Projecting low and extensive dimensional chaos from spatio-temporal dynamics

We review the spatio-temporal dynamical features of the Ananthakrishna model for the Portevin-Le Chatelier effect, a kind of plastic instability observed under constant strain rate deformation conditions. We then establish a qualitative correspondence between the spatio-temporal structures that evolve continuously in the instability domain and the nature of the irregularity of the scalar stress signal. Rest of the study is on quantifying the dynamical information contained in the stress signals about the spatio-temporal dynamics of the model. We show that at low applied strain rates, there is a one-to-one correspondence with the randomly nucleated isolated bursts of mobile dislocation density and the stress drops. We then show that the model equations are spatio-temporally chaotic by demonstrating the number of positive Lyapunov exponents and Lyapunov dimension scale with the system size at low and high strain rates. Using a modified algorithm for calculating correlation dimension density, we show that the stress-strain signals at low applied strain rates corresponding to spatially uncorrelated dislocation bands exhibit features of low dimensional chaos. This is made quantitative by demonstrating that the model equations can be approximately reduced to space independent model equations for the average dislocation densities, which is known to be low-dimensionally chaotic. However, the scaling regime for the correlation dimension shrinks with increasing applied strain rate due to increasing propensity for propagation of the dislocation bands. The stress signals in the partially propagating to fully propagating bands turn to have features of extensive chaos.     

Multiscale modeling approach to acoustic emission during plastic deformation

We address the long-standing problem of the origin of acoustic emission commonly observed during plastic deformation. We propose a framework to deal with the widely separated time scales of collective dislocation dynamics and elastic degrees of freedom to explain the nature of acoustic emission observed during the Portevin-Le Chatelier effect. The Ananthakrishna model is used as it explains most generic features of the phenomenon. Our results show that while acoustic emission bursts correlated with stress drops are well separated for the type C serrations, these bursts merge to form nearly continuous acoustic signals with overriding bursts for the propagating type A bands.     

On the dynamical mechanism of cross-over from chaotic to turbulent states

The Portevin-Le Chatelier effect is one of the few examples of organization of defects. Here the spatio-temporal dynamics emerges from the cooperative behavior of the constituent defects, namely dislocations and point defects. Recent dynamical approach to the study of experimental time series reports an intriguing cross-over phenomenon from a low dimensional chaotic to an infinite dimensional scale invariant power-law regime of stress drops in experiments on CuAl single crystals and AlMg polycrystals, as a function of strain rate. We show that an extension of a dynamical model due to Ananthakrishna and coworkers for the Portevin-Le Chatelier effect reproduces this cross-over. At low and medium strain rates, the model shows chaos with the structure of the attractor resembling the reconstructed experimental attractor. At high strain rates, the model exhibits a power-law statistics for the magnitudes and durations of the stress drops as in experiments. Concomitantly, the largest Lyapunov exponent is zero. In this regime, there is a finite density of null exponents which itself follows a power law. This feature is similar to the Lyapunov spectrum of a shell model for turbulence. The marginal nature of this state is visualized through slow manifold approach.     

The influence of a static magnetic field up to 15 T on the manifestation of the Portevin-Le Chatelier effect in NaCl: Eu crystals

The influence of a static magnetic field on the instability of plastic flow (the Portevin-Le Chatelier effect) is revealed in NaCl: Eu quenched crystals. It is found that, in an external magnetic field, the yield stress of the crystals is reduced, the probability of plastic strain jumps and their amplitude decrease, and the amplitude distribution of the plastic strain jumps becomes random. The number of shear bands formed on the surface of crystals strained in the magnetic field is halved as compared to that observed without a magnetic field.     

Numerical Simulation of the Elastic Shrinkage Induced by Portevin-Le Chatelier Effect

A new one-dimensional phenomenological model based on the dynamic strain aging mechanism is developed. In order to account for the elastic shrinkage induced by the Portevin-Le Chatelier effect, elastic deformation is considered under the boundary conditions of the present model. The simulated results are found to be in good agreement with the experimental observations.     

Scaling behavior of the Portevin-Le Chatelier effect in an Al-2.5%Mg alloy

The scaling behavior of the Portevin-Le Chatelier (PLC) effect was studied by deforming an Al-2.5%Mg alloy for a wide range of strain rates. To reveal the exact scaling nature, the time series data of true stress versus time, obtained during deformation, were analyzed by two complementary methods: the finite variance scaling method and the diffusion entropy analysis. From these analyses we could establish that, in the entire span of strain rates, the PLC effect showed the Levy-walk property.     

Universality and scaling of unstable plastic flow

The statistics of the jumplike plastic deformation of a Cu–Be alloy under the conditions of a low-temperature unstable plastic flow is studied experimentally. At a high strain rate, the parameters of the load jumps are found to be related by power laws, which corresponds to a scale-invariant behavior. A comparison with the data obtained for another mechanism of plastic instability, namely, the Portevin–Le Chatelier effect, points to the existence of universal laws governing the dynamics of a dislocation ensemble in the conditions of plastic instability.     

数字剪切散斑干涉法研究铝合金中Portevin-Le Chatelier带的离面变形行为

作为Portevin-Le Chatelier(PLC)带的重要特征之一,其离面变形仍缺乏实验研究.本文提出使用数字剪切散斑干涉法研究GB6061铝合金中PLC带的离面变形.通过图像相减得到的条纹图,实时观察了PLC带的几何形貌和传播过程,并获得了PLC带离面位移分布.在1/15 s内,PLC带的最大离面位移为245 nm,位置偏向于PLC带传播的前沿.在条纹图中,PLC带传播前沿的亮条纹始终较窄.此外,实验还观察到PLC带位置变更和倾角转向的演化过程.实验表明,数字剪切散斑干涉法具有高灵敏度和防震性,是研究PLC带离面变形简便有效的方法.     

Dynamic mechanism of the temperature dependence of the Portevin-Le Châtelier effect

The influence of temperature on the dynamic and static characteristics of unstable plastic flow due to the Portevin-Le Chatelier effect is investigated experimentally and by numerical modeling. It is concluded that the distinctive features of the complex macroscopic behavior of the plastic instability are determined by collective processes associated with the long-range interaction of dislocations and lead to dynamic effects of mesocopic scale. Fiz. Tverd. Tela (St. Petersburg) 40, 487–492 (March 1998)     

固溶处理对Al-Cu合金中Portevin-Le Chatelier效应空域行为的影响

研究了在定加载速率拉伸条件下，固溶处理对Al-Cu合金材料中PLC(Portevin-Le Chatelier)效应空域行为的影响.在较低拉伸速率时，变形初期PLC变形带在试件宽度方向中央附近某点处“成核”.随着变形的继续，出现多带共存的现象.且带宽较小，带与拉伸轴向的夹角较大.在较大拉伸速率条件下，PLC变形带先连续传播，再随机出现. 关键词： PLC效应 动态应变时效 固溶处理     

Plastic deformation instability in copper alloys

A systematic investigation was made of the temperature and velocity conditions for the appearance of stepwise deformation in copper alloys with a high alloying content (the Portevin-Le Chatelier effect) and the deformation mechanics of crystalline materials was analyzed from the viewpoint of an emsemble of dislocations. A mathematical model of the Portevin-Le Chatelier effect was constructed, based on general ideas concerning the behavior of dislocations and their interaction with alloying elements, and from the viewpoint of the theory of oscillations, taking into account the rigidity of the test-piece-machine system. Mechanical tests were performed on copperalloy test pieces in the temperature range 20–400°C. A characteristic feature found for the oscillation modes was that the peak-to-peak value Δσ was independent of ɛ and that there was a plateau with a weak dependence σ(ɛ) at the upper σ level. The dependences of the oscillation period on the temperature and the given plastic deformation velocity were in good agreement with experiment. State Scientific-Research and Design Institute of Alloys and Working of Nonferrous Metals. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 14–20, February, 1993.     

Current theoretical approaches to collective behavior of dislocations

Plastic deformation is a highly dissipative process that induces a variety of patterns such as the cell structure in multislip conditions, the matrix structure and the persistent slip bands in cyclic deformation, as also the static and propagating bands in constant strain rate conditions. The diversity and the complexity of these patterns with length scales ranging from nanometers all the way to millimeters level, and time scales ranging from picoseconds to a few hours, pose serious challenges for modeling the collective behavior of dislocations. While a large body of knowledge has accumulated on the mechanics of dislocations and their interactions for a long time, describing such patterns has been slow mainly due to lack of methods to deal with the collective behavior of dislocations. The purpose of this review is to present the rich variety of dislocation patterns observed in different deformation conditions along with the recent advances in modeling using borrowed techniques traditionally used in condensed matter physics. These can be classified as statistical and dynamical approaches. The review begins with a summary of different types of patterns and their characterization. Appropriate background material is provided both in terms of basic dislocation mechanisms and theoretical methods. The latter includes the Langevin and distribution function approaches, and a host of standard dynamical methods such as the Ginzburg–Landau approach, methods of characterization of chaos and slow manifold analysis. Statistical models for the cell structure and persistent slip bands are based on Langevin dynamics and distribution function theoretic approaches. Of the dynamical models, the first set addresses the slowly emerging matrix structure and persistent slip bands. The second set of models is devoted to the study of a type of propagative instability called the Portevin–Le Chatelier effect. Generic features of the instability addressed include bistability, negative strain rate sensitivity of the flow stress, different types of bands, the dynamics and statistics of stress drops, and their characterization. Three different models all of which are dynamical in nature are discussed. While these models are quite different with regard to their frameworks, what they seek to describe and the levels of sophistication undertaken, these models capture a good variety of the observed features. The review ends with a summary and outlook.     

Critical dynamics of burst instabilities in the Portevin-Le Ch atelier effect

We investigate the Portevin-Le Chatelier effect (PLC), by compressing Al-Mg alloys in a very large deformation range, and interpret the results from the viewpoint of phase transitions and critical phenomena. The system undergoes two dynamical phase transitions between intermittent (or "jerky") and "laminar" plastic dynamic phases. Near these two dynamic critical points, the order parameter 1/tau of the PLC effect exhibits large fluctuations, and "critical slowing down" (i.e., the number tau of bursts, or plastic instabilities, per unit time slows down considerably).     

Markov property of continuous dislocation band propagation

Tensile tests were carried out by deforming polycrystalline samples of substitutional Al–2.5%Mg alloy at room temperature for a range of strain rates. The Portevin–Le Chatelier (PLC) effect was observed throughout the strain rate regime. The deformation bands in this region are found to be of type A in nature. From the analysis of the experimental stress time series data we could infer that the dynamics of type A dislocation band propagation is a Markov process.     

Portevin-Le Chatelier effect triggered by complex loading paths in an Al–Cu aluminium alloy

Serrated flow has been observed during non monotonic tensile tests of an Al–Cu aluminium alloy in the naturally aged state. The associated propagative localisation bands were observed by digital image correlation (DIC). In particular, the Portevin-Le Chatelier (PLC) effect and also Lüders bands were observed in interrupted tests during which the specimen was held for a length of time and also in tests with partial unloading followed by a holding time. Increasing strain rate jumps also triggered the PLC effect. These observations indicate the existence of the PLC effect in this material which was formerly considered insensitive to it at room temperature under monotonic loading conditions. There is no evidence of PLC serrations during constant strain rate tests. A strain ageing finite element model is used that captures the experimentally found PLC triggering effects.     

On the foundations of stochastic dislocation dynamics

A stochastic approach to dislocation dynamics is proposed that starts off from considering the geometrically necessary fluctuations of the local stress and strain rate caused by long-range dislocation interactions during plastic flow. On a mesoscopic scale, a crystal undergoing plastic deformation is thus considered an effective fluctuating medium. The auto- and cross-correlation functions of the effective stress and the plastic strain rate are derived. The influences of dislocation multiplication, storage and cross slip on the correlation functions are discussed. Various analogies and fundamental differences to the statistical mechanics of thermodynamic equilibrium are outlined. Application of the theory of noise-induced transitions to dislocation dynamics gives new insight into the physical origin of the spontaneous formation of dislocation structures during plastic deformation. The results demonstrate the importance of the strain-rate sensitivity in dislocation patterning.     

Plastic deformation macrolocalization during serrated creep of an aluminum-magnesium Al-6 wt % Mg alloy

The nonlinear dynamics of the space-time structure of macrolocalized deformation is studied by a set of high-speed in situ methods under the conditions of serrated creep in an aluminum-magnesium Al-6 wt % Mg alloy at room temperature. Macroscopic deformation jumps with an amplitude of several percent are detected in the creep curve of this alloy. It is found that a complex space-time structure of macrolocalized deformation bands moving in a correlated manner forms spontaneously in the material during the development of a deformation jump. The difference between the observed picture of deformation bands and the well-known Portevin-Le Chatelier classification of deformation bands is discussed.     

数字散斑法在局域剪切带三维变形研究中的应用

在适当的温度、应变率和预变形下,合金材料的拉伸试验中,将会出现伴随应力锯齿形跌落的雪崩式剪切变形带,即波特文-勒夏特利埃(Portevin-Le Chatelier,PLC)效应。利用高速数字摄像系统(分辨力1000 frames/s)并结合数字散斑干涉法(Digital speckle pattern interferometry,DSPI)和数字散斑相关法设计了一套光学变形测量系统,实现了拉伸试验中对试件表面三维变形的实时、精确测量。利用该光学系统对铝铜合金试件在拉伸试验中产生的跳跃传播的局域剪切带瞬态成核过程进行捕捉。通过结合数字散斑相关法得到的面内变形定量结果和数字散斑干涉法得到的表现离面变形的条纹图,再现了剪切变形带成核和传播瞬间的三维变形过程。     

红外测温法研究Al-Mg合金中的Portevin-Le Chatelier效应

试件塑性变形过程伴随着机械能向热能的转化.利用红外测温法,通过分析红外热像仪采集的温度场图像,系统研究了Al-Mg合金中的Portevin-Le Chatelier （PLC）效应.在不同应变率下,实验得到了三类锯齿形应力-应变曲线,分析了相应情况下试件温度变化曲线的异同及其原因,探讨了三种类型PLC变形带的空间传播特性.研究发现,试件表面的温升随着应变率的增加而增加;PLC带的倾角转向发生在试件的两端或者带外的温度最高处. 关键词： Portevin-Le Chatelier效应 红外测温 Al-Mg合金     

Plastic flow instability: Chernov–Lüders bands and the Portevin—Le Chatelier effect

We have studied the regularities in the evolution of macroscopic nonuniformities in the plastic flow of metals in the form of the Chernov–Lüders bands and the Portevin—Le Chatelier effect. The regularities in the evolution of strain inhomogeneity in these two cases have been established, and the kinetics of motion of the Chernov–Lüders fronts and abrupt deformation in the Portevin–Le Chatelier effect has been analyzed. It has been shown that the Chernov–Lüders fronts and Portevin–Le Chatelier jumpwise straining can be treated as macroscopic autowave processes of switching and excitation, respectively, in deformed media of various origins.