# A Numerical Study of Blast Resistance of Carbon Fiber Reinforced Aluminum Alloy Laminates
## 3 Finite Element Modeling  
### Geometry, Boundary Conditions, and Contact Modeling  
All numerical simulations were carried out with the FE code ABAQUS/Explicit. Only a quarter of the panel was modeled to shorten the simulation time due to the double-symmetry nature of the problem. Corresponding constraints were defined on the two symmetric planes, while the other edges were fully clamped, as shown in Figure 3. [PERSON] et al. [16] compared the difference between the screw fixation constraint and boundary fixation constraint, and the results showed that the constraint mode only had an effect on the vibration characteristics of the structure, but had no effect on the maximum deformation. In this paper, the boundary node fixation constraint was used to replace the screw constraint.  
Aluminum alloy and carbon fiber were meshed using ABAQUS shell element C3D8. The element size was set to be 1 mm, as shown in Figure 4. The normal contact of shell elements was defined using hard contact formulation, while the tangential behavior was described with a penalty with a friction coefficient of 0.3.  
The delamination failure of the fiber-metal composite layer includes impact delamination between layers, peeling off of the fiber layer, and delamination between the fiber layer and aluminum alloy. Two ways of simulating delamination are provided in ABAQUS/Explicit: Cohesion finite elements and Cohesion Contact. As the thickness of the bonding layer between fiber layers was very small, its influence could be ignored [22]. In this paper, failure of Cohesion Contact with a thickness of 0 was selected to simulate the impact of delamination between fiber layers.  
Figure 3: Simplified clamping device.  
Figure 2: Definition of fiber layup direction.  
The viscous behavior contact provided in the Interaction module of ABAQUS uses linear elastic adherence-separation behavior, as shown in Figure 5. In the three-dimensional stress state, the interlayer contact stress \(t\) is defined as:  
\[t=\begin{cases}t_{n}\\ t_{s}\\ t_{t}\end{cases}\right\}=\begin{bmatrix}K_{nn}&K_{ns}&K_{nt}\\ K_{ns}&K_{ss}&K_{st}\\ K_{nt}&K_{st}&K_{tt}\end{bmatrix}\begin{cases}\delta_{n}\\ \delta_{s}\\ \delta_{t}\end{cases}=K\delta \tag{2}\]  
where \(t_{n}\) is normal stress in direction 3 as shown in Figure 6, \(t_{s}\) is shear stress in direction 1, and \(t_{t}\) is shear stress in direction 2. The corresponding separation displacements are \(\delta_{n}\), \(\delta_{n}\), and \(\delta_{k}\), and \(K\) is bond stiffness. When the normal stress or shear stress between layers reaches the corresponding strength limit (point A), the bonding contact begins to fail. The general maximum stress separation criterion can be written as follows:  
Figure 4: Model meshing.  
Figure 5: Linear elastic adhesion–separation viscous behavior of progressive damage failure.  
Figure 6: One-way fiberboard coordinate system.  
\[m\alpha x\left\{\frac{\langle t_{n}\rangle}{t_{n}^{o}},\frac{t_{s}}{t_{s}^{o}}, \frac{t_{t}}{t_{t}^{o}}\right\}=1 \tag{3}\]  
where \(t_{n}^{o}\), \(t_{s}^{o}\), \(t_{t}^{o}\) are the strength limits in the corresponding direction, and the \(<\) \(>\) symbol means absolute value. In the three-dimensional complex stress state, delamination occurs before the normal stress reaches the strength value due to the contribution of shear force. In this paper, the secondary stress separation criterion is adopted:  
\[\left\{\frac{\langle t_{n}\rangle}{t_{n}^{o}}\right\}^{2}+\left\{\frac{t_{s}}{ t_{s}^{o}}\right\}^{2}+\left\{\frac{t_{t}}{t_{t}^{o}}\right\}^{2}=1 \tag{4}\]  
According to the relevant literature [15; 21], in the 3D simulation model, linear stiffness degradation controlled by displacement is adopted. Linear elastic adherence-separation viscosity behavior of the secondary stress separation criterion is shown in Figure 7 and contact parameters are shown in Table 2.