1 Introduction

The singularity behaviors near the interface crack tip for isotropic, orthotropic, or anisotropic bimaterial have been widely studied^[1-28].

If the stress singularity exponent is a complex number, the stresses near the interface crack tip will show oscillatory singularity behaviors. The oscillatory singularity behaviors have been widely investigated by analytical methods, e.g., the complex function method, the eigenfunction expansion method, the stress function method, the weight function method, and the integral transformation method, numerical methods, e.g., the finite element method and the boundary element method, and experimental methods^{[1-12, 26-28]}. The oscillatory character of the stress and interface overlapping of the displacement cannot give ideal interpretation in the physical sense. If the stress singularity exponent is a real number, then the stresses near the interface crack tip will show non-oscillatory singularity behaviors^[16-23]. Zhou and Wang^[19] obtained the displacement functions by the governing equation containing four undetermined functions by the Schmidt method. Li et al.^[20] and Yang et al.^[22] obtained the stress functions by the governing equations containing sixteen undetermined coefficients and two stress singularity exponents with the stress function method and the method of elimination unknowns, respectively. Zhang et al.^[23] obtained the stress functions by the governing equations containing eight undetermined coefficients and a stress singularity exponents with the method of elimination unknowns. However, since the stress singularity exponent is one or two real numbers^[16-23], the stress oscillation and the interface overlapping of displacement never occur. Both the oscillatory and non-oscillatory singularity behaviors are considered in Refs.^[13], ^[14], ^[15], ^[24], and ^[25].

In this paper, the non-oscillatory singularity behaviors of the mode II interface crack subjected to the loading at infinity, i.e., $| y| \to + \infty$ for orthotropic bimaterial, are investigated. The new stress functions are structured to be with eight or ten undetermined coefficients and an unknown singularity exponent. The characteristic equation and the stress singularity exponent are derived by the existence theorem of the non-trival solution for the system of eight homogeneous linear equations. Solving the system of twelve non-homogeneous linear equations with ten unknowns and using the uniqueness of the solutions for this equation system, we can obtain the ten undermined coefficients. The theoretical formulae of the stress intensity factors and the analytic solutions of the stresses near the mode II interface crack tip are obtained.

2 Complex function method and undetermined coefficient method

Suppose that the part when $y>0$ is occupied by the upper orthotropic material $(j=1)$, the part when $y<0$ is occupied by the lower orthotropic material $(j=2)$, the parts when $y=0$ and $| x| < a$ are the crack surfaces, and the part when $y=0$ and $| x |> a$ is the bonded interface of two dissimilar materials with a central crack of the length $2a$. The elastic constants and the compliance coefficients of each orthotropic material $j\;(j=1, \, 2)$ are $(E_1 )_j $, $(E_2 )_j$, $(\nu _{12} )_j$, $(G_{12} )_j$ and $(b_{11} )_j$, $(b_{22} )_j$, $(b_{12} )_j$, $(b_{66} )_j$, respectively. The relationships of these quantities are given by

With the compatibility equation for a linear elastic plane of the orthotropic bimaterial, we can deduce the governing equation as follows^[29-33]:

where $ j=1, \, 2$, $x$ and $y$ are the Cartesian coordinates, and $U_j = U_j (x, y)$ is the stress function.

The fracture behaviors near the mode II interface crack tip for orthotropic bimaterial are discussed in this paper. The stress functions $U_j = U_j (x, y)\; (j = 1, 2)$ for this problem satisfy the governing equation (2) and the following boundary conditions:

We solve the partial differential equation (2) by a complex function method. Let

Then, substituting (6) into (2) yields the following biquadratic equations:

whose discriminants are

For two different orthotropic materials $j=1$ and $j=2$, there exist three cases: (i) $\Delta _1>0$ and $\Delta _2>0$, (ii) $\Delta _1<0$ and $\Delta _2<0$, and (iii) $\Delta _1>0$ and $\Delta _2<0$. In this paper, only Case (i) will be considered. Similarly, we can also discuss Cases (ii) and (iii).

When $\Delta _j>0$, we can obtain the characteristic roots of Eq.(2) as follows:

where $\beta _{j2} > \beta _{j1} > 0.$

From Eqs.(7), (9), and (1), we have

where $j=1, \, 2.$ Introduce the following complex variables:

Then, we have

From Eqs.(10a) and (12), we can reduce Eq.(2) to the generalized biharmonic equations as follows^{[29, 32-33]}:

By use of the undermined coefficient method, we can solve the fracture problem of the mode II interface crack tip in orthotropic bimaterial. The key lies in the determinations of the stress singularity exponent, the stress intensity factors, and the stresses.

The real part and the imaginary part of the analytic function of the complex variable $z_{jk}$ are both harmonic functions^[34] and the solutions of Eq.(13), i.e., Eq.(2).

The stress singularity exponent of the mode II interface crack tip in the orthotropic bimaterial depends on the eight bimaterial self boundary conditions (3) and (4). It is not related to Eq.(5). Therefore, the special stress functions can be chosen as follows:

where $A_{jk, \lambda }$ and $B_{jk, \lambda }\; (j, k = 1, 2)$ are eight undetermined coefficients. $\lambda$ is the stress singularity exponent, which sometimes is called eigenvalue. $z_{jk} \;(j, k = 1, 2)$ are given in Eq.(11). $\overline{\overline U} _{jk, \lambda } (z_{jk} )\;(j, k = 1, 2)$ are the analytic functions, and

Because Eq.(5) holds, the forward two conditions of Eq.(4) hold. Therefore, the boundary conditions (3), (4), and (5) have actually ten conditions. According to the similar reasons with Eq.(14), the solutions of the fracture problems (2), (3), (4), and (5) can be structured as follows:

where $A_{jk, \lambda }$, $B_{jk, \lambda }$, and $C_{j, \lambda }$ are ten undetermined coefficients. $\lambda$ is the stress singularity exponent. $z_{jk}\; (j, k = 1, 2)$ are given by Eq.(11). $\overline{\overline U} _{jk, \lambda } (z_{jk} )\;(j, k = 1, 2)$ are the analytic functions. $(\overline {\overline{\overline U} _{jk, \lambda } (z_{jk} )} )\;(j, k = 1, 2)$ denote the conjugate functions of $\overline{\overline U} _{jk, \lambda } (z_{jk} )\;(j, k = 1, 2)$.

3 Stress singularity exponent and existence of non-trivial solution

From Eqs.(14), (15), and(12), we can deduce the stresses of each material $j$$(j=1, \, 2)$ as follows^[29-33]:

Considering the boundary conditions (3) and (4), we choose $U_{jk, \lambda } (z_{jk} )$ of the right-side of Eq.(17) as follows:

where

In the above equation, $r$ and $\theta$ are the polar coordinates measured from the interface crack tip, and $z_{jk} \;(j, k = 1, 2)$ are expressed by Eq.(11).

From $- 1 = {\rm e}^{{\rm i}( - 1)^{j - 1} \pi} $, we know

Substituting Eqs.(20), (17b), and (17c) into Eqs.(3) and (4), after some calculations, we can obtain the eight equations as follows:

We solve the system of the eight homogenous linear equations in Eqs.(21a)--(22d) by the successive elimination of the unknowns. The eight appropriate elementary transformations are made for the coefficient matrix $A_{8 \times 8}$ in Eqs.(21a)--(22d). Then, we can obtain the determinant as follows:

The bimaterial parameters in Eq.(23) can be expressed by

where $e_j$, $f_j$, $g_j$, and $h_j$ are the parameters of each single material $j\;(j=1, \, 2)$.

By the existence theorem of the non-trivial solution for the equation system (21a)--(22d), from Eq.(23) and $| {A_{8 \times 8} } | = 0$, we can derive the characteristic equation of the mode II interface crack for the orthotropic bimaterial as follows:

or

Incidentally, from Eq.(23) and $| {A_{8 \times 8} } | = 0$, we have $\sin(\lambda\pi) = 0$, where $\lambda = n \;(n = 0, \, \pm 1, \, \pm 2, \, \cdots )$, in independent of $e_{12}$, $f_{12}$, $g_{12}$, and $h_{12}$, and they should be omitted.

Because the factor $\cot (\lambda\pi)$ can be included by Eq.(25), we can choose the stress singularity exponent as follows^[16-24]:

where $\varepsilon$ is the positive real bielastic constant.

The fracture behavior at the crack tip for each material $j$ ($j=1, \, 2)$ directly affect the singularity state near the interface crack tip for the double materials, each of which is a single material. From the classical result for a single material, in the vicinity of the interface crack tip $(z_{jk} \to a, r \to 0) \;, n=0$ in Eq.(27), we have

Substituting Eq.(28) into Eq.(25), we obtain

Substituting the expansion of $\tan (\varepsilon \pi)$ in power series into Eq.(29), omitting the sufficiently small $\varepsilon$ at least the fourth degree, we can find the quadric characteristic equation as follows:

When $e_{12} \ne 0$ and $f_{12} \ne 0$, in Eq.(24a) and (24b), the upper and lower materials are dissimilar. From Eq.(30), we can derive the following discrimination condition of the non-oscillatory singularity:

If the parameters $e_{12}$, $f_{12}$, $g_{12}$, and $h_{12}$ satisfy the condition (31), then the bielastic constant $\varepsilon\; (\varepsilon > 0)$ can be determined by Eq.(30). Putting $\varepsilon$ into Eq.(28), we can obtain the real stress singularity exponent $\lambda$.

Solving Eq.(26), we can derive

By the same causes before Eq.(28), in the vicinity of the interface crack tip, we have $n=0$ in Eq.(32), and then

The values of $\lambda$ in Eqs.(32) and (33) are independent of the bimaterial parameters in Eq.(24). Therefore, the stress singularity exponent of the orthotropic single material of the two orthotropic materials $j=1$ and $j=2$ are equal.

In fact, when the two orthotropic materials $j=1$ and $j=2$ are the same, the double materials will become a single material. Substituting $e_{12}=0$ and $f_{12}=0$ in Eq.(24) into Eq.(25), we can obtain the characteristic equation (26).

From the elastic constants in Refs.^[23] and ^[36]--^[38] and Eqs.(8) and (9), we can obtain the quantities $\Delta _j$ and $\beta _{jk} \;(j, k = 1, 2)$ of each orthotropic material (see Table 1). The mechanical parameters of each material in Table 1 are substituted into Eqs.(24), (28), (30), and (31). Then, we can obtain the bimaterial parameters, the discriminative conditions, the bielastic constant, and the real stress singularity exponent for the three orthotropic bimaterial (see Table 2).

4 Stress function and uniqueness of undetermined coefficient

Substituting Eq.(28) into Eq.(16), we can obtain the stress functions as follows:

From Eqs.(34), (15), (11), and (12), we can obtain $(\sigma _x )_j $, $(\sigma _y )_j$, and $(\tau _{xy} )_j$ of each material $j\;(j=1, \, 2)$ as follows^[29-33]:

Putting Eq.(28) into Eq.(18), we have

From Eq.(36), Eq.(11), and $- 1 = {\rm e}^{{\rm i}( - 1)^{j - 1} \pi }$, we have

Substituting Eqs.(37) and (35) into Eqs.(3), (4), and (5), and using the strain-stress relations and strain-displacement relations, we can derive the system of twelve non-homogeneous linear equations in ten unknowns as follows:

where Eq.(40b) is non-homogeneous.

The coefficient matrix and the augmented matrix for the system of Eqs.(38a)--(40b) are denoted by $A_{12 \times 10}$ and $\overline{A}_{12 \times 11}=({A_{12\times 10} }$ $ b_{{\rm {II}}, 12\times1})$, respectively. Let $r_i$ and $c_j$ be the $i$th row and the $j$th column. To solve the system ${A_{12\times 10}X }=$ $ b_{{\rm {II}}, 12\times1}$ of Eqs.(38a)--(40b), we need to do the twelve elementary transformations for $\overline{A}_{12 \times 11}$. It is known that

where the number of the undetermined coefficients is 10. From this, the system of twelve non-homogeneous linear equations with the ten unknowns (38), (39) and (40) has and only has a unique solution^[36], i.e., the ten undetermined coefficients are uniquely determined.

The augmented matrix for the system of the new ten nonzero rows is denoted by $\overline{A}_{10 \times 11} = (A_{10 \times 10} b_{{\rm {II}}, 10 \times 1} )$. The new system $A_{10 \times 10}X= b_{{\rm {II}}, 10 \times 1}$ of the non-homogeneous linear equations is solved by the inverse sequence back-substitution from down to up. The nine elementary transformations are taken for the new augmented matrix $\overline{A}_{10 \times 11}$. The expressions for the ten coefficients in the stress functions can be obtained as follows:

where $j, k = 1, 2$, $\beta _{jk}$ are the imaginary parts of Eq.(9a) for Eq.(7), $e_j$, $f_j$, and $g_j$ are shown as Eq.(24), and

The fracture problem of the mode II crack for the orthotropic single material can be reduced to solve the following boundary value problem:

Equation (43) can be converted into the generalized biharmonic equation^{[29, 32-33]}. The boundary conditions (44) and (45) have four conditions. We can choose the stress function as follows:

where $A_k$ and $B_k$ are four undetermined coefficients, $\overline{\overline U} _k (z_k )$ are the analytic functions, and

From Eqs.(46) and (47), we can obtain the stresses as follows:

where

From Eqs.(48) and (35), for the orthotropic single material, we have

We think that the single material is a special example of the double material. Putting $\varepsilon = 0$ in Eqs.(33) and (50) into Eq.(41), we have

Obviously, substituting Eqs.(49) and (48) into Eqs.(44) and (45), we can obtain the coefficient formulae (51).

5 Stress intensity factor and one-sided limit

The stress intensity factor depends on the stress, the stress function, the stress singularity exponent, and the applied outer load^{[5, 7, 9, 18]}. Based on the stress expressions (35c) and (35b) and the one-sided limit, we can introduce the definitions on the stress intensity factors of the mode II interface crack in the orthotropic bimaterial for each material $j\;(j=1, \, 2)$ and the double material as follows:

where $j = 1, 2$, and

By Eq.(19), we consider that $z_{jk} \to a^ +$ is the same as $r \to 0$ and $\theta \to ( - 1)^{j - 1} 0$, and $z_{jk} \to a^ -$ is the same as $r \to 0$ and $\theta \to ( - 1)^{j - 1} \pi$, where $j, k=1, 2$. From Eqs.(36) and (11), we have

where $C_{j, \varepsilon } \;(j = 1, 2)$ are given in Eqs.(41e) and (41f).

For the orthotropic single material, substituting $\varepsilon {\rm{ = }}0$ in Eq.(33) into Eq.(58), we can derive the stress intensity factor of the mode I crack as follows:

With the data listed in Tables 1 and 2, From Eqs.(24) and (41e)--(42), we can obtain the related parameters of each orthotropic material (see Table 3).

From Eqs.(56c) and (57c), we can discover that the two one-sided limits create almost the same effects on the stress intensity factors of the materials $(k_2 )_j$ and $(k_1 )_j$ $(j=1, \, 2)$. Similarly, from Eq.(58), we can observe that the effects of the upper and lower materials on the stress intensity factors of the double material $k_2$ and $k_1$ are also nearly the same. These results occur because the coefficients (41) satisfy all the twelve boundary conditions in Eqs.(3), (4), and (5).

From Eqs.(58), (30), (41e), and (41f), we know that the variations of the normalized intensity factors $k_2 /\tau$ and $k_1 /\tau$ depend on the crack length $a$. The variations of the two factors for the mode II interface crack $k_2 /\tau$ and $k_1 /\tau$ with $a$ for the above three bimaterials are illustrated in Fig. 1. The minute variations of the factor $k_1 /\tau$ can be observed from the inner small figure of Fig. 1.

6 Stress and stress intensity factor of each orthotropic material

With Eq.(36), we have

With Eq.(60), we have

Substituting Eqs.(41), (56c), (57c), (61), and (19) into Eq.(35), we can deduce the stresses in the vicinity of the mode II interface crack tip $(z_{jk} \to a, \;r \to 0, \;j, k = 1, 2)$ for the orthotropic bimaterial as follows:

where $j = 1, 2$, $0 < r \ll a, $ $j = 1: 0^ \circ < \theta < 180^ \circ$, $j = 2: -180^ \circ < \theta < 0^ \circ$, and $(k_2 )_j$ and $(k_1 )_j$ are the stress intensity factors of the orthotropic material $j$ shown in Eqs.(56c) and (57c), which can be rewritten as follows:

Substituting Eqs.(56c), (63), and $\varepsilon {\rm{ = }}0$ in Eq.(33) into Eq.(62), considering Eqs.(50) and (59), we can obtain the analytic solutions of the stresses near the mode II crack tip $(z_{jk} \to a$, $r \to 0$, $j, k = 1, 2)$ for the orthotropic single material as follows:

where $0 < r \ll a$, and $- 180^ \circ < \theta <180^ \circ.$

The results of Eqs.(33), (59), and (64) agree well with the classical conclusions in Refs.^[29]--^[31]. This implies that the theoretical deductions are correct.

7 Conclusions

The non-oscillatory singularity behaviors near the mode II interface crack tip for the orthotropic bimaterial are carefully studied. New special stress functions are constructed, containing ten undermined coefficients. The boundary conditions are transformed into the systems of eight homogeneous or twelve non-homogeneous linear equations. By the existence theorem of non-trivial solution for the system of the eight homogeneous linear equations, the characteristic equation and the stress singularity exponent of the orthotropic bimaterials are given, the discriminating condition of the non-oscillatory singularity is obtained. The uniqueness of the solution for the system of twelve non-homogeneous linear equations is proved, and the ten undermined coefficients in the stress function are uniquely determined. The stress intensity factors for the mode II interface crack are defined, and their theoretical formulae are derived. The analytic solutions of the stresses near the mode II interface crack tip for the orthotropic bimaterial are obtained. The results show that the stress intensity factors and the stresses possess mixed crack characteristics. The stress singularity exponent, the stress intensity factors, and the stresses of the mode II crack for the orthotropic single material are obtained. All of the results given in this paper have valuable significance in either academic researches or engineering applications.