1 Introduction

Holes with different shapes are commonly used in many thin-walled structures for numerous applications, where the stress concentration (SC) around the holes may lead to structural failure. In general, three different methods, i.e., analytical, numerical, and experimental methods, can be utilized to investigate SC. One of the most powerful analytical methods is the complex variable theory. In Ref. [1], the Airy stress function was considered as a summation of two complex functions, allowing the application of conformal mapping of such a problem in a planar medium. Savin^[2] presented a comprehensive study for the calculation of the SC around holes with different shapes for both isotropic and anisotropic plates. Theocaris and Petrou^[3] studied the stress distributions and intensities around triangular holes. Lei et al.^[4] investigated the stress and displacement around an elastic artificial rectangular hole. Rezaeepazhand and Jafari^[5] studied the SC in metallic plates with holes. Louhghalam et al.^[6] used a mapping-finite element to study the SC in plates with holes. Zhou et al.^[7] investigated the dynamic SC in thick plates with two holes based on the refined theory. The crack initiation and propagation around the holes, due to the presence of SC, may lead to catastrophic failure in some structures. The problem of cracks emanating from a circular hole was first solved by Bowie^[8], who used a conformal mapping and complex variable method to examine the problem. Tweed and Rooke^[9-10] used singular integral equations to calculate the stress intensity factor (SIFs) numerically for the radial cracks at the edge of a circular hole in an infinite plate under uniform remote loading, and analyzed the SIFs under the condition that there was loading on the cracks and the hole. Kim and Hill^[11] developed accurate weight functions to determine the SIFs for one crack or two equal cracks emanating from a circular hole in a finite width plate. Hasebe and Chen^[12] examined a single crack emanating from a circular hole, and investigated the interaction between the crack and the cracked hole. Miao et al.^[13] studied a centered circular cracked hole in a rectangular plate with the boundary element method.

Unequal cracks emanating from circular holes are also studied by many researchers. Junbiao et al.^[14] combined the Muskhelishvili complex variable theory and the least square method to obtain the SIFs for unequal cracks emanating from circular holes in finite and infinite plates. Abdelmoula et al.^[15] investigated the unequal cracks due to circular holes with a new conformal mapping to transform the upper half plane into the interior of a unit circle.

Saimoto et al.^[16] used the body force method to calculate the SIFs for the asymmetrical radial cracks initiated from a circular hole in a finite plate under tension. For piezoelectric materials, based on the complex variable method, Wang and Gao^[17] examined the mode Ⅲ fracture problem for the edge cracks emanating from a circular hole. Chen et al.^[18] provided a semi-analytical solution for an infinite plate containing a circular hole with curved edge cracks. Zhao et al.^[19] combined the complex variable method with an approximate superposition method to obtain the SIFs for the multiple hole-edge cracks in an infinite plate, and presented a modification for the method to solve problems with small cracks. Hasebe and Horiuchi^[20] investigated a strip with semi-elliptical notches or cracks on both sides with a rational mapping function. Kujawski^[21] presented a simple method to estimate the SIFs for the small cracks emanating from anotch. Philipps et al.^[22] studied the SIF of mode Ⅰ for a crack emanating from the root of a sharp V-notch with the dislocation distribution technique and the singular integral equation. There are also several studies on the cracks emanating from the holes with different shapes. Newman^[23] developed an improved boundary collocation method for the stress analysis of equal cracks emanating from a circular and an elliptical hole in finite and infinite plates. Based on the Muskhelishvili complex theory and boundary collocation method, Wang^[24-25] calculated the SIFs for one crack and two equal cracks emanating from an arbitrary hole. Guo and Liu^[26] studied the anti-plane problem for an elliptical hole with two equal straight cracks in a one-dimensional hexagonal quasicrystal. Hasebe and Ueda^[27] used the Muskhelishvili method and rational mapping function to analyze the crack emanating from a corner of the square hole in an infinite plane under tension. Based on the Muskhelishvili method, Liu and Duan^[28-29] obtained the SIFs for the unequal cracks emanating from an elliptical hole in an infinite plate subject to pure shear and tension at infinity. For small cracks, the discrepancy between their results and other available results in the literature is significant and controversial. Yang et al.^[30] investigated the fracture behavior of an elliptical hole with two asymmetrical cracks in one-dimensional hexagonal quasicrystals under the piezoelectric effect with the conformal mapping technique. With a conformal mapping, Wang and Pan^[31] presented the analytical solutions for a finite slit crack partially penetrating two circular inhomogeneities embedded in an unbounded matrix.

A considerable amount of research has been devoted to investigating the equal and unequal cracks emanating from a circular hole. However, there are only few works on the unequal cracks emanating from an elliptical hole. Moreover, some of the analytical methods presented in the literature are not suitable for the analysis of small cracks. In the present study, an infinite isotropic plane containing an elliptical hole with two unequal aligned cracks is examined. The Muskhlishvili complex variable method is used and a new conformal mapping function is presented to transform the outside region of the unit circle to the outside region of the elliptical hole with two unequal cracks. The plane is subjected to a uniform remote loading in arbitrary directions, and the hole and cracks are assumed to be traction free. Following this, the mapping function is expanded to the sum of fractional expressions to obtain the analytical functions. In this paper, not only SIFs (K_Ⅰ and K_Ⅱ) are calculated for arbitrary loading, but also the stress field around the cracked hole is evaluated. Furthermore, the effects of the crack length and the ratio of the ellipse semi-axes (a/b) on the SIFs are studied. SIFs are calculated for small and long unequal cracks emanating from circular and elliptical holes. To validate the accuracy of the analytical method, the results of some numerical examples are compared with the results of the finite element method (FEM) and the available data in the literature.

2 Theory

An infinite isotropic plane subject to the far-field stresses (N₁ and N₂) is considered, where there is an elliptical hole in the plane and two unequal aligned cracks emanating from the hole (see Fig. 1). a_R and a_L are the lengths of the right and left cracks, respectively. The size of the hole and cracks are assumed to be small enough compared with the plane dimensions. A two-dimensional (2D) stress analysis in the absence of body forces is studied for the linear elastic range.

2.1 Mapping function

The mapping function Z₁, which transforms the outside region of a unit circle (in the ζ-plane) onto the outside region of the unit circle with two unequal radial aligned cracks (in the Z₁-plane), is expressed as follows:

(1)

where

(2a)

(2b)

(2c)

In the above equations, l_i is the length of the crack emanating from the unit circle. Savin^[2] presented a mapping function Z=ω(ζ) for a plane with an elliptical hole as follows:

(3a)

(3b)

where a and b are the semi-axes of the ellipse. Combining Eqs. (3) and (1), the mapping function for the transformation of the outside region of the unit circle in the ζ-plane onto the outside region of the elliptical hole with two unequal aligned cracks in the Z-plane is

(4)

where l_i used in the calculations for the parameters M and N is defined by

(5)

In the above equation, a_i is the crack length. In the special case of equal semi-axes (a=b), ω(ζ) is the mapping function for the circular cracked hole. In Fig. 2, the boundary points P_i, P_i, and D are transferred from the points G_i, G_i, and G, respectively.

It is noted that the conjugate ω(ζ) and the first and second derivatives of ω(ζ) are calculated as follows:

(6a)

(6b)

(6c)

which will be used to calculate the analytical functions in the next sections.

2.2 Linear elasticity theory

For a 2D problem in the absence of body forces, a biharmonic equation is obtained by combining the equilibrium equations and the compatibility conditions^[1], i.e.,

(7)

where U is the stress function. Muskhelishvili^[1] presented the solution to Eq. (7) as follows:

(8)

where φ₁(z) and χ₁(z) are the analytical functions to be obtained. The corresponding stress components can be stated as follows:

(9a)

(9b)

where

Since the hole and the cracks are traction free, the stresses on the contours of the hole and cracks in the polar coordinate system (r, θ) are^[2]

(10a)

(10b)

Applying the transformation rule, the stress components in the Cartesian coordinates are

(11a)

(11b)

where θ is the angle of the r-axis with respect to the x-axis. For simply-connected domain problems, when the hole and cracks are assumed to be traction free, it is possible to express the functions φ₁ and ψ₁ as follows^[2]:

(12a)

(12b)

where B₁, B₂, and C₂ are defined by the stresses at infinity. Then, we have

(13)

where β is the angle between N₁ and the x-axis. Therefore, the stresses at infinity are

(14)

Using the mapping function z=ω(ζ) with the assumptions

we can rewrite Eq. (12) as follows:

(15a)

(15b)

The function φ₀(ζ) is defined by^[1]

(16)

It can be obtained through^[12]

(17)

For simplicity, the following expressions are defined:

(18a)

(18b)

In some previous investigations^{[19, 28-29]}, φ₀(ζ) can be obtained from Eq. (16) while neglecting the first term of the right-hand side of Eq. (16), which is defined as I₁ in Eq. (18a). In Refs. [19], [28], and [29], it was assumed that the expression

was analytical in the interior of the unit circle, where I₁ was considered as zero according to the Cauchy integral theorem. It can be found that the results give inaccurate SIFs for small cracks compared with other approaches^{[14, 23]}. Therefore, in the present study, I₁ is considered in the calculations. Since φ₀(t) and its derivatives are analytical in the exterior of the unit circle, is analytical in the interior of the unit circle. Moreover, according to Eq. (19), the roots of are not inside the unit circle.

(19)

In the present paper, to obtain φ₀(ζ), the function ω(ζ) is expanded to the sum of the fractional terms, and the Q(ζ) term of Eq. (4) is separated into four parts as follows:

(20)

where r_i (i=1, 2, 3, 4) are functions of the crack length, i.e.,

(21a)

(21b)

Each term on the right-hand side of Eq. (20) may be expanded into power series, and can be estimated as follows^[20]:

(22)

To obtain A_k and α_k, the following equations should be solved:

(23a)

(23b)

With the method proposed by Hasebe and Horiuchi^[20], A_j and α_j can be determined.

Substituting Eq. (22) into Eq. (20) and determining the partial fraction decomposition, we can state Eq. (20) as follows:

(24a)

(24b)

(24c)

where the values of ε_{1, k}ⁱ, ε_{2, k}ⁱ, and ε_{3, k}ⁱ are defined in Appendix A. According to Eq. (24), the new approximated mapping function can be written in the form of series as follows:

(25)

This mapping function has one pole on ζ=0 and 48 poles on ζ=α_kr_i in the interior of the unit circle. Considering the approximate mapping function of Eq. (25), we can obtain the expression for I₁ in Eq. (16) as follows:

(26a)

(26b)

The expression for f₁⁰+i f₂⁰ in Eq. (18b) can be obtained from the boundary conditions of the hole and cracks as follows^[2]:

(27)

Substituting Eq. (27) into Eq. (18b), we can formulate the expression for H(ζ) as follows:

(28)

Using the Cauchy integral theorem and the mapping functions of Eqs. (4) and (25), we can obtain the expressions for H₁(ζ) and H₂(ζ) as follows:

(29a)

(29b)

Substituting Eqs. (28) and (26) into Eq. (16), we can derive the function φ₀(ζ) and its first derivative as follows:

(30)

(31)

Substituting ζ=δ_kⁱ into φ'₀(ζ), we can obtain 48 linear complex equations. By solving 96 linear equations, we can determine the values for φ'₀(δ_k), after which φ₀(ζ) and ψ(ζ) can be calculated from Eqs. (30) and (17), respectively. The complex analytical functions φ(ζ) and ψ(ζ) can be obtained by substituting φ₀(ζ) and ψ₀(ζ) into Eq. (15). The derivatives of the analytical functions may be calculated as follows:

(32a)

(32b)

(32c)

Then, the stress components can be determined by means of Eq. (9).

2.3 SIFs

For a mixed mode problem, the SIFs at the right (R) and left (L) cracks can be expressed as follows^[27]:

(33a)

(33b)

From Eqs. (6c), (31), and (33), we can express the SIFs as follows:

(34)

(35)

3 Results and discussion

The analytical equations for the stress field and SIFs, which were presented in the previous section, are applied to Mode Ⅰ, Mode Ⅱ, and the mixed mode. The results are compared with those obtained with the FEM and available in the literature. In the FEM, the triangular and quadrilateral elements are implemented, respectively, for the cracked tips and other regions. To present non-dimensional SIFs, K_Ⅰ0 and K_Ⅱ0 are defined by

(36a)

(36b)

For the elliptical hole, a can be substituted for R. It is noted that the above parameters are the SIFs of an equivalent center crack.

3.1 Stress field for tension loading

The tangential stress (σ_θ) of point D, at the top of the circular hole with two equal cracks in an infinite plane, is shown in Fig. 3. The plane is subjected to the uniaxial tension at infinity, perpendicular to the crackline (N₁=σ, N₂=0, β=π/2).

Figure 3 shows that the results of the analytical formulation accurately agree with those obtained with the FEM. In this figure, when the crack length increases, the tangential stress will decrease from -1 to about -3. Moreover, there are significant differences between the results of the cracked hole and the hole without cracks.

In the case of the elliptical hole with two equal cracks, the tangential stresses of point D for some ratios of the ellipse semi-axes (a/b) are shown in Fig. 4. It is found that the absolute values of the stresses at point D for the cracks with long lengths increase when a/b increases.

The distribution of the tangential stress on the surface of the cracks and the circular hole for the case of equal cracks and a_R/R=1 is presented in Fig. 5(a). From this figure, it is apparent that the stresses tend to infinity at points P₁ and P₂ due to the singularity of these points. Noting that the values of ω'(ζ) at the crack tips are zero, the corresponding stresses are singular. The stresses on the ligament of the right crack are shown in Fig. 5(b). From this figure, it is apparent that the stress at the crack tip has a singular tendency.

3.2 SIF of mode Ⅰ

In this section, K_Ⅰ for an infinite plane containing the cracked hole under tensile loading are obtained. Assume that

The coefficients B₁, B₂, and C₂ are obtained from Eq. (13) as σ/4, σ/2, and 0, respectively. In other words, the above conditions can be stated as follows:

The values of K_Ⅰ obtained from Eq. (34) are compared with the results in Refs. [14] and [23] and the results obtained with the FEM (see Fig. 6). As shown in Fig. 6, the results are in good agreement.

To investigate the effect of I₁ (see Eq. (18a)) on K_Ⅰ, the present formula is applied to two cases, i.e., with and without I₁. The results are shown in Fig. 7. It is apparent that neglecting I₁ causes a considerable error for small cracks. However, for long cracks, a_R/R>2, no considerable difference is found, and the values of K_Ⅰ of both cases converge to 1. It is worth mentioning that for the cases of long crack length, the values of the SIFs are approximately the same as those obtained from the problem for equivalent center cracks.

The analytical results are compared with the results of Ref. [23] for different ratios of the ellipse semi-axes (a/b) in Fig. 8. It is found that the current formulations agree with the results from Newman^[23] who used an improved boundary collocation method. It is observed that, for small cracks, when the ratio of a/b increases, the value of K_Ⅰ increases remarkably.

The values of K_Ⅰ for unequal cracks emanating from the elliptical hole (a/b=2) are presented in Fig. 9. Eq. (34) can be used for the case of one crack emanating from the hole with a_L=0. The results of this case are added to Fig. 9(a).

As can be seen from Fig. 9, both K_{Ⅰ, R} and K_{Ⅰ, L} increase when a_R increases. Moreover, the increase rate in K_{Ⅰ, R} for the cases of right small cracks is considerable. The same results can be found when a_L increases.

3.3 SIFs of mode Ⅱ

For loading

we have

In this case, with Eq. (14), we can obtain

Substituting these coefficients into Eqs. (34) and (35), we can calculate the values of K_Ⅱ for equal cracks (a_R=a_L) emanating from a circular hole.

The results of the present method with and without I₁ are compared with those of the FEM (see Fig. 10). The discrepancy of the results for small cracks between the two cases can be found for modes Ⅱ and Ⅰ.

The values of K_Ⅱ for unequal cracks emanating from the elliptical hole (a/b=2) for pure shear loading at infinity are shown in Fig. 11. According to Fig. 11, we can see that, the variations of K_{Ⅱ, R} and K_{Ⅱ, L} in terms of a_R are similar to the behaviors of K_Ⅰ (see Fig. 9). Contrary to K_Ⅰ, the increase rate in K_{Ⅱ, L} is not significant when a_R increases.

3.4 Mixed mode SIF

To investigate arbitrary direction loading, e.g., β=π/4, the case, in which N₁=σ and N₂=0, is considered. The stresses at infinity for this case are

The coefficients

are calculated according to Eq. (14). Applying these coefficients in Eqs. (34) and (35), we can obtain K_{Ⅰ, R}, K_{Ⅰ, L}, K_{Ⅱ, R}, and K_{Ⅱ, L} for the elliptical cracked hole (a/b=2). The values of K_{Ⅰ, R} and K_{Ⅱ, R} for various unequal cracks are shown in Fig. 12.

From these figures, it can be observed that, for small a_R, the effect of increasing a_L is significant on K_{Ⅰ, R} whereas can be neglected on K_{Ⅱ, R}.

4 Conclusions

The stress field and SIFs in an infinite isotropic plane containing circular or elliptical holes with two unequal aligned cracks are investigated for different remote loading. The Muskhelishvili complex variable theory is combined with a new conformal mapping function to calculate the analytical functions. Based on the Hasebe approach, the mapping function is expanded to the sum of fractional expressions to obtain accurate analytical functions.

In the case of cracks with equal lengths, the values of the tangential stress at the top point of the circular hole change from -1 (without cracks) to about -3 (with large cracks) when the length of the cracks increases. For small cracks, the present analytical solution improves the SIFs, comparing with those in the previous works. It should be noted that the results for large equal cracks (not unequal) are transformed to the equivalent center crack problems in modes Ⅰ and Ⅱ. It is concluded that the new analytical solution not only improves the results of SIFs for all ranges of crack lengths, but also provides the stress field around the cracked hole.

Appendix A

ε_{1, k}ⁱ, ε_{2, k}ⁱ, and ε_{3, k}ⁱ are defined by

(A1)

(A2)

(A3)

where

(A4)

The arrays of the matrices in Eq. (A4) are the indices of r, e.g., r_{p(1, 2)}=r₃.