# The Modified Void Nucleation and Growth Model (MNAG) for Damage Evolution in BCC [PERSON]
## 3 Results and Discussion  
### The NAG Model  
The NAG model, based on the Curran-Seaman-Shockey (CSS) theory [31; 32], attributed the evolution of the total void volume in a system to void nucleation and growth [31; 32]. Without loss of generality, the initial volume was assumed to be unity. At a given time interval \(\Delta t\), the number density of voids due to the nucleation process, \(\Delta N_{Vn}\), is expressed as:  
\[\Delta N_{Vn}=8\pi\Delta t\dot{N}_{o}exp(\frac{\max(\sigma_{m}-\sigma_{n0}, \ 0)}{\sigma_{n}}),\ \sigma_{m}>\sigma_{n0} \tag{1}\]  
\[\Delta N_{Vn}=0,\ \sigma_{m}<\sigma_{n0} \tag{2}\]  
The void volume, \(\Delta V_{n}\), due to the void number density, \(\Delta N_{Vn}\), is thus:  
\[\Delta V_{n}=\Delta N_{Vn}*\left(k_{Vs}*R_{0}\right)^{3} \tag{3}\]  
where \(\dot{N}_{o}\) is the base void nucleation rate, \(\sigma_{m}\) is the tensile stress, \(\sigma_{n0}\) is the threshold stress beyond which voids start to nucleate, \(\sigma_{n}\) is the reference stress for void nucleation, \(R_{0}\) is the void nucleation radius, and \(k_{Vs}\) is a unitless scaling constant that we added here to account for the initial size of voids. It is important to note that this parameter was not present in the original NAG model. The void nucleation term contributed to a steady increase in the total damage at the initial stage where void nucleation was dominant.  
The void volume due to the void growth process, \(\Delta V_{g}\), in the original NAG model is expressed as:  
\[\Delta V_{g}=V_{t}*\left(exp\left[\frac{3}{4}\left(\frac{max(\sigma_{m}- \sigma_{g0},\ 0)}{\eta}\right)\Delta t\right]-1\right) \tag{4}\]  
Figure 1: (**a**) Snapshot of (001) single-crystal Ta at the time associated with a maximum in void number where atoms are colored based on the centrosymmetric parameters [60], (**b**) distribution of the void volume fraction along the length of the simulation cell (Z direction).  
where \(V_{t}\) is the current void volume, \(\sigma_{g0}\) is the threshold stress beyond which voids start to grow, and \(\eta\) is the material viscosity that dictates the void growth rate sensitivity. The void growth term contributed to the exponential growth of damage following void nucleation.  
In the original NAG model, the change in the number density of voids, \(\Delta N_{V}\), was only due to void nucleation:  
\[\Delta N_{V}=\Delta N_{Vn} \tag{5}\]  
And the total change in the void volume, \(\Delta V\), was the sum of \(\Delta V_{n}\), \(\Delta V_{g}\):  
\[\Delta V=\Delta V_{n}+\Delta V_{g} \tag{6}\]  
The damage \(D\), as characterized by the void volume fraction, at time \(t\), is:  
\[D=\frac{V_{t}}{1+V_{t}} \tag{7}\]  
where \(V_{t}\) is the current void volume. Typically, the above parameters in the NAG model were obtained through calibration of the above equations to velocity-time data from experiments, for example, in the continuum damage model DFRACT package [61]. Although the NAG model provided a statistical treatment of void nucleation and growth process, it lacked an explicit treatment of void coalescence, which was a crucial contributor to later stages of damage accumulation. In addition, due to the lack of in-situ evolution of void number density and damage data obtained through experiments, there was no way to validate these parameters in terms of their capability to capture the full-time evolution of void number density and damage.