## 2 Mechanics Model
### Material Damage Model for Intralaminar Failure  
In this section, the advanced CDM model, which can reflect the direction of matrix crack, is introduced. The constitutive model and damage variables are briefly summarized here, and for details for comprehensive understanding, the work in [15] can be referenced. The present damage model, in which the crack direction parameter is used as a major factor along with the damage variable, has a good ability to predict the propagation of the discrete crack well in the composites' failure simulation. Three types of matrix crack were mainly studied to predict the failure behavior of FRP laminates by CDM: transverse crack, longitudinal crack, and delamination [49; 50]. Many researchers focus on the transverse failure in the \(0^{\circ}\) layer and the longitudinal failure in the \(90^{\circ}\) layer, but the general cases such as angled matrix failure are less studied by the anisotropic damage model. In actual engineering applications, matrix cracks are properly angled with the mid-plane under a complex loading state [51; 52], but are almost parallel to the fiber in every layer, so the matrix cracks parallel to the fiber are only considered in this work. An anisotropic damage model that reflects various kinds of matrix cracks and their interaction is presented here. In damaged FRP materials, the complementary free energy density is given as follows. \[ \begin{array}{l}\psi=\frac{\sigma_{11}^{2}}{2E_{1}(1-d_{1})}+\frac{1}{E_{2}} \left(\frac{\sigma_{22}^{2}}{1-d_{2}}+\frac{\sigma_{33}^{2}}{1-d_{3}}\right)- \frac{v_{12}}{E_{1}}(\sigma_{22}+\sigma_{33})\sigma_{11}\\ -\frac{v_{22}}{E_{2}}\sigma_{22}\sigma_{33}+\frac{\sigma_{12}^{2}}{2G_{12}(1-d _{4})}+\frac{\sigma_{23}^{2}}{2G_{23}(1-d_{5})}+\frac{\sigma_{31}^{2}}{2G_{12} (1-d_{6})}\end{array}  \tag{1}\]  
where \(d_{i},i=1,2,3\) are the damage variables for tension or compression failure modes, while \(d_{i},i=4,5,6\) are the damage variables for shear failure modes. According to the second principle of thermodynamics for ensuring positive dissipation of mechanical energy:  
\[\boldsymbol{\varepsilon}=\frac{\partial\psi}{\partial\sigma}=\boldsymbol{S}: \boldsymbol{\sigma} \tag{2}\]  
where \(\boldsymbol{\varepsilon}\), \(\boldsymbol{\sigma}\), and \(\boldsymbol{S}\) are the strain tensor, stress tensor, and flexibility operator, respectively. \[\boldsymbol{S}=\frac{\partial^{2}\psi}{\partial\sigma\bigotimes\partial \sigma}=\left( \begin{array}{ccccc}\frac{1}{E_{1}(1-d_{1})}&-\frac{v_{12}}{E_{1 }}&-\frac{v_{12}}{E_{1}}&0&0&0\\ \frac{1}{E_{2}(1-d_{2})}&-\frac{v_{23}}{E_{2}}&0&0&0\\ &\frac{1}{E_{2}(1-d_{3})}&0&0&0\\ &&\frac{1}{G_{12}(1-d_{4})}&0&0\\ &symm.&&\frac{1}{G_{23}(1-d_{5})}&0\\ &&\frac{1}{G_{12}(1-d_{6})}\end{array} \right) \tag{3}\]  
Among six damage variables, two variables are independent, and their relationships are as follows:  
\[1-d_{4}=(1-d_{1})(1-d_{2}),1-d_{5}=(1-d_{2})(1-d_{3}),1-d_{6}=(1-d_{3})(1-d_{1}) \tag{4}\]  
Damage variable \(d_{1}\) is associated with fiber and matrix damage in the fiber direction, while \(d_{2}\) and \(d_{3}\) are associated with discrete matrix cracks for tension failure. Based on the bilinear softening law, the individual damage variables are determined as follows. \[d=\left\{ \begin{array}{ll}0&\delta_{\varepsilon}<\delta_{\varepsilon}^{*}\\ \frac{\delta_{\varepsilon}^{\prime}(\delta_{\varepsilon}-\delta_{\varepsilon} ^{*})}{\delta_{\varepsilon}(\delta_{\varepsilon}^{\prime}-\delta_{\varepsilon }^{*})}&\delta_{\varepsilon}^{*}\leq\delta_{\varepsilon}<\delta_{\varepsilon}^ {f}\\ 1&\delta_{\varepsilon}\geq\delta_{\varepsilon}^{f}\end{array} \right. \tag{5}\]  
where \(\delta_{\varepsilon}^{*}\) is the equivalent displacement at the damage initiation and \(\delta_{\varepsilon}^{f}\) is the equivalent displacement when the material reaches its ultimate strength. Three coordinates are introduced as shown in Figure 1: global Cartesian coordinates \((\overline{x},\overline{y},\overline{z})\), layer coordinates \((x,y,z)\), and local element coordinates \((x^{\prime},y^{\prime},z^{\prime})\).