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

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

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

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

Elasto-Hydrodynamics of Quasicrystals and Its Applications 437

Fig. 4. Normalized dynamics stress intensity factor (DSIF) versus time

constants and specimen geometry including the shape and size very much.

**dimensional icosahedral quasicrystals** 

results are presented in the following.

that are different.

For the plane problem, i.e.,

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

There are some oscillations of values of the stress intensity factor in the figure. These oscillations characterize the reflection and diffraction between waves coming from the crack surface and the specimen boundary surfaces. The oscillations are influenced by the material

**3. Elasto-/hydro-dynamics and applications to fracture dynamics of three-**

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

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

> ( ) <sup>0</sup> *z*

(11)

Fig. 3. Comparison of the present solution with analytic solution and other numerical solution for conventional structural materials given by other authors


Table 1. The normalized static S.I.F. of quasicrystals for different space steps

that are subjected to a rapidly varying applied load <sup>0</sup> *p*() () *t p f t* , where 0 *p* is a constant with stress dimension and *f* ( )*t* is taken as the Heaviside function. It is well known the coupling effect between phonon and phason is very important, which reveals the distinctive physical properties including mechanical properties, and makes quasicrystals distinguish the periodic crystals. So studying the coupling effect is significant.

The dynamic stress intensity factor *K t*( ) for quasicrystals has the same definition given by equation (10), whose numerical results are plotted in Fig. 4, where the normalized dynamics stress intensity factor 0 0 *K t ap* ( )/ is used. There are two curves in the Fig. 4, one represents quasicrystal, i.e., *R M*/ 0.01 , the other describes periodic crystals corresponding to *R M*/ 0 , the two curves of the Fig. 4 are apparently different, though they are similar to some extends. Because of the phonon-phason coupling effect, the mechanical properties of the quasicrystals are obviously different from the classical crystals. Thus, the coupling effect plays an important role.

In Fig. 4, 0*t* represents the time that the wave from the external boundary propagates to the crack surface, in which <sup>0</sup>*t* 2.6735 μs . So the velocity of the wave propagation is 0 0 *H t* / 7.4807 km/s , which is just equal to the longitudinal wave speed <sup>1</sup>*c LM* ( 2 )/ . This indicates that for the complex system of wave propagation-motion of diffusion coupling, the phonon wave propagation presents dominating role.

Fig. 3. Comparison of the present solution with analytic solution and other numerical

H a0/10 a0/15 a0/20 a0/30 a0/40 K 0.9259 0.94829 0.9229 0.97723 0.99516 Errors 7.410% 5.171% 3.771% 2.277% 0.484%

that are subjected to a rapidly varying applied load <sup>0</sup> *p*() () *t p f t* , where 0 *p* is a constant with stress dimension and *f* ( )*t* is taken as the Heaviside function. It is well known the coupling effect between phonon and phason is very important, which reveals the distinctive physical properties including mechanical properties, and makes quasicrystals distinguish

The dynamic stress intensity factor *K t*( ) for quasicrystals has the same definition given by equation (10), whose numerical results are plotted in Fig. 4, where the normalized dynamics

represents quasicrystal, i.e., *R M*/ 0.01 , the other describes periodic crystals corresponding to *R M*/ 0 , the two curves of the Fig. 4 are apparently different, though they are similar to some extends. Because of the phonon-phason coupling effect, the mechanical properties of the quasicrystals are obviously different from the classical crystals.

In Fig. 4, 0*t* represents the time that the wave from the external boundary propagates to the crack surface, in which <sup>0</sup>*t* 2.6735 μs . So the velocity of the wave propagation is

0 0 *H t* / 7.4807 km/s , which is just equal to the longitudinal wave speed

of diffusion coupling, the phonon wave propagation presents dominating role.

. This indicates that for the complex system of wave propagation-motion

is used. There are two curves in the Fig. 4, one

solution for conventional structural materials given by other authors

the periodic crystals. So studying the coupling effect is significant.

stress intensity factor 0 0 *K t ap* ( )/

<sup>1</sup>*c LM* ( 2 )/

Thus, the coupling effect plays an important role.

Table 1. The normalized static S.I.F. of quasicrystals for different space steps

Fig. 4. Normalized dynamics stress intensity factor (DSIF) versus time

There are some oscillations of values of the stress intensity factor in the figure. These oscillations characterize the reflection and diffraction between waves coming from the crack surface and the specimen boundary surfaces. The oscillations are influenced by the material constants and specimen geometry including the shape and size very much.
