## 3 Results and Discussion
### Strength of Alloys with Size-Distributed and Spatially Distributed Precipitates  
Real alloys contain inclusions of different sizes distributed in the material bulk. The authors of [24] experimentally established the log-normal size distribution of precipitate diameter and thickness for various types of inclusions in aluminum alloys after different aging times. In contrast to previous work, here we consider the case of \(\theta^{\prime}\) phase by means of using the corresponding parameters determined in Section 3.4. We take into account possible strain softening of the material due to a decrease in the precipitate size during prolonged plastic deformation as discussed in Section 3.4. In order to reveal the influence of change in precipitate structure during the multiple interactions with dislocations, for each precipitate distribution, we perform calculation for two cases: the first one, \"uncuttable\", is a virtual case of the constant size of each inclusion, even small ones; the second one is \"cuttable\", accounting for the effective reduction in precipitate diameter in the course of plastic deformation according to Equations (5) and (6). The chains of precipitates are randomly placed in the computation domain. The diameters of precipitates are randomly chosen in accordance with the log-normal distributions established in experiments [24], but with different typical size, which is the value close to the maximum of the size distribution of precipitates--see [57] for details. Because cutting influences on small inclusions, here we restrict ourselves by the typical size no more than 80 nm, while the distributions with the typical sizes up to 130 nm are observed experimentally [24] and analyzed in our previous simulations [57]. The calculated stress-strain curves for different typical sizes and volume fractions of inclusions, which are almost equal to the mass fractions of copper atoms, are presented in Figure 11 for both problem statements with and without accounting for the evolution of precipitate size at multiple interactions. In the approximation of uncuttable inclusions, the shear stress reaches some plateau and holds at this level till the end of the system tracing. The initial growth of shear stress corresponds to the transient stage, during which most dislocations reach strong obstacles in the form of precipitates of large diameter and become fixed. Only a portion of dislocations continues to move along the weakest bands of material, in which there are no strong obstacles. It leads to strong localization of the plastic deformation in the system [57]. The reached level of shear stress increases with the increase in the volume fraction of precipitates, but decreases with the increase in their size at constant volume fraction. This is because the increase in precipitate size leads to a decrease in the precipitate concentration, which increases the probability of the formation of weak bands in the material. One can conclude from Figure 11 that the evolution of precipitate size leads to strain softening, which means diminution in shear stress acting in the system at large strains. Only alloys with a high enough volume fraction of precipitates suffer from the softening. It is explained by the fact that, at small volume fractions, there is already enough weak bands in the material, and further weakening of precipitates in these bands does not influence the mechanical response. On the other hand, a large concentration of precipitates eliminates the weak bands. Dislocations are forced to move along higher strength paths, and a higher level of stress holds in the system. In this case, gradual cutting of precipitates weakens the most frequently walked bands and provides the material softening. The increase in localization of plastic flow due to precipitate cutting is illustrated in Figure 12 with the spatial distributions of plastic strain; cutting forms several weaker paths, which concentrate the most part of the plastic strain.