Analytical Solution for a 1D Hexagonal Quasicrystal Strip with Two Collinear Mode-III Cracks Perpendicular to the Strip Boundaries

Abstract

We considered the problem of determining the singular elastic fields in a one-dimensional (1D) hexagonal quasicrystal strip containing two collinear cracks perpendicular to the strip boundaries under antiplane shear loading. The Fourier series method was used to reduce the boundary value problem to triple series equations, then to singular integral equations with Cauchy kernel. The analytical solutions are in a closed form for the stress field, and the stress intensity factors and the energy release rates of the phonon and phason fields near the crack tip are expressed using the first and third complete elliptic integrals. The effects of the geometrical parameters of the crack configuration on the dimensionless stress intensity factors are presented graphically. The studied crack model can be used to solve the problems of a periodic array of two collinear cracks of equal length in a 1D hexagonal quasicrystal strip and an eccentric crack in a 1D hexagonal quasicrystal strip. The propagation of cracks produced during their manufacturing process may result in the premature failure of quasicrystalline materials. Therefore, it is very important to study the crack problem of quasicrystalline materials with defects as mentioned above. It can provide a theoretical basis for the application of quasicrystalline materials containing the above defects.

1. Introduction

Quasicrystals (QCs) were first discovered by Schechtman et al. [1] in 1984. Quasicrystals are substances that possess a unique atomic structure, as evidenced by their perfect long-range ordering and noncrystallographic rotational symmetry. This discovery has led to a breakthrough in condensed-matter physics in recent years.

Using the cut and projection method, a 3D quasilattice can be obtained by selecting the projection of the respective 6D periodic lattice [2,3]. The properties of QCs at room temperature are characterized as hard and brittle [4]. Quasicrystal defects have also been observed [5]. When mechanical loads are applied to the QC materials, the defects in the materials may cause a premature failure of the QCs. From a theoretical and practical application point of view, it is very important to investigate the crack problem of QCs.

As a special solid material, many studies [6,7,8,9] show the complex structures and unusual properties of QCs that are sensitive to force, heat, and electricity [10]. When it comes to force, QCs differ significantly from conventional crystals in terms of force, electricity, heat, and related physical and chemical properties [11,12]. As a result, the properties of QCs have attracted more attention. The double symmetry of icosahedral quasicrystalline phason is revealed, and dislocations and defects were observed at nearly 135° angles by Li et al. [13]. Zhang et al. studied the thermodynamic stability of icosahedral QCs in annealed alloys [14]. Li and Li [15] obtained the analytical expressions for the effective properties of the 1D hexagonal quasicrystalline materials and discussed the effects of the volume fraction of the inclusion on the elastic properties of the composite materials. An analytical expression for the effective properties of one-dimensional hexagonal quasicrystals was obtained by Li and Li [15], and the influence of the volume fraction of inclusion complexes on the elastic properties of composites is discussed.

In particular, the elasticity theory of QCs has attracted a lot of attention from researchers [16,17,18]. Based on the elastic behaviors of 1D QCs, Fan [19,20] presented the mathematical theory of quasicrystalline elasticity. On this basis, the displacement potential function and stress potential function methods were further developed. Peng et al. [21] obtained the final governing equations of 1D hexagonal QCs. These equations, composed of four simple harmonic equations, were substituted for the 22 equations, which originally represented the elasticity of 1D hexagonal QCs. Li et al. [22,23] explored the problem of straight dislocations and moving spiral dislocations in one-dimensional hexagonal QCs. Liu et al. [24] studied the interaction of dislocations and defects in one-dimensional hexagonal QCs. Subsequently, Liu et al. [25] derived the governing equations for the planar elasticity of one-dimensional QCs and further obtained a general solution for this planar problem. With the help of complex function theory, Guo et al. [26,27,28] solved the problem of hole-edge cracks embedded within one-dimensional hexagonal QCs.

The complex function method is a very effective method to solve the quasicrystal defect problem. The operator method and Stroh method are also widely applied to solve quasicrystal fracture problems. Based on the operator theory, Gao et al. [29] considered the two-dimensional octagonal quasicrystal elastic problem. Gao et al. [30] applied the Stroh method to the solution of the QC elasticity problem. Based on this approach, Radi et al. [31] investigated the problem of straight cracks in two-dimensional QCs, and Li et al. [32] considered the two-dimensional deformation problem of icosahedral QCs. Guo et al. [33] applied the semi-inverse method to the solution of one-dimensional hexagonal QCs with crack problems. As mentioned above, the previous studies are confined to infinite solids with defects. Numerical methods and Green’s function method have been used a lot in recent years to solve quasicrystal mechanics problems. Wang and Ricoeur [34] used the quasicrystal fracture theory, a numerical tool developed to calculate the fracture quantity in the finite element environment. Cheng et al. [35] solved the antiplane fracture problem for one-dimensional hexagonal piezoelectric QCs using the boundary element method. Zhang et al. [36] discussed the three-dimensional fracture problem of 1D hexagonal quasicrystal coating containing interfacial cracks using the integral transformation method. Li and Li [37] studied infinite quasicrystalline composites containing elliptic inclusions by using Green’s function method.

The Fourier transform method of classical elasticity theory is still an important method to solve the quasicrystal mechanics problem, and a series of research results have been obtained. Peng and Fan [38] used the Fourier series method in dealing with the crack and indentation problems of 1D hexagonal QCs. For one-dimensional hexagonal quasicrystals with piezoelectric effects, Zhou and Li [39] also used the Fourier transform technique and discussed the Yoffe-type moving crack problem. Using Fourier transform, the basic solution of unit point displacement discontinuity on the interface was derived by Zhao et al. [40]. Fan et al. [41] obtained an elementary solution of interfacial cracks embedding two dimensional decagonal quasicrystalline bimaterials by means of the spread displacement discontinuity method and Fourier transform.

In practical engineering, structures with defects are very common, such as Griffith cracks and holes. Under different loading conditions, these kinds of defective structures will produce stress concentration phenomena. Stress concentrations can cause fatigue cracks in the material and cause the object to fracture. Therefore, the study of holes and other defects is attracting the attention of many researchers. Xiao et al. [42] studied the fracture of a circular hole with two symmetrical cracks. Ghajar and Hajimohamadi [43] obtained the solution of the fracture problem for quasisquare hole edges with asymmetrical cracks. Li and Gao [44] derived the expressions of elliptic holes and cracks in complex potential elliptic piezoelectric materials. In recent years, research on collinear cracks has been abundant. Shi [45] studied closed-form solutions for periodic cracks in antiplane in one-dimensional hexagonal QCs. For different loading conditions, Liu et al. [46] gave results for the stress intensity factors for collinear cracks. For finite solid with cracks, Li and Fan [47] studied the problem of a one-dimensional hexagonal QC containing two collinear cracks parallel to the boundary of the band type body using the complex variable function method. Guo et al. [48], by using Fourier transformation, analyzed a finite crack parallel to the strip boundaries in a 1D hexagonal QC strip and obtained the analytical solutions of the elastic fields for the phonon and the phason fields. As of now, the literature on 1D hexagonal QCs containing finite cracks perpendicular to the strip boundaries is not extensive. The structure of two collinear cracks perpendicular to the strip boundaries can provide theoretical support for engineering application of QC materials.

The objective of this paper is to focus on the problem of a one-dimensional hexagonal QC strip containing two collinear cracks perpendicular to the boundary of the strip under antiplane shear loading, as shown in Figure 1. The present crack model can be used to solve the problems of a periodic crack consisting of two equal-length, collinear cracks in a 1D hexagonal quasicrystal strip (see Figure 2) and an eccentric crack in a 1D hexagonal quasicrystal strip (see Figure 3). Moreover, the present crack model can show the interaction between cracks. Therefore, it is important to consider the problem of two collinear cracks in an infinitely long 1D hexagonal QC strip of finite width. The cracks are assumed to be perpendicular to the strip boundaries.

The Fourier transformation method was adopted in this study and is a very important method for solving the elastic and fracture problems of QCs, but the solving process is very complicated. Even for simple defective configurations (such as straight cracks), it is necessary to solve complex integral equations, which are very difficult to solve, and numerical solutions are also needed in most cases. Therefore, a lot of complex calculations were carried out in this study, and finally the explicit expressions for the elastic field were obtained. By using the Fourier series method, the boundary value problem was reduced to triple series equations, which were further transformed to a singular integral equation with Cauchy kernel. The analytical solutions of the stress field, the stress intensity factors, and the energy release rate for the phonon and phason fields near the crack tip are expressed by the complete elliptic integrals of the first and third kinds. Finally, the numerical examples are provided to show the influences of the geometrical parameters of crack configuration on the dimensionless stress intensity factors. The obtained results are conducive to understanding the failure mechanism of one-dimensional (1D) hexagonal quasicrystal strips containing two collinear cracks perpendicular to the strip. By using the Fourier series method, the boundary value problem of the considered configuration was transformed into a singular integral equation. The phonon field, phason field, stress intensity factors, and energy release rate at the crack tip are expressed using the first and third complete elliptic integrals. Finally, the effect of the geometric parameters of the configuration on the stress intensity factor and energy release rate is illustrated by numerical examples. The results obtained are useful for understanding the damage mechanism of a 1D hexagonal QC strip containing two collinear cracks perpendicular to the strip.

2. Problem Statement

In this paper, a defect configuration containing two collinear cracks in an infinitely long one-dimensional hexagonal quasicrystal strip was selected for study. The width of the strip is 2h, and the distance between two collinear cracks is 2a, where $-h\le x\le h,\hspace{0.17em}-\infty <y<\infty$, $a\le \left|x\right|\le b,\hspace{0.17em}y=0\hspace{0.17em}(0\le a<b\le h)$, as shown in Figure 1.

One-dimensional hexagonal QCs are three-dimensional structural materials whose atomic arrangements exhibit periodicity in the plane and quasiperiodicity in the direction perpendicular to it. Assume that the z-axis direction is the direction where the atomic arrangement is quasiperiodic, and that the xy-plane is the plane where the atomic arrangement is periodic. Let the collinear crack be placed in the direction parallel to the z-axis. The crack is perpendicular to the edge of the strip, and the edge of the strip is placed parallel to the y-axis. Suppose that the phonon and phason fields of a 1D hexagonal quasicrystal strip are loaded with uniform antiplane shear loadings, i.e.,

$${[{\sigma }_{zy}^{}(x,y),\hspace{0.17em}{H}_{zy}^{}(x,y)]}^{\mathrm{T}}={[{\sigma }_{zy}^{\infty },\hspace{0.17em}{H}_{zy}^{\infty }]}^{\mathrm{T}},\hspace{1em}y\to \pm \infty ,$$

(1)

where the superscript T represents the transpose.

Basic equations of QCs include geometric equations, equilibrium equations, and constitutive equations. The constitutive equations are [16]:

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

(2)

The expression of the equilibrium equations is:

$${\sigma }_{ij,i}=0,\hspace{1em}{H}_{ij,i}=0$$

(3)

The gradient equations are:

$${\epsilon }_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i}),\hspace{1em}{w}_{ij}=v_{i,j}$$

(4)

where the comma means to take the partial derivative; the repeated subscript denotes summation; $H_{ij},\hspace{0.17em}{w}_{ij}$, and $v_i$ represent the stress, strain, and displacement of the phason field, respectively; $C_{ijkl}$ is the phason field elastic constant, $K_{ijkl}$ is the phonon field elastic constant, and $\hspace{0.17em}{R}_{ijkl}$ is the phonon–phason field coupling elastic constant. The equations satisfy the following symmetric relation:

$$C_{ijkl}=C_{jikl}=C_{ijlk}=C_{klij},\hspace{1em}{R}_{ijkl}=R_{jikl},\hspace{1em}{K}_{ijkl}=K_{klij}$$

(5)

We assumed that the cracks are parallel to the quasiperiodic direction of the one-dimensional hexagonal QCs. It follows from the nature of the quasicrystalline materials that the geometric properties of the material do not change with the quasiperiodic direction. That is, the field variables of the phonon field and phason field are independent with respect to the z-axis, and therefore

$$\frac{\partial (\hspace{0.17em})}{\partial x_z}=0$$

(6)

Then, the corresponding elasticity problem can be decomposed into the following two independent problems [19]: the plane elasticity problem for ordinary hexagonal crystals and the antiplane elasticity problem for the coupling of the phonon and phason fields. The former problem can be solved by the general theory of linear elasticity. The latter problem, quite different from the general problem of the elasticity of crystals, is of the greatest interest. So, we considered the second question.

For the antiplane problem of 1D hexagonal QCs, there are only the nonzero displacement of the phonon field $u_z\left(x,y\right)$ and the displacement of the phason field $v_z\left(x,y\right)$ along the $z$-axis for describing quasilattices. According to the elasticity theory of QCs, the phonon field and the phase subfield are coupled and can be expressed by the Equations [19]:

$$\begin{array}{l}{\sigma }_{yz}={\sigma }_{zy}=2C_{44}{\epsilon }_{zy}+R_3{w}_{zy},\hspace{1em}{\sigma }_{zx}={\sigma }_{xz}=2C_{44}{\epsilon }_{zx}+R_3{w}_{zx},\\ H_{zx}=2R_3{\epsilon }_{zx}+K_2{w}_{zx},\hspace{1em}{H}_{zy}=2R_3{\epsilon }_{zy}+K_2{w}_{zy},\end{array}$$

(7)

The geometrical equations:

$${\epsilon }_{zx}=\frac{1}{2}\frac{\partial u_z}{\partial x},\hspace{1em}{\epsilon }_{zy}=\frac{1}{2}\frac{\partial u_z}{\partial y},\hspace{1em}{w}_{zx}=\frac{\partial v_z}{\partial x},\hspace{1em}{w}_{zy}=\frac{\partial v_z}{\partial y},$$

(8)

and the equilibrium equations:

$$\frac{\partial {\sigma }_{zx}}{\partial x}+\frac{\partial {\sigma }_{zy}}{\partial y}=0,\hspace{1em}\frac{\partial H_{zx}}{\partial x}+\frac{\partial {\sigma }_{zy}}{\partial y}=0,$$

(9)

where ${\sigma }_{ij}$, ${\epsilon }_{ij}$, $u_i$ (i = z, j = x or y) are the stress, strain, and displacement of the phonon field, respectively; $H_{ij}$, $w_{ij}$, $v_i$ (i = z, j = x or y) are the stress, strain, and displacement of the phason field, respectively; $C_{44}$ and $K_2$ are the elastic constants of the phonon and the phason fields, respectively; and $R_3$ is the phonon–phason coupling elastic constant.

From Equations (7)–(9), we obtained the following governing equations:

$${\mathbf{B}}_0{\nabla }^2\mathbf{u}=\mathbf{0},$$

(10)

where ${\nabla }^2$ is the two-dimensional Laplace operator, and:

$$\mathbf{u}=[u_z,\hspace{0.17em}{v}_z{]}^{\mathrm{T}},\hspace{0.17em}\begin{array}{l}\hfill \end{array}\begin{array}{l}\hfill \end{array}{\mathbf{B}}_0=\left[\begin{array}{cc}{C}_{44}& R_3\\ R_3& K_2\end{array}\right].$$

(11)

Due to $C_{44}{K}_2-R_3^2\ne 0$, we have ${\mathbf{B}}_0^{-1}=\frac{1}{C_{44}{K}_2-R_3^2}\left[\begin{array}{cc}{K}_2& -R_3\\ -R_3& C_{44}\end{array}\right]$.

Thus, Equation (10) is equivalent to:

$${\nabla }^2\mathbf{u}=\mathbf{0}.$$

(12)

Equation (7) can be can be expressed as:

$${\mathbf{T}}_y=[{\sigma }_{zy},\hspace{0.17em}{H_{zy}]}^{\mathrm{T}}={\mathbf{B}}_0\frac{\partial \mathbf{u}}{\partial y},\begin{array}{l}\hfill \end{array}\begin{array}{l}\hfill \end{array}{\mathbf{T}}_x=[{\sigma }_{zx},\hspace{0.17em}{H_{zx}]}^{\mathrm{T}}={\mathbf{B}}_0\frac{\partial \mathbf{u}}{\partial x}.$$

(13)

The configuration under consideration has geometric symmetry, and the load applied also has symmetry, so, it was simplified to consider only $0\le x\le h,\hspace{0.17em}0\le y<\infty$. There is no loading force on the cracked surface, i.e.,

$$\begin{array}{l}{[{\sigma }_{zy}(x,\hspace{0.17em}0),\hspace{0.17em}{H}_{zy}(x,\hspace{0.17em}0)]}^{\mathrm{T}}=\mathbf{0},\begin{array}{l}\hfill \end{array}\begin{array}{l}\hfill \end{array}a<\left|x\right|<b,\\ {{[\mathrm{u}}_z(x,\hspace{0.17em}0),\hspace{0.17em}{v}_z(x,\hspace{0.17em}0)]}^{\mathrm{T}}=\mathbf{0},\begin{array}{l}\hfill \end{array}\begin{array}{l}\hfill \end{array}\left|x\right|<a,\hspace{0.17em}b<\left|x\right|<h,\end{array}$$

(14)

$${[{\sigma }_{zx}(\pm h,\hspace{0.17em}y),\hspace{0.17em}{H}_{zy}(\pm h,\hspace{0.17em}y)]}^{\mathrm{T}}=\mathbf{0},\begin{array}{l}\hfill \end{array}\begin{array}{l}\hfill \end{array}-\infty <y<\infty .$$

(15)

3. Solution Procedure

To solve the above boundary value problem, the Fourier series method was used to transform the problem to a singular integral equation with Cauchy kernel, and the analytical solutions are then obtained by the complete integral. Taking Equation (15) into account, an appropriate solution to Equation (10) can be expressed by the following Fourier series

$$\mathbf{u}(x,\hspace{0.17em}y)={\displaystyle \sum _{n=1}^{\infty }{\mathbf{C}}_n{\mathrm{e}}^{-2n\beta y}\cos (2n\beta x)}+y{\mathbf{A}}_0,\hspace{1em}0\le x\le h,\hspace{0.17em}0\le y<\infty ,$$

(16)

where $\beta =\pi /2h$; $C_n={\left[A_n,\hspace{0.17em}{B}_n\right]}^{\mathrm{T}}\hspace{0.17em}\left(n\ge 0\right)$; and A₀ are the unknown vectors to be determined from the edge loading conditions. The stress expressions for the phonon field and phase field are calculated as:

$${\mathbf{T}}_y(x,y)={\left[{\sigma }_{zy}\right(x,y),\hspace{0.17em}{H}_{zy}(x,y\left)\right]}^{\mathrm{T}}=-2\beta {\displaystyle \sum _{n=1}^{\infty }n{\mathbf{B}}_0{\mathbf{C}}_n{\mathrm{e}}^{-2n\beta y}\cos (2n\beta x)}+{\mathbf{B}}_0{\mathbf{A}}_0.$$

(17)

From Equations (1) and (17), one finds:

$${\mathbf{A}}_0={\mathbf{B}}_0^{-1}{[{\sigma }_{zy}^{\infty },\hspace{0.17em}{H}_{zy}^{\infty }]}^{\mathrm{T}}.$$

(18)

According to Equations (16) and (17), using the boundary condition (14) yields the unknown coefficients $B_n=0\hspace{0.17em}\left(n\ge 1\right)$ and the system of simultaneous triple series equations for the unknown coefficient $A_n$:

$$\begin{array}{l}{\displaystyle \sum _{n=1}^{\infty }{A}_n\cos (2n\beta x)}=0,\hspace{1em}(x\le a,\hspace{0.17em}b\le x\le h),\\ {\displaystyle \sum _{n=1}^{\infty }n{A}_n\cos (2n\beta x)}=\frac{1}{2\beta }\frac{K_2{\sigma }_{zy}^{\infty }-R_3{H}_{zy}^{\infty }}{C_{44}{K}_2-R_3^2},\hspace{1em}(a<x<b).\end{array}$$

(19)

Further, we introduce the following representation:

$$p(x)=\frac{\partial u_z(x,\hspace{0.17em}0)}{\partial x}.$$

(20)

Using Fourier transform, A_n can be expressed by p(x) as follows:

$$A_n=-\frac{1}{nh\beta }{\displaystyle {\int }_a^{b}p(\xi )\sin (2n\beta \xi )\mathrm{d}\xi }.$$

(21)

Substituting Equation (21) into the second formula in Equation (19), and using the known result [49]:

$${\displaystyle \sum _{n=1}^{\infty }\sin (2n\beta \xi )\cos (2n\beta x)}=-\frac{1}{2}\frac{\sin (2\beta \xi )}{\cos (2\beta \xi )-\cos (2\beta x)},$$

(22)

We can obtain the following singular integral equation for p(x):

$$\frac{1}{h}{\displaystyle {\int }_a^b\frac{\sin (2\beta \xi )p(\xi )}{\cos (2\beta \xi )-\cos (2\beta x)}\mathrm{d}\xi =\frac{K_2{\sigma }_{zy}^{\infty }-R_3{H}_{zy}^{\infty }}{C_{44}{K}_2-R_3^2}}.$$

(23)

Following the method of Li [50], the solution of Equation (23) can be derived as:

$$p(x)=\frac{K_2{\sigma }_{zy}^{\infty }-R_3{H}_{zy}^{\infty }}{C_{44}{K}_2-R_3^2}\frac{[\cos (2\beta x)+1]-[\cos (2\beta b)+1]\chi }{\sqrt{[\cos (2\beta a)-\cos (2\beta x)][\cos (2\beta x)-\cos (2\beta b)]}},$$

(24)

with $\chi =\Pi \left(r,k\right)/K\left(k\right)$, where K(k) denotes the first complete elliptical integral and $\Pi \left(r,k\right)$ denotes the third complete elliptical integral, i.e.,

$$K(k)={\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{1}{\sqrt{1-k^2{\sin }^2\theta }}}\mathrm{d}\theta ,\hspace{1em}\mathsf{\Pi }(r,k)={\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{1}{[1+r{\sin }^2\theta ]\sqrt{1-k^2{\sin }^2\theta }}}\mathrm{d}\theta ,$$

(25)

with:

$$k=\frac{\sqrt{{\tan }^2(\beta b)-{\tan }^2(\beta a)}}{\tan (\beta b)},\hspace{1em}r=\frac{{\tan }^2(\beta a)-{\tan }^2(\beta b)}{{\sec }^2(\beta b)}.$$

(26)

4. Elastic Fields and Analytical Solutions

4.1. Elastic Fields at the Crack Tip

Substituting Equation (24) into Equation (21), then into Equation (17), we can obtain the explicit solutions in closed form for the antiplane shear stresses of the phonon field and the phason field at the crack tip as follows:

$${\left[{\sigma }_{zy}\right(x,0),\hspace{0.17em}{H}_{zy}(x,0\left)\right]}^{\mathrm{T}}=-{[{\sigma }_{zy}^{\infty },\hspace{0.17em}{H}_{zy}^{\infty }]}^{\mathrm{T}}\frac{[\cos (2\beta x)+1]-[\cos (2\beta b)+1]\chi }{\sqrt{[\cos (2\beta a)-\cos (2\beta x)][\cos (2\beta x)-\cos (2\beta b)]}},$$

(27)

for b < x < h, and:

$${\left[{\sigma }_{zy}\right(x,0),\hspace{0.17em}{H}_{zy}(x,0\left)\right]}^{\mathrm{T}}={[{\sigma }_{zy}^{\infty },\hspace{0.17em}{H}_{zy}^{\infty }]}^{\mathrm{T}}\frac{[\cos (2\beta x)+1]-[\cos (2\beta b)+1]\chi }{\sqrt{[\cos (2\beta a)-\cos (2\beta x)][\cos (2\beta x)-\cos (2\beta b)]}},$$

(28)

for 0 < x < a.

We found from the above expressions of ${\sigma }_{zy}\left(x,0\right)$ and $H_{zy}\left(x,0\right)$ that the antiplane shear for the phonon and phason fields shows the usual square-root singularity near the crack tips.

4.2. Stress Intensity Factor and Energy Release Rate

The stress intensity factor based on the linear elasticity theory represents the strength of the stress field, which plays the role of a physical quantity characterizing the fracture phenomena of materials and structures. The stress intensity factors can be defined as:

$$K_L^q=\underset{x\to a^{-}}{\lim }\sqrt{2\pi (a-x)}q(x,\hspace{0.17em}0),\hspace{1em}{K}_R^q=\underset{x\to b^{+}}{\lim }\sqrt{2\pi (x-b)}q(x,\hspace{0.17em}0),$$

(29)

where $q\left(x,\hspace{0.17em}0\right)$ stands for the stresses of the phonon field and the phason field, respectively; the subscript L represents the tip of the crack at $x=a$, and R represents the tip of the crack at $x=b$.

Inserting Equations (27) and (28) into Equation (29), we obtain the stress intensity factors of the phonon and phason fields as follows:

$$\begin{array}{l}{\left[K_L^{\sigma },\hspace{0.17em}{K}_L^H\right]}^{\mathrm{T}}=Y_L\sqrt{\pi a}{[{\sigma }_{zy}^{\infty },\hspace{0.17em}{H}_{zy}^{\infty }]}^{\mathrm{T}},\\ {\left[K_R^{\sigma },\hspace{0.17em}{K}_R^H\right]}^{\mathrm{T}}=Y_R\sqrt{\pi b}{[{\sigma }_{zy}^{\infty },\hspace{0.17em}{H}_{zy}^{\infty }]}^{\mathrm{T}},\end{array}$$

(30)

where $Y_L$ and $Y_R$ denote the dimensionless intensity factors, i.e.,

$$\begin{array}{l}{Y}_L=\sqrt{\frac{2h}{\pi a}\tan \frac{\pi a}{2h}}\frac{{\cos }^2(\pi a/(2h))-\chi {\cos }^2(\pi b/(2h))}{\sin (\pi a/(2h))\sqrt{{\cos }^2(\pi a/(2h))-{\cos }^2(\pi b/(2h))}},\\ Y_R=\sqrt{\frac{2h}{\pi b}\cot \frac{\pi b}{2h}}\frac{\cos (\pi b/(2h))(\chi -1)}{\sqrt{{\cos }^2(\pi a/(2h))-{\cos }^2(\pi b/(2h))}}.\end{array}$$

(31)

The strain intensity factors $K_L^{\epsilon }$, $K_L^w$, $K_R^{\epsilon }$, and $K_R^w$ of the phonon and phason fields can be obtained by the following relationship:

$$\begin{array}{l}{\left[K_L^{\epsilon },\hspace{0.17em}{K}_L^w\right]}^{\mathrm{T}}={\mathbf{B}}_0^{-1}{\left[K_L^{\sigma },\hspace{0.17em}{K}_L^H\right]}^{\mathrm{T}},\\ {\left[K_R^{\epsilon },\hspace{0.17em}{K}_R^w\right]}^{\mathrm{T}}={\mathbf{B}}_0^{-1}{\left[K_R^{\sigma },\hspace{0.17em}{K}_R^H\right]}^{\mathrm{T}}.\end{array}$$

(32)

There, Equation (24) shows that the stress intensity factors of the phonon field and phase field are uncoupled. However, the crack energy and energy release rate are related to the phonon field, phason field, and phonon–phason coupling field. Guo et al. [33] pointed out that the energy release rate can be chosen as a fracture criterion for 1D hexagonal QCs.

The energy release rate at the right crack tip can be determined by:

$$G_R=\frac{\partial }{\partial a}{\displaystyle {\int }_0^a\left[{\sigma }_{zy}(x,\hspace{0.17em}0)u_z(x,\hspace{0.17em}0)+H_{zy}(x,\hspace{0.17em}0)v_z(x,\hspace{0.17em}0)\right]\mathrm{d}x}.$$

(33)

Substituting Equations (27) and (28) into Equation (33), we have:

$$G_R=\frac{K_R^{\sigma }{K}_R^{\epsilon }+K_R^H{K}_R^w}{2}.$$

(34)

Using Equation (32), the energy release rate is also expressed as:

$$G_R=\frac{1}{2}\left[K_R^{\sigma },\hspace{0.17em}{K}_R^H\right]{\mathbf{B}}_0^{-1}{\left[K_R^{\sigma },K_R^H\right]}^{\mathrm{T}}.$$

(35)

Substituting Equation (30) into Equation (35), we finally obtain the energy release rate of the crack tip at x = a:

$$G_R=\frac{h\cot (\pi b/(2h)){\cos }^2(\pi b/(2h)){(\chi -1)}^2}{{\cos }^2(\pi a/(2h))-{\cos }^2(\pi b/(2h))}\frac{K_2{({\sigma }_{zy}^{\infty })}^2+C_{44}{(H_{zy}^{\infty })}^2-2R_3{\sigma }_{zy}^{\infty }{H}_{zy}^{\infty }}{C_{44}{K}_2-R_3^2}.$$

(36)

Similarly, the energy release rate of the crack tip at $x=b$ can be also obtained as:

$$G_L=\frac{2h{[{\cos }^2(\pi a/(2h))-\chi {\cos }^2(\pi b/(2h))]}^2}{\sin (\pi a/h)[{\cos }^2(\pi a/(2h))-{\cos }^2(\pi b/(2h))]}\frac{K_2{({\sigma }_{zy}^{\infty })}^2+C_{44}{(H_{zy}^{\infty })}^2-2R_3{\sigma }_{zy}^{\infty }{H}_{zy}^{\infty }}{C_{44}{K}_2-R_3^2}.$$

(37)

5. Discussion and Results

As mentioned above, we derived the analytical expressions (30) and (31) in closed from of the SIFs (stress intensity factors) for the phonon field and the phason field. When the geometrical parameters of the present defect configuration change, some special crack models are simulated. The details will be given as follows:

(1)

When the constant a approaches to zero, $\chi -1=\tan (\pi b/\left(2h\right))$, Equations (30) and (31) become:

$${\left[K_R^{\sigma },\hspace{0.17em}{K}_R^H\right]}^{\mathrm{T}}=\sqrt{2h\tan \frac{\pi b}{2h}}{[{\sigma }_{zy}^{\infty },\hspace{0.17em}{H}_{zy}^{\infty }]}^{\mathrm{T}},$$

(38)

In this case, the original configuration degenerates into Griffith crack, which has a width of 2b. The crack is embedded in an infinitely long one-dimensional hexagonal QC strip, which has a length of 2h. The stress intensity factor results for the crack tip of this degraded defect are the same as for a classical elastic material [51], as shown in Equation (38).

In this case, the result for an infinite 1D hexagonal QC with a single crack can be obtained by letting $a\to 0$.

$$G=h\tan \left(\frac{\pi b}{2h}\right)\frac{K_2{({\sigma }_{zy}^{\infty })}^2+C_{44}{(H_{zy}^{\infty })}^2-2R_3{\sigma }_{zy}^{\infty }{H}_{zy}^{\infty }}{C_{44}{K}_2-R_3^2}.$$

(39)

(2)

When $b=h$, Equations (30) and (31) are simplified as:

$${\left[K_L^{\sigma },\hspace{0.17em}{K}_L^H\right]}^{\mathrm{T}}=\sqrt{2h\cot \frac{\pi a}{2h}}{[{\sigma }_{zy}^{\infty },\hspace{0.17em}{H}_{zy}^{\infty }]}^{\mathrm{T}},$$

(40)

In case of the $b=h$, the original configuration degenerates into two edge cracks perpendicular to the edges of the 1D hexagonal QC strip. In this example, $b$ is the strip width, and we derived the stress intensity factor of crack tip $x=a$ shown in Equation (40).

The results of the energy release rate at the crack tip $x=a$ are as follows:

$$G=h\cot \left(\frac{\pi a}{2h}\right)\frac{K_2{({\sigma }_{zy}^{\infty })}^2+C_{44}{(H_{zy}^{\infty })}^2-2R_3{\sigma }_{zy}^{\infty }{H}_{zy}^{\infty }}{C_{44}{K}_2-R_3^2}.$$

(41)

(3)

When the width h approaches to infinity, Equation (25) leads to:

$$\mathsf{\Pi }(r,k)=K(k)+\frac{r}{r+k^2}[S(r,k)-K(k)],$$

(42)

where:

$$S(r,k)={\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{\sqrt{1-k^2{\sin }^2\theta }}{[1+r{\sin }^2\theta ]}}\mathrm{d}\theta .$$

(43)

Thus, the stress intensity factors at crack tips $x=a$ and $x=b$ for an infinite hexagonal quasicrystal solid with two collinear cracks are:

$${\left[K_R^{\sigma },\hspace{0.17em}{K}_R^H\right]}^{\mathrm{T}}=\frac{1-\lambda }{\sqrt{1-{(a/b)}^2}}\sqrt{\pi b}{[{\sigma }_{zy}^{\infty },\hspace{0.17em}{H}_{zy}^{\infty }]}^{\mathrm{T}},$$

(44)

$${\left[K_L^{\sigma },\hspace{0.17em}{K}_L^H\right]}^{\mathrm{T}}=\frac{\lambda -{(a/b)}^2}{a/b\sqrt{1-{(a/b)}^2}}\sqrt{\pi a}{[{\sigma }_{zy}^{\infty },\hspace{0.17em}{H}_{zy}^{\infty }]}^{\mathrm{T}},$$

(45)

where:

$$\lambda =\frac{E(\sqrt{1-{(a/b)}^2})}{K(\sqrt{1-{(a/b)}^2})},$$

(46)

in which $K(\cdot )$ and $E(\cdot )$ denote the first complete elliptical integral and the third complete elliptical integral. Especially, if the phason field is neglected, the present results agree well with the purely elastic results [52].

6. Numerical Examples

The material constants of QCs are not easy to measure, but there have been breakthroughs in recent years. Most of the obtained quasicrystalline phasons are icosahedral and decahedral quasicrystals, so the material constants of these two materials are more often used. For 1D hexagonal QCs, we currently considered only the effect of the geometric parameters of the crack on the stress intensity factor.

The influence of crack geometry on the stress intensity factor is considered under the circumstance of the subject to uniform antiplane shear load of the phonon field and phason field at an infinite distance. Figure 4 and Figure 5 show that the stress intensity factors at the right and left crack tips increase as the ratio of $b/h$ becomes large. For the same $b/h$, we found that the stress intensity factors at the right and left crack tips are different when the ratio of $a/b$ is different. As the case of the ratio of $a/b$ becomes r we found that the stress intensity factor at the right is greater than the stress intensity factor at the left when $b/h<0.8$, and the stress intensity factor at the left increases faster than the stress intensity factor at the right when $b/h>0.9$, as shown in Figure 4. When the value of $a/b$ becomes smaller, the situation of the stress intensity factors at the left and right crack tips are opposite, as shown in Figure 5.

Figure 6 and Figure 7 show that the stress intensity factors at the right and left crack tips increase as the ratio of $b/h$ becomes larger, and when the ratio of $b/h$ is constant, we found that the stress intensity factors at the right and left crack tips are larger as the ratio of $a/b$ becomes smaller.

In order to discuss the influence of the geometry size of the defect on the energy release rate, the elastic constants of the materials as shown below were selected: $C_{44}=70 \mathrm{GPa}$, $K_2=120 \mathrm{MPa}$, and $R_3=1.8 \mathrm{GPa}$. Then, we set $a=0.001 \mathrm{m}$, ${\sigma }_{zy}^{\infty }=6 \mathrm{MPa}$, and $H_{zy}^{\infty }=4 \mathrm{MPa}$. The unit of energy release rate is N/m in

Table 1. Energy release rate at the crack tip $x=b$ ($G_R$ in ${10}^{-5} \mathrm{N}/\mathrm{m}$ ).

a/b	b/h = 0.3	b/h = 0.5	b/h = 0.7	b/h = 0.8	b/h = 0.9
0.01	−1.549288	−0.877002	−0.158493	0.169430	0.543670
0.05	−1.588817	−0.984306	−0.373535	−0.136019	0.037799
0.10	−1.593339	−0.996414	−0.397145	−0.168764	−0.014285
0.20	−1.595465	−1.002031	−0.407833	−0.183289	−0.036647
0.50	−1.595465	−1.005044	−0.413315	−0.190470	−0.047091
0.90	−1.596886	−1.005713	−0.414524	−0.191995	−0.049094

and

Table 2. Energy release rate at the crack tip $x=a$ ($G_L$ in ${10}^{-5} \mathrm{N}/\mathrm{m}$ ).

a/b	b/h = 0.3	b/h = 0.5	b/h = 0.7	b/h = 0.8	b/h = 0.9
0.01	0.925691	1.023583	1.231495	1.431593	1.821182
0.05	0.067885	0.073672	0.08558	0.096627	0.11722
0.10	0.023201	0.024864	0.028229	0.031294	0.036893
0.20	0.008029	0.008458	0.009316	0.01009	0.011488
0.50	0.001652	0.001686	0.001759	0.00183	0.001971
0.90	0.000176	0.000176	0.000176	0.000177	0.00018

.

Based on the calculated data shown in Table 1, we can arrive at the conclusion that the narrower the width of the strip containing a fixed length of collinear cracks, the greater the energy release rate at the crack tip $x=b$. This is consistent with the trend shown by the stress intensity factors in Figure 7. This conclusion is consistent with the results shown in Figure 4. When the energy release rate exceeds the critical value, crack extension occurs. Mode-III fracture toughness was obtained experimentally. We fixed the value of $b/h$, and the energy release rate at the crack tip $x=b$ decreases when $a/b$ increases.

Table 2 shows that when the value of $a/b$ is fixed, the strip width decreases as the value of $b/h$ increases, and the energy release rate at $x=a$ increases. This is consistent with the trend of the stress intensity factor at $x=a$ shown in Figure 6. When the value of $b/h$ is fixed, then the $a/b$ decreases, and then the energy release rate at $x=a$ increases consequently, which means that the crack is more likely to expand at $x=a$ in this case. The results are consistent with those of Figure 6.

7. Conclusions and Outlook

The fracture mechanics of two collinear cracks perpendicular to the strip boundaries in a one-dimensional hexagonal quasicrystal strip under longitudinal shear loading was analyzed theoretically by means of the Fourier series method. This method is still an important method for solving the elasticity and fracture problems of quasicrystals. With this method, we reduced the problem to a series of equations, and then transformed them into singular integral equations. The solution is then expressed as the complete elliptic integral of the first and third kinds. By means of the Fourier series method and integral equation method, the closed-form solutions of the stress field, the stress intensity factors, and the energy release rates of the phonon and phason fields near the crack tips were obtained analytically. Specifically, the following conclusions were obtained:

(1)

The results show that the electric field, electric displacement, and phonon and phase stresses near the crack tip are singular. The stresses of the phonon and phason fields show the traditional square-root singularities;

(2)

The effects of the geometrical parameters of dimensionless SIFs near the right crack tip and the left tip are very different. Crack configurations on the dimensionless stress intensity factors are presented graphically. According to the results of the discussion, the trend of the stress intensity factor at $x=a$ and $x=b$ can be seen. Further, the results of the energy release rate of the crack tip were obtained;

(3)

The fracture problem of two collinear cracks perpendicular to the strip boundaries in a one-dimensional hexagonal quasicrystal strip can be used to simulate more of the actual situation.

Based on the above conclusions, a basis for crack expansion is provided for the application of one-dimensional quasicrystalline materials with banded bodies containing co-linear cracks. With the closed-form solutions of the QC crack problem and the parameters related to the crack, the method and accuracy of measuring the fracture toughness of quasicrystal can be improved, and it is of practical significance to reveal the fracture characteristics of the QC crack problem. Crack extension occurs when the energy release rate exceeds a critical value. By further comparing with the fracture toughness of Mode III that can be obtained from experiments, it is possible to know when crack extension occurs. Based on the present conclusions, it is also possible to continue to extend the conclusions to the study of fracture problems in QCs with magnetic, electrical, and thermal effects.