*3.1. Experimental Results*

Figure 2 shows SEM images of the microstructure of A356 aluminum alloys. The structure of the initial A356 alloy does not differ significantly from the alloys with ScF3 nanoparticles. Silicon inclusions are present in the structure of all alloys (Figure 2a,b). At the same time, ScF3 nanoparticles are concentrated in the A356 aluminum alloy around the silicon inclusions (Figure 2c,d) and along the grain boundaries. The nonregular distribution of nanoparticles in the structure of the A356 alloy occurs due to the force of attraction

between nanosized inclusions, as well as under the influence of the solidification front, which displaces them to the boundaries when the melt is cooled.

**Figure 2.** SEM images of the structure A356 alloys (**<sup>a</sup>**,**b**) and A356–1% ScF3 (**<sup>c</sup>**,**d**).

The microstructures of the initial aluminum alloy A356, A356 + 0.2% ScF3, and A356 + 1% ScF3 are shown in Figure 3. It can be seen that the introduction of 0.2% ScF3 nanoparticles into the aluminum alloy led to a decrease in the average grain size from 310 to 190 μm. An increase in the amount of nanoparticles in the A356 alloy to 1% led to a decrease in the grain size to 100 μm.

Figure 4 shows tensile diagrams, and Table 1 shows data on the mechanical properties of aluminum alloys strengthened with ScF3 and Al2O3 nanoparticles. The results were obtained at room temperature (*T*room = 293 K).

**Figure 3.** Microstructure of A356 alloy without treatment (**a**), A356 after ultrasonic treatment (**b**), A356–0.2% ScF3 (**c**), and A356–1% ScF3 (**d**).

**Figure 4.** Loading diagrams of aluminum alloys: 1—A356; 2—A356–0.2% Al2O3; 3—A356–0.2% ScF3; 4—A356–1% Al2O3; 5—A356–1% ScF3.


**Table 1.** Mechanical properties of aluminum alloys.

Figure 4 and Table 1 demonstrate 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.

The approximation of the obtained experimental stress–strain curve allowed us to obtain the function of *σ*(*ε*) with an error not exceeding 0.1%.

*σ* = ⎧⎨ ⎩ *G<sup>ε</sup>*, if *ε* ≤ *τ*0/*G τ*0 + *τ*1 *ε* − *τ*0/*G ε*∗ + *ε* , if *τ*0/*G* < *ε* , (25)

where *τ*0 is the yield strength, *τ*1 = *τ*∞ − *τ*0 is the hardening stress, which characterizes the maximum increase of the flow stress during the plastic deformation, and *ε*∗ is an empirical parameter that determines the rate at which the flow curve reaches the asymptote.

The values of the material constants: *τ*0<sup>∗</sup>, *τ*1<sup>∗</sup>, and *ε*∗ for various volume fractions of scandium fluoride particles are presented in Table 2.


**Table 2.** The material constants *τ*0<sup>∗</sup>, *τ*1<sup>∗</sup>, and *ε*∗.

#### *3.2. Results of Theoretical Investigation*

The theoretical investigations were conducted for a matrix of aluminum Al 5083 alloy hardened by reinforcement particles. The main calculations were performed for the following parameters [51]: *α*m = 2.3 × 10−<sup>5</sup> <sup>K</sup>−1, Young modulus of 73 GPa, matrix shear modulus of *G* m = 28.08 GPa, and yield strength of *σ*Ym = 85 MPa.

A variety of oxides, carbides, borides, and fluorides were utilized as the reinforcement phase in aluminum alloys. Some selected physical and mechanical properties of commonly used reinforcements are summarized in Table 3.


**Table 3.** Characteristics of reinforcement particles.

> Let us consider the contribution of various mechanisms to material hardening.

The intensity of the stresses caused by the transfer of the load from the matrix to the particles *σ*NP is determined by the volume fraction of the hardening particles and, at *f*p = 0.01, is 0.5% of the yield strength of the matrix material. For *σ*Ym = 85 MPa, the calculated value of *σ*NP = 0.425 MPa, which indicates an insignificant contribution of this mechanism to the hardening of the material.

The influence of the average grain size on the stress intensity is be rated by the Hall–Petch law, which describes the hardening of a material through the retardation of dislocations by grain boundaries in a polycrystal. The values were calculated (Equation (4)) taking into account the average grain size *<sup>d</sup>*gr obtained from optical images of the structure (Figure 3). The calculated values *σ*HP for the alloys were as follows: A356—2.5 MPa, A356– 0.2%—3.2 MPa, and A356–1%—3.7 MPa. Despite a significant decrease in the average grain size, this mechanism had little effect on the increase in the mechanical properties of the A356 alloy. Taking this into account, the effect of the difference in CTE should be considered.

Let us proceed to the analysis of the effect of Orowan stress on the material hardening process. For convenience of analysis, we rewrite Equation (5) in the following form:

$$
\sigma\_{\rm Or} = G\_{\rm m} \frac{b}{\Lambda\_{\rm P} - 2R\_{\rm P}} = G\_{\rm m} \frac{b}{\delta\_{\rm P}} \frac{\sqrt[3]{f\_{\rm P}}}{1 - \sqrt[3]{f\_{\rm P}}}.\tag{26}
$$

Figure 5 shows the dependence of Orowan stresses on the volume fraction of hardening particles, calculated according to Equation (26).

**Figure 5.** Dependence of Orowan stresses *σ*Or in alloy A356 strengthened by ScF3 particles on the volume fraction of hardening particles, calculated for Δ*T* = 100 K. Particle size: 1—*δ*p = 20 nm; 2—*δ*p = 40 nm; 3—*δ*p = 80 nm.

It can be seen from Figure 5 that an increase in the volume fraction of hardening particles at the same size led to an increase in Orowan stresses. A similar effect was observed with an increase in the size of particles at the same volume fraction. This effect was associated with a decrease in the minimum distance between particles <sup>Λ</sup>p − *<sup>δ</sup>*p. Dislocations under the influence of the applied external influence bypass the particles, leaving rings around them ("Orowan rings"). If this process occurs when the amount of bend is small, then the required increase in energy will be less than in the case when the dislocation line must completely loop around the particles before it is released. As the distance between the particles decreases, the length of the dislocation line increases significantly. As a result, the efficiency of particles as obstacles to the movement of dislocations increases, and the Orowan stresses increase.

Let us proceed to the analysis of the effect of thermal stresses arising from the difference in the coefficients of thermal expansion, elastic modulus, and shear modulus of the matrix and the particle on the hardening of the material.

In Figure 6, the dependence of the radial stresses *σrr* in the particle and matrix on the radial coordinate is shown when the composite was heated from room temperature (293 K). A region of tensile stresses is formed due to the difference in elastic properties. The radial stresses in the particle material have constant values. In the matrix material, these stresses decrease quite sharply with increasing distance from the particle and matrix interface and become negligible at a distance on the order of 5 *<sup>R</sup>*p.

**Figure 6.** The distribution of the radial stresses *σrr* in the particle ScF3 and matrix for various values of the temperature difference: 1— Δ*T* = 10 K; 2—50 K; 3—100 K; 4—200 K.

Figure 7 shows the radial distribution of tangential stresses when the composite was heated from room temperature (293 K). In a particle, these stresses have constant positive values, which are associated with stretching due to the impact of the matrix. A sharp jump in tangential stress occurs at the boundary between the particle and the matrix. In this case, tangential stresses become compressive. The tangential stresses decrease sharply with an increase in the distance from the interface and become negligible at a distance on the order of 5 *<sup>R</sup>*p.

**Figure 7.** The distribution of the tangential stresses *σrr* in the particle and matrix for various values of the temperature difference: 1—Δ*T* = 10 K; 2—50 K; 3—100 K; 4—200 K.

Note that, due to spherical symmetry, tangential and meridional stresses are equal: *σϕϕ* = *σθθ*.

An analysis of the effect of the temperature difference Δ*T* on the stress state allows us to conclude that, with growth, there is an increase in stresses in the particle and the adjacent part of the matrix. In this case, to a first approximation, the magnitude of the arising stresses is proportional to Δ*T*.

Let us determine the maximum shear stresses *τ*max = 12 *<sup>σ</sup>rr* − *σϕϕ* arising in a dispersion-strengthened material as a result of the temperature change. Figure 8 shows the dependence of maximal shear stresses on the radial coordinate. In the hardened particle, *τ*max = 0. A sharp increase in *τ*max = 0 occurs at the interface between the particle and the matrix. Then, as the distance from the particle boundary increases, the value of the maximum shear stresses *τ*max = 0 monotonously decreases and becomes vanishingly small.

According to the condition of Saint Venant and Tresca, the plastic deformation begins when the maximal shear stress reaches half of the yield strength. The mathematical formulation of this condition has the following form [52]:

$$
\tau\_{\text{max}} = \frac{1}{2} \tau\_0. \tag{27}
$$

It is very critical for an engineer to locate and evaluate the maximum shear stress in a material in order to design the construction in such a way to resist failure.

On the basis of the dependences in Equations (11) and (12), it is possible to determine the magnitude of the temperature difference leading to the occurrence of plastic deformation.

$$
\Delta T\_{\rm pl} = \frac{3K\_{\rm P} + 4\mu\_{\rm m}}{9K\_{\rm P}\mu\_{\rm m}\left(\alpha\_{\rm P} - \alpha\_{\rm m}\right)}\,\tau\_{0}.\tag{28}
$$

According to Equation (22), the plastic deformation due to thermal stresses of the aluminum matrix with strengthening scandium fluoride particles occurs when the temperature difference is approximately equal to 72 K. Therefore, above thermal stress, consideration should be given when designing technological constructions for a considerable temperature range.

Figure 9 shows the dimensionless stress intensity *τ*∗ = *<sup>τ</sup>*int/*<sup>τ</sup>*scale in the vicinity of nine particles. The scale used here is the stress intensity at the particle–matrix interface.

$$
\pi\_{\rm scale} = 6\sqrt{3} \frac{K\_{\rm p} \mu\_{\rm m}}{3K\_{\rm p} + 4\mu\_{\rm m}} \left(\alpha\_{\rm p} - \alpha\_{\rm m}\right) \Delta T. \tag{29}
$$

**Figure 8.** The distribution of the maximal shear stresses σrr in the particle and matrix for various values of the temperature difference: 1—Δ*T* = 10 K; 2—50 K; 3—100 K; 4—200 K.

**Figure 9.** Dimensionless intensity of thermal stresses. Distance between hardening particles <sup>Λ</sup>p = 100 nm: (**a**) *<sup>R</sup>*p = 5 nm; (**b**) *Rp*= 20 nm.

The highest stress values were observed in particles. With distance from the particles, the stress intensity sharply decreased. With an increase in the size *<sup>R</sup>*p of particles at the same distance between them <sup>Λ</sup>p, the region in which thermal stresses were observed caused by the difference in the coefficients of thermal expansion of the matrix and the particle increased. The influence of neighboring particles at *<sup>R</sup>*p < 0.2Λp was insignificant. Thus, with a small volume fraction *f* = 2*R*p/Λp<sup>3</sup> < 0.06, the analysis of the stress–strain state of the dispersion-hardened material caused by the difference in thermal expansion coefficients could be carried out without taking up to the effects of the collective interaction of particles and the matrix.

Figure 10 shows the dependence of *σ*CTE on the volume fraction of hardening particles, calculated for different values of Δ*T*.

**Figure 10.** Dependence of thermal stresses *σ*CTE in the A356 alloy hardened by ScF3 particles on the volume fraction of hardening particles, calculated for various values of Δ*T*. Particle size *δ*p = 80 nm: 1—Δ*T* = 50 K; 2—Δ*T* = 100 K; 3—Δ*T* = 200 K.

Figure 10 shows that, with an increase in the volume fraction of hardening particles, an increase in thermal stresses occurred *σ*CTE. This fact was associated with an increase in the number of hardening particles in the alloy and, accordingly, with an increase in their contribution to the hardening of the material. When the material was heated or cooled, as a result of the mismatch between the thermal expansion coefficients and the elastic constants of the matrix and the particles, thermal stresses increased. The calculation results show that an increase in the volume fraction of the strengthening ScF3 particles in the A356 alloy from 0.1% to 5% at Δ*T* = 50 K led to an increase in *σ*CTE from 0.46 MPa to 11.8 MPa, and, at Δ*T* = 200 K, it led to an increase in *σ*CTE from 1.84 MPa to 46.538 MPa.

Figure 11 shows the dependence *σ*CTE on the volume fraction of particles in alloys, hardened by particles from different materials with a temperature change Δ*T* = 100 K.

The qualitative behavior of all curves in Figure 11 coincides with an increase in the volume fraction of the hardening phase, while an increase in stresses occurs *σ*CTE. However, the intensity of the stresses depends on the material of the hardening particles. Thus, with a volume fraction *f*p = 5% and a temperature difference Δ*T* = 100 in alloys hardened with ScF3 particles, the thermal stresses are *σ*CTE = 23.61 MPa; when hardened with Al2O3 particles, the magnitude of thermal stresses is *σ*CTE = 37.64 MPa; when hardened with TiO2, it is *σ*CTE = 40.53 MPa. A comparison of calculations performed according to Equation (25) with calculations performed according to Equation (7) shows that, for small volume fractions of the hardening phase, Equation (7) underestimates *σ*CTE, and, for large values of *f*p in predicting *σ*CTE using Equation (7), one can consider an average estimate for particles of different composition.

**Figure 11.** Dependence of thermal stresses *σ*CTE in the A356 alloy on the volume fraction of hardening particles; particle size *δ*p = 80 nm, Δ*T* = 100 K. Curves 1–3 correspond to the calculation using Equation (25), while curve 4 corresponds to the calculation using Equation (7): 1—ScF3; 2 —TiO2; 3—Al2O3.

The contributions of the described mechanisms to the yield strength of the composite calculated from Equations (3), (4), (6), and (25) are presented in Figure 12. The results demonstrate that theoretical values are very close to the experimental data. 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.

**Figure 12.** Dependence of different strengthening mechanisms on the mass fraction of the hardening particles: 1—Hall–Petch stresses; 2—Nardon and Prewo stresses; 3—Orowan stresses; 4—thermal stresses; 5—strengthening stresses *σ*Str = *σ*NP + *σ*HP + *σ*Or + *σ*CTE; -—Orowan stresses [53]; -— thermal stresses [53]; —strengthening stresses [53].

The predicted values of the total improvement in yield strength due to various strengthening mechanisms were 31 MPa and 61 MPa for the alloys with 2% and 5% mass fraction of Al2O3. Experimental results [53] showed that the strengthening stresses are equal to 30 MPa and 45 MPa, respectively. The difference between the experimental and theoretical results for the alloy with the 5% mass fraction of Al2O3 may be explained by the agglomeration of the hardening particles in the composite.

Figure 13 shows the dependence of the yield strength on the mass fraction of the second phase in alloys reinforced with particles of different materials with a change in temperature Δ *T* = 100. An increase in the mass fraction of particles led to an increase in the limiting shear stress for all considered cases. The smallest values *τ*0 were observed for alloys hardened with titanium oxide. At low values of the mass fraction (less than 2%), the highest values *τ*0 were achieved in alloys hardened with scandium fluoride; at large values of the mass fraction of the second phase, the highest values were achieved in alloys hardened with alumina.

**Figure 13.** Dependence of the yield stress on the mass fraction of the second phase in alloys strengthened by particles of different materials with temperature difference Δ*T* = 100 K: 1—SiC; 2 —B4C; 3—ScF3; 4—TiB2; 5—Al2O3; 6—TiO2.

#### *3.3. Verification of the Results*

<sup>V</sup><03D

Verification of the modeling results was carried out by comparison with experimental data. Table 4 presents the mechanical properties of the A356 alloy and A356-based composites.

Table 4 shows both experimental results and theoretical predictions of the yield strength *σ*Y for different temperatures. The yield strength of composites was greatly enhanced with an increase in reinforcement ratio for all tested conditions. A considerable improvement in yield strength of the composite was recorded with integration into the matrix of the 1% Al2O3 and 1% ScF3 disperse phase. The enhancement in strain hardening capacity of a composite at elevated temperature led to decreased variations in the yield strength of composites compared with aluminum alloy A356.


**Table 4.** Yield strength of A356 alloy and A356-based composites.

Upon comparing the experimental and theoretical values, one can see that, in general, the results of the predictions were fairly close to the experimental data. The good correlation between the experimental measurements and simulation results validates the correct methods and approaches for the simulation of processes of plastic deformation.
