*4.2. Analysis of Results*

According to the measurement methodology presented in Section 3, the analysis covers the results obtained in two directions perpendicular to each other and their absolute value. The distributions of determination coefficient *R*<sup>2</sup> of the linear correlation between *HV5* hardness and the total number of events *NoETOT* and of resolution coefficient *RECOF* as a function of threshold voltage *Ug* for S235 steel are presented in Figure 9a–c for the parallel direction, the normal direction, and the absolute value, respectively. In all three charts (Figure 9a–c), the curves illustrating the history of determination coefficient *R*2, after the initial phase of a clear increase (in the range of *Ug* from 0 to about 0.4 V), look quite similar; for higher voltage values, there are no essential changes. The distributions of coefficient *RECOF* are characterized by a global maximum falling on *Ug* values ranging from 0.5 to 0.6 V. These maxima decide the selection of the discrimination voltage for which the linear correlations between *NoETOT* and HV5 are developed and presented in Figure 10a for the parallel direction, in Figure 10b for the normal direction, and in Figure 10c for the absolute value. The values of determination coefficient *R*<sup>2</sup> differ considerably for the two directions and total about 0.7 for the parallel direction (which is too low for practical use) and about 0.92 for the normal direction. However, in the analyzed case, the impact of the parallel direction of magnetization on the value of the linear correlation determination coefficient *R*<sup>2</sup> for the absolute value (*R*<sup>2</sup> = 0.914) is very slight, and the correlation can be used in practice.

**Figure 9.** Averaged *NoETOT* distributions and distributions of coefficients *RECOF* and *R*<sup>2</sup> as a function of threshold voltage *Ug*—S235 steel (**a**) parallel direction; (**b**) normal direction; (**c**) absolute value.

**Figure 10.** Linear correlations between *NoETOT* and *HV5* hardness—S235 steel (**a**) parallel direction, *Ug* = 0.6 V; (**b**) normal direction, *Ug* = 0.5 V; (**c**) absolute value, *Ug* = 0.5 V.

The distributions of determination coefficient *R*<sup>2</sup> of the linear correlation between *HV5* hardness and the total number of events *NoETOT* and of resolution coefficient *RECOF* as a function of threshold voltage *Ug* for DC01 steel are presented in Figure 11a–c for the parallel direction, the normal direction, and the absolute value, respectively. In all three charts below, the curves illustrating the history of

determination coefficient *R*2, after the initial phase of a clear increase (in the range of *Ug* from 0 to about 0.5 V), look quite similar; for higher voltage values, there are no essential changes. A characteristic feature of the *RECOF.* distributions is that the first local maximum falls within the range of voltage *Ug* variability where the determination coefficient *R*<sup>2</sup> takes low values. It is only the values of the second local maximum that decide the selection of the discrimination voltage for which the linear correlations between *NoETOT* and *HV5* are developed and presented in Figure 12a for the parallel direction, in Figure 12b for the normal direction, and in Figure 12c for the absolute value.

**Figure 11.** Averaged *NoETOT* distributions and distributions of coefficients *RECOF.* and *R*<sup>2</sup> as a function of threshold voltage *Ug*—DC01 steel (**a**) parallel direction; (**b**) normal direction; (**c**) absolute value

**Figure 12.** Linear correlations between *NoETOT* and HV5 hardness—DC01 steel (**a**) parallel direction, *Ug* = 1.7 V; (**b**) normal direction, *Ug* = 1.1 V; (**c**) absolute value, *Ug* = 1.1 V.

The values of determination coefficient *R*<sup>2</sup> for the two directions of magnetization vary from about 0.73 to about 0.8, which is too low to use in practice. Finding the absolute value improves the situation only slightly, raising coefficient *R*<sup>2</sup> for optimal discrimination voltage *Ug* to about 0.8, which is still just on the limit of possible practical use.

In the presented results of the testing, higher *NoETOT* values correspond to higher hardness (higher degree of plastic strain), which is also observed for the values of other properties of the Barkhausen noise (Figures 7 and 8) [7,16,17]. However, this is not a permanent trend that occurs for every material. S235JRG2 steel is investigated in [29], and it is found that a rise in the plastic strain degree involves a decrease in the BN *URMS* value measured in the load direction and an increase in the value determined in the normal direction. It is also found that the main reason for the observed changes is the number of dislocations causing non-homogeneous microdeformations. The stress field around dislocations interacts with all suitably oriented displacements of domain walls. If the number of displacements is reduced, the values of the BN properties are reduced, too. The authors of [29] state that this makes it possible to account for the drop in the BN property value measured in the load direction, but it still

cannot explain the rise of the value in the normal direction. A similar trend of changes is observed in [13]. In [30], Fe–2%Si alloy is investigated using an enveloping coil, which allows the assumption that the magnetization direction is the same as the direction of the load. It is found that a rise in plastic strain involves a rise in the integrals of *Ub* voltage. This trend is opposite to the one observed in [29]. The results of the testing of plastically deformed Armco iron presented in [31] show a non-uniform trend of changes in *Ub* voltage integrals. After an initial increase with a rise in plastic strain, if the strain value exceeds about 10%, a further rise in strain involves a decrease in the values. Another essential factor is additionally found in [32]: the impact of the specimen loading history (the method of achieving plastic strain). In the case of specimens for which plastic strain was achieved in a single loading cycle, the BN properties decreased with a rise in plastic strain. For specimens where subsequent plastic strain values were achieved gradually to the value of about 1%, the values of the BN properties increased. A further rise in strain caused a gradual drop in the values of the BN properties. However, the drop never reached the level measured for specimens where strain was achieved in a single loading cycle.

The analysis of the phenomena of interaction between a change in the degree of plastic strain and dislocation density is a complex process that requires many more microscopic and macroscopic studies on the changes that arise in the domain structure of polycrystalline ferromagnetic materials due to a complex state of stresses and strains.
