A Modification of Magnetic Adaptive Testing: Progressive Method for Nondestructive Inspection of Microstructural Changes in Ferromagnetic Constructional Materials

Abstract

Magnetic adaptive testing (MAT), thanks to its relative simplicity from the point of view of both hardware and software, appears to be very promising for non-destructive analysis of various ferromagnetic constructional materials used in many industrial applications. In order to make the inspection of tested objects faster, even more straightforward and the processing of data easier, we concentrated our effort both to improve the experimental procedure, based on specific way of the measurement of magnetization curves at piece-wise linear (triangular) exciting field with a constant field rate of change (i.e., slope) as well as to create a universal software tool for MAT data analysis. On contrary to the original implementation of MAT, the hysteresis loops are measured with decreasing maximum field values, starting at sample saturation region; therefore, time consuming sample demagnetization can be skipped completely. In addition, an advanced tool for the processing of experimentally obtained magnetization curves is presented. Using this application allows to find proper non-traditional magnetic parameters (e.g., the differential permeability) being the most sensitive to various types of industrial load (e.g., thermal, mechanical and/or neutron irradiation) and, at the same time, sufficiently correlated with other, traditional magnetic as well as non-magnetic parameters used routinely for the assessment of possible structural changes associated with applied, often long-term, load even on a microscopic scale, prior to any damage manifests on a macroscopic, visible level. The software capabilities were demonstrated on the data representing the material, whose response to acting thermal load is being difficult to analyze, since the dependence of observed parameters (differential permeability) upon defined artificial ageing was rather complicated. Nevertheless, the sensitivity of differential permeability to such a load was found being more than 8 times larger than in case of traditional hysteretic parameter, namely the remanent flux density while the correlation between them was high.

1 Motivation

In order to increase the quality of production as well as the safety of operation in industrial practice, a need for various methods of, preferably nondestructive, structural health monitoring and testing of manufactured components and constructional units permanently grows. Along with older, already well-established and recognized methods, based, e.g., on electromagnetic phenomena (Barkhausen noise analysis, eddy current testing, etc.), also other new, more or less sophisticated, methods for NDE arose in the past years. Recently, an advanced method for non-destructive characterization of microstructural changes in ferromagnetic constructional materials was introduced. This method was named by its creator, Dr. Ivan Tomáš, as Magnetic Adaptive Testing (MAT) [1,2,3]. The comparison of selected nonstandard NDE techniques (including MAT), employed and being under permanent development at our institute for many years, was summarized, e.g., in [4].

One of major efforts in this work was the development of a universal software utility for the purposes of non-destructive analysis of microstructural changes by means of MAT, namely, the routine analysis of experimental data. The approach described later takes advantages of an experimental equipment with recently developed hardware components along with proper hardware/software based feedback used for precise magnetic quantities waveform control as well as the best measurement practice, described in details in [5].

A detailed explanation of MAT principle, needed for describing what the presented software is about and how the known core ideas of MAT are implemented, is given further. The data analysis has newly been enhanced for testing obtained data trends for similarity with the dependencies of conventional parameters, reflecting the status of microstructure, using the sample Pearson’s correlation coefficient. Also, several important differences between original MAT and its modification utilized regularly in our lab are discussed. The key differences are related especially to the experimental part (no sample demagnetization needed, more precise measurements at lower exciting field amplitudes, etc.).

2 Fundamentals of Magnetic Adaptive Testing

MAT is based on the measurement of a series of minor hysteresis loops with triangular exciting magnetic field intensity (i.e., the current flowing through the winding generating that field). From these loops, properly chosen magnetic parameters of various constructional materials used in common practice, are evaluated. Further, these magnetic parameters are compared with different physical parameters obtained by other, usually destructive techniques (Charpy impact tests, Vickers hardness, and many others). Promising results, especially in comparison with the methods using traditional magnetic parameters (such as, e.g., the coercivity, remanence, saturation magnetization, initial permeability, etc.), initiated the effort for further development and improvement, see, e.g., [6,7,8,9,10,11,12,13].

2.1 Differential Permeability

Among various magnetic quantities, the differential permeability $\mu _{diff}$ appears to be one of several potential candidates for further testing of the possibilities of this approach, thanks to its exceptional sensitivity to, e.g., long-term mechanical and/or thermal load as well as the neutron irradiation. All these factors are closely related to microstructural changes and material fatigue that can result even in the failure of vital constructional units. Therefore, reliable detection of such structural damages is being of key importance especially in nuclear power industry and similar fields.

If multichannel data acquisition (DAQ) systems are used for the measurement of magnetization characteristics (or, hysteresis loops), at least two time waveforms of magnetic quantities have to be measured. In order to obtain comparable results, the magnetization curves shall be measured under properly defined excitation conditions, i.e., the amplitudes, frequencies, as well as the waveform shapes of relevant magnetic quantities. Depending on particular situation and applied standards, either the exciting field intensity H(t), or the magnetic flux density (induction) B(t), have to be under control [5].

The relative differential magnetic permeability is defined in general as being proportional to the slope of the tangent to the magnetization curves at any particular (k-th) operating point, corresponding to the Cartesian coordinates ($H_{k},B_{k}$) in two-dimensional HB-space. Since the magnetic quantities vary in time, it is changing with the time as well. Thus, at a concrete time instant $t_{k}$ it is equal to

$$\begin{aligned} {\mu _{diff}(t=t_{k})=\frac{1}{\mu _{0}} \frac{dB(t)}{dH(t)} \Bigg |_{{\begin{array}{l} H=H_{k} \\ B=B_{k} \end{array}}},} \end{aligned}$$

(1)

where $\mu _{0}$ is the magnetic constant (or, permeability of free space), $H_{k}=H(t=t_{k})$, and $B_{k}=B(t=t_{k})$. As a result of general DAQ operation principle, instead of continuous waveforms H(t) and B(t), and/or their representations by analytic functions, only discrete datasets $\lbrace H_{i} \rbrace _{i=0}^{N-1}$ and $\lbrace B_{i} \rbrace _{i=0}^{N-1}$ of equidistantly sampled instantaneous values are available for one period of signals with N data points per period. Therefore, the precise values of differential permeability cannot be found directly applying (1) at any arbitrary operating point, but only as a discrete numerical derivative found, e.g., from the differences between adjacent sampled data-points. Since the signals are usually affected by random noise, this approach standardly brings about various numerical artifacts deteriorating the results. Therefore, numerical differentiation is usually discouraged, especially when N is low. Fortunately, there is an easy way out of this problem. The idea is based on the fact that when the sample is magnetized by properly controlled exciting magnetic field waveform H(t), then the numerical calculation of the differential permeability can be avoided completely. The proof follows directly from the rearrangement of (1),

$$\begin{aligned} \mu _{diff}(t)=\frac{1}{\mu _{0}} \frac{dB(t)}{dH(t)}=\frac{1}{\mu _{0}} \frac{\frac{dB(t)}{dt}}{\frac{dH(t)}{dt}}. \end{aligned}$$

(2)

The main numerator dB(t)/dt is, in accordance with Faraday’s law of induction, proportional to the voltage induced in the secondary winding with $N_{s}$ turns enclosing an area A, $v_{ind}(t)=N_{s} A \frac{dB(t)}{dt}$. Expression (2) can thus be rewritten as

$$\begin{aligned} \mu _{diff}(t)=\frac{1}{\mu _{0} N_{s} A} \frac{v_{ind}(t)}{\frac{dH(t)}{dt}}. \end{aligned}$$

(3)

If, in addition, H(t) is a piece-wise linear (triangular) function of time, the slope of linear sections is constant, with its absolute value determining the exciting field rate of change

$$\begin{aligned} \bigg |\frac{dH(t)}{dt} \bigg |=c_{H}. \end{aligned}$$

(4)

Equation (3) then becomes

$$\begin{aligned} \mu _{diff}(t)=\frac{1}{\mu _{0} N_{s} A c_{H}} |v_{ind}(t) |, \end{aligned}$$

(5)

i.e., the relative differential permeability is directly proportional to the absolute value of induced voltage. Hence, there is no need for numerical determination of the differential permeability, it is enough to measure the time variation of induced voltage.

2.2 Acquisition Procedure and Arrangement of Data

To assess various magnetic properties of the material under investigation (namely, $\mu _{diff}$) in a wide range of operating conditions within HB-space, a lot of data needs to be obtained and processed. For this purpose, a procedure was suggested in [1], consisting of the measurement of a set of minor hysteresis loops at a triangular exciting field waveform with the maximum applied field value $H_{max}$ changing within prescribed interval with a constant step $\Delta H$. As mentioned in original paper, the loops were measured with successively increased amplitude of applied field, which requires the samples to be carefully demagnetized before the measurement. In practice, a problem may be the demagnetization of large objects to be tested; therefore, any possibility to avoid it is welcome. Instead, we normally measure the loops from higher to lower field amplitudes, and prior to each measurement several pre-magnetization cycles, with the same amplitude and frequency (or, field rate of change $c_{H}$) as required for particular loop, are applied in order to “overwrite” any previous history of magnetizing. Thus, well-defined initial conditions before the measurement of any specific loop are guaranteed, and, as verified by many experiments, the demagnetization of sample is not needed anymore. The amplitude of i-th hysteresis loop is

$$\begin{aligned} H_{i}=H_{max,i}=H_{start} - (i-1) \cdot \Delta H, \end{aligned}$$

(6)

for $i=1, 2, \cdots , r$ being an index variable, $H_{start}$ is the highest (or, starting) amplitude corresponding to the first measurement. It is usually chosen as an integer multiple (r) of the step $\Delta$H (i.e., $H_{start}=r \cdot \Delta H$), and the last (lowest) value of amplitude $H_{stop}=\Delta H$. Obviously, r is the total number of measured minor hysteresis loops. Note that, since $c_{H}$ is always kept the same, decreasing amplitude results in decreasing period $T_{i}$ (and thus, increasing frequency $f_{i}$) of the magnetizing signal.

Thanks to the sample pre-magnetization, the measured hysteresis loops are perfectly symmetrical; therefore, the relative differential permeability can be evaluated using (5) from the part of induced voltage waveform related to the descending branch of the loop, i.e., the exciting field changing monotonically from $+H_{max,i}$ to $-H_{max,i}$. This corresponds to the time ranging from the interval $t \in \langle T_{i}/4, 3T_{i}/4\rangle$. Since within this interval the relationship between time t and applied field H(t) is linear and monotonic, the induced voltage can be easily expressed as a function of applied field, $v_{ind}(H)$. Interpolation of this function at properly chosen instantaneous values of exciting field $H_{j}$ (corresponding to the time instants from the above interval) yields the discrete values of $\mu _{diff}(H=H_{j})$. In order to obtain easily processable datasets, the interpolation step is usually chosen to be equal to $\Delta H$ as well. Hence, for the i-th loop with the amplitude $H_{max,i}$

$$\begin{aligned} H_{j}=j \cdot \Delta H, \end{aligned}$$

(7)

where $j=0, \pm 1, \pm 2, \cdots , \pm (r-i)$. Therefore, $2(r-i)+1$ interpolated values can be found. These discrete values of differential permeability can be arranged into the matrix with the number of rows equal to the number of minor loops r and the number of columns $c=2r-1$, whose element is

$$\begin{aligned} \mu _{i,j}= { {\left\{ \begin{array}{ll} \mu _{diff}(H_{j}) &{} \text {if }j=-(r-i), \cdots ,+(r-i) \\ \text {NaN} &{} \text {otherwise} \end{array}\right. } }. \end{aligned}$$

(8)

Here, i and j are the row and column index variables, respectively, NaN is the “Not a Number” data type, used to represent missing (or, not needed) values.

In order to reveal the relationship between any of the types of long-term load mentioned in Sect. 2.1 (resulting expectedly in microstructural changes) and the differential permeability obtained by the above mentioned approach, it is necessary to repeat the measurements after each step of successively applied defined load with the same measuring conditions regarding the amplitudes, frequencies, and exciting field rate of change. This way, several matrices given by (8), with the values of elements reflecting the applied load are available. Obtained matrices are arranged into layered three-dimensional array, whose 3rd coordinate (associated with another, layer index variable, k) represents the influence of previously applied (known) load sequence upon the value of a particular discretized differential permeability localized by certain values of i and j. The number of layers l corresponds to the number of times the load is applied, including the initial state for $k=1$. Putting any of the vectors representing the change of $\mu _{diff}$ (i.e., $\mu _{i,j}$) with respect to k for fixed i and j into the correlation with the changes of selected conventional parameters affected by the same kind of load we can obtain a large set of calibration curves, which can be used, e.g., for the assessment of material fatigue. Since the behavior of such calibration curves sometimes cannot be interpreted clearly and predictably, only a subset of them is often useful for further analyses. To find such a subset is the task of the software tool described later (see Sect. 4).

3 Experimental Hardware and Control Software

A very detailed description of recent advances in the experimental hardware and software along with important peculiarities and tricks, making the measurement of magnetization curves in general easier, faster, and more accurate is given, e.g., in [5]. A universal arbitrary waveform generator with 16-bit basic vertical resolution was used as a source of exciting signal. As discussed above, for the purposes of MAT, triangular waveform of exciting field H(t) [and thus, the current i(t)] is required. The magnetizing circuitry utilizing well-known connection of high-power operational amplifier acting as the voltage-to-current converter with hardware-based feedback introduced from the current sensing calibration resistor accomplishes this task precisely. Recently developed prototype of high-power operational amplifier is shown in Fig. 1. Two identical fast digital multimeters (DMM) with $6\frac{1}{2}$ digits resolution and with maximum sample rate of 50 kHz, synchronized via external trigger inputs, were used to obtain simultaneously sampled datasets of applied field and induced voltage. If needed, faster modular data acquisition (DAQ) devices with higher resolution, available on the market, can be utilized. Main advantage of DMM over standard DAQ-s is that the multimeters usually allow more precise control of sampling interval (changes with $1~\upmu \hbox {s}$ steps possible). All the control upon particular tasks of the experiment, including overall control of instruments, adjusting the measurement parameters, timing, etc. as well as the acquisition, processing and storage of data is taken over by the software developed in commercially available software development package.

Fig. 1

The prototype of high-power operational amplifier

4 Software Tool for MAT Data Processing

The data acquisition itself and MAT analysis can in principle be separated. Therefore, we focused mainly on the second stage, since the measurements are performed routinely, as described above, without any issues to be addressed. Further processing of data using the approach described in Sect. 2.2 is carried out by means of tailor-made software tool, developed using commercially available graphical programming environment allowing to work with large multidimensional data arrays.

4.1 Data Pre-processing and Filtering

An efficient way how to make the assessment of the changes caused by applied load independent of the type of tested parameter and the units of measurement is the evaluation of the differences between arbitrary general and initial values normalized to initial state values prior to any applied load (an array layer corresponding to $k=1$), i.e, relative differences

$$\begin{aligned} \begin{aligned}&\Delta \mu _{i,j,k}=\frac{\mu _{i,j,k}-\mu _{i,j,1}}{\mu _{i,j,1}}=\frac{\mu _{i,j,k}}{\mu _{i,j,1}}-1 \\&i,\,j\text { fixed} \\&k=1, 2, \cdots , l, \end{aligned} \end{aligned}$$

(9)

where $\mu _{i,j,k}$ is a particular value of discretized initial permeability in 3D data array for changing k, while both i and j are fixed. These changes can be expressed in % as well.

Since the induced voltage signal, in accordance with (5) directly related to the differential permeability, is in general affected by (pseudo)random noise associated, e.g., with structure-dependent Barkhausen jumps and other noise sources, the calculation of relative differences [especially, the division of two random values in (9)] is often markedly influenced by numerical artifacts. To minimize such effects, proper data smoothing should be applied to each layer before computation of the relative differences. This task is accomplished by means of applying various 2D low-pass filtering techniques, known, e.g., from bitmap image processing. A well-established powerful tool for this purpose is a discrete 2D convolution of an array of a selected layer [i.e., fixed k in (9)] with a small matrix called convolution kernel. If the convolution kernel element values are calculated by 2D normalized Gaussian distribution, the convolution procedure acts as a linear 2D low-pass filter smoothing the data in the layer array (a.k.a. Gaussian blur). The properties of filter can be adjusted by the size of kernel array (default value is a square of $3\times 3$ elements) and the standard deviation $\sigma$ (default value, related to the kernel size is 1 element). The filter thus simply performs properly weighted averaging of adjacent pixels, including the one in the center of floating kernel array.

4.2 Sorting Data Using Logical Masks

In order to exclude the above mentioned (Sect. 2.2) spurious and misbehaving, usually numerous, calibration curves from further data processing, various logical tests can be accomplished first to save computer processing time. The changes of differential permeability of the sample subjected to known treatment (i.e., artificial aging) along the index k can be tested, e.g., for monotonicity, linearity, as well as for the correlation with any reference curves representing the response of parameters available from other, destructive, and/or nondestructive experimental analyses to the same load sequence. For this purpose, Pearson’s correlation coefficient appears to be a good choice.

The easiest test is checking the vectors for monotonicity. It is enough to judge, if the next successive, $(k+1)$-th value is either larger (or, smaller) than k-th one for the whole range of k. The test can be further enhanced, e.g., for equality of adjacent values (i.e., weak monotonicity) and constantness, 5 cases can be tested in total:

$$\begin{aligned} \begin{aligned}&\mu _{i,j,k}= {\left\{ \begin{array}{ll} \text {strictly increasing} &{} \text {if }\mu _{i,j,k+1} > \mu _{i,j,k} \\ \text {non-decreasing} &{} \text {if }\mu _{i,j,k+1} \ge \mu _{i,j,k} \\ \text {constant} &{} \text {if }\mu _{i,j,k+1} = \mu _{i,j,k} \\ \text {non-increasing} &{} \text {if }\mu _{i,j,k+1} \le \mu _{i,j,k} \\ \text {strictly decreasing} &{} \text {if }\mu _{i,j,k+1} < \mu _{i,j,k} \end{array}\right. } \\&i,\,j\text { fixed} \\&k=1, 2, \cdots , l-1. \end{aligned} \end{aligned}$$

(10)

As a result, 5 binary TRUE/FALSE logical masks, reflecting the areas representing monotonic/non-monotonic parts of the total operating area can be found. These masks implemented in the software are, by the nature, mutually exclusive; thus, maximum 1 out of 5 monotonicity tests can be true. It is worth noting that strict monotonicity is a must for using an inverse function to interpolate the single-valued load corresponding to a particular value of selected investigated parameter.

The linearity can be assessed along with more general testing of similarity of the differential permeability changes and selected reference curve of any arbitrary shape (as mentioned above) using sample Pearson’s correlation coefficient $r_{xy}$. Again, a logical mask can be found, indicating whether $r_{xy}$ of the dependence of $\mu _{i,j,k}$ for fixed i and j exceeds a chosen threshold value $r_{xy,thr}$. A small drawback of using $r_{xy}$ here is a missing information about the slope of curves—it only tells if there is a relationship between tested and reference curve. The slope, in case of linear dependencies reflecting the sensitivity of a particular parameter (given by the row and column index) to applied artificial aging, is a very important information as well. Unfortunately, this can hardly be applied for general, especially for non-monotonical, dependencies. Therefore, an additional logical test was added and used as an extended definition, or measure of sensitivity, finding whether the relative changes, given as

$$\begin{aligned} \begin{aligned}&\Delta _{i,j}=\left( max\{\Delta \mu _{i,j,k}\}-min\{\Delta \mu _{i,j,k}\} \right) \cdot 100\% \\&i,\,j\text { fixed} \\&k=1, 2, \cdots , l \end{aligned} \end{aligned}$$

(11)

are greater, or equal to minimal required value.

As a final step, the results of logical tests can be element-wise OR-ed or AND-ed to create a global logical mask in order to eliminate the parts of total operating area (having in general a triangular shape, given by the range of indices i and j), where the observed changes are unacceptably affected by, e.g., random noise, etc. As a result, the subset of calibration curves satisfying utmost three selected criteria (degree of correlation, relative change, monotonicity) can be found and used for interpolation reflecting the level of microstructural changes under the load applied to any construction unit made of the same material operated in unknown conditions.

5 Results and Discussion

5.1 Samples and Treatment

To minimize the side effects associated with the shape and geometry (e.g., demagnetization effect, field inhomogeneities, etc.), magnetically closed ring shaped specimens made of commercial low-carbon steel (C<0.06%, Mn<0.45%, Si<0.15%, Al<0.02%, P<0.02%, S<0.02%) were used for the verification of fundamental ideas of MAT, see Fig. 2. Average ring dimensions were: outer diameter $D \approx 56$ mm, inner diameter $d \approx 46$ mm, and height $h \approx 5$ mm. The material was subjected to successive annealing at the temperature of 753 K for 2, 4, 6, 20, 40, and 60 h. Thus, an artificial aging, associated with long-term exposure to elevated temperatures, was simulated. The relatively low temperature was chosen in order to test the limits of MAT capabilities, since at such low temperatures (far bellow critical one for the steels) no significant variations in microstructure are expected. Therefore, any changes indicated by MAT parameters are believed to be caused mainly by residual stress removal. Nevertheless, even in the range of 753–793 K (i.e., 480–520 °C) the primary recrystallization in the wires of a similar composition was observed [14]. Between each annealing step, the dimensions of samples were checked again, and a set of magnetization curves at the same excitation conditions (as required for MAT) was measured. In order to keep the winding geometry and, especially, the number of turns the same during the experiments, both primary and secondary coils were wound at the same time (two-wire flat cable was used).

Fig. 2

Samples used for experiments

Fig. 3

Examples of experimentally obtained waveforms of the exciting field H(t) (top), corresponding flux density B(t) (middle), and minor hysteresis loops B(H) (bottom)

Fig. 4

Time variations of induced voltage $v_{ind}(t)$ (top), its absolute values as a function of applied field (middle), and the relative differential permeability $\mu _{diff}$ (so-called butterfly curves, bottom), found numerically using the definition formula (1). In accordance with (5), $\mu _{diff} \propto |v_{ind}(H) |$

5.2 Measurements of Minor Hysteresis Loops

Time waveforms of important magnetic quantities, measured after 2 h of annealing, are displayed in Fig. 3. The magnetic flux density was found as a numerical integration of induced voltage. Also, the set of minor hysteresis loops is shown (bottom). From these B(H) curves, traditional magnetic hysteretic parameters (e.g., coercive field, remanent magnetic flux density, hysteresis loop area, etc.) use to be evaluated. Figure 4 shows the induced voltage as a function of time as well as of the applied field. Since the H(t) relationship is piece-wise linear, the geometrical shape of the middle and bottom curves is strictly correlated. For clarity, not all the measured curves are shown (here, only 20 out of 100); usually several tens, or even hundreds of minor loops are measured. Directly proportional relationship between the field amplitude $H_{max,i}$ and signal period $T_{i}$ (see Sect. 2.2) is clearly visible. The frequencies $f_{i}=1/T_{i}$ were chosen as a compromise between sufficient levels of the induced voltage and the need to eliminate any dynamic effects, associated with, e.g., eddy current losses in conductive materials, being proportional to $f^{2}$, etc. One of the advantages of MAT from the hardware point of view is that full magnetic saturation is not a must, since the largest sensitivity to applied treatment is usually found near the coercivity region (see later, e.g., Fig. 6, right column and Fig. 7, 2nd row). Therefore, the values of important experimental parameters were chosen as follows; starting amplitude $H_{start}=1500~\hbox {Am}^{-1}$, step $\Delta H=15 \hbox {\, Am}^{-1}$, and $c_{H}=1500~\hbox {Am}^{-1}\hbox {s}^{-1}$. Thus, the frequency of triangular exciting field changed from 0.25 to 25 Hz. In order to keep the number of samples per period N always the same, the sampling rate of data acquisition devices, $f_{s,i}= Nf_{i}$, has to be changed as well. It is worth mentioning that the magnetization dynamics deteriorates the effectiveness of this method remarkably, since the hysteresis loops become wider and rounded, the waveforms of H(t) and B(t) can even get out of phase, i.e., the maxima/minima of both waveforms are reached at different time instants. As a result, the sensitivity to any type of applied load significantly decreases.

Fig. 5

3D representation of discretized induced voltage, proportional to $\mu _{i,j,k}$ with fixed k, here k = 2 (layer of data values after 2 h of annealing). Raw values without filtering (left), low-pass filter applied to the same data (right) (Color figure online)

Fig. 6

The matrices of interpolated discretized induced voltage, proportional to $\mu _{i,j,k}$ (left), and relative difference matrices $\Delta \mu _{i,j,k}$ (right) for various annealing times $t_{a}$ = 0, 2, 4, 6, 20, 40, 60 h (Color figure online)

5.3 Data Interpolation and MAT Analysis

Fig. 7

Available logical masks without (left), and with application of 2D low-pass convolution filter (right). Threshold values were $r_{xy,thr}$ = 0.99 for sample Pearson’s correlation coefficient, and minimal relative change $\Delta _{i,j,k}$ = 200%. Only the sample Pearson’s correlation coefficient and relative change masks were logically AND-ed to compute the global mask, since there were almost no monotonic dependencies found. White color outside triangular area means NaN values, light gray within triangle corresponds to FALSE values, and any other color within triangle represents TRUE values, i.e. that the calibration curve at given row and column indices (mapped here to discrete $H_{i}$ and $H_{j}$ values) meets the specific criterion tested by that particular mask (Color figure online)

An example of discretized induced voltage (directly related to the differential permeability $\mu _{i,j,k}$) after 2 h of annealing, displayed as a colored 3D surface plot with contours, is in Fig. 5. The row and column array indices (i and j) were in accordance with (6) and (7) mapped to particular exciting field amplitudes ($H_{i}$) and discrete values of exciting fields used for the interpolation ($H_{j}$), respectively.

The evolution of differential permeability with annealing time $t_{a}$ is shown in Fig. 6. The values of interpolated discretized induced voltage, displayed as colored triangular matrices, were obtained by the procedure described in Sect. 2.2. The relative difference matrices of $\Delta \mu _{i,j,k}$ were calculated using (9), see Sect. 4.1. The upper left matrix representing the initial state for $t_{a}$ = 0 was used as the reference.

The influence of applying convolution filtering on the resulting logical masks is demonstrated in Fig. 7. The dependence of remanent magnetic flux density $B_{r}$ upon the annealing time $t_{a}$ from Fig. 8 (red curve) was used as the reference curve during computing the masks. Also, the data for the coercive field $H_{c}$ and the hysteresis loop area $A_{hl}$ are available for analysis, as they can be easily evaluated from the measured major hysteresis loop (at the highest field amplitude used, $H_{start}$). The “TRUE” values for individual masks were distinguished by color. Any calibration curve to be visualized can be user-selected by the cursor position. A substantial reduction of data to be further taken into account is remarkable, as only the calibration curves fallen within TRUE values of the global mask are assumed. The effect of filtering is especially visible in case of testing the relative changes—in the right part of triangle, where the induced voltage levels (and differential permeability) are relatively low, resulting in lower signal-to-noise ratio. The relative changes reach extremely high values (as discussed in Sect. 4.1), not reflecting the physics behind; just have a look at many randomly placed red rectangles indicating TRUE values. The noise is clearly visible also in the matrices of relative differences (right column in Fig. 6), even if they were filtered. Here, the white-colored areas inside triangles mean out-of-scale values. Fortunately, these are efficiently eliminated by proper masking (choice of threshold values and masks themselves). Note that aggressive filtering (too large kernel window and/or $\sigma$) is discouraged, since some useful information contained in the experimental data can be hidden.

The subsets of masked curves (i.e., those satisfying particular criterion, based on the selected threshold value and/or the monotonicity test), corresponding to three available masks computed from the filtered data (sample Pearson’s correlation coefficient, relative change, and increasing, respectively—see the right column in Fig. 7), along with the global mask found as element-wise logical AND of the first two masks, are shown in Fig. 9. As can be seen, even if the threshold value of sample Pearson’s correlation coefficient $r_{xy,thr}=0.99$ looks to be strict enough, there are many calibration curves (several thousands) of the differential permeability dependencies that meet this request. If also the relative change $\Delta _{i,j,k}\ge 200\%$ is taken into account, the number of calibration curves substantially decreases. Finally, if both criteria are satisfied at the same time, as determined by the global mask, we obtain relatively low number of calibration curves, being very similar to the selected reference curve $B_{r}(t_{a})$. Except for similarity, all of them also exhibit about $8\times$ larger relative changes, than the reference curve with $\approx 25\%$. This means $8\times$ larger sensitivity of properly chosen differential permeability dependencies to thermal treatment than for traditional parameter—remanent flux density while the shape of any globally masked calibration curve precisely follows the reference one, see Fig. 9, bottom. Note that the relative changes of masked curves achieved more than 2200%, but the shape of them is far from the reference one. Further, there is neither monotonically increasing nor non-decreasing curve found in filtered data, as could possibly be expected from the reference curve shape (in general increasing, although not monotonically). Hence, none of monotonicity testing masks was used to find the global mask. It is worth mentioning that research has been reported that found a monotonic correlation between the differential permeability and annealing parameters [15, 16]. However, different materials (Fe—1 wt% Cu, 15Ch2MFA steel alloy, respectively) subjected to different treatment were studied. In the first case, in addition to annealing also a cold rolling was applied. In the second study, a special thermal processing curve was used (see also [3]) and the ductile-brittle transition temperature (DBTT) was used as an independent variable instead of annealing time. Therefore, the comparison with our results is not entirely applicable. Possibly, other parameters, a.k.a. MAT descriptors, could be used in order to find monotonic dependence upon artificial thermal aging, but their relationship to the (non-monotonic) dependencies of traditional magnetic parameters ($H_{c}$, $B_{r}$, etc.), reported in this work, thus would be lost.

If the coercive field dependence $H_{c}(t_{a})$, blue curve in Fig. 8, was used as the reference curve, there was no overlapping found among individual masks. Therefore, the global mask, computed again as a logical AND of the elements of all the partial masks taken into account, contains no TRUE values. The only interesting test is thus based on the correlation coefficient, using which several tens of curves with similar shape were found. In this case, the relative changes achieved about $-25\%$, which is only slightly better comparing to the reference curve with $-20\%$.

Regardless of which reference curve is selected, the behavior of masked as well as standard reference magnetic parameters is qualitatively similar. As can be seen, the largest changes were always observed after the first step. The knowledge described above indicates that, most likely, already 2 h of annealing caused the relieve of most internal residual stresses associated with sample machining, meanwhile further, less pronounced changes can be attributed to the moderate recrystallization and growth of crystalline grains associated with continued annealing. The structure seems to be stabilized after several hours of annealing, since further changes of parameters were almost negligible.

6 Conclusions, Open Problems, Future Plans

A variation of recently established perspective non-destructive evaluation method for monitoring microstructural changes—MAT is introduced from the point of view of measurement procedure optimization, along with detailed explanation of MAT data parsing based on the utilization of newly developed advanced software tool. For this purpose, the series of minor hysteresis loops obtained at piece-wise linear waveforms of exciting field with prescribed field rate of change needs to be measured. The main benefit is that the measurement of the loops starting from the highest exciting field amplitudes downwards allows to omit sample demagnetization entirely. The created software is usable for everyday analysis of experimental data by means of MAT.

Fig. 8

Selected reference curves representing the dependencies of traditional hysteretic parameters ($B_{r}$ and $H_{c}$) upon the annealing time $t_{a}$. The hysteresis loop area $A_{hl}$ (not shown in this figure) can be used as the reference, too (Color figure online)

The applicability of presented software was demonstrated on the data representing the material, whose response to applied thermal load is intentionally being hard to analyze, because of completely missing monotonicity of observed parameter dependencies (e.g., differential permeability) upon defined artificial aging. In case of MAT such a situation was not found to be reported elsewhere. Even in spite of that, the presented software tool allowed to find clear correlation with standard magnetic parameter(s) (remanent magnetic flux as well as the coercive field), whose behavior is non-monotonic as well. This was achieved thanks to the use of the sample Pearson’s correlation coefficient expanding the possibilities of data evaluation also for non-monotonic dependencies of traditional hysteretic parameters as well as MAT descriptors (e.g., differential permeability) upon degradation variables. An easy-to-do simulation of artificial aging of the samples was carried out via long-term annealing.

Fig. 9

The curves masked by the logical masks corresponding to the filtered data (right column in Fig. 7). The plot with symbols represents the selected reference curve, $B_{r}(t_{a})$ (see Fig. 8), normalized to the first (initial) value

The universal software tool for MAT yields substantially narrowed choice of calibration curves needed to be taken into account for the non-destructive investigation of ferromagnetic materials. This is a major success on contrary to the results reported in [4], where conventional approach to data analysis brought no useful outcome. As a result, working with large amount of data has become a routine task allowing fast analysis of obtained data reflecting the changes in materials associated with any kind of load acting even for a long-time. Moreover, if the threshold values are chosen properly, it is possible to find a set of magnetic parameters reflecting assumed stress relieving, and/or microstructural changes, with the sensitivity far better than that observed in case of traditional parameters found from the hysteresis loops.

On the other hand, revealing useful information from the experimental data requires an experienced user, since the masking criteria should be tuned with care. Especially, the response of tested material to the artificial aging represented by long-term annealing as described above, gives the results extremely difficult to interpret. The absence of monotonicity, for instance, is a problem from the point of view of finding an inverse function useful for the interpolation of actual material condition related to applied treatment. In such a case, the whole reference calibration curve(s) can be divided into monotonic subsections for which the inverse functions can be found separately.

Taking into account the presented results one can see that MAT has proven itself to be an efficient tool for structural health monitoring not only from the point of view of the changes in the microstructure (as reported in several studies referenced here), but also the changes in internal stresses associated, e.g., with cold/hot machining of materials for constructional units can be indicated reliably.

An essential plan for the near future is to find a proven causal link between the changes in microstructure directly observed by various destructive as well as non-destructive techniques already adopted.

Further research will also focus on the characterization of structural changes extended to magnetically open constructional units of various shapes with the aim of applying MAT in common industrial practice. This goal is associated with the need to develop appropriate magnetizing and sensing circuitry giving the analysis results, as far as possible, independent of the shape and dimensions of the device under test. This appears to be a big challenge, since there are some natural issues, associated with the measurements on magnetically open samples, to be addressed. In practice, standardly the C-shaped cores with at least two windings are used—one for magnetizing of tested device, and another one for sensing of the induced voltage. Thus, two major problems arise; first, the dependence between the magnetizing current and the magnetic field intensity in the sample needs not to be linear (usually it is not, as widely known and also confirmed by many experiments carried out in our lab). Secondly, the magnetic flux density in the tested material is found indirectly from the voltage induced in the coil placed on magnetizing yoke. Obviously, this might deteriorate the accuracy substantially.