Investigation of Stresses Induced Due to the Mismatch of the Coefﬁcients of Thermal Expansion of the Matrix and the Strengthening Particle in Aluminum-Based Composites

: An experimental and theoretical investigation of the strength properties of aluminum alloys strengthened by dispersed nanoparticles, as well as the determination of the signiﬁcance of various mechanisms responsible for the strengthening of the material, was carried out. Results of experimental investigation demonstrate that the hardening of aluminum alloy A356 by Al 2 O 3 and ScF 3 nanoparticles leads to an increase in the yield strength, ultimate tensile strength, and plasticity. Despite the similar size of Al 2 O 3 and ScF 3 nanoparticles, the physicomechanical properties of nanoparticles signiﬁcantly affect 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 (CTEs) of the matrix and strengthening particles on the basis of the fundamental principles of mechanics of a deformable solid and taking into account the elastic properties of not only the matrix, but also the particle. The forming of thermal stresses induced due to the mismatch of the coefﬁcients of thermal expansion of the matrix and the strengthening particle in aluminum-based composites was investigated. 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. 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. shear-forming dislocations, prismatic dislocation loops of vacancy and interstitial types, dislocation dipoles of vacancy and interstitial types, and interstitial atoms, monovacancies, and bivacancies. Author Contributions: Conceptualization, O.M., O.D. and A.K.; methodology, O.M. and I.Z.; soft-ware, 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. authors read


Introduction
Aluminum-based alloys are widely used in the automobile industry not only because they are easily castable in complex forms but also because of their excellent wear resistance, light weight, and good strength, which make them widely used in the manufacture of numerous automotive parts, such as engine blocks and wheels. Additionally, aluminumbased alloys alloy are used as a matrix for obtaining composites [1,2], which have an enhanced wear resistance, favorable mechanical properties, and higher strength, Young's modulus, and fatigue resistance [3,4] at room and elevated temperatures. Dispersionhardened alloys are characterized by isotropic mechanical properties, high plasticity, and high strength [5,6]. The physical theory of plasticity and strain hardening was developed by Orowan [7], Ashby [8], Hirsch, and Humphreys [9,10].
shear-forming dislocations, prismatic dislocation loops of vacancy and interstitial types, dislocation dipoles of vacancy and interstitial types, and interstitial atoms, monovacancies, and bivacancies.
An approach which combines methods of plasticity physics and mechanics of deformable solid was used in [24][25][26] to explore the limits of elastic and plastic resistance of the tube from dispersion-hardened aluminum alloy subjected to internal and external pressure.
It is known that, in composites, there is a large difference in the coefficients of thermal expansion between the matrix and the strengthening particles [27][28][29]. Thermally induced stresses could be generated as the result of a global heating and cooling of constructions, as well as local heating and cooling effects when a temperature gradient develops.
Therefore, the problem of thermally induced stresses due to a mismatch of the coefficients of thermal expansion between the matrix and strengthening particles has implications for the determination of mechanical properties of composite materials. The stress-strain state in plastic area around an oversized spherical particle which was introduced into a spherical hole in the matrix was analyzed in [30]. The analysis of stress-strain state must consider the contribution of thermally induced stresses in addition to the mechanically induced stresses [31]. These thermal stresses may be higher than the yield strength of the matrix and induce either debonding between particles and matrix or crack propagation leading to damage accumulation in the matrix [32,33].
The thermal stresses may be sufficient to generate new dislocations at the interfaces between the matrix and the reinforcement. In [29,34], the authors proposed that the increased strength observed in Al-SiC composites could be accounted for by a high dislocation density in the aluminum matrix, as observed in transmission electron microscopy (TEM). An increase in the density of newly created dislocations near reinforcement fibers was calculated in [24,35]. According to [36], when the composite is heated or cooled, misfit strains which are sufficient to generate dislocations occur because of differential thermal contraction at the Al-SiC interface. The investigations of the elastic and plastic stresses and strains generated about an inclusion, which has a coefficient of thermal expansion different from that of the matrix as a result of heating or cooling, was carried out in [34]. Furthermore, an in situ transmission electron microscopy investigation was undertaken into dislocation generation at the composite due to the differential thermal contraction. A simple model based on results of experimental investigation was developed to account for the relative dislocation density due to the differential thermal contraction.
Thus, it is very important to analyze the thermal stresses on the particle-matrix interface in order to predict a possible modification of the composite properties [37].
The purpose of this work was to experimentally and theoretically investigate the strength properties of aluminum alloys strengthened by nanoparticles, as well as to determine the significance of various mechanisms responsible for strengthening the material. Particular attention is paid to the development of a physicomathematical model of the occurrence of thermal stresses caused by a mismatch between the coefficients of thermal expansion (CTEs) of the matrix and strengthening particles, according to the fundamental principles of solid mechanics and taking into account, in contrast to previously existing models, the elastic properties of not only the matrix, but also the particle.
(a) (b) (c) As initial components for the synthesis of ScF3 nanoparticles, scandium oxide, hydrofluoric acid, and hydrochloric acid were used, which makes the process safer, without the use of poisonous fluorine, which is used in the direct synthesis of ScF3 from metallic scandium.
For the introduction of nanoparticles, a powder mixture of Al-5 wt.% nanoparticles was prepared. For deagglomeration and distribution of nanoparticles in the powder mixture, stearic acid was used as a surfactant. First, 200 mL of petroleum ether and 1.5 wt.% stearic acid were added to the powder mixture. Then, for 20 min, the powder mixture was mechanically mixed. After mechanical mixing, the aluminum-nanoparticle powder mixture was air-dried (90 °С) and sieved. The resulting powder mixture was introduced into the melt using ultrasonic treatment, which allows degassing and refining the melt, preventing deagglomeration, and evenly distributing nanoparticles in the volume [40].
Aluminum alloy A356 was melted in a graphite crucible with a total melt volume of 500 g. Ultrasonic treatment was carried out using a water-cooled magnetostrictive transducer (power-5 kW, frequency-17.5 kHz, RELTECH, Saint-Petersburg, Russia). The waveguide of the ultrasonic equipment was made in the form of a niobium alloy (VN2AE) cone, the working amplitude of which was 30 μm. At a melt temperature of 730 °C, ultrasonic degassing was carried out for 1 min. After degassing, an aluminum-nanoparticle master alloy (5 wt.%) was introduced into the melt with ultrasonic treatment. After dissolution of the master alloy in the melt, ultrasonic treatment lasted for 2 min (temperature, 730 °C). The melt obtained at a temperature of 710 °C was poured into a steel chill mold. The amount of ScF3 and Al2O3 nanoparticles in the A356 aluminum alloy ranged from 0 to 1 wt.%. The A356 alloy without particles was obtained with similar melt processing parameters. Mechanical tests were performed on a universal testing machine (Instron Europian Headquarters, High Wycombe, UK), Instron 3369, at a speed of 0.2 mm/min. The samples were tensioned at room temperature (25 °C), and the number of samples for each composition was at least four pieces. The structures of the obtained materials were investigated through optical microscopy, Olympus GX71 (Olympus Scientific Solutions Americas, Waltham, MA, USA). Samples were subjected to preliminary mechanical polishing, electrolytical etching, and anodization. The electrochemical oxidation of the metallographic specimen surface in a 5% solution of hydrofluoric acid (HBF4) at a voltage of 20 V and a current of 2 A was carried out to identify grain boundaries. As initial components for the synthesis of ScF 3 nanoparticles, scandium oxide, hydrofluoric acid, and hydrochloric acid were used, which makes the process safer, without the use of poisonous fluorine, which is used in the direct synthesis of ScF 3 from metallic scandium.
For the introduction of nanoparticles, a powder mixture of Al-5 wt.% nanoparticles was prepared. For deagglomeration and distribution of nanoparticles in the powder mixture, stearic acid was used as a surfactant. First, 200 mL of petroleum ether and 1.5 wt.% stearic acid were added to the powder mixture. Then, for 20 min, the powder mixture was mechanically mixed. After mechanical mixing, the aluminum-nanoparticle powder mixture was air-dried (90 • C) and sieved. The resulting powder mixture was introduced into the melt using ultrasonic treatment, which allows degassing and refining the melt, preventing deagglomeration, and evenly distributing nanoparticles in the volume [40].
Aluminum alloy A356 was melted in a graphite crucible with a total melt volume of 500 g. Ultrasonic treatment was carried out using a water-cooled magnetostrictive transducer (power-5 kW, frequency-17.5 kHz, RELTECH, Saint-Petersburg, Russia). The waveguide of the ultrasonic equipment was made in the form of a niobium alloy (VN2AE) cone, the working amplitude of which was 30 µm. At a melt temperature of 730 • C, ultrasonic degassing was carried out for 1 min. After degassing, an aluminum-nanoparticle master alloy (5 wt.%) was introduced into the melt with ultrasonic treatment. After dissolution of the master alloy in the melt, ultrasonic treatment lasted for 2 min (temperature, 730 • C). The melt obtained at a temperature of 710 • C was poured into a steel chill mold. The amount of ScF 3 and Al 2 O 3 nanoparticles in the A356 aluminum alloy ranged from 0 to 1 wt.%. The A356 alloy without particles was obtained with similar melt processing parameters. Mechanical tests were performed on a universal testing machine (Instron Europian Headquarters, High Wycombe, UK), Instron 3369, at a speed of 0.2 mm/min. The samples were tensioned at room temperature (25 • C), and the number of samples for each composition was at least four pieces. The structures of the obtained materials were investigated through optical microscopy, Olympus GX71 (Olympus Scientific Solutions Americas, Waltham, MA, USA). Samples were subjected to preliminary mechanical polishing, electrolytical etching, and anodization. The electrochemical oxidation of the metallographic specimen surface in a 5% solution of hydrofluoric acid (HBF4) at a voltage of 20 V and a current of 2 A was carried out to identify grain boundaries.

Strengthening Mechanism
It is generally accepted that nanocomposite hardening occurs due to the load transfer, Orowan mechanism, and CTE mismatch mechanism [41]. The total strengthening can be calculated as a superposition of the individual strengthening mechanisms. where σ Ym is the yield strength of the matrix, σ NP is the stress caused by the transfer of the load from the matrix to the particles, σ Or is the Orowan stress, σ CTE is the thermal stress arising from the difference in the coefficients of thermal expansion, elastic modulus, and shear modulus of the matrix and the particle. The load transfer mechanism is the most accepted strengthening mechanism. The transfer of the load from the soft matrix to the hard particles when an external load is applied helps to harden the material, as proposed by Nardon and Prewo [42].
where f p is the volume fraction of particles.
The grain size has an influence on the strength of the alloy, since the grain boundaries can impede the movement of dislocations. This is due to the different orientations of adjacent grains and to the high lattice disorder characteristic of these regions, which prevents the glide of dislocations in a continuous slip plane. The Hall-Petch equation relates stress to average grain size. There are empirical models available for predicting the yield strength change due to the change in the grain size in metal matrix composites by extending the Hall-Petch [43,44] relationship as follows: where σ HP is the change in tensile strength due to the contribution of the Hall-Petch law, k Y is the hardening coefficient (constant for each material), and d gr and d gr0 are the average grain sizes of the obtained alloys. This equation assumes that the Hall-Petch parameters k Y ≈ 68 MPa·(µm) 1/2 for Al alloys [45]. Particles perform a fundamental role in the final grain size of the matrix of composites as they can interact with grain boundaries and act as nuclei of crystallization, slowing or stopping grain growth. An increase in the volume fraction f p and a decrease in the particle diameter δ p lead to a finer-grained structure, which is theoretically modeled by the Zener-Smith equation [46].
where k z coefficient values are in the range 0.1 < k z < 1. Equation (5) is theoretically derived to describe the deceleration of migrating grain boundaries by particles. The Orowan mechanism is based on the interaction of nanoparticles with dislocations. Solid "noncutting" particles impede the movement of dislocations, which leads to bending of the dislocation line around the particles (Orowan loops) under the action of an external load [47]. The Orowan mechanism is very important in metal matrix composites, hardening by fine particles when the interparticle distance is not large. An increase in the composite yield strength by Orowan stress may be expressed as follows [47]: where G m is the matrix shear modulus, b is the Burgers vector of the matrix (b = 0.202 nm), and Λ p is the distance between particles. When a composite is subjected to heating or cooling, the difference in the coefficients of thermal expansion, elastic modulus, and shear modulus between the matrix and hardening particles produce internal stress state changes. The improvement in yield strength from thermal mismatch can be calculated using the equation suggested in [48][49][50].
Metals 2021, 11, 279 6 of 20 Equation (6) was obtained on the basis of the simplest geometric estimates under the assumption that the volume mismatch between the matrix and the reinforcing particles, arising from the difference in thermal expansion coefficients, leads to the appearance of geometrically necessary dislocations around the reinforcing particles. The disadvantages of the above approach include the fact that the stresses in Equation (6) do not depend on the elastic properties of the strengthening particles.
A more rigorous assessment of the thermal stresses arising from the difference in the coefficients of thermal expansion, elastic modulus, and shear modulus of the matrix and particle can be carried out using the methods of solid mechanics.

Mathematical Model of the Stresses Caused by the Thermal Expansion Mismatch between the Matrix and Strengthening Particles
Let us consider the stress-strain state that arises as a result of a change in the temperature of a spherical particle with a radius R p surrounded by a matrix. The coefficients of linear thermal expansion of materials are assumed to be different.
The equation of equilibrium of an elastic medium in spherical coordinates under the assumption of spherical symmetry can be written in the following form: The relationship between stresses σ ij and strains ε ij , expressing the generalized Hooke's law, under nonisothermal conditions, according to the von Neumann hypothesis, has the following form: where λ is the the Lame coefficient, G is the shear modulus, K is the bulk strain modulus, α is the coefficient of linear thermal expansion, ∆T = T test − T room is the temperature difference between the testing temperature T test and room temperature T room , and δ ij is the Kronecker symbol. The components of the strain tensor under spherical symmetry are Substituting into the equilibrium equation stresses expressed in terms of displacements, we obtain ∂ ∂r First, we integrate Equation (11) for a spherical particle for the following boundary conditions: As a result of simple calculations, we obtain the displacement field in the particle.
The stress field in a particle using the generalized Hooke's law has the following form: The displacement of the matrix material can be defined as The parameter U is determined from the condition of continuity of the radial stresses at the boundary between the particle and the matrix: σ rr | − = σ rr | + . As a result, we obtain Using the generalized Hooke's law, one can determine the stress field in the matrix.
We now turn from considering the stresses created by a single particle to the stresses caused by an ensemble of particles. The stress intensity characterizing the stress state of the material is determined by the following equation: Taking into account Equations (13)- (15) and (18)- (20), the intensity of stresses caused by the difference in the coefficients of thermal expansion of the strengthening particles and the matrix is equal to The average value of the thermal stress arising from the difference in the coefficients of thermal expansion, elastic modulus, and shear modulus of the matrix and the particle is determined using the traditional procedure of averaging over a spherical volume with a radius Λ p /2.
Thus, the average value of thermal stresses can be estimated as 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 ScF 3 nanoparticles. Silicon inclusions are present in the structure of all alloys (Figure 2a,b). At the same time, ScF 3 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 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. 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. The microstructures of the initial aluminum alloy A356, A356 + 0.2% ScF 3 , and A356 + 1% ScF 3 are shown in Figure 3. It can be seen that the introduction of 0.2% ScF 3 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 ScF 3 and Al 2 O 3 nanoparticles. The results were obtained at room temperature (T room = 293 K). The microstructures of the initial aluminum alloy A356, A356 + 0.2% ScF3, and A + 1% ScF3 are shown in Figure 3. It can be seen that the introduction of 0.2% ScF3 n particles into the aluminum alloy led to a decrease in the average grain size from 3 190 μm. An increase in the amount of nanoparticles in the A356 alloy to 1% led to crease in the grain size to 100 μm.      Figure 4 and Table 1 demonstrate that the hardening of aluminum alloy A356 by 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%.

Experimental Results
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 * , τ 1 * , and ε * for various volume fractions of scandium fluoride particles are presented in Table 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 −5 K −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. 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 d 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: Figure 5 shows the dependence of Orowan stresses on the volume fraction of hardening particles, calculated according to Equation (26). 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 Λ p − δ 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 σ rr 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 5R p .
Metals 2021, 11, x FOR PEER REVIEW 13 of 22 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 5Rp.  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 5Rp.  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 5R p . 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 5Rp.  Note that, due to spherical symmetry, tangential and meridional stresses are equal: An analysis of the effect of the temperature difference ∆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 ∆T.
Let us determine the maximum shear stresses τ max = 1 2 σ rr − σ ϕϕ 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]: 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.
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 * = τ int /τ scale in the vicinity of nine particles. The scale used here is the stress intensity at the particle-matrix interface.
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]: 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.

( )
According to Equation (22), the plastic deformation due to thermal stresses of the aluminum matrix with strengthening scandium fluoride particles occurs when the tem-  The highest stress values were observed in particles. With distance from the particles, the stress intensity sharply decreased. With an increase in the size R p of particles at the same distance between them Λ 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 R p < 0.2Λ p was insignificant.
Thus, with a small volume fraction f = 2R p /Λ p 3 < 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 ∆T.  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 ScF 3 particles in the A356 alloy from 0.1% to 5% at ∆T = 50 K led to an increase in σ CTE from 0.46 MPa to 11.8 MPa, and, at ∆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 ∆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 ∆T = 100 in alloys hardened with ScF 3 particles, the thermal stresses are σ CTE = 23.61 MPa; when hardened with Al 2 O 3 particles, the magnitude of thermal stresses is σ CTE = 37.64 MPa; when hardened with TiO 2 , 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. Metals 2021, 11, x FOR PEER REVIEW 2 of 2 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, ∆T = 100 K. Curves 1-3 correspond to the calculation using Equation (25), while curve 4 corresponds to the calculation using Equation (7) ; -Orowan stresses [53]; -thermal stresses [53]; -strengthening stresses [53]. 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 100 T = . An increase in the mass fraction of particles led to an increase in the limiting shear stress for all considered cases. The smallest values 0  were observed for alloys hardened with titanium oxide. At low values of the mass fraction (less than 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 Al 2 O 3 . 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 Al 2 O 3 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 ∆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.

Verification of the Results
Verification of the modeling results was carried out by comparison with experimental data. Table 4 presents the mechanical properties of the А356 alloy and А356-based composites. Table 4 shows both experimental results and theoretical predictions of the yield strength 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 .

Verification of the Results
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% Al 2 O 3 and 1% ScF 3 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. 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.

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 Al 2 O 3 and ScF 3 nanoparticles led to an increase in the yield strength, ultimate tensile strength, and plasticity. The introduction of 0.2 wt.% Al 2 O 3 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 Al 2 O 3 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.% ScF 3 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 ScF 3 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 Al 2 O 3 and ScF 3 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.