# Scaling and Complexity of Stress Fluctuations Associated with Smooth and Jerky Flow in FeCoNiTiAl High-Entropy Alloy

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Section

_{86}-Al

_{7}-Ti

_{7}(atomic percent) HEA samples were fabricated by arc-melting pure elements under a Ti-gettered high-purity argon atmosphere, using the same process as in Ref. [40] (see also [41] for detail on similar fabrication processes). The elements used for fabrication had a purity of at least 99.99 weight percent. All the alloy ingots were first repeatedly melted at least six times to ensure chemical homogeneity and then drop-cast into a 60 mm × 20 mm × 5 mm copper mold. The ingots were homogenized at 1150 °C for 2 h, water-quenched to room temperature (T), and cold rolled with a total reduction of 70% at room temperature. The sheets were then recrystallized at 1150 °C for ~1 min and furnace-cooled to 700 °C. The samples were then aged at 780 °C for 4 h and cooled to ambient temperature by quenching in water. The heat treatment occurred under vacuum (less than 0.001 MPa) using a rate of 20 °C min

^{−1}.

^{−2}s

^{−1}. Two samples were tested at each temperature. Specimens were first heated to the desired testing temperature at a rate of 40 °C min

^{−1}and then held at the testing temperature for 10 min before tensile tests commenced. The tensile-loading direction was parallel to the rolling direction.

#### 2.2. Analysis of Stress Fluctuations

## 3. Results

#### 3.1. Plastic Deformation Behavior

_{1}phase precipitates (cf., e.g., [41]). However, investigation of the strengthening mechanisms which govern this slow time-scale deformation behavior goes beyond the scope of the present paper and will be discussed elsewhere.

#### 3.2. RCMSE and MF Analysis

_{min}did not exceed 0.06, and the error in f(α

_{min}) was less than 0.12 in all such cases. It is expected that further increases in the time and force resolution will allow for improving the determination of the MF behavior of low-amplitude signals.

_{max}− α

_{min}[Figure 6b], which characterizes the signal heterogeneity regarding its local singularity (cf. [17]): a wide spectrum corresponds to a high inhomogeneity, while Δα would be equal to 0 for the ideal case of a uniform fractal. In the cases when f(α) descends below zero (see Figure 5 and the comments in the previous paragraph), the extreme α values were determined at the level of f = 0.

_{min}) takes on a noticeably positive value for these fluctuations (see Figure 5), which indicates a high degree of uniformity associated with the most singular events [26], which may be attributed to a tendency to a periodic occurrence of such discontinuities. The spectrum width becomes quite high for the part 500 °C (II) preceding the onset of type-C serrations. It can also be seen in Figure 5 and Figure 6a that α

_{min}decreases considerably in comparison with its value in region 500 °C (I), which testifies to an increase in the maximum local singularity of the fluctuations. Therewith, f(α

_{min}) is also positive in region (II), in consistence with the occurrence of large stress fluctuations directly visible on the detrended signal in Figure 3c. Finally, the deformation noise observed at 700 °C is also characterized by a high heterogeneity and local singularity, as reflected in a relatively large Δα and small α

_{min}, and could also be expected from the RCMSE analysis. Overall, the increase in the intensity of stress fluctuations, be it due to a temperature increase (700 °C), or due to changes emerging after some deformation [500 °C (II)], led to stronger heterogeneity and singularity in terms of MF analysis. As far as f(α

_{min}) is concerned, it is zero at 700 °C, which is similar to what was observed at the other two temperatures outside the PLC domain.

_{min}) and heterogeneous (the greatest Δα) behavior was found at 300 °C, which showed oscillations with relatively low amplitudes, so that the deformation noise could have a notable influence on the MF scaling behavior [see Figure 3b]. The oscillations were more intense at 400 °C such that the domination of one kind of signal may explain a narrower spectrum than at 300 °C. This difference between the signals at 300 °C and 400 °C may also be responsible for the overall lower sample entropy curve at 400 °C, as discussed above (Figure 4). In contrast to these macroscopically smooth regions I, the large oscillations corresponding to type-A serrations (regions II) rendered very similar f(α) curves for both temperatures (cf. α and Δα values in Figure 6). Therewith, it can be seen that the heterogeneity and the maximum local singularity diminish upon the onset of type-A serrations, in agreement with their regular character, usually ascribed to a repetitive propagation of deformation bands along the specimen [17,26,27]. Also, the curve traced in Figure 4 for 400 °C (II) illustrates an increasing f(α

_{min}), as could have been expected from the discussion of the above examples.

## 4. Discussion

_{min}) in both regions analyzed at 500 °C (Figure 5), which attracts attention to an additional aspect. Indeed, positive f(α

_{min})-values were also observed at all temperatures within the PLC domain, including 300 °C and 400 °C corresponding to type A behavior. Therefore, this tendency may be intrinsic for the PLC effect that implies a recurrent occurrence of macroscopic stress serrations representing the regions with the strongest discontinuities on the deformation curve [49]. On the contrary, the observation of a comparable trend for the visually irregular stress fluctuations at 500 °C (I) is unexpected because such a trend does not show up for similar “deformation noises” at all temperatures beyond the PLC domain. Therefore, this finding allows for a supposition that the conditions leading to the PLC effect may also promote a tendency to recurring behavior in the fine-scale deformation processes. Vice versa, this peculiarity is a candidate for further investigation as a possible indicator of approaching a macroscopic instability.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Analytical Methods

#### Appendix A.1. Refined Composite Multiscale Entropy Analysis

_{i}the ith point of the detrended time series data, and N is the total number of data points of detrended data. Here, the length of the coarse-grained time series ${y}_{k,j}^{\tau}$ is N/τ [59]. After constructing the coarse-grained time series, create the template vector, ${y}_{k,i}^{\tau ,m}$, of dimension m:

_{th}template vectors of dimension m:

#### Appendix A.2. Multifractal Analysis

^{−D}(the relationship is rigorous in the limit l → 0), where the exponent, D = 1, is the topological dimension of the one-dimensional space on which the interval is defined. For a scale-invariant porous object, the number of the filled sites can be characterized by a similar expression where the exponent is no longer an integer and is instead referred to as the fractal dimension, f. Such a simple approach is, however, insufficient for heterogeneous objects. Besides the possible multiplicity of construction rules for the geometrical support itself, real objects carry nonconstant physical quantities (e.g., mechanical stress in the case of plastic deformation [63]) so that complexity may arise even for a quantity defined on a continuous support. A direct description of such complex objects would be unintelligible, needing a large table of numbers (infinite in the mathematical limit) providing the positions of all non-empty sites and the corresponding values of the physical quantity. However, it occurs that many natural phenomena possess the property of scale invariance in a statistical sense [44]. This feature made it possible to extend the fractal approach to the MF formalism [64]. The latter is based on the examination of the scaling of a local probabilistic measure, µ, which is defined to characterize the intensity of variation of the signal. For detrended time series, like those presented in Figure 2a–c of the paper, the idea is to cover the analyzed time interval with a grid with a step, δt, and determine the singularity strength, α, for each location from the following relationship which describes the scaling of the local measure:

^{α}for δt → 0,

^{α}

^{−1}, indicates a local discontinuity when α < 1. In particular, α

_{min}corresponds to the location with the densest measure.

_{j})|, where the index, j, corresponds to the data points within the considered δt box. Furthermore, as the least measured values are the most overshadowed by undesirable noise, some authors limit the analysis to the positive qs. Nevertheless, Figure 5 of the paper shows that the two branches of the f(α) dependence, which correspond to different signs of the moment q, change with temperature in a similar manner. It can thus be suggested that the right descending branch corresponding to the negative qs is determined correctly, providing that the negative q is not varied in too large of a range. In the present investigation, it was swept between 12 and −7. Finally, Figure A1 illustrates the feasibility of the MF analysis by comparing the scaling of the partition functions, Σ

_{α}(δt, q), for one of the real detrended time series and for a random signal over a time interval of the same length as the typical deformation curve. The necessity of such a precaution is obvious from plot (b). The figure shows that although the partition functions rapidly converge to the same slope, which agrees with the nonfractal structure of the uniform noise corresponding to a singularity spectrum consisting of a single point, (1, 1), the dependences form a narrow fan with decreasing the box size towards the smallest time step. Even though this fan collapses for a long enough random signal, it may still give rise to an apparent singularity spectrum for a short-time series. Accordingly, such an apparent spectrum was calculated and compared to the spectra obtained for the deformation curves [cf. Figure 5 in the main text]. Finally, plot (a) demonstrates that the MF scaling is conveniently determined for the detrended signal.

**Figure A1.**Example of a family of partition functions, whose scaling gives estimates of α values. Different lines correspond to different q in the range from 12 to −7 (

**a**) Results of calculations for a detrended signal at 300 °C (the initial part denoted in the paper as I). (

**b**) Similar calculations for a generated random noise.

## References

- Cantor, B.; Chang, I.T.H.; Knight, P.; Vincent, A.J.B. Microstructural development in equiatomic multicomponent alloys. Mater. Sci. Eng. A
**2004**, 375–377, 213–218. [Google Scholar] [CrossRef] - Yeh, J.-W. Alloy Design Strategies and Future Trends in High-Entropy Alloys. JOM
**2013**, 65, 1759–1771. [Google Scholar] [CrossRef] - George, E.P.; Raabe, D.; Ritchie, R.O. High-entropy alloys. Nat. Rev. Mater.
**2019**, 4, 515–534. [Google Scholar] [CrossRef] - Miracle, D.B.; Senkov, O.N. A critical review of high entropy alloys and related concepts. Acta Mater.
**2017**, 122, 448–511. [Google Scholar] [CrossRef] - Samoilova, O.; Pratskova, S.; Shaburova, N.; Moghaddam, A.; Trofimov, E. Effect of Au addition on the corrosion behavior of Al
_{0.25}CoCrFeNiCu_{0.25}high-entropy alloy in 0.5 M H_{2}SO_{4}solution. Mater. Lett.**2023**, 350, 134932. [Google Scholar] [CrossRef] - O’Brien, S.; Christudasjustus, J.; Delvecchio, E.; Birbilis, N.; Gupta, R. Microstructure and corrosion of CrFeMnV multi-principal element alloy. Corros. Sci.
**2023**, 222, 111403. [Google Scholar] [CrossRef] - Yin, Y.; Tan, Q.; Yang, N.; Chen, X.; Ren, W.; Liu, L.; Chen, H.; Atrens, A.; Ma, N.; Huang, H.; et al. Cost-effective and facile route to ultrafine-microstructure high-entropy alloy for cryogenic applications. Mater. Sci. Eng. A
**2023**, 881, 145408. [Google Scholar] [CrossRef] - Thurston, K.V.S.; Gludovatz, B.; Hohenwarter, A.; Laplanche, G.; George, E.P.; Ritchie, R.O. Effect of temperature on the fatigue-crack growth behavior of the high-entropy alloy CrMnFeCoNi. Intermetallics
**2017**, 88, 65–72. [Google Scholar] [CrossRef] - Liu, K.; Komarasamy, M.; Gwalani, B.; Shukla, S.; Mishra, R.S. Fatigue behavior of ultrafine grained triplex Al
_{0.3}CoCrFeNi high entropy alloy. Scr. Mater.**2019**, 158, 116–120. [Google Scholar] [CrossRef] - Zhang, Y.; Stocks, G.M.; Jin, K.; Lu, C.; Bei, H.; Sales, B.C.; Wang, L.; Béland, L.K.; Stoller, R.E.; Samolyuk, G.D.; et al. Influence of chemical disorder on energy dissipation and defect evolution in concentrated solid solution alloys. Nat. Commun.
**2015**, 6, 8736. [Google Scholar] [CrossRef] - El-Atwani, O.; Alvarado, A.; Unal, K.; Fensin, S.; Hinks, J.A.; Greaves, G.; Baldwin, J.K.S.; Maloy, S.A.; Martinez, E. Helium implantation damage resistance in nanocrystalline W-Ta-V-Cr high entropy alloys. Mater. Today Energy
**2021**, 19, 100599. [Google Scholar] [CrossRef] - Joseph, J.; Haghdadi, N.; Annasamy, M.; Kada, S.; Hodgson, P.D.; Barnett, M.R.; Fabijanic, D.M. On the enhanced wear resistance of CoCrFeMnNi high entropy alloy at intermediate temperature. Scr. Mater.
**2020**, 186, 230–235. [Google Scholar] [CrossRef] - Shim, S.H.; Pouraliakbar, H.; Hong, S.I. High strength dual fcc phase CoCuFeMnNi high-entropy alloy wires with dislocation wall boundaries stabilized by phase boundaries. Mater. Sci. Eng. A
**2021**, 825, 141875. [Google Scholar] [CrossRef] - Chen, B.; Li, X.; Niu, Y.; Yang, R.; Chen, W.; Yusupu, B.; Jia, L. A dual-phase CrFeNbTiMo refractory high entropy alloy with excellent hardness and strength. Mater. Lett.
**2023**, 337, 133958. [Google Scholar] [CrossRef] - Lebyodkin, M.A.; Lebedkina, T.A.; Brechtl, J.; Liaw, P.K. Serrated Flow in Alloy Systems. In High-Entropy Materials: Theory, Experiments, and Applications; Brechtl, J., Liaw, P.K., Eds.; Springer International Publishing: Cham, Switzerland, 2021; pp. 523–644. [Google Scholar] [CrossRef]
- D’Anna, G.; Nori, F. Critical dynamics of burst instabilities in the Portevin-Le Chatelier effect. Phys. Rev. Lett.
**2000**, 85, 4096–4099. [Google Scholar] [CrossRef] - Bharathi, M.S.; Lebyodkin, M.; Ananthakrishna, G.; Fressengeas, C.; Kubin, L.P. Multifractal Burst in the Spatiotemporal Dynamics of Jerky Flow. Phys. Rev. Lett.
**2001**, 87, 165508. [Google Scholar] [CrossRef] - Yang, F.; Luo, H.; Pu, E.; Zhang, S.; Dong, H. On the characteristics of Portevin–Le Chatelier bands in cold-rolled 7Mn steel showing transformation-induced plasticity. Int. J. Plast.
**2018**, 103, 188–202. [Google Scholar] [CrossRef] - Ryu, W.H.; Ko, W.-S.; Isano, H.; Yamada, R.; Ahn, H.; Yoo, G.H.; Yoon, K.N.; Park, E.S.; Saida, J. Sustainable steady-state serrated flow induced by modulating deformation sequence in bulk metallic glass. J. Alloys Compd.
**2023**, 946, 169308. [Google Scholar] [CrossRef] - Lv, J.W.; Wei, C.; Shi, Z.L.; Zhang, S.; Zhang, H.R.; Zhang, X.Y.; Ma, M.Z. The size-dependence of compressive mechanical properties and serrated-flow behavior of Ti-based bulk metallic glass. Mater. Sci. Eng. A
**2022**, 857, 143968. [Google Scholar] [CrossRef] - Knapek, M.; Pešička, J.; Lukáč, P.; Minárik, P.; Král, R. Peculiar serrated flow during compression of an FeAlCrMo medium-entropy alloy. Scr. Mater.
**2019**, 161, 49–53. [Google Scholar] [CrossRef] - Han, Z.; Ding, C.; Liu, G.; Yang, J.; Du, Y.; Wei, R.; Chen, Y.; Zhang, G. Analysis of deformation behavior of VCoNi medium-entropy alloy at temperatures ranging from 77 K to 573 K. Intermetallics
**2021**, 132, 107126. [Google Scholar] [CrossRef] - Brechtl, J.; Chen, S.; Lee, C.; Shi, Y.; Feng, R.; Xie, X.; Hamblin, D.; Coleman, A.M.; Straka, B.; Shortt, H.; et al. A Review of the Serrated-Flow Phenomenon and Its Role in the Deformation Behavior of High-Entropy Alloys. Metals
**2020**, 10, 1101. [Google Scholar] [CrossRef] - Estrin, Y.; Kubin, L.P. Spatial coupling and propagative plastic instabilities. In Continuum Models for Materials with Microstructure; John Wiley & Sons: Hoboken, NJ, USA, 1995; pp. 395–450. [Google Scholar]
- Prigogine, I.; Nicolis, G. Self-organisation in nonequilibrium systems: Towards a dynamics of complexity. In Bifurcation Analysis; Springer: Berlin/Heidelberg, Germany, 1985; pp. 3–12. [Google Scholar] [CrossRef]
- Lebyodkin, M.A.; Lebedkina, T.A.; Jacques, A. Multifractal Analysis of Unstable Plastic Flow; Nova Science: Hauppauge, NY, USA, 2009; Available online: https://books.google.com/books?id=YiQpAQAAMAAJ (accessed on 23 August 2011).
- Kubin, L.P.; Fressengeas, C.; Ananthakrishna, G. Chapter 57 Collective behaviour of dislocations in plasticity. In Dislocations Solids; Nabarro, F.R.N., Duesbery, M.S., Eds.; Elsevier: Amsterdam, The Netherlands, 2002; pp. 101–192. [Google Scholar] [CrossRef]
- Weiss, J.; Rhouma, W.B.; Richeton, T.; Dechanel, S.; Louchet, F.; Truskinovsky, L. From Mild to Wild Fluctuations in Crystal Plasticity. Phys. Rev. Lett.
**2015**, 114, 105504. [Google Scholar] [CrossRef] [PubMed] - Zaiser, M. Scale invariance in plastic flow of crystalline solids. Adv. Phys.
**2006**, 55, 185–245. [Google Scholar] [CrossRef] - Zuev, L.B.; Barannikova, S.A. Autowave physics of material plasticity. Crystals
**2019**, 9, 458. [Google Scholar] [CrossRef] - Maaß, R.; Derlet, P. Micro-plasticity and recent insights from intermittent and small-scale plasticity. Acta Mater.
**2018**, 143, 338–363. [Google Scholar] [CrossRef] - Mudrock, R.N.; Lebyodkin, M.A.; Kurath, P.; Beaudoin, A.J.; Lebedkina, T.A. Strain-rate fluctuations during macroscopically uniform deformation of a solution-strengthened alloy. Scr. Mater.
**2011**, 65, 1093–1096. [Google Scholar] [CrossRef] - Baró, J.; Corral, Á.; Illa, X.; Planes, A.; Salje, E.K.; Schranz, W.; Soto-Parra, D.E.; Vives, E. Statistical similarity between the compression of a porous material and earthquakes. Phys. Rev. Lett.
**2013**, 110, 088702. [Google Scholar] [CrossRef] - Dimiduk, D.M.; Woodward, C.; LeSar, R.; Uchic, M.D. Scale-Free Intermittent Flow in Crystal Plasticity. Science
**2006**, 312, 1188–1190. [Google Scholar] [CrossRef] - Lebedkina, T.A.; Bougherira, Y.; Entemeyer, D.; Lebyodkin, M.A.; Shashkov, I.V. Crossover in the scale-free statistics of acoustic emission associated with the Portevin-Le Chatelier instability. Scr. Mater.
**2018**, 148, 47–50. [Google Scholar] [CrossRef] - Lebyodkin, M.; Bougherira, Y.; Lebedkina, T.; Entemeyer, D. Scaling in the Local Strain-Rate Field during Jerky Flow in an Al-3%Mg Alloy. Metals
**2020**, 10, 134. [Google Scholar] [CrossRef] - Lebedkina, T.A.; Zhemchuzhnikova, D.A.; Lebyodkin, M.A. Correlation versus randomization of jerky flow in an AlMgScZr alloy using acoustic emission. Phys. Rev. E
**2018**, 97, 013001. [Google Scholar] [CrossRef] [PubMed] - Lebyodkin, M.A.; Kobelev, N.P.; Bougherira, Y.; Entemeyer, D.; Fressengeas, C.; Lebedkina, T.A.; Shashkov, I.V. On the similarity of plastic flow processes during smooth and jerky flow in dilute alloys. Acta Mater.
**2012**, 60, 844–850. [Google Scholar] [CrossRef] - Brechtl, J.; Feng, R.; Liaw, P.K.; Beausir, B.; Jaber, H.; Lebedkina, T.; Lebyodkin, M. Mesoscopic-scale complexity in macroscopically-uniform plastic flow of an Al
_{0.3}CoCrFeNi high-entropy alloy. Acta Mater.**2023**, 242, 118445. [Google Scholar] [CrossRef] - Yang, T.; Zhao, Y.L.; Tong, Y.; Jiao, Z.B.; Wei, J.; Cai, J.X.; Han, X.D.; Chen, D.; Hu, A.; Kai, J.J.; et al. Multicomponent intermetallic nanoparticles and superb mechanical behaviors of complex alloys. Science
**2018**, 362, 933–937. [Google Scholar] [CrossRef] - Wang, F.; Wu, J.; Guo, Y.; Shang, X.; Zhang, J.; Liu, Q. A novel high-entropy alloy with desirable strength and ductility designed by multi-component substitution for traditional austenitic alloys. J. Alloys Compd.
**2023**, 937, 168266. [Google Scholar] [CrossRef] - Brechtl, J.; Xie, X.; Liaw, P.K.; Zinkle, S.J. Complexity modeling and analysis of chaos and other fluctuating phenomena. Chaos Solitons Fractals
**2018**, 116, 166–175. [Google Scholar] [CrossRef] - Wu, S.-D.; Wu, C.-W.; Lin, S.-G.; Lee, K.-Y.; Peng, C.-K. Analysis of complex time series using refined composite multiscale entropy. Phys. Lett. A
**2014**, 378, 1369–1374. [Google Scholar] [CrossRef] - Feder, J. Fractals; Springer: Berlin/Heidelberg, Germany, 2013; Available online: https://books.google.com/books?id=mgvyBwAAQBAJ (accessed on 11 November 2013).
- Mandelbrot, B.B. The Fractal Geometry of Nature; Henry Holt and Company: New York, NY, USA, 1983; Available online: https://books.google.com/books?id=SWcPAQAAMAAJ (accessed on 16 July 2021).
- Roth, A.; Lebedkina, T.; Lebyodkin, M. On the critical strain for the onset of plastic instability in an austenitic FeMnC steel. Mater. Sci. Eng. A
**2012**, 539, 280–284. [Google Scholar] [CrossRef] - Chmelı, F.; Ziegenbein, A.; Neuhäuser, H.; Lukáč, P. Investigating the Portevin–Le Châtelier effect by the acoustic emission and laser extensometry techniques. Mater. Sci. Eng. A
**2002**, 324, 200–207. [Google Scholar] [CrossRef] - Mulford, R.; Kocks, U. New observations on the mechanisms of dynamic strain aging and of jerky flow. Acta Met.
**1979**, 27, 1125–1134. [Google Scholar] [CrossRef] - Lebyodkin, M.A.; Estrin, Y. Multifractal analysis of the Portevin–Le Chatelier effect: General approach and application to AlMg and AlMg/Al
_{2}O_{3}alloys. Acta Mater.**2005**, 53, 3403–3413. [Google Scholar] [CrossRef] - Brechtl, J.; Chen, S.Y.; Xie, X.; Ren, Y.; Qiao, J.W.; Liaw, P.K.; Zinkle, S.J. Towards a greater understanding of serrated flows in an Al-containing high-entropy-based alloy. Int. J. Plast.
**2019**, 115, 71–92. [Google Scholar] [CrossRef] - Jiang, W.; Yuan, S.; Cao, Y.; Zhang, Y.; Zhao, Y. Mechanical properties and deformation mechanisms of a Ni
_{2}Co_{1}Fe_{1}V0.5Mo_{0.2}medium-entropy alloy at elevated temperatures. Acta Mater.**2021**, 213, 116982. [Google Scholar] [CrossRef] - Guo, L.; Gu, J.; Gong, X.; Li, K.; Ni, S.; Liu, Y.; Song, M. Short-range ordering induced serrated flow in a carbon contained FeCoCrNiMn high entropy alloy. Micron
**2019**, 126, 102739. [Google Scholar] [CrossRef] - Costa, M.; Goldberger, A.L.; Peng, C.-K. Multiscale Entropy Analysis of Complex Physiologic Time Series. Phys. Rev. Lett.
**2002**, 89, 068102. [Google Scholar] [CrossRef] - Costa, M.; Peng, C.-K.; Goldberger, A.L.; Hausdorff, J.M. Multiscale entropy analysis of human gait dynamics. Phys. A Stat. Mech. Its Appl.
**2003**, 330, 53–60. [Google Scholar] [CrossRef] - Omidvarnia, A.; Zalesky, A.; Mansour, L.S.; Van De Ville, D.; Jackson, G.D.; Pedersen, M. Temporal complexity of fMRI is reproducible and correlates with higher order cognition. NeuroImage
**2021**, 230, 117760. [Google Scholar] [CrossRef] - Costa, M.; Goldberger, A.L.; Peng, C.-K. Multiscale entropy analysis of biological signals. Phys. Rev. E
**2005**, 71, 021906. [Google Scholar] [CrossRef] - Wang, J.; Shang, P.; Xia, J.; Shi, W. EMD based refined composite multiscale entropy analysis of complex signals. Phys. A Stat. Mech. Its Appl.
**2015**, 421, 583–593. [Google Scholar] [CrossRef] - Iliopoulos, A.C.; Nikolaidis, N.S.; Aifantis, E.C. Analysis of serrations and shear bands fractality in UFGs. J. Mech. Behav. Mater.
**2015**, 24, 1–9. [Google Scholar] [CrossRef] - Humeau-Heurtier, A.; Wu, C.; Wu, S.; Mahé, G.; Abraham, P. Refined Multiscale Hilbert-Huang Spectral Entropy and Its Application to Central and Peripheral Cardiovascular Data. IEEE Trans. Biomed. Eng.
**2016**, 63, 2405–2415. [Google Scholar] [CrossRef] [PubMed] - Castiglia, S.; Trabassi, D.; Conte, C.; Ranavolo, A.; Coppola, G.; Sebastianelli, G.; Abagnale, C.; Barone, F.; Bighiani, F.; De Icco, R.; et al. Multiscale Entropy Algorithms to Analyze Complexity and Variability of Trunk Accelerations Time Series in Subjects with Parkinson’s Disease. Sensors
**2023**, 23, 4983. [Google Scholar] [CrossRef] [PubMed] - Wu, S.; Wu, C.; Lin, S.; Wang, C.; Lee, K. Time Series Analysis Using Composite Multiscale Entropy. Entropy
**2013**, 15, 1069–1084. [Google Scholar] [CrossRef] - Costa, M.D.; Goldberger, A.L. Generalized Multiscale Entropy Analysis: Application to Quantifying the Complex Volatility of Human Heartbeat Time Series. Entropy
**2015**, 17, 1197–1203. [Google Scholar] [CrossRef] - Lebyodkin, M.A.; Lebedkina, T.A. Multifractal analysis of evolving noise associated with unstable plastic flow. Phys. Rev. E
**2006**, 73, 036114. [Google Scholar] [CrossRef] - Halsey, T.C.; Jensen, M.H.; Kadanoff, L.P.; Procaccia, I.; Shraiman, B.I. Fractal measures and their singularities: The characterization of strange sets. Phys. Rev. A
**1986**, 33, 1141–1151. [Google Scholar] [CrossRef] - Chhabra, A.; Jensen, R.V. Direct determination of the f(α) singularity spectrum. Phys. Rev. Lett.
**1989**, 62, 1327–1330. [Google Scholar] [CrossRef]

**Figure 1.**Example of (

**a**) optical imagery of the microstructure of the investigated alloy and (

**b**) the corresponding grain size distribution.

**Figure 2.**Representative examples of stress–time curves for four temperature values (200–600 °C) (

**a**) and their magnification (

**b**). The curves were recorded using a sampling time of 10 ms. The vertical dashed lines crossing the deformation curve recorded at 300 °C indicate the conventional way of detection of the critical strain for the onset of the PLC effect according to the first visible type-A serrations.

**Figure 3.**Representative examples of detrended deformation curves. (

**a**) Fluctuations about a smooth deformation curve at 200 °C; (

**b**) Type-A serrations at 300 °C. The vertical dashed line corresponds to its counterparts in Figure 2a,b; (

**c**) A portion of the detrended deformation curve before the first type-C serration at 500 °C. Also shown is the as-recorded idle noise during a short initial time interval before the load starts increasing; (

**d**) Frequency dependence of the power spectral density for the signal from Figure 3c. The spectrum was calculated over the interval t < 27 s, which does not include the ultimate oscillations.

**Figure 4.**Sample entropy dependences on the scale factor, τ, for detrended deformation curves recorded at different temperatures. The mark (I) refers to the analysis of the initial part, t ≤ 27 s, of the deformation curve at 500 °C.

**Figure 5.**“Singularity spectra”, f(α), of the deformation noise representing qualitatively different behaviors. The Roman numerals after the T values indicate the cases when the calculation was performed separately for the initial (I) and later (II) portions of the signals.

**Figure 6.**(

**a**) Position of the edges of the singularity spectra, α

_{min}and α

_{max}, relative to unity. (

**b**) The corresponding width of the spectrum at the level of f(α) = 0. The Roman numerals have the same meaning as in Figure 3.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lebyodkin, M.; Brechtl, J.; Lebedkina, T.; Wen, K.; Liaw, P.K.; Shen, T.
Scaling and Complexity of Stress Fluctuations Associated with Smooth and Jerky Flow in FeCoNiTiAl High-Entropy Alloy. *Metals* **2023**, *13*, 1770.
https://doi.org/10.3390/met13101770

**AMA Style**

Lebyodkin M, Brechtl J, Lebedkina T, Wen K, Liaw PK, Shen T.
Scaling and Complexity of Stress Fluctuations Associated with Smooth and Jerky Flow in FeCoNiTiAl High-Entropy Alloy. *Metals*. 2023; 13(10):1770.
https://doi.org/10.3390/met13101770

**Chicago/Turabian Style**

Lebyodkin, Mikhail, Jamieson Brechtl, Tatiana Lebedkina, Kangkang Wen, Peter K. Liaw, and Tongde Shen.
2023. "Scaling and Complexity of Stress Fluctuations Associated with Smooth and Jerky Flow in FeCoNiTiAl High-Entropy Alloy" *Metals* 13, no. 10: 1770.
https://doi.org/10.3390/met13101770