## 3 Numerical Examples
### Numerical Examples of Heterogeneous Material Containing Inclusions
This paper analyzes two cases in which the inclusion's elastic modulus is either larger or smaller than that of the plate. The results are shown in Figure 11, with the \(x\)-axis being the ratio of the inclusion center-crack tip distance to the inclusion radius. As shown in Figure 11, when the inclusion's elastic modulus exceeds the plate's and approaches the crack tip, the normalized stress intensity factor decreases, indicating a crack-tip \"blunting\" effect. Conversely, when the inclusion's modulus is lower than the plate's and near the crack tip, the factor increases, showing a \"sharpening\" effect. Furthermore, the inclusion-to-plate modulus ratio significantly impacts the crack-tip stress intensity factor, with the \"blunting\" or \"sharpening\" effects becoming more pronounced as the ratio increases or decreases. In actual composite materials, inclusions are usually multiple. As shown in Figure 12, in the two-dimensional rectangular plate of length \(h\) and width \(b\), there are multiple circular inclusions with the radius \(R\) being randomly distributed. An edge crack of length \(a\) exists on one side of the plate, which is subjected to uniform uniaxial tensile loads \(\sigma\ =\ 10\) MPa. The elastic modulus and Poisson's ratio of the plate and inclusions are \(E_{1}\), \(E_{2}\) and \(\
u_{1}\), \(\
u_{2}\), respectively. The data used in this example are as follows: \(h\) = 100 mm, \(b\) = 50 mm, \(R\) = 3 mm, \(a\) = 25 mm, \(E_{1}\) = 10\({}^{5}\) MPa, \(\
u_{1}\)= 0.2, \(E_{2}\) = 100 MPa, and \(\
u_{2}\) = 0.2. The crack tip was meshed with CPS3 elements and properly refined to enhance computational accuracy, while other regions used CPS4 elements. The entire mesh had 26,860 quadrilateral elements, 45 triangular elements, 26,905 elements in total, and 27,247 nodes. Integral domains with different \(R\)c/\(R\)tip ratios were chosen to calculate the stress intensity factor at the crack tip. The normalized factor was chosen as \(K_{0}\ =\ \sigma\sqrt{\tau a}\). Figure 11: The influence of elastic modulus of inclusions on normalized stress intensity factor. Figure 13 compares the stress intensity factors from the present method and those from ABAQUS across different integration domains. When the domain is small, both methods yield similar results. As the domain grows and intersects the material interfaces, ABAQUS's inability to handle the derivatives of material parameters means it fails to accurately reflect the crack-tip state. In contrast, the present method maintains stable values and satisfies domain-independence even when the integration domain crosses interfaces. This further highlights the advantage of the present method in solving the crack problems in heterogeneous materials with complex interfaces. Furthermore, this paper analyzes the more realistic three-dimensional example. As shown in Figure 14, a cuboid of length \(h\), width \(b\), and height \(c\) contained a two-dimensional rectangular crack (length \(a\), width \(c\)) and several cylindrical inclusions (base radius \(R\), height \(c\)) were randomly distributed inside. The top and bottom surfaces of the cuboid were subjected to uniform uniaxial tensile loads \(\sigma~{}=~{}10\) MPa. The material parameters of the cuboid and inclusions were the same as those in the two-dimensional problem, specifically, \(h\) = 100 mm, \(b\) = 50 mm, \(c\) = 5 mm, \(a\) = 25 mm, and \(R\) = 3 mm. The crack tip was meshed with wedge elements and properly refined to enhance computational accuracy, while other regions used hexahedral C3D8 elements. The entire mesh comprises 243 wedge elements, 94,963 hexahedral elements, 95,206 elements in total, and 110,694 nodes. Integral domains with different \(Rc/R\)tip ratios were chosen to calculate the stress intensity factor at the mid-point of the three-dimensional crack front. The normalized factor was chosen as \(K_{0}~{}=~{}\sigma\sqrt{\pi ac}\). Figure 12: (**a**) Numerical example of two-dimensional heterogeneous materials with multiple inclusions; (**b**) model meshes.