## 4 Conclusions  
The energy release rate associated with stress intensity factors under mixed mode loading for aluminum alloy material has been investigated using both a numerical method and theoretical derivation. The present results demonstrate the simpler, accurate calculation process and accurate evaluation of the energy release rate. The following conclusions can be drawn:  
1. A relatively simple and precise numerical method was established to evaluate the energy release rate associated with the stress intensity factors under mixed mode loading based on the concept that the energy release rate is equal to the change rate of the energy difference before and after crack kinking.
2. Based on the numerical method, a series of spatial inclined ellipses in Mode I-II and ellipsoids in Mode I-II-III with different propagation angles computed from non-dimensional value (\(K/\sqrt{EG}\)) were fitted by MATLAB, and the expression of the energy release rate with the crack propagation angle was obtained.
3. A theoretical expression of energy release rate at any propagation angle for a crack tip under I-II-III mixed mode crack was deduced based on the propagation mechanism of the crack tip under the influence of a stress field. It is confirmed that the theoretical expression deduced could provide results as accurately as the present numerical method.
4. The present results are consistent with the experimental data. The error, which is lower than 5%, can be accepted considering that the specimens are not manufactured using ideal elastic material. Consequently, the present results demonstrate the simpler, accurate calculation process and the accurate evaluation of energy release rate.  
Figure 19: Distribution of Stress Intensity Factor on the crack tip line under mode III loading.  
**Author Contributions**: Conceptualization, S.J.C. and Y.Y.; methodology, S.J.C. and Y.Y.; software, W.s.H.; validation, S.J.C. and. W.s.H.; formal analysis, S.J.C. and Y.Y.; investigation, H.C.; resources, H.C.; data curation, Y.Y. and W.s.H.; writing--original draft preparation, Y.Y. and W.s.H.; writing--review and editing, S.J.C. and Y.Y.; visualization, S.J.C. and Y.Y.; supervision, S.J.C.; project administration, S.J.C.; funding acquisition, S.J.C. and H.C. All authors have read and agreed to the published version of the manuscript.  
**Funding**: This work was supported by the Natural Science Foundation of Shanghai (18 ZR1416500); Development Fund for Shanghai Talents (2018053); and \"Shuguang Program\" supported by the Shanghai Education Development Foundation and Shanghai Municipal Education Commission (19 SG51).  
**Conflicts of Interest**: The Authors declare no conflicts of interest.  
**Nomenclature**  
\(G\) Energy release rate  
\(K\) and \(K_{l},K_{ll},K_{lll}\) Stress intensity factor, and for Mode I, Mode II, Mode III.  
\(E\) Elastic modulus  
\(v\) Poisson's ratio  
\(\mu\) Shear modulus  
\(2L\) Length of rectangular plate  
\(W\) Wide of rectangular plate  
\(D\) Thickness of rectangular plate  
\(A\) Length of initial crack  
\(\theta\) Kinking angel of crack  
\(\sigma\) Normal stress  
\(\tau\) Shear stress  
\(\Delta a\) Length of kink crack  
\(U_{1},\,U_{2}\) Strain energy before kink extension and after kink extension  
\(J\) Integral for a kink  
\(a,b,\,\alpha\) Semi-major axis, semi-minor axis and inclination of ellipse  
\(\beta\) Oblique angle of crack  
\(\sigma_{m},\sigma_{\theta\theta},\tau_{r\theta}\) Stress fields in polar coordinate  
\(k^{\prime}(\alpha)\) Local stress intensity factors