## 2 Integral and Crack Propagation Criteria  
### Two-Dimensional Interaction Integral with MFEM  
The energy integral \(I\), called the interaction integral, includes the contour \(\Gamma\) around the crack tip, which is a highly accurate method that has emerged in recent years to solve the two-dimensional fracture problem [29; 30]. It is based on the quasi-static derivation process, assuming that the material has the characteristics of small deformation, linear elasticity and continuity.  
\[I=\int\Gamma[W\delta_{lj}-\sigma_{ij}^{(act)}\frac{\partial(u_{i}( aux))}{\partial x}-\sigma_{ij}(aux)\frac{\partial(u_{i}(act))}{\partial x}]n_{j}d\Gamma \tag{1}\]  
where \(n_{j}\) is the unit external normal vector of the contour \(\Gamma\). In addition, the form of the strain energy density \(W\) for the interaction integral is as follows:  
\[W=\sigma_{ij}^{(act)}\epsilon_{ij}^{(aux)}=\sigma_{ij}^{(aux)}\epsilon_{ij}^ {(act)} \tag{2}\]  
In finite element theory, it is not easy to solve the infinitesimal contour integral near the crack tip. In order to facilitate the calculation in practical applications, the interaction integral formula is usually converted into an equivalent surface integral as shown in Figure 1. There is a new contour C set outside the contour \(\Gamma\), which makes the region S closed. Then, through the divergence theorem, the line integral of Equation (1) can be further written as below [19; 27; 31]  
\[I=\int S\bigg{[}\sigma_{ij}^{(act)}\frac{\partial(u_{i}( aux))}{\partial x}+\sigma_{ij}^{(aux)}\frac{\partial(u_{i}(act))}{\partial x}-W \delta_{lj}\bigg{]}\frac{\partial q}{\partial x}dS-\int C^{+}+C^{-}\bigg{[} \sigma_{l2}^{(act)}\frac{\partial(u_{i}(aux))}{\partial x}+\sigma_{ij}^{( aux)}\frac{\partial(u_{i}(act))}{\partial x}\bigg{]}qm_{j}dC \tag{3}\]  
where \(q\) is defined as the variable weight function, which takes the value of 1 in region S and 0 in other places. In addition, there is a unit normal vector \(m_{j}\), which is perpendicular to profile C, as shown in Figure 1.  
Therefore, the stress intensity factors and T-stress in the real fields can be obtained by selecting the appropriate auxiliary fields. Because the selection of the stress intensity  
Figure 1: The equivalent surface integral of the interaction integral.  
factors and T-stresses in the auxiliary fields is different for the interaction integral, the selection of auxiliary fields in this paper is consistent with that of the Refs. [19; 32; 33; 34].  
When the integral path \(\Gamma\) approaches the tip of crack, the relationship between the stress intensity factors and the interaction integral \(I\) can be obtained as below  
\[I=\frac{2}{E^{*}}(K_{\text{I}}K_{\text{I}}(^{aux})+K_{\text{II}}K_{\text{II}}( ^{aux})) \tag{4}\]  
where \(E^{*}\) is defined as follows:  
\[E^{*}=\left\{\begin{array}{cc}E,&(\text{plan stress})\\ E/(1-v^{2}),&(\text{plan strain})\end{array}\right. \tag{5}\]  
If the SIFs of the auxiliary fields satisfy special conditions [19], the SIFs of the actual fields can be derived as  
\[K_{\text{I}}=\frac{E^{*}}{2}I\Big{(}K_{\text{I}}(^{aux})=1\,,K_{\text{II}}(^ {aux})=0\Big{)} \tag{6}\]  
\[K_{\text{II}}=\frac{E^{*}}{2}I\Big{(}K_{\text{I}}(^{aux})=0\,,K_{\text{II}}(^ {aux})=1\Big{)} \tag{7}\]  
Similarly, T-stress can also be derived by interaction integral and its expression is as follows:  
\[T=\frac{E^{*}}{f}I \tag{8}\]  
where \(f\) is defined as a line load, which is applied on the crack propagation plane and along the line of the crack, as shown in Figure 2.  
In order to facilitate subsequent observation and comparison, the dimensionless forms of the SIFs and T-stress are adopted as [11; 14]  
\[K_{\text{I}}=\frac{P}{\pi BR}\sqrt{\pi a}K_{\text{I}}^{*},K_{\text{II}}=\frac {P}{\pi BR}\sqrt{\pi a}K_{\text{II}}^{*},T=\frac{P}{\pi BR}T^{*} \tag{9}\]