The mechanism of power-law scaling behavior by controlling shear bands in bulk metallic glass

Materials Science & Engineering A 639 (2015) 663–670

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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea

The mechanism of power-law scaling behavior by controlling shear bands in bulk metallic glass Z. Wang a, J.W. Qiao a,b,n, G. Wang c, K.A. Dahmen d, P.K. Liaw e, Z.H. Wang f, B.C. Wang a, B.S. Xu a,b a Laboratory of Applied Physics and Mechanics of Advanced Materials, College of Materials Science and Engineering, Taiyuan University of Technology, Taiyuan 030024, China b Key Laboratory of Interface Science and Engineering in Advanced Materials, Ministry of Education, Taiyuan University of Technology, Taiyuan 030024, China c Laboratory for Microstructures, Shanghai University, 200444 Shanghai, China d Deparement of Physics and Institute of Condensed Matter Theory, University of Illinois at Urbana Champaign, Urbana, IL 61801, USA e Department of Materials Science and Engineering, The University of Tennessee, Knoxville, TN 37996-2200, USA f Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China

art ic l e i nf o

a b s t r a c t

Article history: Received 9 April 2015 Received in revised form 3 May 2015 Accepted 23 May 2015 Available online 27 May 2015

Bulk metallic glasses deform irreversibly under a stress through shear-banding courses that manifest as the serrated flow behavior. The compressive deformation and dynamic serrated flow behavior of Zr52.5Cu17.9Ni14.6Al10Ti5 bulk metallic glass samples with different aspect ratios have been investigated. The yield strength nearly remains a constant value of approximately 2 GPa, while the compressive plasticity increases obviously with decreasing aspect ratio. It is found that the serrated flows display a power-law scaling behavior at different aspect ratios. The power-law scaling behavior is discussed by controlling shear bands in BMG. In addition, a new method was proposed to study the power-law-scaling behavior. When the aspect ratio is small, the friction between the sample and the platen will play a significant role that attributes to a lateral constraint. The uniaxial stress and the lateral constraint will cause a hydrostatic pressure on the sample close to the platen. The shear bands are controlled by the different stress states, which leads to a power-law-scaling behavior in serrated flows. The investigations have a contribution to understanding the plastic-deformation mechanism of BMGs. Crown Copyright & 2015 Published by Elsevier B.V. All rights reserved.

Keywords: Amorphous materials Mechanical test Deformation Fracture

1. Introduction The bulk metallic glasses (BMGs), as a relatively new class of materials, exhibit unique mechanical properties, such as high strength and large elastic limit in comparison with their crystalline counterparts [1,2]. However, their use as structural materials is severely limited due to their poor ductility and premature fracture at room temperature arising from shear localization [3–6]. While the mechanical properties of metallic glasses have been studied for many years, their underlying deformation mechanisms are fundamentally different from those of crystalline solids due to their lack of long-range atomic order and remain the subject of strong debate. As it is known, the plastic deformation of BMGs occurs via the formation of highly-localized shear bands [7]. The n Corresponding author at: Laboratory of Applied Physics and Mechanics of Advanced Materials, College of Materials Science and Engineering, Taiyuan University of Technology, Taiyuan 030024, China. Fax: þ 86 351 6010311. E-mail address: [email protected] (J.W. Qiao).

http://dx.doi.org/10.1016/j.msea.2015.05.074 0921-5093/Crown Copyright & 2015 Published by Elsevier B.V. All rights reserved.

formation of these shear bands is considered as serrations during compression on the stress–strain curve [8,9]. Song et al. [10] have found that the serrated flow was a result of intermittent shearing along the principal shear plane. The plasticity of BMGs relates to some properties [11,12] or testing conditions [13], and the plasticity of BMGs is an intrinsically-dynamical phenomenon and should be closely related to dynamical features. Investigating the sample-size-dependent shear-band operations in BMGs is the key to understanding the physical processes responsible for the strength and ductility of BMGs. In crystalline alloys, the fluctuation phenomenon is called the Portevin–Le Chârtelier (PLC) effect [14] during plastic deformation. Serrated flows also were known as repeated yielding of glassy metals during plastic deformation, which has been associated with the shear-band formation and propagation [15]. There have been a great amount of attempts to describe the mechanism during plastic deformation. Two of the ones are the classical “free volume” model by Turnbull et al. [16] and Speapen [17] and the “shear transformation zone” (STZ) model by Argon and co-workers

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[18]. Johnson and Samwer [19] proposed a cooperative shear model, which took an important role of the influence of shear modulus into account on the plastic flow of BMGs. All of these views were focused on the effects of intrinsic properties of BMGs on their macroscopic-deformation responses. The serrated flow is associated with shear banding. Researchers have focused on the energy change during the single shearbanding behavior. A few studies [13,20] have shown the relationship between the strain energy and the interaction of shear bands. Applying the confining pressure to the BMG samples can also trigger the formation of multiple shear bands. In this case, the effect of hydrostatic pressure on the flow and fracture behavior of the Zr-based BMG has been investigated [21–23]. It is found that BMGs can display the large inelastic deformation of more than 10% under confinement. Moreover, Bruck et al. [24] studied mechanical properties at different aspect ratios, sample height H/diameter D (1:2 and 2:1), and found a slight increase in the yield strength and distinct increase in the compressive plasticity with decreasing aspect ratios. Recently, Zhang et al. [25] and Wu et al. [26] investigated the effect of aspect ratio on the compressive deformation and fracture behavior of Zr-based BMGs and realized that the specimens with small aspect ratios exhibit great plastic strain under compression. In this study, in order to explore the serration dynamics in a Zr-based BMG, the effect of different aspect ratios (1:2, 1:1, and 2:1) on the amplitudes of serrations, compressive plasticity, as well as the formation and evolution mechanisms for a Zr52.5Cu17.9Ni14.6Al10Ti5 [atomic percent (at%)] metallic glass in the serrated flow is conducted.

2. Experimental procedures Alloy ingots with a nominal composition of Zr52.5Cu17.9Ni14.6Al10Ti5 were prepared by arc-melting a mixture of pure metals (weight purity Z99.9%) in a Ti-gettered argon atmosphere. To ensure the compositional homogeneity, each ingot was remelted at least four times. Rodshaped samples with a diameter of 2 mm and a length of about 60 mm were prepared by suction casting into a water-cooled copper mold. Compressive testing specimens about 1, 2, and 4 mm long were cut from the rod-like samples by a diamond saw with the cooling water, and then carefully polished with different aspect ratios (height: diameter) of 1:2, 1:1, and 2:1, respectively, to an accuracy of 5 μm. The uniaxial-compressive tests were conducted at the strain rate of 2  10  4 s  1 at 298 K (room temperature) on the cylindrical specimens using a MTS 809 materials-testing machine. The fracture surface of the specimens was observed with scanning electron microscopy (SEM) to identify fracture mechanisms.

3. Results and discussion The compressive engineering stress–strain (σ − ε ) curves of the Zr52.5Cu17.9Ni14.6Al10Ti5 BMGs with different aspect ratios (H:D) are shown in Fig. 1. The yielding strengths are almost the same value of  2 GPa, while the compressive plasticity is much more different. It is clear that when the sample's aspect ratio is 1:2, its compressive plasticity can reach 57%. However, the sample does not fail with the catastrophic fracture, which is mainly identical with some other BMGs that the aspect ratio is lower than 1 [27]. When the aspect ratios are 1:1 and 2:1, the samples exhibit the similar serrated flow behavior after yielding. It is evident that the repeating cycles of a sudden stress drop are followed by reloading elastically, which accords with the previous results [28]. The compressive plasticity of the sample with the aspect ratio of 1:2 reaches  35%. Actually, a sudden change has been found when the plasticity is 20%, which denotes that the sample has failed.

Fig. 1. Engineering stress–strain curves of the Zr52.5Cu17.9Ni14.6Al10Ti5 BMG compressed at three aspect ratios. Insets are the local amplification figure corresponding the aspect ratio of 1:2 and 1:1. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

However, a distinct separation cannot be found on the surface. The platen will press the fracture section of the sample continuously, which reflects large compressive plasticity in Fig. 1. It is reported that a similar work-hardening phenomenon on engineering stress–strain curves, and the samples presented great workhardening on true stress–strain curves, when the aspect ratio is small [26,29]. However, if the sample's aspect ratio is far lower than 1, the friction between the sample and loading platen will play a vital role and increases dramatically, which results in the much higher deformation force and compressive strength. A serrated flow is known as repeated yielding of glassy metals during plastic deformation. The inset of Fig. 1 shows the amplitudes of stress drops during the course of deformation with aspect ratios of 1:2 and 1:1. It is obvious that the amplitudes range of the sample with an aspect ratio of 1:2 is from 2 to 7 MPa, while the amplitude of other aspect ratios (1:1 and 2:1) is similar with previous research that is 12–40 MPa [28]. Han et al. [13] considered that the energy stored in the testing machine had a strong influence on the mechanical behavior of BMG samples. In order to eliminate the machine vibration error, any serration event with a stress fluctuation less than 5 MPa in jerky flows is excluded in the statistical analysis. Therefore, the sample with an aspect ratio of 1:2 could be regarded as of non-serration. In this study, it should be noted that these stress drops are considered as the lower limit of the effective stress drops occurring in the sample [29]. Kimura and Masumoto [30] considered the damping effects of the machine. The effective stress drop in the machine assembly is 2–3 times larger than that measured per shear event in this study through their evaluation method. Here, we suppose that the measured stress drops as the effective stress drop and take the aspect ratios of 1:1 and 2:1 into account. Fig. 2 shows the stress drop versus strain curves of different samples with aspect ratios of 1:1 and 2:1. The X axis represents the strain, and the increment is 1%. The Y axis value is the average of the amplitude of stress drops between the adjacent strain increments (1%), i.e., the first data of the aspect ratio of 1:1 means the average of all the amplitudes of stress drops between the strains from 2% to 3%. By that analogy, all the data points could be collected from the beginning of the serrated flow to the end. From Fig. 2, it is evident that the average amplitudes of stress drops of the sample with the aspect ratio of 1:1 remain a stable scope approximately from 23 to 32 MPa. This trend means that these serrations distributions are relative homogeneous. The serrated

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Fig. 2. The plot of the amplitude of stress drop versus strain. Inset shows the fraction that before fracture with an aspect ratio of 2:1.

flow behavior cannot break during compressive testing due to the equivalence of the height and diameter. The start and end average amplitudes of stress drops are lower than those of the middle ones, which is in accordance with previous studies [31]. The evolution of serrations is followed by yielding and grows to mature, which will experience a process of the middle transition. Naturally, the amplitudes of the stress drops are smaller. It is presented that an obvious increasing process of the sample with an aspect ratio of 1:1 at the strain of  20% in the stress–strain curves, as shown in Fig. 1. Actually, the material has failed, though a large number of serrations emerge in following fraction. The average amplitudes of the stress drops are somewhat smaller than before. Compared with the sample with an aspect ratio of 1:1, the other of 2:1 displays a clear rising trend, and the average amplitude of stress drops is low before the strain of 8%. Properly speaking, the variation of the amplitudes of stress drops goes through three stages. The rising stage is from the strains of 2% to 4%. As mentioned above, the evolution of serrations will experience a process of the middle transition. Hence, the amplitudes of stress drops present a gradually-increasing stage. Then the serrations turn into a mature stage, i.e., a stable jerky flow stage. In the end, it is shown that another sharply rise appears, particularly at the strains from 9% to 11%. As it is known, BMGs normally fail in a pure shear mode after compressive testing and do not exhibit a catastrophic fracture. For the same reason, the sample will also experience a transitional time, as displayed in the inset of Fig. 2. It is evident that even the last data is lower than the first data in one serration, which leads to the fact that the amplitude of the serration is larger than the former one. Then the amplitudes of stress drops become smaller gradually until fracture. Fig. 3(a) and (b) shows histograms of the amplitudes of stress drops, which exhibit the distributions of the amplitudes of stress drops. In Fig. 3(a), the amplitudes of stress drops are concentrated in a range from 20 to 35 MPa. The serrations that the amplitudes of the stress drops below 20 MPa usually can be captured in the first stage, i.e., the region after yielding in the stress–strain curve. When the aspect ratio is 1:1, the largest amplitudes are approximately 40 MPa, which are far smaller than that of the sample with an aspect ratio of 2:1. For Fig. 3(b), it is found that the scope of the amplitudes is wider than that of the aspect ratio 1:1. It ranges from 10 to 70 MPa. However, the distribution of the serrations that the amplitudes of stress drops exceed 40 MPa is sporadic. Most of the amplitudes of stress drops are similar to those of 15–35 MPa, at the aspect ratio of 1:1. Although the distributions are different, the

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mean values of the amplitudes are so close, which are 28.26 and 27.56 MPa corresponding to the aspect ratios of 1:1 and 2:1, respectively. Recently, it is reported that the amplitudes of different aspect ratios (1:1 and 2:1) are below 100 MPa [32]. The results in this study are identical with their conclusions. Broadly speaking, the amplitudes of stress drops are stable in a range, which is a homogeneous serrated flow behavior. When the aspect ratio is 1:1, the stable serrated flow behavior can retain at a large strain range during deformation. On the contrary, the course of the serrated flow behavior is short due to the low plasticity, which leads to the amplitudes of stress drops are decreasing little by little obviously before fracture. Fig. 3(c) and (d) exhibits the histograms of the rising time of serrations. One serration event includes the processes of strain-energy accumulation and strain-energy release. Usually, the rising region of each serration is regarded as the elastic-energy accumulation, and the falling region is the process of the elastic energy release. It is reported that an exponential decay of the shearing-stress model was proposed during the stress-drop process of the serrated flows [33]. The rising and falling times [34] in each serration are counted. It is shown that the average rising time and falling time are 4.044 s and 0.219 s, when the sample's aspect ratio is 1:1. The rising time is approximately 18 times of the falling time, which is much longer than the falling time, while the other sample's average times with the aspect ratio of 2:1 are 2.513 s and 0.157 s, respectively. Similarly, the rising time is about 16 times of the falling time. It is noted that the stress drop is a sharply-decreasing region, compared with the rising process during one serration. From Fig. 3(c) and (d), one can find that the sample's rising time mainly ranges from 2 s to 5.5 s with an aspect ratio of 1:1 and the other is from 1 s to 4 s with an aspect ratio of 2:1. The role of the rising time in the jerky flow behavior will be discussed next. It is found that shear avalanches could present a power-law scaling behavior [28,35]. The power-law scaling behavior has a significant feature that the internal state is a self-similar or scalefree pattern, i.e., the structures on one scale are almost the same as structures on the other scales. As mentioned above, the serratedflow behavior contains the energy of accumulation and release. The elastic energy is accumulated during the stress rising and quickly relaxed in each serration. The elastic energy cannot be fully relaxed in the limited time in one serration and then promotes the formation of new shear bands, which is in accordance with above results that the rising time is much longer than the falling time in one serration. The stored energy is not relaxed completely, while the serration turns to the next accumulated course, which results in a relatively small serration, as shown by the green curve in the inset of Fig. 1. Consequently, the average amplitude of stress drops is relatively low at the aspect ratio of 1:1, as displayed in Fig. 2. To further investigate the power law scaling behavior of the serrated flows, the distribution of shear-avalanche sizes reflecting the shear avalanche in BMGs is discussed [36,37]. However, the shear-avalanche size cannot be measured directly. Instead, the elastic energy density accumulated is chosen to characterize the shearavalanche sizes. The elastic energy density of a serration event (Δδ ) is

Δδ =

1 Δσ Δε 2

(1)

where Δσ and Δε are the amplitude of the stress drop and the variable quantity of the strain, respectively, as shown in the inset of Fig. 1. Assuming that the plastic strain results in a drift of the elastic-energy-density value, a normalization of the elastic energy density is carried out to eliminate the statistical error [34]. The plots of the elastic energy density versus strain with different aspect ratios are obtained with a baseline energy, f (ε ), through

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Fig. 3. (a, b) The frequency histograms of the amplitudes of stresses with the aspect ratio of 1:1 and 2:1. (c, d) The frequency histograms of the rising time with the aspect ratios of 1:1 and 2:1. Insets are marked the rising time with different aspect ratios.

linear-regression fitting, as displayed in Fig. 4(a) and (c). The elastic energy density is linearly related to the variable quantity of the stress and strain. In Fig. 4(a), the distribution of these scatters exhibits uniformity due to the homogeneous nature of the serrated flow, i.e., the amplitude of the stress drop and the variable quantity of the strain are nearly the same. While for the sample with an aspect ratio of 2:1, it is obvious that the elastic energy density is increasing with the strain. The amplitudes of stress drops at the aspect ratio of 2:1 increase with the strain, as shown in Fig. 2, and the difference of the variable quantity of the strain is small. It is found that the elastic energy density is notably increasing after the strain of about 8%. Once reaching the limitation of the elastic-energy accumulation, the stored energy will be burst out as soon as possible. Some non-adiabatic effects, such as friction, will be released via stress drops causing an energy burst [38]. It is presented in the average elastic energy density, which are 14,428 and 8973 J/m3 at the aspect ratios of 1:1 and 2:1, respectively. Defining a new non-dimensional variable, S = Δδ /f (ε ), the distribution of the normalized elastic energy density with different aspect ratios, as displayed in Fig. 4(b) and (d), is discussed . These two plots are the histograms of N(S) versus S, here N(S) is the number of S. It is noted that the distribution is increasing at first then decreasing gradually in Fig. 4(b) and (d), though the aspect ratios are different, i.e., the plots reveal distinct normal distributions, and peak distributions are observed. These distributions do not eventually lead to a power-law relation. The observations indicate that shear bands may display a power-law-scaling behavior. It is found that a power-law distribution of the shear avalanche will occur spatiotemporally, which suggests a dynamic-behavior transition to the

power-law-scaling behavior, as the temperature decreases or strain rate increases [39]. Lacking dislocations, the ductility of the BMG is based on shear bands. Moreover, the shear band is closely related to the serrated flow event. An elastic strain field will be formed in the elastic-energy-accumulation regime of one serration during the deformation in compressive testing. The rising time is many times of the falling time in each serration, and the rising and falling times in one serrated flow are associated with the energies of accumulation and release. The elastic-strain field can be completely relaxed. In addition, the elastic energy is relaxed, and the strain field subsequently disappears because of the stress drop during the serration event, and the strain field can also interact with each other [39]. In this case, there is no spatial correlation between shear bands during deformation and displays a characteristic of the power-law scaling behavior. Contact the mentioned above, i.e., the rising time and the elastic energy density. Transforming Eq. (1), it is obtained that

Δδi =

vti Δσ 2H

(2)

where Δδi is the elastic energy density of the ith serration, v is the velocity of the compression, ti is the time of the ith serration, H is the height of the sample, and Δσ is the amplitude of the corresponding serration. The average elastic energy density is always discussed to explore the serrated flow behavior. Eq. (2) can also be transformed as −

Δδ =

vt

∑ 2Hi Δσ

i

(3)

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Fig. 4. (a, c) The profiles of the elastic energy density versus strain. The red lines are baselines, f(ε), by linearly regression fitting. (b, d) The statistical distributions of elastic energy densities, N(s), for BMGs deformed at different aspect ratios of 1:1 and 2:1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) −

where Δσ is the average of elastic energy, i is the amount of serrations. Fig. 3(c) and (d) is focused on the rising time in serrations, and it is verified that the rising time is considerably larger than the falling time in each serration. One is 18 times greater and the other is 16 times with the aspect ratios of 1:1 and 2:1, respectively. In order to further simplify Eq. (3), the time of each serration can be replaced by the rising time, and the falling time is negligible though the elastic energy density contains the range of the stress drop. Then Eq. (3) can be replaced as −

Δδ =

vt

∑ 2Hr Δσ (4)

i

where tr represents the rising time in each serration, as shown in the insets of Fig. 3(c) and (d). Therefore, the average elastic energy densities that are calculated by this method are 11,428 and 6927 J/m3, which are close to the results that obtained before. −

Actually, Δδ is a linear correlation with t and Δσ as well. A similar approach can also be carried out to characterize the power-law scaling behavior in the dynamic analysis of the serrated flow behavior. From Eq. (4), it is noted that the elastic energy density is related to v, H, t, and Δσ . Here, v, H, and Δσ are constants, and v is a function of strain rate. Generally, H and v are external conditions, and Δσ is controlled by the materials, which determines the time of serrations. In other words, the power-law scaling behavior is subjected to the external and internal conditions in the dynamic serrated flow behavior. By SEM, the deformation morphologies of the samples with

different aspect ratios were observed to reveal the serrated flow and fracture mechanism. Fig. 5(a)–(c) shows the enlarged images of the lateral surfaces with different aspect ratios. The whole fracture photographs of the failed samples are shown in the insets of Fig. 5(a)–(c). Fig. 5(a), (b), and (c) shows lateral surfaces of the samples with the aspect ratios of 1:2, 1:1, and 2:1, respectively. Each image is the enlarged fraction of the dashed rectangle in each inset. Although the catastrophic fracture does not emerge in the sample with the aspect ratio of 1:2, numerous shear bands can be captured, as shown by the red arrows in Fig. 5(a). Another feature is that a great number of horizontal shear bands can be recognized when the shear bands are subjected to a constraint during deformation. Analogously, a high density of shear bands is distributed on the surface of the sample as well in Fig. 5(b). The right green arrows represent the primary shear bands, and the left ones stand for the secondary shear bands. The primary shear bands form and propagate continually, which gives rise to the eventual fracture, as displayed by the blue arrow in Fig. 5(c). The shear angle (θ) marked in Fig. 5(c) is 43°, which is slightly below 45°. Fig. 5(d) shows the morphology of the fracture surface of the sample with the aspect ratio of 2:1. It is noted that two regions of visible difference appear on the fracture surface. The yellow dashed rectangle corresponds to a stable range of the dominant shear bands before the fracture and serrations that are associated with the stress drop in stress–strain curves are often noticed in this region [11]. Intensive plastic deformation of BMG with a large number of shear bands before intensive crack initiation is supposed to have effects on its properties. The plastically and

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Fig. 5. SEM images of the fractured BMGs deformed at different aspect ratios of 1:2, 1:1, and 2:1. (a, b, c) The profiles of the lateral surfaces of the samples. (d) The fracture surface of a sample. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

elastically deformed areas may release significant amount of extra heat upon structural relaxation [40]. The region that is marked with the pink rectangle possesses most of the fracture surface. Since the shear angles (θ) are approximately 43° that are smaller than 45° in both Fig. 5(b) and (c). The Mohr–Coulomb (MC) yield criterion is generally used to explain the observed compression/ tension asymmetry [41]. The MC yield criterion can well explain the fracture when the shear angle is lower than 45°. The general form of the MC yield criterion is given as τ = τ0 − ασn , here τ and σn are the shear stress and normal stress at yield, respectively, τ0 is a material constant of shear resistance, and α is the normal stress coefficient (or MC coefficient ), which can be expressed as α = − cot (2θ ). Based on the MC yield criterion, a rotation mechanism of the primary shear bands due to a high compressive plasticity was proposed [42]. The value of shear angles is  43° that is lower than 45°, when the samples' aspect ratios are 1:1 and 2:1, which is consistent with the MC yield criterion. It is found that each serration was associated with an individual shear band in an early study of the compressive behavior of Zrbased BMGs [43]. Because of lacking the periodic structure, the ductility of BMGs is dominated by shear bands rather than by the motion of dislocations, compared with crystals. The accumulated elastic energy is used for shear-band sliding. Most BMGs deform by a dominant shear band along a principal shear plane, and a number of secondary shear bands will be formed during deformation in compressive testing. As stresses are increasing, strains are accommodated elastically, until the stress level reaches the value that it can activate a serrated flow. At the first stage, the amplitudes of serrated flows are relative small, because the energy that is stored in early shear bands cannot support them to slide a long way. The elastic energy is accumulated with increasing the stress. When the stored energy is high enough, the mature shear bands will be formed, which reflects the homogeneously serrated flows in the stress–strain curves, as shown in Fig. 1. The initial stage corresponds to nucleation and propagation of new shear bands while the second stage corresponds to the formation of a dominant shear band

throughout the sample [44]. When the applied strain is fully enough accommodated in shear bands, the stress will be relaxed. The origin of the spontaneous strain localization is taken to be shear softening: as a local region is plastically deformed, it becomes softer than the surrounding undeformed regions and, thus, can be more susceptible to the subsequent flow [45]. Sun et al. [46] have investigated the shear-band interactions on affecting the deformation and fracture behavior of BMGs in serrated flows. The BMG sample usually deforms by the formation of a single dominant shear band. The stick-slip model of a single shear band was used to describe the deformation behavior of BMGs, and the kinetics equation was proposed to interpret the effect of shear bands during deformation in compression [46]. It is obvious that the aspect ratio plays a vital role in the serrated flow behavior. There is no doubt that the compressive plasticity, the amplitudes of stress drops, and the formation of multiple shear bands are strongly dependent on the aspect ratios of the BMG samples. The samples with a normal aspect ratio of 2:1 have been studied under different conditions, such as different stain rates, ambient temperatures, and liquid-nitrogen temperatures and so on in previous research [47,48]. Compared with samples whose aspect ratio is 2:1, those with different aspect ratios of 1:1 and 1:2 exhibit various features. When the sample's aspect ratio is 1:2, the fluctuation of the amplitude ranges from 2 MPa to 7 MPa, i.e., the serrated flow behavior is hardly found in the stress–strain curve. What is more, a rising trend is shown obviously, and the compressive plasticity is extremely high, nearly 60%. It is reported that large strains can be obtained for specimens with an aspect ratio lower than 1 [27]. For the sample with an aspect ratio of 1:1, the serrations look more homogenous, i.e., the amplitude of the stress drop and the time of each serration are nearly identical. Another feature is that a large amount of crosswise shear bands can be perceived in Fig. 5(a) and (b). This trend indicates that numerous shear bands turn to horizontal displacement due to the strong stress constraint. A lateral stress that is induced by friction between the platens and the top and bottom surfaces will be formed for the samples

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with small aspect ratios in compression, which leads to a more complex stress field. Besides the uniaxial stress, the friction also plays an important role during deformation. The two kinds of stresses will cause a hydrostatic pressure in the sample close to the platen [26]. The constraint often causes a triaxial stress state and is easy to induce the formation of multiple shear bands. It is reported that due to the formation of numerous shear bands, BMGs usually have certain ductility when subjected to compression or bending loads [25]. If there is a confining pressure acting on a BMG sample, a number of shear bands will form. Recently, Lewandowski et al. [22] studied the effect of hydrostatic pressure on the plastic flow and fracture behavior of Zr-based BMG. It is reported that when the Zr-based BMG samples were subjected to a lateral confinement, the plastic strain would reach over 10% with the formation of numerous shear bands during deformation [23]. To further understand the influence of different aspect ratios on the serrated flow behavior, the deformation process of compressive testing is schematically illustrated in Fig. 6. Fig. 6(a), (b) and (c) shows the schematic diagram of compressive testing for the samples with different aspect ratios of 2:1, 1:1, and 1:2, respectively. The BMG samples generally fail in a pure shear mode, and a large amount of shear bands can be observed on the surfaces of these BMG samples, as shown in Fig. 5(a)–(c). In fact, a thin layer formed in the sample inside approximately 45° because of the motion of the shear bands. Regardless of the value of the aspect ratio, the friction can emerge between the platens and the top and bottom surfaces as long as they have contract with each other in compressive testing, and the friction is the lateral constraint, σL , as displayed in Fig. 6(a)–(c). The effect of the lateral constraint is the strongest at both ends, and the effect is smaller and smaller when it closes to the middle of the samples, as presented in Fig. 6(a) and (b). In Fig. 6(a), the primary shear bands usually form at the position that the effect of the lateral constraint is not strong and propagate along approximately 45°, which leads to shear fracture eventually. It is reported that two regions are formed in the samples: one is a difficult deformation region, which needs large force and much energy to deform, and the other is an easy deformation region, which needs relative low energy to deform [26]. Assume that if the primary shear band is formed from the junction of the lateral surface and the top and bottom surfaces, and the shear angle is 45°. Apparently, two cones will be obtained along the straight line, which reflects the lateral stress gradient, as shown in the red fraction in Fig. 6(b). Since the lateral gradient increases with decreasing the aspect ratio, the hydrostatic pressure in Fig. 6(b) is larger than that of Fig. 6(a), which increases the difficulty of deformation. Due to the smaller aspect ratio, the shear paths become short as well, i.e., the shear displacements is smaller than that of the sample with an aspect ratio of 2:1. From Fig. 5(b), a large number of horizontal shear bands can be found on the surface by SEM. Therefore, the large compressive plasticity and the low amplitudes of stress drops are obtained because of the hydrostatic pressure and the aspect ratio. To further increase the lateral constraint between sample and platen, the shear bands density is sharply increasing, as shown in Fig. 5(a). The diameter of the sample is two times of its height which looks like an amplified BMG ribbon. Conner et al. [49] reported that the thin BMG ribbons displayed ductility without failure in bending. When the sample's aspect ratio is 1:2, the difficult deformation region [26] accounts for most of the samples, and more energy and larger force are required to deform in compression. Due to the merge of the two cones (Fig. 6(b)), the effect of the lateral constraint is the strongest that makes the deformation become more homogeneous. Although the sample has accumulated enough elastic energy for shear bands, the shear bands will be hindered immediately and turn to other orientations as soon as they slide.

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Fig. 6. The illustration of the lateral constraint. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

This trend is the reason that the amplitudes of stress drops are extremely small, as mentioned above when the sample's aspect ratio is 1:2. The aspect ratio is so small, which gives rise to the lateral movement of shear bands, and a large number of secondary shear bands can be found as well, as presented in Fig. 5(a). The present work further demonstrates that the different aspect ratios provide the constraints to some extent, which can prevent the propagation of individual shear band and also improve the proliferation of shear bands, resulting in the definite development of plastic deformation in compressive testing. However, the internal mechanism of the different serrated flows may attribute to the free

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volume [16,17] at the atomic scale, and this trend will be explored in the future research.

4. Conclusions In summary, the dynamics of shear bands and serrated flow in a Zr-based BMG are systematically investigated. The compressive ductility of a Zr52.5Cu17.9Ni14.6Al10Ti5 BMG depends on the aspect ratio strongly, while the yield strength almost remains constant. The compressive plasticity increases obviously with decreasing aspect ratio. Based on the elastic energy density analysis, the power-law scaling behavior for the shear avalanche of plasticity gives evidence that the serrated flow dynamics reaches a new state. The elastic energy density is also calculated by the new method and the results agree well with the theoretical calculation. The friction between the sample and the platen will play a significant role that attributes to a lateral constraint, especially with the aspect ratio of 1:1. The investigations of shear plane after failure provide further evidence that the formation and propagation of multiple shear bands are induced by the lateral constraint. The uniaxial stress and the lateral constraint will cause a hydrostatic pressure on the sample close to the platen. The shear bands are controlled by the different stress states, which leads to a power-law-scaling behavior in serrated flows. Generally, the Zrbased BMG serration dynamics in different views is illustrated, which is possibly valid for other BMG systems during deformation.

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

Acknowledgment J.W.Q. would like to acknowledge the financial support of National Natural Science Foundation of China (No. 51371122), the Program for the Innovative Talents of Higher Learning Institutions of Shanxi (2013), and the Youth Natural Science Foundation of Shanxi Province, China (No. 2015021005). Z.H.W. would like to acknowledge the National Natural Science Foundation of China (No. 11390362).

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