Recurrence analysis of the Portevin-Le Chatelier effect
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Recurrence analysis of the Portevin-Le Chatelier effect 
A. Sarkara*, Charles L. Webber Jr.b, P. Barata, P. Mukherjeea 
a Variable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700064, India 
b Department of Physiology, Loyola University Medical Center, 2160 S. First Avenue, 
Maywood, IL 60153, USA 
Abstract 
 Tensile tests were carried out by deforming polycrystalline samples of Al-
2.5%Mg alloy at room temperature in a wide range of strain rates where the Portevin-
Le Chatelier (PLC) effect was observed. The experimental stress-time series data have 
been analyzed using the recurrence analysis technique based on the Recurrence Plot 
(RP) and the Recurrence Quantification Analysis (RQA) to study the change in the 
dynamical behavior of the PLC effect with the imposed strain rate. Our study revealed 
that the RQA is able to detect the unique crossover phenomenon in the PLC dynamics.  
PACS: 62.20.Fe, 05.40.Fb, 05.45.Tp 
Keywords: Recurrence Plot, Recurrence Quantification Analysis, Portevin-Le 
Chatelier effect 
Introduction 
The Portevin-Le Chatelier (PLC) effect is one of the widely studied 
metallurgical phenomena, observed in many metallic alloys of technological 
importance [1-12]. It is a striking example of the complexity of the spatiotemporal 
dynamics, arising from the collective behavior of dislocations. In uniaxial loading with 
                                                 
* Corresponding author: apu@veccal.ernet.in 
constant imposed strain rate, the effect manifests itself as a series of serrations (stress 
drops) in the stress-time or strain curve. Each stress drop is associated with the 
nucleation of a band of localized plastic deformation, often designated as PLC band, 
which under certain conditions propagates along the sample. The microscopic origin 
of the PLC effect is the dynamic strain aging (DSA) [13-19] of the material due to the 
interaction between mobile dislocations and diffusing solute atoms. At the 
macroscopic scale, this dynamic strain aging leads to a negative strain rate sensitivity 
(SRS) of the flow stress and makes the plastic deformation nonuniform.  
In polycrystals three types of the PLC effect are traditionally distinguished on the 
qualitative basis of the spatial arrangement of localized deformation bands and the 
particular appearance of deformation curves [20, 21]. Three generic types of 
serrations: type A, B and C occur depending on the imposed strain rate. For 
sufficiently large strain rate, type A serrations are observed. In this case, the bands are 
continuously propagating and highly correlated.  The associated stress drops are small 
in amplitude [22,23]. If the strain rate is lowered, type B serrations with relatively 
larger amplitude occur around the uniform stress strain curve. These serrations 
correspond to intermittent band propagation. The deformation bands are formed ahead 
of the previous one in a spatially correlated manner and give rise to regular surface 
markings [22,23]. For even smaller strain rate, bands become static. This type C band 
nucleates randomly in the sample leading to large saw-tooth shaped serration in the 
stress strain curve and random surface markings [22,23]. 
From metallurgical point of view, the PLC effect is usually undesirable since it has 
detrimental influences like the loss of ductility and the appearance of surface markings 
on the specimen. Beyond its importance in metallurgy, the PLC effect is an epitome 
for a general class of nonlinear complex systems with intermittent bursts. The 
succession of plastic instabilities shares both physical and statistical properties with 
many other systems exhibiting loading-unloading cycles e.g. earthquakes. PLC effect 
is regulated by interacting mechanisms that operate across multiple spatial and 
temporal scales. The output variable (stress) of the effect exhibits complex 
fluctuations which contains information about the underlying dynamics. 
The PLC effect has been extensively studied over the last several decades with the 
goal being to achieve a better understanding of the small-scale processes and of the 
multiscale mechanisms that link the mesoscale DSA to the macroscale PLC effect. 
The technological goal is to increase the SRS to positive values in the range of 
temperatures and strain rates relevant for industrial processes. This would ensure 
material stability during processing and would eliminate the occurrence of the PLC 
effect. 
Due to a continuous effort of numerous researchers, there is now a reasonable 
understanding of the mechanisms and manifestations of the PLC effect. A review of 
this field can be found in Ref. [17,18]. The possibility of chaos in the stress drops of 
PLC effect was first predicted by G. Ananthakrishna et. al. [24] and latter by V. 
Jeanclaude et. al. [25].  This prediction generated a new enthusiasm in this field. In 
last few years, many statistical and dynamical studies have been carried out on the 
PLC effect [10-12, 26-32]. Analysis revealed two types of dynamical regimes in the 
PLC effect. At medium strain rate (type B) chaotic regime has been demonstrated [30, 
33], which is associated with the bell-shaped distribution of the stress drops. For high 
strain rate (type A) the dynamics is identified as self organized criticality (SOC) with 
the stress drops following a power law distribution [33]. The crossover between these 
two mechanisms has also been a topic of intense research for the past few years 
[29,33,34-36]. It is shown that the crossover from the chaotic to SOC dynamics is 
clearly signaled by a burst in multifractality [29,33].  
This crossover phenomenon is of interest in the larger context of dynamical 
systems as this is a rare example of a transition between two dynamically distinct 
states. Chaotic systems are characterized by the self similarity of the strange attractors 
and sensitivity to initial conditions quantified by fractal dimension and the existence 
of a positive Lyapunov exponent, respectively. On the contrary, the SOC dynamics is 
characterized by infinite number of degrees of freedom and a power law statistics.  
The general consensus that the dynamic strain aging is the cause behind the 
PLC effect suggests a discrete connection between the stress fluctuation and the band 
dynamics. We do not have a system of primitive equations to describe the dynamics of 
the band, so we must extract as much information as possible from the data itself. We 
use the stress data recorded during the plastic deformation for our analysis. However, 
we do not analyze these data blindly but in the framework of nonlinear dynamics as 
the band dynamics shows intermittency. In this work, we have carried out detailed 
recurrence analysis of the stress time data observed during the PLC effect to study the 
change in the dynamical behavior of the effect with the imposed strain rate. 
Experimental details: 
Substitutional Aluminum alloys with Mg as the primary alloying element are model 
systems for the PLC effect studies. These alloys have wide technological applications 
due to their advantageous strength to weight ratio. They show good ductility and can 
be rolled to large reductions and processed in thin sheets and are being extensively 
used in beverage packaging and other applications. However, the discontinuous 
deformation behavior of these alloys at room temperature rule them out from many 
important applications like in the automobile industry. These alloys exhibit the PLC 
effect for wide range of strain rates and temperatures. Under these conditions the 
deformation of these materials localize in narrow bands which leave undesirable band-
type macroscopic surface markings on the final products.  
 Tensile tests were conducted on flat specimens prepared from polycrystalline 
Al-2.5%Mg alloy. Specimens with gauge length, width and thickness of 25, 5 and 2.3 
mm, respectively were tested in an INSTRON (model 4482) machine. All the tests 
were carried out at room temperature (300K) and consequently there was only one 
control parameter, the applied strain rate. To monitor closely its influence on the 
dynamics of the PLC effect, strain rate was varied from 7.98×10-5 S-1 to 1.60×10-3 S-1. 
The PLC effect was observed through out the range. The stress-time response was 
recorded electronically at periodic time intervals of 0.05 seconds. Fig. 1 shows the 
observed PLC effect in a typical stress-strain curve for strain rate 1.20×10-3 S-1. The 
stress data shows an increasing trend due to the strain hardening effect. The trend is 
eliminated and analyses reported in this study are carried out on the resulting data. The 
inset in the Fig. 1 shows a typical segment of the trend corrected stress-strain curve. In 
the varied strain rate region we could observe type B, B+A and A serrations as 
reported [20,21]. We kept the sampling rate same for all the experiments. 
Consequently the number of data points was not same for different strain rate 
experiments. To analyze the data in similar footing we have carried our analysis on the 
stress data from the same strain region 0.02-0.10 for all strain rate experiments. 
Recurrence Analysis 
 Eckman, Kamphorst and Ruelle [37] proposed a new method to study the 
recurrences and nonstationary behaviour occurring in dynamical system. They 
designated the method as “recurrence plot” (RP). The method is found to be efficient 
in identification of system properties that cannot be observed using other conventional 
linear and nonlinear approaches. Moreover, the method has been found very useful for 
analysis of nonstationary system with high dimension and noisy dynamics. The 
method can be outlined as follows: given a time series {xi} of N data points, first the 
phase space vectors ui={xi,xi+τ,……,xi+(d-1)τ}are constructed using Taken’s time delay 
method. The embedding dimension (d) can be estimated from the false nearest 
neighbor method. The time delay (τ) can be estimated either from the autocorrelation 
function or from the mutual information method. The main step is then to calculate the 
N×N matrix 
)(, jiiji xxR −−Θ= ε ,               i,j=1,2,….,N        (14) 
where εi is a cutoff distance, ||..|| is a norm (we have taken the Euclidean norm), and 
Θ(x) is the Heavyside function. The cutoff distance εi defines a sphere centered at ix . 
If jx  falls within this sphere, the state will be close to ix and thus 1, =jiR . The binary 
values in jiR ,  can be simply visualized by a matrix plot with color black (1) and white 
(0). This plot is called the recurrence plot.  
 However, it is often not very straight forward to conclude about the dynamics of 
the system from the visual inspection of the RPs. Zbilut and Webber [38,39] 
developed the recurrence quantification analysis (RQA) to provide the quantification 
of important dynamical aspects of the system revealed through the plot. The RQA 
proposed by Zbilut and Webber is mostly based on the diagonal structures in the RPs. 
They defined different measures, the recurrence rate (REC) measures the fraction of 
black points in the RP, the determinism (DET) is the measure of the fraction recurrent 
points forming the diagonal line structure, the maximal length of diagonal structures 
(Lmax), the entropy (Shannon entropy of the line segment distributions) and the trend 
(measure of the paling of recurrent points away from the central diagonal). These 
variables are used to detect the transitions in the time series. Recently Gao [40] 
emphasized the importance of the vertical structures in RPs and introduced a 
recurrence time statistics corresponding to the vertical structures in RP. Marwan et. al. 
[41] extended Gao’s view and defined measures of complexity based on the 
distribution of the vertical line length. They introduced three new RP based measures: 
the laminarity, the trapping time (TT) and the maximal length of the vertical structures 
(Vmax). Laminarity is analogous to DET and gives the measure of the amount of 
vertical structure in the RP and represents laminar states in the system. TT contains 
information about the amount as well as the length of the vertical structure. Applying 
these measures to the logistic map data they found that in contrast to the conventional 
RQA measures, their measures are able to identify the laminar states i.e. chaos-chaos 
transitions. The vertical structure based measures were also found very successful to 
detect the laminar phases before the onset of life-threatening ventricular 
tachyarrhythmia [41]. Here we have applied the measures proposed by Marwan et. al. 
along with the traditional measures to find the effect of strain rate on the PLC effect. 
Results and Discussions 
 RP and RQA have been successfully applied to diverse fields starting from 
Physiology to Econophysics in recent years. A review of various applications of RPs 
and RQA can be found in the recent article by Marwan et. al. [42]. Here we extend the 
list of application of RPs and RQA and for the first time apply these methods to study 
the dynamical behavior of the PLC effect in Al-2.5%Mg alloy. In this study, we 
particularly concentrate on the strain rate region of 7.98×10-5 S-1 to 1.60×10-3 S-1.  The 
main goal is to demonstrate the ability of RQA to detect the unique crossover 
phenomenon observed in the PLC dynamics. 
 It has been shown that a RP analysis is optimal when the trajectory is embedded 
in a phase space reconstructed with an appropriate dimension d [38]. Such a 
dimension can be well estimated using a false nearest neighbor technique. The d-
dimensional phase space is then reconstructed using delay coordinate. The time delay 
τ can be estimated using the mutual information or the first zero of an auto correlation 
function. Based on the false nearest neighbor method we have chosen d to be 10 for all 
the strain rates. τ obtained from the mutual information were in the range 1-14 for 
different strain rate data. A parameter specific to the RP is the cutoff distance εi. εi is 
selected from the scaling curve of REC vs, εi as suggested in the literature [43].  Fig. 2 
shows the RPs of the stress fluctuations during the PLC effect at four different strain 
rates. From the visual inspection of the RPs it is easy to understand that the dynamical 
behavior of the PLC effect changes with the strain rate. However, it is wise to go for 
the RQA and quantify the difference in the PLC dynamics with strain rate. Fig. 3 
shows the variation of the various RQA variables with strain rate. It can be seen from 
the Fig. 3 that the RQA variables like DET and laminarity do not show any systematic 
variation with strain rate. Lmax, TT and Vmax decreased rapidly with strain rate and 
reached a plateau. Trend values remained almost constant at lower strain rates and 
decreased at higher strain rates. The variation of entropy with strain rate is rather 
interesting. The entropy initially decreased with strain rate and suddenly reached a 
higher value. However, the most important behavior was observed in the variation of 
REC and a variable derived from REC and DET, i.e. the ratio of DET and REC 
(DET/REC). The REC values decreased initially and reached a low value and then 
again started increasing. This variation is quite appealing in the sense that the REC 
value is very low in the crossover region and hence is able to detect the crossover 
phenomenon of the PLC effect. On the contrary, the DET/REC values showed an 
abrupt jump in the crossover region. 
 It is clearly evident from this study that RQA is able to detect the crossover in 
the PLC dynamics from type B to type A region. However the detailed explanation of 
the results obtained from the study are not straightforward. Further study is necessary 
which will in turn also help to understand the dislocation dynamics involved in the 
PLC effect. 
Conclusions 
 In conclusion, for the first time we have applied the recurrence analysis to study 
the dynamical behavior of the PLC effect. The study revealed that the recurrence 
analysis is efficient to detect the unique crossover, as indicated in the earlier studies, in 
the dynamics of the PLC effect.  
References 
1. F. Le Chatelier, Rev. de Metall. 6, 914 (1909). 
2. A.W. Sleeswyk, Acta Metall. 6, 598 (1958). 
3. P.G. McCormick, Acta Metall. 20, 351 (1972). 
4. J.D. Baird, The Inhomogeneity of Plastic Deformation (American Society of 
Metals, OH, 1973). 
5. A. Van den Beukel, Physica Status Solidi(a) 30, 197(1975). 
6. A. Kalk, A. Nortmann, and Ch. Schwink, Philos. Mag. A 72, 1239 (1995). 
7. M. Zaiser, P. Hahner, Phys Status Solidi (b) 199, 267 (1997). 
8. M. S. Bharathi, M. Lebyodkin, G. Ananthakrishna, C. Fressengeas, and L. P. 
Kubin, Acta Mater. 50, 2813 (2002).  
9. E. Rizzi, P. Hahner, Int. J. Plasticity 20, 121(2004). 
10. P. Barat, A. Sarkar, P. Mukherjee, S. K. Bandyopadhyay, Phys Rev Lett 94, 
05502 (2005). 
11. A. Sarkar, P. Barat, Mod Phys Lett B 20, 1075 (2006) 
12. A. Sarkar, A. Chatterjee, P. Barat, P. Mukherjee, Mat Sci Eng A, in press 
13. A.H. Cottrell, Dislocations and plastic flow in crystals (Oxford University 
Press, London, 1953). 
14. A. Van den Beukel, U. F. Kocks, Acta Metall. 30, 1027 (1982). 
15. J. Schlipf, Scripta Metall. Mater. 31, 909 (1994). 
16. P. Hahner, A. Ziegenbein, E. Rizzi, and H. Neuhauser, Phys. Rev. B 65, 
134109 (2002). 
17. Y. Estrin, L.P. Kubin, Continuum Models for Materials with Microstructure, 
ed. H.B. Muhlhaus (Wiley, New York, 1995, p. 395). 
18. L.P. Kubin, C. Fressengeas, G. Ananthakrishna, Dislocations in Solids, Vol. 
11, ed. F. R. N. Nabarro, M. S. Duesbery (Elsevier Science, Amsterdam, 
2002, p 101). 
19. A. Nortmann, and Ch. Schwink, Acta Mater. 45, 2043 (1997);  
20. K. Chihab, Y. Estrin, L.P. Kubin,  J. Vergnol, Scripta Metall. 21, 203(1987). 
21. K. Chihab, and C. Fressengeas, Mater. Sci. Eng. A 356,102 (2003). 
22. M. Lebyodkin, L. Dunin-Barkowskii, Y. Brechet, Y. Estrin and L. P. Kubin,  
Acta mater 48, 2529 (2000). 
23. P. Hahner, Mat Sci Eng A 164, 23 (1993). 
24. G. Ananthakrishna, M.C. Valsakumar, J. Phys. D 15, L171(1982). 
25. V. Jeanclaude, C. Fressengeas, L.P. Kubin, Nonlinear Phenomena in Materials 
Science II, ed. L.P. Kubin (Trans. Tech., Aldermanndorf, 1992, p. 385). 
26. M. Lebyodkin, Y. Brechet, Y. Estrin, L.P. Kubin, Acta Mater. 44, 
4531(1996). 
27. M.A. Lebyodkin, Y. Brechet, Y. Estrin, L.P. Kubin, Phys. Rev. Lett. 
74,4758(1995). 
28. S. Kok, M.S. Bharathi, A.J. Beaudoin, C. Fressengeas, G. Ananthakrishna, 
L.P. Kubin, M. Lebyodkin, Acta Mater. 51, 3651(2003).  
29. M.S. Bharathi, M. Lebyodkin, G. Ananthakrishna, C. Fressengeas, L.P. 
Kubin, Phys. Rev. Lett. 87, 165508(2001).  
30. S. Venkadesan, M. C. Valsakumar, K. P. N. Murthy, and S. Rajasekar,  Phys. 
Rev. E 54, 611 (1996). 
31. M.A. Lebyodkin and T.A. Lebedkine, Phys. Rev. E 73, 036114 (2006) 
32. D. Kugiumtzis, A. Kehagias, E. C. Aifantis, and H. Neuhäuser, Phys. Rev. E 
70, 036110 (2004)  
33. G. Ananthakrishna, S. J. Noronha, C. Fressengeas, and L. P. Kubin, Phys. 
Rev. E 60, 5455 (1999). 
34. G. Ananthakrishna and M. S. Bharathi, Phys. Rev. E 70, 026111 (2004) 
35. M. S. Bharathi, G. Ananthakrishna, EuroPhys Lett 60, 234 (2002). 
36. M. S. Bharathi and G. Ananthakrishna, Phys. Rev. E 67, 065104 (2003) 
37. J. –P. Eckmann, S. O. Kamphorst, D. Ruelle, Europhys. Lett. 4, 973 (1987).  
38. J. –P. Zbilut and C. L. Webber Jr., Phys. Lett. A 171, 199 (1992). 
39. C. L. Webber Jr. and J. –P. Zbilut, J. Appl. Physiol. 76, 965 (1994). 
40. J. Gao and H. Cai, Phys. Lett. A 270, 75 (2000).  
41. N. Marwan, N. Wessel, U. Meyerfeldt, A. Schirdewan, J. Kurths, Phys. Rev. 
E 66, 026702 (2002).  
42. N. Marwan, M. C. Romano, M. Thiel and J. Kurths, Physics Reports  438, 237 
(2007). 
43. C. L. Webber Jr, J. –P. Zbilut, In: Tutorials in contemporary nonlinear 
methods for the behavioral sciences, Chapter 2, pp. 26-94, 2005. M. A. 
Riley, G. Van Orden, eds. 
Fig. 1 True stress vs. true strain curve of Al-2.5%Mg alloy deformed at a strain rate of                    
1.20×10-3 S-1. The inset shows a typical segment of the trend corrected true stress vs true 
strain curve. 
Fig. 2 Recurrence plots at the strain rate (a) 1.60×10-3 S-1 (b) 7.97×10-4 S-1 (c) 3.85×10-4 
S-1 (d) 1.99×10-4 S-1  
Fig. 3 Variation of the Recurrence Quantification Analysis variables with strain rate. 
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
0.040 0.042 0.044 0.046 0.048
True Strain
True Strain
Fig. 1 
Fig. 2 
0.0 5.0x10-4 1.0x10-3 1.5x10-3
Type AType B
Strain rate (s-1)
Strain rate (s-1)
0.0 5.0x10-4 1.0x10-3 1.5x10-3
0.0 5.0x10-4 1.0x10-3 1.5x10-3
0.0 5.0x10-4 1.0x10-3 1.5x10-3
Strain rate (s-1)
0.0 5.0x10-4 1.0x10-3 1.5x10-3
Strain rate (s-1)
Strain rate (s-1)
0.0 5.0x10-4 1.0x10-3 1.5x10-3
Strain rate (s-1)
0.0 5.0x10-4 1.0x10-3 1.5x10-3
Strain rate (s-1)
0.0 5.0x10-4 1.0x10-3 1.5x10-3
Strain rate (s-1)
0.0 5.0x10-4 1.0x10-3 1.5x10-3
Type AType B
Strain rate (s-1)
Fig. 3
