1 Introduction

For the safety of engineering structures^[1], three-dimensional stress intensity factors (SIFs) are very important parameters in linear elastic fracture mechanics. They can be used to predict crack propagation life and residual strength of the structure. At present, the solutions of three-dimensional SIFs are attracted attentions by many researchers^[2-4]. A brief survey of the methods can be summarized as follows: the finite element method (FEM)^[5-9], the boundary element method (BEM)^[10-12], the mesh-free method, the analytical method^[13-14], the empirical method^[15-16], the hybrid displacement discontinuity method^[17-21], and the weight function method^[22].

Due to the stress concentration around cavity, cracks are easily to initiate and propagate around the surface cavity under the fatigue loading. Consequently, the problem for cracks emanating from a surface cavity in a finite body is discussed in this article. At present, many studies on the liked-plane crack problem have been performed. Stefanescu et al.^[15] by means of empirical methodology made an analysis for asymmetric through-thickness fatigue cracks at the hole. Lin and Smith^[23] used both the quarter-point displacement and $J$-integral methods to make an analysis for corner cracks emanating from fastener holes under tension. Yan^[17-20] used the hybrid displacement discontinuity method to research the problem for three-dimensional cracks emanating from a hole in a finite plate. However, there are few studies^[24-25] for the three-dimensional problem of cracks emanating from cavity, especially for cracks emanating for the surface cavity. Lee^[13-14] proposed an analytical method for cracks emanating from a spherical cavity in a long circular cylinder. Xiao and Yan^[11] performed a numerical analysis for cracks emanating from a surface cavity in an infinite body by the BEM. From what we have discussed, only the BEM of solution for cracks emanating from a surface cavity is available. However, the BEM is very time-consuming and not suitable for the engineering application. Therefore, a time-saving method with the fair accuracy is required for engineering applications.

A semi-analytical and semi-engineering method of a closed form solution is advanced for three-dimensional cracks without surface cavity^[26-28]. In these articles, this theory is extended, and a new semi-analytical and semi-engineering method of a closed form solution for cracks emanating from the surface cavity is developed. To solve the problem of crack emanating from the surface cavity, the displacement mode of the normal slice is established, and the relevant functions about the cavity are introduced. A series of useful results about three-dimensional SIFs are calculated. With this method, the three-dimensional SIFs can be determined if the relevant two-dimensional SIFs are known. The results show that the present method complies well with that of the BEM. The procedure of the present method can be summarized as follows:

(i) Determine the mode expressions $h_1 ({\rho , 0})$ and $h_2 ( {\rho , 2\pi } )$ of the two-dimensional crack opening displacement with given two-dimensional SIFs, by means of the theory of elasticity and the energy release rate method.

(ii) Establish the relationship between the mode expressions $h( {\rho , \varphi } )$ of three-dimensional crack opening displacements and the mode expressions $h_1 ( {\rho , 0} )$ and $ h_2 ( {\rho , 2\pi } )$ of two-dimensional crack opening displacements.

(iii) Establish the relationship between three-dimensional SIFs $K_\mathrm I$ and the amplitudes $\nu _{00}$ of three-dimensional crack opening displacements by means of the near field stress and displacement expressions.

(iv) Determine the amplitudes $\nu _{00}$ of three-dimensional crack opening displacements. Thus, the three-dimensional SIFs $K_\mathrm I$ can be obtained.

Three types of cracks are considered as follows in this paper:

(i) The notch-edge cracks in the plane stress.

(ii) The cavity-edge cracks in the axi-symmetric stress.

(iii) Cracks emanating from a surface semi-spherical cavity in a finite body.

Method of closed form solution

In this paper, the problem for cracks emanating from a surface semi-spherical cavity in a finite body, as shown in Fig. 1, is discussed. At first, the cylindrical coordinate system $({z, r, \varphi } )$, as shown in Fig. 1, is introduced, and the variable $\rho $ is defined as $\rho =r-R$.

2.1 Crack opening displacements and SIFs

In this paper, the modes of three-dimensional crack opening displacements are assumed to be constituted by the modes of the plane notch-edge cracks and the axi-symmetric cavity-edge cracks, as shown in Fig. 2.

2.1.1 Two special cases

(i) $\omega \to 0$

The case with $\omega \to 0$ can be taken as the plane stress situation. The mode of the crack opening displacement is the mode of the notch-edge cracks in the plane stress. The mode of the crack opening displacement of the notch-edge cracks can be obtained^[26].

(ii) $\omega =2\pi $

The case with $\omega =2\pi $ is approximated to the axi-symmetrical situation. For the case with $\omega =2\pi $, there is a normal slice, as shown in Fig. 3. In addition, the normal stress is applied to the normal slice, and there are no shear stresses due to symmetry. However, the normal stresses are parallel to the crack surfaces and give little influence on the three-dimensional crack opening displacement. Furthermore, from the Saint-Venant principle, it only gives effects on the part of the crack which is very next to the normal slice. Therefore, the mode of the crack opening displacements in the case of $\omega =2\pi $ can be assumed as the mode of the cavity-edge cracks in the axi-symmetric stress.

2.1.2 Normal case of $\omega \in (0, \pi )$

In the case of $\omega \in (0, \pi ), $ it is assumed that the mode of the crack opening displacements in the normal sections is constituted by the modes of two kinds of cracks, i.e., the plane notch-edge cracks and the axi-symmetric cavity-edge cracks.

2.1.3 Solution of three-dimensional SIFs

In this section, the expression of three-dimensional crack opening displacements $\nu (\rho , \varphi )$ is firstly discussed. For simplicity, the configuration of the crack front is assumed as a circular arc. A concept of the normal slice would be introduced, as shown in Fig. 4. The normal slice crosses to the originating point $O$ and is perpendicular to the $r\varphi$-plane. The point $P(r, \varphi )$ on the surface crack plane can also be written as the point $P(\rho +R, \varphi )$ due to $r=\rho +R$.

As the modes of three-dimensional crack opening displacements can be expressed by the modes of the plane notch-edge cracks and the axi-symmetric cavity-edge cracks, the three-dimensional crack opening displacement is assumed as

(1)

where $h_1 (\rho , 0) $ is the mode of the crack opening displacement of the plane notch-edge cracks, and $h_2 (\rho , 2\pi )$ is the mode of the crack opening displacement of the axi-symmetric cavity edge cracks, $a$ is the crack length, $\omega $ denotes the angle between two surfaces of the finite body, $\rho +R$ and $ \varphi $ denote the radius and the angular position, respectively, $\nu _{00} $ is the vertical displacement at the origin point of the cavity along the $Z$-axis in the finite body, and $\beta ( {a/R, \varphi } )$ is the cavity displacement function.

The function $\beta ( {a/R, \varphi } )$ is defined as

(2)

where $\beta (a/R, 0) $ is the cavity displacement function when $\omega \to 0$, and $\beta (a/R, 2\pi )$ is the cavity displacement function when $\omega =2\pi $.

The function $f(\varphi -\omega /2) $ is proposed by the author and expressed as

(3)

where

(4)

To build the relationship between three-dimensional SIFs and the crack opening displacement, a kind of normal slice along the crack front is introduced. As shown in Fig. 5, the coordinates of an arbitrary point $Q(r_Q , \varphi _Q )$ along the crack front of this normal slice are

(5)

where $r_Q $ and $\varphi _Q $ are the radius and the angular of a point $Q$ along the crack front, respectively, and $r_P $ and $\varphi _P $ indicate the radius and the angular of a point $P$ at the vicinity of the crack front, respectively.

Then, the SIFs are expressed in terms of the crack opening displacements in the vicinity of the crack front as

(6)

where $E_\mathrm n $ is the generalized Young's modulus of the normal slices and assumed as

(7)

in which the constant $\varepsilon $ is much less than 1 and equals 0.000 1 in this article.

The generalized Young's modulus of the normal slices $E_\mathrm n $ should satisfy three cases $\omega =0, \pi , $ and $2\pi $, respectively. The generalized Young's moduli with $\omega =0, \pi , $ and $2\pi $ are

(8)

From Eqs. (1) , (5) , and (6) , the SIFs of the normal slice can be expressed as follows:

(9)

From Eqs. (3) and (4) , three-dimensional SIFs are obtained when $\omega =\pi $ and expressed as

(10)

Therefore, a series of the results of the SIFs will be obtained along with the different $\varphi $ only if the original displacement $\nu _{00} $ is known. The original displacement $\nu _{00} $ will be obtained from Eq. (29) in Subsection 2.3.3.

2.2 Differential equation of opening displacement of cracks

Suppose that the crack front is a circular. To determine the original displacement $\nu _{00} $, the proportional extension of the virtual crack is assumed, and the potential energy increment d$\Pi$ during the crack growth will be considered.

The proportional extension of crack, as shown in Fig. 6, can be expressed as

(11)

The energy release of three-dimensional cracked body is expressed as

(12)

where $s$ stands for the crack front, $G$ represents the energy release rate of an arbitrary normal slice with the unit thickness along the crack front, and d$a$ and d$s$ represent the amount of crack extension and the thickness of the normal slices, respectively. Then,

(13)

Substituting Eqs. (10) and (13) into Eq. (12) , we can obtain the total energy release rate of three-dimensional cracks,

(14)

in which

(15)

Moreover, according to the principle of superposition in the linear elastic mechanics, the load acting on the body boundary can be transferred onto the surfaces of the crack and can be expressed as

(16)

where $\sigma _0 $ is a generalized force, and $t(\rho /R, \varphi /\omega )$ is the load distribution function. Then, the potential energy of a linear elastic cracked body can be written as

(17)

in which $A$ means the crack area, and $B$ is given by

(18)

Differentiation of Eq. (17) yields

(19)

It is evident that the total energy release rate is equal to the differential rate of the potential energy. From Eqs. (19) and (14) , $\nu _{00} $ can be obtained by Bernoulli's differential equation, i.e.,

(20)

2.3 Closed form solution of ${{\nu }_{00}}$

Then, the solution of Eq. (20) can be expressed as

(21)

in which $\nu_{000} =\nu_{00} | {_{g=0} } , $ and $B_0 =B| {_{g=0} } $.

To obtain the closed form solution of $\nu _{00} $, two extreme cases are considered.

2.3.1 $\omega \to 0\;(\mathrm d(\varphi /\omega )$ is invariable), as shown in Fig. 2(a)

For the special cases of $\omega \to 0$, the three-dimensional crack can be reduced to the notch-edge thickness crack in the plane stress, as shown in Fig. 2(a).

Therefore, from Eq. (21) , we have

(22)

in which $I_1 = I |_{\omega \to 0} , B_1 =B |_{\omega \to 0} , \nu _{001} ={v_{00} } |_{\omega \to 0} $, $B_{01} =B |_{\omega \to 0, g=0} , $ and $\nu _{0001}={\nu _{00} } |_{\omega \to 0, g=0} $.

From Eqs. (1) , (8) , (15) , and (18) ,

(23)

2.3.2 $\omega =2\pi $, as shown in Fig. 2(b)

For the special case of $\omega =2\pi $, the three-dimensional crack can be reduced to the axi-symmetric crack, as shown in Fig. 2(b).

Therefore, from Eq. (21) , we have

(24)

in which $I_2 = I |_{\omega =2\pi } , B_2=B |_{\omega =2\pi } , \nu _{002} = {\nu _{00} } |_{\omega =2\pi } $, $B_{02} =B |_{\omega =2\pi , g=0}$, and $\nu _{0002} = {\nu _{00} } |_{\omega =2\pi , g=0} $.

From Eqs. (1) , (8) , (15) , and (18) ,

(25)

2.3.3 Closed form solution

In the general cases of three-dimensional cracks emanating from the partial spherical surface cavity in the finite body, $I/B^2$ can be assumed in terms of $I_1 /B_1^2 $ and $I_2 /B_2^2 $,

(26)

where the constants $m(\omega )$ and $n(\omega )$ can be determined by the least square method.

Substituting Eq. (26) into Eq. (21) yields

(27)

From Eqs. (22) and (24) , it can also be obtained that

(28)

Therefore, from Eqs. (27) and (28) , consider $\nu _{00} $ as a state function. Then, the following equation can be obtained:

(29)

(30)

in which the magnitude of the crack opening displacement $\nu _{00} $ is the closed form solution.

3 Typical cases of three-dimensional crack problems 3.1 Brief review of two-dimensional crack opening displacements

The expressions of two-dimensional crack opening displacements are obtained by the energy release rate method. The brief results are expressed as follows.

3.1.1 Mode and magnitude of crack opening displacements of normal slice

Referring to the analysis for the modes of crack opening displacement of the longitudinal slice in Refs. [26]-[28], the modes of cracks opening displacements $h(\rho , 0) $ and $h(\rho , 2\pi )$ of the normal slices are proposed.

3.1.3 Notch-edge cracks in plane stress while $\omega \to 0$

For the special case of $\omega \to 0$, the modes of three-dimensional crack opening displacements in finite bodies can be reduced to those of two-dimensional crack opening displacements. The crack opening displacement $\nu (\rho , 0) $ for the notch-edge cracks, as shown in Fig. 2(a), will be explained as

(31)

where

(32)

when $\rho=0$, the displacement $\nu(0, 0) $ is obtained and expressed as

(33)

(34)

(35)

(36)

Furthermore, the displacements $\nu _{001} $ and $\nu (0, 0) $ can be easily obtained by many methods, and the details of the solution will not be discussed in this article. Then, from Eqs. (32) and (33) , $\beta(\frac{a}{R}, 0) $ and $F_1(\frac{a}{R}, \frac{t}{R})$ can be known. The detailed expression about SIFs $K_\mathrm I $ of the notch-edge cracks can be obtained in Ref. ^[29]. From Eqs. (34) , (35) , and (36) , $\alpha_1(\frac{a}{R}, \frac{t}{R})$ can be known. Substitute Eqs. (32) , (33) , (34) , and (35) into Eq. (31) . Then, the mode of crack opening displacement $h_1 ( {\rho , 0} )$ is known and expressed as

(37)

3.1.3 Cavity-edge cracks in axi-symmetric stress while $\omega =2\pi $

For the special case of $\omega =2\pi $, the expression about the mode of the three-dimensional crack opening displacements in the axi-symmetric finite body is similar to that of the two-dimensional crack opening displacement. The crack opening displacement $\nu (\rho , 2\pi )$ for the cavity-edge cracks, as shown in Fig. 2(b), can be expressed as follows:

(38)

in which

(39)

when $\rho=0$, the displacement $\gamma(0, 0) $ is obtained and expressed as

(40)

(41)

(42)

(43)

Furthermore, the displacements $\nu _{002} $ and $\nu (0, 2\pi )$ can be obtained in Ref. ^[14]. Then, from Eqs. (39) and (40) , $\beta(\frac {a}{R}, 2\pi)$ and $F_2(\frac{a}{R}, \frac{t}{R})$ can be known. The detailed solutions about SIFs $K_\mathrm I $ of the cavity-edge cracks are known in Ref. ^[14]. From Eqs. (41) , (42) , and (43) , $\alpha_2(\frac{a}{R}, \frac{t}{R})$ can be known. Substitute Eqs. (39) , (40) , (41) , and (42) into Eq. (38) . Thus, the mode of crack opening displacement $h_2 ( {\rho , 2\pi } )$ is known and expressed as

(44)

3.2 General procedure for solution of three-dimensional crack opening displacements and SIFs

The modes of three-dimensional crack opening displacements are expressed by the modes of the plane notch-edge cracks and the axi-symmetric cavity-edge cracks. Then, we have

(45)

where

(46)

(47)

(48)

The original displacement $\nu _{00} $ should be determined and expressed as

(49)

and the constants $m(\omega )$ and $n(\omega )$ are determined by the following formula:

(50)

in which

(51)

(52)

(53)

(54)

(55)

Finally, the three-dimensional SIF can be obtained by the following formula:

(56)

It should be mentioned that for a circular crack in the determination of limit about $h^2( {\rho , \varphi } )/r_1 $, when $r_1 $ approaches zero, it can be seen from Fig. 7 that

(57)

Then,

(58)

in which $h_1 ( {\rho , 0} )$ and $h_2 ( {\rho , 2\pi } )$ are known from Eqs. (37) and (44) .

From Eqs. (3) and (4) , the three-dimensional SIFs are obtained when $\omega =\pi $ and expressed as

(59)

4 Numerical results and discussion 4.1 Validation analysis of present approach

To prove the accuracy of the present method, the following cases are considered:

For this present method, there are the maximum errors of SIFs at $\varphi =0$^[30], because the order of the stress singularity with $\varphi =0$ is not 1/2. Therefore, the SIFs results are not studied at the points $\varphi=0$.

The results obtained by both the present method and the boundary element approach are compared in Figs. 8(a) and 8(b). It is found that the results provided by this method are in excellent agreement with those of the boundary element software FRANC3D^[11]. At the same time, the energy release rate criterion is used to evaluate the accuracy of this present method. In order to express the energy release rate without the material parameters, the general energy release rate concept is considered and defined as

(60)

The following case of the general energy release rate is discussed:

The general energy release rate solutions for two different methods are shown in Fig. 9. The computational results in Fig. 9 illustrate that this present approach is accurate for calculating three-dimensional SIFs for cracks emanating from the surface cavity.

As the original displacement $\nu _{00} $ is a very important parameter for calculating the three-dimensional SIFs, the comparison of the original displacement between the present method and the FEM is shown in Fig. 10. The results illustrate that the present solutions of the original displacement are in excellent agreement with those of the FEM.

4.2 Series of SIFs

In this section, a complete series of results are evaluated using the present approach. The following examples are considered:

(i) $a/R=0.1, 0.2, 0.3, 0.5, 0.9, 1.3, 2.0, 2.5, t/R=6, \omega =\pi $.

(ii) $a/R=0.1, 0.2, 0.4, 0.6, 1.0, 1.5, 2.5, 3.5, 4.0, 4.5, t/R=10, \omega =\pi $.

The SIFs results of the present method are displayed in Figs. 11 and 13. The SIFs normalized by $\sigma \sqrt {\pi a} $ at $\varphi =\pi /2$ are shown in Figs. 12 and 14. The SIFs results shown in Figs. 8, 11, and 13 are used to reveal the effect of the geometry parameters, $\varphi $ and $t/R$, on the SIFs value. Figures 12 and 14 illustrate the effects of the crack length parameter $a/R$ on the normalized SIFs.

4.3 Survey of effect of geometry size on SIFs

In this section, the effect of the geometry configurations on SIFs is discussed. For convenience, the SIFs are normalized by $\sigma \sqrt {\pi a} $ at $\varphi =\pi /2$ and denoted by $K_\mathrm I|_{\varphi=\frac {\pi}{2}} $ when we study the effect of the geometry configurations on the SIFs. The following conclusions are found:

(I) When $a/R=$ constant and $t/R=6$ and 10, $\varphi $ is a variable

(i) $a/R\le 1$

According to the observation of Figs. 11 and 13, almost the SIFs keep constant with the increase of $\varphi $. Then, the effect of the angular $\varphi $ on the three-dimensional SIFs can be neglected.

(ii) $a/R>1$

According to the observation of Figs. 11 and 13, SIFs increase with the increase of $\varphi $. Then, the angle $\varphi $ has the amplifying influence on the three-dimensional SIFs.

(II) When $t/R=6$ and 10, $a$ is a variable

According to the observation of Figs. 12 and 14, $K_\mathrm I|_{\varphi=\frac {\pi}{2}} $ decreases until it achieves its minimum with the increase of the crack length $a$. $K_\mathrm I|_{\varphi=\frac {\pi}{2}} $ decreases sharply with the increase of the crack length $a$ when $a/R\le 1$, and $K_\mathrm I|_{\varphi=\frac {\pi}{2}} $ almost keeps unchanged with the increase of the crack length $a$ when $a/R>1$. Therefore, the initiation of three-dimensional cracks easily happens around the surface cavity.

The effects of the cavity on three-dimensional SIFs have been discussed in Ref. ^[11].

5 Conclusions

In this article, a new semi-analytical and semi-engineering approach for cracks emanating from the surface semi-spherical cavity in a finite body is developed. From the above derivations and computations, the following conclusions can be obtained:

(i) In order to solve the effect of the surface cavity, the mode of the crack opening displacements of the normal slice is established, and the normal slice relevant functions are introduced.

(ii) Three assumptions used in this paper (the generalized Young's modulus, the mode of crack opening displacement, and the separation of variables) can be verified numerically.

(iii) The calculation is very time-saving, as the main work is only to calculate several integrals numerically. For any given case, the STFs along the crack front can be calculated within several seconds. Hence, a complete series of useful results about the STFs can be obtained.

(iv) The accuracy of the present method is investigated. The results provided by this method are in nice agreement with those obtained by the BEM.

(v) The effects of geometry parameters, $\varphi $, $a/R$, and $t/R$, on three-dimensional SIFs for a surface crack emanating from the surface cavity are investigated.

(vi) As the present method is based on the elastic fracture mechanics, this method can only be used in the case with the small plastic zone around the crack front.

Acknowledgements Thanks for the fracture mechanics investigation group of Cornel University for FRANC3D.