# The Modified Void Nucleation and Growth Model (MNAG) for Damage Evolution in BCC [PERSON]
## 3 Results and Discussion
### The MNAG Model: Comparison to Other Models  
To evaluate the robustness of the MNAG model, its performance was compared to a few commonly used damage models:  
1. The original NAG model [31; 32].
2. The Cocks-Ashby model based on the power-law creep [30; 68].
3. The Meyer-JMAK treatment: the void growth model based on the Curran-Seaman-Shockey (CSS) theory [31; 32] and the Johnson-Mehl-Avrami-Kolmogorov (JMAK) equations [8; 9; 8; 70; 71; 72; 73] proposed by [PERSON] et al. [8].
4. The Tepla model based on the [PERSON] model [35; 41].
5. The Tonks model based on the pore growth solution proposed by [PERSON] [34; 74].  
The parameters for all of these models (that are described in detail in the Supplementary Materials) were obtained by fitting them to the same MD data for void nucleation, growth, and coalescence generated from the shock-loaded (001) single-crystal Ta.  
Figure 6 shows the fitted results of the original NAG model compared to the current MD results. In general, the NAG model behaved similarly to the MNAG model during SI and SII in terms of both void number density and damage. However, it failed to reproduce the quasi-linear growth of voids at the later stage (SIII in Figure 6a), as void coalescence was not accounted for in the original model. Based on the original NAG model, when \(\sigma_{s}\) dropped below both \(\sigma_{n0}\) and \(\sigma_{g0}\), void growth ceased, and the damage showed a saturation instead of a quasi-linear growth in SIII. Additionally, the original NAG model incorrectly showed a continuous increase in the void number density in SII, which reached a steady state in SIII; yet the void number density should actually decrease as voids continue to grow and coalesce in SII and SIII. The Cocks-Ashby model in Figure 6b demonstrated a gradual increase in damage accumulation until it reached \(\sim\)0.1 and then increased exponentially. This was different from the MD results that showed a steady increase in the damage following an initial exponential growth. The Meyer-JMAK treatment in Figure 6c described SI and SII quite well. Specifically, the damage evolution matched well with the MD results  
Figure 5: The MNAG model fitted to nanocrystalline Ta with different grain sizes: (**a**) 20 nm, (**b**) 10 nm, (**c**) 5 nm. The left-hand panels show the void number density and damage based on MD and MNAG model, and the right-hand panels show the contribution from void nucleation, growth, and coalescence term over time based on the MNAG model.  
at \(\dot{N_{o}}=7\times 10^{-7}\) m\({}^{-3}\) s\({}^{-1}\) until -45 ps, after which void coalescence became significant resulting in a large discrepancy between the model and the MD results. Such behavior was similar to the original NAG model as they both were based on the Curran-Seaman-Shockey (CSS) theory.  
Figure 6d shows the results for the Tepla and the Tonks models. These two models were shown to be identical for certain parameter constraints under purely hydrostatic loading and could be fitted to reproduce either the early-stage behavior (SI and SII in case 1) or the later stage (SIII in case 2) behavior, but not all stages simultaneously. The above comparison clearly demonstrated that the MNAG model performed well in terms of reproducing the damage evolution in all stages, unlike the other damage models that excelled in accurately representing damage only at early or late times. More detailed information on the above models can be found in the Supplementary Materials.