## 2 Fracture Parameter Solution Method Based on the Postprocessing Routine of ABAQUS
### Computation of Interaction Integral  
The interaction integral (abbreviated as I-integral), also known as the interaction energy integral or the M-integral, is a two-state integral derived from the J-integral of the actual and specified auxiliary fields. Due to the designability of the auxiliary field in the I-integral, it has been widely used for various homogeneous, heterogeneous, and composite materials to determine critical fracture parameters such as stress intensity factors (SIFs) and T-stresses [26]. For linear elastic materials, stress, strain, and displacement satisfy the principle of superposition. Calculate the J-integral under the combined actual and auxiliary fields using Equation (1) to obtain the following:  
\[J^{S}\ =\ \underset{\Gamma\to 0}{\lim}\int_{\Gamma}[\frac{1}{2}(\sigma_{ij}+ \sigma_{ij}^{\rm aux})(\epsilon_{ij}+\epsilon_{ij}^{\rm aux})\delta_{1i}-( \sigma_{ij}+\sigma_{ij}^{\rm aux})(u_{j,1}+u_{j,1}^{\rm aux})]n_{i}d\Gamma\ =\ J+J^{\rm aux}+I, \tag{9}\]  
After rearrangement, the equation can be divided into three parts: the J-integral for the actual field alone, the J-integral for the auxiliary field alone, and the interaction term between the actual and auxiliary fields. The interaction term is as follows:  
\[I\ =\ \underset{\Gamma\to 0}{\lim}\int_{\Gamma}[\frac{1}{2}(\sigma_{jk} \epsilon_{jk}^{\rm aux}+\sigma_{jk}^{\rm aux}\epsilon_{jk})\delta_{1i}-( \sigma_{ij}u_{j,1}^{\rm aux}+\sigma_{ij}^{\rm aux}u_{j,1})]n_{i}d\Gamma \tag{10}\]  
Equation (10) is defined as the interaction integral. \(\Gamma\) is an infinitesimal closed loop including the crack tip. As the interaction integral defined above is a line integral around an infinitesimal crack tip contour, it cannot be directly solved in finite element calculations. Therefore, we need to convert it into an area integral. There are two primary methods to accomplish this. The first method transforms the line integral in Equation (10) into the difference between a finite contour integral and an area integral of the surrounding region. The second method involves multiplying the integral in Equation (10) by a weighting function to directly convert it into an equivalent area integral. The first method may result in larger computational errors when calculating the line integral [27]. Therefore, this study adopts the second method to convert the line integral into an area integral. The weight function \(q\) is defined as 1 within the integration radius and 0 outside. According to Reference [23], the converted equivalent area integral form is as follows:  
\[I\ =\ \int_{A}(\sigma_{ij}u_{j,1}^{\rm aux}+\sigma_{ij}^{\rm aux}u_{j,1}- \sigma_{jk}^{\rm aux}\epsilon_{jk}\delta_{1i})q_{j,i}dA+\int_{A}(-f_{j}u_{j,1 }^{\rm aux}+\sigma_{ij}u_{j,1}^{\rm aux}-\sigma_{ij,1}^{\rm aux}\epsilon_{ ij})qdA, \tag{11}\]  
In the equation, \(A\) is an area of finite size, including the crack tip, and \(f_{j}\) is the body force of the element. Based on the crack tip strain field, equilibrium equations without body forces \(f_{j}\ =\ 0\), and the constitutive equations, Equation (11) can be expressed as follows:  
\[I\ =\ \int_{A}(\sigma_{ij}u_{j,1}^{\rm aux}+\sigma_{ij}^{\rm aux}u_{j,1}- \sigma_{jk}^{\rm aux}\epsilon_{jk}\delta_{1i})q_{j,i}dA+\int_{A}\sigma_{ij}[S_{ ijjId}^{\rm tip}-S_{ijkl}(\mathbf{x})]\sigma_{kl,1}^{\rm aux}qdA \tag{12}\]  
For heterogeneous materials, the elements within the integration region and the crack tip may be located in different materials, so there are different material parameters. The calculation of Equation (12) requires only the material properties at the crack tip and integration points, without involving the derivatives of material parameters. Thus, this form of interaction integral does not require differentiable material parameters, and compared to the traditional form, it broadens the applicable scope. To enhance its computational efficiency, Equation (12) is transformed into the Gaussian quadrature summation form of Equation (13).