## 3 Finite Element Modeling and Damage Criteria
### Damage Model for CARALL
#### 3.2.2 Damage Model for CFRP Layers  
The Linde criterion is a strain-based progressive damage model which suggests that material failure does not occur immediately after damage. Instead, the material still possesses some residual bearing capacity. This capacity is maintained through matrix failure, a non-fatal failure mode in which layers can withstand increasing loads even after the first layer is cracked. As the stress is redistributed, the damaged material is still able to withstand additional loads. The model takes into account different damage criteria for matrix and fiber failure [23]:  
\[f_{f}=\sqrt{\frac{\epsilon_{11}^{f,T}}{\epsilon_{11}^{f,C}(\epsilon_{11})^{2} +\left[\epsilon_{11}^{f,T}-\frac{\left(\epsilon_{11}^{f,T}\right)^{2}}{ \epsilon_{11}^{f,C}}\right]\epsilon_{11}>\epsilon_{11}^{f,T}}} \tag{7}\]  
\[f_{m}=\sqrt{\frac{\epsilon_{22}^{f,T}}{\epsilon_{22}^{f,C}(\epsilon_{22})^{2} +\left[\epsilon_{22}^{f,T}-\frac{\left(\epsilon_{22}^{f,T}\right)^{2}}{ \epsilon_{22}^{f,C}}\right]\epsilon_{22}+\left(\frac{\left(\epsilon_{22}^{f,T }\right)}{\epsilon_{12}^{f}}\right)^{2}(\epsilon_{12})^{2}}{>\epsilon_{22}^{f,T}}} \tag{8}\]  
Figure 4: Finite element model of three-point bending test.  
where \(f_{f},f_{m}\) represent the failure modes of the fiber and matrix, and \(\varepsilon\) represents the strain of each side, where the superscript \(T\) represents the tensile direction and \(C\) represents the compression direction.  
The Hashin criterion is a progressive damage model based on stress, similar to the Linde model. This model posits that the material still has some bearing capacity even after damage occurs, and considers the shear effect on fiber tensile failure. The Hashin criterion predicts four different failure modes: fiber tensile failure, fiber compression failure, matrix tensile failure, and collective compression failure. Therefore, it is more accurate to assess the degree of damage:  
\[F_{ft}=\frac{\sigma_{11}^{2}}{X_{T}^{2}}+\alpha(\frac{\sigma_{12}^{2}+\sigma_{ 13}^{2}}{S_{12}^{2}})^{2}=1(\sigma_{11}\geq 0) \tag{9}\]  
\[F_{fc}=(\frac{\sigma_{11}}{X_{C}})^{2}=1,(\sigma_{11}<0) \tag{10}\]  
\[F_{mt}=(\frac{\sigma_{22}+\sigma_{33}}{Y_{T}^{2}})^{2}+\frac{\sigma_{12}^{2}+ \sigma_{13}^{2}}{S_{12}^{2}}+\frac{\sigma_{23}^{2}+\sigma_{22}\sigma_{33}}{S_ {23}^{2}}=1,(\sigma_{22}+\sigma_{33}\geq 0) \tag{11}\]  
\[F_{mc}=(\frac{\sigma_{22}+\sigma_{33}}{2S_{23}})^{2}+\left[\left(\frac{Y_{C}} {2S_{23}}\right)^{2}-1\right]\frac{\sigma_{22}+\sigma_{33}}{Y_{C}}+\frac{ \sigma_{12}^{2}+\sigma_{13}^{2}}{S_{12}^{2}}+\frac{\sigma_{23}^{2}+\sigma_{22 }\sigma_{33}}{S_{23}^{2}}=1,(\sigma_{22}+\sigma_{33}<0) \tag{12}\]  
Similar to the Linde criterion, \(F_{ft},F_{fc},F_{mt},F_{mc}\) represent fiber tensile failure, fiber compression failure, matrix tensile failure, and matrix compression failure. \(\sigma\) represents the upward stress of each side, where the superscript \(T\) represents the tensile direction and \(C\) represents the compression direction. \(Y\) represents the strength in the direction of the fiber, \(X\) represents the strength in the direction of the matrix, and \(S\) represents the shear strength of each side.  
Two different progressive damage models including the Linde criterion and Hashin criterion are compared in the current work. For the Linde criterion, the failure modes are classified into fiber failure and matrix failure. On the other hand, the Hashin criterion includes four failure modes including fiber tensile failure, fiber compression failure, matrix tensile failure, and matrix compression failure. However, the Hashin criterion does not always match the experimental results in the case of damage assessment, particularly for matrix compression failure. When moderate transverse compression impedes the interlaminar shear fracture of the specimen, the outcome using the Linde criterion might be different.