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

The normalized dynamic stress intensity factor can be denoted as ( )/ *static Kt K I I* , in which *static KI*

dynamic initiation of crack growth in classical fracture dynamics there is the only exact analytic solution— the Maue's solution (refer to Fan's monograph [1]), but the configuration of whose specimen is quite different from that of our specimen. Maue studied a semi-infinite crack in an infinite body, and subjected to a Heaviside impact loading at the crack surface. While our specimen is a finite size rectangular plate with a central crack, and the applied stress is at the external boundary of the specimen. Generally the Maue's model cannot describe the interaction between wave and external boundary. However, consider a very short time interval, i.e., during the period between the stress wave from the external boundary arriving at the crack tip (this time is denoted by <sup>1</sup>*t* ) and before the reflecting by external boundary stress wave emanating from the crack tip in the finite size specimen (the time is marked as <sup>2</sup>*t* ). During this special very short time interval our specimen can be seen as an "infinite specimen". The comparison given by Fig. 3 shows the numerical results are in excellent agreement with those of Maue's solution within the

Our solution corresponding to case of 0 *w w x y* is also compared with numerical solutions of conventional crystals, e.g. Murti's solution and Chen's solutions (refer to Fan [1] and Zhu and Fan [9] for the detail), which are also shown in Fig. 3, it is evident, our solution

The mesh size or the space step of the algorithm can influence the computational accuracy too. To check the accuracy of the algorithm we take different space steps shown in Table 1, which indicates if *h a* <sup>0</sup> /40 the accuracy is good enough. The check is carried out through static solution, because the static crack problem in infinite body of decagonal quasicrystals has exact solution given in Chapter 8 of monograph given by Fan [1], and the normalized static intensity factor is equal to unit. In the static case, there is no wave propagation effect, *La Ha* / 3, / 3 0 0 the effect of boundary to solution is very weak, and for our present specimen *La Ha* / 4, / 8 0 0 , which may be seen as an infinite specimen, so the normalized static stress intensity factor is approximately but with highly precise equal to unit. The table

The dynamic crack problem presents two "phases" in the process: the dynamic initiation of crack growth and fast crack propagation. In the phase of dynamic initiation of crack growth, the length of the crack is constant, assuming <sup>0</sup> *at a* ( ) . The specimen with stationary crack

shows that the algorithm is with a quite highly accuracy when *h a* <sup>0</sup> /40.

 

(10)

*a p* **.** For the

0

is the corresponding static stress intensity factor, whose value here is taken as 0 0

dynamic stress intensity factor, which is defined by

short interval in which the solution is valid.

**2.3.3 Influence of mesh size (space step)** 

**2.4 Results of dynamic initiation of crack growth** 

presents very high precise.

Fig. 2. (c) Displacement component of phason field *wx* versus time

and the phonon-phason coupling effect. From Fig. 2(c), in the phason field we find that the phason mode presents diffusive nature in the overall tendency, but because of influence of the phonon and phonon-phason coupling, it can also have some characters of fluctuation. So the model describes the dynamic behaviour of phonon field and phason field in deed. This also shows the mathematical modeling of the present work is valid.
