# The Modified Void Nucleation and Growth Model (MNAG) for Damage Evolution in BCC [PERSON]
## 3 Results and Discussion
### Parameterization of the MNAG Model
#### 3.3.1 Single-Crystal Ta  
Table 1 shows a set of parameters used as initial values for the MNAG model. With these initial values, the above equations were integrated over time to derive the full evolution of void number density and damage, allowing the adjustable parameters to vary in order to fit the MD data. Here, we fixed the base void nucleation rate \(\dot{N}_{o}\) at \(1.5\times 10^{22}\) m\({}^{-3}\) ps\({}^{-1}\), void nucleation threshold \(\sigma_{n0}\) at \(\sigma_{spall}\times 0.7\), material viscosity \(\eta\) at 20 GPa ps, and critical void number density \(k_{V}\) at \(1.5\times 10^{24}\) m\({}^{-3}\). These parameters were derived based on a preliminary fitting of the model based on data from single-crystal Ta. In addition, the material viscosity chosen was close to the value reported in Ref. [64]. Such treatment reduced the number of fitting parameters and resolved the problem of multiple solutions during fitting. The fitting of the MNAG parameters was carried out using the Extended Kalman Filter (EKF; [65]), where, starting from the initial set of parameters in Table 1, the integration was performed with a time step to get the final damage and the errors were propagated backward to update the parameters until convergence was reached (as indicated by no appreciable change in the parameters). An integration time step of 0.2 ps was shown to be small enough to achieve convergence, as demonstrated in Figure S4 in the Supplementary Materials. Thus, there were seven adjustable parameters remaining in the MNAG model (highlighted in bold in Table 1). There were two adjustable parameters for the void nucleation term: void nucleation reference stress \(\sigma_{n}\) and void size scaling constant \(k_{VS}\); one adjustable parameter for the void growth term: void growth threshold stress \(\sigma_{g0}\); three adjustable parameters for the void coalescence term: critical void coalescence radius scaling constant \(k_{R}\), void coalescence reference radius \(R_{c}\), base void coalescence rate \(V_{c0}\); and one adjustable parameter for stress evolution: offset time for peak stress \(t_{offset}\). Among all these parameters:  
* \(\sigma_{n}\) most significantly affected the void number density and the resulting damage due to void nucleation: the lower the value of \(\sigma_{n}\), the higher the rate of void nucleation;
* \(\sigma_{g0}\) most significantly affected the growth of existing voids at tensile stress above \(\sigma_{g0}\): the lower the value of \(\sigma_{g0}\), the higher the rate of void growth;
* \(V_{c0}\) most significantly affected the slope of the quasi-linear growth during the void coalescence stage: the greater value of \(V_{c0}\) led to a higher rate of void coalescence. The fitted number density and damage in single-crystal Ta along the (001), (110), and (111) orientations are shown in Figure 3. Three stages of void evolution could be observed in these plots: SI, SII, and SIII. SI (35-42 ps): the void number density increased rapidly until it peaked; SII (42-45 ps): exponential growth of existing voids under high tensile stresses; SIII (from 45 ps on): multiple coalescence events due to a high number density of the voids (beyond the critical number density \(k_{V}\)), which experienced significant growth in SII. The contribution from void coalescence to the void volume increased smoothly until \(\sim\)48 ps. However, when the stress dropped below \(\sigma_{n0}\) and \(\sigma_{g0}\), the void nucleation and growth became negligible. As such, \(R_{t}\) plateaus, and thus \(\Delta V_{c}\) in Equation (10) was reduced to a quasi-linear contribution term, i.e., asymptotic growth of voids [38]. It was noted here that although we divided the void evolution into distinct stages, as discussed above, the void nucleation and growth took place in both SI and SII, and void coalescence took place in all stages, as could be observed from the right-hand panels in Figure 3. MD snapshots showing damage distribution at various stages were provided in Figure S3 in the Supplementary Materials.