## 2 Development of Progressive Failure Analysis
### Damage Initiation and Evolution  
During PFA, failure initiation should be detected to apply the damage evolution model. In this study, damage initiation was evaluated using the Hashin failure criterion [22; 23]. Using this criterion, failures at the \"fiber tension/compression\" and \"matrix tension/compression\" modes can be evaluated. The following equations are the Hashin's criteria at each failure mode:  
\[\text{Fiber tension failure }:f_{ft}=\left(\frac{\widetilde{\sigma}_{11}}{X_{t}} \right)^{2}+\left(\frac{\widetilde{\sigma}_{12}}{S}\right)^{2}+\left(\frac{ \widetilde{\sigma}_{13}}{R}\right)^{2}\geq 1\]  
<Fiber compression failure mode>  
\[\delta_{eq}^{fc}=L_{c}\langle-\varepsilon_{11}\rangle\]  
\[\sigma_{eq}^{fc}=\ \frac{L_{c}\langle-\sigma_{11}\rangle\langle-\varepsilon_{11} \rangle}{\delta_{eq}^{fc}}\]  
<Matrix tension failure mode>  
\[\delta_{eq}^{mt}=L_{c}\sqrt{\langle\varepsilon_{22}\rangle^{2}+\langle \varepsilon_{33}\rangle^{2}+\gamma_{12}^{2}+\gamma_{23}^{2}+\gamma_{13}^{2}}\]  
\[\sigma_{eq}^{mt}=\ \frac{L_{c}\langle(\langle\sigma_{22}\rangle\langle \varepsilon_{22}\rangle+\langle\sigma_{33}\rangle\langle\varepsilon_{33} \rangle+\tau_{12}\gamma_{12}+\tau_{23}\gamma_{23}+\tau_{13}\gamma_{13})}{ \delta_{eq}^{mt}}\]  
<Matrix compression failure mode>  
\[\delta_{eq}^{mc}=L_{c}\sqrt{\langle-\varepsilon_{22}\rangle^{2}+\langle- \varepsilon_{33}\rangle^{2}+\gamma_{12}^{2}+\gamma_{23}^{2}+\gamma_{13}^{2}}\]  
\[\sigma_{eq}^{mc}=\ \frac{L_{c}(\langle-\sigma_{22}\rangle\langle-\varepsilon_{22 }\rangle+\langle-\sigma_{33}\rangle\langle-\varepsilon_{33}\rangle+\tau_{12} \gamma_{12}+\tau_{23}\gamma_{23}+\tau_{13}\gamma_{13})}{\delta_{eq}^{mc}} \tag{4}\]  
where \(L_{c}\) is the characteristic length and \"\(\langle\rangle\)\" refers to the Macaulay brackets. The definition of the Macaulay brackets is as follows:  
\[\langle x\rangle=\left\{ \begin{array}{ll}0,&x\leq 0\\ x,&x\geq 0.\end{array} \right. \tag{5}\]  
To calculate the damage variables at each failure mode, the \"equivalent displacement and equivalent stress at the initial failure state\" and the \"equivalent displacement at the final failure state\" should be obtained. Using the failure criterion parameter, \(f_{I}\), the equivalent displacement and stress at the initial failure state can be obtained as follows:  
<Equivalent displacement at the initial failure state>  
\[\delta_{I,eq}^{0}=\frac{1}{\sqrt{f_{I}}}\delta_{I,eq} \tag{6}\]  
Figure 2: Stress-displacement relationship for progressive failure analysis. \(<\)Equivalent stress at the initial failure state\(>\)  
\[\sigma_{l,\alpha q}^{0}=\frac{1}{\sqrt{f_{l}}}\sigma_{l,\alpha q} \tag{7}\]  
The superscript \"0\" implies the failure initial state. The equivalent displacement at the final failure state can be calculated using the crack band model [6]. \(<\)Crack band model\(>\)  
\[\varepsilon^{f}=\ \frac{2G_{c}}{\sigma_{0}L_{c}} \tag{8}\]  
\(<\)Equivalent displacement at the final failure state\(>\)  
\[\delta_{l,\alpha q}^{f}=\frac{2G_{l}^{c}}{\sigma_{l,\alpha q}^{0}} \tag{9}\]  
Using the equivalent stress and equivalent displacement, a damage variable at each failure mode can be calculated based on the following equation:  
\[d_{l}=\frac{\delta_{l,\alpha q}^{f}\bigg{(}\delta_{l,\alpha q}-\delta_{l, \alpha q}^{0}\bigg{)}}{\delta_{l,\alpha q}\bigg{(}\delta_{l,\alpha q}^{f}- \delta_{l,\alpha q}^{0}\bigg{)}}\bigg{(}\delta_{l,\alpha q}^{0}\leq\delta_{l, \alpha q}\leq\delta_{l,\alpha q}^{f}\bigg{)} \tag{10}\]  
\[\left\{ \begin{array}{l}d=0\quad:\text{\it undamaged\ status}\\ 0<d<1\ :\text{\it damage\ propagating}\\ d=1\quad:\text{\it complete\ damage\ status}\end{array} \right\}\]  
As the damage initiates and propagates in the material, the variable gets incremented from 0 to 1. Damage variables are applied to the stiffness matrix based on each failure mode while considering the effective stress condition. \[d_{f}=\left\{ \begin{array}{l}d_{ft}\quad\text{\it if}\ \ \overline{\sigma}_{11}\geq 0 \\ \dot{d}_{fc}\quad\text{\it if}\ \ \overline{\sigma}_{11}<0\end{array}