13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Consider an elastic plane domain with a crack, as shown in Fig.1. Let the domain studied be Ω, and assume the configuration boundary of the domain to be L, not including the crack surface. The body forces within the domain are assumed to be F(l) (l=1, 2), and the lengths of the inner crack and the edge crack are taken to be 2a and a, as shown in Fig.1(a) and Fig.1(b), respectively. Embed Ω into an infinite plane domain with a crack, the crack length being 2a, and apply unknown fictitious loads X(l) (l=1, 2) along a fictitious boundary S outside Ω, whose shape is similar to that of the real boundary L, as also shown in Fig.1. Due to the use of the Erdogan fundamental solutions [18], not only the governing differential equations within Ω but also the stress boundary conditions on the crack surface are satisfied automatically. Therefore, only the boundary conditions along the contour of Ω need to be considered. Under the combined action of the body force F(l) and the fictitious loads X(l) (l=1, 2), the boundary conditions along L can be expressed as ( ) ( ) ( ) ( ) ( ) ( ) 2 2 ( ) ( ) ( ) ( ) L S S L Ω Ω L S 1 1 ; d ; d 1,2 l l l l k k k l l G z z X z s G z z F z V H z k Ω = = + = = ∑ ∑ ∫ ∫∫ , (1) where zL∈L, zS∈S, zΩ∈Ω; k=1,2 denotes that two boundary conditions exist along L for plane problems; Hk are the known boundary functions along L; and ( )l kG are the kernel functions consisting of the Erdogan fundamental solutions. Eq. (1) are nonsingular fictitious boundary integral equations because the source points will never coincide with the field points in the kernel functions. However, analytic solutions to Eq. (1) are normally not available, and the integral equations should be solved on a numerical baisis. For this purpose, the unknown fictitious loads X(l) are expressed in terms of a set of B-spline functions, and the boundary-segment technique is used to eliminate the resulting boundary residues [10]. Then Eq. (1) turns into the following numerical equation as [ ]{ } { } { } A X B C + = , (2) where { }X is the column matrix consisting of the unknown spline node parameters of the fictitious loads along S; [ ]A is influence matrix of { }X ; and { }B and { }C are the known column matrices depending on the body forces F(l) within Ω and the boundary condition functions Hk along L, respectively. Usually Eq. (2) needs to be solved on a least-squares basis as generally overdeterminate collocation is conducted to achieve a better solution with more boundary segments while keeping the number of fictitious boundary elements to be at a lower level. Once the spline node parameter { }X is determined, the mode-I and II SIFs of the cracked problem can be obtained from the discrete forms of the following equations: ( ) ( ) ( ) ( ) ( ) 2 2 ( ) ( ) ( ) ( ) S S Ω Ω S 1 1 d d I,II l l l l j j j l l K K z X z s K z F z V j Ω = = = + = ∑ ∑ ∫ ∫∫ , (3) where ( )l j K (j=I,II; l=1,2) are the Erdogan fundamental solutions of SIFs [18]. 3. Crack growth-based fatigue life prediction 3.1. Paris law Crack growth can occur under cyclic loading. Using Paris equation, the crack growth rate can be expressed as the function of the SIF range, and can be written as
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