## 3 Numerical Examples
### Numerical Examples of Heterogeneous Material Containing Inclusions  
Composite materials commonly contain particulate inclusions within their internal structure, making them heterogeneous materials characterized by complex interfaces. This section validates the method for heterogeneous materials with inclusions and compares the results with the finite element results. As shown in Figure 9, a rectangular plate of length \(h\) and width \(b\) has an edge crack of length \(a\) on one side. It contains a circular inclusion of radius \(R\), whose center lies on the crack extension line, with a horizontal distance \(x\) between the inclusion center and the crack tip. The plate is subjected to uniform uniaxial tensile loads \(\sigma~{}=~{}10\)MPa on the upper and lower sides. The elastic moduli and Poisson's ratios of the plate and inclusion are \(E_{1}\), \(E_{2}\) and, respectively. The data used in this example are as follows: \(h\) = 100 mm, \(b\) = 50 mm, \(R\) = 3 mm, \(x/R\) = 3, \(a\) = 25 mm, \(E_{1}\) = 35 MPa, \(E_{2}\) = 17.5 MPa, and \(\
u_{1}~{}=~{}\
u_{2}\) = 0.495. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Rc/Rtip** & **5** & **8** & **10** & **15** & **20** \\ \hline \(K_{1}/K_{0}\) & 2.818 & 2.820 & 2.820 & 2.821 & 2.820 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Normalized SIFs for different integral domains. Figure 8: The definition of integral domains. Crack tips are meshed with CPS3 elements and refined to boost computational precision, while other regions use CPS4 elements. The mesh comprises 25,731 quadrilateral and 66 triangular elements, totaling 25,797 elements and 26,125 nodes. The stress intensity factors at crack tips are calculated by choosing integration domains of varying \(R\)c/\(R\)tip sizes, with the normalized factor defined as \(K_{0}\ =\ \sigma\sqrt{\pi a}\). In Figure 10, when the integration domain is small and excludes the inclusion or is large enough to fully contain it, both methods yield similar normalized stress intensity factors. However, when the integration domain intersects the inclusion, the stress intensity factors calculated by ABAQUS fluctuate significantly due to the unhandled interface derivative terms of the material parameter. This leads to inconsistent results across different domains, making accurate crack assessment difficult. In contrast, the method proposed in this paper ensures that the normalized stress intensity factor remains domain-independent, even when the domain intersects with the inclusion. Thus, the present method demonstrates unique advantages in handling heterogeneous materials with complex interfaces. Figure 10: Comparison of computational results between the present method and ABAQUS, providing a two-dimensional example with a single inclusion. Figure 9: (**a**) Numerical example of two-dimensional heterogeneous materials with a single inclusion; (**b**) model meshes. To analyze the influence of the inclusion's properties on the stress intensity factor, the rectangular plate model with a circular inclusion and an edge crack shown in Figure 9a was selected. The aim was to study how inclusion parameters affect the stress intensity factor at the crack tip. The model dimensions were \(h\) = 100 mm, \(b\) = 50 mm, \(a\) = 25 mm, and \(R\) = 3 mm. The plate's material properties were \(E_{1}\) = 35 MPa and \(\
u_{1}\) = 0.495, while the inclusion's elastic modulus was \(E_{2}\), with a Poisson's ratio of \(\
u_{2}\) = 0.495. By keeping the crack position and plate material properties constant and varying the inclusion's horizontal position and elastic modulus, the study explores the influence of inclusion position and inclusion-to-plate modulus ratio on the stress intensity factor. The normalized stress intensity factors at the crack tips were calculated for different inclusion-to-plate modulus ratios and inclusion positions. The normalized factor was the stress intensity factor when the plate has no inclusion.