## 3 Results and Discussion
### Accounting for Precipitate Softening in the Equation of Dislocation Motion  
The single parameter of the model in Equations (1) and (2) that describes dislocation-precipitate interaction is the energy of dislocation core \(\epsilon_{p}\) for the segments contacting the precipitate. The criterion of its choosing is the best coincidence between the model and MD in the level of oscillations during the first three interactions. Other parameters follow our previous works with other types of precipitates [57; 58]. The used parameters are the following: \(\epsilon_{1}=0.2\) eV/\(b\), \(R_{0}=6b\), \(\epsilon_{p}=0.8\) eV/\(b\). Multiple interactions of dislocation with an obstacle lead to a decrease in the average shear stresses in the MD system, as it is shown in Section 3, devoted to MD study. Copper atoms, which initially form the lattice of \(\theta^{\prime}\) phase, are spread out in space, and the dislocation shifts to an edge of the precipitate. Corresponding softening of the precipitate can be described by the decrease in the effective diameter of inclusion, which enters Equation (1). Based on the obtained MD data for two diameters of precipitates, we propose the following  
Figure 9: Volume fraction \(\alpha\) of voids by dependence on time: (**a**) for a single system with two different radii of probe sphere of “Construct surface mesh” algorithm; and (**b**) for different systems with additional linear approximation. dependence to describe the change in the properties of the systems containing small and large inclusions during prolonged plastic deformation:  
\[d=d_{0}\exp[-k(d_{0})N], \tag{5}\]  
where \(d_{0}\) is the initial diameter of precipitate before the interactions with dislocation, \(N\) is the number of interactions and \(k(d_{0})\) is the exponent dependent on the initial diameter of inclusion. We find the following values for the considered initial diameters: \(k(4.6\ \mathrm{nm})=0.03\) and \(k(18\ \mathrm{nm})=0.0015\). The change in diameter of inclusion is introduced into the model, and then calculations are carried out, repeating the MD statement of the loading. The modeled time dependences of the shear stresses averaged over the system for two diameters and two distances between precipitates are shown in Figure 10 together with MD data. The usage of Equation (5) for decrease in the effective diameter of inclusions allows us to obtain dependences, which well reproduce the general trends towards the decrease in stresses in the MD system with the accumulation of plastic deformation. However, this approach cannot describe the features of the multiple dislocation-precipitate interactions in all details. Since here we do not take into account the interaction of dislocation with vacancies emitted after previous interactions, the modeled dependences do not reflect the temporal increase in the stress curves in the MD system. As noted in Section 3.2, for the larger inclusion, the initial stage up to 8 ns is similar to the initial stage of evolution up to 1 ns for smaller inclusions, after which the disordering of the inclusion structure is activated, and a decrease in the average stress becomes more pronounced. In the experimental work [32], it was shown that upon deformation of the Al-Cu sample containing strengthening particles of \(\theta^{\prime}\) and \(\theta\) phases, the mechanism of damage of the \(\theta^{\prime}\) precipitates was size-dependent. Cutting was observed for sufficiently small particles with a diameter not exceeding 60 nm, while larger particles acquired only bending. This result is consistent with our data obtained in MD calculations, where it is shown that for the large inclusion of 18 nm in diameter, the decrease in the acting stresses occurs much slower than that for 5 nm ones. Since MD simulation of large systems for a time sufficient  
Figure 10: Time dependences of shear stresses averaged over system for precipitates with diameters \(d\) of 4.6 nm (**a**) and 18 nm (**b**) and different distances \(D\) between inclusions: 9.87 and 17.6 nm (**a**) and 27 nm (**b**).