**4. Conclusions**

An experimental and theoretical investigation of the strength properties of aluminum alloys strengthened by nanoparticles, as well as a determination of the significance of various mechanisms responsible for the strengthening of the material, was carried out.

Results of experimental investigation demonstrated that the hardening of aluminum alloy A356 by Al2O3 and ScF3 nanoparticles led to an increase in the yield strength, ultimate tensile strength, and plasticity. The introduction of 0.2 wt.% Al2O3 nanoparticles made it possible to increase the yield strength, ultimate tensile strength, and plasticity from 85 to 100 MPa, from 130 to 180 MPa, and from 3.5% to 4.1%, respectively, and an increase in the content of Al2O3 nanoparticles allowed an increase in the yield strength and ultimate tensile strength of the alloy to 113 MPa and 195 MPa, respectively.

The use of 0.2 wt.% ScF3 nanoparticles increased the yield strength, ultimate tensile strength, and ductility of the A356 aluminum alloy to 98, 190 MPa, and 4.3%, respectively, and an increase in the content of ScF3 nanoparticles made it possible to increase the yield strength and ultimate tensile strength of the alloy to 109 MPa and 250 MPa, respectively. Despite the similar size of Al2O3 and ScF3 nanoparticles (~80 nm), the physicomechanical properties of nanoparticles significantly affected the possibility of increasing the mechanical properties of the A356 aluminum alloy.

A physicomathematical model of the occurrence of thermal stresses was developed caused by the mismatch of the coefficients of thermal expansion of the matrix and strengthening particles, on the basis of the fundamental principles of mechanics of a deformable solid, and in contrast to existing models, taking into account the elastic properties of not only the matrix, but also the particle.

In the case of thermal deformation of dispersion-hardened alloys, when the CTE of the matrix and particles noticeably differ, an additional stress field is created in the vicinity of the strengthening particle. Thermal stresses increase the effective particle size. This phenomenon can significantly affect the result of the assessment of the yield strength.

In the particle, the stresses are constant. In the matrix material, these stresses decrease quite sharply. As the distance from the particle increases, the value of the shear stresses monotonously decreases and becomes vanishingly small, when the distance from the particle boundary exceeds five particle diameters.

The growth of temperature difference leads to an increase in contact pressure between the matrix and the particle. On the inner surface of the matrix, plastic flow begins when the maximum shear stresses exceed the yield strength of the material at a given temperature. The plastic deformation due to thermal stresses of the aluminum matrix with strengthening scandium fluoride particles occurs when the temperature difference is equal to approximately 72 K.

The strengthening caused by thermal mismatch makes the largest contribution to the yield strength improvement. The yield strength increments due to Nardon–Prewo and Orowan mechanisms are much lower.

**Author Contributions:** Conceptualization, O.M., O.D. and A.K.; methodology, O.M. and I.Z.; software, O.M.; validation, O.D. and T.K.; formal analysis, O.D. and T.K.; investigation, O.D. and A.K.; writing, O.M. and O.D.; supervision and funding acquisition, A.V. All authors read and agreed to the published version of the manuscript.

**Funding:** This work was financially supported by Grant N 17-13-01252 from the Russian Science Foundation and Ministry of Science and Higher Education of the Russian Federation, project No 0721- 2020-0028.

**Data Availability Statement:** The data presented in this study are available in the article.

**Acknowledgments:** The authors acknowledge the Russian Science Foundation under Grant N 17- 13-01252. I.Z. and A.K. acknowledge the financials support by the Ministry of Science and Higher Education of the Russian Federation, project No 0721-2020-0028.

**Conflicts of Interest:** The authors declare no conflict of interest.
