Dynamic response of an icosahedral quasi-crystalline medium with a Griffith crack under mechanical loadings

Abstract

Elastodynamic response of three-dimensional icosahedral quasi-crystals under mechanical loadings is presented in this article. Two kinds of cases are considered here, in which the plane problem and anti-plane problem are included. The phonon and phason elastic fundamental fields along with their coupling effect in crack analysis are explicitly presented in terms of the theoretical and numerical analyses. Using the finite difference method, some numerical results of stresses, displacements, and normalized dynamic stress intensity factor are obtained. By comparing the results of quasi-crystals, this article reveals the influence of phonon and phason elastic fundamental fields, which occupy an important position in dynamic deformation behavior of quasi-crystals. Of course, phonon–phason coupling effect also should not be neglected.

Introduction

Quasi-crystals (QCs) are viewed as a new structure of solids, which is different from crystals, and possess noncrystallographic rotational symmetry and quasi-periodic long-range orientational symmetry. ¹ According to the quasi-periodicity, similar to crystals, QCs have a phonon degree of freedom; meanwhile, QCs have a special type of elastic degree of freedom, which is named as phason degrees of freedom, and this fact has been investigated widely extensively and profoundly. On the issue of the generalized static elasticity theory, almost all of the scholars agree on its form, but they are unable to agree on the elasto-/hydrodynamic deformation of QCs. There are different theoretical viewpoints.^2,3 Namely, almost all of the scholars have the same viewpoint for the phonon modes, which represent wave propagation. However, as the study of this topic has progressed, different viewpoints have been postulated regarding the role of phason modes in the dynamic deformation process. We can basically summarize the relevant viewpoints as follows: Similar to the phonon field $\mathbf{u}$ in QCs, the phason field $\mathbf{w}$ represents long-wavelength propagation. This idea originated with Bak, ⁴ who indicated that phasons correspond to particular structural fluctuations (or structural disorders) in QCs, and he formulated the phonon and phason fields based on a six-dimensional space description, as has been discussed in many works. Because of six continuous symmetries, Bak believed that there exist six hydrodynamic vibration modes in QCs. According to this viewpoint, many authors, for example, Ding et al., ⁵ Hu et al., ⁶ Fan and Mai, ⁷ Li and Liu, ⁸ Rochal and Lorman, ⁹ have claimed that $\mathbf{u}$ and $\mathbf{w}$ occupy equally important position in the deformation of its dynamics. It is evident that this is a generalized perspective of elastodynamics because the equations of motion for both phonons and phasons are wave equations describing long-wavelength propagation. Some researchers have discussed applications to the defect dynamics and thermodynamics of QCs. The phason field $\mathbf{w}$ represents diffusion rather than wave propagation; thus, phasons play different roles than phonons in the hydrodynamics of QCs. This idea has been supported by Rochal and Lubensky, and they claim that phason modes denote the relative motion of the density waves, and the phason field $\mathbf{w}$ is not sensitive to spatial translations, such that the phasons are not oscillatory; instead, they are diffusive with very large diffusion times. ¹⁰ Furthermore, some researchers have explained that the motion of the phasons occurs by atomic jumps. Following this model, Rochal and Lorman, ¹¹ Fan et al., ¹² Zhu and Fan, ¹³ and Wang et al. ¹⁴ have carried out extensive work in terms of analytical and numerical methods, resulting in many solutions for the dislocation and crack dynamics of different QC systems. Li ¹⁵ gives a general solution of elasto-/hydrodynamics of two-dimensional (2D) QCs based on these models. In contrast to the generalized elastodynamics, this area of study may be termed the elasto-/hydrodynamics of QCs because the equations of phonon motion are elastodynamic equations, while the equations of motion of the phasons are diffusion equations originating from hydrodynamics.^16–18 The motion of a phason mode $\mathbf{w}$ corresponds to the dynamics of long-wavelength fluctuations and has been observed relatively recently in i-Al-Pd-Mn QCs by Francoual et al., ¹⁹ Letoublon et al., ²⁰ Coddens, ²¹ and so on. This model differs from both model 1 and model 2 mentioned above. Because this is a relatively new observation, information on its mathematical formulation has not yet been reported.

More recently, the phonon–phason elasticity of QCs also attracted a lot of attention. For example, Li et al.^22,23 provided fundamental solutions for thermo-elasticity based on half infinite plane cracks embedded in one-dimensional hexagonal QCs. Colli and Mariano ²⁴ and Radi and Mariano ²⁵ have described linear elasticity of QCs and obtained some meaningful results for dislocations in QCs. Wang and Pan ²⁶ obtained some analytical solutions for some defect problems in one-dimensional hexagonal and 2D octagonal QCs. Of course, dislocation and plate theory of QCs have been developed by some researchers, for example, Caillard et al., ²⁷ Yang et al., ²⁸ Sladek et al., ²⁹ Li and Liu, ³⁰ Liu et al., ³¹ and Li and Fan. ³²

There exist about 200 QCs observed to date, in which about 100 are icosahedral QCs, so these QCs present major importance in the material. So far, people research dynamic initial growth and fast propagation of the crack for conventional structural materials, the results of approximate analysis and numerical analysis and experimental measurement are obtained based on the finite size specimen.^33,34 In order to accomplish our subject, this work may put forward some new insights into the deformation behavior of the kind novel quasi-crystalline materials by extending the practice to the investigation. In this similar research topic, Zhu and Fan ¹³ and Wang et al. ¹⁴ have used the cracked decagonal Al-Ni-Co QC to simulate dynamic behavior of QCs, and Tupholme ³⁵ obtained a solution of a crack moving in one-dimensional hexagonal QCs under anti-plane shear loadings. Hakan and Aklncl ³⁶ use the matrix transformations and Fourier transform for dynamic plane elasticity problems of 2D QCs and provide some significant results.

It is hoped that the dynamics problem of QCs could be better convinced if they have considered the dynamic fracture behavior of the finite specimen of icosahedral QCs. So, we extend their method to the dynamic behavior of QCs based on the finite difference method (FDM). Two kinds of cases are considered in this article: one is a plane problem and the other is an anti-plane problem. On the basis of the finite specimen, the results for both kinds of cases of the stresses and displacements and stress intensity factors are calculated using the FDM.

Basic equations and simplified model

If a quasi-crystalline medium is subjected to given loadings, there will yield two displacement fields. One is $\mathbf{u}(\mathbf{r},t)$ , which is named for displacement fields of phonon. The other is $\mathbf{w}(\mathbf{r},t)$ named for displacement fields of phason. Consequently, it also yields the elastic strain tensors ${\epsilon }_{\mathrm{ij}}$ for phonon field and $w_{\mathrm{ij}}$ for phason field ⁵

$${\epsilon }_{\mathrm{ij}}=\frac{{\partial }_j{u}_i+{\partial }_i{u}_j}{2},w_{\mathrm{ij}}={\partial }_j{w}_i,{\partial }_j=\frac{\partial }{\partial x_j}$$

(1)

In linear elasticity of QCs, the stress tensors related to the strain tensors can be expressed by ⁵

$${\sigma }_{\mathrm{ij}}=C_{\mathrm{ijkl}}{\epsilon }_{\mathrm{kl}}+R_{\mathrm{ijkl}}{w}_{\mathrm{kl}},H_{\mathrm{ij}}=R_{\mathrm{klij}}{\epsilon }_{\mathrm{kl}}+K_{\mathrm{ijkl}}{w}_{\mathrm{kl}}$$

(2)

where ${\sigma }_{\mathrm{ij}}$ and ${\epsilon }_{\mathrm{ij}}$ are the phonon stresses and strains, respectively; $u_i$ and $w_i$ are the phonon and phason displacements, respectively; $H_{\mathrm{ij}}$ and $w_{\mathrm{ij}}$ are the phason stresses and strains, respectively; $C_{\mathrm{ijkl}}$ , $K_{\mathrm{ijkl}}$ , and $R_{\mathrm{ijkl}}$ are the phonon, phason, phonon–phason coupling elastic constants, respectively. For the icosahedral QCs, we have the elastic constants

$$C_{\mathrm{ijkl}}=\lambda {\delta }_{\mathrm{ij}}{\delta }_{\mathrm{kl}}+\mu ({\delta }_{\mathrm{ik}}{\delta }_{\mathrm{jl}}+{\delta }_{\mathrm{il}}{\delta }_{\mathrm{jk}})$$

(3)

and $[\mathbf{K}]$ and $[\mathbf{R}]$ are expressed by

$$\begin{array}{c}\left[\mathbf{K}\right]=\left[\begin{array}{ccccccccc}{K}_1\hfill & 0\hfill & 0\hfill & 0\hfill & K_2\hfill & 0\hfill & 0\hfill & K_2\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& K_1\hfill & 0\hfill & 0\hfill & -K_2\hfill & 0\hfill & 0\hfill & K_2\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & K_2+K_1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & K_1-K_2\hfill & 0\hfill & K_2\hfill & 0\hfill & 0\hfill & -K_2\hfill \\ \multicolumn{1}{c}{K_2}& -K_2\hfill & 0\hfill & 0\hfill & K_1-K_2\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & K_2\hfill & 0\hfill & K_1\hfill & -K_2\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & -K_2\hfill & K_1-K_2\hfill & 0\hfill & -K_2\hfill \\ \multicolumn{1}{c}{K_2}& K_2\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & K_1-K_2\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & -K_2\hfill & 0\hfill & 0\hfill & -K_2\hfill & 0\hfill & K_1\hfill \end{array}\right]\hfill \\ \left[\mathbf{R}\right]=\left[\begin{array}{ccccccccc}R\hfill & R\hfill & R\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & R\hfill & 0\hfill \\ \multicolumn{1}{c}{-R}& -R\hfill & R\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & -R\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -2R\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & -R\hfill & R\hfill & 0\hfill & -R\hfill \\ \multicolumn{1}{c}{R}& -R\hfill & 0\hfill & 0\hfill & R\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & -R\hfill & 0\hfill & -R\hfill & 0\hfill & 0\hfill & R\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & -R\hfill & R\hfill & 0\hfill & -R\hfill \\ \multicolumn{1}{c}{R}& -R\hfill & 0\hfill & 0\hfill & R\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & -R\hfill & 0\hfill & -R\hfill & 0\hfill & 0\hfill & R\hfill \end{array}\right]\hfill \end{array}$$

(4)

where $\lambda =C_{12}\mathrm{and}\mu =(C_{11}-C_{12})/2$ are the Lamé constants of the material. As everyone knows, stress intensity factor is the basic physical quantity, so the dynamic stress intensity factor and static stress intensity factor for mode I and mode III crack problems can be defined, respectively, by

$$\begin{array}{c}{K}_{\mathrm{I}}^{\parallel }\left(t\right)=\underset{x\to a^{+}}{lim}\left\{{\left[2\pi (x-a)\right]}^{1/2}{\sigma }_{\mathrm{yy}}(x,0,t)\right\},\hfill \\ K_{\mathrm{I}\mathrm{I}\mathrm{I}}^{\parallel }\left(t\right)=\underset{x\to a^{+}}{lim}\left\{{\left[2\pi (x-a)\right]}^{1/2}{\sigma }_{\mathrm{yz}}(x,0,t)\right\}\hfill \end{array}$$

(5a)

$$K_{\mathrm{I}}^{\parallel ,\mathrm{static}}=\sqrt{\pi a}p,K_{\mathrm{I}\mathrm{I}\mathrm{I}}^{\parallel ,\mathrm{static}}=\sqrt{\pi a}q$$

(5b)

The dimensionless stress intensity factor can be introduced as follows ³⁷

$$\frac{K^{\parallel }\left(t\right)}{K^{\parallel ,\mathrm{static}}}$$

(6)

For the elastodynamics, we adopt the viewpoint raised by Lubensky et al. ³

$$\rho \frac{{\partial }^2{u}_i}{\partial t^2}=\frac{\partial {\sigma }_{\mathrm{ij}}}{\partial x_j},\frac{1}{{\Gamma }_w}\frac{\partial w_i}{\partial t}=\frac{\partial H_{\mathrm{ij}}}{\partial x_j}$$

(7)

in which ${\Gamma }_w$ represents the kinetic coefficient of phason field and $\rho$ is the mass density. It is worth noting that this theory used to explore dynamics of the prospective materials is completely different from traditional continuum dynamics.

A cracked specimen of icosahedral QCs of plane problem based on FDM

Referring to Figure 1, we consider a specimen of icosahedral QCs with a Griffith crack of 2D plane problem. Suppose the Griffith crack penetrates through the specimen of icosahedral QCs along the z-axis direction, that is, the quasi-periodic direction. If the deformation of the specimen of icosahedral QCs is induced by a uniform tensile stress at upper and lower surfaces, it is obvious that all variables of elastic field are independent of z, that is, $\partial /\partial z=0$ . Then, all the equations can be simplified.

Figure 1.

The specimen for icosahedral QCs with a Griffith crack.

From these equations (3)–(7), let us take displacement variables of equilibrium as the unknown variables of the governing differential equations for 2D problem, then the equation can be easily expressed by

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{{\partial }^2{u}_x}{\partial t^2}=}& c_1^2\frac{{\partial }^2{u}_x}{\partial x^2}+(c_1^2-c_2^2)\frac{{\partial }^2{u}_y}{\partial x\partial y}+c_2^2\frac{{\partial }^2{u}_x}{\partial y^2}\hfill \\ \multicolumn{1}{c}{}& +c_3^2(\frac{{\partial }^2{w}_x}{\partial x^2}+2\frac{{\partial }^2{w}_y}{\partial x\partial y}-\frac{{\partial }^2{w}_x}{\partial y^2})\hfill \end{array}$$

(8a)

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{{\partial }^2{u}_y}{\partial t^2}=}& c_2^2\frac{{\partial }^2{u}_y}{\partial x^2}+(c_1^2-c_2^2)\frac{{\partial }^2{u}_x}{\partial x\partial y}+c_1^2\frac{{\partial }^2{u}_y}{\partial y^2}\hfill \\ \multicolumn{1}{c}{}& +c_3^2(\frac{{\partial }^2{w}_y}{\partial x^2}-2\frac{{\partial }^2{w}_x}{\partial x\partial y}-\frac{{\partial }^2{w}_y}{\partial y^2})\hfill \end{array}$$

(8b)

$$\frac{{\partial }^2{u}_z}{\partial t^2}=c_2^2{\nabla }^2{u}_z+c_3^2(\frac{{\partial }^2{w}_x}{\partial x^2}-\frac{{\partial }^2{w}_x}{\partial y^2}-2\frac{{\partial }^2{w}_y}{\partial x\partial y}+{\nabla }^2{w}_z)$$

(8c)

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{\partial w_x}{\partial t}=}& d_1{\nabla }^2{w}_x+d_2(\frac{{\partial }^2}{\partial x^2}-\frac{{\partial }^2}{\partial y^2})w_z\hfill \\ \multicolumn{1}{c}{}& +d_3(\frac{{\partial }^2{u}_x}{\partial x^2}-2\frac{{\partial }^2{u}_y}{\partial x\partial y}-\frac{{\partial }^2{u}_x}{\partial y^2}+\frac{{\partial }^2{u}_z}{\partial x^2}-\frac{{\partial }^2{u}_z}{\partial y^2})\hfill \end{array}$$

(8d)

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{\partial w_y}{\partial t}=}& d_1{\nabla }^2{w}_y-d_2\frac{{\partial }^2{w}_z}{\partial x\partial y}\hfill \\ \multicolumn{1}{c}{}& +d_3(\frac{{\partial }^2{u}_y}{\partial x^2}+2\frac{{\partial }^2{u}_x}{\partial x\partial y}-\frac{{\partial }^2{u}_y}{\partial y^2}-2\frac{{\partial }^2{u}_z}{\partial x\partial y})\hfill \end{array}$$

(8e)

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{{\partial }^2{w}_z}{\partial t^2}=}& (d_1-d_2){\nabla }^2{w}_z\hfill \\ \multicolumn{1}{c}{}& +d_2(\frac{{\partial }^2{w}_x}{\partial x^2}-\frac{{\partial }^2{w}_x}{\partial y^2}-2\frac{{\partial }^2{w}_y}{\partial x\partial y})+d_3{\nabla }^2{u}_z\hfill \end{array}$$

(8f)

where ${\nabla }^2$ is 2D Laplacian operator, $c_1=\sqrt{(\lambda +2\mu )/\rho }$ , $c_2=\sqrt{\mu /\rho }$ , $c_3=\sqrt{R/\rho }$ , $d_1=K_1/\kappa$ , $d_2=K_2/\kappa$ , and $d_3=R/\kappa$ . It needs to be noted that the known constants $c_1$ , $c_2$ , and $c_3$ represent elastic wave speeds of the phonon fields, while $d_1$ , $d_2$ , and $d_3$ are diffusive coefficients that do not represent wave speeds. The length of the specimen is $2L$ , and the height is $2H$ ; the length of the crack is $2a(t)$ . Suppose that the crack is stable for dynamic initial growth of the crack, that is, $a(t)=a_0$ . But for fast propagation of the crack, $a(t)$ is not a constant; instead it is unstable. The symmetry of the specimen is taken into account. So, it is sufficient to analyze the elastic field in the upper right quarter. The boundary conditions can be stated as follows

$$\begin{array}{c}{u}_x=0,{\sigma }_{\mathrm{yx}}=0,{\sigma }_{\mathrm{zx}}=0,w_x=0,\hfill \\ H_{\mathrm{yx}}=0,H_{\mathrm{zx}}=0x=0,0\le y\le H\hfill \end{array}$$

(9a)

$$\begin{array}{c}{\sigma }_{\mathrm{xx}}=0,{\sigma }_{\mathrm{yx}}=0,{\sigma }_{\mathrm{zx}}=0,H_{\mathrm{xx}}=0,\hfill \\ H_{\mathrm{yx}}=0,H_{\mathrm{zx}}=0x=L,0\le y\le H\hfill \end{array}$$

(9b)

$$\begin{array}{c}{\sigma }_{\mathrm{yy}}=p\left(t\right),{\sigma }_{\mathrm{xy}}=0,{\sigma }_{\mathrm{zy}}=0,H_{\mathrm{yy}}=0,\hfill \\ H_{\mathrm{xy}}=0,H_{\mathrm{zy}}=0y=H,0\le x\le H\hfill \end{array}$$

(9c)

$$\begin{array}{c}{\sigma }_{\mathrm{yy}}=0,{\sigma }_{\mathrm{xy}}=0,{\sigma }_{\mathrm{zy}}=0,H_{\mathrm{yy}}=0,\hfill \\ H_{\mathrm{xy}}=0,H_{\mathrm{zy}}=0y=0,0\le x\le a\left(t\right)\hfill \end{array}$$

(9d)

$$\begin{array}{c}{u}_y=0,{\sigma }_{\mathrm{xy}}=0,{\sigma }_{\mathrm{zy}}=0,w_y=0,\hfill \\ H_{\mathrm{xy}}=0,H_{\mathrm{zy}}=0y=0,a\left(t\right)<y\le L\hfill \end{array}$$

(9e)

where $p(t)$ is denoted to simulate a dynamic loading.

Similar to Zhu and Fan ¹³ and Wang et al., ¹⁴ we suppose the initial displacements and velocities of the phonon field and phason field are equal to zeroes, that is

$$\begin{array}{c}{u_i(x,y,t)|}_{t=0}=0,w_i(x,y,t){|}_{t=0}=0,\hfill \\ {\frac{\partial u_i}{\partial t}(x,y,t)|}_{t=0}=0,(i=x,y,z)\hfill \end{array}$$

(10)

We now pay attention in investigating the effect of the phonon and phason fields in the icosahedral Al-Pd-Mn QCs. The related parameters are used, in which the density $\rho =5.1\mathrm{g}/\mathrm{c}{\mathrm{m}}^3$ and the elastic constants $\lambda =75\mathrm{GPa}$ and $\mu =65\mathrm{GPa}$ for the phonon fields; meanwhile, the elastic constants $K_1=300\mathrm{MPa}$ and $K_2=-45\mathrm{MPa}$ for phason fields and the relevant constant of diffusion coefficient ${\Gamma }_w=1/\kappa =4.8\times {10}^{-10}\mathrm{c}{\mathrm{m}}^3\mu \mathrm{s}/\mathrm{g}$ for phason fields. ³⁸ In light of contrasting with the measured parameters of the decagonal QCs, we take $R/\mu =0.01$ for icosahedral QCs, when phonon–phason coupling constant is not measured so far.

First, the response of the cracked specimen can be declared by FDM for several different dynamic loadings. The related parameters $H=20\mathrm{mm}\mathrm{and}L=52\mathrm{mm}$ are used, and the initial length of the crack is $a_0=12\mathrm{mm}$ . A total of three kinds of dynamic loadings are considered in Figure 2.

Figure 2.

(a) Triangle pulse, (b) rectangular pulse, and (c) zigzag pulse.

It is obvious that when the periodicity $T_0$ of three dynamic loads increases, the family of curves is also gradually approaching accordingly at the load curve, the curve in Figure 3(a)–(c) reflects the trend. In dynamic loading, because of the stress wave propagation, the effect of the refraction and reflection, the stress intensity factors are close to the maximum values when the loads approach the maximum values. It always lags for a period of time generally. This phenomenon not only is related to the geometry structure of materials but also is closely related to the loading of the wave. The comparisons of these figure curves can declare and reflect that the load pulses have a significant impact on dynamic stress intensity factors. Of course, the phason field and phonon–phason coupling effect play an important role.

Figure 3.

The schematic diagram of the response of (a) a triangle pulse, (b) a rectangular pulse, (c) and a zigzag pulse.

Second, we will discuss the dynamic stress intensity factor in the dynamic loadings for the following cases: (a) Poisson coefficients are different, while the density of material and the elastic constants of phason field remain unchanged; (b) the elastic constant of the phason field $K_1$ takes different values, $K_2$ takes a fixed value of $-45\mathrm{MPa}$ , and Poisson coefficients remain unchanged (see Table 1).

Table 1.

The related mechanical properties of icosahedral quasi-crystals.

	$\lambda$ (GPa)	$\mu$ (GPa)	ν
Al-Li-Cu	30.4	40.9	0.213
Al-Cu-Fe-Ru	48.4	57.9	0.228
Al-Pa-Mn	74.9	72.4	0.254

The results declared by Figure 4(a) can be summarized as follows: the change in wave is caused by the change in $\nu$ value, which can lead to the refraction and reflection of stress wave with different velocities in the solid. So, it is predictable that the difference will become very remarkable when the loading sustains. Assume the phonon field of material constants $\lambda =75\mathrm{and}\mu =65\mathrm{GPa}$ , then the dynamic stress intensity factors are depicted in Figure 4(b). It can be seen that when the elastic constant of the phason field is a fixed value $K_2=-45\mathrm{MPa}$ and $K_1$ takes different values, the dynamic stress intensity factors fluctuate at the start of a period of time; the dynamic stress intensity factor decreases with the increase in $K_1$ ; and after a period of time, the dynamic stress intensity factor increases with the increase in $K_1$ . Also, if $K_1$ is taken as a fixed value and $K_2$ takes different values, the dynamic stress intensity factors are hardly altered. Thus, it can be declared that $K_1$ occupies a central role in the deformation of icosahedral QCs, and this effect cannot be ignored. In addition, we can infer that in three-dimensional icosahedral QCs, the two elastic constants of phason field, $K_1$ plays a leading role.

Figure 4.

Normalized dynamic stress intensity factors for (a) different Poisson coefficients and (b) different $K_1$ .

The fast crack propagation also belongs to nonlinear problem in the deformation of icosahedral QCs, in which the movement rule of crack is unknown in advance. The researchers always make some simplifying assumptions to the movement rule of crack when they give the analysis for the propagation of crack in the field of fracture mechanics research. We suppose that the crack extends along the x-axial direction and is related to the y-axial direction with symmetry. The parameters of the specimen are given by $L=50\mathrm{mm}\mathrm{and}H=20\mathrm{mm}$ , and the initial length of the crack is $a_0=12\mathrm{mm}$ ; the parameters of material can be seen in the foregoing section, and $p_0=80\mathrm{MPa}$ . Suppose the pull stress in the vicinity of crack tip is the parameter to control the fracture of QCs. By applying stress criterion, the fast crack propagation of icosahedral QCs specimen with a center crack is discussed in this section. In the mathematical iteration, the pull stress perpendicular to the crack surface of a point in the vicinity of the crack tip is calculated at any time. When its value is beyond stress threshold ${\sigma }_c$ of fracture, the crack expands forward a grid. Here, the calculation is based on ${\sigma }_c=450,550,\mathrm{and}650\mathrm{MPa}$ (Figures 5–7).

Figure 5.

(a) Dynamic stress intensity factor () and crack propagation speed () for ${\sigma }_c=450\mathrm{MPa}$ and (b) the stress and displacement near the crack tip for ${\sigma }_c=450\mathrm{MPa}$ .

Figure 6.

(a) Dynamic stress intensity factor () and crack propagation speed () for ${\sigma }_c=550\mathrm{MPa}$ and (b) the stress and displacement near the crack tip for ${\sigma }_c=550\mathrm{MPa}$ .

Figure 7.

(a) Dynamic stress intensity factor () and crack propagation speed () for ${\sigma }_c=650\mathrm{MPa}$ and (b) the stress and displacement near the crack tip for ${\sigma }_c=650\mathrm{MPa}$ .

From these figures, we can summarize as follows: (1) When stress threshold ${\sigma }_c$ increases, the speed rate of crack propagation becomes smooth and has a tendency to decrease; (2) the crack opening displacement near the crack tip has the same tendency with the stress in the crack propagation; and (3) one can find that the stress intensity factor and stress near the plate location suddenly become large in the process of simulation, and the extension rate of crack has a tendency to decline, in which we only describe the crack extension to have a short distance from the plate end location.

A specimen with a crack of anti-plane problem of icosahedral QCs based on FDM

In this section, we consider the anti-plane longitudinal shear problems of icosahedral QCs, in which the nonzero stress components are $u_z=u_z(x,y)$ and $w_z=w_z(x,y)$ and the nonzero strain components are ${\epsilon }_{\mathrm{yz}}={\epsilon }_{\mathrm{zy}}$ , ${\epsilon }_{\mathrm{zx}}={\epsilon }_{\mathrm{xz}}$ , $w_{\mathrm{zy}}$ , and $w_{\mathrm{zx}}$ . The relationship between stress and strain in the anti-plane longitudinal shear problems of icosahedral QCs is

$$\{\begin{array}{c}\multicolumn{1}{c}{{\sigma }_{\mathrm{zx}}={\sigma }_{\mathrm{xz}}=\mu \frac{\partial u_z}{\partial x}+R\frac{\partial w_z}{\partial x},H_{\mathrm{zx}}=(K_1-K_2)\frac{\partial w_z}{\partial x}+R\frac{\partial u_z}{\partial x}}\\ \multicolumn{1}{c}{{\sigma }_{\mathrm{zy}}={\sigma }_{\mathrm{yz}}=\mu \frac{\partial u_z}{\partial y}+R\frac{\partial w_z}{\partial y},H_{\mathrm{zy}}=(K_1-K_2)\frac{\partial w_z}{\partial y}+R\frac{\partial u_z}{\partial y}}\end{array}$$

(11)

The differential equations of equilibrium for anti-plane problem can be expressed by

$$\frac{\partial {\sigma }_{\mathrm{zx}}}{\partial x}+\frac{\partial {\sigma }_{\mathrm{zy}}}{\partial y}=\rho \frac{{\partial }^2{u}_z}{\partial t^2},\frac{\partial H_{\mathrm{zx}}}{\partial x}+\frac{\partial H_{\mathrm{zy}}}{\partial y}=\kappa \frac{\partial w_z}{\partial t}$$

(12)

Substituting equation (11) into equation (12), one can obtain

$$\begin{array}{c}\mu {\nabla }^2{u}_z+R{\nabla }^2{w}_z=\rho \frac{{\partial }^2{u}_z}{\partial t^2},\hfill \\ R{\nabla }^2{u}_z+(K_1-K_2){\nabla }^2{w}_z=\kappa \frac{\partial w_z}{\partial t}\hfill \end{array}$$

(13)

If $c_1=\mu /\rho$ , $c_2=R/\rho$ , $c_3=R/\kappa$ , $c_4=(K_1-K_2)/\kappa$ , $u_z=u$ , and $w_z=w$ , then the equations can be simplified as

$$\frac{{\partial }^{2}u}{\partial t^2}=c_1{\nabla }^{2}u+c_2{\nabla }^{2}w,\frac{\partial w}{\partial t}=c_3{\nabla }^{2}u+c_4{\nabla }^{2}w$$

(14)

Consider a three-dimensional QC with a crack. It can be shown in Figure 8.

Figure 8.

A schematic diagram of a mode III crack penetrates along the z-axial direction.

Without loss of generality, we only consider the upper right quarter of the cracked specimen based on the symmetry. Due to the lack of the elastic constants of the phason field, we first assume the stress of phason field $H_{\mathrm{zy}}=0$ in the boundary. Then, the boundary conditions of the phonon and phason fields can be easily written such that

$$\{\begin{array}{c}{\sigma }_{\mathrm{zy}}=\mathrm{qf}\left(t\right),H_{\mathrm{zy}}=0,y=H,\left|x\right|\le H\hfill \\ \multicolumn{1}{c}{{\sigma }_{\mathrm{zx}}=0,H_{\mathrm{zx}}=0,x=-H,x=H,0\le y\le H}\\ \multicolumn{1}{c}{{\sigma }_{\mathrm{zy}}=0,H_{\mathrm{zy}}=0,\left|x\right|\le a,y=0}\\ \multicolumn{1}{c}{u_z=0,w_z=0,\left|x\right|>a,y=0}\end{array}$$

(15)

The initial displacements and velocity of the phonon field and phason field must be taken into account. We suppose that they equal to zeroes, that is

$$u_z(x,y,0)=0,w_z(x,y,0)=0,{\frac{\partial u_z}{\partial t}|}_{t=0}=0$$

(16)

In the following, we take a = 0.2 to simulate the length of the crack and take H = 1 to simulate the length of the specimen. Meanwhile, the stress loading is deemed to be 10 MPa. The related elastic constants are shown in Table 2.

Table 2.

The related parameters used in the numerical results.

$\mu$	R	$K_1-K_2$	$\kappa$	$\rho$
$0.65\mathrm{g}/\mathrm{cm}\mu {\mathrm{s}}^2$	$0.0066\mathrm{g}/\mathrm{c}{\mathrm{m}}^3\mathrm{\mu }\mathrm{s}$	$0.31\mathrm{g}/\mathrm{cm}\mu {\mathrm{s}}^2$	$1/4.8\times {10}^{10}\mathrm{g}/\mathrm{c}{\mathrm{m}}^3\mathrm{\mu }\mathrm{s}$	$5.1\mathrm{g}/\mathrm{c}{\mathrm{m}}^3$

Figure 9 declares that the normalized dynamic stress intensity factor (NDSIF) is equal to zero before the longitudinal wave induced by the dynamic loading propagates to the surface of the crack. The time is exactly equal to that of the wave that propagates to the surface of the crack.

Figure 9.

Variation of normalized dynamic stress intensity factor with time.

It is obvious that the difference between the displacement of a point at the crack tip of the phonon and phason field is tremendous (Figure 10). The influence of the altered loadings for icosahedral QCs with a crack will be studied as shown in Figure 11. Three kinds of cases are considered. The first one is the loading stress equal to 2 MPa. The other two kinds of cases require that the loading stress is equal to 2 or 10 MPa, respectively.

Figure 10.

The displacement of a point at the crack tip of (a) phonon field and (b) phason field.

Figure 11.

Variation of normalized dynamic stress intensity factor with time.

The NDSIF does not alter along with the change in the loading stress, so the stress at the crack tip increases when the loading stress increases. A total of three displacements are depicted in Figures 12 and 13, and we can see when the load stress increases, the displacements fluctuate drastically.

Figure 12.

The displacement of a point at the crack tip of (a) phonon field and (b) phason field.

Figure 13.

The displacement of a point at the crack tip of (a) phonon field and (b) phason field.

In anti-plane problem of icosahedral QCs, we apply the stress criterion as the fracture criterion which is similar to the plane problem. Suppose that the stress value of a point near the crack tip is to control the size of the fracture parameters. On the basis of stress criterion, this article made a rapidly expanding analysis of crack. In this problem, three kinds of cases will be considered ${\sigma }_{\mathrm{fractrue}}/{\sigma }_{\mathrm{initial}}=0.5,0.7,\mathrm{and}0.9$ , in which ${\sigma }_{\mathrm{fractrue}}$ is the initial fracture stress and ${\sigma }_{\mathrm{initial}}=100\mathrm{MPa}$ (Figures 14 and 15).

Figure 14.

Variation of normalized dynamic stress intensity factor with time for (a) ${\sigma }_{\mathrm{fractrue}}=50\mathrm{MPa}$ , (b) ${\sigma }_{\mathrm{fractrue}}=70\mathrm{MPa}$ , and (c) ${\sigma }_{\mathrm{fractrue}}=90\mathrm{MPa}$ .

Figure 15.

Different speeds of extension versus different fracture stresses.

It is evident that when the initial fracture stresses are equal to 50, 70, and 90 MPa, the times of crack extending to the boundary of the material are 5.6, 6.2, and 7.2 µs, respectively. The dynamic stress intensity factors have a uniform trend based on the different initial fracture stresses. When the fracture stress becomes larger, it is inferred that the icosahedral QCs (Al-Mn-Pd) are prone to fracture. The speed of crack propagation increases as the fracture stress reduces, and the fracture resistance of materials becomes stronger. It can be inferred that the higher speed of the crack tip can be achieved in the expansion, and the dynamic fracture process brings into correspondence with ordinary materials. It shows that even though there exist the phason field and phonon–phason coupling effects in QCs, a lot of similarities exist between the fracture process of QCs and ordinary materials. This phenomenon does not conflicts the basic principle of the fracture dynamics.

Conclusion and discussion

The dynamic problem of icosahedral QCs is a mathematically daunting problem. For the statics problem of QCs, systematical and direct methods of mathematical physics and complex variable functions are developed to solve the equations under appropriate boundary value and initial value conditions, and many exact analytical solutions are constructed. But it is more difficult for the dynamic case. Based on previous theory for dynamic problem of QCs, this article mainly presents for elastodynamic response of three-dimensional icosahedral QCs using FDM. Two kinds of cases are considered: one is the plane problem of dynamic initial growth and fast propagation of the crack for QCs, and the results of approximate analysis and numerical analysis are obtained based on the finite size specimen and the other is the anti-plane problem of QCs. The phonon and phason elastic fundamental fields along with their coupling effect in crack analysis are explicitly presented in terms of numerical analyses. Using the FDM, some numerical results of stresses, displacements, and NDSIF are obtained. By comparing the results of QCs, this article reveals that the influence of phason field and phonon–phason coupling should not be neglected, which occupy an important position in dynamic deformation behavior of QCs.