## 5 Application of the MVD Algorithm and Discussion  
This section addresses the challenge of generating multilinear curves to characterize surface-hardened steel, as discussed in the introduction. When simulating surface-hardened components in finite element analysis (FEA), the aforementioned multilayer approach, which segments the geometry into layers and applies homogeneous properties to each layer, proves effective [12]. To illustrate the generation of multilinear curves for fatigue analysis of surface-hardened components using the MVD algorithm, AISI 4140 or 42 CrMo4 low-alloy steel is chosen due to its widespread use in machinery and automotive engine applications [11]. After undergoing heat treatment, this steel exhibits desirable properties such as high fatigue strength and wear resistance. For the purpose of illustrating the generation of multiple multilinear curves, the typical hardness profile of surface-hardened components is discretized into five layers, each with uniform hardness distribution (Figure 2) and represented with one multilinear stress-strain curve. To generate a distinct multilinear curve for each layer, equations correlating hardness measurements taken at various cross-section points with Ramberg-Osgood cyclic material parameters are utilized. In the literature, a strong correlation between the cyclic strength coefficient \(K^{\prime}\) and hardness is identified, while a good correlation between hardness and the cyclic strain-hardening exponent \(n^{\prime}\) is absent [15; 36]. Consequently, a regression equation correlating cyclic strength coefficient \(K^{\prime}\) with Brinell hardness \(HB\) of 42 CrMo4 steel is employed, while the strain-hardening exponent \(n^{\prime}\) is taken as a constant value [15]:  
\[K^{\prime}=0.009 HB^{2}+0.1173 HB+376.75,\] \[n^{\prime}=0.1087. \tag{11}\]  
Arbitrarily selected hardness values ranging from 550 HB (surface) to 420 HB (core), as observed in the distribution shown in Figure 2, are employed to represent a surface-hardened component. From these hardness values, using Equation (8), the cyclic strength coefficient \(K^{\prime}\) is calculated, and the cyclic strain-hardening exponent \(n^{\prime}\) is constant. The conventional starting point of the plastic strain, \(\epsilon_{\text{p,start}}=0.002\), is chosen, from which the yield stress \(R_{\text{p}0.2}\) is determined. The proposed MVD algorithm, characterized by a predetermined stress deviation value \(SD_{\text{lim}}\), ensures that each layer's multilinear curve adheres to the specified stress deviation threshold. This way, the accuracy of stress amplitudes for any observed strain amplitude remains consistent regardless of the geometry's position. Figure 10 provides multiple multilinear curves constructed for the selected hardness values with the maximum predefined stress deviation \(SD_{\text{lim}}\) of 1% and maximum plastic strain amplitude of 0.015 mm/mm (1.5%). Subsequent calculation of the final stress deviation from the original Ramberg-Osgood curve for each multilinear curve reveals a consistent \(SD_{\text{fin}}\) of 0.82% across all curves, confirming the adherence to the specified stress deviation threshold. Each multilinear curve comprises five points, discretized at equal strain intervals. When the strain-hardening exponent is not held constant, the resulting multilinear curves may display varying slopes and potentially overlap. Consequently, not all curves will share the same optimal strain values. This variability can pose challenges, particularly in FEA, wherein convergence issues may arise due to differing intervals of strain values between multilinear curves [37]. When modeling temperature-dependent materials, the same challenge arises wherein each temperature level requires its own material curve, similar to the situation with varying hardness levels, wherein each hardness level necessitates its own material curve [37].