**1. 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].

**Citation:** Matvienko, O.; Daneyko, O.; Kovalevskaya, T.; Khrustalyov, A.; Zhukov, I.; Vorozhtsov, A. 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. *Metals* **2021**, *11*, 279. https://doi.org/ 10.3390/met11020279

Received: 9 December 2020 Accepted: 29 December 2020 Published: 5 February 2021

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According to continuum linear elastic dislocation theory, Orowan [7] proposed that the interaction between dislocation and hardening particles leads to the formation of shear loops around the particles. This dislocation interaction scenario leads to what is known as Orowan strain hardening. Humphreys and Hirsch [11] proposed an alternative mechanism in which, if all dislocations are edge-oriented, each dislocation/particle interaction leads to two pairs of prismatic loops, one on each side of the particle. This multi-dislocation interaction, which involves the cross-slip of screw components of dislocations, is known as Hirsch looping. The material hardening depends on the types of particles, their volume fraction and homogeneous distribution in the matrix, and the interaction between matrix and strengthening particles.

The stress–strain curves at room temperature and at 400 ◦C for various types of dispersion-hardened aluminum alloys were determined in [12]. The results of investigations demonstrated that the effect of particles on the flow stress is in agreemen<sup>t</sup> with Orowan's theory [7]. A model to describe the creep of FCC metals has been used [13] for alloys and composites strengthened by the dispersion of nanosized particles. According to investigation results, the volume fraction and size of the nanosized particulate assumes a key role in determining the creep response of these materials. The results of investigations of the mechanical properties of aluminum alloys indicate that the hardness, yield strength, and ultimate tensile strength of the Al–2 vol.% C60 nanocomposites are higher than those of the monolithic Al samples.

The investigation in [14] dealt with aspects of the deformation behavior of dispersionhardened aluminum materials during uniaxial creep in a broad temperature range. Results of the investigation demonstrated that the threshold stress decreases considerably with increasing temperature.

The authors of [15] focused on the examination of materials of submicron matrix grain size strengthened with nanosized particles. The results obtained within the scope of this work showed the possibility for significant improvement of the microcomposite properties.

The results of the studies of the effect of nanodiamond addition on the physicomechanical properties of the drilling tool metal matrix were discussed in [16]. The optimal concentration of nanodiamond powder additives in the composition of a hard alloy mixture was defined. The addition of 1–2 wt.% nanodispersed diamonds was shown to contribute to binding oxygen in the thermo-oxidation of diamond in the course of the tool manufacture, which makes it possible to reduce the number of pores in the matrix material structure and to increase its density by 1.7% and hardness by 24.3%.

In [17], the effect of the aspect ratio of carbon nanotubes (CNTs) on strengthening aluminum metal matrix composites (Al MMCs) was investigated. The tensile results showed that the CNTs exhibited a strong strengthening effect in the composites regardless of their aspect ratios. However, the post-loading examination and quantitative analysis indicated that there was a strengthening mechanism transition for CNTs, which was closely associated with the aspect ratio or length of CNTs. The origin of such a transition was explored from the viewpoint of the dislocation–CNT interaction under loading.

The effect of Al4C3 formation on the tensile properties of multiwalled carbon nanotubes which reinforce an Al metal matrix was investigated in [18]. It was concluded that the formation of Al4C3 could effectively enhance the load transfer in composites.

Mathematical modeling of composites hardened by nanoparticles with a metal matrix was considered in [19]. The authors covered in detail the effect of the bulk content, distribution, and morphology of nanoparticles per average path length of a dislocation in the alloy (L). The authors revealed that the uniform distribution of grains in the matrix gave the smallest value of L. An increase in the bulk content of nanoparticles from 1 to 7 vol.% decreased the average path length of the dislocation by 50%, and, if their fraction exceeded 7 vol.%, the value of L was almost invariable.

In [20–23], a physical model of plastic deformation was suggested. The modeling of plastic deformation and strain hardening is based on the concept that the plastic slip in FCC alloys with incoherent nanoparticles leads to the formation of a defect structure with 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 [24–26] 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 [27–29]. 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.
