# The Modified Void Nucleation and Growth Model (MNAG) for Damage Evolution in BCC [PERSON]
## 1 Introduction  
Under dynamic loading, ductile metals fail by nucleation of voids at weak regions in the microstructure followed by growth and coalescence of these voids until complete failure. In this process, many contributing factors, including the exact loading condition (which determines the stress history), the weak spots (heterogeneities such as pre-existing and as-nucleated defects and grain boundaries that act as void nucleation sites), as well as the required stress at which voids start to nucleate and grow, dictates where, when, and at what rate damage accumulates in the microstructure and the overall failure behavior. Understanding the micro-mechanisms and kinetics of the above processes is critical to predicting damage evolution in a given microstructure, as well as establishing guidelines for the design of damage-resistant microstructures. As a result, many efforts have been devoted to understanding damage nucleation and evolution under such loading conditions [1; 2]. For example, [PERSON] et al. investigated the orientation and size effects on void growth behavior in both single-crystal [3; 4] as well as polycrystalline microstructures [5]. In this work, we focused on a special case of failure that occurs under uniaxial tensile strain and high strain rate, i.e., spill failure. Spall fracture is a dynamic failure process characterized by nucleation, growth, and coalescence of voids leading up to the complete failure of the material [6; 7]. During spall failure, the generated strain rates in the material are typically in the order of \(10^{4}\)-\(10^{8}\) s\({}^{-1}\), depending on the loading platform, resulting in damage initiation and growth in a very short time. The fine temporal and spatial resolution involved with measuring void nucleation and growth rates make it a challenging problem for measuring experimentally. For example, in laser-induced shock, under an average strainrate of 10\({}^{7}\)-10\({}^{8}\) s\({}^{-1}\), spallation takes place in a matter of a few nanoseconds [8]. While there had been work done for understanding the origins of void nucleation in metals [9; 10; 11; 12], only a handful of works investigated the rate of void growth and coalescence as a function of microstructure under shock loading [13; 14; 15]. A number of theories have been proposed in the past few decades to hypothesize the mechanisms responsible for void growth under such loading. The majority of these models fell loosely into two categories: diffusion-transport-based models and dislocation-emission-based models. [PERSON] and [PERSON] proposed a pipe-diffusion mechanism for void condensation to account for void growth [16]. It was shown that the above diffusion-transport-based models were only applicable under low to medium strain rate loading, such as the ones generated during creep deformation, as the typical time required for voids to grow to nanometer (nm) size through diffusion was in the order of microseconds to seconds [17]. Under high strain rate loading, such as laser-shock loading, which typically lasts for a few ns [18], there was not enough time for voids to grow to even nm size based on these diffusion models. Under such high strain rate loading, dislocation-emission-based models are used to describe void growth [19; 20; 21]. For example, [PERSON] et al. [17] proposed the generation of geometrically necessary dislocations (prismatic and shear loops) to be responsible for void growth, which was reminiscent of the dislocation-emission-governed crack blunting mechanism proposed by [PERSON] et al. [22]. In general, prismatic loops were generated at the surface of a void and transported material away from the void surface, leading to void growth, whereas shear-loops could lead to an octahedral shape of the void because they nucleated on the slip planes that intersected the void surface. The observation of slip bands emanating from void surfaces in Cu rendered support to the shear-loop-emission-governed void growth [23; 24].