Void growth by dislocation adsorption

ABSTRACT We propose a dislocation adsorption-based mechanism for void growth in metals, wherein a void grows as dislocations from the bulk annihilate at its surface. The basic process is governed by glide and cross-slip of dislocations at the surface of a void. Using molecular dynamics simulations we show that when dislocations are present around a void, growth occurs more quickly and at much lower stresses than when the crystal is initially dislocation-free. Finally, we show that adsorption-mediated growth predicts an exponential dependence on the hydrostatic stress, consistent with the well-known Rice-Tracey equation. GRAPHICAL ABSTRACT IMPACT STATEMENT Void growth during ductile rupture is mediated by dislocation adsorption, not dislocation emission as previously theorized. The mechanism is consistent with Rice-Tracey scaling and observations of voids near cell walls.


Introduction
The ductile fracture process in metals is relevant to a number of failure scenarios including quasi-static tearing [1,2], dynamic spall [3], creep rupture [4], irradiation creep [5], and wear debris generation [6]. Typically, the fracture surfaces of these metals exhibit a characteristic pitted 'ductile dimple' appearance indicative of the microvoid coalescence rupture process [7]. Often ductile fracture is due to the nucleation, growth, and coalescence of microscale voids [7]. It is usually assumed that the material contains a pre-existing population of hard particles or inclusions made from secondary phases. Void nucleation then occurs when these particles crack or the interface between the particles and matrix delaminates [8][9][10]. The voids grow under the action of plastic deformation around the voids, driven by hydrostatic stresses [11,12]. Final rupture occurs when the voids coalesce, which typically occurs when ligaments between This basic picture for rupture in ductile solids has been around for nearly 50 years. Despite this fact, many of the details surrounding it are poorly understood, including the micromechanics of void nucleation and growth. For example, ductile rupture is still observed in materials that do not contain hard particles or preexisting voids, making it unclear how the voids nucleate in the first place [13,14]. There are numerous competing continuum-scale phenomenological models for the ductile rupture process [15] including the Gurson model [16], an adaptation by Tvergaard and Needleman [17], modifications to account for shear localization [18], and stress triaxiality models [19]. The Sandia Fracture Challenge has illustrated the ongoing disagreement with regard to the most appropriate failure models and their difficulties in blindly predicting rupture scenarios [20,21]. Many of these difficulties arise from assumptions made about the underlying micromechanical mechanisms that drive void nucleation and growth; for example, many models assume a pre-existing void density and hence neglect void nucleation altogether. At lower length scales such as the scale of individual grains, there is even broader disagreement as to the proper models to describe the rupture process. These discrepancies are due in part to a lack of clear fundamental understanding of the detailed unit mechanisms governing nucleation, growth, and coalescence.
In this work, we focus on obtaining a detailed micromechanical understanding of the void growth process. Much of the literature on void growth focuses on continuum theories which characterize the void growth rate as a function of the stress state [12,22]. Of these, perhaps the most famous is the work by Rice and Tracey [11], later corrected by Huang [23], who solved for the growth rate of an isolated void in a rigid-perfectly plastic solid loaded with applied strain rate tensor˙ ij and hydrostatic stress σ H . They obtained the simple result for the void growth rate thatṘ where R is the void radius, σ Y is the yield strength, eq = 2 3˙ ij˙ ij is the equivalent plastic strain rate (using Einstein notation), and α = 0.427 is a numerically obtained constant [23]. Hence, the continuum theory predicts an exponential dependence on the hydrostatic stress. The Rice-Tracey equation was recently validated against detailed X-ray tomographic measurements of void growth [24]. According to Rice-Tracey, voids grow as plastic strain accumulates around them, but the detailed micromechanics are unclear.
In an effort to fill this micromechanical knowledge gap, numerous atomistic studies of void growth have been performed in the last decade [26][27][28][29][30][31]. The key finding in all of these studies is that voids grow by nucleating dislocations at the surface of the void. However, such a mechanism poses a fundamental dilemma. In particular, Nguyen and Warner [32] showed using molecular dynamics and transition state theory calculations, that dislocation nucleation from the surface of a void in pure aluminum is a tremendously slow process. For example, the nucleation rate under 0.78 GPa of shear stress is one dislocation per year! In order to obtain nucleation rates on the order of 1 s −1 , applied stresses need to exceed 1 GPa. While stresses of this magnitude may be relevant to high loading rates (e.g. shock loading [33]), they are well beyond the strengths of many structural metals under quasi-static loading.
In this work, we resolve this problem by proposing that void growth via adsorption of dislocations from the surrounding bulk is the primary mechanism of growth. Adsorption-mediated growth was recently shown to be a viable mechanism for small amounts of void growth (<0.1% by radius) by Chang et al. [34]. While Chang et al. 's prior work opens the possibility that dislocation adsportion may be an important growth mechanism, their use of discrete dislocation dynamics was limited to low dislocation densities below those experimentally observed at relevant cell wall boundaries [14,35], and suggested that the adsorption mechanism may be limited to only the very early stages of void growth. Many important questions remain to be answered before the importance of adsorption-mediated growth can be understood, however. Namely: what are the key dislocation processes underlying the mechanism?; what is the connection, if any, between nucleation-mediated and adsorption-mediated growth?; and, what growth rates are predicted by adsorption-mediated growth and how do they compare with experiments? In this work, we address these questions, demonstrating that (1) the dislocation processes enabling adsorption-mediated growth (glide and cross-slip) occur rapidly, (2) even in a scenario where nucleation-mediated growth occurs (no dislocations present initially), adsorption-mediated growth quickly takes over as the mechanism of growth, and (3) growth rates predicted by adsorption-mediated growth are consistent with the experimentally validated Rice-Tracey equation. Lastly, we argue that our results elucidate recent experimental observations of void growth at regions with elevated dislocation density (cell block boundaries) [14]. Combined together, we believe our results demonstrate that adsorption-mediated growth is an important growth mechanism.

Methods and results
We study void growth at a temperature of 300 K in face-centered cubic aluminum using the embedded atom method potential of Mishin et al. [36] and the molecular dynamics code LAMMPS [37,38] (see Supplementary online material for simulation details).
To begin our studies, we focus on the interaction of isolated dislocations with a single void in order to elucidate underlying mechanisms. We introduce a void of radius R 0 and impose a constant hydrostatic tensile stress on the simulation cell. Treating the void as a spherical cavity in an isotropically elastic matrix under a pure hydrostatic load, we can compute the equivalent Eshelby eigenstrain [39,40] to obtain a stress field around the void of (see Supplementary online material) where δ ij is the Kronecker delta and r = √ x i x i . The resulting maximum shear stress at the surface of the void is 3 4 σ H . We study two dislocation geometries: a prismatic (meaning the initial plane of the loop is orthogonal to the Burgers vector) dislocation loop and a straight screw dislocation. Note that if a void were not present, there would be no driving forces acting on the dislocation lines since the system is under a pure hydrostatic stress state. Figure 1 depicts the response of the dislocation lines. In both cases, the dislocation lines are attracted to the void. After initially colliding with the void, the dislocations quickly glide across its face, and then cross-slip onto another glide plane where they are able to glide again. After this process repeats several times, the end result is that closed dislocation loops are traced out on the void's surface. The volume change of the void after the dislocation loops are traced out is [41,42] where b is the Burgers vector, n is the outward normal of the void surface, and S is the area traced out by the dislocation line. Note that the exact same volume change would occur if instead a loop with an oppositely signed Burgers vector and the same shape were nucleated from the surface of the void [43]. Hence the underlying mechanics is exactly the same between adsorption-mediated and nucleation-mediated growth.
The key point is that adsorption-mediated growth can occur much more rapidly since it relies on cross-slip and glide, which are known to occur readily over typical experimental timescales. Furthermore, the externallysourced dislocations can be produced much more quickly (via conventional multiplication processes) than surfacebased nucleation. Since we are constrained to atomistic timescales in these simulations, the stresses need to be large (≥2 GPa), however, to accelerate the processes. Demonstration of the basic unit-mechanisms underlying adsorption-mediated growth is the first major contribution in this work. Next, we demonstrate the behavior of a crystal containing voids under an applied hydrostatic tensile strain rate, a case that has been considered by several researchers [28,30,31]. We focus on two cases: a) no dislocations initially present and b) n prismatic dislocation loops with edges of length 20 nm and 21.2 nm (same shape as shown in Figure 1(a)) initially present. A comparison between these two cases has not been made before. For reference, when n = 18 the initial dislocation density is about 4.3 × 10 16 m −2 . This high initial dislocation density may be representative of a dislocation cell wall given the experimental observations of densities  in excess of 10 15 m −2 locally in highly deformed aluminum [35,44]. Figure 2 presents resulting stress-strain curves under a strain rate of 3 × 10 8 s −1 with voids of initial radii R 0 = 2.5 and 7.5 nm, and a simulation box that is L = 32.5 nm on each edge. Figure 2(a) reproduces the same behavior as previous studies when dislocations are not initially present, with a large spike at yield, followed by a rapid drop in stress and subsequent softening. This initial spike corresponds to the nucleation of one or more dislocations from the surface of the void, which then enables plastic growth of the void. The new result shown here is the influence of pre-seeding the system with dislocations. We show that as the number of initial dislocation loops increases, the stress spike at yield is greatly reduced. This indicates that void growth occurs much more readily when dislocations are present, since void growth initiates at much lower stresses. Beyond initial yield, however, the stress-strain curves are independent of the number of dislocation loops.
The observation that the post-yield behavior is insensitive to the initial dislocation content motivated additional simulations, where we increased the void radius while concurrently increasing the box dimensions so that the number of atoms was kept approximately constant at 1.9 million. In this way, all of the initial conditions can be related to each other and the influence of dislocations-nucleated or initially introduced-can be evaluated. For instance, at 10.5% strain the simulation with an initial void radius 5 nm has grown to an effective void radius of R eff = ((3/4π) V + R 3 0 ) 1/3 = 10 nm, where V is the volume change of the cell. Hence, the simulation with initial void radius of 10 nm emulates the R 0 = 5 nm void case at 10.5% strain. We can then directly determine the influence that dislocations have at that strain by starting the simulation with and without dislocation loops introduced. Figure 2(b) presents the resulting stress-strain curves with and without dislocation loops, all translated along the x-axis so that the instantaneous void sizes match. Interestingly, beyond initial yield where a stress spike is observed in dislocationfree systems, all simulations collapse onto a single universal stress-strain curve. This fact indicates that post-yield, where the bulk of the void growth occurs, voids in simulations with and without dislocation loops grow by the same mechanism: dislocation adsorption. Specifically, under the influence of the void's stress field, dislocations multiply and adsorb at its surface. These results indicate that void growth does not occur by dislocation nucleation alone; void growth may be initiated by nucleation, but the growth itself occurs when the nucleated dislocations multiply and interact with the void's stress field. The finding that adsorption-mediated growth dominates even in dislocation-free systems is the second major contribution of this work.
These results provide details about the adsorptionmediated growth process, but do not provide a means for computing void growth rates akin to Equation (1). To this end, we have computed void growth rates under a fixed tensile hydrostatic stress. By examining the volume history of the system we can estimate the void growth rate. Figure 3(a) shows the volume history expressed in terms of the effective void radius with a void of initial radius 5 nm under a stress of 2.8 GPa. Curves are shown for n = 0, 18, 36 and 54 dislocation loops. When no dislocation loops are present, no void growth is observed because the stress is insufficient to drive nucleation. As the number of loops increases, the void growth rate increases. We can approximate the effective void growth rateṘ eff using linear fits to these time histories. Figure 3(b) summarizes the results from 360 MD simulations; 20 random loop configurations for each value of n with hydrostatic stresses ranging from 2.3 to 2.8 GPa. Since void growth is highly sensitive to the specific dislocation configuration around the void, growth rates vary significantly for the same stress and number of loops (see Supplementary online material). Hence, in Figure 3(b) we plot the median growth rate among the 20 replicates at each set of conditions. To demonstrate that our results are not biased by the use of prismatic dislocation loops, we have also performed simulations using 12 shear loops, giving an initial dislocation density of 5.2 × 10 16 m −2 (see Supplementary online material). These results are also presented in Figure 3(b), showing more scatter but the same overall trend. Figure 3(b) demonstrates that the void growth rate increases with dislocation density and that the adsorption-mediated void growth rate is exponentially dependent on the hydrostatic stress. The latter result is significant because it is consistent with the experimentally validated Rice-Tracey relation, Equation (1). The fact that adsorption-mediated growth follows the same stress dependence as is observed in quasi-static experiments is the third major contribution of this work. Note that the same statement cannot be made about nucleation-mediated growth. Analyzing the exponential curve fits further, we find that the growth rates with prismatic loops at all dislocation densities exhibit an exponent of about 4.4σ H , which according to Equation (1) implies a 'yield strength' of about 341 MPa; this is not an unreasonable order of magnitude for local stresses in highly work hardened aluminum which has been shown to exhibit a macroscopic ultimate strength as high as 250 MPa [45]. Hence, our results predict a growth rate of the formṘ where A(ρ) is a dislocation-density-dependent prefactor.

Discussion
The current work suggests that the dislocationadsportion mechanism proposed by Chang et al. [34] can continue to operate at much higher dislocation densities, relevant to dislocation cell walls where voids are now understood to proliferate. Moreover, the current results suggest that the adsorption mechanism can account for extensive void growth, well beyond 0.1% indicated in the previous studies [34]. These current results, and the observed consistency with Rice-Tracey scaling, suggest that the adsorption mechanism is a more comprehensive explanation for void growth than previously indicated, strengthening the argument that dislocation emission is not a controlling factor. We note that in Rice and Tracey's analysis [11], a hydrostatic stress and farfield strain rate were simultaneously imposed, making Equation (1) depend on both σ H and˙ eq . However, in MD it is not possible to impose a strain rate and a stress state concurrently, which is why we have only focused on the influence of σ H . Overall, we believe the dislocation adsorption mechanism acts to enhance the kinetics of void growth; the driving force for growth is still hydrostatic stress with adsorption of pre-existing dislocations accelerating the growth rate and eliminating the need for nucleation or diffusive processes. The fundamental dislocation processes involved in adsorption-mediated growth-glide and cross-slip of dislocations-occur readily in typical experimental conditions. Cross-slip is the slower of the two processes, however it has been shown that the energy barrier for cross-slip at free surfaces is extremely low (<0.1 eV), making the cross-slip rate high [46].
The incipient nanoscale voids considered here are distinct from the micrometer scale voids typically investigated experimentally, e.g. [47]. The present results suggest that incipient nanoscale voids can grow under high local stresses, but only if in the presence of extremely high dislocation densities, such as those found in a dislocation cell wall; this assertion is supported by recent experimental evidence showing that voids prefer to reside in regions of high dislocation density [14,48]. Additional research is needed to confirm that such behavior occurs not only at the timescales of MD, but also under quasi-static loading conditions and under lower stress conditions at slower timescales. Because the dislocations enable growth without the need for nucleation or diffusive processes, we expect that strain-rate will play only a minor role in this phenomenon. Combined together, the evidence presented here suggests an intimate, mechanistic linkage between void growth and strain-induced dislocation networks which may be influential to theories of ductile fracture.