Fig. 5 Shear stress and dislocation density of the model under monotonic loads and the cyclic load settings of the model. (a) Motion diagram of the boundary layers and the calculation formula of shear strain. (b) Shear Stress-d curves under monotonic loads. Linear shear stress tensor-yz and linear shear stress tensor-xz represent the results of loading along the y -axis and the x -axis, respectively. Open symbols indicate the designed cyclic load levels. Both curves are respectively divided into three stages: elastic, elastic-plastic and plastic. (c) Dislocation density-d bars under monotonic loads. (d) d-time curves for cyclic deformation with different period and load amplitude. Periods are 43.2ps (d1), 50.4ps (d2) and 68.4ps (d3).
Statistical results of dislocation density within the model are shown in Fig. 5c. Dislocation density formula is written as (unit:\(m^{-2}\)):
\begin{equation} \rho=\frac{L}{V}\nonumber \\ \end{equation}
Where ρ is the total length of dislocation lines in the crystal per unit volume, L is the total dislocation length, and V is the volume of crystal. Fig. 5c presents that only a few dislocations exist in the model when d is less than 10Å. Then the dislocation densities both increase rapidly when the model is transformed into the elastic-plastic stage and remain high in the plastic stage. These two types of dislocation density are distinguished by the loading directions and affected by the lattice orientation.36
To explore the elastic-plastic transition characteristics and plastic accumulation principle of the simulation model, we selected three shear stress states for the following cyclic simulations as shown in Fig. 5b. The three different displacement loads named d1, d2 and d3 were applied values of 7.2 Å, 8.4 Å and 11.4 Å, respectively (shown in Fig. 5d). According to the formula of γ(shown in Fig. 5a), the shear strains corresponding to d1, d2 and d3 are 0.097rad, 0.113 rad and 0.153 rad, respectively. All the three load curves shown in Fig. 5d are sinusoidal curves, and the initial velocities of the cyclic loads are similar with the velocity of the monotonic load above. As shown in Fig. 5b and d, the model is in the elastic stage when the load is applied along the y -axis and thex -axis under d1. When the load is d2, the model is in the elastic stage under load along the y -axis and is in the elastic-plastic stage under load along the x -axis. When the load goes up to d3, the model is in the plastic stage along both loading directions. Actually, the variation of the alternating shear stress on the subsurface during RCF is similar to the impulse waveform.25 The alternating shear stress in a certain region is close to zero in the most time of a cycle. However, we ignored the period when the model was at a low alternating shear stress level to reduce the simulation cost. The cyclic shear movement of the boundary layers was set to be continuous. Fig. 5d shows only one load cycle, and we conducted 10 cycles for all three load amplitudes to explore the plastic cumulative damage mechanism of the model. The OVITO software package37 was used to visualize the different defects in the system. Dislocations were analyzed by using the dislocation extraction algorithm (DXA) tool.38 Common neighbor analysis (CNA)39,40 was used to classify atoms in the crystalline structure.