## 4 Conclusions  
Our molecular dynamics study reveals the complex character of the interaction of dislocations with \(\theta^{\prime}\) phase precipitates. It is shown that the first several interactions occur in accordance with the Orowan looping mechanism, while the subsequent interactions provoke a gradual degradation of the precipitate structure up to its scattering into individual copper atoms in the aluminum matrix. \(\theta^{\prime}\) phase shape at various stages of this process obtained in our calculations repeats the experimentally observed one for cut inclusions of \(\theta^{\prime}\) phase [31; 32]. Structural changes in \(\theta^{\prime}\) phase lead to a gradual decrease in the strength of the obstacle to dislocation that is expressed in a decrease in shear stresses acting in the MD system. This softening of \(\theta^{\prime}\) phase occurs with different rates, depending on the diameter of precipitate: studied inclusions of 15 nm in diameter demonstrate a significantly slower decrease in strength in comparison with 5 nm ones. During interactions with inclusion of \(\theta^{\prime}\) phase, the dislocation can leave the initial slip plane due to the cross-slip of segments elongated around the inclusion, which acquire a screw character. After separation from the precipitate, the dislocation moves for some time with a segment ejected into the adjacent slip plane; however, this state is unstable and the dislocation tends to move into one slip plane. Transition of dislocation segments between slip planes is accompanied by emission of vacancies. Volume fraction of accumulated vacancies \(\alpha\) increases significantly during the prolonged plastic deformation of the MD system and can be estimated by the following approximating relation with the plastic strain \(w\) as \(\alpha=2.4\cdot 10^{-3}\)\(w\). Estimations of the volume fraction of free space are important because microvoids can facilitate fracture of material in the plastic localization bands under additional action of tensile stresses [30]. The obtained results of MD calculations are used for fitting the dislocation-precipitate interaction model proposed earlier in [34; 57; 58]. The softening of \(\theta^{\prime}\) phase is introduced into the model through reducing the effective diameter of precipitate. We additionally introduce a dependence of the softening coefficient on the diameter of precipitates into the model. This dependence is based on the different rates of softening of \(\theta^{\prime}\) inclusions depending on its diameter registered in our MD calculations, and the experimental data on the existence of a threshold diameter of \(\theta^{\prime}\), after which no cutting is observed [32]. The model of dislocation-precipitate interaction, which is modified to account for the precipitate softening, is coupled with a 2D discrete dislocation dynamics model [57], which makes it possible to predict the flow stress of alloy depending on the size distribution of precipitates. We compare the predictions of shear strength of Al-Cu alloy obtained with 2D DDD and the experimental data on the dependence of flow stress on the size distribution for the case of \(\theta^{\prime}\) phase precipitates [37]. Taking into account the rather significant error in determining the typical diameter of precipitates and their volume fraction in the experiment, we can conclude that our results are in satisfactory agreement with the experimental data and reproduce the main experimental trend of decrease in flow stress, together with the increase in the mean diameter of precipitates at a constant volume fraction of them. The performed 2D DDD simulation made it possible to identify the region of typical particle sizes and concentrations of \(\theta^{\prime}\) precipitates, in which the softening of alloy during prolonged plastic deformation can be expected. This boundary with a good accuracy is above the 600 MPa shear strength for all considered model alloys.