Multifractal Analysis of Unstable Plastic Flow

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MULTIFRACTAL ANALYSIS OF UNSTABLE PLASTIC FLOW No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

MULTIFRACTAL ANALYSIS OF UNSTABLE PLASTIC FLOW

M. A. LEBYODKIN T. A. LEBEDKINA AND

A. JACQUES

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175

Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Library of Congress Cataloging-in-Publication Data ISBN: 978-1-60741-278-6 (E-Book)

Published by Nova Science Publishers, Inc.   New York

CONTENTS Preface

vii

Chapter 1

Introduction

1

Chapter 2

Unstable Plastic Flow

3

Chapter 3

Multifractal Analysis

11

Chapter 4

Experimental Technique and Analysis of Deformation Curves

29

Chapter 5

Experimental Investigations of Plastic Instability

35

Chapter 6

Conclusion

67

Acknowledgements

71

References

73

Index

79

PREFACE The interest of the application of multifractal analysis to the plastic flow instability is twofold. On the one hand, the unstable, or jerky, flow is a selforganization phenomenon which exhibits a great wealth of behavior. It may be associated to various microscopic instability mechanisms, whereas the same microscopic mechanism may result in various dynamic regimes including deterministic chaos and self-organized criticality. On the other hand, the study of the concomitant dynamics may shed light on the collective behavior of dislocations and their interaction with other crystal defects. The investigations of the fractal properties of serrated deformation curves started several years ago on the case of the Portevin-Le Chatelier (PLC) effect – the jerky flow of alloys, caused by the dislocation-solute interaction. Specifically, it was found that the multifractal analysis makes possible a quantitative characterization of the distinct dynamical regimes of the PLC effect, which are related to its traditional classification based on the kinetics of the deformation bands giving rise to the serrations, and on the resulting shape of the deformation curves. This contribution reports the recent progress in the experimental investigation of the PLC effect by multifractal analysis. A great attention is paid to its practical implementation and to pitfalls which may appear in the processing of real deformation curves. The mathematical basics of the multifractal approach are first presented with an accent put on the experimental point of view. Recent studies of the effect of the material strain hardening and the experimental noise are summarized. A multifractal behavior is shown to exist in the still unexplored range close to the low strain rate boundary of the PLC instability, where the deformation bands were believed to be uncorrelated. Using the results of the recent investigations, the overall pattern of the PLC effect at various strain rates is

viii

M. A. Lebyodkin, T. A. Lebedkina and A. Jacques

discussed within the framework provided by the multifractal analysis. The first results of the analysis of the jerky flow of austenitic steels, which may be due to a mixture of PLC instability, twinning, and martensitic transformation, are presented. Multifractal Cantor sets are used to test various kinds of experimental behavior.

Chapter 1

INTRODUCTION The plastic flow of solids, which is governed by the motion and interaction of crystal defects, is an example of a dynamical problem that is specific of nonlinear systems composed of a large number of interacting elements. The dynamics of such systems may show unusual properties that result from the self-organization of elementary motions [1, 2]. It is often very complex and, moreover, exhibits a continuous power spectrum. As a consequence, the theory of the linear dynamical systems, which relates the deterministic complexity of a system to a superposition of harmonic oscillators, is unable to distinguish it from a stochastic behaviour. The development of the nonlinear theory and discrete mathematics showed that a complex dynamics may be associated with other kinds of order [3-6]. This order may not be apparent in the records of the system evolution (called signals from here on) but manifests itself in the underlying scaling laws [7, 8]. The scale invariance was found both in the temporal structure of signals and in various spatial structures. Well-known examples include turbulent flow, currency exchange rates, earthquakes, Barkhausen effect in magnetics, dynamics of superconducting vortices, dendrite crystal growth, seashores, clouds, snow flakes, and many others. The elaboration of the adapted mathematical methods led to a conclusion that such a behavior is not less frequent in the natural phenomena than periodicity. Moreover, in spite of the diversity of the specific mechanisms, the dynamics of the nonlinear systems obeys universal laws accounting for selforganization. Thus, every particular problem presents both a specific and a general interest. Such investigations were visibly retarded in the material science in comparison with the pioneering works in chemistry, biology, economy, and physics. However, the material scientists also recently turned to these problems. Recent works showed convincingly how the application of the concepts of self-

2

M.A. Lebyodkin, T.A. Lebedkina, A. Jacques

organization can clarify questions that concern well-known phenomena but were unanswered within the microscopic theory of plasticity (see reviews [9-13] and references therein). Quite a number of mathematical methods have been developed to characterize complex spatial or temporal structures on the basis of thermodynamic and statistical approaches, stability analysis, spectral features, or geometric properties (see [14-38] for a nonexhaustive list). To squeeze through this jungle, it should be realized that they are based on well-known statistical principles and aim at making the interpretation of complex behaviours easier. Basically, the correlation between events within a signal or elements of a structure can be revealed by the traditional statistical techniques involving calculation of statistical moments [39], i.e., certain sums of powers of the respective data – a signal record or a structure image. However, these general methods are convenient when a few first moments, often only the mean value and the variance provide a comprehensive description. As the behavior of the nonlinear systems may need considering higher moments, the data interpretation becomes a difficult task. This requires techniques making use of specific statistical features of the analyzed structures. In particular, the multifractal analysis is based on the identification of scaling laws characterizing a nonuniform pattern associated with a physical property. The present work is an attempt to apply the multifractal analysis to describe an unstable plastic flow.

Chapter 2

UNSTABLE PLASTIC FLOW 2.1. The Nature of Plastic Instability The synergy of crystal defects may not show up in the standard mechanical tests. Indeed, the deformation curve results from the motion of a huge number of defects, most commonly, dislocations – the main carriers of plasticity [40]. While the dislocation density is small at the beginning of plastic deformation, their elastic interaction is weak. Under these conditions, the averaging of the weakly coupled elementary dislocation motions leads to a macroscopically smooth plastic flow. A nonuniformity caused by self-organization may be limited to short range at the level of small groups of dislocations. It can be detected by appropriate techniques, such as surveying fine slip lines on a specimen surface, or the observation of jump-like motion of dislocation pile-ups through obstacles by in situ transmission electron-microscopy [41]. As the dislocation density increases with straining, the self-organization shows up at coarser levels. Generally, two groups of collective phenomena are discussed. The most studied is the phenomenon of dislocation patterning, i.e., progressive formation of specific dislocation structures upon straining [9-13]. Such a spatial instability of the uniform plastic flow stands beyond different stages of the work hardening – the increase of the flow stress with the dislocation density upon straining [42, 43]. These investigations proved the aptitude of the concept of self-organization to describe the dislocation dynamics. The second group refers to catastrophe-like processes leading to emergence of macroscopic serrations on the deformation curves (see Fig. 1). The serrations indicate that the plastic strain rate ε& of the sample repeatedly exceeds the value ε& a imposed by the straining device.

Moreover, the high-rate deformation does not uniformly pass in the entire

4


specimen but is localized, so that the local strain rate may exceed ε& a by many orders of magnitude. This spatiotemporal instability has been a subject of an intense study during the last two decades (see review [11]) and is the main scope of the present paper. The serrated deformation, also referred to as jerky, discontinuous, or jumplike flow, can be caused by various microscopic mechanisms that involve different kinds of crystal defects: dynamical strain aging of dislocations by impurities [11, 44-47], low-temperature thermomechanical instability [48-51], mechanical twinning [52], phase transformations [53], or cracking [54]. The most of examples in this chapter are taken from investigations of the Portevin-Le Chatelier (PLC) instability in alloys, which is controlled by the dynamical strain aging. This choice is partly due to a remarkable richness of the behaviour observed under conditions of the PLC effect. Besides, its microscopic mechanism has been well understood. This provides a solid basis for the study of collective effects. An additional example will illustrate preliminary studies of the jerky flow in conditions where both the PLC effect and twinning take place.

Type C 200

σ, MPa

Type B

Type A

100

0

0.0

0.1

ε

0.2

Figure 1. Examples of deformation curves for Al-3at.% Mg samples, corresponding to three commonly distinguished types of the PLC effect. The samples were deformed at room temperature and strain rates of 4×10-6 s-1 (type C), 2×10-4 s-1 (type B), and 6×10-3 s-1 (type A). The two lower curves are deliberately shifted downwards to clearly separate stress jumps on the chart.


5

2.2. Portevin-Le Chatelier Effect 2.2.1. Microscopic Mechanism The PLC instability is caused by the interaction of mobile dislocations with impurity atoms. A convenient way to understand the occurrence of serrations on the deformation curves is to convert the microscopic mechanism of the plastic flow to a relation between the stress and the plastic strain rate [55]. At microscopic scale, the dislocations move discontinuously because they are repeatedly pinned by obstacles. Every new act of motion requires a thermally activated unpinning of the dislocation from the obstacles. The corresponding waiting time is commonly much longer than the free-flight time of the dislocation between obstacles and, therefore, determines the plastic strain rate. At a given temperature, the waiting time decreases and, therefore, the strain rate increases when the stress σ is increased (Fig. 2(a), curve A). Such a monotonous dependence results in a stable plastic deformation with the values of σ and ε& prescribed by the straining conditions. In an experiment under a constant imposed strain rate, such a model gives a flat deformation curve at a constant stress, as sketched by curve A in Fig. 2(b). A real deformation curve can then in principle be deduced taking into account the work hardening, and, eventually, the effects of the specimen geometry, the heat release during deformation, and so on. The presence of impurities changes the behaviour of individual dislocations. The solute atoms diffuse to dislocations temporarily arrested at obstacles and create additional barriers for their thermally activated motion. The corresponding stress increment depends on the time available for diffusion, which is limited by the waiting time [56]. Therefore, the additional stress decreases when the strain rate increases. The competition of two factors, the thermal activation and the dynamic aging of dislocations, leads to a nonmonotonous (N-shaped) dependence of the stress on the strain rate with a range of a negative slope (Fig. 2(a), curve B). Such a shape of the material characteristic is a well-known source of relaxation oscillations [57] that must occur when the imposed strain rate finds itself in this range. Indeed, it is easy to see from Fig. 2(a) that during experiments under a constant imposed strain rate, the forth and back jumps in the plastic strain rate are associated with a drastic unloading and much slower reloading (note the logarithmic ε& scale). In other words, the elastic reaction of the deformation machine converts the rate jumps into a toothed deformation curve shown schematically by curve B in Fig. 2(b). Such a cyclic behaviour is similar, for

6


example, to the instability in electronic devices characterized by a negative differential resistance [58].

σ

(a)

(b)

σ

A B B A

ε&a

ln ε&

ε

Figure 2. (a) Schematic drawing of the distortion of the function of strain rate sensitivity of stress, σ (ε& ) , under conditions of the PLC effect. The monotonous curve A (dashed line) stands for a thermally activated motion of dislocations. The N-shaped curve B (solid line) illustrates the effect of the dynamic strain ageing of dislocations. The arrows describe a cyclic behavior that occurs when the imposed strain rate

ε&a

finds itself in the range of the

negative slope. (b) Corresponding shapes of deformation curves.

The discussed model of the instability is local: it assumes that all dislocations behave equally, so that the strain rate of the whole specimen synchronously follows the same cycle. Taking into account the spatial coordinate along the specimen length, the nonlocal models predict localization of the high-rate plastic flow in deformation bands that can be immobile or propagate as solitary waves [46]. The spatial pattern of slip bands is another salient feature of the PLC effect. Thus, the unstable deformation is intrinsically nonuniform. However, while the heterogeneity of the real material is not considered, different parts of the specimen remain equivalent. As a result, the nonlocal models do not modify the conclusion on the periodic oscillations of the average strain rate that determines the toothed shape of the deformation curve.

2.2.2. Real Behaviour It can be seen from Fig. 1 that the real deformation curves demonstrate essentially more complex shapes than the periodic oscillations predicted by the


7

microscopic model. The figure illustrates three major types of the PLC instability, which are usually distinguished, depending on the shapes of the deformation curves and on the character of spatial correlations between slip bands [59]. Close to the lower strain-rate boundary of the PLC effect [see Fig. 2(a)], each stress drop is believed to be associated with the occurrence of a deformation band. The optical survey of the specimen surface does not show a noticeable correlation between bands [60]. It is generally concluded that the bands occur at random sites (type C effect). As ε& a is increased, the band correlation becomes stronger: they successively appear in the neighboring regions, leading to a hopping propagation of strain (type B). A quasi-continuous propagation of the deformation bands along the crystal is observed upon further increase of the strain rate close to the upper boundary of the instability (type А). In this case, the stress drop amplitudes display a high variance. In contrast to types B and C, not each drop may be associated with the occurrence of a new band. The sequences of drops can rather be ascribed to fluctuations in the velocity and width of a propagating band. Finally, some additional types of PLC effect are sometimes singled out [61]. These particular cases will not be dealt with here. This simplified classification seizes the most prominent features of the PLC effect but by no means, exhausts the variety of the spatiotemporal patterns observed experimentally. The strain heterogeneity in real samples depends on both internal factors, such as the material microstructure and composition, and on the experimental conditions, including the strain rate, the temperature, and the test scheme. Particularly, it is not always possible to ascribe a deformation curve to a specific type, which causes contradictions in the literature. Moreover, this diversity impedes the verification (or falsification) of numerical models of the PLC effect, as the irregular character of the deformation curves often makes their visual comparison meaningless. These problems justify the need for a quantitative analysis to ensure further progress in the understanding of the collective dynamics of dislocations.

2.2.3. Dynamical Regimes Associated with the PLC Types During mechanical testing, the evolution of only one or few parameters is followed, for example, the stress and the overall strain. It is not known a priori how many independent variables are needed for a full description of the dislocation dynamics. It is obvious that a system of noninteracting dislocations possesses an infinite number of degrees of freedom, so that the statistical approaches must be appropriate. On the other hand, a self-organization might

8


essentially reduce the number of the effective degrees of freedom. It goes without saying that various methods of analysis, adapted to different situations, should be explored [62-78]. Recently, the application of several methods of the theory of nonlinear dynamical systems made it possible to relate types A and B of the PLC instability to specific dynamical regimes. The statistical investigations showed that the distributions of the stress drop amplitudes, durations, and intervals between drops are described by power laws under type A conditions [62-71]. The power law indicates the absence of a characteristic scale of the processes and mathematically expresses the scale invariance. Indeed, this follows from the underlying property of the power-law function to remain unchanged upon scaling: (kl)a ~ la. The power-law statistical distributions are reminiscent of the phenomenon of self-organized criticality (SOC) that was first proposed to describe earthquakes [79, 80]. The detection of such a dynamics looks rather natural in the case of plastic flow since SOC is characteristic of extended systems with an infinite number of degrees of freedom [6, 79-82]. A fundamentally different regime, associated with a finite dimensionality, was found at lower strain rates with the aid of the method of reconstruction of the phase trajectory (dynamical analysis [14, 17]). This method treats the σ(t) dependence as a one-dimensional projection of a higher-dimensional trajectory that develops in a phase space with a dimension n determined by the number of the independent variables of the system. Schematically, the procedure consists in the embedding of the recorded time series {σj} into phase spaces with a gradually increasing dimension d. This is done by taking d equidistant points of the deformation curve as d components of a vector and considering its evolution when moving along the curve. If the underlying dynamics is low-dimensional, the generated phase trajectory would no more change for large enough d ≥ n. The trajectories reconstructed from type B curves were found to be embedded into 4 to 6 dimensional phase spaces [67, 69, 70, 72-74]. They were identified with strange attractors – the trajectories characteristic of the deterministic chaos, a paradigm of the turbulent flow. In this case, too, the dynamics of the plastic flow is associated with a scale-invariant behavior which manifests itself in the self-similar geometry of strange attractors. The occurrence of qualitatively different regimes does not seem surprising since the characteristic scales of stress fluctuations, as well as the velocity and the width of the slip bands, may change by orders of magnitude when the experimental conditions are varied. The parallel use of both analyses proved that two dynamic regimes may occur in the same material under different conditions. Evidence was found that the corresponding transition is related to the transition from localized to propagating deformation bands [67, 69, 70]. These results call


9

up for a method of analysis apt to provide an overall characterization of the unstable plastic flow. Taking into account that both SOC and chaos are related to self-similarity, the multifractal analysis was tested as a general framework for the study of the plastic instability [69, 70, 75, 78]. The multifractality was detected for various deformation curves, with both scaling exponents and scaling ranges depending on the experimental conditions. In particular, a drastic widening of the multifractal spectra was observed at the localization-propagation transition. Such a behavior is similar to the multifractality predicted for the Anderson transition from localized to delocalized electron wave functions [83]. This adds to the interest of the multifractal analysis of plastic instability, which can be seen as representative of a broader class of phenomena. In particular, there are indications that a similar dynamical behavior may show up for different microscopic mechanisms of plastic instability and at finer scale levels [84-92].

Chapter 3

MULTIFRACTAL ANALYSIS The notion of fractals refers to an unusual non-Euclidian geometry of selfsimilar structures looking the same at any scale, the ubiquity of which among natural objects or natural phenomena was discovered during the last forty years [93]. The reader interested in a thorough description of fractals may address himself to a vast choice of books and original articles (e.g., [15, 23]). A practical consideration with an accent put on the problems met by experimenters (the finiteness of the data set, the superposition of the signal with the measurement noise, the system evolution during the test) will be given in this chapter. The mathematical basics of the multifractal approach will be illustrated on the example of Cantor sets – fractal geometrical objects embedded in a one dimensional space. Their visual similarity with the sequences of stress jumps will help making clearer the motivation of the application of the multifractal analysis to the unstable plastic flow. Moreover, the comparison of the multifractal characteristics of the model Cantor sets and the experimental data will justify the use of the Cantor sets for “modeling” the correlations involved in the processes of the unstable plastic flow.

3.1. Fractal Dimensions Let us first consider a trivial (non-fractal) self-similar object – a line – and measure the length of a line segment. For this, a grid with a division l is superimposed on the segment and the number Nl of the divisions needed to cover the whole segment is counted. Obviously, Nl is inversely proportional to l in the limit l→0: N l ~ l

−1

, which expresses the scale invariance in question. Therefore,

12


the length L = lim N l l does not depend on l and is a pertinent geometric l →0

characteristic of the segment. This consideration can be generalized for the cases of higher dimensions. In particular, covering a surface with squares or a volume with cubes reveals the scaling law

Nl ~ l −D

(1)

Δσ, relative units

where l is the square side or the cube edge and D is the respective topological dimension of the analyzed object.

100

t, s

150

Figure 3. A portion of a sequence of stress jump amplitudes Δσ for an AlMg alloy deformed at a strain rate of 1.4×10-3 s-1. The Δσ values are normalized to allow for the work hardening as described in Sec. 4.2.1.

In contrast to these trivial geometrical objects, the stress jumps often form an intermittent sequence of clustered events, which does not uniformly fill the time interval representing the test duration. An example of such a sequence is presented in Fig. 3. To characterize it thoroughly, the analysis should capture both the nonuniform filling of the interval and the variations of the jump amplitudes. Let us first consider the nonuniform filling, i.e., look for a characteristic apt to describe a porous segment. Creation of holes in the segment by removing subsegments would in general break relationship (1). However, if the sequence


13

filled segments

shown in Fig. 3 were scale invariant, some power law would be preserved. The way to disclose it may be understood by considering a porous geometric object having built-in fractality guaranteed by its construction rule.

0.25

0.0

0.30

0.5

1.0

position

Figure 4. Schematic drawing of the Cantor set. Columns indicate filled segments. The bottom drawing is obtained at the fourth step of the set construction. The magnification of a part of it in the upper drawing demonstrates how each segment reproduces the same structure as the construction proceeds.

The simplest example of such an object is given by the classical Cantor set. It is constructed starting from a unit interval [0; 1] and using a simple recursion rule. Namely, the middle third of the interval is removed at the first step, the middle thirds of two remaining intervals are removed at the second step, and so on. At the mth step, the set contains 2m segments with the same length equal to (1/3)m (Fig. 4). The continuation of this process ad infinitum generates a structure that is porous everywhere and has a zero topological dimension (for rigorous definitions of various dimensions, see [93, 94]). Making a convenient choice of l diminishing as powers of 1/3, it is easy to deduce that the Cantor set is self-similar:

Nl = l

−Df

,

(2)

where Df = ln2/ln3. Unlike Eq. (1), the exponent Df takes on a fractal value that is greater than the corresponding topological dimension (i.e., zero) and less than the topological dimension (one) of the space into which the set is embedded. Such sets are called fractals and Df is referred to as a fractal or capacity dimension.

14


They possess unusual geometrical properties. In particular, the result of the length 1− D

measurement for the Cantor set depends on the grid box size l: L = l f and tends to zero when l is reduced. It is worth pointing out that contrary to such an ideal fractal, the refinement of experimental data is always limited because of either physical mechanisms involved or the finite measurement resolution. Moreover, the experimental system size imposes an upper scaling bound, whereas the Cantor set may be seen as an element of a coarser structure that can be reconstructed ad infinitum according to the same recursion rule. This distinguishes real physical fractals from their mathematical idealization. Namely, the real objects possess characteristic scales so that scaling law (2) is only approximately valid. The fractal dimension Df quantifies the self-similarity of the given set. The modification of the recursion rule would produce different structures characterized by different Df values. Thus, the calculation of the fractal dimension of a deformation curve must allow not only checking whether the intermittent sequence of the instants of stress jumps is self-similar and, therefore, indicates a certain correlation between plastic bursts, but also characterizing it quantitatively. In particular, the non-fractal random or periodic arrangement of gaps in the segment would reduce its average linear density and impose a characteristic scale of the order of the least gap but would not change (above this characteristic length scale) the exponent D = 1 of the scaling law (1) deduced for the continuous interval. Thus, once a scaling law has been found, the value of Df can discriminate clustered events from random or periodic sequences. At the same time, the fractal dimension is a global characteristic that detects clustering but disregards its local variations. In particular, the comparison of figures 3 and 4 shows that the Cantor sets considered above can mimic neither a nonuniform clustering nor the variations of the event amplitudes. Indeed, in the above calculations, each grid box is either occupied or not but its occupancy is not measured. This problem is the subject of the multifractal approach.

3.2. Multifractals The way to consider a fluctuating physical property of a real object, for example, the probability of occurrence of an event during a time interval, the amplitudes of events, the masses of structural elements, the electric or magnetic moments, and so on, can be clarified by assigning weights to the segments of the Cantor set. The modified procedure starts from a unit segment with a weight μ = 1 and includes an additional rule to handle the weight repartition. The current


15

weight μi of each of 2m segments of the mth construction is shared on equal terms between the two nonempty segments generated from it at the next step. That is, the weight of the new segments is obtained by multiplying μi by a factor p = 1/2. Therefore, μi = (1/2)m at the mth iteration and obeys a power law

μ i ~ liα

(3)

with α = ln2/ln3 = Df for the Cantor set. The exponent α (Lipshitz-Hölder index) is often called singularity strength because α < 1 indicates the divergence of the local density:

μ i / li ~ liα −1 when li tends to 0 and, therefore, reveals a

discontinuity. The weight μ is an example of a probability measure of a physical quantity distributed on a geometrical support. The property of scale invariance provides an opportunity to describe it via the corresponding scaling indices Df and α. This consideration should be further generalized to be applied to natural self-similar objects that are usually nonuniform and require more scaling indices. A heterogeneous self-similar set may be subdivided into subsets, each subset having its own scaling indices. In general, the “weight” of an element in the ith grid box with size l, which is given by an integral of the local measure dμ over the box, obeys relationship (3)

μ i (l ) = ∫ dμ ( x) ~ l α , where α can take on a range of values corresponding to different regions of the set. The subsets formed of the boxes corresponding to close singularity values between α and α+dα can be characterized by counting their number N(α) and calculating the respective fractal dimension f(α) of the subset: N(α) ~ l-f(α),

(4)

The dependence f(α), also called singularity spectrum, is a continuous function that provides a comprehensive description of a self-similar object carrying a physical property: it characterizes both the corresponding measure and the underlying fractal geometry of its support. It can be illustrated by a nonuniform generalization of the Cantor set, which is based on a multiplicative procedure mimicking a cascade process. This time, the segment is iteratively divided into (generally unequal) parts that are ascribed certain weights. Each

16


segment with a length li and a weight μi remaining at the mth iteration step is divided into k pieces in such a way that the size of the kth peace is li multiplied by a factor λk and its weight is μi multiplied by a factor pk. Thus, the actual values of li and μi are obtained as multiples of different combinations of the corresponding rescaling factors. The condition Σ pk = 1 provides normalization of the generated measure. For the sake of generality, the removal of a subsegment is expressed by p = 0 so that Σ λk = 1. For example, the above Cantor set corresponds to λ1 = λ2 = λ3 = 1/3, p1 = p3 = 1/2, and p2 = 0.

(a) 0.0001

μ 0.0000

0.570

0.575

(b)

0.0002

μ

0.0000

0.235

0.240

x Figure 5. Two examples of multifractal Cantor sets generated by repeating the respective recurrent procedure fourteen times. μ denotes the local probabilistic measure of the generated elements; x is the coordinate within the initial segment [0; 1].

Figure 5 shows two examples of heterogeneous Cantor sets. The rescaling factors λk = 1/4 (k = 1 to 4), p1 = 1/6, p2 = p3 = 1/4, and p4 = 1/3 produce a set that entirely fills the initial segment, i.e., has a continuous support. The set generated with rescaling factors λ1 = 0.25, λ2 = 0.35, λ3 = 0.4, p1 = 0.6, p2 = 0, and p3 = 0.4 appears very clustered and containing large gaps. These sets allow analytical calculation of the f(α) spectra (see, e.g. [15]). The corresponding dependences are represented in Fig. 6(a). They clearly show the multifractal character of the generated sets and reveal the presence of singularities (α < 1). In particular, the clustered set [Fig. 5(b)] is almost everywhere singular. Returning back to the


17

comparison with nonfractal sets, the ideal cases of infinite random or periodic sequences would give spectra consisting of a single point α = 1 and f = 1. This clarifies the multifractal approach to the analysis of the deformation curves, the dynamical nature of which is unknown a priori: any spectrum which is not limited to this single point reveals a nontrivial power law.

1.0

(a)

f 0.5

0.0

0.5

α

1.0

(b) 1.0

D

0.5 -20

-10

0

10

20

q Figure 6. Analytical prediction of the shape of (a) the singularity spectra and (b) the generalized dimension spectra for the multifractal Cantor sets of Figure 5(a) (solid lines) and Figure 5(b) (dashed lines).

18


As follows from the definition, the function f(α) has a well defined physical meaning. However, the proposed description does not give a practical way to calculate it numerically for a given (e.g., experimentally measured) data set. Nor does it clearly demonstrate the usefulness of the multifractal approach since the meaning of different parts of the singularity spectrum remains uninterpreted. These difficulties can be overcome if the above description is compared to a mathematically equivalent approach based on the calculation of the spectrum of generalized dimensions D(q) [Fig. 6(b)]. Introducing the qth moments of the measure of a multifractal set, Zq(l) =

∑μ i

q i

(q ≠1) and Z1(l) =

∑μ i

i

ln μ i , it

can be demonstrated [23] that

lim Z q (l ) ~ l ( q −1) D ( q ) and lim Z 1 (l ) ~ D(1) ln l . l →0

l →0

(5)

These sums can be calculated for various q, and D(q) can be evaluated. The f(α) spectrum is then given by the Legendre transform of the function τ(q) = (q1)D(q) [23]: f(α(q)) = qα(q) –τ(q) ; α(q) = dτ(q)/dq.

(6)

Equation (5) explains the mathematical meaning of the multifractal approach. Namely, when q is varied from q = -∞ to q = ∞, different μ values (from the lowest to the highest weights), associated with different data subsets, consecutively become dominant in the above sums. This property is known as a “mathematical microscope”. Some of the generalized dimensions (from now on designated as Dq) are well known and allow a direct interpretation as illustrated in Fig. 7. Particularly, as follows from relationship (5), Z0 is the number of boxes with nonzero measure. Therefore, the capacity D0 is simply the fractal dimension of the geometrical support of the measure. Obviously, it is the largest subset of the data set, i.e., fmax = D0. The information dimension D1 (D1 = α(1) = f(α(1)) characterizes how fast the information contained in the signal increases with increasing resolution. The correlation dimension D2 reflects pairwise correlations of the events [22]. The extreme values D∞ = αmin and D-∞ = αmax characterize subsets of segments having the highest weight and segments having the lowest nonzero weight, respectively. In this sense, the span of the spectra quantifies the signal heterogeneity. The heterogeneity is also reflected by the capacity f∞: larger f∞ values indicate a higher degree of homogeneity within the largest events.


1.0

f

19

fo=Do

(a)

0.5

≅ f∞ 0.0

αmax≅ D−∞

α1=D1 αmin≅ D∞

1.0

1.5

α

D- ∞ (b)

1.5

D 1.0

D2 D∞ -20

0

20

40

q Figure 7. Examples of spectra of (a) singularities and (b) generalized dimensions for an experimental data set. The designations explain the meaning of some multifractal parameters.

To summarize, it is worth recalling that the classical statistical analysis describes the overall distribution of the events probability (i.e., the distribution of amplitudes and frequencies) but cannot reveal their clustering. The calculation of the fractal dimension of a signal or a spatial structure allows detecting the selfsimilar clustering. However, the local fluctuations are inaccessible to the simple

20


fractal approach that only characterizes the global scaling properties. The above examples, illustrating the multifractal approach, show how the scanning of the fractal properties of various subsets by varying the exponent q helps uncovering both the global clustering (D0 < 1) and the presence of localized fluctuations that result in an extended spectrum. At the same time, the multifractal analysis does not describe their locations within the original data set. This question may be addressed to by some other kinds of data processing, e.g., wavelet analysis [16], which are beyond the scope of the present paper.

3.3. Numerical Implementation of the Multifractal Analysis The calculation of the generalized dimensions with the use of relationship (5) is direct. A more difficult task is to calculate the singularity spectrum because Eq. (6) implies derivation of the function τ(q) that is represented by a discrete data set. That is, obtaining a sufficient precision necessitates a detailed tracing of this function, which requires a huge computer power. For this reason, a direct method using

a

normalized

~ (l , q ) = measure μ i

μ iq / ∑ j μ qj

was

proposed

for

calculation of f(α) [95]. The scaling relationships

Σ α (l , q) = ∑i μ~i (l , q ) ln μ i (l ) ~ α (q ) ln l Σ (l , q) = μ~ (l , q) ln μ~ (l , q ) ~ f (q ) ln l f

∑

i

i

(7)

i

for the new measure implicitly define the f(α) function. The results of calculations for the considered Cantor sets are shown in Figs. 8 and 9. Both sets contained 214 segments. The grid division length was varied as a power of 2. Figure 8 presents examples of scaling dependences Σq(l) for several q values [Eq. (5)]. It can be seen that the linear trend is satisfactory in the range approximately from 2-13 to 2-3 for both sets. That is, the scaling law is valid almost everywhere, with the exception for the ranges close to the limit of the structure refinement and to the system size. Figure 8(b) also reveals an artifact interfering with the scaling law for the clustered set. Obviously, the location of gaps within the data set due to the recurrent rule causes undulations that may impede the slope determination if the data set is not long enough. This might be a serious problem in many real situations with a poor data collection. However, such an artifact can be corrected by randomly choosing the starting point for the grid generation and


21

repeating calculations several times. The results of averaging over ten trials are also shown in Fig. 8(b). It can be seen that the averaging effectively reduces the undulations.

log2(Σq)

0

q=0 q=5 q=20

(a) -200 -10

log2 l

-5

0

log2(Σq)

0 q=0 q=20

(b)

-100 -10

log2 l

-5

0

Figure 8. Examples of scaling [see Eq. (5)] for the Cantor sets of Figure 5. Labels (a) and (b) correspond to Figure 5. Open symbols in chart (b): the origin of the grid coincides with the left boundary of the segment [0; 1]. Dark symbols show the suppression of undulations for the clustered set by averaging over ten runs with a randomly chosen starting point for the grid generation.

22

M.A. Lebyodkin, T.A. Lebedkina, A. Jacques 1.0

f

0.5

0.0 0.5

α

1.0

Figure 9. Results of numerical calculation of the singularity spectra for the multifractal Cantor sets of Figure 5(a) (dark circles) and Figure 5(b) (light circles). Solid lines trace the analytical curves (cf. Figure 6).

The power-law exponents f(q), α(q), and Dq are determined as slopes of the linear portions of the respective scaling dependences. The results of calculation of the singularity spectra f(α) are represented in Fig. 9. The error bars show the leastsquare estimate of the standard deviation of the slopes. The comparison with the analytical dependences (solid lines reproducing Fig. 6) shows a perfect coincidence of the analytical and calculated curves in the range q ≥ 0 (left branch of the spectrum) for the dense set and a good fitting for the clustered set. In contrast, systematic deviations are observed for negative qs (right branches of the spectra). The deviation is small for the dense set but quite significant for the intermittent set. Moreover, the error bars quickly grow in the latter case when q moves to large negative numbers. It should be recalled that the behaviour in the negative q range is determined by the low μi values that usually correspond to the dilute data subsets, and the numerical procedure may introduce a bias in the evaluation of α. It can be concluded that the observed imprecision is caused by the poor statistics accumulated for negative qs when the same grid size is used in the dense and dilute regions of the data set (fixed-size box-counting technique). This imperfection is especially important in the case of clustered objects containing depleted regions. The use of a variable grid size was proposed to accumulate a sufficient statistics for every q value (fixed-mass box-counting technique [96]). However, even this approach often does not allow a good


23

precision in processing of experimental data. For this reason, the quantitative conclusions will be based henceforth on the trends in the positive q range, i.e., the low α branch of the spectra.

3.4. Effect of Noise and Data Truncation Before considering the analysis of experimental results, it is worth testing the effect of the random noise and the data truncation on the Cantor sets. Strictly speaking, the set composed of a multifractal set and a non-multifractal noise is not multifractal because the noise addition breaks the scale invariance. The present paragraph concerns the general problem of unmasking the deterministic structure in a real signal that is inevitably noisy [37, 38]. It is also associated with the problem of the limited experimental resolution that confines the dynamical range of the signal in the absence of noise. The results of such tests will serve as a basis for understanding the structure of the experimental data.

3.4.1. Unclustered Set It occurs that the dense Cantor set [Fig. 5(a)] is extremely robust with regard to noise. Figure 10(a) illustrates the shape of a signal composed of the Cantor set and a noise with the amplitude of 10% of the maximum value of the Cantor measure. It was found that the noise addition narrows the scaling intervals and increases the data scatter. Namely, the upper inflection point shifted downward from about 2-3 for the initial set to about 2-4 for the noisy set (the lower scaling bound remained practically unmodified). Nevertheless, the comparison of the respective singularity spectra (curves A and B in Fig. 11) proves that the underlying multifractal structure persists in the entire q range. Particularly, the positive q branches of both spectra coincide. Moreover, curve C shows that the presence of the multifractal structure is still detectable when the noise reaches the level of 50 %. In this case, however, the uncertainty of the spectrum determination was very significant. The scaling range shrank to the interval of 2-10.5 to 2-5 and the spectrum became distorted, so that the dimension f fell into an unphysical negative range for large |q|. For a practice benchmark, the detection of the multifractality in the considered set was rather satisfactory for the noise up to one third of the maximum signal level. Taking into account the complexity of the problem of denoising in the presence of discontinuities, such opportunity makes

24


the multifractal analysis a convenient tool for detecting the scale invariance in noisy objects. (a)

0.0001

μ 0.0000

0.570

0.575

(b)

0.0002

μ

0.0000

0.235

0.240

x

Figure 10. The multifractal sets of Figure 5 superimposed with a random noise comprising (a) 10% and (b) 5% of the maximum measure value. 1.0

0.5 A B C

f

0.0

0.8

1.0

α

1.2

Figure 11. Effect of random noise on the singularity spectra for the multifractal Cantor set of Figure 5(a). Curve A – initial set; curve B – after addition of a 10% noise; curve C – after addition of a 50% noise. The dashed line shows zero level, below which the dimension values are senseless.


25

The effect of noise could be additionally diminished in a simple way by cutting off the low-amplitude part of the noisy signal using a threshold equal to the noise amplitude. The tests showed that while the noise and, respectively, the threshold did not exceed 10%, both the positive q branch and the position of the spectrum maximum remained almost intact. Only some deterioration of the negative q branch of the spectrum was observed. As the noise was further increased, the changes caused by the truncation progressively grew: the scaling interval shrank, the left branch of the spectrum shifted to lower α (stronger singularity), and its right branch became unsmooth. Nevertheless, the left edge (q ≥ 1) remained close to the spectrum of the pure Cantor set up to a 50% threshold. This observation provides an additional conclusion concerning the analysis of real signals. Namely, it follows that the multifractal spectrum of the dense set is robust not only with respect to the presence of noise but also to the resolution limit. Indeed, such a small dynamical range of the processed set appeared sufficient for revealing its self-similarity. The latter conclusion was also confirmed by the analysis of thinned Cantor sets that were obtained using the same multiplicative procedure as above (see Sec. 3.2) but stopping it after a smaller number of iterations.

3.4.2. Clustered Set In the case of the clustered Cantor set, the spectrum part corresponding to high positive q values was also found to persist against noise. In this case, however, the spectrum was strongly distorted even by a low-amplitude noise. Figures 10(b), 12, and 13 show the results of such tests for the noise amplitude at a 5 % level. It can be seen in Fig. 12 that the noise does not noticeably deteriorate the linear scaling dependences for large enough positive qs, but that a slight bending can be detected for small positive q values. At the same time, the scaling exponents change: the corresponding slopes coincide with the initial slopes at the largest q and are progressively modified when q is decreased. The overall changes become clear when the singularity spectra (Fig. 13) of the initial Cantor set (curve A) and the perturbed set (curve B) are compared. It can be seen that the noise addition leads to the spectrum extension and distortion. These modifications can easily be understood. Indeed, the geometrical support of the pure Cantor set is a fractal with the fractal dimension D0 ≈ 0.62, which determines the position of the maximum of the respective singularity spectrum (curve A). In contrast, the support of the noisy signal is the entire segment. This imposes D0 equal to 1. Further, the uniform filling of the segment with the noise depresses the signal

26


singularity and, therefore, the spectrum is shifted to the right along the α-axis. It seems likely that the robustness of the left branch of the spectrum is due to the property of the “mathematical microscope”: taking a high positive power of the local measure makes the higher-amplitude Cantor measure dominate against the lower-amplitude noise. In contrast, the Cantor set and the noise are mixed in the statistical sums at small nonzero qs. This gives rise to the observed bending of the scaling dependences. 0

(a) -5

Σα -10

-15 -10

-5

0

-5

0

log2 l 0

(b) -5

Σα -10

-15 -10

log2 l

Figure 12. Examples of scaling dependences [see Eq. (7)] for (a) the clustered Cantor set and (b) the same set superimposed with the 5% random noise. A unique scale is chosen for both graphs to make the comparison easier; q takes on the values of 20, 3, 1.5, 1, 0.5, and 0 from the upper curve downwards.


1.0

27

A B C

f 0.5

0.0 0.5

α

1.0

Figure 13. Effect of random noise on the singularity spectra for the clustered Cantor set. Curve A – initial set; curve B – after addition of a 5% noise; curve C – after cutting off the part of the noisy set below the 5% level.

In Fig. 13, the scaling exponents were calculated in the same l interval for all q values. This was a general practice in the present work. Its aim was to avoid arbitrariness in the detection of the multifractality. In the case considered, this leads to a peculiar shape at the top of curve B because the scaling in the negative q range is mostly determined by the small-amplitude noise, while the scaling at positive qs is governed by the high-amplitude Cantor set. A satisfactory linearity for negative qs was found in a narrow interval approximately from 2-9 to 2-5. Calculation in this range led to shrinking of the right branch of the spectrum towards the spectrum top, thus confirming that this branch is controlled by the noise component. These tests show that the underlying clustered structure can be detected by the multifractal analysis due to the robustness of scaling at (large) positive qs. The comparison of the results in Figs. 11 and 13 shows a principle difference between the influence of noise on the processing of dense and clustered structures. Indeed, the noise does not involve qualitative changes of the spectra of the dense set but may introduce fake scaling laws for the clustered structure. This effect, however, can be cured by the data truncation technique, as illustrated by Curve C in Fig. 13, which shows that the multifractal spectrum can be effectively restored. The application of a threshold is analogous to suppressing the negative q sums. It thus results in a declination of the right branch of the singularity spectrum. However, the remaining part of the measure provides a sufficient precision for the

28


determination of the statistically significant left branch. The comparison with the results of similar tests for the dense set testifies that the data truncation may be a useful check when the analysis provides evidences of a multifractal behaviour but does not give a smooth spectrum.

Chapter 4

EXPERIMENTAL TECHNIQUE AND ANALYSIS OF DEFORMATION CURVES 4.1. Recording of Deformation Curves The mechanical tests were performed using a typical setup for tensile tests. Only its general features will be described here, while some additional details will be provided when required. Flat specimens for observation of the PLC effect were cut from polycrystalline cold-rolled sheets of Al-Mg alloys with 2.5 or 3 atomic percent of magnesium. The working part of the samples was about half as narrow as their heads fastened in the grips. The typical gauge length, width, and thickness varied in the ranges of 20-30 mm, 5-7 mm, and 1.5-2.5 mm, respectively. The specimens were usually annealed at 400oC for several hours and quenched in water to provide a uniform solute concentration and, therefore, avoid an extrinsic source of heterogeneity. The tests were carried out at room temperature with a constant imposed strain rate ε& a in the range of 4×10-6 s-1 to 1.4×10-2 s-1. The stress was recorded at a sampling rate of 0.5 to 500 Hz, depending on the strain rate, so that the data file contained at least several thousand points available for analysis. The typical number of data points in the analyzed file was ten thousand. A special attention was paid to the sufficient sampling of stress jumps. In addition, the persistency of the analysis was checked by thinning out the original data sets twice of four times. It was verified that it did not significantly modify the results of calculation of the multifractal characteristics.

30


4.2. Preparation of Time Series for the Analysis 4.2.1. Removal of the Work Hardening A general difficulty in studying the dynamic properties of a dislocation ensemble, as compared to many other natural systems, is caused by the strain hardening due to the evolution of the dislocation microstructure. The hardening can be seen in Fig. 1 as a gradual increase of stress during deformation. As a result, the average amplitude of the stress jumps can evolve, too. Such a systematic trend may alter the results of the analysis of the associated dynamics (cf. [97]). Usually, it is taken into account by a procedure involving the data normalization (e.g., [11]). As will be seen in Sec. 5, the multifractal analysis is only slightly affected by the changing average intensity of the fluctuations. However, the treatment aimed at removing the strain hardening effect was a general rule in the present investigations. The essence of such a treatment is explained below. The average stress jump depth usually changes fast at the beginning of deformation (ε ≤ 5-10%) and stabilizes afterwards. For this reason, short enough portions at the late quasistationary stages of the deformation curves, which correspond to an almost saturated work hardening, are usually allotted for the analysis. The remaining work hardening is then taken into account by normalization of the selected portion. The aim of the data normalization is twofold. First, it allows extending the amount of data for the analysis by selecting a longer portion at the quasistationary stage. Second, the normalization makes it possible to analyze the initial transient stage, as well, and, therefore, test the effect of the microstructure evolution on the dislocations dynamics. It should be noted that the discrimination of two stages does not concern the common nomenclature of the work-hardening stages [42]. As can be seen in Fig. 1, the plastic instability observed in the present work mostly corresponded to parabolic hardening (the socalled stage III). Various normalization procedures were proposed in literature, depending on the specific behavior observed. Some techniques imply an in-phase increase of σ and Δσ upon straining. Such a procedure sounds physically based because both parameters are related to the microstructure state. In this case, it is sufficient to divide the current stress value by a fit of either the deformation curve

σ (t ) or the

time dependence Δσ (t ) of stress jump amplitudes. However, the relation between σ and Δσ is often not straightforward. Particularly, the average Δσ value sometimes decreases along the quasistationary portion. The more general

Experimental Technique and Analysis of Deformation Curve

31

technique includes two stages. At first, the slow trend is removed from the deformation curve by subtracting the corresponding fit

σ (t ) . The resulting curve

oscillates around zero level. Next, the amplitudes of the oscillations are leveled off by normalization with regard to the average trend Δσ (t ) . The data fitting also admits different approaches. For example, calculation of the running average

Δσ (t ) was applied in [69, 70]. However, it should be taken into account that averaging the stress on too narrow intervals would smooth out the stress fluctuations in question. For this reason, a coarse fitting with a polynomial was usually applied in the present paper.

4.2.2. Extraction of Time Series After compensation for strain hardening, the deformation curve is represented by a time series {σj}. Here, the designation σ is kept for the normalized stress value. The most rigorous way to detect an ordered behavior of a deformation curve would be to analyze the corresponding pseudophase trajectory reconstructed from the {σj} series as described in Sec. 2.2.3. This approach, however, has several inconveniences. Particularly, this would make a procedure quite complex and, besides, involve additional approximations. More important, it would require a low dimensionality of the trajectory, i.e., a small number of collective degrees of freedom of the dislocation ensemble. In order to handle any kind of the dynamic behavior, the multifractal analysis of the one-dimensional time series itself was proposed in [69]. In doing so, it should be taken into account that the multifractal analysis deals with a singular measure behavior, whereas the deformation curve contains not only drastic stress drops but also smooth reloading portions. The use of a series of stress jumps, which resembles nonuniform Cantor sets (cf. Figs. 3 and 5), seems to be the most natural procedure. However, the jumps extraction is associated with a considerable ambiguity. First, it requires discriminating the small PLC stress jumps from the experimental noise and, probably, from stress fluctuations of different nature, e.g., those caused by the breakthrough of small dislocation pile-ups. Second, it imposes replacing each jump with a deltafunction, i.e., it does not take the jump shape and duration into account. This would inevitably lead to a significant loss of information, especially, under conditions of type A serrations when the stress fluctuations of all possible amplitudes and durations are observed. An alternative way is based on the idea that the drastic jumps can be amplified by taking the time derivative of the deformation curve. As suggested in

32


[69], this can be implemented via the analysis of the time series obtained by calculating the absolute value of the finite difference approximant ψj(tj) of the time derivative dσ/dt of the entire normalized curve. A similar approach is based on the use of a small-scale gradient field formed by the absolute values of stress increments, ψj(tj) = |σj - σj-1| (e.g., [75]). Such time series were used in the present work. It was checked that both kinds of series give close results. The drawback of this approach is that the analyzed signal is almost everywhere nonzero. This imposes the D0 value close to 1 so that the mere fact of the intermittence of the jump nucleation cannot be uncovered because of the presence of noise. As was seen in Sec. 3.4.2, this can impede the detection of the multifractal spectra of the clustered signals. Various methods used to analyze a deterministic structure corrupted by a stochastic noise are discussed in literature (see, e.g., [37, 38] and references therein). It should be stressed that the standard procedures of the noise suppression based on the linear Fourier analysis are of little use because both the stochastic noise and the discontinuous events have continuous Fourier spectra. In particular, the wavelet technique is often combined with the multifractal analysis [98]. As will be seen below, the truncation of the low-amplitude part of the signal using a variable threshold, as was suggested in Sec. 3.4.2 for the corrupted Cantor set, allows a simple practical way for discrimination of the random and the multifractal components of an experimental signal.

4.2.3. Calculation of the Multifractal Spectra To calculate the spectra, the data set ψj(tj) was covered with grids with the division δt varied as a power of 2. The probability measure μi(δt) was defined as a sum of amplitudes of ψ-spikes in the ith interval δt, normalized with respect to the sum over the entire series:

μ i (δt ) = ∑k =1ψ k / ∑ j =1ψ j , n

N

(8)

where N is the whole number of the data points and n the number of points in the ith interval. Calculations were performed by the box-counting fixed-size technique using relationships (5) and (7) where l should be replaced by δt. The spectra were traced by varying q in the range from, typically, q = -20 to q = 40. The starting point for covering the time interval with a grid was chosen at random,

Experimental Technique and Analysis of Deformation Curve

33

and the estimate of fractal dimensions was usually obtained by averaging over ten trials.

Chapter 5

EXPERIMENTAL INVESTIGATIONS OF PLASTIC INSTABILITY 5.1. Multifractal Structure of Type C Curves As was said in the Introduction, two dynamic regimes were detected using the dynamical and the statistical analysis of the PLC effect: the deterministic chaos under conditions of type B effect at intermediate strain rates and the selforganized criticality for type A effect at high strain rates. The same techniques of analysis did not provide characterization of the dynamics associated with type C serrations that are specific of the low strain rate deformation and are generally believed to occur randomly [11]. Ideally, the hypothesis of randomness would be consistent with a periodic occurrence of stress jumps according to the model of the PLC effect in the spirit of the relaxation oscillations. Indeed, the PLC bands would nucleate at random sites each time the necessary stress level is attained (see Fig. 2). The irregular character of the stress fluctuations could then be tentatively related to a stochastic heterogeneity of strain in the sample. However, such a speculation is not supported by the experimental data. Indeed, the deformation curves show a great variety of shapes at low strain rates, which seems to be difficult to explain by a random factor. To check whether type C behavior is associated with stochastic or correlated deformation processes, the multifractal analysis was applied to the deformation curves recorded at low strain rates.

36


5.1.1. Geometry of Deformation Curves The Al-3 at. % Mg polycrystalline alloy (grain size of 50-100 μm) used in the present tests exhibited a transition from type C to type B serrations when the imposed strain rate was increased from 10-5 s-1 to more than 2×10-5 s-1. A mixture of both types was observed in the tests at these strain rates. No type B serrations were detected below 10-5 s-1. To guarantee the conditions of type C instability, the tests carried out at ε& a = 4×10-6 s-1 will be considered in this paragraph. The stress was recorded at a sampling rate of 0.5 Hz. The files for the multifractal analysis contained from 10000 to 40000 data points. The measurement noise did not exceed 0.1 MPa.

(a) σ, MPa

200

180

(b) |dσ/dt|

5

0

30000

35000

t, s

Figure 14. (a) Example of a portion of type C deformation curve of an Al-3%Mg polycrystalline sample deformed at a strain rate of 4×10-6 s-1 and (b) the corresponding time derivative of the normalized stress.

All the observed curves possessed common features characteristic of type C behavior, namely, deep stress drops separated by smoother reloading portions (cf.


37

Fig. 1). In spite of the qualitative similarity, the exact shapes of the curves were quite diverse and displayed stress jump clustering to a variable degree. Figures 14 and 15 present two distinct examples of the deformation curves and respective time series. The stress jumps of Fig. 14 display a variety of amplitudes up to 30 MPa and are not considerably clustered. In Fig. 15, the large jumps with the size in the range from a few MPa to about 30 MPa form clusters. The observation of clustering bears evidence to a correlation of the deformation processes. The reloading portions contain smaller stress fluctuations (≤ 1 MPa) that do not visibly manifest clustering. The analysis revealed multifractality for most of the deformation curves. It was found that the curves illustrated in Figs. 14 and 15, which were recorded under identical conditions, are characterized by multifractal spectra that are respectively similar to those discussed above for the dense and the clustered Cantor sets corrupted by noise (cf. Figs. 11-13). These examples represent two extreme cases of the behavior observed in the experiments and will be discussed here in detail.

5.1.2. Dense Time Series Figure 16(a) presents examples of scaling dependences (5) for the deformation curve shown in Fig. 14. In contrast to the above considered cases of the multifractal Cantor sets, the nontrivial scaling laws do not cover the whole scale. In a typical plot, log(Zq) first increase with log(δt) roughly following lines with slopes equal to (q-1). This indicates a random behavior of the stress drops at a short time scale, which corresponds to Dq = 1 [cf. Eq.(5)] and a f(α) spectrum condensed at the (1, 1) point. Then, the log(Zq) plots cross over into a second set of lines in typical intervals between 200-500 s and 20000-30000 s, with slopes different from q-1. In Fig. 16(a), such behavior, marked by vertical dashed lines, starts around 500 s and persists almost up to the test duration (37600 s). Then, the dependences saturate. Within the dashed lines, the dimensions Dq differ from unity, and a nontrivial f(α) spectrum can be found [Fig. 17(a)]. A multifractal behavior is thus detected for these time intervals. The range of the linear portion – about 1 or 2 orders of magnitude – is typical of laboratory experiments [99]. The scaling confinement may be due to a number of factors, either physical (the nonstationary character of the instability, a superposition of various microscopic mechanisms, the finite dislocation free path confined to the specimen size, the shortness of the data set accumulated before the specimen fracture, and so on) or related to the measurement noise and the limited

38


experimental resolution. As will be seen below, both the scaling and its limitations may give information on the underlying physical mechanisms (see [99] for the discussion of the intrinsic cutoffs in empirical fractals). Similarly to the treatment of the model Cantor sets, the data were truncated by cutting off small ψ spikes to check the steadiness of the spectrum. In contrast to the model situations, the level of the random component in the experimental signal is a priori unknown. Indeed, only the instrumental noise can be measured directly, whereas nothing can be said about the possible contribution of real stochastic processes of plastic flow to the stress fluctuations. For this reason, the calculations were repeated for several successively increasing magnitudes of the threshold. No noticeable influence on the spectra of the truncated signals was found while the threshold height ψthr was inferior to approximately 0.03ψmax, where ψmax is the maximum spike amplitude. The subsequent increase of the threshold led to a continuous but rather slow narrowing of the scaling interval and depression of the singularity spectrum (compare curves A and B in Fig. 17(a)). For example, D0 was reduced to the value of 0.95 for ψthr = 0.03ψmax in the example considered. As expected, the left-hand branch was also weakly affected. Strong – but not significant – changes were only observed on the low-statistics right branch that corresponds to the least measure, that is, to the small-amplitude bursts that are gradually cut off. Such a continuous deterioration of the spectra is obviously caused by the depletion of the jump statistics. As a whole, the data truncation tests confirm the conclusion on the multifractal structure of the deformation curve. The threshold of 0.03ψmax corresponds to the stress drop amplitude about 1 MPa, i.e., an order of magnitude higher than the experimental noise. This is consistent with the observation that the stress drops with amplitude between 0.1 MPa and 1 MPa correspond to short time scales, for which random type scaling was found.

5.1.3. Intermittent Time Series The results of the analysis of the deformation curves characterized by a stronger intermittence were found to be similar to that for the clustered model set. The examples of scaling dependences and the f(α) spectrum of the deformation curve of Fig. 15 are presented in Figs. 16(b) and 17(b) (curve A). Three different domains can be seen in Fig. 16(b). The short initial portion (δt < 50 s), tending to a (q-1) slope, reflects the lack of correlation of the stress fluctuations. A crossover to multifractal behavior is detected at intermediate scales (50 s < δt < 2000 s), with the upper scaling bound roughly corresponding to the maximum length of the stress jump clusters. At larger time scales, the plots return to a (q-1) slope, i.e.,


39

an uncorrelated behavior. The f(α) spectrum obtained for the intermediate time scales [Fig. 17(b)] is markedly different from the spectrum in Fig. 17(a). On the contrary, it resembles the case of the noisy clustered Cantor set (Fig. 13), with the data points around q = 0 tending to cluster near the point α ≈ 1.1, f ≈ 1.

(a) σ, MPa

220

200

(b)

|dσ/dt|

5

0 30000

40000

t, s Figure 15. The same as Figure 14 for another sample. The stress jumps look clustered and the variation of their amplitudes is less pronounced. Note that the time interval is twice as long in comparison with Figure 14.

The data truncation tests showed that the spectra were largely unaffected by the thresholds below approximately the same value of 0.03ψmax as for the dense set. Therefore, this value indicates the amplitude level of the stress fluctuations that do not possess a multifractal structure. This testifies to a nonfractal nature of a part of dislocation glide processes participating in the plastic flow.

40


(a)

log2(Σq)

0

-40

q-1

q=10 q=5 q=2 q=0

-80

(b)

log2(Σq)

0

q-1

-40 q-1 -80

5

10

log2(δt)

15

Figure 16. Log-log plot of the statistical sums Zq vs grid box length δt for several q values [see Eq. (5); arbitrary units] for the time series shown in (a) Figure 14 and (b) Figure 15. Vertical dashed lines show scaling intervals used to calculate singularity spectra. Solid lines trace the respective slopes. The (q-1) slope corresponding to an uncorrelated behavior is shown by inclined dashed lines for q=10 only.

The spectra drastically changed and acquired a smooth shape indicating multifractality when ψthr exceeded this level (Fig. 17(b), curve B). It can be seen that the maximum (fmax = D0) of the new spectrum is considerably reduced and shifted to the left, testifying to an essentially discontinuous behavior (α < 1) on a fractal support (D0 < 1). The positive q branch is weakly influenced by cutting off the small events: both the initial and the modified spectra are close to each other


41

in this region and converge for the highest magnitudes of q. A further increase in ψthr was not accompanied by significant changes in the new spectrum until the threshold reached approximately 0.05ψmax. Such stability bears evidence to an inherent character of the spectrum presented by curve B with regard to the underlying structure of the time series. Finally, the subsequent increase in the threshold led to a progressive shrinking of the scaling interval and distortion of the spectrum because of the lack of the jump statistics. 1.0 (a)

A B

f 0.5

0.0

1.0

(b)

f

0.5

0.0

0.5

1.0

α

1.5

2.0

Figure 17. Singularity spectra for the time series shown in (a) Figure 14 and (b) Figure 15. A - entire data set; B - after cutting off bursts below ψthr = 0.03ψmax.

42


The observed changes in the shape of the singularity spectrum and the comparison with the results for the intermittent Cantor set suggest that the large stress jumps in Fig. 15 form a self-similar structure that is revealed when the small-scale irregularities are removed. This conclusion is also consistent with the closeness of the upper bound of the scaling interval to the maximum length of the stress jump clusters. Besides, it was checked that the time series composed of the jumps below the threshold give broken f(α) dependences near the point (α, f) = (1, 1), similar to the spectra obtained for trial random sets. Thus, it may be assumed that the discontinuity on the deformation curve in Fig. 15 is a superposition of a multifractal structure related to the clustered stress jumps and an uncorrelated stochastic component. The parameters of the found spectra characterize various features of the multifractal structure. Specifically, the low magnitude of D0 (D0 ≈ 0.45 in Fig. 17(b)) quantifies the clustering degree of the stress jumps.

5.1.4. Correlation of Type C Stress Fluctuations The data obtained show that the macroscopic features of plasticity: the flow stress level, the characteristic type C shape of the deformation curves, i.e., deep stress serrations separated by portions of smoother flow, and the range of stress drop amplitudes, were similar for all samples deformed at the imposed strain rate of 4×10-6 s-1. At the same time, a closer examination of the shapes of the deformation curves showed some variation reflected in the degree of clustering of the stress drops. In spite of this diversity, the results of the multifractal analysis clearly reveal a self-similarity of the temporal structure associated with stress discontinuities. Although the underlying dynamic mechanism cannot be deduced from the global characteristics provided by the fractal dimensions, it is essential that these data unambiguously prove a correlated behavior of type C bands. However, two typical multifractal spectra could be distinguished among the whole set of data. Most of the observed spectra were close to the low-clustering behavior shown in Fig. 17(a), which reveals a continuous support (D0 = 1) of the set of serrations. The scaling dependences [Fig. 16(a)] covered a wide δt range which was bound at small scales (below 200-500 s) but extended up to the test duration. The other typical (but less frequent) behavior is shown in Fig. 17(b) and corresponds to a strong clustering: D0 varied from the lowest value of 0.45 (Fig. 17) to about 0.6. In this case, the upper scaling limit [Fig. 16(b)] was typically much lower and roughly corresponded to the maximum duration of the stress drop clusters. At larger scales, the f(α) spectrum condensed to the (1, 1) point, which indicates the absence of a correlation between clusters.


43

The diversity of the deformation curves indicates a transitory state of the dislocation dynamics. Taking into account the weakening of the band correlation with the decreasing strain rate, the type C domain can be ascribed to a gradual transition from the dynamic chaos detected for type B effect to periodic relaxation oscillations predicted for a uniform material. Alongside with the qualitative conclusions on a nonvanishing correlation of type C bands, the above data provide a quantitative characterization of the variety of deformation curves. As a result, they also allow for some hypotheses on the nature of the underlying dislocation processes, the verification of which must become possible with the development of high-frequency techniques of local extensometry [100, 101]. The results of the data truncation tests bear evidence that the multifractal behavior is associated with relatively large stress jumps ranging from 1 MPa to 30 MPa. This is also confirmed by the observation of their clustering. No correlation was found for the stress fluctuations with amplitudes below approximately 1 MPa. This may be partly due to the masking effect of the experimental noise. However, since the measurement noise does not exceed 0.1 MPa, it is reasonable to suppose an inherently stochastic nature of the small events. Particularly, those may be caused by the motion of individual dislocation groups (pile-ups) in subcritical conditions when the stress level is elevated enough to initiate the breakthrough of the pile-ups accumulated at obstacles. In some sense, these events can be considered as precursors of the catastrophic dislocation avalanches associated with the deep stress jumps. The memory about such catastrophes decays because of the plastic relaxation during reloading but may not fully relax before the next catastrophe, thus leading to formation of a self-similar structure of the deformation curve.

5.2. Multifractal Analysis of Type B Serrations 5.2.1. Geometry of Deformation Curves Type B serrations are usually observed in a wide range of strain rates, often covering about two orders of magnitude for Al alloys. Their visual examination shows that the shapes of the deformation curves are somewhat modified when the strain rate is changed, although the curves possess typical features distinguishing the type B behavior. This qualitative observation gives an additional argument that in this case, too, there is a need for a quantitative approach to the evaluation of the plastic flow reflected in the deformation curves.

44


(a)

σ, MPa

210

200

(b)

|dσ/dt|

50

0

350

t, s

400

Figure 18. (a) Example of a portion of type B deformation curve of an Al-3%Mg polycrystalline sample deformed at a strain rate of 6×10-4 s-1 and (b) the corresponding plot of the time derivative of the normalized stress.

Figure 18 presents a typical example of type B serrations. It can be seen that the curve has an intermittent character: the irregular stress jumps alternate with packets of rather regular oscillations showing characteristic scales of stress amplitudes and frequency. This conforms to the results evidencing the dynamic chaos under type B conditions. Thus, it is not surprising that a multifractal behavior was also found in this regime. At the same time, the analysis of type B serrations confronts another difficulty related to work hardening. Namely, not only the average stress jump amplitudes can evolve during deformation, but also the average jump frequency, or, more precisely, the intervals between the neighboring jumps. This problem required a special study that was undertaken in [78] and is highlighted in the current paragraph.


45

The normalization of the deformation curves, aimed at mimicking the stationary behavior by taking into account the growth or, sometimes, the reduction of the stress jump depth, was described in Sec. 4.2.1. The possible evolution of the jump frequency has not been discussed in literature under this angle. Such a disregard may be caused by various reasons. First, the strain hardening effect on the jump frequency is often weaker in comparison with the influence on their size. Besides, in contrast to the size normalization, there are no obvious physical grounds for the time reconstruction. Finally, the time reconstruction is a more difficult numerical task. Indeed, when the entire time series and not only the series of stress jumps are to be analyzed, this would imply a step-by-step reconstruction of the deformation curve and, therefore, accumulation of numerical errors. Despite all this, the problem of the evolution of the interjump intervals cannot be disregarded because many authors reported significant frequency changes in the course of deformation (e.g., [102]).

5.2.2. Preparation of Time Series From the first sight, the use of the plastic strain ε instead of time, which may not be a proper variable accounting for the microstructure evolution, comes to mind. However, the strain dependence of strain increments accumulated between stress jumps manifests qualitative trends similar to those observed for the time delays. For this reason, the evaluation of the influence of the slow evolution of jump frequencies on the results of the multifractal analysis of the original deformation curves was undertaken by applying a direct time reconstruction. The multifractal properties of the original and reconstructed time series were compared using deformation curves recorded at different strain rates in order to provide a basis for contrasting physical effects with numerical artifacts. In addition, the conclusions were tested with the use of model signals generated by stretching or compressing multifractal Cantor sets. The Al-Mg alloy used in this study had nominally the same (as in the previous section) Mg content, a similar polycrystalline microstructure, and demonstrated a similar behavior. The type B curves were observed in the range from approximately 5×10-5 s-1 to 2×10-3 s-1. Figure 19 represents time dependences of stress jump amplitudes and interjump intervals for one of the samples deformed at ε&a = 6×10-4 s-1 (cf. Fig. 2 in Ref. [78]). The dependences display two distinct portions, thus confirming the discrimination between the transient and the stabilized behavior, which was suggested in Sec. 4.2.1. It can be seen that both portions on average behave in a roughly linear manner but are characterized by

46


different rate of growth of Δσ and Δt . The transient stage is characterized by the fast increase of the moving averages, whereas the quasistationary portion displays almost no time dependence of Δσ and a slowed down increase in Δt . The weaker dependence at the stabilized stage justifies the general neglect of the nonstationary character of the jump frequency in literature. However, considerably stronger dependences of the jump parameters at the saturation stage, either ascending or descending, were observed, too: in some cases, the changes reached 50 %. Therefore, it is useful to define a procedure taking the effect of strain hardening into account, which would be valid for both the quasistationary and transient stages of the stress-strain curve. Particularly, this would give a handle to the study of the transient behavior, which is almost unexplored up to now.

Δσ, MPa

(a)

5

0

(b)

Δt, s

1

0 200

t, s

400

Figure 19. Example of time dependences of (a) the stress jump size Δσ and (b) intervals Δt between neighboring jumps. The dashed lines show regions with different values of the average rate of growth of the jump parameters (cf. example in [78]).


47

The ψj(tj) time series were constructed using three kinds of data files: portions of the original deformation curves, the same portions after leveling off the average stress jump size, and, finally, after additional reconstruction allowing for the variable jump frequency. Recalculation of the initial data files was based on the linear trends in the evolution of the jump size and frequency. As in the above examples, such a rough approximation aimed at preserving the natural variance of the jump parameters. The jump size variation was eliminated by normalization using a linear regression fit through the Δσ (t ) -dependence. The constant stress jump frequency was mimicked by gradual rescaling of the sampling intervals dt in proportion to the linear regression fit f(t) through the time dependence Δt(t) of the interjump intervals:

dt j = t *j +1 − t *j = *

dt , f (t j )

(9)

where j is the serial number of the data point. To mimic the alteration of the stress jump frequency using Cantor sets, those were proportionally stretched (or compressed) according to Eq. (9), where the time variable should be replaced with the coordinate l. The stretching function, which appears in the denominator in Eq. (9), had the form

f (l ) = 1 + β l ,

(10)

with the stretching factor β. In addition, the effect of stretching along the ordinate axis, which corresponds to the stress jump amplitudes, was also tested using the same form of the proportionality function.

5.2.3. Analysis of Experimental Curves The multifractal structure was revealed for a part of experimental curves. In contrast to a strong variation of the multifractal spectra of type C curves, the spectra found for type B effect were close to each other. Particularly, the changes observed upon the strain rate variation were weak, thus indicating that the differences in the detailed shapes of the corresponding deformation curves were not related to qualitative changes in the dislocation dynamics. This confirms the results of the first investigations of the multifractality under type B conditions [69, 70, 75]. Figure 20 presents examples of scaling dependences for the transient

48


stage of one of the deformation curves. The corresponding ψj(tj) series was obtained by leveling off the stress jump size only. It can be recognized that scaling may be clearly detected for these data in spite of the jump frequency evolution that was not taken into account by the curve preprocessing.

log2(Zq)

0

q=0 q=5 q=10 q=20 q=40

-200

4

log2(δt)

8

Figure 20. Example of log-log Zq(δt) dependences [see Eqs. (5)] for a time series obtained upon eliminating the growth of the stress jump size at the transient stage of the deformation curve of an Al-3%Mg sample ( ε& a = 6×10-4 s-1).

The influence of various types of the deformation curve reconstruction on the corresponding multifractal structure is illustrated in Fig. 21. It compares the singularity spectra f(α) of the original and modified time series for both the quasistationary plastic flow (curves A and B) and transient deformation (curves CE) of one of the samples tested at ε& a = 6×10-4 s-1. Also shown are spectra for two samples deformed under conditions of type C effect at 6×10-4 s-1 (curves F and G; cf. Fig. 17). Figure 21 testifies that the positive q wings of the spectra are weakly affected by the unsteadiness of the plastic instability. Indeed, curve A obtained for the original deformation curve and curve B found after leveling off both the jump amplitudes and the interjump intervals practically coincide in this q range. The influence is more visible but also small at the transient stage. Indeed, although the jump size normalization (curve D) followed by the complete data reconstruction (curve E) do lead to a progressive shift of the multifractal spectrum found for the unmodified data (curve C), the value of the shift is considerably smaller than the


49

changes observed under different experimental conditions (cf. curves F and G). Particularly, the horizontal shift between curves C and E in Fig. 21 comprises approximately 0.045 ± 0.002 for α(40) ≈ αmin and 0.007 ± 0.0003 for α(0). In the whole set of experiments, the maximum change in αmin, caused by reconstruction of the transient stage, did not exceed 0.05, whereas the shifts observed upon transitions between different dynamical regimes of the PLC effect could reach the value of 0.6 even at the stabilized stage. It can be concluded that that the unsteadiness of the deterministic noise related to the PLC effect does not hide its multifractal structure for q ≥ 0. It is also not surprising that the results for negative qs are more sensitive to the curve reconstruction.

1.0

f

A B C D E F G

0.5

0.0 0.5

1.0

Figure 21. Examples of the singularity spectra for -6

-1

α

ε&a -4

1.5

= 6×10-4 s-1 (curves A to E) and

ε&a

-1

= 4×10 s (curves F and G). The data for the 6×10 s were separately processed at the quasistationary stage (curves A and B) and the transient stage (curves C to E). A and C unmodified data; B and E – full reconstruction including stress drops normalization and time recalculation; D, as well as F and G – stress drop normalization.

It is worth noting that the vertical error bars on f(α) for α approaching αmin (i.e., at high q values) are much larger than for fmax : as only the largest and the rarest stress drops have a significant weight at high q, the statistics becomes poor. Changing the amplitude of these events by normalization may thus disproportionately affect the calculated value of f(α). As can be seen from Eq. (6),

50


df(α)/dα = q and takes on infinite values at q = ± ∞. Thus, the f(α) plot approaches a vertical asymptote at α = αmin (or α = αmax), and a small variation in α results in a strong variation in f(α). Nevertheless, it will be seen below that this parameter might also provide a useful piece of information in some cases. The curve reconstruction usually did not noticeably improve the scaling that typically covered a range of 1 to 1.5 orders of δt magnitude for the original curves. In some cases, however, the extension of the scaling range was significant 1 and reached half a decade . Such an improvement justifies the reconstruction used. However, it should be kept in mind that the physical grounds for the time reconstruction are unclear. As a compromise, the following procedure was proposed in [78]. To avoid physical arbitrariness and numerical errors, the signal preprocessing should be limited to leveling off the stress drop size. If no scaling is found, a time reconstruction may be undertaken in order either to corroborate the nonfractal nature of the deformation curve or to reveal its possible multifractal structure.

5.2.4. Distortion of the Cantor Sets Taking into account that the experimental spectra of type B curves reveal a dense filling (D0 = 1) of the time axis with stress jumps, the work-hardening effect was mimicked by distorting the dense Cantor set [Fig. 5(a)]. It was found that the influence of stretching may be neglected for the trial Cantor set when β takes on the values corresponding to real experimental situations. However, the determination of the scaling indices would be less obvious in the case of experimental curves, both because of the experimental noise and because the slopes of the scaling dependences are not known a priori. For this reason, unrealistically high values of β = 3 and β = 10 were deliberately chosen to test the work-hardening effect. Examples of scaling dependences for the undistorted Cantor set and the same set after stretching in the horizontal direction (β = 3) are compared in Fig. 22. It can be seen that even for such a high β value, only the large-scale parts of the dependences are considerably influenced. The comparison of the dependences for two q values illustrates that the deviation from linearity is negligible at small q and becomes stronger when q is increased. However, the inflection point does not move with changing q for a given β value, so that the multifractal characteristics

1

It must be remembered that the experimental noise is also affected by the transformations. This may shift the lower bound of the scaling dependences.


51

can be conveniently determined. It can be seen that at small δt, the scaling dependences have approximately the same slope before and after stretching. Consequently, the multifractal spectra almost coincide for both sets for q ≥ 0 (curves A and B in Fig. 23).

0

Σα -5

q=40 q=5

-10

β=0 β=3

-10

-5

0

log2(δl)

Figure 22. Examples of scaling dependences used to extract the singularity strength α(q) for the Cantor set of Figure 5(a) (β = 0) and for the same set after stretching of the intervals between segments in proportion to the segment coordinate with the stretching factor β = 3 [see Eq. (10)]. Σα is given by Eq. (7), δl is the grid size (arb. units).

Moreover, close results were obtained after stretching the Cantor set with the same factor β = 3 in both the horizontal and normal directions, which was aimed at mimicking the gradual increase of both Δσ and Δt. The essential changes in the spectra manifested themselves at further increasing of β, as illustrated by curve C (β = 10). Such a strong deformation of the initial set impeded a correct determination of the slopes of the scaling dependences and the resulting spectrum became noticeably narrowed. Even so, the correction was not crucial. In Fig. 23, for example, the corresponding increase in αmin reaches the value of 0.04, which is small in comparison with the meaningful changes observed experimentally upon variation of the strain rate. These results bear evidence that the endurance of the calculated multifractal spectra is a consequence of the scale invariance of the analyzed time series. Namely, when the signal is stretched or compressed, the new elements, replacing the old ones at a given scale level, have the same multifractal structure. This preserves the local structure of the signal in the limit δt → 0. Obviously, this also

52


leads to a contraction of the scaling range, unless the data set is infinite and is measured with an infinite resolution. Conversely, the reconstruction of the experimental curves may extend the scaling range when the scale invariance is not corrupted by a strong nonfractal noise. 1.0

(a)

f 0.5 A B C 0.0

0.8

1.0

1.2

α

(b)

1.2

D 1.0

0.8 -20

0

q

20

40

Figure 23. Spectra of (a) singularities and (b) generalized dimensions for the dense multifractal set. The solid line shows the analytical dependence. A – spectrum calculated for the original set, B – results of calculation after stretching in the horizontal direction (β = 3), C – after horizontal stretching with β = 10.

5.2.5. Temporal Structure of Type B Curves The data obtained prove that the smooth trends in the stress serration parameters only lead to a contraction of the range of the scale invariant behavior but do not influence on the scaling indices and, therefore, do not prevent from


53

evaluating the deterministic structure associated with the PLC instability. By consequence, the self-similar character of type B deformation curves, first disclosed by the techniques of reconstruction of the phase trajectory [72-74], is confirmed by the multifractal analysis. The corresponding spectra are found to be much narrower than those for type C curves. Together with the observation of the scaling range approaching the limit of the data acquisition time, it testifies to a more uniform behavior and, therefore, more intense correlation of type B bands. These results are also consistent with the detection of the dynamic chaos for type B serrations. Indeed, the ideal strange attractor is a uniform fractal characterized by a single fractal dimension. Figure 21 shows that the multifractal characteristics of the deformation curve are close for the transient and quasistationary deformation stages. This bears evidence that the dislocation processes involved in the PLC effect remained qualitatively the same over the entire deformation curve in the tests performed. Such uniqueness is not a general property of the PLC effect that manifests a great variety of behavior. In particular, qualitative changes in the deformation curve shape, which indicate transitions between different dynamical regimes, may occur in the course of deformation without need to change the strain rate (or temperature) [62-64]. The similarity observed in the present tests illustrates the capacities of the multifractal analysis as a tool that allows revealing essential dynamical features of the deformation curves. The present data validate application of the multifractal analysis to characterization of the complex structure of stress serrations, which reflects the collective dislocation dynamics, and detection of the changes that occur in the process of deformation or are induced by variation of experimental conditions. Herewith, the robustness of the multifractal analysis makes it possible either to avoid the unnecessary pretreatment of deformation curves or to justify a certain reconstruction procedure. At the same time, it should be noted that these results also show the necessity of an extreme caution in the analysis of experimental data. Particularly, reconstruction of the transient deformation stage was accompanied by detectable changes in the multifractal spectra (see Fig. 21). This imposes a great care when interpreting small variations of the fractal dimensions, often reported in literature.

54


5.3. The Overall Behavior of the PLC Instability 5.3.1. Summary of Experimental Data The previous paragraphs described the multifractal analysis of the PLC instability at low and intermediate strain rates. The type A curves appearing at higher ε&a values are usually very irregular (see Fig. 1). Consequently, their visual evaluation brings yet less information than in the above considered case and does not disclose modifications of their temporal structure, caused by variation of the experimental conditions. The first quantitative results giving information on this behavior were based on the statistical analysis of various parameters of stress jumps [62-68]. It showed that the statistics of jump amplitudes and durations progressively acquires a power-law shape when the strain rate is increased in the range corresponding to the B → A transition. As was pointed out in the Introduction, the power law indicates disappearance of the inherent event scales and is akin to self-organized criticality [79-82]. At not very high strain rates, the histograms display intermediate shapes between the monotonous power law and the peaked shape characteristic of type B serrations. In this case, the statistical analysis does not provide comprehensive information. In contrast, as will be seen below, the multifractal analysis allows a continuous characterization of the heterogeneous behavior in the entire interval of strain rates. At high ε&a values, the determination of the multifractal spectra is usually a comfortable task. First, the average stress jump parameters do not noticeably evolve during deformation, except for a short initial portion of the deformation curve, which is usually cut off before the analysis. Second, the scaling intervals are rather wide and often cover two orders of magnitude. An example of a singularity spectrum for a Al-3% Mg alloy deformed at a high strain rate is shown in Fig. 24. It can be noted that its width is considerably larger than the width of the spectra of type B curves for a similar alloy (cf. Fig. 21). An example of the rate dependence of the width of the singularity spectra in a wide ε& a range covering three PLC types is presented in Fig. 25(a) [75]. All specimens were prepared of the same ingot of a Al-3% Mg alloy. The width was measured on the positive branch of the singularity spectra: Θ = α1 – α20. It can be seen that the dependence is nonmonotonous. The changes in the trends are observed in the strain rate intervals corresponding to the transitions between the instability types. In particular, Θ slightly depends on ε& a in the middle range from approximately 10-5 s-1 to 10-3 s-1 and exhibits a minimum around 10-4 s-1. This


55

confirms the above conclusion that type B behavior is characterized by a high level of homogeneity. The conclusion on homogeneity also conforms to the observation of enhanced values of the fractal dimension of the most concentrated subsets, f80 ≈ f∞, in the same ε& a range [Fig. 25(b)]. Moreover, in spite of the strong uncertainty of the experimental determination of this parameter (see Sec. 5.2.3), the data in Fig. 25(b) help completing the scheme of the occurrence of the chaotic dynamics in the PLC effect. Indeed, the decrease of f∞ with increasing ε& a leads to an assumption that the events ordering is stronger near the lower bound of the strain rate interval of type B behavior and weakens towards its upper bound. Together with the characteristic intermittent shape of type B deformation curves, this supports the well-known scenario of the transition from periodic motion to chaos via intermittence (e.g., [5]), i.e., alteration of intervals of almost periodic oscillations with bursts of chaotic fluctuations. In the framework of this hypothesis, the decrease in f∞ may reflect the decreasing fraction of regular oscillations. It should also be noted that the competition between the close-toperiodic oscillations and chaos might explain why a part of type B deformation curves gave scaling dependence plots which were nontrivial, but for which a unique scaling law could not be determined. 1.0

f

0.5

0.0 0.5

1.0

1.5

α

Figure 24. Example of a singularity spectra for

ε&a

= 6×10-3 s-1.

56


(a)

α1-α20

0.3

0.0

(b) 0.2

f80

0.0

1E-5

1E-4

.

εα , s

1E-3

0.01

-1

Figure 25. Strain-rate dependences of (a) the singularity spectrum width Θ = α1 – α20 and (b) the fractal dimension f∞ determined as f(q = 80) for a Al-3% Mg alloy.

The Θ raising towards low ε& a , although less pronounced in Fig. 25 than in the case of similar alloys discussed above (cf. Fig. 21), confirms the results of comparison of the multifractality of type B and type C serrations. Similarly, its growth towards high strain rates indicates the growing irregularity of the deformation curves, which is associated with the type B to type A transition. Type A behavior is assumed to be related to SOC, i.e., power-law distributions of both the amplitudes of stress drops and the time intervals between them. In that case, the stress drops are not clustered and their probability does not depend on time. By analogy with the uniform Cantor set, it may be supposed that the time support of the events is continuous, i.e., D0 = 1, and, because of the homogeneous nonclustered distribution, the width of the spectrum should be low. A sharp decrease of Θ may thus be expected after the type B to type A transition. Such a


57

behavior was indeed observed on a slightly different alloy, Al-2.5% Mg [69, 70]. The description of these results will complete the schematic pattern of the PLC behavior. On this alloy, a complex of techniques including the dynamical, the statistical, and the multifractal analysis was applied to provide a direct mapping of the multifractal spectra onto the dynamical regimes. Three groups of specimens were prepared using different pretreatment to modify their microstructure. The first group was cut from the as-rolled material with an average grain size of 30–40 μm along the rolling direction and a grain aspect ratio approximately equal to 5. The microstructure slightly changed after annealing for 4 h at 320°C and quenching in water: the grains became coarser, about 50 μm in length, and acquired a less anisotropic shape with an aspect ratio of 3. Finally, the third set of specimens was produced by additional annealing for 3 h at 460°C and water quenching. These samples had a nearly equiaxed microstructure with coarse grains about 1 mm in diameter.

2

1 2 3

type B, sets 1 and 2

α-5-α5

type B, set 3

type A SOC

chaos 1

1E-5

1E-4

.

εα , s

1E-3

0.01

-1

Figure 26. The multifractal range Θ = α-5 – α5 as a function of the applied strain rate for a Al-2.5% Mg alloy. 1 – as-rolled specimens; 2 – after annealing at 320°C; 3 – after additional annealing at 460°C.

58


Figure 26 presents the strain rate dependences of the singularity spectra width for all kinds of specimens. Alongside with the Θ( ε& a ) dependences, the PLC types and the dynamic regimes found by the appropriate techniques are indicated. The quantitative confrontation of the dependences in Figs. 25 and 26 is complicated because of the different Θ definition in the corresponding works. Namely, in [69, 70], the spectrum width refers to the whole spectra including the negative branches, Θ = α-5 – α5. This imposes an essential bias in the absolute Θ values in two figures. Nevertheless, it does not prevent from a qualitative comparison of the changes caused by the strain rate variation. The comparison of the results testifies that the sequence of the dynamic regimes is similar for both alloys, although the respective strain rate intervals are shifted relative to each other so that Figs. 25 and 26 look complementary. Indeed, only type B and A serrations were observed in the tests on the Al-2.5% Mg alloy. No type C behavior was found down to the strain rate of 5.6×10-6 s-1. This is reflected in the absence of an upward drift on the left side of the Θ( ε& a ) dependences in Fig. 26. On the other hand, the respective dependence in Fig. 25 looks incomplete at high strain rates in the sense that the spectrum widening upon B to A transition is not followed by its narrowing similar to that observed in Fig. 26. As both alloys demonstrated type A deformation curves in this ε& a range, the nature of the difference needs additional investigations. For example, the strong heterogeneity of type A serrations in Al-3% Mg samples may be tentatively ascribed to the uncompleted transition to SOC. Indeed, the statistical analysis showed that the respective distributions of stress jump parameters can be fitted by power laws in certain intervals [75]. Whatever the explanation, the observed difference illustrates the deficiency of the visual evaluation of the discontinuous curves, describing all curves as a unique type A, and the need of the quantitative analysis. The data of Fig. 26 reveal that the distinct dynamical regimes of the PLC instability are characterized by different sensitivity to the specimen microstructure. Type A serrations and the associated SOC dynamics, detected by the statistical analysis, were observed in the same strain rate range, ε& a ≥ 5×10-3 s1

, for all kinds of Al-2.5% Mg specimens. Accordingly, all three Θ( ε& a )

dependences are close to each other in this ε& a interval. It can be thus concluded that the self-organized critical state reached is microstructure insensitive. Herewith, the low Θ values are indicative of the system being in a unique state, which is consistent with the SOC hypothesis.


59

In contrast, type B effect was detected in different ε& a ranges: below 8.3×10-4 s-1 for two sets of small-grain Al-2.5% Mg specimens (curves 1 and 2) and below 1.4×10-4 s-1 for large-grain specimens (curve 3). This difference is grasped by the multifractal analysis and is manifested by the extended range of the transitory behavior characterized by a strong heterogeneity (high Θ values) on curve 3. The sensitivity to microstructure can be qualitatively understood by taking into account the relationship between type B effect and the deterministic chaos, which is associated with a small number of degrees of freedom. Such conditions are more likely to occur in a small-grain material having a more uniform microstructure than in a large grained one, where deviations from the average behavior are stronger. The multifractal analysis allows defining this connection more accurately. Indeed, it can be seen in Fig. 26 that the chaos is observed in narrower ε& a intervals than the intervals of type B instability. Actually, both the shapes of the deformation curves and the kinetics of deformation bands, used as a basis for the phenomenological nomenclature of PLC types, only provide a rough criterion for the associated dynamics. In contrast, Figure 26 reveals a relationship between the chaos and the degree of heterogeneity expressed by the width of the multifractal spectra. Namely, the domain of the chaotic behavior is located below a certain Θ level showed by the dashed horizontal line.

5.3.2. Mechanism Governing the Succession of PLC Types These examples of the analysis of the PLC effect illustrate the potential and the deficiencies of the multifractal approach to the study of the plastic instability. Although it is not apt to determine the dynamical mechanism responsible for a specific spatiotemporal behavior, it allows detecting and characterizing the selfsimilarity that is often associated with nonlinear processes. Particularly, in the case of the PLC effect, the multifractal analysis provides a continuous quantitative evaluation of the changes in the temporal structure of stress fluctuations, which are induced by varying experimental conditions. A simple explanation of the changes of PLC types with the imposed strain rate was first developed in [63, 6670]. This concept will be shortly outlined here. So far, the main attention in the multifractal analysis was concentrated on a single parameter, namely, the width of the multifractal spectra, which characterizes the heterogeneity of the deformation curves. Even so, the data obtained make more precise the understanding of the succession of PLC types.

60


The proposed approach consists of completing the nonlocal microscopic models of the PLC effect with consideration of the heterogeneity of plastic flow in real samples. This is done by assuming a microstructure-sensitive spatial coupling of local strains in the heterogeneously deforming material. The dynamics follows from the competition between the repetitive reproduction of the heterogeneity, governed by the N-shaped function of the stress-rate sensitivity, and the uniformization controlled by the coupling between differently deformed regions. Various physical mechanisms of coupling are discussed in literature [11]. To specify the discussion, the coupling via internal elastic stresses will be considered below, although the overall scheme is general and may be adapted to specific coupling mechanisms. The internal stresses originate from geometric incompatibilities between differently strained regions of the specimen. This activates plastic relaxation processes which homogenize the strain and, therefore, lead to recovery of the internal stresses. That is, the pattern of stress serrations is associated with a dynamic equilibrium between the generation and the relaxation of plastic incompatibilities. It is reasonable to ascribe the observed transitions between PLC types to the changing conditions of the balance between the concomitant dynamical scales: the characteristic time necessary for plastic relaxation and the available time, which is represented by the reloading between stress jumps. The latter is a complex quantity because the reloading is not purely elastic (it is especially remarkable for type C serrations). However, it monotonously decreases when the strain rate is increased. In its turn, the plastic relaxation time depends on numerous factors including the deformation conditions and the specimen microstructure. Nevertheless, as only the strain rate was varied in the tests, the whole sequence of observations may be explained qualitatively. The slow reloading at the lowest strain rates (type C effect) leads to efficient relaxation of plastic incompatibilities and, therefore, weakly correlated bands nucleated almost randomly when the local stress reaches at some point the threshold level for the onset of instability. Were the uniformization perfect, the deformation curves would represent ideal relaxation oscillations. The data presented in Sec. 5.1 show that the incompatibilities generated upon large stress jumps of type C may not be completely relaxed. This results in a strong heterogeneity, as documented in wide multifractal spectra and their considerable 2 variation from sample to sample . Under B type conditions, the faster reloading 2

The particular behaviour is supposed to depend on the ratio between the local stress increase upon band generation and the variation of internal stresses within the material. In a material with a homogeneous microstructure, and a narrow range of internal stresses, all parts of a specimen will be nearly in the same state. The occurrence of a slip band at a random site will put the


61

leaves less time to equalize the strain gradients generated by a localized deformation band. As a result, the next band nucleates at a neighboring site (relayrace propagation). In other words, the inhibition of the relaxation of the local strain heterogeneity is the reason for the enhanced nonhomogeneity of band nucleation and more uniform deformation curves. The respective strain rate range is characterized by narrow multifractal spectra and a nonzero capacity f∞ of the most concentrated set of plastic events, which is associated with the regular stress oscillations. The correlation of the band nucleation is consistent with a small number of collective degrees of freedom required for the emergence of chaos. When the strain rate is further increased, the plastic relaxation becomes inefficient so that a deformation band will trigger plasticity in its neighborhood with a high probability, making band propagation over large distances possible. This leads to a new increase of heterogeneity of the strain localization pattern, an increase in the width of the multifractal spectra, and a decrease in f∞. Finally, such a strain rate is attained that the dislocation system constantly finds itself close to the instability threshold. This gives rise to plastic events of all sizes. Since such behavior is related to the self-organized criticality, the observed decrease of the heterogeneity degree is no surprise. To conclude this discussion, it is worth noting that the essential role of spatial coupling in the PLC dynamics unifies this phenomenon with cascade processes and justifies the similarity between the properties of the deformation curves and the multifractal Cantor sets generated by a multiplicative procedure.

5.4. Plastic Instability in Austenitic Fe-Mn-C Steels We conclude the discussion by an example of a currently performed study of plastic instability in austenitic steels, which involves interplay between several microscopic mechanisms. Austenitic Fe-Mn-C steels exhibit a potentially interesting combination of high tensile stress (900 MPa to 1400 MPa at room temperature) and high ductility (0.7 engineering strain at necking) [103], which presents some analogies with the behavior of 300 series stainless steels [104]. At low temperatures (between 77 K and approximately 250 K), the plastic strain is associated to the formation of ε martensite. Between 250 K and 673 K, it takes place by a combination of dislocation glide and twinning. At high temperatures, only dislocation glide is active. These changes of deformation mechanism are neighbouring sites under instability conditions, and the stress drops will tend to cluster. In the case of a heterogeneous material, and a wide range of internal stresses, the instability conditions may not be reached at neighbouring sites. The correlation between bands will then be weaker.

62


thought to be due to the variations of the (chemistry dependant) Stacking Fault Energy (SFE) from less than 18 mJm-2 at low temperature to more than 50 mJm-2 at high temperature. At room temperature, the easy twin formation results in a reduction of the effective travel distance of dislocations, similar to a continuous grain size refining. However, Fe-Mn-C specimens show stress-strain curves typical of plastic instabilities. This is illustrated in Fig. 27(a), where the specimen load, the traverse displacement Δl1, and the specimen elongation Δl0 measured on a part of the gauge length are plotted versus time. The load curve displays serrations of increasing amplitude, and, while the traverse displacement is regular, the Δl0 plot is made of straight segments, either nearly horizontal, or with the same slope as Δl1. The observed dependences can be easily interpreted by suggesting that that the major part of plastic strain takes place by motion of single slip bands along the gauge length [Fig. 27(b)]. Each band carries a finite true strain

ε b , and moves with a velocity vb . This results in an apparent true strain

rate ε&t ≈ vb Δε b l ≈ 4×10-3 s-1, where l is the specimen length l1 + Δl1 , when the band moves between the arms of the strain gauge (Fig. 28), and about 10-4 s-1 otherwise. Thus, under the applied stress, the specimen apparently has two equilibrium states, with notably different strain rates. The true strain rate

ε&t remains of the same order of magnitude during the whole test, but the time Δti necessary to the ith band to travel between the arms of the strain gage increases strongly at the end of the test. The average band velocity can be estimated by dividing the distance between the arms of the strain gage

l0 + Δl0 ≈ (l1 + Δl1 )l0 l1 by Δti . It is first nearly constant, but decreases

strongly at the end of the test. On the other hand, the plastic strain cumulated as the ith band travels along the gage length (i.e., Δε b ) increases from 10-3 to nearly 10-1. In spite of the strong change in the average kinetics of the propagating zone of localized strain during testing, the parameters of the accompanying stress fluctuations evolve slightly and can be treated, after common amplitude normalization (see Sec. 4.2.1), by the multifractal analysis. In terms of the above scheme of band propagation, these fluctuations can be considered as associated to irregularities in the band width, plastic strain, and velocity. Physically, the band propagation is the result of the development of plastic flow into neighboring regions along the specimen axis. That is, the stress fluctuations reflect plasticity

Experimental Investigations of Plastic Instability 5

(b)

Δl, mm

(a)

F, KN

63

30 4

3

20

l1

Δl1

l0

2 Δl0

10

1

0

0 0

t, s

200

400

Figure 27. (a) Raw experimental data from a tensile test on a Fe-Mg-C specimen ( ε&a = 7×10-4 s-1; T = 300 K): Load (left hand scale), traverse displacement (Δl1) and elongation of part of the specimen gauge length (Δl0) (right hand scale). (b) Schematic drawing illustrating a propagating slip band.

6

. εt (10

-3

s-1)

4

2

0

-2 0

100

300

200

400

t, s Figure 28. Average strain rate, computed from the traverse displacement (thick line) and from the gage displacement (thin line).

64


processes at a finer scale in comparison with the macroscopic band. The first results of the multifractal analysis bear evidence that the dynamics of plastic instabilities strongly depend on the temperature and strain rate. At a reference strain rate of 7×10-4 s-1, and in the domain where plastic strain takes place by martensitic transformation (low T), the stress fluctuations have a non-fractal stochastic character associated with a f(α) spectrum reduced to (1, 1), but scaling laws may sometimes be observed in a limited range (from about 3 s to 50 s). The corresponding spectra are narrow, with 0.8 ≤ αmin ≤ 0.9. The intermediate T range corresponding to the combination of twinning and dislocation glide is characterized by undoubted correlated behavior. Figure 29 presents scaling dependences for the late portion (ε > 15%) of one of the stress-time curves recorded at room temperature. It shows a high quality of linearity that extends over more than two orders of magnitude. The corresponding singularity spectra found in the interval of log2(δt) ≈ 1 to 8 are represented in Fig. 30 (curve A). The initial portion of the deformation curve also displays multifractal properties but with a narrower scaling interval and a narrower spectrum (curve B; αmin ≈ 0.7, to be compared with αmin ≈ 0.55 for curve A). At high temperatures corresponding to dislocation glide, no scaling law could be found. Metallography of the deformed samples at different strains shows that a system of twins parallel to the primary slip planes is formed at the beginning of the test, whereas the later stage is characterized by formation of a secondary (intersecting) twin system. At the same time, the measurement of acoustic emission accompanying deformation shows that the twinning itself is associated with local bursts in the strain rate and contributes to the irregular structure of the deformation curves. A first analysis suggests that the heterogeneous plastic deformation by both twinning and dislocation motion is responsible for the narrow spectrum (curve B). Larger and larger slip bands then begin to form and propagate, leaving a hardened material in their wake. Once a band has traveled to the end of the specimen, the stress needs to increase before a new band can be formed. This results in larger and larger stress steps and an increase of the clustering of stress variations, i.e., a widening of the f(α) spectrum. It follows from these first trials that the understanding of the interplay of twinning, dislocation motion, and perhaps, PLC effect due to interstitial diffusion, which leads to the instability of plastic deformation, requires utilization of various experimental designs enforced with the quantitative analysis of stress fluctuations. Such investigation is now in prospect.


65

log2(Σq)/(q-1)

0

-5

-10 5

log2(δt)

10

Figure 29. Examples of scaling dependences [see Eq. (5)] for a Fe-Mg-C specimen deformed at 7×10-4 s-1 at room temperature.

1.0

f 0.5

A B

0.0

1.0

α

1.5

Figure 30. Singularity spectra for the stabilized plastic instability (curve A) and for the initial stage of the deformation (curve B).

Chapter 6

CONCLUSION The data presented in this paper show that under unstable plastic flow conditions, the self-organization of dislocations gives rise to various self-similar structures of serrated deformation curves. This makes it possible to use the multifractal analysis to detect the underlying collective phenomena in the dislocation dynamics and to put severe constraints on the possible missing links between the microscopic nature of the plastic flow and the macroscopic behavior of a deformed sample. Indeed, this is the self-organization that prevents from a simple description of the plastic flow as a stochastic process in a “gas” of mobile dislocations moving through a field of obstacles. It leads to quite complex behavior often looking like random noise but governed by nonlinear dynamic equations. Such a deterministic “noise” depends on the experimental conditions and can appear via qualitatively different dynamical modes requiring different theoretical approaches. For this reason, the understanding of the associated dynamics necessitates a set of techniques able to characterize and model the collective behavior. The examples discussed above show that the multifractal analysis is a tool to measure the complexity of stress-strain curves, which is valid within the whole domain of plastic instabilities. In particular, the multifractal analysis supplies a tool for checking theoretical and computational models of plastic instabilities. Indeed, the models pretending to capture the multiscale nature of plastic flow in real materials have to predict not only the emergence of serrations but also their temporal structure in varying experimental conditions. The above discussion of the role of spatial heterogeneity in the sequence of PLC types bears witness that the richness of the PLC effect might be reproduced in the framework of a unique dynamical model incorporating spatial coupling of local strains and allowing for their relaxation. Several efforts

68


x, rel. units

in this direction showed that such models are able to reproduce all types of PLC bands and statistics [47, 62-66, 68]. Moreover, the deterministic chaos and SOC were found in a model of the PLC effect in polycrystals, which directly considered strain incompatibilities caused by the dislocation glide in various slip systems [105]. Although a comprehensive quantitative description, including the multifractal analysis, has not been elaborated to date, this suggests that such models have the right basis for reproducing the dynamics of the PLC effect. The understanding of the dynamics associated with a single mechanism of plastic instability would be the first step in handling more complicated situations, an example of which was discussed in the paragraph devoted to mechanical tests on austenitic steels. When dealing with real data, the quantitative analysis is complicated by several factors of general character: the presence of random noise that is superposed on the deterministic noise associated with the collective dynamics, the systematic trends changing the average scale of the deterministic signal, the shortness of the experimental signal, and, eventually, the presence of nonfractal events caused by different microscopic mechanisms. The data obtained show that the multifractal analysis is robust with regard to nonfractal contributions and allows handling short time series.

t, rel. units Figure 31. Fragments of displacement curves recorded with a CCD camera at neighbouring sites in an AlMg sample.

Conclusion

69

0.9

D40 0.8

0.7

0

2

4

6

8

10

x0, mm Figure 32. Dependence of the generalized dimension D40 of the displacement curves on the initial position along the sample.

The investigations performed represent the first approach to the multifractal analysis of the phenomenon of plastic instability. The comprehensive study would require understanding the changes involved not only upon variation of the applied strain rate, but also temperature, composition, specimen geometry and size. Moreover, although the stress evolution reflects the spatiotemporal pattern of the plastic instability, the relation between the temporal and spatial behaviour may not be direct. This imposes application of the multifractal analysis to the study of spatial structures formed in the course of deformation. These investigations are becoming possible with the increasing time and space resolution of the techniques of local extensometry. An example of such experiments, currently performed under conditions of the PLC effect in AlMg alloys [106], is presented in Figs. 31 and 32 and concludes the discussion of the perspectives of the multifractal approach to the study of plastic instabilities. Using a CCD camera [107], displacements of ten to twenty benchmarks painted on the specimen surface were recorded with a frequency of 1000 Hz (Fig. 31). The preliminary multifractal analysis showed a self-similar character of both the stress-time curves and the displacement signals. Moreover, due to the global character of the multifractal characteristics, which characterize every signal as a whole, the analysis revealed the existence of a spatial modulation of various multifractal parameters with a millimeter-scale period (Fig. 32). This phenomenon might have a general character. Particularly, spatial waves of strain localization were earlier observed

70


for materials showing macroscopically smooth plastic flow [108]. It should be emphasized that the detection of the modulation masked by PLC serrations became possible due to application of the multifractal analysis and provides a spectacular illustration of its capacities. It was then confirmed by a direct comparison of short portions of signals from different points on the sample surface. These results, which have a preliminary character and require additional studies, may reveal another level of heterogeneity associated with the phenomenon of strain localization.

ACKNOWLEDGEMENTS One of the authors (M. L.) acknowledges support from CNRS for a one year stay at LPM, Ecole des Mines de Nancy. The support of the Russian Foundation for Basic Research (grant 04-02-17140) is greatly appreciated.

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INDEX A accounting, 2, 56 acoustic, 81 acoustic emission, 81 aging, 6, 8, 9 aid, 11 alloys, xvii, 6, 37, 55, 70, 72, 86 alternative, 40 ambiguity, 40 amplitude, 29, 31, 32, 34, 38, 41, 47, 49, 62, 77, 78 Amsterdam, 89, 91 anisotropic, 71 annealing, 71, 72 application, xvii, 2, 11, 15, 35, 66, 86 aptitude, 6 argument, 55 aspect ratio, 71 atoms, 7, 8 attention, xvii, 37, 74 attractors, 12 averaging, 5, 26, 27, 39, 41

B barriers, 8

behavior, xvii, xviii, 2, 9, 12, 39, 44, 45, 46, 48, 50, 53, 54, 55, 56, 57, 65, 66, 67, 68, 70, 73, 74, 76, 77, 80, 83 behaviours, 2 benchmark, 30 benchmarks, 86 bending, 32 bias, 28, 72 biology, 2

C Cantor set, xviii, 15, 17, 18, 19, 20, 21, 22, 26, 27, 28, 29, 31, 32, 33, 34, 40, 41, 46, 47, 48, 52, 57, 59, 62, 63, 71, 77 capacity, 18, 23, 76 catastrophes, 54 chaos, xvii, 12, 43, 54, 56, 66, 68, 74, 76, 84, 90 chaotic, 68, 74 chaotic behavior, 74 chemistry, 2, 77 classical, 17, 25, 89 classical mechanics, 89 classification, xvii, 10 clouds, 1 clustering, 19, 25, 45, 53, 54, 81 clusters, 45, 48, 52, 54 compensation, 39 competition, 8, 68, 75

80

Index

complementary, 72 complexity, 1, 30, 83 components, 12, 41 composition, 10, 86 computer, 25 concentration, 37 confinement, 46 confrontation, 72 Congress, xii constraints, 83 construction, 17, 19 controlled, 6, 34, 75 correlation, 2, 10, 18, 23, 45, 48, 54, 66, 76 correlations, 10, 15, 23 coupling, 75, 77, 84 covering, 16, 41, 55, 68 cracking, 6 CRC, 90 critical state, 73 crystal, xvii, 1, 5, 6, 10 crystal growth, 1 currency, 1

depression, 47 detection, 11, 30, 34, 40, 66, 86 deterministic, xvii, 1, 12, 29, 40, 43, 61, 65, 74, 83, 84, 90 deviation, 28, 63 diffusion, 8, 81 dimensionality, 11, 39 discontinuity, 19, 53 discrete data, 25 discrimination, 38, 41, 57 dislocation, xvii, 5, 7, 11, 38, 40, 46, 49, 54, 59, 66, 76, 77, 80, 83, 84 dislocations, xvii, 5, 6, 7, 8, 9, 11, 38, 77, 83 displacement, 77, 79, 80, 85, 86 distribution, 25, 71 divergence, 19 diversity, 2, 10, 53, 54 division, 16, 26, 41 ductility, 77 duration, 17, 40, 46, 53 dynamical system, 1, 11 dynamical systems, 1, 11

D

E

data collection, 26 data processing, 25 data set, 15, 23, 24, 25, 26, 28, 37, 41, 46, 52, 64 defects, xvii, 1, 5, 6 deficiency, 73 definition, 23, 72 deformation, xvii, xviii, 5, 6, 7, 8, 9, 10, 12, 18, 21, 38, 39, 40, 43, 45, 46, 47, 53, 54, 55, 56, 57, 58, 59, 60, 62, 64, 65, 66, 67, 68, 70, 73, 74, 75, 77, 81, 82, 83, 86 deformation stages, 66 degree, 24, 45, 53, 74, 76 degrees of freedom, 11, 40, 74, 76 delays, 57 delta, 40 dendrite, 1 denoising, 30 density, 5, 18, 19 dependant, 77

economy, 2 elaboration, 2 electron, 5, 13 electronic, xii, 8 electrostatic, xii elongation, 77, 79 endurance, 64 engineering, 77 equilibrium, 75, 78 equilibrium state, 78 Euclidian geometry, 15 evidence, 45, 51, 54, 64, 66, 80 evolution, 1, 11, 12, 15, 38, 56, 57, 58, 59, 86 exchange rate, 1 exchange rates, 1 experimental condition, 10, 12, 61, 66, 67, 74, 83, 84 experimental design, 81 expert, xii extraction, 40

Index extrinsic, 37

F flight, 7 flow, xvii, 1, 3, 5, 6, 7, 9, 11, 12, 15, 47, 49, 53, 55, 60, 75, 78, 83, 84, 86 fluctuations, 10, 12, 25, 38, 39, 40, 43, 46, 47, 48, 49, 54, 68, 74, 78, 80 Fourier, 41 Fourier analysis, 41 fractal dimension, 18, 19, 20, 23, 25, 32, 41, 53, 66, 67, 68, 70 fractal geometry, 20, 93 fractal properties, xvii, 25 fractality, 17 fractals, 15, 18, 47 fracture, 46 France, 94 freedom, 11

G gas, 83 gauge, 37, 77, 79 generalization, 20 generation, 75, 76 grain, 44, 71, 73, 77 grains, 71 grid generation, 26, 27 grids, 41 groups, 5, 54, 71 growth, 56, 57, 58, 60, 70

high-frequency, 54 histogram, 67 Holland, 90, 91 homogeneity, 24, 68 homogeneous, 71, 76 hypothesis, 43, 68, 73

I idealization, 18 identification, 2 implementation, xvii impurities, 6, 8 in situ, 5 independent variable, 11, 12 indices, 20, 62, 65 infinite, 11, 21, 62, 64 inhibition, 76 injury, xii instabilities, 77, 80, 83, 84, 86 instability, xvii, xviii, 5, 6, 7, 8, 9, 10, 11, 12, 38, 44, 46, 60, 65, 67, 68, 73, 74, 75, 76, 77, 81, 82, 84, 86 intensity, 38 interaction, xvii, 1, 5, 7 interpretation, 2, 23 interstitial, 81 interval, 17, 19, 30, 31, 34, 41, 47, 49, 51, 52, 67, 68, 73, 80 intrinsic, 47 Investigations, xv, 43 IOP, 91 iteration, 19, 20

J

H handling, 84 heat, 8 heat release, 8 height, 47 heterogeneity, 9, 10, 24, 37, 43, 73, 74, 75, 76, 86 heterogeneous, 20, 21, 67, 76, 81 high temperature, 77, 81

81

jerky flow, xvii, xviii, 6 Jung, 91

K kinetics, xvii, 74, 78 Kolmogorov, 91

82

Index

L large-scale, 63 law, 11, 17, 19, 22, 26, 67, 81 laws, 11, 73 lead, 40, 60, 65, 75 limitations, 46 linear, 1, 18, 26, 28, 32, 41, 46, 57, 58 linear regression, 58 links, 83 literature, 10, 39, 40, 56, 57, 67, 75 localization, 9, 12, 76, 86 location, 26 London, 90 low temperatures, 77 low-temperature, 6 LTD, 91

M magnesium, 37 magnetic, xii, 19 magnetic moment, 19 mapping, 71 masking, 54 mathematical, xviii, 2, 15, 18, 23, 32 mathematical methods, 2 mathematics, 1, 93 measurement, 15, 18, 44, 46, 54, 81 mechanical, xii, 5, 6, 11, 37, 84 mechanical testing, 11 memory, 54 microscope, 23, 32 microscopy, 5 microstructure, 10, 38, 39, 56, 57, 71, 73, 75, 76 mimicking, 20, 56, 63 modeling, 15 models, 9, 10, 75, 83 modulation, 86 Moscow, 89 motion, 1, 5, 7, 8, 9, 54, 68, 78, 81 motivation, 15

multifractal, xvii, 2, 12, 15, 19, 21, 22, 23, 24, 25, 28, 29, 30, 31, 34, 38, 40, 44, 46, 47, 48, 49, 53, 54, 56, 57, 59, 60, 62, 63, 64, 65, 66, 67, 71, 72, 73, 74, 76, 78, 80, 83, 84, 85 multifractality, 12, 30, 34, 46, 50, 59, 70 multiples, 20

N natural, 2, 11, 15, 20, 38, 40, 58 neglect, 57 New York, xi, xiii, 89, 90, 91, 93 noise, xviii, 15, 29, 30, 31, 32, 33, 34, 40, 44, 46, 47, 54, 61, 62, 64, 83, 84 nonlinear, 1, 2, 11, 74, 83, 89 non-linear dynamics, 89 nonlinear systems, 1, 2 normal, 63 normalization, 20, 38, 39, 56, 58, 60, 61, 62, 78 nucleation, 40, 76

O observations, 75 one dimension, 15 optical, 10 organization, 2, 5, 83 oscillations, 8, 9, 10, 39, 43, 54, 56, 68, 76

P packets, 56 paper, 6, 25, 39, 83 parabolic, 39 parameter, 62, 68, 74 patterning, 5 periodic, 9, 10, 18, 21, 43, 54, 68 periodicity, 2 phase space, 12 phase transformation, 6 physical mechanisms, 18, 47, 75 physics, 2, 89

Index plastic, xvii, 1, 3, 5, 7, 8, 9, 11, 12, 15, 18, 38, 47, 49, 54, 55, 56, 60, 74, 75, 77, 78, 80, 82, 83, 84, 85 plastic deformation, 5, 8, 81 plastic strain, 6, 7, 8, 56, 77, 78, 80 plasticity, 2, 5, 53, 76, 78 PLC, xvii, 6, 7, 9, 10, 11, 37, 40, 43, 61, 65, 66, 67, 68, 71, 72, 73, 74, 75, 77, 81, 84, 86 polycrystalline, 37, 44, 45, 55, 57 polynomial, 39 poor, 26, 28, 62 porous, 17, 18 Portevin-Le Chatelier, xvii, 6, 7 power, 1, 11, 17, 19, 22, 25, 26, 28, 32, 41, 67, 70, 73 power-law, 11, 28, 67, 70 powers, 2, 18 prediction, 22 preparation, xii preprocessing, 59, 62 probability, 19, 20, 25, 41, 70, 76 procedures, 39, 40 progressive, 5, 51, 60 propagation, 10, 12, 76, 78 property, xii, 2, 11, 19, 20, 23, 32, 66 proportionality, 59

R race, 76 random, 10, 18, 21, 29, 30, 31, 33, 34, 41, 43, 46, 47, 53, 76, 83, 84 randomness, 43 range, xviii, 5, 8, 9, 20, 26, 28, 29, 31, 34, 37, 41, 45, 46, 53, 55, 57, 60, 62, 64, 65, 67, 68, 72, 73, 76, 80 recalling, 25 reconstruction, 11, 56, 57, 58, 60, 61, 62, 64, 66 recovery, 75 recursion, 17, 18 reduction, 56, 77 refining, 77 regression, 59 regular, 56, 68, 76, 77

83

relationship, 17, 20, 23, 25, 74 relationships, 41 relaxation, 8, 43, 54, 75, 84 relaxation process, 75 relaxation processes, 75 relaxation time, 75 reproduction, 75 resistance, 8 resolution, 18, 23, 29, 31, 46, 64, 86 robustness, 32, 34, 66 rolling, 71 room temperature, 7, 37, 77, 80, 82 Russian, 87

S sample, 6, 43, 45, 49, 55, 60, 76, 83, 85, 86 sampling, 37, 44, 59 saturation, 57 scaling, 1, 2, 11, 12, 16, 18, 19, 20, 25, 26, 27, 28, 29, 31, 32, 33, 34, 46, 47, 50, 51, 52, 53, 59, 62, 63, 64, 65, 67, 68, 80, 82 scaling law, 1, 2, 16, 18, 19, 26, 34, 46, 68, 80 scaling relations, 25 scaling relationships, 25 scatter, 29 science, 2, 89 scientists, 2 selecting, 38 Self, 89, 92 self-organization, xvii, 1, 5, 11, 83 self-organized criticality, xvii, 11, 43, 67, 76, 89 self-similar objects, 20 self-similarity, 12, 18, 31, 53, 74 sensitivity, 9, 73, 75 series, 39, 40, 41, 56, 59, 77 services, xii shape, xvii, 8, 9, 22, 29, 34, 40, 50, 52, 53, 66, 67, 68, 71 signals, 1, 31, 40, 47, 57, 86 similarity, 15, 45, 66, 77 singular, 21, 40 singularities, 21, 24, 65 sites, 10, 43, 76, 85

84

Index

SOC, 11, 12, 70, 73, 84 sounds, 39 spatial, 1, 2, 5, 9, 10, 25, 75, 76, 84, 86 spatial heterogeneity, 84 spatiotemporal, 6, 10, 74, 86 spectra, 12, 21, 22, 24, 28, 29, 31, 32, 34, 40, 41, 46, 47, 49, 50, 52, 53, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 74, 76, 80, 82 spectrum, 1, 20, 22, 23, 25, 28, 29, 31, 32, 34, 35, 46, 47, 48, 50, 52, 54, 60, 64, 65, 68, 70, 71, 72, 73, 80 speculation, 43 stability, 2, 51 stages, 5, 38, 39, 57 stainless steel, 77 stainless steels, 77 standard deviation, 28 statistical analysis, 25, 43, 67, 73 statistics, 28, 47, 51, 62, 67, 84 stochastic, 1, 40, 43, 47, 53, 54, 80, 83 stochastic processes, 47 strain, xviii, 6, 7, 8, 9, 10, 11, 16, 37, 38, 39, 43, 44, 45, 53, 54, 55, 56, 57, 59, 64, 66, 67, 68, 70, 72, 73, 74, 75, 77, 78, 80, 84, 86 strains, 75, 81, 84 strange attractor, 12, 66 strength, 19, 63 stress, 5, 7, 8, 9, 10, 11, 12, 15, 16, 17, 18, 37, 38, 39, 40, 43, 44, 45, 46, 47, 48, 49, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 70, 73, 74, 75, 76, 77, 78, 80, 83, 86 stress level, 43, 53, 54 stress-strain curves, 77, 83 stretching, 57, 59, 62, 63, 65 superconducting, 1 superposition, 1, 15, 46, 53 suppression, 27, 40 surprise, 76 symbols, 27 systematic, 28, 38, 84 systems, 1, 11, 38, 84

T technology, 89

temperature, 8, 10, 66, 77, 80, 86 temporal, 1, 2, 53, 67, 74, 84, 86 tensile, 37, 77, 79 tensile stress, 77 theoretical, 83 theory, 1, 11 thermal, 8 thermal activation, 8 thermodynamic, 2 threshold, 31, 35, 41, 47, 51, 53, 75 threshold level, 75 thresholds, 49 time, 7, 8, 12, 17, 19, 20, 25, 32, 39, 40, 41, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 56, 57, 58, 59, 60, 61, 62, 64, 66, 70, 75, 76, 77, 80, 84, 86 time series, 12, 39, 40, 45, 50, 51, 52, 53, 56, 57, 58, 60, 64, 84 topological, 16, 18 topology, 93 trajectory, 12, 39, 66 transformation, xviii, 80 transformations, 62 transition, 12, 44, 54, 67, 68, 70, 73 transitions, 61, 66, 68, 75 transmission, 5 travel, 77 trend, 26, 38, 39 trial, 53, 62 turbulence, 90 turbulent, 1, 12 twinning, xviii, 6, 77, 80 twins, 81

U UK, 90, 91, 92 uncertainty, 29, 68 uniform, 5, 32, 37, 54, 66, 71, 74, 76 universal law, 2 unmasking, 29

Index

V values, 8, 16, 18, 20, 23, 26, 28, 31, 32, 33, 34, 40, 50, 58, 61, 62, 63, 67, 68, 72, 73 variable, 29, 41, 45, 56, 58, 59 variance, 2, 10, 58 variation, 49, 53, 58, 59, 62, 64, 66, 67, 72, 76, 86 vector, 12 velocity, 10, 12, 78 visible, 60

85

visual, 10, 15, 55, 67, 73 vortices, 1

W water, 37, 71 wavelet, 25, 41 wavelet analysis, 25 wealth, xvii witness, 84

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