## 2 Materials, Experiments, and Numerical Methods
### Finite Element Modeling of Hardness Measurement  
Indentation techniques are widely used for the assessment of hardness and hardening or softening phenomena in various materials. During the measurement, an indenter with a known geometry is pushed into the surface of the material under a known load, and the size of the indent is measured with a microscope, while the indentation depth is calculated from the displacement values of the indenter [20]. The most common geometries of the indenter are spherical, triangular, or rectangular pyramidal (for Rockwell, Berkovich, and Vickers methods) [20; 21]. In the case of the Brinell method, the measurement is done with a spherical indenter [20]. Given the geometric dimensions and the diameter of the indent, the hardness value can be calculated by using Equation (9) [22]:  
\[HB=\frac{2F}{\pi\Big{(}D^{2}-\sqrt{D^{2}-d^{2}}\Big{)}}\,, \tag{9}\]  
where \(HB\) is the Brinell hardness, \(F\) is the load force, \(D\) is the diameter of the indenter (sphere), and \(d\) is the diameter of the indentation, observed on the surface of the investigated material. The geometrical model used for the FEM simulation is shown in Figure 2a,b. Figure 1: Geometry used for the modeling of tensile test. Figure 2: (**a**) Geometric set-up used for the simulation of Brinell hardness; and (**b**) fine mesh elements in the vicinity of an indent. The diameter \(d\) can be assessed from the indentation depth \(h\) by Equation (10); however, the measured and calculated values can be somewhat different due to the relaxation of the material after deformation. In our study, we used the back-calculated values to determine both the surface pressure and hardness values. \[d=\sqrt{D^{2}-\left(D-2h\right)^{2}}, \tag{10}\]  
Knowing the load \(F\) and the indentation size \(d\), the value of the surface pressure \(P_{m}\) can be estimated as [20]:  
\[P_{m}=\frac{4F}{\pi d^{2}}, \tag{11}\]  
The pressure can be fitted with a function expressed by Equation (12) [20]:  
\[P_{m}=k\left(\frac{d}{D}\right)^{p}, \tag{12}\]  
where \(k\) is a proportionality factor, and \(p\) is an exponent, expressing the slope of the load curve. According to the results of [PERSON] and [PERSON] [20], the values of \(p\) in Equation (12) and \(n\) in Equation (1) are nearly identical. [PERSON] et al. [6] showed that there is a link between the hardness and tensile test parameters. The value of \(k\) can be determined by the model of [PERSON] [23]:  
\[P_{m}=3\mathrm{K}\left(0.1\frac{\mathrm{d}}{\mathrm{D}}\right)^{n}, \tag{13}\]  
In engineering practice, mainly finite element (FEM, [6,16]) and discrete element (DEM, [24,25]) methods are widely used for the simulation of hardness tests. In the present case, FEM was employed, and the calculation method is based on the minimization of the virtual plastic work according to Equation (14) [26]. \[\pi=\int_{V}\vec{\sigma}\cdot\dot{\tilde{x}}dV+\int_{V}\lambda\cdot\dot{ \epsilon}_{V}dV+\int_{S}\overline{F}_{i}\cdot\overline{v}_{i}dS, \tag{14}\]  
where \(\pi\) is the value of virtual work, \(\overline{\sigma}\) is the mean stress, \(\dot{\tilde{\epsilon}}\) is the mean strain rate, \(\lambda\) is the average stress value in the cell, \(\dot{\epsilon}_{V}\) is the volumetric strain rate, \(\overline{F}_{i}\) is the vectorial form of force acting on the surface of the element, and \(\overline{v}_{i}\) is the sliding velocity of the surface element. The simulations of Brinell hardness were carried out by means of DEFORM 2D\({}^{\text{\textcircled{\textcircled{\textcircled{\textcircled{\textcircled{ \textcircled{\  
lower rolls was different, while the roll diameters are identical. In the case of symmetric rolling, the velocities of the upper and lower rolls are typically equal, so the introduced shear component is concentrated within a very thin surface layer and is significantly lower compared to the case of asymmetric rolling [29].