*4.3. Implications for Damage and Rupture of Materials*

The present simulations bear direct relevance to spall formation under shock loading conditions; for example, spall by microvoid coalescence has been observed in 1100-O aluminum at shock stresses as high as 6 GPa [33]. However, it is reasonable to ask if the present observations also relate to quasi-static ductile rupture of aluminum alloys. The main gaps between our simulation conditions and those of typical experiments are that our applied stresses are much higher (for example, the yield strength of 2xxx series aluminum alloys is in the range of 150–350 MPa depending on alloy and heat treatment) and our particles sizes are smaller.

First, we address dislocation-mediated growth. The reality is that aside from high rate loading scenarios where high stresses are attained [52] and highly localized loading (e.g., under a nano-indenter), dislocation nucleation is typically far too slow to meaningfully impact material behaviors [51,53]. Hence, we do not expect that dislocation-mediated growth, as observed here through nucleation of dislocations, will be relevant under most loading scenarios. Indeed, even within the present spectrum of simulations, the dislocationmediated growth only occurred at the highest stresses approaching the athermal critical stress. On the other hand, Sills and Boyce recently showed that dislocation-mediated growth processes are greatly accelerated when dislocations are already present in the material, and thus do not require nucleation [54]. Kinematically speaking, it is equivalent to nucleate a dislocation at a crack tip or to adsorb an existing dislocation of opposite sign. We believe that dislocation-mediated growth via adsorption of *existing* dislocations is likely to play an important role in particle delamination, just as Sills and Boyce argue it does for void growth.

In terms of the relevance of lattice-trapped delamination, we are fortunate to have a fully characterized thermal activation law for the lattice-trapped delamination rate. This means that we can extrapolate the model to quasi-static conditions since the physics of thermal activation is equally valid at lower stresses and larger particle sizes. Importantly, the process of lattice-trapped delamination is bookended by two conditions on the stress intensity factor; namely, *Kc* < *K* < *K*at. Below the critical stress intensity factor, *Kc*, delamination is not energetically favorable. And above the athermal stress intensity factor, *K*at, thermal activation is not operable. Equation (9) provides an estimate for the athermal hydrostatic stress *σ<sup>H</sup>* ath at which *K*at is reached. For brittle fracture, *Kc* = *γsm* + *γsp E*

where *γsm* and *γsp* are the surface energies for the matrix and precipitate, respectively. The surface energy for Al is known experimentally to be *γsm* = 0.98 J/m<sup>2</sup> [55]. To estimate the surface energy for *θ*-phase, we use the average value for the interatomic potential used here (computed for (100) and (110) surfaces), giving *γsp* = 1.38 J/m2 [38]. These values give a critical stress intensity factor of *Kc* = 0.457 MPa·m1/2. Using these results, we estimate the stress and particle size range where lattice-trapped delamination may operate (i.e., where *Kc* < *K* < *K*at) when *φ* = 22.5◦ in Figure 12a. The plot shows that in the particle size range considered here, stresses must exceed 2.8 GPa for latticetrapped delamination to operate. On the other hand, for particles with a radius of 1 μm, lattice-trapped delamination can occur in the range of 200 to 550 MPa, which falls within the stress range relevant to quasi-static loading conditions, especially considering that inhomogeneous microstructural stresses can far exceed the homogeneous far-field applied stresses. Using this stress range, we plot the predicted lattice-trapped delamination rate . *a* as a function of stress for particle radii of 1 and 10 μm at temperatures of 300, 500, and 700 K. Interestingly, the resulting delamination rates are large enough that they may be relevant to applications. For example, full delamination of a 1 μm particle requires ~3 μm of crack growth (half of the circumference), and for this to occur in 1 years' time would require a delamination rate of just . *<sup>a</sup>* ≈ <sup>10</sup>−<sup>13</sup> m/s. We emphasize that because we have made a number of approximations in our analysis, these delamination rates should be interpreted only in terms of their rough order of magnitude. Nonetheless, these results indicate that lattice-trapped delamination via thermally activated brittle crack growth may be broadly relevant to void nucleation.

While we have only considered the influence of hydrostatic loading on the delamination behavior here, shear stresses are also expected to affect delamination. For example, experimental work by Croom et al. [56] and Achouri et al. [57] has demonstrated void nucleation under shear-dominated conditions in pure Cu and particle-containing Al, respectively. In the case of particle-mediated nucleation, in addition to the applied shear stress the non-uniform plastic strain accumulation around the particle provides a driving force for delamination. In the context of lattice-trapped delamination, this driving force is difficult to quantify since it requires an elastic-plastic analysis of deformation with a significant accumulation of plastic strain. None-the-less, lattice-trapped delamination could also be operative under shear-dominated loading, but additional research is necessary to quantify the influence of shear stresses and shear deformation.

Perhaps the most confounding aspect of our simulations is that we set out to study crack nucleation and instead wound-up studying crack growth. One might have expected that the rate controlling step of void nucleation would be the appearance of an interfacial crack, in the sense that a crack spontaneously appeared along the interface. However, this was not the behavior we observed. Instead, we found that excess free volume in the form of vacancy-type defect clusters at the interface appeared quickly and then grew by lattice-trapped delamination. The majority of the simulation time was then spent growing the crack by lattice-trapped delamination until dislocation nucleation occurred. Hence, for our simulations, the rate controlling step for void nucleation was actually lattice-trapped delamination. Once the first dislocations appeared, it is true that the void fully nucleated in a catastrophic manner more consistent with a true nucleation event, but we believe that this may be an artifact of our high stresses; in our lower stress simulations, the dislocationmediated delamination stage never occurred. We note that additional research is necessary to determine the mechanism for vacancy-type cluster nucleation so that a comprehensive picture for void nucleation can be assembled. While the present observations are directly applicable to the strain-rates associated with shock-induced spallation, extrapolation of the thermally activated process of lattice-trapped delamination suggests that the process is also relevant at quasi-static timescales for micrometer-sized particles.
