# Denouement of the Energy-Amplitude and Size-Amplitude Enigma for Acoustic-Emission Investigations of Materials

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{1/2}, where S is the avalanche area), averaged for fixed S, should be the same, independently of the type of materials or avalanche mechanisms. However, there are experimental evidences that the temporal shapes of avalanches do not scale completely in a universal way. The self-similarity also leads to universal power-law-scaling relations, e.g., between the energy, E, and the peak amplitude, A

_{m}, or between S and A

_{m}. There are well-known enigmas, where the above exponents in acoustic emission measurements are rather close to 2 and 1, respectively, instead of $E~{A}_{m}^{3}$ and $S~{A}_{m}^{2}$, obtained from the mean field theory, MFT. We show, using a theoretically predicted averaged function for the fixed avalanche area, $U\left(t\right)=at\mathrm{exp}\left(-b{t}^{2}\right)$ (where a and b are non-universal, material-dependent constants), that the scaling exponents can be different from the MFT values. Normalizing U by A

_{m}and t by t

_{m}(the time belonging to the A

_{m}: rise time), we obtain ${t}_{m}~{A}_{m}^{1-\phi}$ (the MFT values can be obtained only if φ would be zero). Here, φ is expected to be material-independent and to be the same for the same mechanism. Using experimental results on martensitic transformations in two different shape-memory single-crystals, φ = 0.8 ± 0.1 was obtained (φ is the same for both alloys). Thus, dividing U by A

_{m}as well as t by ${A}_{m}^{1-\phi}$ (~t

_{m}) leads to the same common, normalized temporal shape for different, fixed values of S. This normalization can also be used in general for other experimental results (not only for acoustic emission), which provide information about jerky noises in materials.

## 1. Introduction

_{m}, size, S, energy, E, or duration, T; η is the characteristic exponent, and x

_{c}is the cut-off value) [1,2,3,4,5,6,7,8,9]. The power-law distributions reflect a self-similar behaviour spanning wide range of the parameter, x (e.g., the temporal shape of an avalanche looks the same at different time scales). Examples for such behaviour can be the classical Barkhausen noise, sand piles, fracture, martensitic transformations in shape memory materials, plastic deformations, etc. In many cases, the avalanches are jerky responses to slowly changing driving force or field. Thus, considerable efforts were devoted to predict how the corresponding exponents of the above distributions can be grouped into universality classes [5,10,11]. In addition, power-law-scaling relations between the exponents of the above parameters were obtained (e.g., the energy, E, is related to the amplitude, ${A}_{m}$, as $E~{A}_{m}^{\chi}$) with predictions that these should be the same within one universality class [1,6,12,13,14]. Furthermore, the self-similarity leads not only to power laws, but to universal-scaling functions, which can have predictive power, and in recent publications the authors have gone beyond the power laws and focused on the universal, (properly normalized) temporal shape of avalanches [1,2,4,5,10].

_{a}is the characteristic attenuation time of the signal, and ω is the resonant frequency. Thus, it was concluded in [25] that the measured AE spectrum does not reflect the temporal shape of avalanches (i.e., the v(t) distribution) nor the model predictions. Therefore, a detailed analysis of the observed AE jerk profiles only reveals information about the transfer function of the measuring system (material properties + detector: see Figure 1 in [25]) and says little about the local avalanche mechanism. It was also shown, from the convolution of the transfer function with different model functions for the source [25], that while the characteristic exponents of the energy and size PDF’s were invariant, the detected duration time, $D$, was significantly distorted compared to the true duration time, T. Furthermore, the so-called energy-size enigma was exposed: while the mean field theory (MFT) predicts χ = 3 for the scaling exponent between the energy and amplitude, their model simulations for AE results provided χ = 2. It is worth mentioning that in a set of papers by Barcelona’s group [17,18,19] a less pessimistic conclusion was drawn. It was argued that if one considers the convolution of a simple rectangular-signal source, then for signals with long duration times, $T$, as compared to ${\tau}_{a}$, i.e., $\frac{{\tau}_{a}}{T}\ll 1$, the detected duration time can be close to the true duration time, $T$. On the other hand, for $\frac{{\tau}_{a}}{T}\gg 1$, the results provide information about the attenuation time, ${\tau}_{a}$ [17,18,19], and the scaling exponents between the energy and duration time or the energy and amplitude are considerably different from the values predicted by the MFT. For intermediate values of $\frac{{\tau}_{a}}{T}$ a transition between the above two limits can be observed. On the other hand, surprisingly, as it was also mentioned in [25], it was obtained that even for $\frac{{\tau}_{a}}{T}\ll 1$, $E~{A}_{m}^{2}$ was observed.

_{m}, and the maximum time (raising time), t

_{m}. We assume that their ratio, $\frac{{A}_{m}}{{t}_{m}}$, instead of being constant, is given as $\frac{{A}_{m}}{{t}_{m}}~{A}_{m}^{\phi}$. φ is material-independent, and the same is the same for the same mechanism. It appears, as a correction term, in the scaling exponents, and, thus, provides the denouement of the enigmas. Thus, we will illustrate, using experimental data obtained during martensitic transformation in two ferromagnetic shape-memory alloys, that this indeed leads to deviations from the predicted scaling exponents and, e.g., the slopes of the logS versus logA

_{m}or logE versus logA

_{m}are given by $3-\phi $ and $2-\phi $, respectively, i.e., they can be much smaller than the predicted MFT values (2 and 3). Furthermore, dividing U(t) by A

_{m}as well as t by ${A}_{m}^{1-\phi}$ (~t

_{m}) leads to the same common, normalized temporal shape for different fixed values of S. Our results can also be valid in general for scaling relations between the experimentally determined parameters from other types of measurements of avalanches (magnetic emission [26], high-resolution detection of the deformation or stress drops, etc.) and not only for AE.

## 2. Expressions for the Exponents of Scaling Relations

^{b}, then the scaling function, ${u}_{o}\left(\frac{t}{T}\right)$, is a universal theoretical prediction (for large sizes and long times) [1,2]:

_{S}= const.) for universal ${u}_{o}\left(\frac{t}{T}\right)$, we arrive at the scaling relation

_{av}, and T, we can write from (4)

_{av}is independent of γ.

_{av}, the maximal value of the voltage (the peak value), U

_{m}, is commonly determined, so the relation between them should be considered. It was argued in [6] that the peak amplitude is a good measure of U

_{av}, but the relation between these two parameters was not checked experimentally. We will show in the next section that they are indeed interrelated, but instead a linear relation

_{m}and t

_{m}are linearly related to each other if a (B) is constant. This is in accordance with the result of [25], where this relation was analysed by simulations for the fixed value of a (~B), and ${U}_{m}~{t}_{m}^{\xi}$, with ξ = 0.95, was obtained. We will show below that the value of B has a definite dependence on U

_{m}, $a~B~{U}_{m}^{\phi}$. Dividing both sides of (12) by U

_{m}, and using (14), we obtain the dimensionless (reduced) form of U with the two scaling parameters (recommended also in [25], since they are not distorted by transfer effects) U

_{m}and t

_{m}(${U}^{*}=\frac{U}{{U}_{m}}$ and ${t}^{*}=\frac{t}{{t}_{m}}$, respectively) as

_{av}and U

_{m}are proportional to each other (as predicted by [6]), only if ${T}^{*}$ is constant. Now, we can calculate the reduced duration time as the difference of the start and finish times (${t}_{s}^{*}$ and ${t}_{f}^{*}$, respectively) given by a fixed threshold value, C, from (15)

_{e}, and log corresponds to log

_{10}, respectively. Thus, (22a) can also be written as

_{10}2.718 = 0.434.

_{m}as:

_{m}and t

_{m}, we have

_{m}and T, using (22a) and (24) in the form $T={t}_{m}{T}^{*}=\frac{{U}_{m}}{B}{T}^{*}=\frac{{U}_{m}^{1-\phi}}{\alpha}\sqrt{2ln\frac{{U}_{m}}{C}}$ as;

_{m}.

_{AE}, S

_{AE}, A

_{m}, A

_{av}, and D, instead of $E,S,{U}_{m},{U}_{av}$, and T), we have to take into account that the transfer effects can distort the values of A

_{av}and D. Thus, taking that ${A}_{av}=\frac{{S}_{AE}}{D}$, instead of (28), we have (using that ${A}_{av}=\frac{{S}_{AE}}{D}=\frac{S}{T}\frac{T}{D}={U}_{av}\frac{T}{D}$ and that U

_{m}~ A

_{m}, i.e., U

_{m}= δA

_{m}with $\delta \cong 1$ for $\frac{{\tau}_{a}}{T}\ll 1$ [19])

_{m}only in a given experiment, where ${\tau}_{a}$ and C are constant. Thus, we can write that

_{m}, i.e., ${T}^{*}>0$). Indeed, for T* > 2 we can take into account that erf2 $\cong $ 1, so the first term can be neglected as compared to $\frac{\sqrt{\pi}}{4}$ (T* exp(−T*

^{2}) = 0.037 for T* = 2). Thus, ${T}^{*}>2$ also means that in $\frac{\sqrt{\pi}}{4}\left(1-\frac{2}{\sqrt{\pi}}{T}^{*}exp\left(-{T}^{*2}\right)\right)$ the second term is less than 0.05, and we obtain $I\cong const=\frac{\sqrt{\pi}}{4}$. The ${T}^{*}>2$ requirement leads also to the condition that $\frac{{U}_{M}}{C}>8$ (see also (22)). Thus, we have

_{av}and A

_{m}, as given by Equation (29a). Since the duration time approaches its true value (and $\frac{1}{2ln\frac{{A}_{m}}{C}}$ goes to zero), only asymptotically for very large values of $\frac{{A}_{m}}{C}$, $\frac{1}{2ln\frac{{A}_{m}}{C}}$ can also appear in all relations where the duration time is present. The deviation is less than 5% only if $\frac{{A}_{m}}{C}\cong {10}^{6}$, and in most of the experiments $\frac{{A}_{m}}{C}$ can be much less than this limiting value. In addition, since the definition of A

_{av}contains the duration time, there is a transfer correction term, $\theta $, in (29a), (30), and (31), which depends on A

_{m}(and, of course, goes to zero for increasing A

_{m}, i.e., by decreasing the $\frac{{\tau}_{a}}{T}$ ratio). This term can be calculated from the slope of the experimental $log{A}_{m}$ versus the $logD$ experimental plot. Relation (24), which expresses the U

_{m}(~A

_{m})-dependence of the a (in Equation (12)) or B (in (14) and (23)) proportionality factors, with the exponent φ, leads to (25), (32), and (37), which can offer a denouement of the enigmas if $\phi $ is close unity. Equation (39) shows that the slope of the $ln\frac{{E}_{AE}}{{S}_{AE}}$ versus $ln{A}_{m}$ indeed should be close to unity (i.e., independent of the parameters $\frac{1}{2ln\frac{{A}_{m}}{C}}$, φ, and θ) in accordance with the prediction (10), while the slope of the $ln\frac{{E}_{AE}}{{S}_{AE}}$ versus the $ln{A}_{av}$ plot should be $\frac{1}{z}$ times larger (see Equation (40)). On the contrary, the slope of the $ln{E}_{AE}$ versus the $ln{A}_{av}$ plot has a larger slope than that of the $ln{E}_{AE}$ versus the $ln{A}_{m}$ plot, providing a smaller deviation than the expected value of 3. It will be shown in the next chapter that when choosing properly the centre of the window of fit on the A

_{m}axis and keeping it fixed for the different scaling plots, the value of θ, and, thus, z, can be kept constant. Thus, we will obtain that the conclusions drawn from different experimentally determined scaling exponents are in good agreement with each other and are consistent with a γ = 2 MF value.

## 3. Analysis and Discussion of Experimental Data on Scaling Exponents

_{45}Co

_{5}Mn

_{36.6}In

_{13.4}as well as Ni

_{49}Fe

_{18}Ga

_{27}Co

_{6}compositions (denoted by alloy A and B, respectively, in the following), during martensitic transformations, will be analysed. The details of the AE measurements on Ni

_{45}Co

_{5}Mn

_{36.6}In

_{13.4}are described in [26]. A very similar setup and data acquisition were applied for the AE measurements on the Ni

_{49}Fe

_{18}Ga

_{27}Co

_{6}single crystal (the results of which have not been published yet [27]). In both cases, the AE measurements were carried out with Sensophone AED 404 Acoustic Emission Diagnostic Equipment (Geréb and Co., Ltd., Budapest, Hungary) with a piezoelectric sensor (MICRO-100s from Physical Acoustics Corporation, Princeton Junction, NJ, USA). The sampling rate was 16 MHz, and the setup had a band-pass from 30 KHz to 1 MHz. A 30 dB preamplifier and a main amplifier (logarithmic gain) with a 90 dB dynamic range were used. The threshold level was 38 dB, and logarithmic data binning was used. We will just reuse the data obtained in [26,27] for the analysis of the relations predicted in the previous chapter.

#### 3.1. Relations between A_{m} and A_{av}, A_{m} and t_{m}, and A_{m} and D

_{m}and t

_{m}, since it provides the experimental check of the A

_{m}(U

_{m})-dependence of the B parameter in (14). Since the A

_{m}and t

_{m}parameters are free of threshold and transfer effects, in this case the fitting can be made from the beginning, up to the upper bound of the fitting window. Figure 1 shows the $log{A}_{m}$ versus $log{t}_{m}$ for cooling in alloy A (a) at small, constant, external magnetic field (B = 250 mT) and for heating in alloy B (b) (at B = 0). The slope of this plot is different from unity and is given by Equation (25). It can be seen that, indeed, straight lines are obtained up to certain upper bounds, A

_{ub}= 40 mV and A

_{ub}= 20 mV for alloys A and B, respectively. These values will be used on the A

_{m}axes in the following fits as well. The upper bound, and the scatter of points above it, is most probably caused by the possible overlap of avalanches and by the small numbers of hits at large amplitudes. The rise time, as compared to the duration time, is very short due to the long exponentially decaying tail of the expression (12) and the overlapping of avalanches can result in a reduced effective A

_{m}and increased t

_{m}. From the slopes of the straight lines (2.4 ± 0.1 and 2.2 ± 0.2, for A and B, respectively) $\phi =0.6\mp 0.1$ and $\phi =0.6\pm 0.1$ are obtained.

_{m}axes in the following fits as well.

_{m}-dependence of parameter θ (see Equation (30)). It is worth noting that the above three plots in Figure 1, Figure 2 and Figure 3 already provide the values for all the three fitting parameters (z, φ, and θ) used in the previous chapter, for both alloys: $z=0.74\pm 0.07$, $\phi =0.6\pm 0.1$, and $\theta $ = 0.77 ± 0.08 as well as $z=0.62\pm 0.07$, $\phi =0.6\pm 0.1,$ and $\theta =0.6\pm 0.1$, for alloys A and B, respectively. It can be seen that they, taking also into account the error bars of the original exponents, fulfil nicely the predicted relation (31). In the following, from the exponents of other scaling relations, we can collect more data on the above parameters and can obtain their average values too.

#### 3.2. Scaling Relations between E_{AE}, S_{AE}, $\frac{{E}_{AE}}{{S}_{AE}}$, and the Amplitude, A_{m}

_{m}) and, thus, can used as a check of the reliability of the AE measurements.

#### 3.3. Scaling Relation between $\frac{{E}_{AE}}{D}$ and the Amplitude as Well as between ${S}_{AE}$ and the Duration Time, D

_{m}-dependence of the proportionality factor between the two scaling parameters A

_{m}and t

_{m}(see Equations (14) and (23)), and the average values of these for the investigated two alloys are summarized in Table 1. It can be seen that they are the same in both alloys. It is important to emphasize that φ = 0.8 can also give account for the observed enigma for the energy-amplitude- and size-amplitude-scaling relations. Furthermore, the obtained results are in very good agreement with the $\gamma =2$ MF value: in Equations (32) and (37), the values of 2 and 3 belong to this.

## 4. Temporal Shape of Avalanches

_{m}and t

_{m}as the two scaling parameters, which are not distorted by the transfer properties. Furthermore, as one can expect from Equation (14), these are not independent from each other. Thus, we investigate and compare two cases:

- (i)
- assume that B is constant in (14), and both the voltage and time scales will be normalized by A
_{m}; - (ii)
- assume that the scaling parameters are not proportional to each other, but the $\frac{{A}_{m}}{{t}_{m}}~{A}_{m}^{\phi}$ relation holds (see Equations (14), (23) and (24)), i.e., the voltage scale will be reduced by A
_{m}and the time scale by ${A}_{m}^{1-\phi}$.

## 5. Relation between the Energy and Amplitude for Analysis of Multi-Avalanche Processes

_{m}related to overlap of the avalanches and/or to the small number of events per box. Thus, for alloy B, the dashed line is not the line fitted to the points, as it just shows a line with slope 0.17, as expected from φ = 0.83 (see Table 1).

## 6. Conclusions

_{m}, and maximum time (raising time), t

_{m}, then the two scaling parameters are interrelated by ${U}_{m}\left(~{A}_{m}\right)=B{t}_{m}$ (Equation (14)). Here, the parameter B is not constant but can be dependent on A

_{m}. From the analysis of AE measurements on the martensitic transformations in two different single-crystalline shape memory alloys, it was obtained that

- (i)
- from the relation between measured maximum amplitude (A
_{m}~ U_{m}) and t_{m}, the value of φ could be determined, and $\phi =0.73$ was obtained (the same values in both alloys); - (ii)
- the φ parameter appears in the expression of the power exponents for the relation between the energy and A
_{m}as well as between the area and A_{m}; these are $3-\phi $ and $2-\phi $, respectively, which provide a denouement of the enigma; - (iii)
- experimental values of exponents of different scaling relations between the measured AE parameters (energy, area, amplitudes, duration time) are consistent with the above relations;
- (iv)
- using A
_{m}and ${A}_{m}^{1-\phi}$ parameters for reducing the voltage and time scales, respectively, nice, common temporal-avalanche shapes were obtained for different bins of area.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**$log{A}_{m}$ versus $log{t}_{m}$ plots for cooling in alloy A (

**a**) at small, constant, external magnetic field (B = 250 mT) and for heating in alloy B (

**b**) (at B = 0).

**Figure 2.**Relation between A

_{av}and A

_{m}for cooling (austenite to martensite) transformation in alloy A (

**a**) at small, constant, external magnetic field (B = 250 mT) and for heating in alloy B (

**b**) (at B = 0). The slopes and the values of the centre of the fits are $z=0.74\mp 0.07$ and $\frac{{A}_{m}}{C}=30$ (C = 3.5 mV), as well as $z=0.62\mp 0.07$ and $\frac{{A}_{m}}{C}=40$ (C = 0.4 mV), for A and B, respectively.

**Figure 3.**$log{A}_{m}$ versus $logD$ plots for cooling in alloy A (

**a**) at small, constant, external magnetic field (B = 250 mT) and for heating in alloy B (

**b**) (at B = 0). The slopes (1.3 ± 0.1 and 1.9 ± 0.1, respectively), using again the same fitting windows with the same mid values of A

_{m}as in Figure 2, provide $\theta =\frac{1}{1.3}=0.77\mp 0.08$ and $\theta =0.6\pm 0.1$ for alloys A and B, respectively.

**Figure 4.**$log\frac{{E}_{AE}}{{S}_{AE}}$ versus $log{A}_{m}$ plots for cooling in alloy A (

**a**) at small, constant, external magnetic field (B = 250 mT) and for heating in alloy B (

**b**) (at B = 0).

**Figure 5.**$log\frac{{E}_{AE}}{{S}_{AE}}$ versus $log{A}_{av}$ plot for heating in alloy B (at B = 0).

**Figure 6.**$log{S}_{AE}$ versus $log{A}_{m}$ plots for cooling in alloy A (

**a**) at small, constant, external magnetic field (B = 250 mT) and for heating in alloy B (

**b**) (at B = 0).

**Figure 7.**$log{E}_{AE}$ versus $log{A}_{m}$ plots for cooling in alloy A (

**a**) at small, constant, external magnetic field (B = 250 mT) and for heating in alloy B (

**b**) (at B = 0).

**Figure 9.**$log\frac{{E}_{AE}}{D}$ versus $log{A}_{m}$ plots for cooling in alloy A (

**a**) at small, constant, external magnetic field (B = 250 mT) and for heating in alloy B (

**b**) (at B = 0). The slopes are 1.6 ± 0.1 and 1.65 ± 0.05, respectively.

**Figure 10.**$log\frac{{E}_{AE}}{D}$ versus $log{A}_{av}$ plot for heating in alloy B (at B = 0). The slope is 2.30 ± 0.05.

**Figure 11.**$log{S}_{AE}$ versus $logD$ plot for heating in alloy B (at B = 0). The slope is sensitive to the window of fit: e.g., fitting between 0.2 $\mathrm{mS}$ and 3 $\mathrm{mS}$, the slope is $2.14\pm 0.17$.

**Figure 12.**Normalized ${U}^{*}\left({t}^{*}\right)$ functions obtained by scaling both the voltage and time scales by the peak amplitude, A

_{m}, for heating in alloy B (at B = 0) at different bins of avalanche area, S.

**Figure 13.**Normalized ${U}^{*}\left({t}^{*}\right)$ functions obtained by scaling the voltage with A

_{m}and time scales by ${A}_{m}^{1-\phi}$ (φ = 0.74: see Table 1) for heating in alloy B (at B = 0) at different bins of avalanche area, S.

**Figure 14.**$log\frac{{E}_{AE}}{{A}_{m}^{2}}$ versus $log{A}_{m}$ functions for cooling in alloy A (

**a**) at small, constant, external magnetic field (B = 250 mT) and for heating in alloy B (

**b**) (at B = 0 mT).

**Table 1.**Values of $\phi $ calculated from different relations for alloy A and B, respectively. It can be seen that they are very similar, and the average value for the two alloys is φ

_{av}≅ 0.8 ± 0.1.

Equation | Value of φ | |
---|---|---|

Alloy A | Alloy B | |

(25) | 0.6 ± 0.1 | 0.6 ± 0.1 |

(32) | 0.8 ± 0.1 | 0.90 ± 0.08 |

(37) | 0.9 ± 0.1 | 1.0 ± 0.1 |

Average | 0.77 ± 0.11 | 0.83 ± 0.13 |

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**MDPI and ACS Style**

Kamel, S.M.; Samy, N.M.; Tóth, L.Z.; Daróczi, L.; Beke, D.L.
Denouement of the Energy-Amplitude and Size-Amplitude Enigma for Acoustic-Emission Investigations of Materials. *Materials* **2022**, *15*, 4556.
https://doi.org/10.3390/ma15134556

**AMA Style**

Kamel SM, Samy NM, Tóth LZ, Daróczi L, Beke DL.
Denouement of the Energy-Amplitude and Size-Amplitude Enigma for Acoustic-Emission Investigations of Materials. *Materials*. 2022; 15(13):4556.
https://doi.org/10.3390/ma15134556

**Chicago/Turabian Style**

Kamel, Sarah M., Nora M. Samy, László Z. Tóth, Lajos Daróczi, and Dezső L. Beke.
2022. "Denouement of the Energy-Amplitude and Size-Amplitude Enigma for Acoustic-Emission Investigations of Materials" *Materials* 15, no. 13: 4556.
https://doi.org/10.3390/ma15134556