13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- 3.3 Numerical implementation of the dual BIEs In the context of the dual boundary integral equations, we apply four- to eight-node isoparametric elements to discretize the uncracked boundary and three types of nine-node quadralateral curved elements to discretize the crack surface. Once these quantities are determined, the displacements, tractions and CODs on boundary are known everywhere. Thus, we can now rewrite the dual boundary integral equations in a discretized form in terms of these parameters to be determined using the shape functions. According to the nature of the kernel and the relative position of the source point with respect to the element on which the integration is carried out, the integrals in the discretized dual BIEs are regular or non-regular. All these integrals have been calculated carefully [9]. Based on the numerical method, computer programs have been written in Fortran to calculate displacements, tractions and the CODs of a multilayered dissimilar elastic solid of finite or infinite extent containing cracks. The stress intensity factors (SIFs) can be calculated by using the CODs on the crack surface. We have examined the accuracy of the proposed DBEM in [9]. 4 A square crack in the FGM interlayer In Fig. 2, a square crack is located in the FGM interlayer bonded to the two homogeneous half-spaces and parallel to the interfaces between a homogeneous half-space and the FGM interlayer. The square crack has the side length 2c. The crack surfaces are subjected to uniform compressive stress p. Among three materials, materials 1 and 3 are homogeneous media and material 2 is a gradient medium. The elastic modulus of the materials is approximated by ( ) z E z E e α 1 2 = and ( ) 3 1 / 1/ ln h E E =α (11) where h is the thickness of the interfacial layer, the constant α can be positive or negative, and E1, E2 and E3 are the elastic moduli of three layers. Let 2=cα , / 0.5 = h c and ν1=ν2=ν3=0.3. The five cases, / 0.05, 0.15, 0.25, 0.35, 0.45 = d c , are analyzed. The crack surface is discretized into 100 nine-node elements. For the FGM described in expression (11), the FGM is closely approximated by n bonded layers of elastic homogeneous media. Each layer has the thickness equal to h/n and shear modulus equal to E2(z) at the top depth of the layer, i.e. for the i-th layer, z=Hi, where Hi=ih/n, ( n i 1, 2, , = Λ ). Two homogeneous materials bonded through the FGM are considered as semi-infinite domains for the layers H0 and Hn+1 respectively. For all the layers, the Poisson’s ratios are the same and equal to 0.3. It can be observed that a close approximation of the elastic modulus variation can be obtained using a large number of n [5]. Figs. 3 and 4 illustrate the variations of the SIF values with the crack distance d to the FGM interlayer. In these figures, KI and KII are symmetrical to the x′ -axis. It can be found that the crack distance d increasing, the KI and KII values increase. From 0= d to d h= , the elastic modulus on Fig. 2 A square crack in the FGM interlayer O′ x′ O z x or y E3=E1exp(αh) E1 d h E2=E1exp(αz) 2c
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