# Dynamic Analysis of Cracked Octagonal Quasicrystals

Authors: Wu Li; Tian You Fan
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

We focus on the dynamic fracture problem of octagonal quasicrystals by applying a rectangular sample with a Griffith crack which is often used in classical elastic media based on the method of finite difference. This paper mainly is to investigate the variation of phonon, phason fields, and stress singularity around the crack tip including the stress intensity factor. In addition, the moving boundary due to the crack propagation has also been treated by introducing an additional condition for determining solution. The influence of wave propagation and diffusion in the dynamic process is also discussed in detail. Through comparing the results of octagonal quasicrystals with the results of crystal, this paper proclaims the influence of phonon-phason coupling in dynamic fracture problem of octagonal quasicrystals which should not be neglected.


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## Body

## 1. Introduction

Quasicrystals are a form of solids different from both crystals and glassy solids, which possess quasiperiodic long-range translational symmetry and noncrystallographic rotational symmetry [1]. Due to the quasiperiodicity, quasicrystals have a special type of elastic degrees of freedom, termed as phason degrees of freedom apart from phonon degrees which exist too for crystals. Under given load conditions, they are giving rise to two displacement fields u(r,t) for phonon field and w(r,t) for phason field, which also yield the elastic strain tensors εij and wij, respectively [2–4],(1.1)εij≡12(∂ui∂xj+∂uj∂xi),wij≡∂wi∂xj(wij≠wji).
In linear elasticity the stress tensors are related to the strain tensors obtained by [2](1.2)σij=Cijklεkl+Rijklwkl,Hij=Rklijεkl+Kijklwkl,
where ui, wi are phonon and phason displacements, σij and εij phonon stresses and strains, Hij and wij phason stresses and strains, Cijkl, Kijkl, and Rijkl the phonon, phason, phonon-phason coupling elastic constants, respectively. But for the elasto-/hydro-dynamics, there are different theoretical points of view, for example, those being put forward by Bak [5, 6] and by Lubensky et al [7]. Here we adopt the viewpoint raised by Fan [2]:(1.3)ρ∂2ui∂t2=∂σij∂xj,1Γw∂wi∂t=∂Hij∂xj,
in which Γw=1/κ denotes the kinetic coefficient of phason field.For this subject, Zhu and Fan and Wang et al. have used the icosahedral Al-Pd-Mn quasicrystal and the decagonal Al-Ni-Co quasicrystal to simulate dynamic behaviour of quasicrystals [8, 9]. However, little attention has been devoted to the octagonal quasicrystals which occupy a very important position in this solids. Of course, there are many theoretical and experimental results reported up to now [10–33]. Their results could be better convinced if they have considered the dynamic behaviour of octagonal quasicrystals.In the following sections, we will reveal the dynamic fracture behaviour of octagonal quasicrystals by introducing a rectangular sample with a Griffith crack based on the method of finite difference. In the end, we explain physical significance of the results obtained in the paper. Of course, further studies of the quasicrystals should be consummated by the future results. It is hoped that the dynamics question of quasicrystals will be observed by experiments.


## 2. Basic Formulas and Simplified Model

We focus on the solution for the plane elasticity of the octagonal quasicrystals that is, the atom arrangement along thez-direction is periodic and along the x-y plane is quasiperiodic. For the octagonal quasicrystals, we have the elastic constants matrix [CKR]:(2.1)[CKR]=[L+2ML00RR00LL+2M00-R-R0000MM00-RR00MM00-RRR-R00K1K200R-R00K2K10000-R-R00K1+K2+K3K300RR00K3K1+K2+K3].
In this case, the stress-strain relations are reduced to (2.2) from (2.1),(2.2)σxx=L(εxx+εyy)+2Mεxx+R(wxx+wyy),σyy=L(εxx+εyy)+2Mεyy-R(wxx+wyy),σxy=σyx=2Mεxy+R(wyx-wxy),Hxx=K1wxx+K2wyy+R(εxx-εyy),Hyy=K1wyy+K2wxx+R(εxx-εyy),Hxy=(K1+K2+K3)wxy+K3wyx-2Rεxy,Hyx=(K1+K2+K3)wyx+K3wxy+2Rεxy,
where L=C12, M=(C11-C12)/2 are the phonon elastic constants, K1, K2, and K3 are the phason elastic constants, and R is phonon-phason coupling elastic constant.Substituting (2.2) into (1.3), we obtain the equations of motion of octagonal quasicrystals given by displacement components as follows:(2.3)∂2ux∂t2=c12∂2ux∂x2+(c12-c22)∂2uy∂x∂y+c22∂2ux∂y2+c32(∂2wx∂x2+2∂2wy∂x∂y-∂2wx∂y2),∂2uy∂t2=c22∂2uy∂x2+(c12-c22)∂2ux∂x∂y+c12∂2uy∂y2+c32(∂2wy∂x2-2∂2wx∂x∂y-∂2wy∂y2),∂wx∂t=d12∂2wx∂x2+d22∂2wx∂y2+(d22-d12)∂2wy∂x∂y+d32(∂2ux∂x2-2∂2uy∂x∂y-∂2ux∂y2),∂wy∂t=d12∂2wy∂x2+d22∂2wy∂y2+(d22-d12)∂2wx∂x∂y+d32(∂2uy∂x2+2∂2ux∂x∂y-∂2uy∂y2),
in which(2.4)c1=L+2Mρ,c2=Mρ,c3=Rρ,d1=K1κ,d2=K1+K2+K3κ,d3=Rκ,
note that constants c1, c2, and c3 have the meaning of elastic wave speeds, while d12, d22, and d32 do not represent wave speed; they are diffusive coefficients.To ensure the uniform configuration along the periodic direction physically and geometrically, we consider such model that a rectangular octagonal quasicrystal containing a Griffith crack in the center is subjected a dynamic tensile stress on both upper and lower boundaries. Assume that the Griffith crack penetrates through the solid along the periodic direction. As shown in Figure1(a), the size of the specimen is that ED equals to 2L and EF equals to 2H, and a is the length of the crack, which is constant, and p(t) is the dynamic load, which varies with time. This represents the condition of the initiation of crack growth. As shown in Figure 1(b), the conditions are almost the same to that in Figure 1(a) except that the length of the crack a(t) is a function of time instead of constant and the dynamic load p(t)becomes constant.Sample of cracked quasicrystal under tension: (a) Crack stable with a dynamic load, (b) Crack fast propagation with a constant load.
(a)(b)As shown in Figures1(a) and 1(b), only the upper right quarter needs to consider due to the symmetry. Referring to the upper right part and considering a fix grips case, the following boundary conditions should be satisfied:(2.5)ux=0,σyx=0,wx=0,Hyx=0onx=0for0≤y≤H,σxx=0,σyx=0,Hxx=0,Hyx=0onx=Lfor0≤y≤H,σyy=p(t),σxy=0,Hyy=0,Hxy=0ony=Hfor0≤x≤L,σyy=0,σxy=0,Hyy=0,Hxy=0ony=0for0≤x≤a(t),uy=0,σxy=0,wy=0,Hxy=0ony=0fora(t)≤x≤L,
in which p(t) is the dynamic load.In addition, the initial conditions including the initial displacement and velocity are equal to zeroes:(2.6)ux(x,y,t)∣t=0=0,uy(x,y,t)∣t=0=0,wx(x,y,t)∣t=0=0,wy(x,y,t)∣t=0=0,∂ux∂t(x,y,t)∣t=0=0,∂uy∂t(x,y,t)∣t=0=0.In the following section, we let the related parameters2H=40mm and 2L=20mm, the initial length a0=2.4mm of the crack, and elastic constants can be found in Table 1.Table 1
The related parameters used in numerical computation [24, 26, 28] (unit: 1012 dyn/cm2).
C11C12K1K2K3κρ2.34300.57411.20000.24000.12001/4.8×1010 g/cm3 · μs54.186×10-3 g/cm3


## 3. Demonstrations of the Finite Difference Method and the Results

The final governing equation (2.3) along with boundary conditions equation (2.5) and initial conditions equation (2.6) are very complicated, analytic solution for the boundary-initial value problem is not available at present, which has to be solved by numerical method. Here we extend the method of finite difference of Shmuely and Alterman [22] scheme for analyzing crack problem for conventional engineering materials to quasicrystalline materials. A grid is imposed on the upper right of the specimen shown in Figure 2, the grid is extended beyond the half step.Figure 2
Finite difference scheme quarter specimen.Taking the finite difference scheme proposed by [22], (2.3) and the conditions equations (2.5) and (2.6) can be reduced to certain difference equations to solve. The details on numerical computation are listed in the Appendix. To check the validity of the model and the accuracy of the algorithm, at first we give a numerical example for phason displacement. Figure 3 depicts displacement component of phason field wx versus time. According to the present model phasons represent diffusion, the computational results indicate this is true, and the component wx is identical to the fundamental solution of pure diffusion in mathematical physics, that is,(3.1)w~1t-t0e-(x-x0)2/Γw(t-t0),
where t is the time t0 is a special value of t, x is a distance, and x0 is a special value of x. Though there is small fluctuation around the fundamental solution due to phonon-phason coupling, this checking confirms that the model is reasonable and the finite difference method and its computer implementation are effective and of high accuracy. In addition the phonons present the character of wave propagation this is natural and easily understood, and need not to give additional discussion and graphic illustration.Figure 3
Displacement component of phason fieldwx versus time and comparison with the fundamental solution of pure diffusion equation in mathematical physics.In the “phase” of dynamic initiation of crack growth, the specimen with stationary crack assuminga(t)=constant is subjected to a rapidly varying load p(t)=pf(t), where f(t) is taken as the Heaviside function. The stresses at the crack tip present singularity of order r-1/2, where r denotes the distance from the crack tip, so that we can obtain the dynamic stress intensity factor for phonon stress field such as(3.2)KI(t)=lim⁡x→a+2π(x-a)σyy(x,0,t)
and the normalized dynamics stress intensity factor; that is, KI(t)/πap (πap denotes the static stress intensity factor of a Griffith crack) versus time is depicted in Figure 4.Figure 4
Normalized S.I.F. for phonon stress field versus time.In Figure4, there are two curves in the figure: one represents quasicrystal, that is, R/M=0.01, and the other describes periodic crystals corresponding to R/M=0. Because of the phonon-phason coupling effect, the mechanical properties of the quasicrystals are obviously different from the classical crystals. But the phonon wave propagation plays dominating role in dynamics of quasicrystals.The follows are the cases that the specimen with a stationary crack are subjected to two most common types of pulse dynamic load. The pulses possess different periodT0, respectively.From Figure5, it can be easily concluded that S.I.F. is influenced greatly by the shape of the load of the pulse imposed. Also it is interesting that there is a time of lag between the variables and the load getting extremum. The reason for that is the influence of stress wave propagation as well as the effect of refraction and reflection. It is obviously seen that the shape of these sets of waves in the initial is like sawtooth. there will be of a bit fluctuations in the process when the wave reaches the second peak value. From the above analysis it can be concluded that the curve of S.I.F. is influenced greatly by the shape of the load of the pulse imposed, and the shapes of both are similar.Figure 5
Normalized S.I.F. versus time for different wave length of zigzag pulse dynamic loads.From Figure6, we conclude that the curved line of S.I.F. has relationships with the energy of the relative load. The larger the time of the pulse continues, the more mildly the S.I.F. varies also in the same conditions S.I.F. is larger on average in the long run.Figure 6
Normalized S.I.F. versus time for different wave length of rectangular pulse dynamic loads.Now we focus on the discussion for the “phase” of fast crack propagation. To explore the inertia effect caused by the fast crack propagation, the specimen are designed under the action of constant loadp(t)=p rather than time-varying load, but the crack grows with high speed. The problem for fast crack propagation is a nonlinear problem, because one part of the boundaries’ crack is of unknown length beforehand. For this moving boundary problem, we must give additional condition for determining solution. That is, we must give a criterion checking crack propagation or crack arrest at the growing crack tip. This criterion can be imposed in different ways, for example, the critical stress criterion or critical energy criterion. The stress criterion is used in this paper: σyy<σc represents crack arrest, σyy=σc represents critical state, and σyy>σc represents crack propagation. Here σc=450 Mpa is adopted as this criterion. The detailed introduction is cited from [2].The crack velocity for quasicrystals and periodic crystals is constructed in Figure7, from which, we can see that the velocity in quasicrystals is lower than that of the periodic crystals. The simulation reveals the dominating role of highly coordinate atomic environment as structure-intrinsic obstacles for crack propagation. For explaining this phenomenon we can explain a specific crack propagation mechanism: because the phonon-phason coupling makes the quasicrystals different from periodic crystals, the crack tip propagation velocity maybe hindered by phonon-phason coupling effect. Though the term seems like that introduced by [32], the meaning of them is different.Figure 7
The crack velocity versus normalized crack size with different coupling constants withp=1 Mpa.Next we will explore the velocity under different loads in quasicrystal. The above described procedure was conducted, keeping the same geometry and material constants. With various loads, the relation between velocity and crack growth is depicted in Figure8. It is understandable, while the load is increasing, the time to reach the stress criterion is short; so that the velocity becomes fast.Figure 8
Variation of crack velocity versus crack growth size for different load levels.For the profile of growing crack shown in Figure9, presents roughness when load level is growing, but there is no experimental observation for fast crack propagation though Ebert et al. had made some observation by scanning tunneling microscopy for quasistatic crack growth [33]. Because the fast propagation and quasistatic crack growth belong to two different regimes, the comparison cannot be done.Figure 9
Normalized crack growth size(a-a0)/a0 of crack tip versus time for different load levels.In Figure10, we can see that the fluctuation of normalized S.I.F. is very large when the criterion σc=450 Mpa, that is, because the surface of the crack is not flat.Figure 10
Normalized S.I.F. for quasicrystal in the criterionσc = 450 Mpa.


## 4. Conclusion and Discussion

The dynamic fracture process of octagonal quasicrystals occupies an important position in dynamic response of quasicrystals. Of course, many analytic solutions which are related to the static mechanics of octagonal quasicrystals have been obtained. But obtaining the analytic solution of dynamic fracture process of octagonal quasicrystals is very difficult. In this paper, we simulate dynamic fracture process of octagonal quasicrystals. Because the analytic solution about it is not available, we develop the finite difference procedure based on argument of Bak as well as argument of Lubensky et al., which can also be seen as a collaborating model of wave propagation and motion of diffusion. Numerical results reveal the validity of wave propagation behavior of phonon field and behavior of motion of diffusion of phason field; the interaction between phonons and phasons is also recorded. The dominant physical quantities (i.e., stress intensity factors) have been discussed in detail. The formulism is applied to analysis of dynamic initiation of crack growth and crack fast propagation for two-dimensional octagonal quasicrystals of point group 8 mm the displacement and stress fields around the tip of stationary and propagating cracks are revealed. The results show that the phonon-phason coupling plays an important role in dynamic fracture behaviour of octagonal quasicrystals, which is different from crystals.



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