# A Numerical Study of Blast Resistance of Carbon Fiber Reinforced Aluminum Alloy Laminates
## 3 Finite Element Modeling
### Damage Model of Composite Material  
In engineering application, the impact of CARALL is a transient three-dimensional stress problem, so it is necessary to establish a three-dimensional model to analyze it. When the 3D damage model is adopted, the initial damage is usually taken as the judgment basis, that is, when the element in the calculation meets the initial damage condition, the element is judged as the failing element and deleted. According to the interpretation of the judgement method in the ABAQUS user manual, although this treatment can improve the calculation speed, the failure process of the fiber material during the damage is ignored. The test results show that, when the fiber composite material is damaged, it still has a certain bearing capacity, and adopting the material stiffness degradation technology can more accurately reflect the objective damage process of the material. In this paper, the impact failure criterion of unidirectional fiber adopts the three-dimensional Hashin failure criterion improved by [PERSON] and [PERSON] et al. (2017), taking into account the shear failure when the center sag of the curved surface structure is deformed, and the impact damage criterion includes five failure modes (the unidirectional fiber coordinate system is shown in Figure 6): fiber tension and compression failure (direction 1), matrix tension and compression failure (directions 2 and 3), matrix shear failure (directions 1-2), out-of-plane shear failure (directions 1-3 and 2-3), and lamination failure. Tensile failure of fiber in direction 1: It was found through the test that the tensile failure in the fiber layout direction is mainly a tensile fracture:  
\[{f_{1}}^{2}=(\frac{\sigma_{11}}{X_{T}})^{2}+\alpha[(\frac{\tau_{12}}{S_{12}}) ^{2}+(\frac{\tau_{31}}{S_{31}})^{2}](\sigma_{11}>0) \tag{5}\]  
where 1, 2, and 3 represent the distribution direction (1), in-plane vertical fiber direction (2), and out-of-plane vertical fiber direction (3) of the carbon fiber; \(X_{T}\), \(S_{12}\), and \(S_{31}\) are the tensile strength in direction 1, the shear strength in plane 1-2, and the shear strength in plane 1-3, respectively; \({f_{1}}^{2}\) is the damage parameter. Fiber compression failure in direction 1: Compression failure in the fiber layout direction is mainly the compression fracture of the carbon fiber:  
\[{f_{2}}^{2}=(\frac{\sigma_{11}}{X_{C}})^{2}(\sigma_{11}<0) \tag{6}\]  
The matrix tensile failure in directions 2 and 3 is in-plane and out-of-plane in the vertical fiber direction, and the tensile failure of fiber in the one-way plate is mainly a matrix tensile failure, that is, the material failure is judged when the tensile stress in directions 2 and 3 reaches the tensile strength of the matrix. Therefore, the matrix tensile \begin{table}
\begin{tabular}{l c} \hline \hline  & **Value** \\ \hline \(\rho\) (kg/m\({}^{3}\)) & 1750 \\ E11 (GPa) & 134 \\ E22 = E33 (GPa) & 5.2 \\ \(v12=v31\) & 0.25 \\ \(v23\) & 0.38 \\ G12 = G13 = G23 (GPa) & 3.1 \\ Xt (MPa) & 2160 \\ Xc (MPa) & 1470 \\ Yt = Zt (MPa) & 71 \\ Yc = Zc (MPa) & 1030 \\ S12 (MPa) & 450 \\ S31 = S32 (MPa) & 325 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: T700s mechanical properties of unidirectional carbon fiber plate [21]. failure criterion of the Hashin criterion can be used in accordance with the tensile failure criterion assumed in directions 2 and 3:  
\[f_{3}^{2}=\left(\frac{\sigma_{22}+\sigma_{33}}{Y_{T}}\right)^{2}+\frac{\tau_{12} ^{2}+\tau_{13}^{2}}{S_{12}^{2}}+\frac{\tau_{23}^{2}-\sigma_{22}\sigma_{33}}{S_{2 3}^{2}}(\sigma_{22}+\sigma_{33}>0) \tag{7}\]  
After the matrix fails in the in-plane and out-of-plane vertical fiber directions, the fiber and the fractured matrix are subjected to continuous compression loads until the unidirectional carbon fiber cracks and fails:  
\[{f_{4}}^{2}=(\frac{\sigma_{22}+\sigma_{33}}{Y_{C}})^{2}(\sigma_{22}+\sigma_{33 }<0) \tag{8}\]  
where \(Y_{C}\) is the compression strength in direction 2, \(Y_{C}\) = \(Z_{C}\), and \(f_{4}\) is the damage parameter.