Plastic Flow Instability: Chernov Lüders Bands and the Portevin Le Chatelier Effect

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1 ISSN , Technical Physics, 2017, Vol. 62, No. 3, pp Pleiades Publishing, Ltd., Original Russian Text V.V. Gorbatenko, V.I. Danilov, L.B. Zuev, 2017, published in Zhurnal Tekhnicheskoi Fiziki, 2017, Vol. 87, No. 3, pp SOLID STATE Plastic Flow Instability: Chernov Lüders Bands and the Portevin Le Chatelier Effect V. V. Gorbatenko a *, V. I. Danilov a,b, and L. B. Zuev a,c a Institute of Strength Physics and Material Science, Siberian Branch, Russian Academy of Sciences, Akademicheskii pr. 2/4, Tomsk, Russia b National Research Tomsk Polytechnic University, Leninskii pr. 30, Tomsk, Russia c National Research Tomsk State University, Leninskii pr. 36, Tomsk, Russia * gvv@ispms.tsc.ru Received March 23, 2016 Abstract 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. DOI: /S INTRODUCTION Macroscopic inhomogeneities and instability of a plastic flow have been manifested on scales on the order of specimen length L as interconnected variations of stress strain curve σ() and pattern (spatiotemporal structure) of localized deformation. Among various localization patterns, the macroscopic waist [1] and the Chernov Lüders bands (CLBs) [1, 2], as well as the Portevin Le Chatelier (PLC) [3, 4] and Savart Masson effects [5] are the most familiar. The first two effects are spatial (localization) and the third and fourth effects are temporal (interrupted fluidity) inhomogeneities of the plastic flow process. Traditional explanations of the nonuniform strain regularities are based on dislocation models (see, e.g., [6 8]), which are distinguished by a compelled transition from the macroscale effect (~L) to the microscale of the Burgers dislocation vector b L. In this transition, some important macroscopic aspects of the effect are inevitably lost. According to the idea put forth in [9, 10], the localization of a plastic flow is a universal property of the deformation of materials of any origin and is observed during the entire process. The localization patterns with a characteristic scale λ < L are interpreted as modes of self-sustained waves (autowaves) of a localized plastic flow, the type of which is determined by the strain hardening stages [9]. For this reason, we must consider the above-mentioned variants of macroscopic instability of a plastic flow in terms of the concepts of autowave processes [11, 12], supplementing them with the concepts of self-organization and structuring of deformable media in the context of those proposed in [13]. For this reason, we will try to analyze the formation and evolution of the above instabilities using the autowave approach to the plasticity problem, which was developed in monograph [9]. MATERIALS AND EXPERIMENTAL PROCEDURES We studied the CLB evolution on hot-rolled steel (08 ps, % C, % Mn, and % Si) with a nearly ferrite structure and a mean grain size of ~20 μm. The PLC effect was studied on duralumin (naturally aged D1 alloy; % Cu, % Mn, and % Mg) with the grain size (~30 μm) of the Al-based solid solution and with strengthening precipitation of the submicrometer size. The composition of alloys is given in wt %. Specimens of the dog-bone shape with a working part size of mm were prepared from the sheets and then stretched at rate V mach = m/s (strain rate = s 1 ) on the Walter + Bai AG testing machine of series LFM-125 at 300 K. Typical stress strain σ() curves for 08 ps steel (with a sharp yield point and a plateau) and D1 alloy (serrated yield) are shown in Fig

2 396 GORBATENKO et al. σ, MPa (a) Interval of measurement σ, MPa (b) σ, MPa t, s Interval of measurement Fig. 1. Stress strain curves: (a) steel; (b) D1 alloy. Inset shows the shape of the sharp yield point on a magnified scale. The planar surface of specimens intended for observations was diffusively reflective, which was determined by the requirements of the experimental technique used. In this study, the visualization of plastic strain localization zones and the recording of their kinetics were performed using the method of digital statistical speckle photography [14] developed based on speckle photography [15]. The specimen subjected to tension was illuminated by the coherent radiation of a semiconductor laser with a wavelength of 635 nm and a power of 15 mw. The images of the specimen being deformed, which were obtained with such illumination, were recoded by PixeLink PL-B781 digital video camera with a frequency of 10 Hz, digitalized, and saved in the memory. For each point of the image, a sequence of counts that characterize the time variations of its brightness was formed, and the variance and mathematical expectation were calculated (the relation between these quantities were used for mapping of strain localization zones). Using this technique, it is possible to register the regions in which the deformation of the material is localized almost in situ for a preset increment of the total elongation of the specimen. On the photographs of specimens given below, these regions of strain localization have the form of narrow dark lines. Fig. 2. Nucleation and growth of a CLB nucleus. The time interval between recording of images is 7 s. 1 mm EVOLUTION OF THE CHERNOV LÜDERS BANDS Stress strain curves σ() for steel specimens (Fig. 1a) demonstrate the proportionality limit σ pl and the upper ( σ ( u) y ) and lower ( σ () l y ) yield stresses, as well as the yield plateau of length ~0.03, on which oscillations of the deforming stress can be seen. For a complete investigation of the nucleation and evolution of CLBs in steel, in experiments, we recorded speckle images beginning with stress σ σ pl < σ ( u) y as shown in Fig. 1a and terminated the recording at the end of the yield plateau and a transition to the strain-hardening stage. Experiments show that plastic deformation is localized first in the form of a CLB nucleus appearing at σ σ pl, i.e., at the stage of microplastic deformation. It was shown earlier in [16] that its formation is accompanied by acoustic emissions. It can be seen from Fig. 2 that a nucleus in the form of a narrow edge of the deformed material grows across the sample with a rate V nucl ( ) 10 3 m/s. In the σ() diagram, the growth in the nucleus corresponds to the formation of ascending and descending branches of the sharp yield point along the path σ pl σ ( u) y σ () l y. At the instant when the nucleus crosses the entire cross section of the specimen, the CLB formation terminates, and its expansion begins. This process corresponds to the yield plateau on the σ() curve at the level σ = σ () l y. The CLB formed is bounded by the pair of fronts propagating in the opposite direction along the specimen axis with velocities ±V f. The single fronts observed in some cases are usually associated with the CLB nucleation at the clamp of the testing machine, when the other front is not seen because it immediately leaves the observed part of the specimen almost. The velocity of propagation of CLB fronts is an important characteristic of the process. It turned out that during the nucleation of a single CLB, two its fronts move with almost the same velocity V f ±8

3 PLASTIC FLOW INSTABILITY mm Fig. 3. Propagation of two CLBs. Time interval between recording images is 60 s m/s so that 25V mach V f 0.1V nucl. As one of the fronts approaches to the machine clamp, its velocity decreases to zero, while the velocity of the other front increases to ( ) 10 4 m/s. In the simultaneous nucleation of two bands (Fig. 3), all four fronts first move with close but lower velocities of ±(3 5) 10 5 m/s. When fronts 1 and 4 approaching the clamps slow down and then stop, the remaining fronts 2 and 3 move towards each other with the velocity twice as high as the initial velocity. The slopes of the fronts to the axis of elongation of the specimen are ~π/3, but when the fronts approach the clamps, they usually turn to ~π/2 to the specimen axis. Therefore, the following rule is observed for CLB fronts in the samples being deformed for a preset value of V mach on the yield plateau as follows: () i N i= 1 () i V = V = const, f f (1) where V f is the modulus of the velocity of propagation of the ith CLB front, N is the number of simultaneously propagating fronts, and V f = m/s. Regularity (1) ensures the constancy of the growth rate of the area of the plastically deformed zone in the sample on the yield plateau holds, so that 10V mach V f 10 2 V mach ; i.e., 10 V f 10 2 [17, 18]. Vmach Digital statistical speckle photography has made it possible to obtain information on the details of the structure of CLB fronts and their motion. As shown in Fig. 4, different segments of fronts move with different velocities so that the front line can locally be distorted and split. In front of each edge, precursors with a configuration resembling a CLB nucleus can appear in the undeformed part of the sample. Expanding precursors form a new front at a distance of mm from the initial front, after which both fronts concordantly move in the same direction so that the trailing edge passes through the part of the sample, which has already been plastically deformed. The meeting of propagating fronts during the evolution of two CLBs in the specimen can follow one of the following two scenarios. In the first scenario, the fronts penetrate into the regions of adjacent bands; i.e., plastic information that previously occurred in one band continues for a certain time in the deformed region corresponding to the other band. In the second scenario, the zone between the fronts is fragmented with the formation of secondary fronts connecting the The relation between velocities V mach and V f can be determined from the equality of two estimates of the deformation time recorded in the form L/ V f δl/v mach, where δl is the absolute elongation of the specimen on the yield plateau. In this case, we have Vmach V = L f Vmach = > V δl pl mach, (2) where pl = δl/l is the length of the yield plateau in the units of strain. Usually, the inequality 10 2 pl Fig. 4. Evolution and splitting of the CLB front. Region of nucleation of a new front is indicated.

4 398 GORBATENKO et al X, mm X, mm Fig. 5. Sequential images of the movable localized deformation band in the PLC effect. Time interval between recording of images is 5 s mm/s 1.3 mm/s 1.5 mm/s t, s 400 Fig. 6. Chronogram of motion of localized deformation bands for serrated yield of the specimen of D1 aluminum alloy. DEVELOPMENT OF SERRATED YIELD Plastic deformation of the D1 alloy is characterized by the PLC effect [3, 4, 19]. Beginning from the yield stress, serrations of intermittent yield are observed on the entire σ() curve shown in Fig. 1b; the shape of serrations continuously evolves upon a decrease in the strain rate because of the elongation of the specimen. In the PLC effect, sharp yield points of types A, B, and C are sequentially observed on the σ() curve [20]. At the linear strain-hardening stage, sharp yield points of predominantly C type are formed in the D1 alloy in the strain interval [21], which appear on the σ() curve as single strong stress jumps (see the inset to Fig. 1b) and are associated with the formation of the morphologically simplest single localized deformation bands. For this reason, the recording and anal- primary fronts. This process is accompanied with the above-mentioned oscillations of the stress relative to level σ () l y on the yield plateau. In this case, the primary fronts continue their propagation over the previously deformed part of the specimen ysis of speckle images for the D1 alloy were carried out in this study precisely at this stage as shown in Fig. 1b. As in the case of CLBs, the PLC bands nucleate on the lateral surface of the specimen near the clamps or in the middle part of the working zone and then grow through the entire cross section of the specimen. However, nuclei in this case grow with the rate ~1 m/s, which is three orders of magnitude higher than the growth rate of CLB nuclei. The band formed after the growth of a nucleus contains the leading and trailing edges separated by a distance of ~1.7 2 mm from each other (Fig. 5) and is inclined to the specimen elongation axis by an angle of π/3. The band moves [22] in the self-similar way along the specimen to one of the clamps of the test machine with velocity ±V sb. During nucleation and motion of the band, a C-type sharp point of serrated yield appears on the σ() diagram, followed by an increase in the deforming stress. When the band reaches the clamp, the deforming stress falls again, and a new band is formed in the middle part of the specimen and moves over it in the direction opposite to that of the previous band. As this band also reaches the clamp, the above-described process is repeated so that each sharp serrated yield point corresponds to the run of a PLC band through half the working part of the specimen. Therefore, in the case of serrated yield, the PLC bands pass through the working part of the specimen many times; this is demonstrated on the chronogram of the process shown in Fig. 6. This chronogram can be used to estimate the magnitude of the velocity of bands, m/s V sb m/s, which is higher than the velocity of CLB fronts by more than an order of magnitude. The periodic motion of PLC fronts continues during strain hardening up to the beginning of the formation of the fracture waist. COMPARISON OF LOCALIZED STRAIN AT CLB FRONTS AND IN THE PLC EFFECT The main results of the observation of the kinetics of evolution of CLB and PLC fronts obtained in this study indicate the similarity and difference of these effects. With regard to the similarity, it should be noted above all that the CLB fronts and bands in the PLC effect are macroscopic manifestations of plastic strain localization, which are narrow moving zones in which the plastic flow is concentrated. In both cases, the kinetics of their formation is associated with the nucleation of localized plasticity on the lateral surface of the working part of the specimens, after which the nuclei grow through the entire cross section. Considerable differences in the behaviors of the processes under investigation are traced at the stage of developed localized deformation associated with CLBs and with the PLC effect. For example, the CLB evolution after the nucleation is the CLB expansion within the working part of the specimen. The entire

5 PLASTIC FLOW INSTABILITY 399 plastic deformation in this case is localized in fronts moving in opposite directions. The CLB fronts move independently, but so that rule (1) of the constancy of the sum of their velocity moduli is observed. If more than one CLB expand simultaneously in the specimen, their fronts can meet and interact. In the evolution of the PLC effect, plastic deformation at each instant exists in only one band in which plastic flow is completely localized so that the events of the meeting of the bands in serrated yield are ruled out. Finally, one more considerable difference between these effects is that the CLB fronts can run through the sample only once, while PLC bands multiply pass through the working length of the specimen. The reasons for this difference are clarified by the autowave model of a plastic flow [9], which introduces an intermediate autowave scale b < λ < L. In constructing this model, a natural assumption has been made that the medium being deformed is active (i.e., contains energy sources distributed over the volume) [11, 13, 23]. In analysis of plasticity, it is expedient to consider elastic stress raisers as sources [10] that can relax with the nucleation of defects in the crystal structure. The properties of the active medium that correspond to specific materials (including those used in our experiments) are determined by the structure of raisers and the kinetics of their relaxation and may change in the course of deformation. Let us try to associate the differences in the CLB and PLC effects with the possibility of the existence of two types of active media that can generate different autowave modes of localized plasticity. In the case of CLB, in accordance with the familiar microscopic models [1, 2], we assume that elements of the active medium on a movable front pass from the elastic to the plastically deformed state due to relaxation of stress raisers. In the general theory of autowave processes [11], this medium is referred to as bistable, and the perturbation front in it is a switching autowave in the system that consists of bistable elements [11, 23]. The latter can exist in the metastable (elastic) or stable (plastically deformed) state. The irreversibility of the switching event of the states, i.e., transitions from the metastable to stable state, excludes the repeated passage of CLB fronts through the same region of the specimen. In other words, the energy spent on the generation of autowaves in the bistable active medium is not restored [23]. For this reason, the general rule for switching autowaves is the annihilation of their fronts upon meeting [11, 23], which is observed quite frequently on the yield plateau during deformation. In the case of the PLC effect, we can assumed from the data on microscopic mechanisms of the phenomena [3, 8, 19] that another sequence of changes in the state of active media elements takes place during the motion of a localized plasticity band. The difference in the behavior of the material at the leading and trailing edges of the localized deformation band is of fundamental importance. We can assume that the leading edge of the band is associated with switching elements in a way analogous to that described above. However, to explain the periodic generation of a new deformation band in the PLC effect (unlike in the CLB) in the specimen, we must assume that the relaxed elements at the trailing edge of the propagating band do not return to the initial state but pass to a third state in which they lose the ability to change for a certain time. In view of the irreversibility of the plastic flow, this transition cannot be the reverse transformation, which leads to the complete recovery of the medium to the initial state after the passage of the band. Active media of this type are well known in the theory of autowave processes and are referred to as excitable [23]. These media can restore the energy spent on generating an autowave and consist of excitable elements that can exist in states of rest, excitation, and refractoriness. In these media, the generation and propagation of single pulses with steep leading and trailing edges, a slowly declining plateau between them, and slow relaxation to the initial state is possible [23]. In fact, this pulse is equivalent to a pair of conjugate switching fronts. At the leading edge of the pulse, the transition of excitable elements from the metastable state to the excited state occurs. At the trailing edge of the pulse, the elements try to restore the initial situation, which, however, is impossible in view of the changed state of the medium. The process terminates at the refractoriness stage characterized by period τ ref during which the elements of the medium are not sensitive to external action. The new quality of the active medium appears in this case because the excitation of the elements of the medium, which ruled out for t < τ ref, becomes possible again for t > τ ref. The excitation emerging in this medium is known as the excitation autowave, which is a pulse that multiply traverses the system [23]. This is obviously true of the localized deformation band in the PLC effect, which can serve as an example of the deformation excitation autowave that multiply propagates through the specimen. Developing the above arguments based on generally accepted concepts [24], let us consider some aspects concerning the microscopic nature of bistable and excitable elements as applied to deformation involving the CLB mechanisms and the PLC effect. In the case of CLB, metastable elements on the band front are elastically stressed crystallites, while stable elements are grains in which plastic deformation has been already initiated [1, 2]. The transition of elements from the metastable to the stable state is irreversible in this case because the sharp yield point disappearing after preliminary deformation is restored only after annealing [1, 24]. In the deforming D1 alloy under strain, an active medium that consists of excitable elements is formed. As these elements, we can consider local regions asso-

6 400 GORBATENKO et al. ciated with thermally activated overcoming of dispersed precipitates by movable dislocations [24]. The restoration of the distributed energy sources (raisers) in this medium can be associated with an increase in the deforming stress under strain hardening. Estimating τ ref, we can formally attach to it the meaning of the interval between sequential events of nucleation of localized plasticity bands, which is equal to L/2V 15 s. Assuming that the restoration of the initial configuration of obstacles to dislocation slip is a thermally activated process, we can write the following expression for refractoriness time τ ω 1 ref D exp U (3) kt, B where U 0.8 ev is the height of the potential barrier, which corresponds to the energy of interaction of dislocations with local precipitates in the Al Cu alloy [24], ω D is the Debye frequency, k B is the Boltzmann constant, and T is the temperature. The calculation for k B T 1/40 ev gives τ ref 15 s, which coincides with the above estimate of this quantity. Thus, CLBs and deformation in the PLC effect are similar to a considerable extent (at least, at the nucleation stage). The sharp yield point in the case of CLB and the serrations in the PLC effect are also similar. However, in the course of further deformation, the alloys exhibit different behaviors due to the fact that the alloys in the PLC effect becomes harder, and the directions of propagation of the bands become nonequivalent, while the propagation of CLBs corresponds to the yield plateau on which there is no strain hardening and the directions of motion of the fronts are equivalent. CONCLUSIONS It has been shown that CLBs and deformation bands in the PLC effect nucleate in deformable media in accordance with the same mechanism by the intergrowth of the nucleus formed on the lateral surface of the specimen through the entire cross section. It has been established that the velocities of propagation of CLB fronts and the deformation bands in the PLC effect differ by an order of magnitude. The VLB fronts and deformation bands in the PLC effect are switching and excitation autowaves, respectively, which emerge in active deformable media of different origins. ACKNOWLEDGMENTS This study was performed under the Program of Fundamental Research of the State Academies of Sciences of the Russian Federation for and is supported in part by the Russian Foundation for Basic Research (project no ). Experiments were performed on the equipment of the Collective Usage Center Nanotech at the Institute of Strength Physics and Material Science, Siberian Branch, Russian Academy of Sciences REFERENNCES 1. J. Pelleg, Mechanical Properties of Materials (Springer, Dordrecht, 2013). 2. P. Hähner, Appl. Phys. A 58, 41 (1994). 3. E. Rizzi and P. Hähner, Int. J. Plast. 20, 121 (2004). 4. M. Zaiser and E. C. Aifantis, Int. J. Plast. 22, 1432 (2006). 5. J. F. Bell, Experimental Foundations of Solid Mechanics, Vol. 1 of Mechanics of Solids, Ed. by C. Truesdell (Springer, New York 1973; Nauka, Moscow, 1984). 6. G. A. Malygin, Phys. Solid State 53, 363 (2011). 7. H. B. Sun, F. Yoshida, and X. Ma, Mater. Lett. 57, 4535 (2003). 8. A. A. Shibkov, A. E. Zolotov, and M. A. Zheltov, Izv. Ross. Akad. Nauk, Ser. Fiz. 76, 97 (2012). 9. L. B. Zuev, V. I. Danilov, and S. A. Barannikova, Physics of Macrolocalization of a Plastic Flow (Nauka, Novosibirsk, 2008). 10. L. B. Zuev, Izv. Ross. Akad. Nauk, Ser. Fiz. 78, 1206 (2014). 11. V. A. Vasil ev, Yu. M. Romanovskii, and V. G. Yakhno, Autowave Processes (Nauka, Moscow, 1987). 12. V. A. Davydov, N. V. Davydov, V. G. Morozov, M. N. Stolyarov, and T. Yamaguchi, Condens. Matter Phys. 7, 565 (2004). 13. A. Seeger and W. Frank, Non-Linear Phenomena in Material Science (Trans. Tech. Publ., New York, 1987), pp L. B. Zuev, V. V. Gorbatenko, and K. V. Pavlichev, Meas. Sci. Technol. 21, (2010). 15. R. Jones and C. Wykes, Holographic and Speckle Interferometry (Cambridge Univ. Press, Cambridge, 1983; Mir, Moscow, 1986). 16. T. V. Murav ev and L. B. Zuev, Tech. Phys. 53, 1094 (2008). 17. O. A. Plekhov, O. B. Naimark, N. Saintier, and T. Palin-Luc, Tech. Phys. 54, 1141 (2009). 18. Yu. V. Petrov and I. N. Borodin, Phys. Solid State 57, 353 (2015). 19. M. M. Krishtal, Phys. Met. Metallogr. 92, 293 (2001). 20. L. J. Cuddy and W. C. Leslie, Acta Met. 20, 1157 (1972). 21. V. I. Danilov, A. V. Bochkareva, and L. B. Zuev, Phys. Met. Metallogr. 107, 616 (2009). 22. G. I. Barenblatt, Automodel Phenomena: Dimensional Analysis and Scaling (Intellekt, Dolgoprudnyi, 2009). 23. V. I. Krinskii and A. M. Zhabotinskoi, Autowave Processes in Systems with Diffusion (Inst. Prikl. Fiz. Akad. Nauk SSSR, Gor kii, 1981), pp M. A. Shtremel, Strength of Alloys, Part 2: Deformation (Metallurgiya, Moscow, 1997). Translated by N. Wadhwa