## 3 Results and Discussion
### Accumulation of Vacancies in the MD System during Prolonged Deformation  
The generated vacancies increase the defectiveness of material and can promote fracture under additional tensile stresses. The accumulation of microvoids in shear bands was previously assumed to be the initiator of the fracture of material [30]. On the other hand, the accumulation of vacancies can be important for a multi-stage alloy manufacturing process as described in [79]. A large number of vacancies in aluminum alloys after preliminary cyclic deformation allowed the authors to obtain precipitation of very fine particles and to register an increase in strength and elongation properties of alloy relative to traditional aging.  
Therefore, we estimate the fraction of free volume in the system associated with the generation of vacancies during plastic deformation. For this purpose, we use the \"Construct surface mesh\" algorithm implemented in the OVITO package [75]. The probe sphere radius is the parameter of the algorithm; a decrease in this parameter makes the algorithm more sensitive to cavities in the system, but, at the same time, starting from a certain radius of the probe sphere, the algorithm begins to determine the space between the aluminum and copper atoms in the Al\({}_{2}\)Cu intermetallic compound as voids. We chose two radii, 0.24 and 0.25 nm; the larger of them is not sensitive to the space inside \(\theta^{\prime}\) phase, but underestimates the volume of vacancies in the system; the smaller one more accurately takes into account free volume inside the vacancies, but sometimes detects the volume in inclusion as free one. Figure 9a demonstrates the curves of the free volume fraction by dependence on time corresponding to two radii of 0.24 and 0.25 nm. Note that the difference between these two estimates at the end of MD calculations is 18%. The average curve for two obtained estimates of free volume is additionally given in Figure 9a; this average curve is presented in Figure 7b together with the vertical position of dislocation.  
Figure 8: Time dependence of averaged shear stress in a system containing \(\theta^{\prime}\) phase of 18 nm in diameter. (**a**) View of inclusion after multiple interactions with dislocation (**b**) Thickness of system is 27 nm, displacement velocity is 3 m/s and temperature is 300 K.  
Since the accumulation of the free volume of vacancies in the shear bands can act as the cause of fracture of deformed samples, it is necessary to estimate the rate of vacancy accumulation in the system during the plastic deformation for further construction of the models of material fracture. Figure 9b demonstrates the dependences of the fraction of free volume on time for the systems with inclusion of 5 nm in diameter. The obtained dependences show similar tendencies to the multi-step growth of free volume in the systems. We cannot distinguish explicit dependences of these curves on temperature and system size. Moreover, the curves show a significant scatter in the final value of the free volume fraction. Therefore, we use the same linear approximation for all of these data in order to fix the main tendency towards the accumulation of vacancies. For further use in continuum fracture models, it is necessary to give the dependence of the free volume fraction on the plastic deformation. During the prolonged deformation, the plastic deformation can be considered approximately equal to the total deformation of the system, because the remaining elastic part is much smaller. Finally, the relationship between the fraction of free volume \(\alpha\) in MD system and the plastic deformation \(w\) can be written as \(\alpha=2.4\cdot 10^{-3}\,w\).