13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- 2.1. Basic principle According to the weight function theory, for a crack subjected to an arbitrary pressure σ(x) distributed at the crack faces, the non-dimensional stress intensity factor f can be determined by a simple quadrature [11,12]. 0 ( ) ( , ) d a x m a x f x a σ σ π = ⋅ ∫ ' ( , ) ( , ) ( ) r r u a x E m a x a f a a σ π ∂ = ∂ (1) where m(a, x) is the weight function for the crack body, E’=E for plane stress, E’=E/(1- ν 2) for plane strain，σ is a reference stress, a and x are the non-dimensional crack length and coordinate along the crack normalized by the characteristic length W (often taken as unity), here W refers to half plate width for the finite width panel containing a center crack. fr(a) and ur(a, x) are the stress intensity factor and crack opening displacement, respectively, for a reference load case. The corresponding crack opening displacements can also be easily determined when the relevant weight function, m(a, x), is available. From Eq. (1), we have 0 ( , ) [ ( ) ] ( , )d ' a a u a x f s a m s x s E σ π = ⋅ ∫ (2) where the non-dimensional stress intensity factor f (s) is obtained using Eq. (1). It should be emphasized that, the σ(x) in Eq. (1) refers to the stress distribution at the prospective crack line, and is determined from stress analysis for the same configuration but without crack. This implies that once the weight function is known for a given crack geometry and σ(x) is determined, the stress intensity factors and crack opening displacements for any crack length can be obtained by simple integration through Eqs.(1) and (2). The advantages are especially useful for stress intensity factor and strip yield model analysis of multiple site damage. 2.2. Weight function method for special collinear cracks The weight function method for single crack is applied for a special multiple site damage case[7]: one large center crack formed by coalescing three un-equal length center cracks in a panel of finite width, with compressive yield stress σs uniformly acting along the un-cracked ligament and in the crack tip region, Fig.1a. The analysis for this case is conducted by assuming the coalesced three un-equal length cracks as one single fictitious crack subjected to segment pressure distribution in plastic zones, in addition to the applied external load, Fig.1b. Essentially, the Dugdale[13] strip yield model is the superposition of two linear elastic solutions. One is for remote uniform tension stress, which is available in Ref. [14]. Another is for segment uniform compressive yield stress acting in the plastic zones. The stress intensity factor and crack opening displacement for this load case can be determined by using WFM, equations (1) and (2). The weight functions for a center crack in a finite sheet were given in Ref.[12]. For the three coalesced cracks, the critical stress and the fictitious crack length are determined based on two conditions [4]: i) Vanishing of the stress singularity at the fictitious crack tips shown in Fig.1b; and ii) Zero of the minimum crack opening displacement at the ligament [a1, a1+d].
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