# Variation of Fracture Toughness with Biaxial Load and T-Stress under Mode I Condition
## 2 Theoretical Model and Verification Method
### The Calculation Method of Critical Distance  
Comparing Equations (1) and (2) with Equations (3) and (4), \(K_{\rm I}\) is equal to \(A_{1}\sqrt{2\pi}\) and is the stress intensity factor. It can be concluded from Equations (3) and (4) that under this assumption, the stress intensity factor \(K_{\rm I}\) is the only parameter that affects the fracture behavior. However, if considering the stress-strain field at a finite critical distance r_c from the crack tip, the higher term of Williams series should not be ignored. For the mode I loading method, the stress and strain field in front of the crack, that is, \(\theta=0\), need to be examined. Assuming that two nonvanishing terms in Equations (1) and (2) are sufficient to characterize the stress-strain at the critical distance, one arrives at:  
\[\sigma_{yy}=\frac{K_{\rm I}}{\sqrt{2\pi r}}+3A_{3}\sqrt{r} \tag{5}\]  
\[\varepsilon_{yy}=\frac{1-\mu^{*}}{E^{*}}\frac{K_{\rm I}}{\sqrt{2\pi r}}-\frac {\mu^{*}}{E^{*}}T \tag{6}\]  
where \(E^{*}=E\) and \(\mu^{*}=\mu\) for plane stress and \(E^{*}=\frac{E}{1-\mu^{2}}\) and \(\mu^{*}=\frac{\mu}{\Gamma-\mu}\) for plane strain conditions, \(T=4A_{2}=\sigma_{xx}\) (\(A_{2}\) is \(A_{n}\) when \(n=2\)) and is the so-called T-stress or the stress parallel to the crack. For an infinite plate, the T-stress is applied to the infinitely far end of the specimen parallel to the direction of the crack. However, in a specimen of a certain shape, the T-stress may come from the geometric effect of the specimen itself or the load parallel to the direction of the crack. Equation (5) shows that the opening stress is controlled by two mechanics parameters \(K_{\rm I}\), and \(A_{3}\). \(K_{\rm I}\) can be interpreted as the applied loading level, and \(A_{3}\) as a function of specimen geometry and loading configuration. Accordingly, Equation (6) can be interpreted as two parameters, \(K_{\rm I}\), and \(T\), that control the opening strain. Note that both \(A_{3}\) and T are proportional to \(K_{\rm I}\) and the applied load for a given geometry due to the linearity of the problem. According to the above analysis, consider that the fracture event is controlled by the open stress at a certain critical distance \(r_{c}\) from the crack tip, and one can obtain:  
\[\sigma_{c}=\frac{K_{\rm c}}{\sqrt{2\pi r}}+3A_{\rm c}\sqrt{r_{\rm c}} \tag{7}\]  
\[\varepsilon_{c}=\frac{1-\mu^{*}}{E^{*}}\frac{K_{\rm c}}{\sqrt{2\pi r_{\rm c}} }-\frac{\mu^{*}}{E^{*}}T_{\rm c} \tag{8}\]  
where \(K_{\rm c}\), \(T_{\rm c}\) and \(A_{\rm c}\) are the critical \(K_{\rm I}\), \(T\) and \(A_{3}\), \(\sigma_{c}\) and \(\varepsilon_{c}\) are critical open stress and critical open strain. Assuming that the critical open strain \(\sigma_{c}\) is an intrinsic parameter of the material, it can be obtained from Equation (7) that \(K_{\rm c}\) is linearly related to \(A_{\rm c}\). For a specimen of a certain shape, the magnitude of the \(A_{3}\) and T-stress depends only on the load parallel to the crack. In the biaxially loaded cross-shaped specimen fracture test, the magnitude of \(A_{3}\) and the T-stress can be effectively changed by changing the horizontal load, which can be used to verify the reliability of the above theoretical analysis.