- Email: [email protected]
Power-law scaling behaviors for shear band intersections in Zr64.13Cu15.75Ni10.12Al10 bulk metallic glass
Journal of Non-Crystalline Solids xxx (xxxx) xxx–xxx
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol
Power-law scaling behaviors for shear band intersections in Zr64.13Cu15.75Ni10.12Al10 bulk metallic glass Bo Shia,⁎, Yuanli Xub, Fuan Weia, Shiyu Luana, Peipeng Jina,⁎ a Qinghai Provincial Key Laboratory of New Light Alloys, Qinghai Provincial Engineering Research Center of High Performance Light Metal Alloys and Forming, Qinghai University, Xining 810016, China b Institute of Materials Science and Engineering, Lanzhou University, Lanzhou 730000, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Bulk metallic glass Compressive deformation Shear band intersection Power-law
At the large compressive deformation levels, shear band offsets at intersection sites and the stress drop magnitudes of serrated flow follow power-law distributions for Zr64.13Cu15.75Ni10.12Al10 bulk metallic glass. In addition, it was revealed that shear band interactions evolve into a process of large degree cooperation of many shear bands via investigating the relation between shear band intersections and serrated flow. To clarify the evolution rules of shear band intersections, quantitatively statistical works were performed. With increasing the plastic strain, the distribution of shear band orientations becomes wider and displays a multi-peak distribution. The average shear band spacing varies as a power function of plastic strain and decreases to a very low level in large deformation stages. Small spacing and arbitrary orientations of shear bands greatly enhance the intersecting probability and lead to the density of shear band intersections varying as a power function of plastic strain. Enormous amount of undeformed regions enclosed by shear bands provide many places for shear bands intersecting.
1. Introduction At low temperatures and high strain rates, the plastic deformation of metallic glasses is highly localized into shear bands [1]. Because of work softening, a single shear band usually propagates rapidly and results in catastrophic failure. Hence, the macroscopic plasticity of metallic glasses is very low [2]. Some constrained loading conditions can make metallic glasses undergo significant plastic deformation via the formation of multiple shear bands [3,4]. In the aspect of plastic deformation mechanism, with the number of shear bands increasing during plastic deformation, the intersections of shear band will play a more remarkable role, especially in the large deformation stage [5]. Besides, in the aspect of mechanical properties, enhancing the shear band numbers can increase the plasticity [6], and further lead to the number of intersections increasing. It is found that the mechanical performances of metallic glasses are strongly correlated to shear band intersections. For example, strong shear band intersections can lead to geometric hardening behaviors in Cu47.5Zr47.5Al5, Zr64.13Cu15.75Ni10.12Al10 and Pd77.5Cu6Si16.5 bulk metallic glasses (BMGs) [7–10]. Thus, it is much more significant to investigate the evolution of shear band intersections. It may also provide some insights for understanding the plastic deformation mechanism and toughening
⁎
BMGs. The variation of the number of shear band intersections with plastic strain has been studied previously. The rapid increase of shear band intersections induced hardening, after counteracting free volume induced softening, may lead to an increase in hardness [11]. The density of shear band intersections and the shear band offsets at intersection sites can reflect the degree of interactions between shear bands. However, the evolution laws of them have not been directly and quantitatively studied so far. In addition, theoretical calculations indicate that strong shear band interactions have great influences on serrated flow behaviors [12,13]. For ductile BMGs, stress drop magnitudes of serrations in strain-stress curve follow power-law distribution, which can be attributed to strong shear band interactions. However, in different deformation stages, the relation between shear band intersections and serrated flows remains to be clarified by further experiments. In the present work, Zr64.13Cu15.75Ni10.12Al10 BMG was compressed to different plastic strains. The evolutions of shear band spacing, shear band orientations, and shear band offsets at intersection sites have been quantitatively studied. The average shear band spacing and the density of shear band intersections vary as power functions of plastic strain. In the large deformation stage, shear band orientations display a multiplepeak distribution, shear band offsets and serrated flows follow power-
Corresponding authors. E-mail addresses: [email protected] (B. Shi), [email protected] (P. Jin).
http://dx.doi.org/10.1016/j.jnoncrysol.2017.07.005 Received 8 May 2017; Received in revised form 28 June 2017; Accepted 4 July 2017 0022-3093/ © 2017 Elsevier B.V. All rights reserved.
Please cite this article as: Shi, B., Journal of Non-Crystalline Solids (2017), http://dx.doi.org/10.1016/j.jnoncrysol.2017.07.005
Journal of Non-Crystalline Solids xxx (xxxx) xxx–xxx
B. Shi et al.
stages, shear bands are separate and free, and the shear band intersections are very few. For the sample with a plastic strain of 83% (Fig. 1b), a large number of fine shear bands form and their orientations are widely distributed. These shear bands form a homogeneous network pattern. Uniformly distributed shear band intersections were observed, indicating severe interactions between shear bands. Shear bands with intersections divide the deformed BMG into two components: one is shear bands, and the other is undeformed amorphous regions enclosed by shear bands with intersections. To quantitatively investigate the evolution of shear band orientations and spacing, the orientation angles of shear bands were statistically analyzed. Fig. 2a–c show the histograms of orientation angles. For the deformed sample with a plastic strain of 6%, the orientation angles are mainly centered at around φ = 45° (Fig. 2a), which are in agreement with the maximum shear stress direction (~ 45° from the load axis). With the plastic strain increasing, the orientation angles distribute in a wider range (Fig. 2b and c). For the deformed sample with a plastic strain of 83%, the orientation angles display a multi-peak distribution. The orientation peaks in the distribution histograms can be identified at around φ = − 30°, 0° and 30°, respectively. Interestingly, the absolute values of orientation peaks for the sample with a plastic strain of 83% are not in agreement with the maximum shear stress direction. It implies that the formation and propagation of numerously new fine shear bands are strongly depended on inhomogeneous local stress states. Firstly, with increasing plastic strain, a large amount of uniformly distributed shear band intersections can hinder shear band propagation and further alter their propagating directions [7]. Secondly, shear bands have long range stress field whose magnitude is close to the yield stress of BMG itself [14]. With shear band spacing decreasing, a strongly overlapping internal stress fields form. Then the overlapping internal stress fields can help to trigger the formation of different orientated shear bands. They can also limit shear band propagation and further alter the propagating directions of shear bands. Finally, with the plastic strain increasing, the ratio of height/width for the deformed sample was reduced. In this case, the lateral constraints from friction may increase the complexity of the stress states [15]. Hence, the complex local stress states make the orientations of shear bands deviate the maximum shear stress direction and distribute more widely. In summary, with the plastic strain increasing, the distribution range of orientation angles is more extensive, and the orientation angle distributions gradually exhibit a multi-peak distribution. These features would promote intersecting probability. In addition, the change of the average shear band spacing with plastic strain was quantitatively analyzed. Fig. 2d shows that the measured average shear band spacing is logarithmically plotted versus plastic strain. As can be seen, the average shear band spacing decreases linearly with plastic strain in log-log plot. It indicates that the average shear band spacing correlates the plastic strain via a power-law relation d ~ εp− λ in ductile Zr64.13Cu15.75Ni10.12Al10 BMG, where the index λ = 1.22 (R2 = 0.981). The power-law relation means that shear band spacing can reduce to a very low level in the large deformation stages. This may lead to strong cooperative interactions of multiple shear bands in a short spatial range. Moreover, the index value in the present
law distributions. 2. Experimental procedure Master alloy ingots with nominal composition of Zr64.13Cu15.75Ni10.12Al10 were prepared by arc-melting the mixtures of highly pure Zr (99.8%), Cu (99.99%), Ni (99.99%), and Al (99.99%) in a Ti-gettered argon atmosphere. Each ingot was remelted more than six times to ensure chemical homogeneity. BMG rods with a square cross section of 2 × 2 mm2 were prepared by arc-melting master alloy ingot and subsequent suction-casting into a water-cooled copper mold. The compression test specimens were cut from the as-cast rod. Before compression test, both ends of these specimens were polished to be parallel. The compression tests were performed on a Shijin WDW-100D universal testing machine. In all compression tests, strain rate and aspect ratio are 1 × 10− 3 s− 1 and 1, respectively. The plastic strain εp was denoted as εp = (h0 − h) / h0, where h0 is the original height, and h is the height after deformation. The shear band patterns were analyzed by scanning electron microscopy (SEM) using a Hitachi S-4800 field emission scanning electron microscope operating at an acceleration voltage of 5 kV. The shear band spacing was determined by the following steps: (i) SEM images were taken from thirty random regions for each sample. (ii) A line paralleled to the loading axis was drawn on each SEM image, and this line intersected with shear bands. (iii) The distance between two adjacent crosspoints of this line and shear bands is considered as the shear band spacing. (iv) The total line length was divided by the total number of crosspoints, and this obtained result is regarded as the average shear band spacing. The density of shear band intersections was determined by the following steps: (i) The number of shear band intersections in ten observation regions (each region has an equal area of 113 μm2) for one sample were counted. (ii) The plane density of intersections for each region was calculated. (iii) The density of shear band intersections is a mean value of all densities for each observation regions. The orientation angle (φ, the angle between an observed shear band line on side surface and horizontal line, ranging from −90° to 90°) distribution was obtained by the following steps: (i) A line paralleled to the loading axis was drawn on each SEM image (ten images were counted for each sample). (ii) Orientation angles of shear bands located on the line were measured. (iii) The frequency of orientation angles who fall into the interval of (φ – δφ / 2, φ + δφ / 2) was counted, and the distribution histogram of orientation angles ranging from − 90° to 90° was plotted. 3. Results and discussion To investigate the shear band evolution with plastic strain, the SEM observations and statistical works were performed. The representative SEM images of shear bands for deformed Zr64.13Cu15.75Ni10.12Al10 BMG samples with plastic strains of 6% and 83% are shown in Fig. 1, respectively. Only a few shear bands formed in the deformed sample with a plastic strain of 6% (Fig. 1a). The angle between the observed shear band and loading axis is about 47°. Obviously, in small deformation
Fig. 1. SEM images of shear bands for the deformed Zr64.13Cu15.75Ni10.12Al10 BMG samples with plastic strains of 6% (a) and 83% (b).
2
Journal of Non-Crystalline Solids xxx (xxxx) xxx–xxx
B. Shi et al.
Fig. 2. (a-c) Histograms of orientation angles for the deformed samples with plastic strains of 6%, 47% and 83%. (d, e) The average shear band spacing d and the density of shear band intersections ND are logarithmically plotted versus plastic strain εp. Each data point corresponds to the mean value and the error bar indicates the calculated standard deviation. (f) The ratios of shear bands with spacing below 2 μm, 1 μm and 0.5 μm for the deformed samples with different plastic strains.
density of shear band intersections (ND) is a power function of plastic strain εp, ND ~ εpβ, where β = 1.32 (R2 = 0.963). It indicates that shear band intersections will dramatically increase in large plastic deformation stages. Fig. 2f shows the ratios of shear bands with small spacing for the deformed Zr64.13Cu15.75Ni10.12Al10 BMG with different plastic strains. Obviously, the percentages of shear bands with small spacing (≤ 2 μm, 1 μm and 0.5 μm) increase rapidly with plastic deformation (Fig. 2f). Additionally, among all the shear bands with spacing below 2 μm, the percentage of shear bands with spacing at sub-micron level (≤ 1 μm) accounted for the majority. This reveals that the sizes of undeformed amorphous regions enclosed by shear bands with intersections become gradually smaller with plastic deformation. The density of shear band intersections and shear band offsets (When a shear band cut through another one, shear displacement on the side surface will be produced at the intersection sites [4,16], as denoted by arrow in Fig. 1b. The shear displacement is defined as shear band offset can reflect the degree of interactions between shear bands. According to the above results, the density of shear band intersections increases greatly at the large deformation levels. To further investigate the shear band interactions, shear band offsets were statistically studied. Fig. 4a–c shows the distribution histograms of shear band offsets in the deformed BMG samples. As can be seen, the narrowing of shear band offset distribution with increasing the plastic strain is gradually clear. The shear band offsets shift to small value, indicating that plastic deformation is mainly carried by the formation and sliding of multiple shear bands, rather than a few dominating shear bands. Moreover, the density of shear band intersections greatly increases with increasing plastic strain. At the large deformation levels (εp ≥ 47%), the distribution histograms display a monotonically decreasing long-tailed distribution, which is often subjected to power-law relation. According to the SEM and statistical results (Figs. 1 and 2), the features of homogeneous distribution, small spacing and various orientations for shear bands enhance the opportunity for shear bands intersecting with each other. Therefore, uniformly distributed shear band intersections were observed at the large deformation level (Fig. 1b). This indicates that the shear band interactions evolve into a process of large degree cooperation of many shear bands. To further quantitatively analyze the distribution histograms of shear band offsets, the probability distribution density of shear band offsets, P(s) = (1 / Ns)[δNs(s) / δs], was calculated. Here, s is the shear band offset, δNs is the number of the shear band offsets who fall into the interval of (s – δs / 2, s + δs / 2), and Ns is the total number of the shear band offsets. Fig. 4d and e shows the probability distribution density P (s) versus shear band offset s for the deformed Zr64.13Cu15.75Ni10.12Al10 BMG samples with plastic strains of 47% and 83%. As can be seen, the P
Fig. 3. The determined values of the shear band spacing in the present work are compared with the data in Refs [4,16]. The dashed line indicates that the variation of shear band spacing with plastic strain displays a monotonically decreasing trend.
work is lower than that in Zr64.13Cu15.75Ni10.12Al10 BMG sample with an aspect ratio of 2 [5], indicating that the dynamics of plastic deformation may be greatly affected by the aspect ratio. However, under the same aspect ratio of 1, the index λ = 1 for the relatively brittle Zr52.5Cu17.9Ni14.6Al10Ti5 BMG [16]. It indicates that the cooperative interactions between shear bands in brittle BMG may be much weaker than that in plastic BMG. It is found that Zr64.13Cu15.75Ni10.12Al10 BMG can undergo super plastic deformation [5]. Previous works demonstrate that a higher Poisson's ratio can lower the barrier energy density for activation of shear transformation zones (STZs) and further promote the nucleation rate of shear bands [17,18]. Here, the Poisson's ratio of Zr64.13Cu15.75Ni10.12Al10 BMG is 0.377, higher than 0.370 for Zr52.5Cu17.9Ni14.6Al10Ti5 BMG [5,19]. Therefore, the nucleation rate of shear bands in plastic Zr64.13Cu15.75Ni10.12Al10 BMG may be higher than that in the Zr52.5Cu17.9Ni14.6Al10Ti5 BMG, and thus the shear band spacing in the Zr64.13Cu15.75Ni10.12Al10 BMG is much lower (shown in Fig. 3). As can be seen from Fig. 3, the dashed line indicates that the variation of shear band spacing with plastic strain displays a monotonically decreasing trend for Zr-based BMGs. In large deformation stages, small shear band spacing makes shear bands easily interact with each other in a short spatial range. Therefore, the probability of interactions becomes larger. As shown in Fig. 2e, the 3
Journal of Non-Crystalline Solids xxx (xxxx) xxx–xxx
B. Shi et al.
Fig. 4. (a–c) The distribution histograms of shear band offsets produced at the shear band intersection sites for the deformed samples with plastic strains of 6%, 47%, and 83%. The solid lines in (b) and (c) indicate monotonously decreasing trends. (d, e) The log-log plots of the probability distribution density P(s) versus shear band offset s for the deformed Zr64.13Cu15.75Ni10.12Al10 BMG samples with plastic strains 47% and 83% fitted by power-law relations with α = 1.17 and 1.69.
(s) can be well described by the power-law relations P(s) ~ s− α with exponents α = 1.17 and 1.69 (R2 = 0.907 and 0.910), respectively. The power-law relation is often an indicator of the self-organized critical (SOC) state, which means a system can buffer against large changes throughout the cooperation of its participants with strong interactions [12,13,20]. Hence, the plastic deformation process of the Zr64.13Cu15.75Ni10.12Al10 BMG evolves into a SOC state when plastic strain is larger than or equal to 47%. The emergence of the SOC state suggests that there are strong interactions between the multiple shear bands. In addition, the power exponent α increases with plastic strain, indicating that the system of shear band interactions in the Zr64.13Cu15.75Ni10.12Al10 BMG becomes more complex and the degree of shear band interactions becomes stronger. Theoretically, serrated flow behavior is closely related to shear band interactions. Stress drop magnitudes of serrations in stress-strain curve follow power-law distribution when considering the strong interactions between shear bands for ductile Zr64.13Cu15.75Ni10.12Al10 BMG [12]. Fig. 5a shows the compressive engineering stress-strain curve of Zr64.13Cu15.75Ni10.12Al10 BMG. The regions A and B respectively represent initial stage and larger deformation stage, and their enlargements are shown in Fig. 5b and c. As can be seen, the serrations of region A display relatively uniform sizes, but the serrations of region B display various sizes and a more complex pattern. The average
magnitudes of stress drop are about 11.4 and 4.4 MPa for regions A and B, respectively. Fig. 5d and e shows the frequency histograms of the stress drop magnitude for regions A and B and their nearby regions. As can be seen, the histogram exhibits a monotonically decreasing trend in the large deformation stage. And the solid line in Fig. 5e indicates a fitting curve with power function. This further implies that shear band interactions become remarkable and evolve into a SOC state with plastic deformation. Such phenomenon has also been discovered in Portevin-Le Chatelier (PLC) effect of CuAl crystalline alloy, where a crossover from chaotic state to SOC state happened [21]. BMGs have no dislocation-like defects, and their plastic deformation is carried by shear bands. In small deformation stage, very few shear bands formed and propagated. Due to lacking of shear band interactions, the continuously regular serration pattern is similar to the serrations caused by one single shear band [2]. In large deformation stage, shear band intersections increase greatly. Shear band intersections can hinder shear band propagation, therefore the sliding of shear band may become harder. Due to the hindering effect from a large number of shear bands with cooperative interactions, the serration pattern becomes more complex and the average stress drop becomes smaller. Actually, in the present work, shear bands are confined within small space, and these shear bands can easily interact with many other shear bands. At the large deformation levels, shear band offsets and serrated flows
Fig. 5. (a) The compressive engineering stress-strain curve of Zr64.13Cu15.75Ni10.12Al10 BMG. The regions marked with A and B represent different deformation stages. (b, c) The enlargements of regions A and B in (a). Frequency histograms of the stress drop magnitudes for boxes A (d) and B (e) and their nearby regions.
4
Journal of Non-Crystalline Solids xxx (xxxx) xxx–xxx
B. Shi et al.
that shear band interactions evolve into a process of large degree cooperation of many shear bands at the large deformation levels. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 51661028). References [1] A.S. Argon, Plastic deformation in metallic glasses, Acta Metall. 27 (1979) 47–58. [2] J. Pan, Q. Chen, L. Liu, Y. Li, Softening and dilatation in a single shear band, Acta Mater. 59 (2011) 5146–5158. [3] Z.F. Zhang, H. Zhang, X.F. Pan, J. Das, J. Eckert, Effect of aspect ratio on the compressive deformation and fracture behaviour of Zr-based bulk metallic glass, Philos. Mag. Lett. 85 (2005) 513–521. [4] J.W. Liu, Q.P. Cao, L.Y. Chen, X.D. Wang, J.Z. Jiang, Shear band evolution and hardness change in cold-rolled bulk metallic glasses, Acta Mater. 58 (2010) 4827–4840. [5] Y.H. Liu, G. Wang, R.J. Wang, D.Q. Zhao, M.X. Pan, W.H. Wang, Super plastic bulk metallic glasses at room temperature, Science 315 (2007) 1385–1388. [6] L. He, M.B. Zhong, Z.H. Han, Q. Zhao, F. Jiang, J. Sun, Orientation effect of preintroduced shear bands in a bulk-metallic glass on its “work-ductilising”, Mater. Sci. Eng. A 496 (2008) 285–290. [7] J. Das, M.B. Tang, K.B. Kim, R. Theissmann, F. Baier, W.H. Wang, J. Eckert, “Workhardenable” ductile bulk metallic glass, Phys. Rev. Lett. 94 (2005) 205501. [8] Q.P. Cao, J.W. Liu, K.J. Yang, F. Xu, Z.Q. Yao, A. Minkow, H.J. Fecht, J. Ivanisenko, L.Y. Chen, X.D. Wang, S.X. Qu, J.Z. Jiang, Effect of pre-existing shear bands on the tensile mechanical properties of a bulk metallic glass, Acta Mater. 58 (2010) 1276–1292. [9] S. Takayama, Drawing of Pd77.5Cu6Si16.5 metallic glass wires, Mater. Sci. Eng. 38 (1979) 41–48. [10] S. Takayama, Serrated plastic flow in metallic glasses, Scr. Metall. 13 (1979) 463–467. [11] B. Shi, Y. Xu, C. Li, W. Jia, Z. Li, J. Li, Evolution of free volume and shear band intersections and its effect on hardness of deformed Zr64.13Cu15.75Ni10.12Al10 bulk metallic glass, J. Alloys Compd. 669 (2016) 167–176. [12] B.A. Sun, H.B. Yu, W. Jiao, H.Y. Bai, D.Q. Zhao, W.H. Wang, Plasticity of ductile metallic glasses: a self-organized critical state, Phys. Rev. Lett. 105 (2010) 035501. [13] B.A. Sun, S. Pauly, J. Tan, M. Stoica, W.H. Wang, U. Kühn, J. Eckert, Serrated flow and stick-slip deformation dynamics in the presence of shear-band interactions for a Zr-based metallic glass, Acta Mater. 60 (2012) 4160–4171. [14] R. Maaß, P. Birckigt, C. Borchers, K. Samwer, C.A. Volkert, Long range stress fields and cavitation along a shear band in a metallic glass: the local origin of fracture, Acta Mater. 98 (2015) 94–102. [15] Z. Wang, J.W. Qiao, G. Wang, K.A. Dahmen, P.K. Liaw, Z.H. Wang, B.C. Wang, The mechanism of power-law scaling behavior by controlling shear bands in bulk metallic glass, Mater. Sci. Eng. A 639 (2015) 663–670. [16] H. Bei, S. Xie, E.P. George, Softening caused by profuse shear banding in a bulk metallic glass, Phys. Rev. Lett. 96 (2006) 105503. [17] J.J. Lewandowski, W.H. Wang, A.L. Greer, Intrinsic plasticity or brittleness of metallic glasses, Philos. Mag. Lett. 85 (2005) 77–87. [18] Y.H. Liu, K. Wang, A. Inoue, T. Sakurai, M.W. Chen, Energetic criterion on the intrinsic ductility of bulk metallic glasses, Scr. Mater. 62 (2010) 586–589. [19] W.H. Wang, The elastic properties, elastic models and elastic perspectives of metallic glasses, Prog. Mater. Sci. 57 (2012) 487–656. [20] P. Bak, C. Tang, K. Wiesenfeld, Self-organized criticality: an explanation of the 1/f noise, Phys. Rev. Lett. 59 (1987) 381. [21] G. Ananthakrishna, S.J. Noronha, C. Fressengeas, L.P. Kubin, Crossover from chaotic to self-organized critical dynamics in jerky flow of single crystals, Phys. Rev. E 60 (1999) 5455.
Fig. 6. The schematic illustration of shear band intersections at the large deformation levels for Zr64.13Cu15.75Ni10.12Al10 BMG.
behaviors follow power-law distribution, which is typical characteristic for a system containing strong interacting constituents. Based on the above results, the schematic illustration of shear band intersections was drawn in Fig. 6. In large deformation stage, previously formed shear bands have endured repeated sliding and finally developed into coarse shear bands. Subsequently, new shear bands formed in the vicinity of previously coarse shear bands. This leads to the sizes of undeformed amorphous regions enclosed by shear bands with intersections becoming smaller. Thus shear bands only have small space to propagate in the large plastic deformation stage. So the shear bands are not free, and they will be restricted within the grid formed by previous shear bands. Thus, the new fine shear band can easily intersect with each other in a small space, resulting in a great increase of the intersection density, and shear bands evolve into a process of large degree cooperation of many shear bands. 4. Conclusions In conclusion, Zr64.13Cu15.75Ni10.12Al10 BMG was compressed to different plastic strain. Shear band spacing and the density of shear band intersections are found to vary as power functions of plastic strain (d ~ εp− 1.22 and ND ~ εp1.32, respectively). With increasing the plastic strain, the distribution of shear band orientations become wider, the orientation angles display a multi-peak distribution. Small spacing and arbitrary orientations of shear bands enhance the intersecting probability. At the large deformation levels, the density of shear band intersections increases greatly. The interactions of shear bands become stronger due to the reason that shear bands are restricted within the network formed by previously coarse shear bands. The shear band offsets and serrated flows display power-law distributions. It indicates
5
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol
Power-law scaling behaviors for shear band intersections in Zr64.13Cu15.75Ni10.12Al10 bulk metallic glass Bo Shia,⁎, Yuanli Xub, Fuan Weia, Shiyu Luana, Peipeng Jina,⁎ a Qinghai Provincial Key Laboratory of New Light Alloys, Qinghai Provincial Engineering Research Center of High Performance Light Metal Alloys and Forming, Qinghai University, Xining 810016, China b Institute of Materials Science and Engineering, Lanzhou University, Lanzhou 730000, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Bulk metallic glass Compressive deformation Shear band intersection Power-law
At the large compressive deformation levels, shear band offsets at intersection sites and the stress drop magnitudes of serrated flow follow power-law distributions for Zr64.13Cu15.75Ni10.12Al10 bulk metallic glass. In addition, it was revealed that shear band interactions evolve into a process of large degree cooperation of many shear bands via investigating the relation between shear band intersections and serrated flow. To clarify the evolution rules of shear band intersections, quantitatively statistical works were performed. With increasing the plastic strain, the distribution of shear band orientations becomes wider and displays a multi-peak distribution. The average shear band spacing varies as a power function of plastic strain and decreases to a very low level in large deformation stages. Small spacing and arbitrary orientations of shear bands greatly enhance the intersecting probability and lead to the density of shear band intersections varying as a power function of plastic strain. Enormous amount of undeformed regions enclosed by shear bands provide many places for shear bands intersecting.
1. Introduction At low temperatures and high strain rates, the plastic deformation of metallic glasses is highly localized into shear bands [1]. Because of work softening, a single shear band usually propagates rapidly and results in catastrophic failure. Hence, the macroscopic plasticity of metallic glasses is very low [2]. Some constrained loading conditions can make metallic glasses undergo significant plastic deformation via the formation of multiple shear bands [3,4]. In the aspect of plastic deformation mechanism, with the number of shear bands increasing during plastic deformation, the intersections of shear band will play a more remarkable role, especially in the large deformation stage [5]. Besides, in the aspect of mechanical properties, enhancing the shear band numbers can increase the plasticity [6], and further lead to the number of intersections increasing. It is found that the mechanical performances of metallic glasses are strongly correlated to shear band intersections. For example, strong shear band intersections can lead to geometric hardening behaviors in Cu47.5Zr47.5Al5, Zr64.13Cu15.75Ni10.12Al10 and Pd77.5Cu6Si16.5 bulk metallic glasses (BMGs) [7–10]. Thus, it is much more significant to investigate the evolution of shear band intersections. It may also provide some insights for understanding the plastic deformation mechanism and toughening
⁎
BMGs. The variation of the number of shear band intersections with plastic strain has been studied previously. The rapid increase of shear band intersections induced hardening, after counteracting free volume induced softening, may lead to an increase in hardness [11]. The density of shear band intersections and the shear band offsets at intersection sites can reflect the degree of interactions between shear bands. However, the evolution laws of them have not been directly and quantitatively studied so far. In addition, theoretical calculations indicate that strong shear band interactions have great influences on serrated flow behaviors [12,13]. For ductile BMGs, stress drop magnitudes of serrations in strain-stress curve follow power-law distribution, which can be attributed to strong shear band interactions. However, in different deformation stages, the relation between shear band intersections and serrated flows remains to be clarified by further experiments. In the present work, Zr64.13Cu15.75Ni10.12Al10 BMG was compressed to different plastic strains. The evolutions of shear band spacing, shear band orientations, and shear band offsets at intersection sites have been quantitatively studied. The average shear band spacing and the density of shear band intersections vary as power functions of plastic strain. In the large deformation stage, shear band orientations display a multiplepeak distribution, shear band offsets and serrated flows follow power-
Corresponding authors. E-mail addresses: [email protected] (B. Shi), [email protected] (P. Jin).
http://dx.doi.org/10.1016/j.jnoncrysol.2017.07.005 Received 8 May 2017; Received in revised form 28 June 2017; Accepted 4 July 2017 0022-3093/ © 2017 Elsevier B.V. All rights reserved.
Please cite this article as: Shi, B., Journal of Non-Crystalline Solids (2017), http://dx.doi.org/10.1016/j.jnoncrysol.2017.07.005
Journal of Non-Crystalline Solids xxx (xxxx) xxx–xxx
B. Shi et al.
stages, shear bands are separate and free, and the shear band intersections are very few. For the sample with a plastic strain of 83% (Fig. 1b), a large number of fine shear bands form and their orientations are widely distributed. These shear bands form a homogeneous network pattern. Uniformly distributed shear band intersections were observed, indicating severe interactions between shear bands. Shear bands with intersections divide the deformed BMG into two components: one is shear bands, and the other is undeformed amorphous regions enclosed by shear bands with intersections. To quantitatively investigate the evolution of shear band orientations and spacing, the orientation angles of shear bands were statistically analyzed. Fig. 2a–c show the histograms of orientation angles. For the deformed sample with a plastic strain of 6%, the orientation angles are mainly centered at around φ = 45° (Fig. 2a), which are in agreement with the maximum shear stress direction (~ 45° from the load axis). With the plastic strain increasing, the orientation angles distribute in a wider range (Fig. 2b and c). For the deformed sample with a plastic strain of 83%, the orientation angles display a multi-peak distribution. The orientation peaks in the distribution histograms can be identified at around φ = − 30°, 0° and 30°, respectively. Interestingly, the absolute values of orientation peaks for the sample with a plastic strain of 83% are not in agreement with the maximum shear stress direction. It implies that the formation and propagation of numerously new fine shear bands are strongly depended on inhomogeneous local stress states. Firstly, with increasing plastic strain, a large amount of uniformly distributed shear band intersections can hinder shear band propagation and further alter their propagating directions [7]. Secondly, shear bands have long range stress field whose magnitude is close to the yield stress of BMG itself [14]. With shear band spacing decreasing, a strongly overlapping internal stress fields form. Then the overlapping internal stress fields can help to trigger the formation of different orientated shear bands. They can also limit shear band propagation and further alter the propagating directions of shear bands. Finally, with the plastic strain increasing, the ratio of height/width for the deformed sample was reduced. In this case, the lateral constraints from friction may increase the complexity of the stress states [15]. Hence, the complex local stress states make the orientations of shear bands deviate the maximum shear stress direction and distribute more widely. In summary, with the plastic strain increasing, the distribution range of orientation angles is more extensive, and the orientation angle distributions gradually exhibit a multi-peak distribution. These features would promote intersecting probability. In addition, the change of the average shear band spacing with plastic strain was quantitatively analyzed. Fig. 2d shows that the measured average shear band spacing is logarithmically plotted versus plastic strain. As can be seen, the average shear band spacing decreases linearly with plastic strain in log-log plot. It indicates that the average shear band spacing correlates the plastic strain via a power-law relation d ~ εp− λ in ductile Zr64.13Cu15.75Ni10.12Al10 BMG, where the index λ = 1.22 (R2 = 0.981). The power-law relation means that shear band spacing can reduce to a very low level in the large deformation stages. This may lead to strong cooperative interactions of multiple shear bands in a short spatial range. Moreover, the index value in the present
law distributions. 2. Experimental procedure Master alloy ingots with nominal composition of Zr64.13Cu15.75Ni10.12Al10 were prepared by arc-melting the mixtures of highly pure Zr (99.8%), Cu (99.99%), Ni (99.99%), and Al (99.99%) in a Ti-gettered argon atmosphere. Each ingot was remelted more than six times to ensure chemical homogeneity. BMG rods with a square cross section of 2 × 2 mm2 were prepared by arc-melting master alloy ingot and subsequent suction-casting into a water-cooled copper mold. The compression test specimens were cut from the as-cast rod. Before compression test, both ends of these specimens were polished to be parallel. The compression tests were performed on a Shijin WDW-100D universal testing machine. In all compression tests, strain rate and aspect ratio are 1 × 10− 3 s− 1 and 1, respectively. The plastic strain εp was denoted as εp = (h0 − h) / h0, where h0 is the original height, and h is the height after deformation. The shear band patterns were analyzed by scanning electron microscopy (SEM) using a Hitachi S-4800 field emission scanning electron microscope operating at an acceleration voltage of 5 kV. The shear band spacing was determined by the following steps: (i) SEM images were taken from thirty random regions for each sample. (ii) A line paralleled to the loading axis was drawn on each SEM image, and this line intersected with shear bands. (iii) The distance between two adjacent crosspoints of this line and shear bands is considered as the shear band spacing. (iv) The total line length was divided by the total number of crosspoints, and this obtained result is regarded as the average shear band spacing. The density of shear band intersections was determined by the following steps: (i) The number of shear band intersections in ten observation regions (each region has an equal area of 113 μm2) for one sample were counted. (ii) The plane density of intersections for each region was calculated. (iii) The density of shear band intersections is a mean value of all densities for each observation regions. The orientation angle (φ, the angle between an observed shear band line on side surface and horizontal line, ranging from −90° to 90°) distribution was obtained by the following steps: (i) A line paralleled to the loading axis was drawn on each SEM image (ten images were counted for each sample). (ii) Orientation angles of shear bands located on the line were measured. (iii) The frequency of orientation angles who fall into the interval of (φ – δφ / 2, φ + δφ / 2) was counted, and the distribution histogram of orientation angles ranging from − 90° to 90° was plotted. 3. Results and discussion To investigate the shear band evolution with plastic strain, the SEM observations and statistical works were performed. The representative SEM images of shear bands for deformed Zr64.13Cu15.75Ni10.12Al10 BMG samples with plastic strains of 6% and 83% are shown in Fig. 1, respectively. Only a few shear bands formed in the deformed sample with a plastic strain of 6% (Fig. 1a). The angle between the observed shear band and loading axis is about 47°. Obviously, in small deformation
Fig. 1. SEM images of shear bands for the deformed Zr64.13Cu15.75Ni10.12Al10 BMG samples with plastic strains of 6% (a) and 83% (b).
2
Journal of Non-Crystalline Solids xxx (xxxx) xxx–xxx
B. Shi et al.
Fig. 2. (a-c) Histograms of orientation angles for the deformed samples with plastic strains of 6%, 47% and 83%. (d, e) The average shear band spacing d and the density of shear band intersections ND are logarithmically plotted versus plastic strain εp. Each data point corresponds to the mean value and the error bar indicates the calculated standard deviation. (f) The ratios of shear bands with spacing below 2 μm, 1 μm and 0.5 μm for the deformed samples with different plastic strains.
density of shear band intersections (ND) is a power function of plastic strain εp, ND ~ εpβ, where β = 1.32 (R2 = 0.963). It indicates that shear band intersections will dramatically increase in large plastic deformation stages. Fig. 2f shows the ratios of shear bands with small spacing for the deformed Zr64.13Cu15.75Ni10.12Al10 BMG with different plastic strains. Obviously, the percentages of shear bands with small spacing (≤ 2 μm, 1 μm and 0.5 μm) increase rapidly with plastic deformation (Fig. 2f). Additionally, among all the shear bands with spacing below 2 μm, the percentage of shear bands with spacing at sub-micron level (≤ 1 μm) accounted for the majority. This reveals that the sizes of undeformed amorphous regions enclosed by shear bands with intersections become gradually smaller with plastic deformation. The density of shear band intersections and shear band offsets (When a shear band cut through another one, shear displacement on the side surface will be produced at the intersection sites [4,16], as denoted by arrow in Fig. 1b. The shear displacement is defined as shear band offset can reflect the degree of interactions between shear bands. According to the above results, the density of shear band intersections increases greatly at the large deformation levels. To further investigate the shear band interactions, shear band offsets were statistically studied. Fig. 4a–c shows the distribution histograms of shear band offsets in the deformed BMG samples. As can be seen, the narrowing of shear band offset distribution with increasing the plastic strain is gradually clear. The shear band offsets shift to small value, indicating that plastic deformation is mainly carried by the formation and sliding of multiple shear bands, rather than a few dominating shear bands. Moreover, the density of shear band intersections greatly increases with increasing plastic strain. At the large deformation levels (εp ≥ 47%), the distribution histograms display a monotonically decreasing long-tailed distribution, which is often subjected to power-law relation. According to the SEM and statistical results (Figs. 1 and 2), the features of homogeneous distribution, small spacing and various orientations for shear bands enhance the opportunity for shear bands intersecting with each other. Therefore, uniformly distributed shear band intersections were observed at the large deformation level (Fig. 1b). This indicates that the shear band interactions evolve into a process of large degree cooperation of many shear bands. To further quantitatively analyze the distribution histograms of shear band offsets, the probability distribution density of shear band offsets, P(s) = (1 / Ns)[δNs(s) / δs], was calculated. Here, s is the shear band offset, δNs is the number of the shear band offsets who fall into the interval of (s – δs / 2, s + δs / 2), and Ns is the total number of the shear band offsets. Fig. 4d and e shows the probability distribution density P (s) versus shear band offset s for the deformed Zr64.13Cu15.75Ni10.12Al10 BMG samples with plastic strains of 47% and 83%. As can be seen, the P
Fig. 3. The determined values of the shear band spacing in the present work are compared with the data in Refs [4,16]. The dashed line indicates that the variation of shear band spacing with plastic strain displays a monotonically decreasing trend.
work is lower than that in Zr64.13Cu15.75Ni10.12Al10 BMG sample with an aspect ratio of 2 [5], indicating that the dynamics of plastic deformation may be greatly affected by the aspect ratio. However, under the same aspect ratio of 1, the index λ = 1 for the relatively brittle Zr52.5Cu17.9Ni14.6Al10Ti5 BMG [16]. It indicates that the cooperative interactions between shear bands in brittle BMG may be much weaker than that in plastic BMG. It is found that Zr64.13Cu15.75Ni10.12Al10 BMG can undergo super plastic deformation [5]. Previous works demonstrate that a higher Poisson's ratio can lower the barrier energy density for activation of shear transformation zones (STZs) and further promote the nucleation rate of shear bands [17,18]. Here, the Poisson's ratio of Zr64.13Cu15.75Ni10.12Al10 BMG is 0.377, higher than 0.370 for Zr52.5Cu17.9Ni14.6Al10Ti5 BMG [5,19]. Therefore, the nucleation rate of shear bands in plastic Zr64.13Cu15.75Ni10.12Al10 BMG may be higher than that in the Zr52.5Cu17.9Ni14.6Al10Ti5 BMG, and thus the shear band spacing in the Zr64.13Cu15.75Ni10.12Al10 BMG is much lower (shown in Fig. 3). As can be seen from Fig. 3, the dashed line indicates that the variation of shear band spacing with plastic strain displays a monotonically decreasing trend for Zr-based BMGs. In large deformation stages, small shear band spacing makes shear bands easily interact with each other in a short spatial range. Therefore, the probability of interactions becomes larger. As shown in Fig. 2e, the 3
Journal of Non-Crystalline Solids xxx (xxxx) xxx–xxx
B. Shi et al.
Fig. 4. (a–c) The distribution histograms of shear band offsets produced at the shear band intersection sites for the deformed samples with plastic strains of 6%, 47%, and 83%. The solid lines in (b) and (c) indicate monotonously decreasing trends. (d, e) The log-log plots of the probability distribution density P(s) versus shear band offset s for the deformed Zr64.13Cu15.75Ni10.12Al10 BMG samples with plastic strains 47% and 83% fitted by power-law relations with α = 1.17 and 1.69.
(s) can be well described by the power-law relations P(s) ~ s− α with exponents α = 1.17 and 1.69 (R2 = 0.907 and 0.910), respectively. The power-law relation is often an indicator of the self-organized critical (SOC) state, which means a system can buffer against large changes throughout the cooperation of its participants with strong interactions [12,13,20]. Hence, the plastic deformation process of the Zr64.13Cu15.75Ni10.12Al10 BMG evolves into a SOC state when plastic strain is larger than or equal to 47%. The emergence of the SOC state suggests that there are strong interactions between the multiple shear bands. In addition, the power exponent α increases with plastic strain, indicating that the system of shear band interactions in the Zr64.13Cu15.75Ni10.12Al10 BMG becomes more complex and the degree of shear band interactions becomes stronger. Theoretically, serrated flow behavior is closely related to shear band interactions. Stress drop magnitudes of serrations in stress-strain curve follow power-law distribution when considering the strong interactions between shear bands for ductile Zr64.13Cu15.75Ni10.12Al10 BMG [12]. Fig. 5a shows the compressive engineering stress-strain curve of Zr64.13Cu15.75Ni10.12Al10 BMG. The regions A and B respectively represent initial stage and larger deformation stage, and their enlargements are shown in Fig. 5b and c. As can be seen, the serrations of region A display relatively uniform sizes, but the serrations of region B display various sizes and a more complex pattern. The average
magnitudes of stress drop are about 11.4 and 4.4 MPa for regions A and B, respectively. Fig. 5d and e shows the frequency histograms of the stress drop magnitude for regions A and B and their nearby regions. As can be seen, the histogram exhibits a monotonically decreasing trend in the large deformation stage. And the solid line in Fig. 5e indicates a fitting curve with power function. This further implies that shear band interactions become remarkable and evolve into a SOC state with plastic deformation. Such phenomenon has also been discovered in Portevin-Le Chatelier (PLC) effect of CuAl crystalline alloy, where a crossover from chaotic state to SOC state happened [21]. BMGs have no dislocation-like defects, and their plastic deformation is carried by shear bands. In small deformation stage, very few shear bands formed and propagated. Due to lacking of shear band interactions, the continuously regular serration pattern is similar to the serrations caused by one single shear band [2]. In large deformation stage, shear band intersections increase greatly. Shear band intersections can hinder shear band propagation, therefore the sliding of shear band may become harder. Due to the hindering effect from a large number of shear bands with cooperative interactions, the serration pattern becomes more complex and the average stress drop becomes smaller. Actually, in the present work, shear bands are confined within small space, and these shear bands can easily interact with many other shear bands. At the large deformation levels, shear band offsets and serrated flows
Fig. 5. (a) The compressive engineering stress-strain curve of Zr64.13Cu15.75Ni10.12Al10 BMG. The regions marked with A and B represent different deformation stages. (b, c) The enlargements of regions A and B in (a). Frequency histograms of the stress drop magnitudes for boxes A (d) and B (e) and their nearby regions.
4
Journal of Non-Crystalline Solids xxx (xxxx) xxx–xxx
B. Shi et al.
that shear band interactions evolve into a process of large degree cooperation of many shear bands at the large deformation levels. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 51661028). References [1] A.S. Argon, Plastic deformation in metallic glasses, Acta Metall. 27 (1979) 47–58. [2] J. Pan, Q. Chen, L. Liu, Y. Li, Softening and dilatation in a single shear band, Acta Mater. 59 (2011) 5146–5158. [3] Z.F. Zhang, H. Zhang, X.F. Pan, J. Das, J. Eckert, Effect of aspect ratio on the compressive deformation and fracture behaviour of Zr-based bulk metallic glass, Philos. Mag. Lett. 85 (2005) 513–521. [4] J.W. Liu, Q.P. Cao, L.Y. Chen, X.D. Wang, J.Z. Jiang, Shear band evolution and hardness change in cold-rolled bulk metallic glasses, Acta Mater. 58 (2010) 4827–4840. [5] Y.H. Liu, G. Wang, R.J. Wang, D.Q. Zhao, M.X. Pan, W.H. Wang, Super plastic bulk metallic glasses at room temperature, Science 315 (2007) 1385–1388. [6] L. He, M.B. Zhong, Z.H. Han, Q. Zhao, F. Jiang, J. Sun, Orientation effect of preintroduced shear bands in a bulk-metallic glass on its “work-ductilising”, Mater. Sci. Eng. A 496 (2008) 285–290. [7] J. Das, M.B. Tang, K.B. Kim, R. Theissmann, F. Baier, W.H. Wang, J. Eckert, “Workhardenable” ductile bulk metallic glass, Phys. Rev. Lett. 94 (2005) 205501. [8] Q.P. Cao, J.W. Liu, K.J. Yang, F. Xu, Z.Q. Yao, A. Minkow, H.J. Fecht, J. Ivanisenko, L.Y. Chen, X.D. Wang, S.X. Qu, J.Z. Jiang, Effect of pre-existing shear bands on the tensile mechanical properties of a bulk metallic glass, Acta Mater. 58 (2010) 1276–1292. [9] S. Takayama, Drawing of Pd77.5Cu6Si16.5 metallic glass wires, Mater. Sci. Eng. 38 (1979) 41–48. [10] S. Takayama, Serrated plastic flow in metallic glasses, Scr. Metall. 13 (1979) 463–467. [11] B. Shi, Y. Xu, C. Li, W. Jia, Z. Li, J. Li, Evolution of free volume and shear band intersections and its effect on hardness of deformed Zr64.13Cu15.75Ni10.12Al10 bulk metallic glass, J. Alloys Compd. 669 (2016) 167–176. [12] B.A. Sun, H.B. Yu, W. Jiao, H.Y. Bai, D.Q. Zhao, W.H. Wang, Plasticity of ductile metallic glasses: a self-organized critical state, Phys. Rev. Lett. 105 (2010) 035501. [13] B.A. Sun, S. Pauly, J. Tan, M. Stoica, W.H. Wang, U. Kühn, J. Eckert, Serrated flow and stick-slip deformation dynamics in the presence of shear-band interactions for a Zr-based metallic glass, Acta Mater. 60 (2012) 4160–4171. [14] R. Maaß, P. Birckigt, C. Borchers, K. Samwer, C.A. Volkert, Long range stress fields and cavitation along a shear band in a metallic glass: the local origin of fracture, Acta Mater. 98 (2015) 94–102. [15] Z. Wang, J.W. Qiao, G. Wang, K.A. Dahmen, P.K. Liaw, Z.H. Wang, B.C. Wang, The mechanism of power-law scaling behavior by controlling shear bands in bulk metallic glass, Mater. Sci. Eng. A 639 (2015) 663–670. [16] H. Bei, S. Xie, E.P. George, Softening caused by profuse shear banding in a bulk metallic glass, Phys. Rev. Lett. 96 (2006) 105503. [17] J.J. Lewandowski, W.H. Wang, A.L. Greer, Intrinsic plasticity or brittleness of metallic glasses, Philos. Mag. Lett. 85 (2005) 77–87. [18] Y.H. Liu, K. Wang, A. Inoue, T. Sakurai, M.W. Chen, Energetic criterion on the intrinsic ductility of bulk metallic glasses, Scr. Mater. 62 (2010) 586–589. [19] W.H. Wang, The elastic properties, elastic models and elastic perspectives of metallic glasses, Prog. Mater. Sci. 57 (2012) 487–656. [20] P. Bak, C. Tang, K. Wiesenfeld, Self-organized criticality: an explanation of the 1/f noise, Phys. Rev. Lett. 59 (1987) 381. [21] G. Ananthakrishna, S.J. Noronha, C. Fressengeas, L.P. Kubin, Crossover from chaotic to self-organized critical dynamics in jerky flow of single crystals, Phys. Rev. E 60 (1999) 5455.
Fig. 6. The schematic illustration of shear band intersections at the large deformation levels for Zr64.13Cu15.75Ni10.12Al10 BMG.
behaviors follow power-law distribution, which is typical characteristic for a system containing strong interacting constituents. Based on the above results, the schematic illustration of shear band intersections was drawn in Fig. 6. In large deformation stage, previously formed shear bands have endured repeated sliding and finally developed into coarse shear bands. Subsequently, new shear bands formed in the vicinity of previously coarse shear bands. This leads to the sizes of undeformed amorphous regions enclosed by shear bands with intersections becoming smaller. Thus shear bands only have small space to propagate in the large plastic deformation stage. So the shear bands are not free, and they will be restricted within the grid formed by previous shear bands. Thus, the new fine shear band can easily intersect with each other in a small space, resulting in a great increase of the intersection density, and shear bands evolve into a process of large degree cooperation of many shear bands. 4. Conclusions In conclusion, Zr64.13Cu15.75Ni10.12Al10 BMG was compressed to different plastic strain. Shear band spacing and the density of shear band intersections are found to vary as power functions of plastic strain (d ~ εp− 1.22 and ND ~ εp1.32, respectively). With increasing the plastic strain, the distribution of shear band orientations become wider, the orientation angles display a multi-peak distribution. Small spacing and arbitrary orientations of shear bands enhance the intersecting probability. At the large deformation levels, the density of shear band intersections increases greatly. The interactions of shear bands become stronger due to the reason that shear bands are restricted within the network formed by previously coarse shear bands. The shear band offsets and serrated flows display power-law distributions. It indicates
5