Slip statistics for a bulk metallic glass composite ... - AIP Publishing

JOURNAL OF APPLIED PHYSICS 124, 185101 (2018)

Slip statistics for a bulk metallic glass composite reflect its ductility Wendelin J. Wright,1,2,a),b) Alan A. Long,3,a) Xiaojun Gu,1 Xin Liu,3 Todd C. Hufnagel,4 and Karin A. Dahmen3 1

Department of Mechanical Engineering, Bucknell University, One Dent Drive, Lewisburg, Pennsylvania 17837, USA 2 Department of Chemical Engineering, Bucknell University, One Dent Drive, Lewisburg, Pennsylvania 17837, USA 3 Department of Physics and Institute of Condensed Matter Theory, University of Illinois at Urbana Champaign, 1110 West Green Street, Urbana, Illinois 61801, USA 4 Department of Materials Science and Engineering, Johns Hopkins University, 3400 N Charles Street, Baltimore, Maryland 21218, USA

(Received 11 August 2018; accepted 20 October 2018; published online 9 November 2018) Serrations in the stress-time curve for a bulk metallic glass composite with microscale crystalline precipitates were measured with exceptionally high temporal resolution and low noise. Similar measurements were made for a more brittle metallic glass that did not contain crystallites but that was also tested in uniaxial compression. Despite significant differences in the structure and stress-strain behavior, the statistics of the serrations for both materials follow a simple mean-field model that describes plastic deformation as arising from avalanches of slipping weak spots. The presence of the crystalline precipitates reduces the number of large slips relative to the number of small slips as recorded in the stress-time data, consistent with the model predictions. The results agree with meanfield predictions for a smaller weakening parameter for the composite than for the monolithic metallic glass; the weakening parameter accounts for the underlying microstructural differences between the two. © 2018 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5051723 A MEAN-FIELD MODEL FOR INTERMITTENT PLASTICITY

A degree of universality1 has been observed recently in the noise that occurs during plastic deformation in a diverse set of materials such as small volumes of crystalline metals,2,3 granular media,4–6 metallic glasses,7–9 and high entropy alloys,10 among others.11–13 A simple model1 provides predictions for the detail-independent aspects of the statistics of the noise and intuition for the underlying micromechanical processes. The fundamental premise of the model is the weak spots in the material slip, and since the weak spots are elastically coupled to one another, they can cause other weak spots to slip in a cascade of slips known as a slip avalanche.1 These avalanches manifest as stress drops when compression testing is performed under displacement control or displacement bursts when testing is performed under load control. In the model, the statistics of the slips (e.g., the size distribution, duration distribution, duration as a function of size, etc.) have been demonstrated to show tuned critical behavior, meaning that the statistics produce predicted power-law exponents over a scaling regime that is limited in extent by a cutoff that is a function of tunable parameters such as the specimen size, applied stress level, strain rate, extent of weakening in the material, and stiffness of the mechanical test system. Furthermore, the dynamics of

a)

W. J. Wright and A. A. Long contributed equally to this work. Author to whom correspondence should be addressed: wendelin@ bucknell.edu

b)

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the slips in the scaling regime show a universal temporal shape (on a material-specific scale).1 In contrast to tuned critical behavior, self-organized critical behavior produces predicted power-law exponents at all length scales with a cutoff determined solely by the system size. Some early experimental papers reported self-organized critical behavior for plasticity of metallic glasses,14 but further studies on the effects of the tuning of various parameters in this paper and others (e.g., Refs. 2, 5–9, and 11) support a tuned critical phenomenon with a tunable size of the scaling regime. We have previously demonstrated7 that the statistics and dynamics of the slip avalanches in a monolithic metallic glass follow twelve different statistical properties predicted by the mean-field model.1 Here, we extend our preliminary analysis of a metallic glass composite with in situ crystalline precipitates using the same data set11 and show that the results are consistent with mean-field predictions for smaller weakening parameters for the composite than for monolithic bulk metallic glasses (BMGs). The presence of the crystalline precipitates effectively lowers the weakening, thereby reducing the number of large slips relative to the number of small slips, consistent with the model predictions. For a monolithic metallic glass, as a shear band propagates, it dilates and weakens; the overall stress level remains nominally constant as plastic deformation proceeds. In contrast, in the BMG composite, there is an increase in the overall stress level with strain between large slip events. The weakening parameter in the mean-field model can be used to describe this range of behavior observed in single and multiphase materials.

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WEAKENING PARAMETER

During constant displacement-rate deformation, the failure stress at a point r that slipped during an avalanche decreases from the local failure stress τ s,r to a diminished value τ d,r .1 The weakening parameter ε is defined in terms of these stresses as ε¼

τ s,r  τ d,r , τ s,r  τ a,r

(1)

where τ a,r is a “sticking” stress at which a failing cell resticks after it has slipped. Often, this sticking stress is assumed to be zero because the universal predictions of the model do not depend on its value. Figure 1 shows plots of stress versus strain from simulations of the mean-field model for two different values of ε, demonstrating the avalanche behavior that can be expected as the weakening parameter ε varies from a large positive value of ε1 (i.e., large weakening) to a smaller value of ε2 (i.e., less weakening than the case of ε1 ).1 A large positive value of ε corresponds to a material that shows weakening, e.g., a typical monolithic metallic glass. A positive value of ε means that slipping regions in a material are weakened by

FIG. 1. Simulations of the mean-field theory for stress as a function of strain for two values of ε during plastic deformation: (a) a large positive value of ε1 (i.e., large weakening); and (b) 0  ε2 , ε1 (i.e., a value of ε2 with less weakening than the case of ε1 ). The beginning of each large avalanche is marked with a dotted vertical line. A few small avalanches are visible in (a) prior to the first and last large avalanche in the figure. In (b), numerous small avalanches are observed, and the large avalanches are smaller compared to those in (a). Again the beginning of each large avalanche is marked with a vertical line. As will be discussed later, for the case of ε2 , the small avalanches themselves can be classified as “smaller” or “larger” based on the size, which manifests as different power-law exponents in the size distribution for the smaller sizes than for the larger ones. The smaller avalanches cannot be distinguished at the scale used in (b). Notice that there is no curvature between points marked with arrows in (a), whereas there is clear curvature between the points marked in (b); the curvature gives rise to an overall increase in stress with strain. See Appendix for simulation details.

the process of slip, such that cells that have already slipped are easier to move again thereby causing large avalanches to occur. A negative value of ε corresponds to hardening. The distribution of avalanche sizes for a material with a positive value of ε is expected to show a limited size-scaling regime in which small avalanches are self-similar and have a broad size distribution with a cutoff that includes large avalanches that show dynamics different from the small avalanche dynamics.1 The large avalanches with sizes outside of this scaling regime recur quasi-periodically while the small avalanches cluster together at short timescales and occur randomly at long timescales.15 As shown in the mean-field simulation in Fig. 1(a) for ε1 , both small and large avalanches are observed. The avalanches fluctuate around a steady-state stress from the onset of plasticity. The distribution of avalanche sizes for a material with a smaller value of ε (i.e., ε2 ) also shows both small and large avalanches as in Fig. 1(b). The large avalanches again recur quasi-periodically, but typically with longer periods between them and with a broader width in the peaked distribution of interevent times.1 A key difference between Figs. 1(a) and 1(b) is that for the smaller value of ε2 , there is more curvature in the stress-strain curve between the minimum stress at the end of any large avalanche to the maximum stress at the start of the next large avalanche. This curvature arises from the plastic strain accumulated by small avalanches that occur between the large avalanches. The corollary of this curvature is that the size of the small avalanches increases with strain. The avalanches fluctuate around an increasing value of stress until the steady state is eventually obtained (the steady state is shown in the figure). Smaller values of ε are correlated with higher values of ductility compared to larger values of ε. Finally, for the case of a negative value of ε (not shown), the average stress increases and small (with a broad size distribution) and large avalanches are observed; as ε decreases to even more negative values, no large avalanches are observed. The range of sizes of the small avalanches decreases for increasing |ε| for both positive and negative values of ε. Not only do the small and large avalanches show distinct size distributions, they also show different propagation dynamics. The small avalanches propagate in a jerky and progressive manner whereas the large avalanches propagate simultaneously across the entire specimen (i.e., they are “system spanning”).7,8 The stress drop rates during large avalanches are approximately one order of magnitude higher than the stress drop rates during small avalanches.7,8 Here, we use data from high-speed, high-resolution testing of a (Zr70Ni10Cu20)82Ta8Al10 composite metallic glass11 to demonstrate the effects on the mean-field statistics of a smaller ε compared to a larger ε for a monolithic metallic glass (Zr45Hf12Nb5Cu15.4Ni12.6Al10). Specifically, this work concerns itself with differences in ε analogous to the differences between ε1 and ε2 : A smaller value of ε (i.e., ε2 ) means that the small slip avalanches will occur over a wider range of stresses as stress increases. Along with an increasing stress level, the maximum avalanche size increases with stress (since at higher stresses, more weak spots are subjected to stresses above their failure stresses causing slip avalanches to persist). Consequently, the stress-integrated distribution of

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avalanche size decays with a more negative (stress-integrated) power-law exponent for the composite (smaller ε) than for the monolithic metallic glass (larger ε). As this illustrates, the power-law exponents for certain statistical distributions are different for the two-phase metallic glass compared to the monolithic metallic glass (i.e., for different values of ε ), but the mean-field theory predicts this difference via the smaller value of ε for the two-phase metallic glass. We also show a scaling collapse for the avalanche distribution that was achieved for the composite material with small weakening, using the critical stress τ c as a single tuning parameter. A scaling collapse is a much more stringent test of the meanfield theory than the power-law exponents of the size distributions of slips alone. Note that while formally both the critical stress τ c and the weakening parameter ε are tuning parameters, the weakening does not need to be known precisely in the mean-field model in order to predict these trends; the only fitting parameter used in the collapses for small or zero weakening is the critical stress τ c .1,2 For zero weakening, and at the critical stress τ c , the material cannot sustain a load and deforms via continuous flow. The critical stress at zero weakening constitutes a continuous nonequilibrium phase transition. MATERIALS AND METHODS

Metallic glasses are amorphous metals that in their monolithic form show high strengths, but typically no strain hardening and limited ductility. A major advance in the development of these materials occurred with the discovery of metallic glasses that solidify with in situ crystalline precipitates [e.g., 16–20]. During processing of these composites, the high temperature melt undergoes partial crystallization by nucleation and growth of a second phase as the temperature decreases. Upon further cooling, the remaining liquid freezes to the glassy state producing a two-phase microstructure of crystalline dendrites or particles in a glassy matrix. Varying the cooling rate affects the size and number density of the crystals. This microstructure results in a more uniform distribution of shear bands throughout deformed specimens compared to the inhomogeneous shear banding that occurs during deformation of fully amorphous metals. Consequently, larger plastic strains are achieved in the composite material under both tension and compression albeit, in some cases, with somewhat reduced strength. (Zr70Ni10Cu20)82Ta8Al10 was cast as 3-mm-diameter rods in copper molds from a master ingot using an arcmelting furnace.16 Microscopy indicates that the crystallites consist of a ductile Ta-rich solid solution with a bodycentered cubic structure and range in diameter from 10 to 30 μm.16,17 Cylindrical specimens were centerless ground to a diameter of 2.7 mm. Electrode discharge machining was used to cut two of the specimens to a length of 8.1 mm with polishing on the ends to ensure a parallelism within 1.5 μm. The third specimen was cut with a diamond saw, and the ends were polished to less exacting tolerances. Specimens were deformed in uniaxial compression at a constant displacement rate to achieve a nominal strain rate of 10−3 s−1 using an Instron 5584 mechanical test system. Data from a

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piezoelectric load cell located beneath the specimen were acquired at 100 kHz21 and Wiener filtered.7 To accomplish the Wiener filtering, a sample of the noise was acquired during a hold segment past the yield point on a specimen of the same geometry and composition, and the fracture data were used as the unit impulse response. Essentially, these data provide the input to the Wiener filter of a general noise profile and the response of the system electronics. The Wiener filter functions as a low-pass filter to attenuate noisy signal components and approximate the original noise-free signal. The Zr45Hf12Nb5Cu15.4Ni12.6Al10 specimens were cast in a similar fashion and tested using the same mechanical test system and protocol, also at a nominal strain rate of 10−3 s−1. The dimensions of the specimens were slightly different with a diameter of 2.5 mm and a length of 7.5 mm, maintaining the 3:1 aspect ratio. RESULTS AND DISCUSSION

Figure 2 comprises representative engineering stressstrain curves for one of the three (Zr70Ni10Cu20)82Ta8Al10 BMG composite specimens and one of the two Zr45 Hf12Nb5Cu15.4Ni12.6Al10 monolithic BMG specimens. The composite specimen sustained 4.5% plastic strain with a yield strength of 1.7 GPa, while the monolithic specimen sustained 1.4% plastic strain with a yield strength of 1.8 GPa. Figure 3 shows a magnification of the plastic flow in both, demonstrating the larger stress increase that occurs with strain in the BMG composite compared to the monolithic BMG. Figure 4 is a plot of a portion of the serrated flow as engineering stress versus time for the BMG composite. The Wiener-filtered data are also shown. The inset shows a rectangle identifying the 40 s of data that are magnified in the main figure. For materials with small weakening (i.e., the case of ε2 ), the model predicts that the distribution D(S, τ) of avalanche sizes S within a small stress binaround stress τ should scale
S , where κ ¼ 1:5 and the as D(S, τ)
S –κ exp – Smax cutoff Smax ¼ Smax (τ) depends on the value τ of the applied stress at which the avalanches are measured. For zero weakening, the model predicts Smax (τ)
f –1=σ , where σ ¼ 0:5, f ; τ c – τ, and τ c is the flow stress (or failure stress) called

FIG. 2. Representative engineering stress-strain curves for Zr45Hf12 Nb5Cu15.4Ni12.6Al10 (monolithic; green) and (Zr70Ni10Cu20)82Ta8Al10 (composite; red).

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FIG. 3. Engineering stress versus strain for Zr45Hf12Nb5Cu15.4Ni12.6Al10 (upper; monolithic; green) and (Zr70Ni10Cu20)82Ta8Al10 (lower; composite; red) for the plastic flow region of Figure 2. Both vertical scales span 100 MPa, and the horizontal axes are the same. The insets show magnifications that span strain values from 0.022 to 0.032. Note that the stress scale is different for the two insets. These are the same data used to perform the avalanche analyses, but the data have been downsampled to 100 Hz in this figure to facilitate ease of plotting.

the critical stress. Generally, Smax increases with the applied stress. If the avalanches are collected over a range of applied stresses from τ ¼ τ 1 to τ ¼ τ 2 and we define f1 ; (τ c – τ 1 ) and f2 ; (τ c – τ 2 ), then the stress integral over the avalanche size distribution for a material with (negligibly) small weakening such as the BMG composite gives the distribution Dint (S) as τð2

ðf2

Dint (S)


D(S, τ) dτ
τ1

Sκ exp(Sf 1=σ ) df

f1 (κþσ)


S

σ

½g(S f2 )  g(Sσ f1 ),

(2)

where g(x) is a universal scaling function. For the case of large S and large f1 and vanishing f2, the second term tends to a constant so that Dint (S) ∼ S(κþσ) . For the case of small S, we can Taylor expand g(Sσ f2 )  g(Sσ f1 )
Sσ ( f2  f1 ) þ . . . ,

so that for the small avalanches, Dint (S)
Sκ . Note that in this context for a small weakening parameter (i.e., the case of ε2 ), both the “smaller” and “larger” avalanches are located within the scaling regime, which yields a different power-law exponent in the size distribution for smaller sizes than for the larger ones. These avalanches all fall within a size-scaling regime (with different power-law slopes) and are distinct from the large avalanches that have sizes larger than those in the scaling regime and that represent the system-spanning slip. A similar argument holds for the complementary cumulative distribution function C(S), which is the integral of Dint (S) from S to infinity, such that for large S, it scales as C(S)
S(κþσ1)
S1 , while for small S, it scales as C(S)
S(κ1)
S0:5 . At sufficiently large weakening (i.e., the case of ε1 ), the model predicts that the avalanche size distribution becomes less dependent on stress and associated regions in the stress-strain curve that have zero curvature. In this case, the model predicts that for all avalanche sizes in the power-law scaling regime C(S)
S(κ1)
S0:5 .1,7 The details for large weakening are given elsewhere.2,4 The results from three composite specimens and two monolithic specimens were used to produce the plots that follow. The stress-integrated complementary cumulative distribution functions (CCDFs) of the stress-drop sizes are shown in the log-log plot of Fig. 5. The monolithic data are shown here to serve as a reference for the composite data that were acquired at the same strain rate. The monolithic data show a single power-law scaling regime with a slope −0.5 as in Ref. 7. The data from the composite show two power-law regimes. The first has a slope of −0.5 and is nearly indistinguishable from the monolithic data as it extends from 0.4 MPa to 1 MPa. The second regime has a slope of −1 and extends from 1 MPa to 3 MPa. These experimental findings are consistent with the model predictions discussed above for materials with small weakening (i.e., ε2 ), with C(S)
S(κþσ1)
S1 for the larger avalanches in the scaling regime while for smaller S, the scaling is C(S)
S(κ1)
S0:5 .1,2

(3)

FIG. 4. Engineering stress versus time for (Zr70Ni10Cu20)82Ta8Al10 (composite) for the plastic portion identified by the rectangle in the inset with the same units. Unfiltered data are shown in black, and Wiener-filtered data are shown in red.

FIG. 5. Complementary cumulative distribution function (CCDF) of stress drop sizes for Zr45Hf12Nb5Cu15.4Ni12.6Al10 (monolithic; green) and (Zr70 Ni10Cu20)82Ta8Al10 (composite; red). The dashed lines are guides to represent the power-law predictions from the mean-field model.

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The data in Fig. 5 span less than a decade in size partly because mechanical testing of metallic glasses poses some unique challenges. As the specimen size increases, the ductility of metallic glasses decreases for a fixed test frame stiffness.22 As ductility decreases, the number of avalanches decreases, thereby limiting the statistics that can be obtained. There is also a practical limit to the size of the largest metallic glass specimen that can be produced, which is imposed by the minimum required critical cooling rate to achieve an amorphous structure for a particular composition. Thus, it is difficult to extend the scaling regime by testing larger specimens. It is also difficult to extend the scaling regime to smaller avalanche sizes due to a combination of factors that include finite data resolution and noise at the low end of the scaling regime. Figure 6 is a log-log plot of the avalanche duration as a function of avalanche size. The power-law slope of this distribution is consistent with the predicted exponent of 0.5 for both materials within the entire scaling regime.1 The avalanche duration in the BMG composite is longer than the duration in the monolithic BMG. Every avalanche event for the three composite specimens was identified, assigned to one of three stress ranges according to (1 – τ=τ c ), and then logarithmically binned in size. For this purpose, τ is the stress at the beginning of a particular avalanche, and τ c is the (non-universal) critical stress, which is given by the maximum stress τ max attained during the test multiplied by a fitting parameter of 1.003, i.e., τ c = 1.003 τ max . The fitting parameter of 1.003 is chosen such that the CCDFs collapse as shown in Fig. 8. The values for τ c for the three specimens under consideration are 1.7946 GPa, 1.8204 GPa, and 1.8856 GPa, which are all within 5% of each other. The maximum stress of the bin with the largest stress values is less than the critical stress by 0.3%. The fact that the maximum stress is less than the critical stress is

FIG. 6. A log-log plot of avalanche duration as a function of avalanche size for Zr45Hf12Nb5Cu15.4Ni12.6Al10 (monolithic; green) and (Zr70Ni10Cu20)82 Ta8Al10 (composite; red). Data without log binning (open symbols) and logbinned data (filled symbols) are shown. The dashed lines are guides to represent the power-law prediction from the mean-field model with a slope of 0.5.

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consistent with the model because a nonzero weakening usually causes the stress to drop in a macroscopic slip avalanche before it can actually reach the critical stress obtained from the scaling collapse for the small avalanches.1 Furthermore, in a finite specimen, the correlation length cannot extend beyond the specimen diameter, further contributing to the same effect.2 The CCDFs for avalanche size for each of the three stress ranges are shown in Fig. 7. Figure 7 demonstrates that for the composite, the avalanche size increases as the applied stress increases, as predicted by the mean-field model and as recently observed in strain-hardening nanocrystals.2,3 Again as the applied stress increases, the maximum avalanche size and the cutoff of the distribution increase. Integrating over a range of stress values to include all avalanches effectively gives a CCDF with a steeper slope for the case of a smaller ε compared to a larger ε. A similar analysis of the avalanche size distributions in different stress windows for the monolithic BMG7 shows that the avalanche size distribution does not change with stress, as is predicted for non-hardening materials, i.e., materials with a larger ε.4 The mean-field model predicts that the CCDFs for different stress ranges should collapse onto a single curve using the mean-field exponents. The only fitting parameter in such a collapse is the material-dependent critical stress τ c . The three distributions in Fig. 7 collapse onto a single scaling function, C(S)(1 – τ=τ c ) –(κ –1)=σ versus the quantity S (1 – τ=τ c )1=σ , as shown in Fig. 8. Here, the value of the material-specific critical stress τ c was tuned to obtain the best collapse. The collapse was achieved using the mean-field exponents κ ¼ 3=2 and σ ¼ 1=2. The black line on top of the collapsed experimental curves shows the predicted scaling function, which is independent of material details.1,2 Figure 9 is a plot of the second moment of stress drop size

FIG. 7. Complementary cumulative distribution function (CCDF) of stress drops for three specimens of (Zr70Ni10Cu20)82Ta8Al10 binned according to the applied stress level; the legend gives the corresponding ranges for (1 – τ=τ c ) , where τ is the stress at the beginning of a particular avalanche and τ c is the (non-universal) critical stress. As the applied stress increases, the avalanche size increases. The error bars represent 95% confidence intervals.

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J. Appl. Phys. 124, 185101 (2018) TABLE I. The average value of the percentage of large avalanches of the total number of avalanches (total = large + small) in each material with the upper and lower bounds of the 95% confidence interval. The composite material has fewer large avalanches than the monolithic samples, as predicted by the model. Only the composite specimens with machined ends were included in the calculation.

FIG. 8. A scaling collapse of the data shown in Figure 7 using κ ¼ 3=2 and σ ¼ 1=2. The solid line shows the predicted scaling function from the Ð1 mean-field theory1,2 given by g(x)
dt eAt tκ . Here, A is a non-universal x

constant. The error bars represent 95% confidence intervals.

versus (1 – τ=τ c ). The second moment follows the predicted power law with a slope of −3 as indicated by the dashed guide line, further confirming the predicted scaling forms for the avalanche statistics. The collapse of the CCDFs and the moment plot constitute a more stringent test of the mean-field model than extracting the correct power-law exponents for data acquired at sufficiently high temporal resolution.23 They also demonstrate that the mean-field model can describe plasticity in a material with both an amorphous phase (which strain softens) and a ductile crystalline phase (which strain hardens). Future work in this area should compare results for a monolithic glass with the same composition but no precipitates to the corresponding two-phase composite to fully

FIG. 9. The second moment of avalanche size in each stress range versus distance from the critical stress. The dashed line is a guide to represent the power-law prediction from the mean-field model with a slope of −3. The error bars represent 95% confidence intervals.

Specimen

Large avalanches (%)

Upper bound (%)

Lower bound (%)

Monolithic Composite

26.41 13.24

31.49 24.85

22.11 7.75

separate the effects of composition and a second phase. The size, amounts, distribution, and composition of the crystalline particles can serve as tuning parameters for the weakening;24–27 similarly, other microstructural phenomenon such as twinning and martensitic phase transformations should serve as tuning parameters in different systems.27,28 Having demonstrated that the statistics of the slip avalanches in the composite BMG agree with the predictions of the mean-field model for materials with small weakening, we can consider the implications for the underlying micromechanisms of deformation. The crystallites in the amorphous matrix likely act as stress concentrators,24–26,29–33 thereby increasing the overall density of shear bands. Indeed the fraction of avalanches that are large is smaller in the composite material than in the monolithic material as shown in Table I (i.e., there are relatively more small avalanches in the composite), consistent with the notion that the crystallites nucleate many small shear bands and thereby increase ductility.24–26,29–33 Furthermore, the mean-field model suggests that inhomogeneities in a material, such as the crystalline precipitates in the BMG composite, lower stress concentrations at the tip of cracks or of slipping regions, so that they cannot easily run away and fail the system.1 By this mechanism, an increased inhomogeneity density for the composite compared to the monolithic BMG widens the damaged region and increases the nucleation size for the catastrophically large avalanches, thereby making the material more ductile. For events of the same size, the stress drop rate is lower for the composite and the duration is longer, which implies that the presence of the crystallites also causes the avalanches to propagate more slowly. Thus, the crystallites nucleate more shear bands that accommodate the plastic strain, but then the crystallites also hinder the propagation of the shear bands that formed. It is the combination of these two factors that delay runaway failure in the composite material. This study highlights an intriguing possibility for the development of new materials test methods. This is particularly relevant for materials that show small weakening for which the size of the largest slip avalanche grows with the applied stress as predicted by the mean-field model.2,11 Furthermore, comparing the slip statistics and slip dynamics to the model predictions can confirm or exclude certain assumptions regarding materials deformation, particularly in systems that pose significant experimental challenges. For example, the deformation of both BMGs and earthquakes has been shown to display tuned critical behavior according to

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the mean-field model.11 Examining subtle features of the dynamics of avalanches in BMGs may prompt new analyses of earthquake catalogs and lead to fresh insights. CONCLUSIONS

Experimental measurements of slip avalanches in a metallic glass with crystalline precipitates agree with the predictions of a mean-field model. In the stress-integrated CCDF of avalanche size, the mean-field theory predicts that the power-law exponent will increase from −0.5 to −1 as material weakening decreases, consistent with the experimental results. Furthermore, a scaling collapse of the stressbinned avalanche size distributions was achieved using a mean-field scaling form with the associated critical exponents κ ¼ 3=2 and σ ¼ 1=2. ACKNOWLEDGMENTS

We thank James Antonaglia, Braden Brinkman, Michael LeBlanc, Will McFaul, and Mo Sheikh for helpful conversations. W.J.W. gratefully acknowledges NSF DMR 1042734 and the Heinemann Family Professorship at Bucknell University. K.A.D. thanks MGA and NSF CBET 1336634 and NSF DMS 1069224 for support. T.C.H. gratefully acknowledges support from NSF DMR 1408686. W.J.W. and K.A.D. thank the Kavli Institute for Theoretical Physics for hospitality and support through the National Science Foundation under NSF PHY17-48958. APPENDIX: SOME DETAILS OF THE SIMULATIONS

The simulation is an exact implementation of the model as described with the ability to add additional stochastic processes into the dynamics of the cell failures. The stress–strain series for typical sections of the simulation signal from the mean-field model4 are shown in Fig. 1 for two values of the weakening parameter ε. The simulation parameters used were number of cells N = 105; ε = 0.006 for Fig. 1(a) and ε = 0 for Fig. 1(b); conservation parameter c = 0.9968; disorder parameter d = 0.1; and stress rate = 0 (for the adiabatic case). The number of cells represents the crosssectional area of the shear band. When a cell fails, its local stress drops. The conservation parameter is the fraction of the stress drop that is distributed to the other cells. The disorder parameter is the fixed spread of the arrest stresses, representing the inhomogeneity of the system. The parameter values were chosen such that the simulation results most resemble the experimental results obtained for monolithic BMGs. When a cell fails, its stress drops to a random arrest stress. This arrest stress is chosen randomly at each failure to model inhomogeneity in the system. In our simulations, the failure stress is set to 1, and the dynamic arrest stress is randomly chosen from the interval [τa – 0.5, τa + 0.5], where τa is the average arrest stress. τa is chosen from the interval [– d/2, + d/2], where the disorder d is a simulation parameter. τa remains fixed throughout the run. The average arrest stress represents inhomogeneity inherent to each cell, whereas the random arrest stress represents the randomly changing environment as a weak spot keeps slipping.

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