## 2 Fracture Parameter Solution Method Based on the Postprocessing Routine of ABAQUS
### Construction of Auxiliary Fields  
The interaction integral is the interaction part of the J-integral under actual and auxiliary fields. Choosing a proper auxiliary field can improve the efficiency of the derivation of the interaction integral. This section elaborates on the mathematical definition of the auxiliary field and develops a relevant calculation program. Table 1 lists the meanings of symbols in the formulas of the article. Figure 4: Node and integration point locations in a two-dimensional isoparametric linear element. Figure 3: Flowchart of the calculation of interaction integral using the ABAQUS–Python postprocessing routine. For isotropic heterogeneous materials, the auxiliary field is typically defined in three ways: (1) equilibrium equation residuals; (2) compatibility equation residuals; (3) constant constitutive tensor equation. In this study, the second form is chosen, where the low-order terms of Williams' crack-tip field for an infinite plate are used as the auxiliary field [25]. The auxiliary strain is derived from the actual material properties and the auxiliary stress field. The specific form is as follows:  
\[u_{i}^{\text{aux}}\ =\ K_{\text{I}}^{\text{aux}}\sqrt{\frac{r}{2\pi}}f_{i}^{ \text{I}}(\theta,\mathbf{S}^{\text{tip}})+K_{\text{II}}^{\text{aux}}\sqrt{\frac{r} {2\pi}}f_{i}^{\text{II}}(\theta,\mathbf{S}^{\text{tip}}), \tag{3}\]  
\[\sigma_{ij}^{\text{aux}}\ =\ \frac{K_{\text{I}}^{\text{aux}}}{\sqrt{2\pi r}}g_{ij}^{ \text{I}}(\theta,\mathbf{S}^{tip})+\frac{K_{\text{II}}^{\text{aux}}}{\sqrt{2\pi r} }g_{ij}^{\text{II}}(\theta,\mathbf{S}^{tip}), \tag{4}\]  
\[\varepsilon_{ij}^{\text{aux}}\ =\ S_{ijkl}(\mathbf{x})\sigma_{kl}^{\text{aux}},\ \ \ (i,j,k,l\ =\ 1,2), \tag{5}\]  
In these equations, \(r\) and \(\theta\) are polar coordinates. \(S_{\text{tip}}\) and \(S_{ijkl}\) represent the compliance tensor and its components, respectively. For isotropic materials, the functions \(f_{i}^{\text{I}}\),\(f_{i}^{\text{II}}\), \(g_{i}^{\text{I}}\) and \(g_{i}^{\text{II}}\) are as follows:  
\[ \begin{array}{lcl}f_{1}^{\text{I}}&=&\frac{1}{2\mu_{\text{tip}}}\cos\frac{ \theta}{2}\Big{(}\kappa_{\text{tip}}-1+2\sin^{2}\frac{\theta}{2}\Big{)},&f_{1} ^{\text{II}}\ =\ \frac{1}{2\mu_{\text{tip}}}\sin\frac{\theta}{2}\Big{(}\kappa_{\text{tip}}+1+2 \cos^{2}\frac{\theta}{2}\Big{)},\\ f_{2}^{\text{I}}&=&\frac{1}{2\mu_{\text{tip}}}\sin\frac{\theta}{2}\Big{(} \kappa_{\text{tip}}+1-2\cos^{2}\frac{\theta}{2}\Big{)},&f_{2}^{\text{II}}\ =\ -\frac{1}{2\mu_{\text{tip}}}\cos\frac{ \theta}{2}\Big{(}\kappa_{\text{tip}}-1-2\sin^{2}\frac{\theta}{2}\Big{)}^{ \prime},\\ &&\\ g_{11}^{\text{I}}&=&\cos\frac{\theta}{2}\Big{(}1-\sin\frac{\theta}{2}\sin\frac{ \theta}{2}\Big{)},&g_{11}^{\text{II}}\ =\ -\sin\frac{\theta}{2}\Big{(}2+\cos\frac{ \theta}{2}\cos\frac{\theta}{2}\Big{)}\\ g_{22}^{\text{I}}&=&\cos\frac{\theta}{2}\Big{(}1+\sin\frac{\theta}{2}\sin\frac{ \theta}{2}\Big{)},&g_{22}^{\text{II}}\ =\ \sin\frac{\theta}{2}\cos\frac{ \theta}{2}\cos\frac{\theta}{2}\\ g_{12}^{\text{I}}&=&g_{21}^{\text{II}}\ =\ \frac{g_{22}^{\text{II}}}{g_{12}^{ \text{II}}}\ =\ \frac{g_{21}^{\text{II}}}{g_{11}^{\text{II}}}\ =\ \frac{g_{21}^{\text{II}}}{g_{21}^{ \text{II}}}\ =\ \frac{g_{21}^{\text{II}}}{g_{21}^{\text{II}}}\ \ \,\\ \end{array}  \tag{6}\]