# The Modified Void Nucleation and Growth Model (MNAG) for Damage Evolution in BCC [PERSON]
## 4 Conclusions  
The performance of a few common damage models was compared to the void number density and damage calculated from MD simulations using shock-loaded single-crystal and nanocrystalline Ta as model systems. Damage models including, the NAG and Meyer-JMAK treatment, performed well in reproducing the void nucleation and growth stages of damage evolution. However, due to the lack of explicit terms related to void coalescence, these models failed to accurately reproduce the long-time evolution of the void number density and damage. The Tepla and Tonks models were able to reproduce either the void nucleation and growth stag or the void coalescence stage alone but not all stages, without the addition of other physical insights. In light of this, a modified void nucleation and growth (MNAG) model was proposed, with the addition of a new term to represent void coalescence. The form of the MNAG model was able to represent all stages of the void evolution compared to the models shown above. It was important to note that the MNAG parameters listed in this paper were specific to the simulation data used to fit them. If the loading conditions or the materials were changed, it would alter the void nucleation stress and, eventually, the total damage evolution. In that case, the MNAG model would need to be reparametrized for those specific cases. In general, when the damage models were used in hydrocode simulations, they were specifically parameterized to represent a narrow set of data. There was no expectation that the same parameter sets could be used interchangeably, irrespective of loading conditions or materials. While there was much effort to develop more physics-based predictive damage models, current analytical models that were parameterized based on experimental and modeling data continue to be used widely to describe the damage in materials.  
**Supplementary Materials:** The following are available online at [[https://www.mdpi.com/article/10.3390/app11083378/s1](https://www.mdpi.com/article/10.3390/app11083378/s1)]([https://www.mdpi.com/article/10.3390/app11083378/s1](https://www.mdpi.com/article/10.3390/app11083378/s1)): Figure S1--Damage (void volume fraction) evolution based on Cocks-Ashby model with different choices of \(n\), as compared to MD results for [001] single-crystal Ta;  
Figure 7: The MNAG model fitted to (001) single-crystal Ta with cell dimensions of: (**a**) 63 \(\times\) 63 \(\times\) 225 nm, (**b**) 95 \(\times\) 95 \(\times\) 338 nm. The left-hand panels show the evolution of the void number density and damage based on the MD and MNAG model, and the right-hand panels show the evolution of stress (\(\sigma_{m}\)) and the contribution from void nucleation, growth and coalescence term over time based on MNAG model.  
Figure S2--Damage (void volume fraction) evolution based on Meyer-JMAK treatment, with different choices of \(N_{o}\), as compared to MD results for [001] single-crystal Ta.; Figure S3--Snapshots of [001] single crystal Ta at different times.; Figure S4--Evolution of number density and volume fraction of voids for [001] single-crystal Ta.  
[PERSON], [PERSON] and [PERSON] conceptualized the idea, [PERSON] performed the simulations, analysis and wrote the paper. [PERSON] and [PERSON] secured funding, reviewed the work and helped edit the manuscript. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the US Department of Energy through the Los Alamos National Laboratory.  
Not applicable.  
This data can be obtained by emailing the corresponding author.  
The MD simulations utilized resources provided by the LANL Institutional Computing Program. This work was supported by the US Department of Energy through the Los Alamos National Laboratory. The Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218 CNA000001).  
The authors declare no conflict of interest.