“…The anticipated model affords, in addition, an applicable foundation for a detailed analysis of the effects of the scanning form design constraints on the specifications of the hardened zone. However, many complications were documented as the major limitations of the use of the thermal analysis, such as the complexities of heat removal from the adjacent faces of the moving instrumental parts, and the shift of negative stability of the surfaces of the various coating materials, and hence favorite the mathematical modeling approach that adapts criterial equations for the determination of the boundary conditions on the understudy surfaces, instead (Jos é Manuel et al, 2017, Makarova et al, 2016. This research aims to design an outline frame for a mathematical model to explain, evaluate, and manage the nanostructure outer surfaces of the coating constituents as a consequential result of the thermal hardening method, given the critical phase of the negative stability`s lowest temperature for the working machine part.…”