- Email: [email protected]
On the mechanism of unstable plastic flow in an austenitic FeMnC TWIP steel
Materials Science and Engineering A 519 (2009) 147–154
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
On the mechanism of unstable plastic flow in an austenitic FeMnC TWIP steel T.A. Lebedkina a , M.A. Lebyodkin b,∗ , J.-Ph. Chateau c , A. Jacques c , S. Allain d a
Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Russia LPMM, Université Paul Verlaine – Metz/CNRS, Ile du Saulcy, 57045 Metz Cedex 01, France c LPM, Ecole des Mines/CNRS, Parc de Saurupt, 54042 Nancy Cedex, France d Arcelormittal Maizières Research SA, Voie Romaine, BP 30320, 57283 Maizières les Metz, France b
a r t i c l e
i n f o
Article history: Received 19 December 2008 Received in revised form 26 April 2009 Accepted 28 April 2009 Keywords: High manganese austenitic steel Jerky flow Deformation inhomogeneities TWIP Dynamic strain aging
a b s t r a c t The complex character of plastic deformation of the austenitic steel Fe22Mn0.6C is studied at room temperature with the aid of high-frequency local extensometry. It is shown that the plastic flow instability, associated with fluctuations of the flow stress, results from quasi-continuous propagation of deformation bands along the specimen axis. This propagation mode is dominant in the entire range of the applied strain-rate from 2.1 × 10−5 s−1 to 10−1 s−1 . Such behavior differs from that of various alloys deforming via dislocation glide under conditions of dynamic strain ageing (Portevin-Le Chatelier effect), which is characterized by a transition from a repetitive occurrence of static deformation bands at lower strain rates to a relay-race and, finally, quasi-continuous deformation band propagation at higher strain rates. The unusual behavior of the deformation bands bears evidence to a particular kind of instability in the investigated steel. A possible role of deformation twins in the observed dynamics of plastic instability is discussed. © 2009 Elsevier B.V. All rights reserved.
1. Introduction The plastic deformation of steels with high manganese content, such as Fe22Mn0.6C (wt.%), exhibits macroscopic plastic instabilities in a wide range of strain-rate and temperature and is characterized by stress fluctuations of various shapes on the stress–strain curve. The resulting deformation curves present different morphologies depending on temperature. Such variance is consistent with the temperature dependence of the microscopic mechanisms of plastic flow of this material, which depend on stacking fault energy (SFE) [1–3]. Indeed, when the deformation temperature is decreased from 673 K down to 77 K, dislocation glide, twinning plus glide, and martensitic transformation plus glide (-martensite with h.c.p. lattice crystallographic structure) are successively observed in accordance with the diminishing SFE [4]. Therefore, it is natural to suggest that the instabilities observed under various deformation conditions can be governed by different microscopic mechanisms. These mechanisms are not known yet, perhaps, with an exception for a temperature interval above 150–200 ◦ C, where the dynamical strain aging of dislocations due to carbon atoms may lead to the Portevin-Le Chatelier effect [5,6]. However, this problem is of interest for both practical applications, since the instabilities impede the processing of these steels, which
∗ Corresponding author. Tel.: +33 03 87315401. E-mail addresses: [email protected], [email protected] (M.A. Lebyodkin). 0921-5093/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2009.04.067
otherwise possess remarkable mechanical features [7,8], and for understanding the basic problem of the deformation mechanisms leading to instability of the uniform plastic flow. At room temperature, the studied FeMnC steel is fully austenitic, with a sufficiently low SFE to promote twinning, which enters into the competition with dislocation glide, as reflected in the notion of twinning-induced plasticity (TWIP) [7]. As a matter of fact, this peculiar feature is at the origin of this steel attraction for industry because it provides a high strain hardening rate leading simultaneously to high ductility and tensile strength. It is known that the formation of twins may be accompanied with pronounced stress fluctuations, although it usually concerns low-temperature deformation [9]. However, the instabilities observed in this steel at room temperature are often tentatively ascribed to the PLC effect [10]. This assumption is based on the observation of the negative strain-rate sensitivity (SRS) of the flow stress and a thermally activated character of plastic deformation. Nevertheless, the diffusion coefficient of carbon in austenite appears to be too low at room temperature to account for the PLC effect [11]. On the other hand, it is known that the negative SRS is not a peculiar feature of the PLC effect only. It also emerges for distinct mechanisms, e.g. for thermo-mechanical instability, which is known to be caused by local overheating of specimens deformed at low temperatures [12]. Moreover, the thermo-mechanical instability demonstrates features in common with the PLC effect, as far as the morphology of the deformation curves and the statistics of stress jumps are concerned [13]. This example proves that neither the behavior of the
148
T.A. Lebedkina et al. / Materials Science and Engineering A 519 (2009) 147–154
SRS nor the similarity of the morphology of deformation curves is a sufficient criterion for distinguishing between various instability mechanisms. In order to provide a further insight into the mechanism of plastic instability in austenitic TWIP steels at room temperature, we study the local strain distribution during tensile deformation of FeMnC samples using high-rate extensometry technique [14]. This paper is organized as follows. Section 2 outlines experimental details and data processing. Section 3 describes experimental data on the macroscopic tensile behavior of the studied steel and the concomitant evolution of the local strain heterogeneity. The overall discussion, including the comparison of the evolution of the strain distribution to salient features of PLC bands, is given in Section 4. Finally, Section 5 highlights hypotheses on which mechanisms are able to lead to the observed plastic instabilities. 2. Experimental procedure Flat tensile specimens with a gauge section 60 mm × 12.6 mm × 1.25 mm in size were prepared from a Fe–22 wt.% Mn–0.6 wt.% C steel, with approximately 3 m grain size. Mechanical tests were performed at room temperature with a constant crosshead speed. The nominal value of the imposed strain-rate ε˙ a , corresponding to the initial specimen length, was varied in the range from 2.08 × 10−5 s−1 to 2.5 × 10−2 s−1 . Besides, several tests were performed on samples fabricated separately but with the same nominal composition and grain structure. These specimens with dimensions 80 mm × 5.4 mm × 0.6 mm were tested in a strain-rate range from 2.2 × 10−5 s−1 to 8 × 10−3 s−1 [2]. Using such specimens, one test was performed with the aid of a pyrometry technique to visualize strain localization at the strain-rate ε˙ a = 10−1 s−1 [11]. In order to conduct local extensometry, one surface of the specimen was painted black, and somewhat 10 white marks of 1 mm in width were superimposed on the black layer normal to the longitudinal axis as illustrated in Fig. 1. The distance between the marks was approximately equal to 1 mm. The resulting sequence of black and white stripes formed a set of about twenty 1 mm long extensometers, in the center of the 60 mm gauge length. The positions of the transitions between black and white marks during the sample deformation were followed using a high resolution 1D CCD camera with a recording frequency of 103 Hz and a pixel size of 1.3 m. The camera was fixed relative to the fixed crosshead. It aimed at the specimen axis in order to avoid any cross-section rotation effect, although negligible in the case of polycrystals. As a matter of fact, the deformation of the samples appeared to be uniform on average in all tests, so that no noticeable rotation effect was observed before ultimate necking. The camera allowed continuous recording during an interval of 600 s, which covered the entire test duration at higher strain rates of ε˙ a > 1.5 × 10−3 s−1 . Consequently, the strain dependence of various parameters of the strain pattern could be built. For lower ε˙ a values, it was possible to survey only portions of tests. In this case, the measurements were repeated several times during the test in order to evaluate the local deformation kinetics qualitatively. To estimate the continuous work hardening rate, = ˙ ∂(ε, ε)/∂ε, the deformation curves obtained at different imposed strain rates were fitted with polynomials in the strain intervals from 0.005 ≤εlog ≤ 0.5–0.6, to eliminate stress fluctuations. The estimation of evolution with strain was then obtained by deriving the fitting function. The evolution of the local strain values were calculated from the displacements measured by the CCD camera: εi (t) = ln
xi+2 (t) − xi (t) , xi+2 (0) − xi (0)
(1)
where xi is the coordinate of the ith transition. In this equation, every second section is taken in order to avoid a possible arte-
Fig. 1. Scheme of the experimental setup. The black and white stripes depict the set of extensometers painted on its surface. A deformation band (B) propagates from the top to the bottom of the specimen.
fact due to unequal sensitivity of the camera to black–white and white–black transitions. During performing the mechanical test, some extensometers leave the camera’s field of vision, some others will enter into it. For this reason, the initial time in Eq. (1) was usually not selected at the beginning of deformation, but at a later instant, in order to seize the largest number of extensometers. In this case, only the relative changes in the local strain values were meaningful. In addition to εi (t) local strain rates ε˙ i (t) were calculated by deriving the εi (t) dependence after preliminary reduction of digital noise with the aid of a running average technique. The construction of strain maps, (t, xi (0), εi (t)), and the respective strainrate maps, (t, xi (0), ε˙ i (t)), allowed visualization of the evolution of the local strain field and, particularly, provided a simple tool to distinguish between the band propagation and localization. These maps, together with the εi (t) dependence, provide evaluation of the band velocity, Vb , band width, w, and the associated strain, εb . As the maps refer to the gauge length of the local extensometers at the initial time, the values of Vb and w are corrected taking the sample elongation into account. Other kinds of data representations, though less illustrative, were also used to evaluate the deformation band parameters. For example, the axial strain distribution profiles εi (xi ) were calculated at subsequent time instants. All methods provided close values for the deformation band parameters. 3. Results 3.1. Macroscopic behavior Fig. 2 shows some representative true stress–strain tensile curves recorded at different imposed strain rates. A negative SRS occurs in the entire strain-rate range investigated as described in Ref. [11] for strain rates up to 1 s−1 . A rough evaluation of the SRS using a relationship S ≈ / ln(ε˙ 2a /ε˙ 1a ) for pairs of deformation
T.A. Lebedkina et al. / Materials Science and Engineering A 519 (2009) 147–154
Fig. 2. True stress–logarithmic strain curves recorded with various applied strain rate: from upper to lower curves, 2.08 × 10−5 s−1 , 5.56 × 10−4 s−1 , 2.22 × 10−3 s−1 , 7.78 × 10−3 s−1 , and 2.5 × 10−2 s−1 . For the sake of clarity, some of ε˙ a values are marked by arrows which indicate the respective values of the critical strain for the onset of plastic instability. Inset: dependence of the critical strain on the imposed strain rate.
curves recorded at different strain rates, i.e., assuming that strain hardening is not considerably dependent on ε˙ a , leads to values that increase monotonously from a few MPa at the beginning of plastic deformation to about 30 MPa at εlog ≈ 0.5. This range of SRS does not essentially differ from the literature data for C–Mn steels under PLC conditions (e.g. [5,6]). The strain-rate dependence of the critical strain for the onset of visible instability appears to have a minimum at intermediate strain rate as illustrated by the inset of Fig. 2. Such a non-monotonic behavior of the critical strain is often observed for the PLC instability [15,16]. In contrast to these macroscopic characteristics, which show some similarity with the PLC effect, the morphology of the serrated stress–strain curves observed presents specific features. The PLC effect shows remarkable dependence on ε˙ a as will be discussed in Section 4. On the contrary, the persistence of the shape of the true stress–strain curves of Fig. 2 attracts attention. This observation leads to suggestion that the same dynamical mode of plastic flow may be active in the entire ε˙ a range. Moreover, the shape of the curves allows for certain assumptions by analogy with the known kinds of unstable plastic flow, in particular, with the highrate regimes of the PLC behavior and thermo-mechanical instability, which are characterized by deformation band propagation [17,18]. Namely, typical sequences are repetitively observed, which consist of a stress rise followed by a steep stress drop and a subsequent increase of the stress at a rate close to the average work hardening rate. This is usually considered as a signature of a recurrent nucleation and propagation of deformation bands: an increase in the applied stress is necessary to nucleate a band which propagates at a lower stress level. The other peculiar feature appears in the behavior of the work hardening rate curves of Fig. 3. First, the deformation curves look roughly linear on the scale of Fig. 2, i.e., the work hardening seems to correspond to a unique stage all over the test. Such work hardening differs from the behavior of typical dynamically strain ageing polycrystalline alloys, which usually exhibit parabolic work hardening in the strain range corresponding to the unstable plastic flow. Quantitative evaluation reveals some variation in . Except for the highest rates, it is found that the initial decrease in is followed by an increase in some strain interval starting at 0.15 ≤ εlog ≤ 0.2 (Fig. 3). The increase in diminishes with increasing ε˙ a and is not observed at a strain rate of ε˙ a = 2.5 × 10−2 s−1 .
149
Fig. 3. Work hardening rate as a function of strain for different strain rates as indicated in Fig. 1.
3.2. Local strain patterns and deformation band characterization Fig. 4 represents an example of synchronized sections of a stress–time curve recorded at ε˙ a = 2.22 × 10−3 s−1 with the respective evolutions of the recorded positions and local strains as described by Eq. (1). Fig. 4a and b shows that stress drops occur synchronously with an acceleration of all the displacements as marked by vertical dotted lines. This indicates that the specimen is deformed between its fixed head and the upper marker as shown in Fig. 1. The displacements subsequently slow down one after another as shown by the deviation from the vertical dashed lines illustrated by the inclined dotted line at t ≈ 160 s. This results in local strain jumps of substantially the same height as illustrated in Fig. 4c. The curves of Fig. 4c are thus made of short intervals of high-rate plastic flow indicated by steps and extended intervals of slower deformation. The delay between the initial acceleration of the displacements and the strain jumps suggests that a localized deformation band moves from the upper part of the specimen through the set of extensometers. This is confirmed by Fig. 5 which presents enlargements
Fig. 4. Representative experimental records as a function of testing time at strain rate of ε˙ a = 2.22 × 10−3 s−1 : force applied to the tensile specimen (a), positions of neighboring sections of the sample (b), and local strain values (c) calculated using the data of (b). Vertical dotted lines indicate instants when a deformation band is nucleated. Dashed lines are approximately marking the instants when the band leaves the first of these sections. The inclined dotted line at t ≈ 160 s marks the edge of a plateau, where the deformation band leaves the respective sections.
150
T.A. Lebedkina et al. / Materials Science and Engineering A 519 (2009) 147–154
of the curves of Fig. 4c and clearly shows that the local strain jumps recorded for various i are shifted in time with regard to each other. Such progressions prove that deformation is mainly achieved by the repetitive propagation of deformation bands along the tensile axis as drawn schematically in Fig. 1. Further observations follow from Figs. 4 and 5. The comparison of Fig. 5a and b shows that the magnitude of the time shift between the local strain jumps in the adjacent sections of the specimen increases with deformation. As shown below, this is due to both the specimen elongation and a decrease in the true band velocity, see Fig. 8, determined for the actual specimen gauge length. At the same time, the slope of the steps increases with deformation, thus indicating an increase in the local strain rate in the deformation band. The average slope of each curve of Fig. 4c is determined by the imposed strain rate. It is worth noting that the step-plateau shape of the εi (t)-curves testifies that the nucleation of the next deformation band takes place practically immediately after the previous one has completed its propagation. Otherwise, periods of homogeneous deformation at the imposed strain rate should also be observed. The εi (t)-curves and the strain-rate maps constructed for various strain rates ε˙ a disclose a combination of different kinds of strain inhomogeneity, including deformation band propagation and in some cases localization. The main features of the deformation bands’ behavior are illustrated in Fig. 6. Importantly, the band propagation regime persists in the entire ε˙ a range investi-
Fig. 5. Evolution of the local strain εloc for four juxtaposed 2 mm extensometers at (a) an early and (b) a later stage of deformation with ε˙ a = 2.22 × 10−3 s−1 . The frames indicating indexes of the respective sections reveal a consecutive passage of a band along the sample.
gated, in agreement with the morphology of deformation curves. Indeed, it can be recognized that the band propagation is dominant at higher strain rates of ε˙ a ≥ 2 × 10−3 s−1 as illustrated in Fig. 6(a)–(d). This conclusion was also confirmed by the visualization of the local straining by the pyrometry technique [11]. A more complicated behavior was observed at an intermediate strain rate of ε˙ a = 5.56 × 10−4 s−1 , see Fig. 6(e). In this case, the observed patterns included band propagation over shorter distances, static bands, and coexistence of two or more bands. However, the propagation remains an essential mode in these conditions too. Moreover, propagation is found to be dominant also at ε˙ a = 1.0 × 10−4 s−1 , see Fig. 6(f). Similar results were observed in the control tests on differently sized specimens. The comparison of the examples of Fig. 6 shows that the deformation band velocity diminishes when ε˙ a is decreased. Noting the limited interval of recording with the CCD camera, this makes difficult the clear detection of propagation at the lowest strain rate, ε˙ a = 2.08 × 10−5 s−1 . A full propagation cycle has not been seized. Nevertheless, the sections recorded at this strain rate testify that the respective deformation bands develop during hundreds of seconds. This observation confirms the persistence of the propagation mode down to very low strain rates [19]. As will be discussed in Section 4, such behavior is remarkably different from that of the dynamics of the PLC effect. Although velocity, width, and strain of a band may fluctuate, cf. Figs. 4 and 5, the records of the evolution of the local strain field allow quantitative estimates. The band velocity increases approximately linearly with ε˙ a as illustrated in Fig. 7. This linear trend is roughly consistent with the power–law dependences of the deformation band velocity on the strain rate, which are usually observed for the so-called type A PLC bands [20,21]. However, the exponents reported in these works take on lower values of n ≈ 0.8. Furthermore, such propagating PLC bands only occur in the range of the highest strain rates within the entire strain-rate range corresponding to the PLC instability. Last, the velocity of the deformation bands in the investigated steel reaches rather low magnitudes at low strain rates (tenths of mm/s), as compared to literature data for type A PLC bands, usually above several mm/s. The strain dependences of the deformation band parameters appear to behave similarly to those for the type A PLC bands. Fig. 8 presents examples of such behaviors for ε˙ a = 2.22 × 10−3 s−1 . The strain in the band increases with the overall strain, whereas the band velocity decreases, so that their product, εb Vb , is roughly constant during the test and close to the crosshead speed V corrected to take elastic straining into account. Therefore, it can be concluded that at a given time, the plastic strain is mostly concentrated in a deformation band, whereas the estimates of the strain outside the band typically give values about ten per cent of the whole strain. It should be noted that the value of εb Vb obtained at different strains varies noticeably. This variation testifies to a strong inhomogeneity of strain: evidently, the deviations are found when a considerable straining takes place outside the band passing through the extensometer set, e.g., when multiple deformation bands occur. The conclusion on the pronounced inhomogeneity of plastic flow is also confirmed by the estimations of the bands width. The values obtained typically vary in the range from several to ten millimetres. However, considerable fluctuations of the band width and velocity prevent from determining the trends in the dependence of the band width on either strain or strain rate. The shape of the curves of Figs. 4(c) and 5 makes possible a comparison between the local strain rate within a band, between bands, and the imposed strain rate. Fig. 9 illustrates such estimations for ε˙ a = 2.22 × 10−3 s−1 . While the average strain rate slowly decreases because of the specimen elongation, the strain rate within the bands gradually increases more than one order of magnitude, and the strain rate between strain bands decreases by a similar
T.A. Lebedkina et al. / Materials Science and Engineering A 519 (2009) 147–154
151
Fig. 6. Strain rate maps calculated from the local strain fields as illustrated in Fig. 3 and showing moving deformation bands at various strain rates; x0 stands for the initial coordinates of the edges of the painted black and white bands. (a and b) 2.5 × 10−2 s−1 ; (c) 7.8 × 10−3 s−1 ; (d) 2.22 × 10−3 s−1 ; (e) 5.56 × 10−4 s−1 ; (f) 1.0 × 10−4 s−1 . The strainrate scale is converted into grey levels (vertical bars). The time origin corresponds to the switching on the CCD camera and should not be confused with the beginning of deformation.
factor, during the first hundred seconds of the test. These strain rates then remain more or less constant. It should be stressed that approximately the same ratios are found for all ε˙ a values. Namely, the strain rate within the bands and that between bands are about one order of magnitude higher, respectively, lower, than ε˙ a .
4. Discussion As shown above, some similarities between the unstable plastic flow of the FeMnC steel and the PLC effect are observed for both the macroscopic parameters of the deformation curves and the kine-
152
T.A. Lebedkina et al. / Materials Science and Engineering A 519 (2009) 147–154
Fig. 7. Deformation band velocity Vb as a function of the applied strain-rate ε˙ a at a strain of ε = 0.33. The slope of the dashed line, determined by least-squares fit, is 0.99.
Fig. 8. Examples of strain dependences of the band velocity Vb , and strain carried by the band εb , at ε˙ a = 2.22 × 10−3 s−1 .
Fig. 9. Local strain rate within (solid circles) and outside (open circles) the deformation bands, estimated by measuring the slopes of the steps and the adjacent “plateaus” on the local strain vs. time curve for one of the 2 mm extensometers. The instants are approximately chosen in the middle of the steps. The open circles are plotted against the corresponding solid circles, although the respective time intervals are located further on the right. The dashed line marks the initial total strain rate; the solid line represents the true total strain rate corresponding to the actual gauge length.
matics of the propagation of the deformation bands. Indeed, the negative SRS, the strain-rate dependence of the critical conditions for the instability onset, and the power–law relationship between the velocities of the type A PLC bands and the applied strain rate are typical of the PLC instability. However, these analogies are not sufficient for drawing conclusions on the instability mechanism. The most striking difference with the PLC instability is the persistent propagation character of the deformation bands, which points to a dynamic nature completely different from that for the PLC bands. This behavior is associated with other peculiarities, such as the linear work hardening and the propagation of deformation bands with extremely low velocities. In order to compare the respective spatiotemporal patterns, let us first summarize the salient features of the PLC effect. The character of the heterogeneous strain distribution is well documented for this effect due to the occurrence of deformation bands. Their traces on the specimen surface can be clearly seen and were recorded using a number of experimental techniques (e.g. [20,22,23]). The strain patterns observed in tensile tests performed at constant applied strain rate and the concomitant shapes of deformation curves reveal similar characteristic features for all materials which exhibit the PLC mechanism of plastic instability. Explicitly, nucleation of a deformation band, usually close to the end of the gauge length, followed by its propagation along the gauge length, is observed at high strain rates only, typically at ε˙ a = 10−3 s−1 . The propagation velocity increases with ε˙ a and usually takes on values from several to hundreds mm/s. This process repeats many times, leading to type A deformation curves which are qualitatively similar to the curves of Fig. 2 in the sense that the stress rises and the subsequent smoother sections, respectively correspond to the nucleation and propagation of deformation bands. When the strain rate is progressively decreased, the PLC bands run shorter distances, and finally, only static bands are occurring. They are strongly correlated, so that the subsequent band nucleation gives an impression of a hopping propagation. The notion of type B bands was initially associated with such hopping propagation [17] but is currently used in a larger context: it refers to a wide strain-rate range including hopping bands, multiple band nucleation, and a transition behavior between the hopping and the quasi-continuous band propagation as proposed in [22,24]. In this entire strain-rate interval similar stress–strain curves are observed which are reminiscent of the so-called relaxation oscillations [25]: fast stress drops due to occurrence of deformation bands, which are followed by slower reloading periods after the deformation band exhaustion. The stress usually drops in less than a few tenths of a second, which provides an upper estimate for the band lifetime. In the conventional tension tests, the time resolution is limited by the elastic stiffness of the tensile testing machine, typically about 0.1 s. An effective velocity of the hopping propagation is sometimes determined for type B bands. As the product of the band velocity and strain is controlled by the applied strain rate, the so determined effective velocity can take on very low values at sufficiently low strain rates ε˙ a , of the order of some tenths of mm/s. These values are similar to the estimated band velocity in the investigated steel. However, it should not be forgotten that in contrast to this steel, each type B band is static. It has a short lifetime corresponding to the duration of the stress drop and a longer reloading time is required before the next band appears. The average velocity measured for type B bands is thus determined by the hopping distance divided by the reloading time and is not a proper band velocity. When the applied strain rate is decreased to about ε˙ a = 10−5 s−1 , the correlation between the static PLC bands decays, so that they become visually uncorrelated and a quantitative analysis is needed to determine the residual correlation [26,27]. This behavior corresponds to deformation curves with drastic stress
T.A. Lebedkina et al. / Materials Science and Engineering A 519 (2009) 147–154
drops separated by long intervals of smooth plastic flow (type C instability). This pattern principally differs from the behavior of the investigated steel, in the sense that the quasicontinuous band propagation is observed in a wide strain-rate range, with a progressively decreasing propagation velocity. The band propagation was not well established for the strain rate of ε˙ a = 2.08 × 10−5 s−1 , usually corresponding to type C behavior of the PLC instability. However, the slow development of a deformation band during hundreds of seconds is consistent with the hypothesis of propagation with an extremely low velocity and qualitatively differs from pronounced type C bands. This qualitative observation is supported by the above-described estimates of the ratios between the applied and the local strain rates. Indeed, the factor of about ten is consistent with the values found for type A PLC bands as discussed in [20]. However, it drastically increases with a decrease in ε˙ a in the case of the PLC effect. For example, a factor as high as 7250 between ε˙ b within a type B deformation band and ε˙ a was found in [28]. The authors of [20] ascribed such a difference to the influence of the applied strain rate on the height of the nonlinear SRS-function under conditions of the PLC effect [29]. It is reasonable to suggest that the difference between the observed behavior of the deformation bands and PLC bands may be associated with twinning which occurs in this low stacking fault energy steel as mentioned in the Introduction. Indeed, its other salient feature is the behavior of the work hardening curve which shows a bump in the range of 0.2 ≤ εlog ≤ 0.4 as described in Fig. 3. This can be related to interaction between dislocation glide and twinning within the room temperature to 400 ◦ C range [2]. Although the estimates of the contribution of twinning to the total strain remain below 10% [2], even a small volumetric fraction of twins can contribute to a high strain hardening. Indeed, it was shown that two secant {1 1 1}1 1 2 twinning systems were activated sequentially in individual grains in the course of deformation [2,30,31]. Stacks of microtwins formed along two non parallel sets of {1 1 1} planes are as strong obstacles as grain boundaries and decrease the mean free path of dislocations, resulting in a dynamical Hall–Petch effect. A correlation was found between the bump in hardening and the occurrence of the second twinning system [2,31]. It can be noted that a hump on the strain dependence of the work hardening rate was also observed for stainless steels deformed below room temperature and ascribed to the occurrence of twins and martensite lamellae [32]. Another important consequence of this peculiar hardening mechanism is that twins themselves interact with grain boundaries and other twins in secant planes. As the front of a twin blocked at an interface can be regarded as a dislocation pile-up, it generates internal stress concentrations. The formation of twins in a given place will then initiate twinning and dislocation glide in the neighboring sites. This may explain the persistent propagation of the deformation bands.
5. Summary and concluding remarks
steel uncovers an important role of twining as suggested earlier [1–3,30,31]. The presence of twinning results in two major effects: it locally hardens the material and generates internal stresses. This process is able to promote the propagation of plastic deformation within the sample. Nevertheless, the microscopic mechanism of the observed plastic instability remains uncertain. The similarities with the PLC effect, discussed in Section 4, do not warrant the same physical mechanism because distinct complex systems may manifest similar dynamics governed by generic factors. In mechanical tests, in particular, the parameters of the deformation bands must obey the constraints imposed by the applied strain rate, the crystallography of the plastic flow, and the internal stresses due to local strain incompatibilities caused by the nucleation and propagation of the deformation bands [33]. Two major processes can be suggested. First, the instability may be triggered by twinning only. The twinning starting at some point will initiate twin nucleation in neighboring sites and, therefore, advance into unoccupied regions of the specimen. Experimental evidence for such a process has been found from acoustic-emission measurements on a Cu–Ge alloy [34]. However, the estimates of the volume fraction of twins in this austenitic steel [2] indicate that twinning cannot allow alone for the entire strain accumulated during strain jumps. An alternative hypothesis may utilize the PLC effect modified by the TWIP effect. It is worth noting that stress oscillations similar to type B PLC behavior were observed for a steel with the same nominal composition in a range of higher temperatures [2], which indicate a change from propagation to static strain localizations. A study of the temperature effect on the strain inhomogeneity may be useful for further discrimination of the mechanisms of plastic instability in this material. Most likely, the mechanism of plastic instability has a complex nature and includes dislocation glide, twinning, and dynamic interaction of solutes with both dislocations and twins. Acknowledgements T.A. Lebedkina would like to acknowledge the support from ArcelorMittal research, as well as from Région Lorraine, Institut National Polytechnique de Lorraine (Nancy), and Université Paul Verlaine – Metz (university cooperation program ARCUS, Project Lorraine/Russia) for her stays in Metz and Nancy during 2007/2008. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
The observations with the use of the local extensometry technique show that at room temperature, the austenitic FeMnC TWIP steel exhibits plastic instabilities analogous to the type A PLC behavior which is marked by the propagation of deformation bands. However, in contrast to the PLC effect, the band propagation regime is observed over four orders of magnitude in the applied strain rate, without transition to localized deformation bands at lower strain rates. This fact in association with the low diffusivity of carbon in austenite excludes the classical mechanism of the PLC effect. On the other hand, the unusual work hardening behavior of this
153
[11] [12] [13] [14]
[15] [16] [17] [18]
S. Allain, J.P. Chateau, O. Bouaziz, Steel Res. 73 (2002) 299–302. S. Allain, Ecole des Mines de Nancy, Nancy, PhD thesis, 2004. L. Rémy, Université Paris Sud, Paris, PhD thesis, 1975. V.H. Schumann, Neue Hütte 17 (1972) 605–609. J. Belotteau, C. Berdin, S. Forest, A. Parrot, C. Prioul, Local Approach to Fracture. EMMC9, Ecole des Mines de Paris, 2006, 187–192. D. Wagner, N. Roubier, C. Prioul, Mater. Sci. Technol. 22 (3) (2006) 301–307. O. Grässel, L. Krüger, G. Frommeyer, L.W. Meyer, Int. J. Plast. 16 (2000) 1391–1409. C. Scott, S. Allain, M. Faral, N. Guelton, Revue de Métallurgie 6 (2006) 293–302. M.V. Klassen-Nekludova, Mechanical Twinning of Crystals, Consultants Bureau, NY, 1964. L. Chen, H.-S. Kim, S.-K. Kim, B.C. De Cooman, ISIJ Int. Online 47 (12) (2007) 1804–1812. S. Allain, Ph. Cugy, C. Scott, J.-Ph. Chateau, A. Rusinek, A. Deschamps, Int. J. Mater. Res. 99 (7) (2008) 734–738. G.A. Malygin, Fizika Metallov I Metallovedenie 63 (5) (1987) 864–875. M.A. Lebyodkin, L.R. Dunin-Barkovskii, T.A. Lebedkina, JETP Lett. 76 (2002) 612–615. J. Weiss, T. Richeton, F. Louchet, F. Chmelik, P. Dobron, D. Entemeyer, M. Lebyodkin, T. Lebedkina, C. Fressengeas, R.J. McDonald, Phys. Rev. B 76 (2007), 224110(8). R. Král, P. Lukáˇc, Mater. Sci. Eng. A 234–236 (1997) 786–789. T.A. Lebedkina, M.A. Lebyodkin, Acta Mater. 56 (2008) 5567–5574. P. Rodriguez, S. Venkadesan, Solid State Phenom. 42–43 (1995) 257–266. V. Gröger, V. Stelzhammer, Z. Trojanová, Mater. Sci. Eng. A 324 (2002) 208–213.
154
T.A. Lebedkina et al. / Materials Science and Engineering A 519 (2009) 147–154
[19] T. Lebedkina, M. Lebyodkin, J.P. Chateau, A. Jacques, S. Allain, Book of Abstracts, Colloque Plasticité, Nancy, France, 10–12 Mars 2008, p. 46, http://www.mines.inpl-nancy.fr/plasticite2008/Plasticite 08.pdf. [20] H. Ait-Amokhtar, P. Vacher, S. Boudrahem, Acta Mater. 54 (2006) 4365–4371. [21] A. Ziegenbein, P. Hähner, H. Neuhäuser, Comput. Mater. Sci. 19 (2000) 27–34. [22] R. Shabadi, S. Kumara, H.J. Roven, E.S. Dwarakadasa, Mater. Sci. Eng. A 364 (2004) 140–150. [23] Q. Zhang, Z. Jiang, H. Jiang, Z. Chen, X. Wu, Int. J. Plast. 21 (2005) 2150–2173. [24] M.S. Bharathi, M. Lebyodkin, G. Ananthakrishna, C. Fressengeas, L.P. Kubin, Acta Mater. 50 (2002) 2813–2824. [25] F.L. Vernon, R.J. Pedersen, J. Appl. Phys. 39 (1968) 2661–2664. [26] M.A. Lebyodkin, Y. Estrin, Acta Mater. 53 (2005) 3403–3413.
[27] M.A. Lebyodkin, T.A. Lebedkina, Phys. Rev. E 77 (2008), 026111(8). [28] H. Ait-Amokhtar, S. Boudrahem, C. Fressengeas, Scr. Mater. 54 (2006) 2113–2118. [29] C. Fressengeas, A.J. Beaudoin, M. Lebyodkin, L.P. Kubin, Y. Estrin, Mater. Sci. Eng. A 400 (2005) 226–230. [30] S. Allain, J.-P. Chateau, D. Dahmoun, O. Bouaziz, Mater. Sci. Eng. A 387–389 (2004) 272–276. [31] S. Allain, J.-P. Chateau, O. Bouaziz, Mater. Sci. Eng. A 387–389 (2004) 143–147. [32] T.S. Byun, N. Hashimoto, K. Farell, Acta Mater. 52 (2004) 3889–3899. [33] S. Kok, A.J. Beaudoin, D.A. Tortorelli, M. Lebyodkin, Model. Simul. Mater. Sci. Eng. 10 (2002) 745–763. [34] A. Vinogradov, D.L. Merson, V. Patlan, S. Hashimoto, Mater. Sci. Eng. A 341 (2003) 57–73.
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
On the mechanism of unstable plastic flow in an austenitic FeMnC TWIP steel T.A. Lebedkina a , M.A. Lebyodkin b,∗ , J.-Ph. Chateau c , A. Jacques c , S. Allain d a
Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Russia LPMM, Université Paul Verlaine – Metz/CNRS, Ile du Saulcy, 57045 Metz Cedex 01, France c LPM, Ecole des Mines/CNRS, Parc de Saurupt, 54042 Nancy Cedex, France d Arcelormittal Maizières Research SA, Voie Romaine, BP 30320, 57283 Maizières les Metz, France b
a r t i c l e
i n f o
Article history: Received 19 December 2008 Received in revised form 26 April 2009 Accepted 28 April 2009 Keywords: High manganese austenitic steel Jerky flow Deformation inhomogeneities TWIP Dynamic strain aging
a b s t r a c t The complex character of plastic deformation of the austenitic steel Fe22Mn0.6C is studied at room temperature with the aid of high-frequency local extensometry. It is shown that the plastic flow instability, associated with fluctuations of the flow stress, results from quasi-continuous propagation of deformation bands along the specimen axis. This propagation mode is dominant in the entire range of the applied strain-rate from 2.1 × 10−5 s−1 to 10−1 s−1 . Such behavior differs from that of various alloys deforming via dislocation glide under conditions of dynamic strain ageing (Portevin-Le Chatelier effect), which is characterized by a transition from a repetitive occurrence of static deformation bands at lower strain rates to a relay-race and, finally, quasi-continuous deformation band propagation at higher strain rates. The unusual behavior of the deformation bands bears evidence to a particular kind of instability in the investigated steel. A possible role of deformation twins in the observed dynamics of plastic instability is discussed. © 2009 Elsevier B.V. All rights reserved.
1. Introduction The plastic deformation of steels with high manganese content, such as Fe22Mn0.6C (wt.%), exhibits macroscopic plastic instabilities in a wide range of strain-rate and temperature and is characterized by stress fluctuations of various shapes on the stress–strain curve. The resulting deformation curves present different morphologies depending on temperature. Such variance is consistent with the temperature dependence of the microscopic mechanisms of plastic flow of this material, which depend on stacking fault energy (SFE) [1–3]. Indeed, when the deformation temperature is decreased from 673 K down to 77 K, dislocation glide, twinning plus glide, and martensitic transformation plus glide (-martensite with h.c.p. lattice crystallographic structure) are successively observed in accordance with the diminishing SFE [4]. Therefore, it is natural to suggest that the instabilities observed under various deformation conditions can be governed by different microscopic mechanisms. These mechanisms are not known yet, perhaps, with an exception for a temperature interval above 150–200 ◦ C, where the dynamical strain aging of dislocations due to carbon atoms may lead to the Portevin-Le Chatelier effect [5,6]. However, this problem is of interest for both practical applications, since the instabilities impede the processing of these steels, which
∗ Corresponding author. Tel.: +33 03 87315401. E-mail addresses: [email protected], [email protected] (M.A. Lebyodkin). 0921-5093/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2009.04.067
otherwise possess remarkable mechanical features [7,8], and for understanding the basic problem of the deformation mechanisms leading to instability of the uniform plastic flow. At room temperature, the studied FeMnC steel is fully austenitic, with a sufficiently low SFE to promote twinning, which enters into the competition with dislocation glide, as reflected in the notion of twinning-induced plasticity (TWIP) [7]. As a matter of fact, this peculiar feature is at the origin of this steel attraction for industry because it provides a high strain hardening rate leading simultaneously to high ductility and tensile strength. It is known that the formation of twins may be accompanied with pronounced stress fluctuations, although it usually concerns low-temperature deformation [9]. However, the instabilities observed in this steel at room temperature are often tentatively ascribed to the PLC effect [10]. This assumption is based on the observation of the negative strain-rate sensitivity (SRS) of the flow stress and a thermally activated character of plastic deformation. Nevertheless, the diffusion coefficient of carbon in austenite appears to be too low at room temperature to account for the PLC effect [11]. On the other hand, it is known that the negative SRS is not a peculiar feature of the PLC effect only. It also emerges for distinct mechanisms, e.g. for thermo-mechanical instability, which is known to be caused by local overheating of specimens deformed at low temperatures [12]. Moreover, the thermo-mechanical instability demonstrates features in common with the PLC effect, as far as the morphology of the deformation curves and the statistics of stress jumps are concerned [13]. This example proves that neither the behavior of the
148
T.A. Lebedkina et al. / Materials Science and Engineering A 519 (2009) 147–154
SRS nor the similarity of the morphology of deformation curves is a sufficient criterion for distinguishing between various instability mechanisms. In order to provide a further insight into the mechanism of plastic instability in austenitic TWIP steels at room temperature, we study the local strain distribution during tensile deformation of FeMnC samples using high-rate extensometry technique [14]. This paper is organized as follows. Section 2 outlines experimental details and data processing. Section 3 describes experimental data on the macroscopic tensile behavior of the studied steel and the concomitant evolution of the local strain heterogeneity. The overall discussion, including the comparison of the evolution of the strain distribution to salient features of PLC bands, is given in Section 4. Finally, Section 5 highlights hypotheses on which mechanisms are able to lead to the observed plastic instabilities. 2. Experimental procedure Flat tensile specimens with a gauge section 60 mm × 12.6 mm × 1.25 mm in size were prepared from a Fe–22 wt.% Mn–0.6 wt.% C steel, with approximately 3 m grain size. Mechanical tests were performed at room temperature with a constant crosshead speed. The nominal value of the imposed strain-rate ε˙ a , corresponding to the initial specimen length, was varied in the range from 2.08 × 10−5 s−1 to 2.5 × 10−2 s−1 . Besides, several tests were performed on samples fabricated separately but with the same nominal composition and grain structure. These specimens with dimensions 80 mm × 5.4 mm × 0.6 mm were tested in a strain-rate range from 2.2 × 10−5 s−1 to 8 × 10−3 s−1 [2]. Using such specimens, one test was performed with the aid of a pyrometry technique to visualize strain localization at the strain-rate ε˙ a = 10−1 s−1 [11]. In order to conduct local extensometry, one surface of the specimen was painted black, and somewhat 10 white marks of 1 mm in width were superimposed on the black layer normal to the longitudinal axis as illustrated in Fig. 1. The distance between the marks was approximately equal to 1 mm. The resulting sequence of black and white stripes formed a set of about twenty 1 mm long extensometers, in the center of the 60 mm gauge length. The positions of the transitions between black and white marks during the sample deformation were followed using a high resolution 1D CCD camera with a recording frequency of 103 Hz and a pixel size of 1.3 m. The camera was fixed relative to the fixed crosshead. It aimed at the specimen axis in order to avoid any cross-section rotation effect, although negligible in the case of polycrystals. As a matter of fact, the deformation of the samples appeared to be uniform on average in all tests, so that no noticeable rotation effect was observed before ultimate necking. The camera allowed continuous recording during an interval of 600 s, which covered the entire test duration at higher strain rates of ε˙ a > 1.5 × 10−3 s−1 . Consequently, the strain dependence of various parameters of the strain pattern could be built. For lower ε˙ a values, it was possible to survey only portions of tests. In this case, the measurements were repeated several times during the test in order to evaluate the local deformation kinetics qualitatively. To estimate the continuous work hardening rate, = ˙ ∂(ε, ε)/∂ε, the deformation curves obtained at different imposed strain rates were fitted with polynomials in the strain intervals from 0.005 ≤εlog ≤ 0.5–0.6, to eliminate stress fluctuations. The estimation of evolution with strain was then obtained by deriving the fitting function. The evolution of the local strain values were calculated from the displacements measured by the CCD camera: εi (t) = ln
xi+2 (t) − xi (t) , xi+2 (0) − xi (0)
(1)
where xi is the coordinate of the ith transition. In this equation, every second section is taken in order to avoid a possible arte-
Fig. 1. Scheme of the experimental setup. The black and white stripes depict the set of extensometers painted on its surface. A deformation band (B) propagates from the top to the bottom of the specimen.
fact due to unequal sensitivity of the camera to black–white and white–black transitions. During performing the mechanical test, some extensometers leave the camera’s field of vision, some others will enter into it. For this reason, the initial time in Eq. (1) was usually not selected at the beginning of deformation, but at a later instant, in order to seize the largest number of extensometers. In this case, only the relative changes in the local strain values were meaningful. In addition to εi (t) local strain rates ε˙ i (t) were calculated by deriving the εi (t) dependence after preliminary reduction of digital noise with the aid of a running average technique. The construction of strain maps, (t, xi (0), εi (t)), and the respective strainrate maps, (t, xi (0), ε˙ i (t)), allowed visualization of the evolution of the local strain field and, particularly, provided a simple tool to distinguish between the band propagation and localization. These maps, together with the εi (t) dependence, provide evaluation of the band velocity, Vb , band width, w, and the associated strain, εb . As the maps refer to the gauge length of the local extensometers at the initial time, the values of Vb and w are corrected taking the sample elongation into account. Other kinds of data representations, though less illustrative, were also used to evaluate the deformation band parameters. For example, the axial strain distribution profiles εi (xi ) were calculated at subsequent time instants. All methods provided close values for the deformation band parameters. 3. Results 3.1. Macroscopic behavior Fig. 2 shows some representative true stress–strain tensile curves recorded at different imposed strain rates. A negative SRS occurs in the entire strain-rate range investigated as described in Ref. [11] for strain rates up to 1 s−1 . A rough evaluation of the SRS using a relationship S ≈ / ln(ε˙ 2a /ε˙ 1a ) for pairs of deformation
T.A. Lebedkina et al. / Materials Science and Engineering A 519 (2009) 147–154
Fig. 2. True stress–logarithmic strain curves recorded with various applied strain rate: from upper to lower curves, 2.08 × 10−5 s−1 , 5.56 × 10−4 s−1 , 2.22 × 10−3 s−1 , 7.78 × 10−3 s−1 , and 2.5 × 10−2 s−1 . For the sake of clarity, some of ε˙ a values are marked by arrows which indicate the respective values of the critical strain for the onset of plastic instability. Inset: dependence of the critical strain on the imposed strain rate.
curves recorded at different strain rates, i.e., assuming that strain hardening is not considerably dependent on ε˙ a , leads to values that increase monotonously from a few MPa at the beginning of plastic deformation to about 30 MPa at εlog ≈ 0.5. This range of SRS does not essentially differ from the literature data for C–Mn steels under PLC conditions (e.g. [5,6]). The strain-rate dependence of the critical strain for the onset of visible instability appears to have a minimum at intermediate strain rate as illustrated by the inset of Fig. 2. Such a non-monotonic behavior of the critical strain is often observed for the PLC instability [15,16]. In contrast to these macroscopic characteristics, which show some similarity with the PLC effect, the morphology of the serrated stress–strain curves observed presents specific features. The PLC effect shows remarkable dependence on ε˙ a as will be discussed in Section 4. On the contrary, the persistence of the shape of the true stress–strain curves of Fig. 2 attracts attention. This observation leads to suggestion that the same dynamical mode of plastic flow may be active in the entire ε˙ a range. Moreover, the shape of the curves allows for certain assumptions by analogy with the known kinds of unstable plastic flow, in particular, with the highrate regimes of the PLC behavior and thermo-mechanical instability, which are characterized by deformation band propagation [17,18]. Namely, typical sequences are repetitively observed, which consist of a stress rise followed by a steep stress drop and a subsequent increase of the stress at a rate close to the average work hardening rate. This is usually considered as a signature of a recurrent nucleation and propagation of deformation bands: an increase in the applied stress is necessary to nucleate a band which propagates at a lower stress level. The other peculiar feature appears in the behavior of the work hardening rate curves of Fig. 3. First, the deformation curves look roughly linear on the scale of Fig. 2, i.e., the work hardening seems to correspond to a unique stage all over the test. Such work hardening differs from the behavior of typical dynamically strain ageing polycrystalline alloys, which usually exhibit parabolic work hardening in the strain range corresponding to the unstable plastic flow. Quantitative evaluation reveals some variation in . Except for the highest rates, it is found that the initial decrease in is followed by an increase in some strain interval starting at 0.15 ≤ εlog ≤ 0.2 (Fig. 3). The increase in diminishes with increasing ε˙ a and is not observed at a strain rate of ε˙ a = 2.5 × 10−2 s−1 .
149
Fig. 3. Work hardening rate as a function of strain for different strain rates as indicated in Fig. 1.
3.2. Local strain patterns and deformation band characterization Fig. 4 represents an example of synchronized sections of a stress–time curve recorded at ε˙ a = 2.22 × 10−3 s−1 with the respective evolutions of the recorded positions and local strains as described by Eq. (1). Fig. 4a and b shows that stress drops occur synchronously with an acceleration of all the displacements as marked by vertical dotted lines. This indicates that the specimen is deformed between its fixed head and the upper marker as shown in Fig. 1. The displacements subsequently slow down one after another as shown by the deviation from the vertical dashed lines illustrated by the inclined dotted line at t ≈ 160 s. This results in local strain jumps of substantially the same height as illustrated in Fig. 4c. The curves of Fig. 4c are thus made of short intervals of high-rate plastic flow indicated by steps and extended intervals of slower deformation. The delay between the initial acceleration of the displacements and the strain jumps suggests that a localized deformation band moves from the upper part of the specimen through the set of extensometers. This is confirmed by Fig. 5 which presents enlargements
Fig. 4. Representative experimental records as a function of testing time at strain rate of ε˙ a = 2.22 × 10−3 s−1 : force applied to the tensile specimen (a), positions of neighboring sections of the sample (b), and local strain values (c) calculated using the data of (b). Vertical dotted lines indicate instants when a deformation band is nucleated. Dashed lines are approximately marking the instants when the band leaves the first of these sections. The inclined dotted line at t ≈ 160 s marks the edge of a plateau, where the deformation band leaves the respective sections.
150
T.A. Lebedkina et al. / Materials Science and Engineering A 519 (2009) 147–154
of the curves of Fig. 4c and clearly shows that the local strain jumps recorded for various i are shifted in time with regard to each other. Such progressions prove that deformation is mainly achieved by the repetitive propagation of deformation bands along the tensile axis as drawn schematically in Fig. 1. Further observations follow from Figs. 4 and 5. The comparison of Fig. 5a and b shows that the magnitude of the time shift between the local strain jumps in the adjacent sections of the specimen increases with deformation. As shown below, this is due to both the specimen elongation and a decrease in the true band velocity, see Fig. 8, determined for the actual specimen gauge length. At the same time, the slope of the steps increases with deformation, thus indicating an increase in the local strain rate in the deformation band. The average slope of each curve of Fig. 4c is determined by the imposed strain rate. It is worth noting that the step-plateau shape of the εi (t)-curves testifies that the nucleation of the next deformation band takes place practically immediately after the previous one has completed its propagation. Otherwise, periods of homogeneous deformation at the imposed strain rate should also be observed. The εi (t)-curves and the strain-rate maps constructed for various strain rates ε˙ a disclose a combination of different kinds of strain inhomogeneity, including deformation band propagation and in some cases localization. The main features of the deformation bands’ behavior are illustrated in Fig. 6. Importantly, the band propagation regime persists in the entire ε˙ a range investi-
Fig. 5. Evolution of the local strain εloc for four juxtaposed 2 mm extensometers at (a) an early and (b) a later stage of deformation with ε˙ a = 2.22 × 10−3 s−1 . The frames indicating indexes of the respective sections reveal a consecutive passage of a band along the sample.
gated, in agreement with the morphology of deformation curves. Indeed, it can be recognized that the band propagation is dominant at higher strain rates of ε˙ a ≥ 2 × 10−3 s−1 as illustrated in Fig. 6(a)–(d). This conclusion was also confirmed by the visualization of the local straining by the pyrometry technique [11]. A more complicated behavior was observed at an intermediate strain rate of ε˙ a = 5.56 × 10−4 s−1 , see Fig. 6(e). In this case, the observed patterns included band propagation over shorter distances, static bands, and coexistence of two or more bands. However, the propagation remains an essential mode in these conditions too. Moreover, propagation is found to be dominant also at ε˙ a = 1.0 × 10−4 s−1 , see Fig. 6(f). Similar results were observed in the control tests on differently sized specimens. The comparison of the examples of Fig. 6 shows that the deformation band velocity diminishes when ε˙ a is decreased. Noting the limited interval of recording with the CCD camera, this makes difficult the clear detection of propagation at the lowest strain rate, ε˙ a = 2.08 × 10−5 s−1 . A full propagation cycle has not been seized. Nevertheless, the sections recorded at this strain rate testify that the respective deformation bands develop during hundreds of seconds. This observation confirms the persistence of the propagation mode down to very low strain rates [19]. As will be discussed in Section 4, such behavior is remarkably different from that of the dynamics of the PLC effect. Although velocity, width, and strain of a band may fluctuate, cf. Figs. 4 and 5, the records of the evolution of the local strain field allow quantitative estimates. The band velocity increases approximately linearly with ε˙ a as illustrated in Fig. 7. This linear trend is roughly consistent with the power–law dependences of the deformation band velocity on the strain rate, which are usually observed for the so-called type A PLC bands [20,21]. However, the exponents reported in these works take on lower values of n ≈ 0.8. Furthermore, such propagating PLC bands only occur in the range of the highest strain rates within the entire strain-rate range corresponding to the PLC instability. Last, the velocity of the deformation bands in the investigated steel reaches rather low magnitudes at low strain rates (tenths of mm/s), as compared to literature data for type A PLC bands, usually above several mm/s. The strain dependences of the deformation band parameters appear to behave similarly to those for the type A PLC bands. Fig. 8 presents examples of such behaviors for ε˙ a = 2.22 × 10−3 s−1 . The strain in the band increases with the overall strain, whereas the band velocity decreases, so that their product, εb Vb , is roughly constant during the test and close to the crosshead speed V corrected to take elastic straining into account. Therefore, it can be concluded that at a given time, the plastic strain is mostly concentrated in a deformation band, whereas the estimates of the strain outside the band typically give values about ten per cent of the whole strain. It should be noted that the value of εb Vb obtained at different strains varies noticeably. This variation testifies to a strong inhomogeneity of strain: evidently, the deviations are found when a considerable straining takes place outside the band passing through the extensometer set, e.g., when multiple deformation bands occur. The conclusion on the pronounced inhomogeneity of plastic flow is also confirmed by the estimations of the bands width. The values obtained typically vary in the range from several to ten millimetres. However, considerable fluctuations of the band width and velocity prevent from determining the trends in the dependence of the band width on either strain or strain rate. The shape of the curves of Figs. 4(c) and 5 makes possible a comparison between the local strain rate within a band, between bands, and the imposed strain rate. Fig. 9 illustrates such estimations for ε˙ a = 2.22 × 10−3 s−1 . While the average strain rate slowly decreases because of the specimen elongation, the strain rate within the bands gradually increases more than one order of magnitude, and the strain rate between strain bands decreases by a similar
T.A. Lebedkina et al. / Materials Science and Engineering A 519 (2009) 147–154
151
Fig. 6. Strain rate maps calculated from the local strain fields as illustrated in Fig. 3 and showing moving deformation bands at various strain rates; x0 stands for the initial coordinates of the edges of the painted black and white bands. (a and b) 2.5 × 10−2 s−1 ; (c) 7.8 × 10−3 s−1 ; (d) 2.22 × 10−3 s−1 ; (e) 5.56 × 10−4 s−1 ; (f) 1.0 × 10−4 s−1 . The strainrate scale is converted into grey levels (vertical bars). The time origin corresponds to the switching on the CCD camera and should not be confused with the beginning of deformation.
factor, during the first hundred seconds of the test. These strain rates then remain more or less constant. It should be stressed that approximately the same ratios are found for all ε˙ a values. Namely, the strain rate within the bands and that between bands are about one order of magnitude higher, respectively, lower, than ε˙ a .
4. Discussion As shown above, some similarities between the unstable plastic flow of the FeMnC steel and the PLC effect are observed for both the macroscopic parameters of the deformation curves and the kine-
152
T.A. Lebedkina et al. / Materials Science and Engineering A 519 (2009) 147–154
Fig. 7. Deformation band velocity Vb as a function of the applied strain-rate ε˙ a at a strain of ε = 0.33. The slope of the dashed line, determined by least-squares fit, is 0.99.
Fig. 8. Examples of strain dependences of the band velocity Vb , and strain carried by the band εb , at ε˙ a = 2.22 × 10−3 s−1 .
Fig. 9. Local strain rate within (solid circles) and outside (open circles) the deformation bands, estimated by measuring the slopes of the steps and the adjacent “plateaus” on the local strain vs. time curve for one of the 2 mm extensometers. The instants are approximately chosen in the middle of the steps. The open circles are plotted against the corresponding solid circles, although the respective time intervals are located further on the right. The dashed line marks the initial total strain rate; the solid line represents the true total strain rate corresponding to the actual gauge length.
matics of the propagation of the deformation bands. Indeed, the negative SRS, the strain-rate dependence of the critical conditions for the instability onset, and the power–law relationship between the velocities of the type A PLC bands and the applied strain rate are typical of the PLC instability. However, these analogies are not sufficient for drawing conclusions on the instability mechanism. The most striking difference with the PLC instability is the persistent propagation character of the deformation bands, which points to a dynamic nature completely different from that for the PLC bands. This behavior is associated with other peculiarities, such as the linear work hardening and the propagation of deformation bands with extremely low velocities. In order to compare the respective spatiotemporal patterns, let us first summarize the salient features of the PLC effect. The character of the heterogeneous strain distribution is well documented for this effect due to the occurrence of deformation bands. Their traces on the specimen surface can be clearly seen and were recorded using a number of experimental techniques (e.g. [20,22,23]). The strain patterns observed in tensile tests performed at constant applied strain rate and the concomitant shapes of deformation curves reveal similar characteristic features for all materials which exhibit the PLC mechanism of plastic instability. Explicitly, nucleation of a deformation band, usually close to the end of the gauge length, followed by its propagation along the gauge length, is observed at high strain rates only, typically at ε˙ a = 10−3 s−1 . The propagation velocity increases with ε˙ a and usually takes on values from several to hundreds mm/s. This process repeats many times, leading to type A deformation curves which are qualitatively similar to the curves of Fig. 2 in the sense that the stress rises and the subsequent smoother sections, respectively correspond to the nucleation and propagation of deformation bands. When the strain rate is progressively decreased, the PLC bands run shorter distances, and finally, only static bands are occurring. They are strongly correlated, so that the subsequent band nucleation gives an impression of a hopping propagation. The notion of type B bands was initially associated with such hopping propagation [17] but is currently used in a larger context: it refers to a wide strain-rate range including hopping bands, multiple band nucleation, and a transition behavior between the hopping and the quasi-continuous band propagation as proposed in [22,24]. In this entire strain-rate interval similar stress–strain curves are observed which are reminiscent of the so-called relaxation oscillations [25]: fast stress drops due to occurrence of deformation bands, which are followed by slower reloading periods after the deformation band exhaustion. The stress usually drops in less than a few tenths of a second, which provides an upper estimate for the band lifetime. In the conventional tension tests, the time resolution is limited by the elastic stiffness of the tensile testing machine, typically about 0.1 s. An effective velocity of the hopping propagation is sometimes determined for type B bands. As the product of the band velocity and strain is controlled by the applied strain rate, the so determined effective velocity can take on very low values at sufficiently low strain rates ε˙ a , of the order of some tenths of mm/s. These values are similar to the estimated band velocity in the investigated steel. However, it should not be forgotten that in contrast to this steel, each type B band is static. It has a short lifetime corresponding to the duration of the stress drop and a longer reloading time is required before the next band appears. The average velocity measured for type B bands is thus determined by the hopping distance divided by the reloading time and is not a proper band velocity. When the applied strain rate is decreased to about ε˙ a = 10−5 s−1 , the correlation between the static PLC bands decays, so that they become visually uncorrelated and a quantitative analysis is needed to determine the residual correlation [26,27]. This behavior corresponds to deformation curves with drastic stress
T.A. Lebedkina et al. / Materials Science and Engineering A 519 (2009) 147–154
drops separated by long intervals of smooth plastic flow (type C instability). This pattern principally differs from the behavior of the investigated steel, in the sense that the quasicontinuous band propagation is observed in a wide strain-rate range, with a progressively decreasing propagation velocity. The band propagation was not well established for the strain rate of ε˙ a = 2.08 × 10−5 s−1 , usually corresponding to type C behavior of the PLC instability. However, the slow development of a deformation band during hundreds of seconds is consistent with the hypothesis of propagation with an extremely low velocity and qualitatively differs from pronounced type C bands. This qualitative observation is supported by the above-described estimates of the ratios between the applied and the local strain rates. Indeed, the factor of about ten is consistent with the values found for type A PLC bands as discussed in [20]. However, it drastically increases with a decrease in ε˙ a in the case of the PLC effect. For example, a factor as high as 7250 between ε˙ b within a type B deformation band and ε˙ a was found in [28]. The authors of [20] ascribed such a difference to the influence of the applied strain rate on the height of the nonlinear SRS-function under conditions of the PLC effect [29]. It is reasonable to suggest that the difference between the observed behavior of the deformation bands and PLC bands may be associated with twinning which occurs in this low stacking fault energy steel as mentioned in the Introduction. Indeed, its other salient feature is the behavior of the work hardening curve which shows a bump in the range of 0.2 ≤ εlog ≤ 0.4 as described in Fig. 3. This can be related to interaction between dislocation glide and twinning within the room temperature to 400 ◦ C range [2]. Although the estimates of the contribution of twinning to the total strain remain below 10% [2], even a small volumetric fraction of twins can contribute to a high strain hardening. Indeed, it was shown that two secant {1 1 1}1 1 2 twinning systems were activated sequentially in individual grains in the course of deformation [2,30,31]. Stacks of microtwins formed along two non parallel sets of {1 1 1} planes are as strong obstacles as grain boundaries and decrease the mean free path of dislocations, resulting in a dynamical Hall–Petch effect. A correlation was found between the bump in hardening and the occurrence of the second twinning system [2,31]. It can be noted that a hump on the strain dependence of the work hardening rate was also observed for stainless steels deformed below room temperature and ascribed to the occurrence of twins and martensite lamellae [32]. Another important consequence of this peculiar hardening mechanism is that twins themselves interact with grain boundaries and other twins in secant planes. As the front of a twin blocked at an interface can be regarded as a dislocation pile-up, it generates internal stress concentrations. The formation of twins in a given place will then initiate twinning and dislocation glide in the neighboring sites. This may explain the persistent propagation of the deformation bands.
5. Summary and concluding remarks
steel uncovers an important role of twining as suggested earlier [1–3,30,31]. The presence of twinning results in two major effects: it locally hardens the material and generates internal stresses. This process is able to promote the propagation of plastic deformation within the sample. Nevertheless, the microscopic mechanism of the observed plastic instability remains uncertain. The similarities with the PLC effect, discussed in Section 4, do not warrant the same physical mechanism because distinct complex systems may manifest similar dynamics governed by generic factors. In mechanical tests, in particular, the parameters of the deformation bands must obey the constraints imposed by the applied strain rate, the crystallography of the plastic flow, and the internal stresses due to local strain incompatibilities caused by the nucleation and propagation of the deformation bands [33]. Two major processes can be suggested. First, the instability may be triggered by twinning only. The twinning starting at some point will initiate twin nucleation in neighboring sites and, therefore, advance into unoccupied regions of the specimen. Experimental evidence for such a process has been found from acoustic-emission measurements on a Cu–Ge alloy [34]. However, the estimates of the volume fraction of twins in this austenitic steel [2] indicate that twinning cannot allow alone for the entire strain accumulated during strain jumps. An alternative hypothesis may utilize the PLC effect modified by the TWIP effect. It is worth noting that stress oscillations similar to type B PLC behavior were observed for a steel with the same nominal composition in a range of higher temperatures [2], which indicate a change from propagation to static strain localizations. A study of the temperature effect on the strain inhomogeneity may be useful for further discrimination of the mechanisms of plastic instability in this material. Most likely, the mechanism of plastic instability has a complex nature and includes dislocation glide, twinning, and dynamic interaction of solutes with both dislocations and twins. Acknowledgements T.A. Lebedkina would like to acknowledge the support from ArcelorMittal research, as well as from Région Lorraine, Institut National Polytechnique de Lorraine (Nancy), and Université Paul Verlaine – Metz (university cooperation program ARCUS, Project Lorraine/Russia) for her stays in Metz and Nancy during 2007/2008. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
The observations with the use of the local extensometry technique show that at room temperature, the austenitic FeMnC TWIP steel exhibits plastic instabilities analogous to the type A PLC behavior which is marked by the propagation of deformation bands. However, in contrast to the PLC effect, the band propagation regime is observed over four orders of magnitude in the applied strain rate, without transition to localized deformation bands at lower strain rates. This fact in association with the low diffusivity of carbon in austenite excludes the classical mechanism of the PLC effect. On the other hand, the unusual work hardening behavior of this
153
[11] [12] [13] [14]
[15] [16] [17] [18]
S. Allain, J.P. Chateau, O. Bouaziz, Steel Res. 73 (2002) 299–302. S. Allain, Ecole des Mines de Nancy, Nancy, PhD thesis, 2004. L. Rémy, Université Paris Sud, Paris, PhD thesis, 1975. V.H. Schumann, Neue Hütte 17 (1972) 605–609. J. Belotteau, C. Berdin, S. Forest, A. Parrot, C. Prioul, Local Approach to Fracture. EMMC9, Ecole des Mines de Paris, 2006, 187–192. D. Wagner, N. Roubier, C. Prioul, Mater. Sci. Technol. 22 (3) (2006) 301–307. O. Grässel, L. Krüger, G. Frommeyer, L.W. Meyer, Int. J. Plast. 16 (2000) 1391–1409. C. Scott, S. Allain, M. Faral, N. Guelton, Revue de Métallurgie 6 (2006) 293–302. M.V. Klassen-Nekludova, Mechanical Twinning of Crystals, Consultants Bureau, NY, 1964. L. Chen, H.-S. Kim, S.-K. Kim, B.C. De Cooman, ISIJ Int. Online 47 (12) (2007) 1804–1812. S. Allain, Ph. Cugy, C. Scott, J.-Ph. Chateau, A. Rusinek, A. Deschamps, Int. J. Mater. Res. 99 (7) (2008) 734–738. G.A. Malygin, Fizika Metallov I Metallovedenie 63 (5) (1987) 864–875. M.A. Lebyodkin, L.R. Dunin-Barkovskii, T.A. Lebedkina, JETP Lett. 76 (2002) 612–615. J. Weiss, T. Richeton, F. Louchet, F. Chmelik, P. Dobron, D. Entemeyer, M. Lebyodkin, T. Lebedkina, C. Fressengeas, R.J. McDonald, Phys. Rev. B 76 (2007), 224110(8). R. Král, P. Lukáˇc, Mater. Sci. Eng. A 234–236 (1997) 786–789. T.A. Lebedkina, M.A. Lebyodkin, Acta Mater. 56 (2008) 5567–5574. P. Rodriguez, S. Venkadesan, Solid State Phenom. 42–43 (1995) 257–266. V. Gröger, V. Stelzhammer, Z. Trojanová, Mater. Sci. Eng. A 324 (2002) 208–213.
154
T.A. Lebedkina et al. / Materials Science and Engineering A 519 (2009) 147–154
[19] T. Lebedkina, M. Lebyodkin, J.P. Chateau, A. Jacques, S. Allain, Book of Abstracts, Colloque Plasticité, Nancy, France, 10–12 Mars 2008, p. 46, http://www.mines.inpl-nancy.fr/plasticite2008/Plasticite 08.pdf. [20] H. Ait-Amokhtar, P. Vacher, S. Boudrahem, Acta Mater. 54 (2006) 4365–4371. [21] A. Ziegenbein, P. Hähner, H. Neuhäuser, Comput. Mater. Sci. 19 (2000) 27–34. [22] R. Shabadi, S. Kumara, H.J. Roven, E.S. Dwarakadasa, Mater. Sci. Eng. A 364 (2004) 140–150. [23] Q. Zhang, Z. Jiang, H. Jiang, Z. Chen, X. Wu, Int. J. Plast. 21 (2005) 2150–2173. [24] M.S. Bharathi, M. Lebyodkin, G. Ananthakrishna, C. Fressengeas, L.P. Kubin, Acta Mater. 50 (2002) 2813–2824. [25] F.L. Vernon, R.J. Pedersen, J. Appl. Phys. 39 (1968) 2661–2664. [26] M.A. Lebyodkin, Y. Estrin, Acta Mater. 53 (2005) 3403–3413.
[27] M.A. Lebyodkin, T.A. Lebedkina, Phys. Rev. E 77 (2008), 026111(8). [28] H. Ait-Amokhtar, S. Boudrahem, C. Fressengeas, Scr. Mater. 54 (2006) 2113–2118. [29] C. Fressengeas, A.J. Beaudoin, M. Lebyodkin, L.P. Kubin, Y. Estrin, Mater. Sci. Eng. A 400 (2005) 226–230. [30] S. Allain, J.-P. Chateau, D. Dahmoun, O. Bouaziz, Mater. Sci. Eng. A 387–389 (2004) 272–276. [31] S. Allain, J.-P. Chateau, O. Bouaziz, Mater. Sci. Eng. A 387–389 (2004) 143–147. [32] T.S. Byun, N. Hashimoto, K. Farell, Acta Mater. 52 (2004) 3889–3899. [33] S. Kok, A.J. Beaudoin, D.A. Tortorelli, M. Lebyodkin, Model. Simul. Mater. Sci. Eng. 10 (2002) 745–763. [34] A. Vinogradov, D.L. Merson, V. Patlan, S. Hashimoto, Mater. Sci. Eng. A 341 (2003) 57–73.