# The Modified Void Nucleation and Growth Model (MNAG) for Damage Evolution in BCC [PERSON]
## 1 Introduction
It was further shown that the critical stress required for dislocation emission from the surface of a void decreased with its radius [17; 25], indicating that void growth governed by dislocation emission should proceed exponentially with time. [PERSON] et al. showed that, under laser shock compression and release, spallation occurred primarily at the grain boundaries in poly- and ultra-fine-grained Ta and proposed a dislocation-emission-based model to explain the faster intergranular void growth rate [8]. Another dislocation-based crystal-plasticity-based model proposed by [PERSON] et al. suggested that local stress concentration generated by elastic and plastic anisotropies across a Grain boundary (GB) was a dominant factor in determining its propensity to nucleate and grow voids at GBs [15]. Moreover, continuous void growth upon nucleation was governed by the retarding effects of dislocation kinetics and micro-inertia of the material, incorporated through dislocation-based plasticity and relativistic drag-based visco-plasticity, respectively [26]. For example, [PERSON] et al. proposed a dislocation-kinetics-based void growth model [27] that was able to reproduce the void size distribution in spall surfaces of laser-shocked Cu as observed experimentally [28]. In addition to these works, there were also efforts to combine the above two models to investigate both creep and ductile failure [29]. Based on these theories, a few analytical models were developed to explicitly address the kinetics of void nucleation and growth processes, thus allowing for the prediction of damage evolution over time. For example, [PERSON] and [PERSON] formulated evolution laws governing damage rate and strain rate, assuming void growth through the power-law creep mechanism [30]. However, the Cocks-Ashby model predicted void growth at low strain rates as the ones observed during creep only and did not deal with void nucleation kinetics. In general, Cocks-Ashby model was only applicable to the growth of a uniform dilute distribution of identical voids under low strain rate loading. Based on macroscopic observations of ductile fractures in Cu and Al during 1-D flyer-plate impact experiments, [PERSON] et al. proposed void nucleation and growth model (NAG model), wherein evolution laws for both nucleation and growth were incorporated based on a statistical treatment of void distribution, rather than a single void [31; 32; 33]. A newer model developed by [PERSON] [34] extended the Cocks-Ashby model for application to high strain rate loading, including a nucleation model and early void growth behavior intended to represent the dynamic stress release upon nucleation. Under a narrow set of stress states, the driving force for void growth behavior in this model was equivalent to the tensile plasticity (TEPLA) model introduced by [PERSON] and [PERSON] [35]. The latter model was an extension of the Gurson-based pore growth threshold surface to dynamic conditions [36]. Along the same lines, a few spallation models were proposed to predict the variation in spall strength with the strain rate based on void nucleation and growth dynamics [37, 38]. Additionally, a variety of constitutive models were proposed to investigate the effect of void size and shape, stress triaxiality, and anisotropy on damage nucleation [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]. Most of these models were parameterized to specific experimental data to reproduce results in that narrow range of loading conditions only [50, 51, 52]. Efforts were also made to apply the aforementioned analytical models to void evolution data extracted from molecular dynamics (MD) simulations for Face Centered Cubic (FCC) metals such as Cu and Al. For example, [PERSON] et al. studied the temperature sensitivity of the NAG model parameters fitted to MD-predicted void evolution in single-crystal Cu under triaxial expansion [53].