## 3 Calculation Results and Discussion  
### Mechanical Behavior Analysis of Interface Pull-Up  
According to the model and method described above, the mechanical behavior of the pull-up of the graphene-copper interface at temperatures of 100 K, 300 K, 500 K, 700 K, 900 K, and 1100 K was simulated, respectively, and the relationship between the pull-up force and pull-up displacement of the interface was obtained, as shown in Figure 4.  
Figure 4: Pull-up force–displacement curves at different temperatures. Figure 3: Tensile stress–strain curves of pristine graphene. As can be seen from the figure, the pull-up force at different temperatures first increased and then decreased with the increase in pull-up displacement. The whole process can be divided into two stages: the pull-up force rapidly increases with the increase in the displacement of the upper copper atom from the graphene layer. In the descending stage, the pulling force decreases rapidly to a lower state and then slowly to 0. This is because when the pull-up displacement is small, the interatomic van der Waals forces have to be overcome for interfacial separation, resulting in a rapid increase in pull-up forces. When the pull-up displacement further increases, the van der Waals forces between copper and carbon atoms begin to decrease, leading to a decrease in the pull-up force until it reaches 0. In addition, the maximum pull-up force and maximum pull-up displacement are negatively correlated with temperature. The maximum pull-up force decreases less in the temperature range of 100 K-700 K, while the maximum pull-up force decreases more in the temperature range of 700-1100 K.  
The peak value of pulling force at different temperatures was selected to calculate the interface failure stress, as shown in Formula (4). \[\sigma=\frac{F}{S}=\frac{F_{\text{max}}}{wl} \tag{4}\]  
where \(\sigma\) is failure stress, Pa; the \(F_{\text{max}}\) is the maximum pulling force shown in Figure 4, N; \(w\) and \(l\) are the width and length of the contact surface between graphene and copper, respectively, m.  
According to Formula (4), the interface pull-up failure stress at different temperatures can be obtained. That is, the normal strength changes with temperature, as shown in Figure 5.  
As can be seen from Figure 5, the normal strength gradually decreases with the increase in temperature. The rate of decline is low in the temperature range of 0-700 K, and the rate and range of decline increase significantly in the temperature range of 700-1100 K. The normal strength decreases to the lowest at 1100 K, which is 1.87 GPa, only 73.7% of that at 100 K. The displacement of the maximum pulling force also decreases with the increase in temperature, and the maximum pulling force at 1100 K corresponds to the minimum displacement, indicating that the interface is easier to separate at high temperatures. In order to further study the effect of graphene content on the mechanical properties of the interface, the graphene content was controlled by adjusting the X-direction dimension to change the embedding depth. The mechanical properties of pull-up were studied at 300 K when the ratio of the graphene lamellar area to the cross-sectional area of the copper matrix in the same direction was 62.5%, 75%, 87.5%, and 100%, as shown in Figure 6a-d, corresponding to graphene content of 6.71 at%, 8.39 at%, 10.07 at%, and 11.75 at%, respectively. The pull-out force-displacement curves under different graphene contents were obtained, as shown in Figure 7.  
Figure 5: Maximum pull-out force–displacement/strength–temperature curves. As can be seen from Figure 7, the maximum pull-up force and maximum pull-up displacement both increased significantly with the increase in graphene content. Compared with the maximum pull-up force (25.28 eV/A)) at 6.71 at% content. The maximum pulling force of 8.39 at%, 10.07 at%, and 11.75 at% increased by 19.4%, 29.84%, and 61.82%, respectively, while the increase in carbon atoms was only 12.5%, 25%, and 37.5%.