ELSEVIER Materials Science and Engineering A2344236 (1997) 115- 118

Kinetics and statistics of jerky flow: experiments and computer simulations

M. Lebyodkin a, L. Dunin-Barkovskii a, Y. Br&het b, L. Kubin c, Y. Estrin d,* a Institute of Solid State Physirs, Russian Academy of Sciences, 142432 Chernogolovka, Russia

b LTPCM, Domaine Unioersitaire de Grenoble, B.P. 75, 38402 Saint-Martin d’Heres Cedex, France c LEM, CNRSIONERA (Oh4), B.P. 72, 92322 Chdtilion Cedeex, France

d Department of Mechanical and Materials Engineering, Unioersity of Western Australia, Nedlands, WA 6907, Australia

Received 14 January 1997; received in revised form 26 March 1997

Abstract

The paper presents an overview of experimental and theoretical studies of the Portevin-Le Chatelier effect with regard to the statistics of stress discontinuities and spatio-temporal behaviour of strain localisations. It is shown that the statistics of stress drops changes progressively from peak-shaped to monotonic distributions when the strain rate or temperature are increased. A mesoscopic model accounting for strain inhomogeneity and spatial coupling is proposed to explain the statistical behaviour observed. The model reproduces the character of deformation curves and the corresponding deformation band kinetics. It also predicts the characteristics of the bands for both constant strain rate and constant stress rate experiments. 0 1997 Elsevier Science S.A.

Keywords: Deformation patterns; Plastic instability; Portevin-Le Chatelier effect: Statistics

1. Introduction

In recent years, the Portevin-Le Chatelier (PLC) effect has received renewed attention (see [l] for re- view). The spatial features [2] and the temporal aspects of the effect [3] have been investigated using physically based deterministic constitutive equations [4], the em- phasis being on the statistical behaviour of an ensemble of elements of a deforming sample [5]. The microscopi- cally based equations have allowed the calculation of band velocities [3] and the prediction of deterministic chaos [6,7], which has been observed experimentally [6]. The modelling approach we present here reproduces the richness of the phenomenon with regard to the appear- ance of discontinuous yielding [8], the spatial character- istics of PLC bands [9] and the stress drop statistics [9]. As will be shown below, the statistics of stress drops in the PLC effect have similarities with the earthquake statistics [lO,l 11, but also show important differences. Another aspect of the PLC phenomenon is the prop-

* Corresponding author. Tel.: + 61 9 3803115; fax: + 61 9 3801024; e-mail: estrin@mech.uwa.edu.au

0921-5093/97/$17.00 0 1997 Elsevier Science S.A. All rights reserved. PIISO921-5093(97)00179-2

agative character of deformation bands. Measurements of the band velocity lead to contradictions as to its dependence on the imposed stress rate [12-141. This paper reviews our experimental studies of the statistics of serrated yielding and of mesoscale computer simula- tions of the PLC effect. It also addresses the systematics of deformation curves and the spatial features (includ- ing the velocity) of deformation bands.

2. Materials and methods

Experiments were performed on Al-Mg single and polycrystals. Here we shall focus on the results for single crystals only. Specimens of an Al-4.5at.%Mg alloy were deformed in tension with a constant imposed strain rate in the range from 3.2 x lop6 to 1.3 x lo-” S ~ ’ at temperatures of 300, 400 and 450 K. In order to eliminate a systematic dependence of the stress drop magnitude, Aa, on strain E due to work hardening, an appropriate normalisation [9] was used. Distributions of normalised, non-dimensional stress drops, s, were then analysed.

116 M. Lebyodkin et al. /Materials Science and Engineering A234-236 (1997) 115-I 18

1

A co

x

0 ! 0

s I \

I \ (4

/ \

d 1p I \

d/ \

9 \

s, .

. -0

I I I 1

1 2 S

WI

- 1 z 0.1

w \ 0‘0,

0 -Jo 0 2 ,oo

Fig. 1. Examples of distribution density function D(S) of normalised stress drop magnitude s for a specimen deformed at 300 K: (a) $=1.3x 10W5 SC’; and(b) ka=2.6x 10m4 s-l.

3. Statistics of the PLC effect

Serrations on deformation curves were relatively ir- regular in the initial stage for all choices of testing conditions. After a transient stage, the statistics of normalised stress drops stabilised and then remained invariant while straining proceeded. The influence of the straining conditions on the statistics was studied for these steady-state distributions. Within the range inves- tigated, distributions change from peak-shaped to monotonically decreasing histograms as the strain rate or temperature increase. This is illustrated by Fig. 1 which presents examples of a bell-shaped and a monotonic histogram for two values of &. The latter can be described by a power law, albeit with a non-uni- versal exponent. A transient distribution may be monotonically decreasing even when the steady-state distribution is bell-shaped. For polycrystals, bell-shaped distributions exist in a wider range of straining condi- tions than for single crystals.

4. Computer modelling

The constitutive equation relates the stress g and the plastic strain rate i: through an N-shaped strain-rate sensitivity function F(i) (Fig. 2):

a=h&+F(i) (1)

where h is the strain hardening coefficient. Supple- mented with a machine equation & = 6/M + & (constant strain-rate experiments) or 0 = d-t (constant stress-rate experiments), it yields a solution in the form of periodic jumps (relaxation oscillations) of the plastic strain rate [4]. In order to allow for different strains in different material elements, a specimen was considered as a one-dimensional chain of IZ ‘slices’ perpendicular to the axial direction coupled by springs of strength K. The

coupling constant K accounts for the incompatibility stresses between the neighbouring slices with different strains, which tend to homogenise the strain within the specimen. The local stress-strain rate behaviour in a slice i is then given by:

0 = h&i + F(ii) + K[(Ei- cl- 1) + (EL- &I+ I)] (2)

As the velocity of sound is much larger than that of propagating deformation bands, the stress is assumed constant along the sample. The plastic strain rate in the machine equation is now replaced by the average: i=ci iJn.

The dependence of the F(i) curve on the imposed strain rate was included in our model: as shown in Ref. [4], an increase in & leads to a decrease in the height and in the width of the maximum of the F(8) curve. Temperature influences the simulated statistics in the same way & does. For simplicity, the F(8) curve was taken as a piece-wise linear function independent of E [9]. Other parameters were chosen to represent a typical material. The combined machine-specimen elastic mod- ulus was taken to be A4 = lo5 MPa for a hard machine (constant & case) and zero or 500 MPa to mimic an

Fig. 2. Strain rate sensitivity of the flow stress for two values of $.

M. Lebyodkin et al. /Materials Science and Engineering A234-236 (1997) 115-I 18 117

10

10 -zI 10 -' ' s 10

Fig. 3. Example of simulated distribution density function D(s) for K =3;2,% iad: ;‘; =3-,:-x 10:~ 4 =.- ‘. s$.3wingz s p3tw;a,3>; !m~mv;miT

id& or a ‘real’ so31 mackme 3 consian‘r b>. The s%Gn- hardening coefficient was varied from zero to 2 x lo3 MPa, the number of slices n from 25 to 600, the coupling constant K from 6 x lo3 to 10’ MPa. The amplitude of initial randomly chosen fluctuations in the strain-rate values $ was taken to be O.OS&,,. In the constant & case, the system was then allowed to evolve naturally, without any additional randomness. In the constant 8 case, the rate of evolution of the system state (0, i) towards the maximum of the F(i) curve is much smaller, so that fluctuations decay effectively. Here, randomisa1ion was repealed at each ca\cu\ation St&$3 w,ti(, t+Lc s~w&A T-z& of ctlnc 0-l cht -tYkcks jwq!@d frv the right branch of the F(i) curve.

5. Results of computer simulations

5.1. Constant strain rate

Three types of serrations were obtained in the simu- lations &pen&ng on the imposed strain rate7 w&n are strength of coupling was chosen in the range from 0.5M to M. Stress drop distributions for type C serrations have a peaked shape. An increase in the strain rate results in a monotonic distribution. Fig. 3 presents a power-law distribution function of normalised stress drops for type A serrations (& = 3 x 10 p4 s- ‘). How- ever, this power-law behaviour is not universal. Strong coupling tends to produce a Gaussian distribution (6- function in the limit case of relaxation oscillations), while high strain rates destroy the scaling. The experi- mentally observed behaviour is reproduced when K is off the order 0E the eastic mudu~u~. whik% cunrfims a pcos1etioti DUT inlegxela4r3n ~5 coxfi& in t-3 6 incompatibility stresses. The simulated spatial patterns off plastic waves are in good a.reement with the cx-ueri- mental observations [15]. With increasing &, a transi-

tion is observed from non-propagating deformation bands occurring at random at each stress jump (type C) to a ‘relay-race’ proliferation of plastic activity along the specimen (type B) and, finally, to true band propa- gation (type A serrations). Decreasing K suppresses the correlation between blocks and, therefore, band propa- gation. However, for K smaller than O.l5M, the stress jump distributions exhibit a power law also in the & range where deformation bands no longer propagate. The predicted behaviour in this regime is analogous to that of ‘earthquakes’ [lO,l 11: deformation bands (avalanches) of all possible sizes occur at random sites. The dependence of V and of the deformation band width on & shows qualitative agreement with experi- mental data [15]. Examples of the V versus &, depen- dence for three K-values are presented in Fig. 4a.

At high stress rates (s/h corresponding to the & range of real band propagation), a similar picture of propagating bands was obtained. The behaviour of static bands is different, however. Since C? is kept con- stant, no unloading stops the band from spreading over the sample once it has been nucleated. It results in propagation of a deformation front in the specimen. Earlier experimental work reporting the band velocity measurements did not distinguish between these two &ferent regimes but rather related all measurements to the band propagation phenomenon. However, a quali- tatively different V(5) dependence can be expected for the front propagation: the smaller the value of 8, the smaller is the difference between strain rates in neigh- bouring material elements and, therefore, the higher is the front propagation velocity. The simulation results (Fig. 4b) confirm this qualitative consideration.

6. Conclusions

The experimental studies show that, under certain deformation conditions, the statistics of the PLC effect are of the power-law type. This is in accord with the results of a mesoscopic model which combines a local constitutive law reflecting the temporal aspects of the PLC effect with spatial coupling. We found that the model also describes the character of the deformation curves and of the spatio-temporal behaviour of strain localisations as a function of the deformation condi- tkms. -lt was shown that unaer constant stress rate, a %7&i&n f-r~m a &%nrna&n f-rQn\ pxQpaga+&n t% a localised band propagation may result in the occur- rence of a i-i-i1nunum in the V&k?dlty versus stress rate dependence.

118 M. Lebyodkin et al. /Materials Science and Engineering A2344236 (1997) 115-118

I 1 t

lo-' ' 1 b/h, s-’

-3

Fig. 4. Simulated band propagation velocity: (a) constant strain rate conditions, h = 10” MPa, K = 105, 5 x IO4 and 2.7 x lo4 MPa (curves 1, 2 and 3, respectively); (b) constant stress rate case, h = 10’ MPa, K = lo5 MPa.

Acknowledgements

This work has been possible due to a post-doctoral fellowship (M.L.) from the Minis&e des Affaires Etrangkres and to a research grant from the Russian Fundamental Science Foundation. One of the authors (Y.E.) acknowledges funding from the Australian Re- search Council.

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