## 4 Modeling of the Hardening of an Aluminum Metal Matrix Composite Reinforced with CNTs
This article proposes a model of a three-phase compound cylinder with new boundary conditions. This model studies the interaction of CNT edge dislocations with a coating containing the interface effect. [PERSON] et al. [142], using Tsai-Halpin's equation, showed that with an increase in the concentration of CNTs in MMC, the elastic modulus of composites increases. The tensile strength of composites is significantly increased compared to the matrix alloy, which is explained by the mechanisms of load transfer, grain refinement, and misfit dislocation due to the addition of CNTs. Modifications are proposed to consider the mechanisms of hardening according to Orowan due to the redistribution of the load and the refinement of the grain of the matrix material, as well as the discrepancy between the thermal expansion coefficients. The simulation results are in good agreement with the experimental ones. [PERSON] et al. [143] proposed a dislocation-based model to investigate the synergistic effect arising from the grain size of the matrix and the size of the reinforcement on the strengthening mechanisms of nano/ultrafine-grained CNT-reinforced MMCs. The model based on the three-phase composite material approach is suitable for studying the elastoplastic stress-strain relationships of MMCs. The overall flow strength of nano/ultrafine grain MMC depends on the grain size of the matrix or CNT, as well as the volume fraction and microstructure of the reinforcement. It is assumed that dislocations formed by a mismatch in the thermal expansion coefficient are redistributed only in a strongly dislocated volume that surrounds a separate inclusion. This zone, as a matrix phase in the calculation model, has a higher strength than the rest of the metal matrix. Therefore, the size of the zone is an additional scale of the microstructure and requires more attention. All these results are explained by the strengthening effect that is caused by thermal expansion mismatch weakening as the grain size of the matrix decreases, which leads to the less pronounced role of reinforcement in the overall response of MMC. The model can provide some insight into the strengthening mechanisms of CNT-reinforced nano/ultrafine-grained MMCs. [PERSON] et al. [144] studied the influence of the length-to-diameter ratio of CNTs on hardening aluminum MMCs. To this end, aluminum samples reinforced with CNTs with different aspect ratios were fabricated by three different powder metallurgy methods. The microstructural study showed that CNTs were uniformly dispersed in materials ranging in aspect ratios from 6.5 to 55. Tensile results showed that CNTs exhibited a strong strengthening effect in composites regardless of their aspect ratios. Post-load studies and quantitative analysis demonstrated a transient hardening mechanism for CNTs that is closely related to the aspect ratio or length of the CNTs. The origin of such a transition has been studied from the point of view of the interaction of dislocation with a CNT under load. These data may provide new insights into the enhancement behavior of CNT-reinforced MMCs [145]. [PERSON] et al. [146] studied the properties of the matrix composite AZ61 reinforced with microscale SiC and CNTs. Ultra-high strength and good ductility (tensile strength: 412 MPa, yield strength: 345 MPa and elongation: 8.0%) were achieved in the AZ61-5 SiC-0.5 CNTs composite. The greatly improved composite strength is mainly due to the synergistic effects of grain refinement, load transfer, and increased dislocation density in the matrix due to thermal mismatch, while good ductility can mostly be attributed to a weakened base texture. It is noted that the formation of clusters and damage to CNTs during processing can reduce the strengthening effect of CNTs. The distribution of a part of the CNTs along the grain boundaries leads to a weakening of the strengthening effect of the Orowan mechanisms.