1 Introduction

With the application of the composite material in engineering, the static and dynamic behaviors of composite materials have attracted the attention of more and more researchers ^[1-5]. For instance, a constant crack moving along the interface of magnetoelectroelastic and functionally graded elastic layers under anti-plane shear and in-plane electric and magnetic loading was investigated by Hu and Chen ^[1]. In recent years, the orthotropic composite material has attracted attention due to their high strength and relative lightness. Therefore, the orthotropic properties ^[6-9] should be considered in studying. For example, Liu and Zhou ^[6] investigated a plane rectangular crack in a three-dimensional (3D) infinite orthotropic elastic material by the Schmidt method. The dynamic stress intensity factors (DSIFs) in an orthotropic strip with functionally graded materials coating weakened by multiple arbitrary cracks under time-harmonic excitation were solved by Monfared and Ayatollahi ^[8]. However, it is found that these solutions have stress singularity in the above studies ^[6-9], and this phenomenon is not reasonable. As a result, starting from the studies of Eringen and Kim ^[10] and Eringen ^[11-12], the non-local theory was used to study the fracture problem based on the maximum stress.

Some fracture problems in functionally graded materials and functionally graded piezoelectric materials were investigated by Zhou and Wang ^[13] and Zhou et al. ^[14] using the non-local theory and the Schmidt method ^[15-16]. Liu et al. ^[17] and Liu and Zhou ^[18] investigated the static and dynamic problems of a rectangular crack in transversely isotropic elastic materials by the non-local theory. The non-local theory solution of a 3D rectangular permeable crack in piezoelectric composite materials under a normal stress loading was analyzed by Liu et al. ^[19]. It is noted that the above mentioned works ^[10-14] focused on anti-plane or two-dimensional (2D) plane problems. To the best of the authors' knowledge, the dynamic behavior of a rectangular crack in a 3D infinite orthotropic medium based on the non-local theory has not been studied.

In this paper, we consider a rectangular crack in a 3D infinite orthotropic medium under a harmonic stress wave based on the non-local theory, the generalized Almansi's theorem, and the Schmidt method. Utilizing the 2D Fourier transform, the mixed boundary value problem is reduced to three pairs of dual integral equations. The non-local theory solutions on the stress field near the crack edges are obtained.

2 Model description

In the present paper, it is assumed that the time-harmonic elastic P-wave is vertical to the crack plane. All the field quantities can be expressed in the following form:

(1)

where $X_0(x, y, z, t)$ denotes the variable which contains a time-factor ${\rm e}^{-{\rm i}\omega t}$, and $X(x, y, z)$ denotes the variable which does not contain the time-factor ${\rm e}^{-{\rm i}\omega t}$. For brevity, the time dependence of ${\rm e}^{-{\rm i}\omega t}$ will be suppressed.

Consider an orthotropic elastic medium with a symmetric rectangular crack located at $z=0$ along the $x$-axis from $-l_1 $ to $l_1 $ and along the $y$-axis from $-l_2 $ to $l_2 $, as shown in Fig. 1. Assume that a distributed normal harmonic stress wave $\sigma _{zz}^\ast (x, y, 0, t)=-\sigma _0 {\rm e}^{-{\rm i}\omega t}$ ($\omega $ denotes the circular frequency of the incident wave) is directly applied on the rectangular crack surfaces. Hence, the boundary conditions can be stated as

(2)

(3)

(4)

where $\sigma _0 $ is a magnitude of the incident wave, $\sigma _{lk}^{\ast (j)} (x, y, z)\, (l=x, y, z$, $k=x, y, z$, $j=1, 2)$ are the non-local stresses, and $u^{(j)}(x, y, z)$, $v^{(j)}(x, y, z)$, and $w^{(j)}(x, y, z)$ are the displacement components in the $x$-, $y$-, and $z$-directions, respectively. $l_1$ and $l_2$ are the half-length and the half-width of the rectangular crack, respectively. The superscripts $j\, (j=1, 2)$ denote the upper space 1 and the lower space 2 in Fig. 1, respectively.

3 Basic equations

With vanishing body forces, the basic equations of the linear, homogeneous, 3D non-local orthotropic elastic medium are

(5)

(6)

(7)

where

and $\rho _0 $ is the density of the elastic materials. The non-local stresses are denoted by $\sigma _{ik}^\ast $, and the local stresses are expressed as $\sigma _{ik}$. A subscript comma denotes partial differentiation with respect to the coordinate. In the orthotropic elastic medium, $c_{11}$, $c_{12}$, $c_{13}$, $c_{22}$, $c_{23}$, $c_{33}$, $c_{44}$, $c_{55}$, and $c_{66} $ are the classical elastic constants. The non-local stresses $\sigma _{ik}^\ast $ at a point $(x', y', z')$ depend on the strains $u_{, k} , \;v_{, k} $, and $w_{, k} (k=x, y, z)$ at all points $(x, y, z)$ in the body, and $\alpha (| {x'-x} |, | {y'-y} |, | {z'-z}|)$ is the influential function. As discussed in Ref. ^[20], the influential function is assumed as follows:

(8)

where $\beta $ and $a$ are the material constants.

Substituting (6) and (7) into (5) and using the Green-Gauss theorem, we get

(9)

(10)

(11)

Here, because the perturbation displacement field in the infinite distance tends to zero, the surface integrals in (9)-(11) may be dropped.

4 Solution procedures

As discussed by Eringen ^[12], it can be obtained that $\sigma _{zz}^{(1)} (x', y', 0)-\sigma _{zz}^{(2)} (x', y', 0)=0$, $\sigma _{xz}^{(1)} (x', y', 0)-\sigma _{xz}^{(2)} (x', y', 0)=0$, and $\sigma _{yz}^{(1)} (x', y', 0)-\sigma _{yz}^{(2)} (x', y', 0)=0$. Therefore, (9)-(11) become

(12)

It remains to solve the integrodifferential (12) for the functions $u(x, y, z)$, $v(x, y, z)$, and $w(x, y, z)$. Similar to Nowinski's ^[21] discussion, as adequate functions, it is decided to select the terms $\delta _n ({z}'-z)\;(n=1, 2, \cdots)$ of the so-called $\delta $-sequences. A $\delta $-sequence, as is generally known, is a one-dimensional (1D) Dirac delta function $\delta _n ({z}'-z)$. In the present case, the non-local interaction in the $z$-direction can be ignored, and only the non-local interaction in the $x$- and $y$-directions are considered. Under the assumption, it can be expressed as

(13)

where the material constant $\beta $ may be determined experimentally. Normally, $a$ is a known constant as the characteristic length. In this paper, $a$ is taken to be the lattice parameter. From (12), we obtain

(14)

Solving the problem, the 2D Fourier transform of (14) respect to $x'$ and $y'$ can be given as

(15)

where

Equation (15) can be solved by Almansi's theorem ^[22], and it is rewritten as

(16)

where the differential operator matrix $M_D$ is

(17)

in which

The determinant of matrix $M_D$ is

(18)

where

The cofactors $A_{lk}$ of det $M_D\, (l, k=1, 2, 3)$ were solved by Chen et al. ^[23] and Ding et al. ^[24]. The general solution of (16) is

(19)

with $\overline {f}$ satisfying

(20)

In the following analysis, only ($A_{31}, A_{32}, A_{33})$ is used for the present problem. For brevity, the details are omitted.

Using the symmetry with respect to the $x$-axis and the $y$-axis and employing the 2D inverse Fourier transform with respect to $x$ and $y$, the solution of $\overline {f}(\varepsilon , \eta , z)$ is

(21)

Substituting (21) into (20) yields

(22)

where $a=b_1 $, $b=b_2 $, $c=b_3 $, and $d=b_4 $.

Because (22) is a homogeneous equation, the solution of $\overline {f}(\varepsilon , \eta , z)$ is a function of $\exp (\lambda z)$, in which $\lambda $ is the root of the algebraic equation

(23)

Let $\lambda ^{\ast 2}=\lambda ^2+b/3a$. Then, (23) becomes

(24)

with

whose roots $\lambda ^{\ast 2}$ are

(25)

where $\omega =(-1+{\rm i}\sqrt 3)/2$, and ${\rm i}=\sqrt {-1} $.

Since the three roots of (23) are $\lambda _1^2 \ne \lambda _2^2 \ne \lambda _3^2 >0$ for the 3D orthotropic elastic medium, the general solutions of the function $\overline {f}^{(j)}(\varepsilon , \eta , z)$ are

(26)

where $A_k (\varepsilon , \eta )$ and $B_k (\varepsilon , \eta )$ ($k=1, 2, 3)$ are unknown functions of $\varepsilon $ and $\eta $, which need to be solved by the boundary conditions.

The general expressions of the displacements and stresses fields satisfying (4) are

(27)

(28)

where

From (6), using (13), we get

(29)

(30)

(31)

where

(32)

(33)

in which

Substituting $\alpha $ from (13) and using (32) and (33), the integration may be performed with respect to $x'$ and $y'$ in (29)-(31) by integrals ^[25],

Therefore, (29)-(31) can be expressed as

(34)

(35)

Solving the problem, the displacement jumps across the crack surfaces are defined by

(36)

Substituting (27) and (28) into (36) and performing the 2D Fourier transform along with (34) and (35) and the boundary conditions (2) and (3), we have

(37)

(38)

Solving (37) and (38) with six unknown functions, taking results into (34), and applying the boundary conditions (2) and (3), we get

(39)

(40)

(41)

(42)

where $g_k (a, \beta , \varepsilon , \eta )\, (k=1, 2, 3, 4, 5)$ are known functions in Appendix A. The dual integral aligns (39)-(42) are solved by the Schmidt method ^[15-16].

The displacement jumps across the crack surfaces are expanded by the following series:

(43)

(44)

(45)

where $a_{mn} $, $b_{mn} $, and $c_{mn}^\ast $ are unknown coefficients, and $P_n^{(1/2, 1/2)} (x)$ is a Jacobi polynomial ^[25]. The 2D Fourier transforms of (43)-(45) can be written as follows ^[26]:

(46)

(47)

(48)

where

and $\Gamma (x)$ and $J_n (x)$ are the Gamma and Bessel functions of order $n$, respectively.

Substituting (46)-(48) into (39)-(42), (42) is automatically satisfied, and (39)-(41) can be reduced as

(49)

(50)

(51)

Thus, (49)-(51) can be solved with the coefficients $a_{mn} $, $b_{mn}$, and $c_{mn}^\ast $ by the Schmidt method ^[15-16]. For brevity, (49)-(51) can be rewritten as

(52)

(53)

(54)

where $A_{mn}^\ast (x, y)$, $B_{mn}^\ast (x, y)$, $C_{mn}^\ast (x, y)$, $D_{mn}^\ast (x, y)$, $E_{mn}^\ast (x, y)$, and $U_0 (x, y)$ are known functions and $a_{mn} $, $b_{mn} $, and $c_{mn}^\ast $ are unknown coefficients. From (53)-(54), we can obtain the unknown coefficients $a_{mn} =b_{mn} =0$. Now, the Schmidt method ^[15-16] can be applied to solve the unknown coefficients $c_{mn}^\ast $. Therefore, the double infinite series in (52) must be reduced to a single infinite series. Namely, (52) is changed to the following form:

(55)

where $c_k^\ast =c_{mn}^\ast $, and $F_k^\ast (x, y)=G_{mn}^{(3)} A_{mn}^\ast (x, y)$. The relations between $k$ and $(m, n)$ must be satisfied one-to-one. Therefore, the coefficients $c_k^\ast $ can now be solved in (55) by the Schmidt method ^[15-16].

When the coefficients $c_{mn}^\ast $ are known, the entire stress fields are obtained. In this study, $\sigma _{zz}^{\ast (1)} $, $\sigma _{xz}^{\ast (1)} $, and $\sigma _{yz}^{\ast (1)} $ along the crack plane are shown as

(56)

(57)

5 Numerical calculations and discussion

In this paper, the material constants are shown in Table 1 ^[27]. The numerical results are discussed in Figs. 2-10. Some conclusions are drawn as follows:

(i) When the lattice parameter $a\to 0$, the stresses along the crack edges become infinite. For $\mathop {\lim }\limits_{a\to 0} \overline {\alpha }_0(a, \beta , \varepsilon , \eta )=1$, (39)-(41) can be reduced to the dual integral equations in classical elasticity.

(ii) The maximum stress does not occur at the crack edges, but in its immediate vicinity in Figs. 2-6. The similar phenomenon is thoroughly confirmed by Eringen ^[28]. It is noted that the distance between the crack edge and the maximum stress point is very small and depends on the geometry of rectangular crack, the circular frequency of the incident wave, and the lattice parameter.

In Figs. 2 and 4, the stress achieves its largest value at the middle of the crack edge ($x=l_1$), and decreases along the edge gradually towards the corner of the crack. Meanwhile, Figs. 5 and and 6 indicate that the value of stress achieves its largest at the middle of the crack edge ($y=l_2$), and it also decreases along the edge gradually towards the corner of the crack.

(iii) Figures 7 and 8 show that the values of stress at the middle of crack edges tend to decrease with the increase of the lattice parameter for $l_1 =1.0$, $l_2 =1.0$, and $y=0$ or $x=0$. It can be easily found that the values of stress at the middle of crack edge ($y=l_2$) are larger than those at the middle of rectangular crack edge ($x=l_1$).

(iv) For the case of $a/\beta =$\, constant, the values of the stress at the middle of crack edge increase with the increase of $l_2 /l_1 $ for $l_1 =1.0$ and $a/\beta =0.000~1$, as shown in Fig. 9. It is also noted that the values of stress at the middle of crack edge ($y=l_2$) are always larger than those at the middle of crack edge ($x=l_1$).

(v) From Fig. 10, the normalized stress at the middle of crack ($y=l_2 )$ with the increase of $\omega l_1 /c_z $ for $l_1=l_2 =1.0$ and $a/\beta =0.000~1$ tends to the first peak, and then the decrease in magnitude reaches a nadir. The dynamic stress field will enhance crack propagation hinging on the circular frequency of the incident wave in a 3D orthotropic elastic medium.

(vi) To verify the correctness of the present results, the orthotropic elastic material in the present problem can be reduced to the transversely isotropic elastic material ^[18] and the case of static fracture problem ^[29], as shown in Fig. 11. We can find that the values of the stress in transversely isotropic elastic material have the same tendency as those of the orthotropic elastic material. Meanwhile, when $\omega l_1 /c_z =0.0$, $l_1 =1.0$, and $l_2 $ tends to infinity, the problem of 3D rectangular crack can be approximately regarded as one of 2D Mode-I Griffith crack. For the 2D Mode-I Griffith crack problem, the resulting solution is obtained by the non-local theory and the Schmidt method. The solution procedure is the same as the present paper, as shown by Yang ^[30]. Therefore, from these results, it can be seen that the dynamic analysis process in the present paper is reasonable and effective.

6 Conclusions

The dynamic behavior of a rectangular crack in a 3D orthotropic medium under a time-harmonic elastic P-wave is analyzed by utilizing the non-local theory. The dynamic stress field near the rectangular crack edges is obtained. Numerical computations illustrate that the influence of the circular frequency of the incident waves and the lattice parameter is adopted. Unlike the classical solution, no stress singularity is presented near the crack edges in the orthotropic medium. The non-local solution yields a finite stress along the crack edges. Thus, we can use the maximum stress hypothesis as a fracture criterion.

Appendix A

Equations (37) and (38) are rewritten as

(A1)

where

We obtain

(A2)

The unknown functions $A_k $ and $B_k $ are expressed by

(A3)

where $[m_{kj}]_{3\times 3} =M^{-1}$, and $[n_{kj}]_{3\times 3} =N^{-1}$.

Taking (A3) into (A1), it can be obtained as

(A4)

(A5)

Substituting (A3) into (34) and applying (A4) and (A5), we get

(A6)

(A7)

(A8)