## 4 Modeling of the Hardening of an Aluminum Metal Matrix Composite Reinforced with CNTs  
The observed hardening of metal matrix composites is the result of the interaction of several hardening mechanisms:  
1. Due to the effective transfer of the load between the matrix and the reinforcing elements;
2. Due to the internal thermal stresses caused by a mismatch in the coefficients of thermal expansion between the matrix and carbide particles; \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Characteristic** & **Graphite** & **Carbon Fibers** & **MWCNTs** & **SWCNTs** & **Steel** \\ \hline Tensile strength, GPa & 100 & 3–7 & 300–600 & 300–1500 & 0.4 \\ Modulus of elasticity, GPa & 1000 & 200–800 & 500–1000 & 1000–5000 & 2000 \\ Maximum tensile elongation, \% & 10 & 1–3 & 20–40 & 20–40 & 26 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical properties of carbon materials [84,130]. Figure 8: SEM images of the worn surface of pure aluminum alloy and CNT-reinforced aluminum alloy and wear rate data under a load of 2 kg at 500 rpm [134]. By grinding the grain of the matrix (Hall-Petch law);  
Due to a mismatch between the elastic moduli of the matrix materials and the inclusion;  
Hardening caused by the dispersed phase, according to [PERSON];  
Hardening caused by the appearance of an interfacial layer between the reinforcing particle and the matrix. The mechanisms that are implemented in each particular case are determined both by the geometry, size, and concentration of particles, as well as the conditions for introducing reinforcing particles into the matrix. The existing methods for modeling the hardening of metal matrix composites can be conditionally divided into two groups:  
Micromechanical modeling based on empirical or semi-empirical models of hardening mechanisms. Computer modeling, including modeling by molecular dynamics methods, finite element modeling, and multiscale modeling, representing some combination of molecular dynamics and finite element modeling. The main problems affecting metal matrix composite hardening are the uneven particle distribution in the bulk, the aggregation of reinforcing particles, and the formation of defects at the particle-matrix interface. One of the important tasks is to prevent these problems [103; 136; 137]. Unlike macroscale descriptions of hardening mechanisms such as the Hall-Petch or Orowan ratios, where the differential increase in strength associated with a single hardening mechanism is added to the base strength of the material, the microscale considers dislocations to interact directly with various obstacles independently and uses the absolute critical allowed shear stress for each type of obstacle. [PERSON] et al. [62] reviewed the theoretical and experimental foundations related to bulk MMCs and the main results achieved in this area. The structural properties and mechanical characteristics caused by the addition of nanoparticles and nanotubes to base metals are presented, and modern methods for the synthesis of MMCs are described. [PERSON] et al. [138] studied the mechanisms of MMCs hardening with CNTs-reinforced Al matrix, including load transfer, dispersion strengthening, dislocation formation caused by thermal mismatch, fine-grained hardening, etc., have been studied. [PERSON] et al. [139] developed a microstructure-based work hardening model by modifying and combining the Cox-Mecking approach to dislocation density evolution, reverse stress models for grain boundaries, and reverse stress models for second phases. The expected strain hardening characteristics of the composites were successfully matched with the experimental results. It has been shown that the effects of CNT reinforcement and grain refinement can significantly increase the strength of composites. Authors in [140; 141] discussed the formation of coated CNTs (there is an amorphous interfacial surface between the reinforcement and the metal matrix), which have a great influence on the force acting on dislocations.