## 2 Materials and Methods  
S235 steel samples (hardness of 182 \(\pm\) 11 HV0.5, tensile yield strength 305 MPa, ultimate tensile strength 390 MPa, and elongation at break 41%, hot-rolled) were used as-received [12]. The matrix of the S235 steel is composed of predominant ferrite and minor pearlite islands. Chemical composition, and metallographic images of the microstructure, shape, and size of the samples can be found in a previous study [12]. The samples were deformed in tension by an Instron 5985 with the deformation direction parallel with the rolling direction (RD). The transverse direction is referred to as TD. The samples were investigated post-mortem in the unloaded state. Samples were subjected to the elastic strain \(\epsilon\) = 2.5%, homogenous plastic straining \(\epsilon\) = 5, 10, 15, 20, and 25%, and finally the non-homogenous plastic straining (necking) \(\epsilon\) = 30, 35, and 40% (for the stress-strain curves, check [12]).  
The true interpretation of MBN signals is associated with the investigation of the interrelation between the MBN, residual stresses, and microstructure. For this reason, the non-destructive XRD and EBSD measurements were carried out as well (in the normal direction plane in the sample's centre with respect to both length and width). The samples for EBSD observations and XRD measurements were cut from the central part of the deformed samples and electrochemically polished using Struers Lectropol and AC2 solution.  
Crystallite size \(t\) (the size of coherently diffracted domains) and micro-deformation were determined from the XRD patterns obtained by an X'Pert PRO diffractometer using the Co-K\(\alpha\) radiation, and the subsequent Rietveld refinement performed in MStruct software [23; 24; 25; 26] (used for the fitting of the measured XRD patterns). The effective penetration depth of the XRD measurements was in this case 1-5 \(\upmu\)m. Using the XRD data, crystallite size \(t\) was calculated via the Scherrer formula (Equation (1))  
\[t=(K.\lambda)/(B.\cos\theta) \tag{1}\]  
where \(K\), \(\lambda\), \(B\), \(\theta\) are the shape factor, X-ray wavelength, peak broadening and half of the diffraction angle, and the Bragg angle, respectively.  
EBSD analysis was performed by a Zeiss Auriga Compact equipped with an EDAX EBSD camera. An area of 400 \(\times\) 400 \(\upmu\)m was selected for the EBSD measurement with a step size of 0.4 \(\upmu\)m. The data were analysed by OIM TSL 7.3. The raw data were partially cleaned using one step of confidence index (CI) standardisation, and one step of grain dilatation, and only points with a confidence index of CI \(>\) 0.1 were used.  
MBN was measured by the use of a RollScan 350. The angular dependence of MBN against RD was analysed with the step 22.5\({}^{\circ}\) (RD corresponds to the angle 0\({}^{\circ}\) whereas TD 90\({}^{\circ}\)). The angular dependence was investigated by the samples' rotation (via the self-made positioning system) and usage of the bipolar surface sensor S1-18-12-01 (produced by StressTech, further details about the shape of the sensor in [9]). MBN signals were processed using MicroScan 600 software (nominal mag. voltage 5V and frequency 175 Hz, frequency range of MBN emission from 20 up to 700 kHz). MBN asymmetry was also investigated as a function of magnetising voltage in the range 2-16 V. MBN represents the effective (rms) value of the MBN emission. The MBN envelopes and the corresponding _PP_ (Peak Position) were extracted from the raw MBN signal. _PP_ refers to the strength of the magnetic field in which the MBN envelope attains the maximum. Note that the samples were fully demagnetised before the tensile test, and MBN signals in this condition did not exhibit asymmetry. The significant asymmetry of MBN which occurred after the tensile test also prevailed in the samples after subsequent demagnetisation.