# Complexity Modeling of Steel-Laser-Hardened Surface Microstructures
## 1 Introduction  
Laser thermal hardening is a recent technique that has been effectively used to harden the surface layers of metallic materials [1; 2; 3]. During the thermal hardening of metals and alloys, the heating of the required area of the irradiation surface causes instant cooling due to the removal of heat from the internal cold layers of the metal. The reduction in the additional processing volume of the material, the absence of a separate cooling process, and the ability to minimize inhomogeneous three-dimensional parts and complex geometries are the main advantages of this process over other common techniques, making it suitable for numerous industrial applications [4; 5; 6; 7]. [PERSON] et al. [8] investigated laser hardening, identifying its advantages over conventional hardening techniques. Recent advances in laser technologies have given rise to new research insights [9; 10; 11]. The increased capacity of data processing in real time has allowed the updating of the laser radiation parameters during the thermal hardening phase, significantly improving the robotic systems above all as to the positioning and speed of the pieces in industrial applications. Indeed, the automatic and proportional adjustment of the irradiation power avoids unwanted overheating, which would affect the results of the thermal hardening process. The hardening and tempering of steel tools, on the one hand, improve the mechanical properties of the microstructure; on the other hand, they determine the final residual stressesand the ultimate dimensions of the piece [12; 13]. Some microstructural investigations performed on hardened materials have revealed a close link between microstructure, micromagnetic parameters, and wear rate [14; 15; 16]. Due to the physical interaction of the microstructure with the magnetic field, significant changes in the microstructure have been observed. These were associated with wear, such as phase transformation and development of residual stresses. Finally, recent research has correlated internal stresses and distortions with the thermo-mechanical-metallurgical interactions that occur during heat treatment [17; 18; 19]. By analyzing the influence that some robotic process parameters have on the geometry of the final microstructures and on the surface roughness of heat-treated materials, this research work aims to improve the laser processing of these materials through the application of robotic technologies. Special attention was focused on robot laser hardening (RLH), which makes it possible to produce many welding configurations accurately by continuously varying both the angles relative to the shaping of the laser beam with the piece and the surface temperature during heat treatments. In the present study, it has been assumed that the microstructure of laser-hardened materials depends on complex structural geometry and the modeling of this complexity can be performed through fractal geometry [20]. Recent studies show that fractal geometry can be successfully applied to describe fracture surfaces on various engineering materials (concrete, steel, etc. ), molybdenum, and rocks, but also on composite materials in different length scale ranges and even in plasmonic metamaterials [21]. Numerous independent process parameters, such as laser power, laser beam speed and temperature, laser beam impact angles, etc., determine the geometric complexity of the microstructure. Since each of these parameters is related in a different way to the surface properties of the material (in particular to hardness, wear resistance, and residual stresses), it is difficult to identify their effects on complexity. In this regard, some authors have shown in previous research work that hardness increases with temperature at low speed [22]. However, they have not found a direct relationship between temperature and complexity.