Experimental evidence for both progressive and simultaneous shear during quasistatic compression of a bulk metallic glass

Two distinct types of slip events occur during serrated plastic flow of bulk metallic glasses. These events are distinguished not only by their size but also by distinct stress drop rate profiles. Small stress drop serrations have fluctuating stress drop rates (with maximum stress drop rates ranging from 0.3–1 GPa/s), indicating progressive or intermittent propagation of a shear band. The large stress drop serrations are characterized by sharply peaked stress drop rate profiles (with maximum stress drop rates of 1–100 GPa/s). The propagation of a large slip is preceded by a slowly rising stress drop rate that is presumably due to the percolation of slipping weak spots prior to the initiation of shear over the entire shear plane. The onset of the rapid shear event is accompanied by a burst of acoustic emission. These large slips correspond to simultaneous shear with uniform sliding as confirmed by direct high-speed imaging and image correlation. Both small and large slip events occur throughout plastic deformation. The significant differences between these two types require that they be carefully distinguished in both modeling and experimental efforts.


INTRODUCTION
The macroscopically brittle behavior of metallic glasses is caused by the microstructural phenomenon known as shear banding.In quasistatic, constant displacement-rate testing of metallic glasses at temperatures well below the glass transition, this shear banding behavior manifests as a series of sudden stress drops.Materials scientists refer to this behavior as "serrated flow," whereas the physics community describes this general phenomenon of discrete behavior as "avalanches."[3][4][5][6][7][8][9][10][11][12] Metallic glasses, by virtue of their amorphous structure, have spatial variations in the amount of free volume.Regions of excess free volume nucleate shear transformation zones when the metallic glass is subjected to stress, and the collective propagation of shear transformation zones leads to shear banding/avalanche behavior.Recently, a mean field model for plastic deformation based on the premise of elastically coupled slipping weak spots 3,4 has been applied to high temporal-resolution data for the stress drops during constant displacement-rate compression of Zr 45 Hf 12 Nb 5 Cu 15.4 Ni 12.6 Al 10 bulk metallic glass. 5Because the model predictions for both the scaling statistics of the stress drops for avalanche events and the dynamics of individual avalanche events agree with high temporalresolution stress measurements of serrations, that work provides experimental evidence for shear transformation zones as the mechanism of deformation in metallic glasses.It also establishes the existence of two regimes for the avalanches: small stress drop serrations with a power law size distribution and self-similar dynamics and less frequent but almost regularly recurring large stress drop serrations that do not show a power law size distribution.While we previously focused on the small avalanches, 5 we consider the large avalanches in depth here.Note that in this work, we use the term serration to refer to stress data obtained using a load cell, whereas we use the term avalanche to refer to statistical measures of serrations from the mean field model.
The nature of shear band propagation, i.e., progressive versus simultaneous, has been a topic of investigation for some time.Initially, the literature presented an either/or philosophy, i.e., investigating whether shear bands propagate progressively or simultaneously; however, as pointed out by some including Schuh et al. 13 and more recently by Homer, 14 another viable possibility is that shear bands first propagate in a progressive fashion during shear band initiation followed by uniform sliding simultaneously across the entire shear band after initiation is complete or during subsequent activation of the same shear band.High-speed imaging 15,16 has captured the simultaneous propagation of shear bands and demonstrated that uniform shear band sliding occurs with velocities on the order of 1 mm/s during quasistatic deformation, but the images have only captured the largest serrations with the largest stress drops.As suggested by Homer, these experiments may not have the resolution to track the small strains of progressive shear band events. 14dditionally Qu et al. have recently shown that shear bands in metallic glasses that have not fully traversed the specimen show linearly decreasing strain from the ends to the tips, indicative of progressive propagation. 17he use of Wiener filtering to eliminate noise from the stress data reveals that two distinct types of slip events occur during serrated plastic flow.These events are distinguished by their size and stress drop rate profiles.For the relevant figures shown in this work, we plot the stress drop rate as a positive quantity that reflects the rate at which the total applied stress on the specimen is decreasing; this is the avalanche temporal profile or shape.By choosing large events of the same duration, we highlight the sharply peaked stress drop rate profile of the large serrations (with maximum stress drop rates of 1-100 GPa/s).The stress drop rate profiles for events of the same duration are notably similar to each other.The large serrations have stress drop sizes larger than 3.3 MPa for the Zr-based composition under consideration. 5e also demonstrate using high-speed imaging and image correlation that the large avalanches are in fact simultaneous in nature by showing stress drop rate data and images for the same shear band event.In contrast, the small events have stress drop sizes less than 3.3 MPa.These shear bands propagate intermittently so the stress drop rate fluctuates, never decreasing below zero until the avalanche is completed.The maximum stress drop rates for the small avalanches range from 0.3-1 GPa/s.This behavior strongly suggests that these serrations correspond to the progressive propagation of shear bands that do not fully transect the specimen.The small size of these events cannot be resolved by the high-speed cinematography that has been employed to date, which is likely the reason why only simultaneous events have been observed thus far.The significant differences between these two types of events merit separate consideration in both modeling and experimental efforts.

METHODS
3 mm diameter rods of Zr 45 Hf 12 Nb 5 Cu 15.4 Ni 12.6 Al 10 were prepared using arc melting and suction casting and verified to be fully amorphous using x-ray diffraction.Rectangular parallelepiped specimens with nominal dimensions of 6 mm along the loading axis and a cross-sectional area of 1.5 mm Â 2 mm were loaded under constant displacementrate compression at a nominal strain rate of 10 À4 s À1 using an Instron 5584 mechanical test system.A custom loading fixture 18 ensured uniaxial loading of the specimens and increased the stiffness of the test frame since the ductility of metallic glasses is known to depend on both the misalignment of the specimen 19 and the ratio of the specimen to frame stiffnesses. 20The load data were acquired using a Kistler 9031A piezoelectric load cell (60 kN), a Kistler 5010B charge amplifier (180 kHz low pass filter), and a Hi-Techniques Synergy P data acquisition system (40 kHz low pass filter) with a data collection rate of 100 kHz.The data from the fracture event were used as the unit impulse response for the purposes of Wiener filtering.Since highspeed imaging of the fracture event demonstrates that fracture occurs with a timescale on the order of 12.5 ls or less for specimens of this size (which is sufficiently fast that the electronics cannot accurately track the temporal features of the load data during fracture), 16 the data for load as a function of time for the fracture event represent the temporal characteristics of the system electronics rather than true material behavior.Data reflecting the noise of the signal were collected by loading specimens of the same composition and dimensions past the point of yielding and recording under constant load.For more details on the Wiener filtering process, see the supplementary material of Ref. 5.
Avalanche statistics are determined from the stress À time data of serrated flow.The filtered load data are converted to engineering stress values.The beginning of an avalanche is defined mathematically as the point in time at which the derivative of the magnitude of the applied stress with respect to time falls below zero.When the derivative rises above zero, the avalanche ends.The size of the avalanche is given by the difference in stress at these two times, and the avalanche duration is given by the difference in time.Again we note that the time derivative of the magnitude of the stress during the avalanche (the stress drop rate), plotted as a positive quantity and as a function of time, constitutes the temporal shape of the avalanche.For details of the analysis, see the supplementary material of Ref. 5.
Specimens of the same composition and dimensions as those used to generate avalanche statistics were imaged during compression to view shear band propagation at the same strain rate of 10 À4 s À1 .A Vision Research Phantom v310, one-megapixel digital high-speed camera was used.
Images were acquired at a rate of 12.5 kHz with an exposure time of 10 ls.Each specimen nearly fully occupied the 224 pixel Â 624 pixel field of view.The load data and images are synchronized.For more details on the imaging process, see Ref. 16.
Additional specimens of the same composition but with a nominal diameter of 3 mm, an aspect ratio of 2:1, and ascast surfaces were monitored for acoustic emission (AE) activity during compression using two MISTRAS Group PCI-2 cards.One card acquired the AE, and the other card (with a "low frequency modification") acquired the load data from the piezoelectric load cell.The cards were connected via a real-time daisy chain cable to synchronize acquisition of the signals.The MISTRAS S9225 sensor was applied directly to the specimen using hot glue.The S9225 sensor has a bandwidth of 300-1800 kHz and a resonant frequency of 250 kHz.The load and AE data were acquired simultaneously at a data acquisition rate of 2 MHz for the AE data and 100 kHz for the load data using the AEwin software from MISTRAS Group, Inc. Figure 1(b) shows a magnification of the serrated flow with unfiltered and Wiener-filtered data.The pattern of a series of small stress drops followed by a large stress drop is repeated throughout the plastic region.Figures 1-4 are constructed using the same data set as Ref. 5.

RESULTS
The purpose of Figure 2 is three-fold: to establish the existence of the small avalanche scaling regime, to show that the avalanches in the small regime follow power law scaling (with an exponent in accordance with the mean field model), and to demonstrate that the large and small avalanches have different propagation dynamics.Figure 2 where t is time, T is the avalanche duration, and _ S is the stress drop rate.This quantity has been used in simulations of the deformation of crystals as a metric for the mean field model predictions. 21,22The quantity C is proportional to the plastic work per unit volume dissipated as a shear band in a metallic glass propagates (as derived in the supplementary material). 23igure 2 (2) The data show the expected mean field behavior C (S) $ S  2 suggests that the small avalanche regime may in fact extend to slightly higher energies; therefore, as in Ref. 5, we will refer to 3.3 MPa as the upper bound of the small avalanche regime, recognizing that this is not a fixed cutoff because there is a transition region between the small and large avalanche regimes.Small avalanches with energies less than those in the scaling regime are observed; these may be due to small local rearrangements that are avalanche-like, or they may be due to noise in the stress data.The critical feature of Figure 2 is that power law scaling is evident in the small avalanche regime, but not for avalanches of a larger size.This observation alone indicates that there is an important difference between small and large avalanches.The fact that the exponent of the power law scaling agrees with the mean field model gives further confidence in the importance of this distinction and provides motivation to explore other predictions of the mean field model in the context of metallic glasses; however, the subsequent results and analysis presented in this paper do not depend on mean field theory.The stress drop rate for a small avalanche (0.7 MPa) is shown as a function of time in Figure 3(a).This particular avalanche lasts 4.4 ms.The avalanche propagates intermittently as the stress drop rate fluctuates, never decreasing below zero until the avalanche is completed at 4.4 ms.The unfiltered stress and Wiener-filtered stress are also shown as functions of time (the compressive stress is plotted as positive).The maximum stress drop rate is 0.34 GPa/s.A similar plot is shown in Figure 3(b) for an avalanche with a stress drop size of 3.3 MPa.This avalanche shows the characteristics of small avalanches at the upper limit of the small avalanche regime.
When the temporal profiles of large avalanches of varying duration are plotted, the similarity in the shapes of the events is obscured (e.g., see Figure S4 in Ref. 5).For this reason, Figure 4 presents the individual large avalanche profiles (in red) as well as the averaged profile (in black) of those large avalanches in the time bin of 7 À 0.7 ms T 7 þ 0.7 ms but with sizes greater than 10 MPa, of which there are 11 avalanches.Notice the slowly increasing stress drop rate for the large avalanches for the approximately first 3 ms of this avalanche duration.Other work [e.g., Refs.16 and 18] has used a larger threshold for the stress drop rate to define the beginning of an avalanche/serration.Here the smaller threshold results in a longer duration avalanche for the events in the large avalanche regime, but the value of the maximum stress drop rate is consistent with other studies.The supplementary material 23 presents the stress drop rate profile and corresponding image correlation results (Figures S1 and S2) for a serration with a stress drop of 28 MPa.Together these definitively show that the sharply peaked profiles represent simultaneous shear.The large avalanches have maximum stress drop rates ranging from 1-100 GPa/s; for those avalanches shown in Figure 4, the stress drop rates range from 16-45 GPa/s.Figures S3 and  S4 show the stress drop rate profile and image correlation for a serration presented in Ref. 16 and for which a video of the shear band propagation is posted online.

DISCUSSION
The mean field model predicts two different regimes for avalanche behavior [Refs.3 and 4, and references therein]: (1) small events due to propagating fronts that result in minimal plastic strain and (2) large events that proceed simultaneously across the entire shear plane (typically oriented at approximately 45 to the loading axis in metallic glasses).The profiles of avalanches in the small and large regimes are strikingly different both in magnitude and shape suggesting that the modes of propagation in these two regimes are different as predicted by the mean field model.
A small avalanche occurs when a shear band nucleates at a stress concentration (such as the specimen-platen interface, an internal pore, or a surface flaw) and propagates incrementally as a front away from the concentration into a region of lower stress.The reduced driving stress causes the shear band to arrest after only a limited amount of plastic strain and a correspondingly small stress drop.As shown in Figure 5, microscopy of deformed specimens reveals that there are shear bands that do not span the entire specimen and, therefore, presumably formed in a progressive fashion (although to our knowledge, such behavior has not yet been directly captured by a high-speed camera, presumably because of insufficient resolution of available imaging techniques).A specimen that did not fracture is deliberately shown in Figure 5 to avoid confusion between shear bands that formed prior to failure and shear bands that may have formed during the fracture process. 24Small avalanches occur throughout the entire test, i.e., they are not limited to the early stages of plastic deformation.
During a large avalanche, many weak spots slip many times, leading to simultaneous propagation of the slip avalanche and uniform sliding over the entire shear plane, i.e., all points in the shear band are displaced at the same rate.These are system-spanning avalanches, meaning that the shear band transects the entire specimen.Shear stops when the applied stress drops below a critical stress at which point the shear band arrests.The incubation time for the large avalanche events, which can be seen in Figure 4, is presumably due to the time required for the slipping weak spots to percolate and move cooperatively as a single shear plane, leading to larger plastic displacements and thus larger stress drops.A single AE burst is recorded for large avalanches across multiple specimens as shown in Figure 6.Such an AE burst marks the end of the incubation period and the beginning of the rapid displacement event in the shear band; when the AE burst is emitted, the shear band is fully formed.We have not observed an AE burst for every large avalanche (and in rare instances, an avalanche appears to have a burst that occurs later), but the majority of bursts that are observed occur during the rapid increase in stress drop rate.Note that prior work regarding AE activity in metallic glasses 25 does not show the temporal relationship between the AE and stress data as clearly as we show here; we believe this difference is due to the nearly perfect synchronization of our data.
These large stress drops are the ones that are observed when conventional load cells are used to record them although their temporal features are often distorted by low pass filters in the signal conditioning electronics of conventional load cells (hence, a piezoelectric load cell was used here to eliminate the distortion).The high signal to noise ratio achieved for the load signal and the use of Wiener filtering in this work enable the observation of serrations in the small avalanche regime.Large serrations are observed when samples are notched to favor shear band propagation on a single shear plane. 26,27If only large avalanches are recorded, the power law scaling evident in the small avalanche regime is not present because the large avalanches are system spanning, i.e., they transect the system and therefore strongly feel the effects of specimen size.This work demonstrates the existence of two markedly different stress drop rate profiles for shear band propagation in bulk metallic glasses.These profiles differ in both shape and magnitude.Direct confirmation of the large events as simultaneous in combination with their distinct stress drop rate profile implies a different mode of shear for the smaller slips.The existence of two modes of shear has important implications for modeling efforts and provides an impetus for future experiments with sufficient resolution to characterize the progressive shear.The high temporal resolution and low noise of these data combined with Wiener filtering to further reduce noise and to minimize the influence of signal conditioning electronics serve as a model for designing similar experiments for other systems of interest.

CONCLUSIONS
In these experiments, metallic glass specimens were subjected to uniaxial compression at quasistatic strain rates.The results suggest the existence of two types of shear as evidenced by the different stress drop rate profiles.The simultaneous slip of large avalanches is confirmed by direct video imaging, and the occurrence of progressive slip is suggested by microscopy of specimens.A burst of acoustic emission is coincident with the onset of rapid shear during the large avalanche events.

Figure 1 (
Figure 1(a) is a plot of engineering stress versus engineering strain for two specimens of Zr 45 Hf 12 Nb 5 Cu 15.4 Ni 12.6 Al 10 .Figure1(b) shows a magnification of the serrated flow with unfiltered and Wiener-filtered data.The pattern of a series of small stress drops followed by a large stress drop is repeated throughout the plastic region.Figures1-4are constructed using the same data set as Ref.5.The purpose of Figure2is three-fold: to establish the existence of the small avalanche scaling regime, to show that the avalanches in the small regime follow power law scaling (with an exponent in accordance with the mean field model), and to demonstrate that the large and small avalanches have different propagation dynamics.Figure2(a) is a log-log plot Figure 1(a) is a plot of engineering stress versus engineering strain for two specimens of Zr 45 Hf 12 Nb 5 Cu 15.4 Ni 12.6 Al 10 .Figure1(b) shows a magnification of the serrated flow with unfiltered and Wiener-filtered data.The pattern of a series of small stress drops followed by a large stress drop is repeated throughout the plastic region.Figures1-4are constructed using the same data set as Ref.5.The purpose of Figure2is three-fold: to establish the existence of the small avalanche scaling regime, to show that the avalanches in the small regime follow power law scaling (with an exponent in accordance with the mean field model), and to demonstrate that the large and small avalanches have different propagation dynamics.Figure2(a) is a log-log plot FIG. 1.(a) Engineering stress versus engineering strain for the two specimens of Zr 45 Hf 12 Nb 5 Cu 15.4 Ni 12.6 Al 10 deformed under quasistatic uniaxial compression.(b) Engineering stress versus time showing serrated flow in Zr 45 Hf 12 Nb 5 Cu 15.4 Ni 12.6 Al 10 .The inset shows a magnification of one stress drop.The black curves indicate unfiltered stress versus time, and the red curves indicate the Wiener-filtered stress.This is the same data as presented in Ref. 5.

FIG. 3 .
FIG. 3. The stress drop rate _ S versus time for a single avalanche in the small avalanche regime after Wiener filtering is performed.The unfiltered and Wiener-filtered data for stress are also shown.The axes scales are different for the plots.(a) This avalanche lasts approximately 4.4 ms with a stress drop size of 0.7 MPa and a maximum stress drop rate of 0.34 GPa/s.(b) This avalanche lasts approximately 8.6 ms with a stress drop size of 3.3 MPa and a maximum stress drop rate of 0.88 GPa/s.

FIG. 5 .
FIG. 5.An optical micrograph of a shear band that terminates within a metallic glass specimen and thus presumably propagated in a progressive fashion.The specimen edge is at the right.All other edges of the micrograph are enclosed within the specimen.The inset shows a scanning electron micrograph of the tip of the shear band where the shear band terminates.