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Nucleation properties of isolated shear bands

Shear banding, or localization of intense strains along narrow bands, is a plastic instability in solids with important implications for material failure in a wide range of materials and across length-scales. In this paper, we report on a series of experiments on the nucleation of single isolated shear bands in three model alloys. Nucleation kinetics of isolated bands and characteristic stresses are studied using high-speed in situ imaging and parallel force measurements. The results demonstrate the existence of a critical shear stress required for band nucleation. The nucleation stress bears little dependence on the normal stress and is proportional to the shear modulus. These properties are quite akin to those governing the onset of dislocation slip in crystalline solids. A change in the flow mode from shear banding to homogeneous plastic flow occurs at stress levels below the nucleation stress. Phase diagrams delineating the strain, strain rate and temperature domains where these two contrasting flow modes occur are presented. Our work enables interpretation of shear band nucleation as a crystal lattice instability due to (stress-assisted) breakdown of dislocation barriers, with quantitative experimental support in terms of stresses and the activation energy.


Introduction
When ductile metals are subjected to large plastic strains, particularly at high strain rates ( 10 3 /s), it is often found that the deformation pattern is not smooth and homogeneous, but shows marked localization of strain along narrow bands [1]. This phenomenon, called shear banding, has important implications since strain localization often acts as a prelude to material failure. This understandably has adverse repercussions for material's ductility and integrity in deformation processing [2][3][4] and high-rate structural applications such as penetration and impact [5][6][7]. Clearly, a detailed understanding of shear band onset characteristics is of critical importance both for quantifying conditions required for banding and designing practical means to control it [6,8,9]. While most extensively studied in the context of polycrystalline metals, shear banding has been also widely observed in other material systems such as metallic glasses [10][11][12][13], polymers [14,15] and granular media [16,17].
The actual appearance of shear bands are dependent to some extent on the material itself, and comprehensive reviews of shear bands can be found in Refs. [1, 18,19]. Nevertheless, recent in situ observations [20,21] suggest that the sequence of steps leading to a fully developed band with high strain localization in its vicinity can be demarcated into distinct stages of nucleation and growth, irrespective of the material system. During the nucleation stage, a strain inhomogeneity nucleates in an otherwise homogeneous strain field, followed by rapid propagation of its front to establish a thin and well-defined shear band interface across the whole specimen. The subsequent growth stage is characterized by accumulation of large plastic strains in the immediate vicinity of this interface, while regions away from the band essentially slide as rigid bodies. In this picture, the minimum shear stress required to nucleate a thin interface may be regarded as the nucleation stress, and stress that drives the localized flow development as the growth stress, of shear bands themselves [22].
The present paper is concerned with understanding the nucleation properties of shear bands in ductile metals, with a specific focus on nucleation stresses. Whereas considerable progress has been made in modeling shear band nucleation and propagation aspects [23][24][25][26], direct experimental characterization of band nucleation has proved a difficult problem. This is primarily because high strain-rate loading configurations such as impact, dynamic compression-shear testing, punch testing and explosive loading of thick-walled cylinders typ-ically used to study shear banding are not suited for isolation of single bands. Furthermore, the small timescales (typically a few µs) involved in band nucleation pose additional experimental challenges. As a result, measurements of band nucleation properties (dynamics, stresses, etc.) in metals have been quite limited. It may be also noted that, even in metallic glasses where shear band nucleation stresses are often reported, stresses are not directly measured but usually extracted from the nanoindentation tests [27][28][29].
In our prior work, we have devised a novel 2D shear configuration wherein isolated shear bands can be formed and examined independently without interfering effects from other bands [20,22]. This has enabled in situ observations of single shear bands, and elucidated several key features of shear banding such as band nucleation, propagation kinetics and kinematics of localized flow (boundary layer) evolution during the secondary growth stage [21].
In the present study, using this configuration, we probe nucleation properties of isolated shear bands in three different model, low melting-point alloys. In situ high-speed imaging and force measurements are used to decouple shear band nucleation from the subsequent growth stage, and thereby accurately determine equivalent stresses. The principal finding from our study is that the band nucleation stress τ C is a material-dependent constant that is insensitive to external loading conditions and ≈ 0.05µ 0 , with µ 0 being the shear modulus.
A transition in the flow pattern from shear banding to homogeneous-type at stress levels below τ C is also unambiguously demonstrated using our in situ flow observations. Based on the results, phase diagrams demarcating shear banding and homogeneous flow modes are constructed.
The remainder of the paper is organized as follows. The shear loading setup, imaging methods and material systems used in the study are described in Sec. 2. In situ observations of shear band dynamics and calculation of nucleation stresses are presented in Sec. 3. The transition between shear banding and homogeneous flow modes, and phase diagrams showing where each of these regimes occur, are also presented in this section. Implications of the findings and a microscopic interpretation of shear band nucleation as a mechanical instability involving sudden breakdown of dislocation barriers are discussed in Sec. 4. Concluding remarks are presented in Sec. 5.

Experimental
The loading configuration used to study shear bands is a 2D (plane-strain) cutting configuration, as shown in Fig. 1. In this configuration, material is subjected to nominal simple shear in a narrow deformation zone (red shaded area OA, Fig. 1) as a thin layer of material of predefined thickness t 0 is removed in the form of a chip by a sharp wedge-shaped tool.
The tool inclination α (Fig. 1) controls the level of imposed shear strain (γ), while strain rate (γ) in the deformation zone is primarily determined by the deformation velocity V 0 . For instance, shear strains in the range of 1-5 and strain rates from 10 to 10 5 /s can be conveniently accessed by controlling these parameters. More importantly, for the present purposes, this shear configuration allows formation of isolated shear bands, i.e., one at a time, and at an a priori known location (tool tip) [21,22]. This feature is central to the present study as it allows shear band nucleation to be directly examined in situ. Furthermore, a large number of shear bands (> 200) can be generated in a single experiment or "cut", which offers additional benefits in regards to data repeatability verification and statistical analysis of band nucleation attributes such as stresses and propagation velocity. Some of these unique aspects of the cutting geometry in relation to study of shear bands appear to have been recognized more than 50 years ago by Recht [30]. Several recent studies, e.g., Refs. [31][32][33], also suggest a renewed interest in utilizing cutting-deformation framework for studying large-strain deformation phenomena, including shear banding, flow instability and ductile fracture. Similar 2D configurations have also proved invaluable for isolating frictional wave phenomena at soft adhesive interfaces [34].
The material systems studied are three low melting-point (T m ) bismuth-based eutectic alloys (labeled Alloys 1-3) with melting points of 47 • C, 70 • C and 138 • C, respectively. The nominal composition and thermophysical properties of these alloys are given in Table 1. The choice of these alloys was governed by the fact that they exhibit shear banding at strain rates multiple orders smaller than the typical rates (> 10 3 /s) at which banding occurs in engineering alloys such as Ti and steels [21]. This means that shear band dynamics can be studied in situ at high resolution, both spatially and temporally, which is highly difficult, if not impossible, to achieve in conventional high T m materials. Furthermore, the deformation behavior of these alloys under ambient temperature conditions is such that the flow stress is highly sensitive to strain rate, with little dependence on strain; for example, see compression test data provided in Fig. S1 (Supplementary Material). In all three alloys, the ambient temperature flow stress σ varies with effective strain rate (ε) as: σ ∝ε m , where m is in the range of 0.06-0.13 (depending on the alloy). It is important to note that this type of highly rate-dependent behavior (with negligible strain hardening) is a common characteristic of metals deforming at very high strain rates, typically > 10 3 /s [35]. Therefore, low T m alloys are attractive model materials to experimentally 'simulate' the most essential features of high strain-rate behavior of metals (including shear banding) at low deformation speeds. The shearing experiments were carried out using a freshly ground high-speed steel tool (edge radius ∼ 5 µm) at different V 0 in the range of 0.005-10 mm/s. This corresponds to a strain rate variation over 4 orders of magnitude from ∼ 0.1 to 10 3 /s. The t 0 was kept in the range of 150 µm to 500 µm, and α between −20 • to +40 • . The effects of ambient temperature were studied by performing experiments at different pre-heat/cool temperatures (T 0 ) from −20 • C to 65 • C. In these experiments, both the sample and tool-holder assembly were maintained at the desired temperature using dry ice or a heat gun.
In all the experiments, force components parallel and perpendicular to V 0 direction ( Fig. 1) were measured using a piezoelectric force sensor (Kistler 9129AA) mounted directly under the tool. These measurements were used for stress calculations.
Direct time-resolved observations of the plastic flow were made using a high-speed CMOS camera (pco dimax HS4), synchronized with the force sensor. The plane-strain condition at the specimen side-surface being imaged was ensured by lightly constraining this side using a transparent sapphire plate and imaging through this plate [21]. The spatial resolution of our imaging was 0.98 µm (per pixel). Although the camera is capable of recording image sequences at up to 50,000 frames per second, frame rates with a 100-200 µs inter-frame time (depending on V 0 ) were found to be adequate for capturing the shear band dynamics, especially given the low deformation speeds under which shear bands can be produced in low T m alloys [21]. Quantitative full-field displacement data were obtained by analyzing the high-speed image sequence using a correlation-based image processing method called as Particle Image Velocimetry (PIV) [36]. Using PIV displacement data, the plastic flow field characteristics were analyzed using streaklines, grid deformation, strain and strain rate maps.
This imaging and image analysis also enabled us to establish direct correlations between the macroscopic force traces and shear band dynamics at the mesoscale. A detailed description of our experimental setup and full-field deformation analysis can be found in Ref. [21].

Results
In situ imaging, image analysis and force measurements have provided complete characterization of the shear band dynamics, and enabled characterization of nucleation stresses across different strain rates and temperatures.

General attributes of shear banding
We begin with a synopsis of the primary experimental observations pertaining to shear band attributes and their dynamics. Figure 2 shows a snapshot of a typical experiment (α = 0 • , t 0 = 250 µm) where the flow is dominated by periodic shear banding. In this experiment, V 0 = 1.5 mm/s, which corresponds to a nominal strain rate of ∼ 50 /s. Figure 2(a) shows the PIV-measured shear strain field and streaklines (white lines) superimposed on a high-speed image. The deformed material (chip) is clearly characterized by two distinct regions -thin bands of high strain separated by low-strain segments. The typical shear band strain is ∼ 8, an order of magnitude higher than in the surrounding regions. The large strain localization within shear bands can be also seen from Fig. 2(b), which shows an artificial deformed grid on top of the image. This was obtained by initially overlapping a square grid (40 µm × 40 µm) on the undeformed workpiece and tracking/updating each point on the grid lines during the deformation process. The stark contrast in the extent of deformation between shear bands and surrounding regions is evident. While the band material is highly sheared, as can be judged based on relative positions of streaklines (cyan color, Fig. 2(b)) on either side of the band, material segments in between the bands are only slightly distorted. It should be also noted that shear bands seen here are macroscopic, in the sense that they traverse across the entire specimen width (dimension normal to the viewing plane in Fig. 2) [21].
The force vs. time plot during shear banding is shown in Fig. 2(c), with F 1 and F 2 being the horizontal and vertical forces, respectively. Synchronous high-amplitude oscillations are seen in both the directions, where each oscillation corresponds to formation of a single band.
That the oscillations (therefore, shear bands) are highly periodic and equally spaced in time suggests that banding is not a stochastic process. Importantly, the fact that shear bands form in a sequential manner, one at a time, affords opportunities for making time-resolved observations of single band initiation and development.  It is also important to note that, during this sliding process, the material remains continuous across the band despite large strains. It is only during the later stages of sliding that a crack forms at the free surface; for example, see at arrow in frame E, where the broken streakline indicates material separation at this location. By frame F , it can be seen that large strain levels > 6 develop around the shear band, with the band also developing a characteristic thickness of ∼ 20 µm. The shear-steps at the free surface are also developed during this sliding stage.
The sequence of steps leading to formation of a fully developed band is now clear. While the nucleation stage establishes band interface and orientation, majority of the localized strains (80%) develop during the subsequent sliding/growth phase. Once the growth stage comes to a halt, marked by minimum in F 1 , another band initiates in the neighboring workpiece region, and the process is repeated.
We also note that the above two-stage mechanism of shear banding is not specific to current model material systems, but has been observed also in other alloys including titanium, cold-worked brass and nickel-base superalloys [20]. In the present study, the two-stage mechanism was also confirmed under different conditions of V 0 , α and t 0 . In situ observations of banding at α = 20 • can be found in Fig. S4 (Supplementary Material), for comparison.

Nucleation stresses
From the above in situ observations, it is clear that shear band nucleation, marking the onset of plastic instability, can be decoupled from the strain-intensive growth stage. Moreover, the fact that nucleation of a thin, well-defined band OO (see frame C, Fig. 3) coincides with the peak in the primary force F 1 allows us to define and measure the characteristic nucleation stress (τ C ) required for its formation. τ C calculation is illustrated using the force plot in Fig. 4, where force values at band initiation, F 1,C and F 2,C , are marked using red • and × respectively. It may be noted here, that while band initiation strictly coincides with the maximum in F 1 , the peak in the tangential force F 2 occurs a little after band has initiated. Similar observations were made also in experiments at different α (see Fig. S4, Supplementary Material). Based on F 1,C and F 2,C , τ C for each band can be thus obtained from the resultant shear force parallel to band and band area. The normal stress σ C acting on the band during its initiation can be similarly derived from the resultant normal force. Mathematically: where F shear and F norm are the resolved shear and normal forces respectively, A S = bt 0 / sin φ is area of a freshly nucleated band, b is the thickness dimension of the sample, and φ as before is band orientation with respect to V 0 (Fig. 3). Measurements over some 200+ bands produced under identical conditions as in Fig. 3 show that τ C is 71 ± 2 MPa, while σ C is 91 ± 3 MPa. Note that in Eq. 1, we do not consider the inertial force components that usually arise in cutting due to momentum changes across the deformation zone (red shaded area OA, Fig. 1) [37,38]. While this approximation is justified at relatively low speeds used in the present work (since the inertial forces are about 8-10 orders of magnitude lower than the deformation forces, F 1 and F 2 ), one should be careful to include the inertial effects at high cutting speeds 10 m/s [39].
Additional experiments over a wide range of testing parameters show that τ C is likely a material constant. This is demonstrated in it is startling to observe that all the points from various experiments fall on a single master line (black dashed line), whose slope is given by τ C = 70 MPa. It is also seen that any deviation in the linearity from this line is of the same order of the scatter in the force/area data itself. These observations make clear that τ C is an intrinsic material characteristic.
The dependence of F norm on A S is shown in Fig. 5(b). It is evident that no single line can be drawn through the data, although it appears that points align themselves along different families, depending on α. In contrast to single τ C , σ C varies over a 50-125 MPa range.
Therefore, it appears that τ C also bears no relationship to the normal stress component acting on the band plane. This is clearly akin to the critical resolved shear stress (CRSS) concept used in crystal plasticity for describing the dislocation slip onset [40]. In fact, additional experiments in the temperature range of -30 • C to 60 • C showed that τ C remains independent of the ambient temperature and is about 0.05-0.06 times the corresponding shear modulus (µ 0 ) value, except close the melting point where τ C /µ 0 ratio was found to be somewhat higher at 0.08.
It is important to note that the above observations pertaining to τ C are no mere coincidence or peculiar to the idiosyncrasies of Wood's metal, since shear banding in the other two alloys investigated in this study was also characterized by a unique τ C . Stress data for Alloy 1 (T m = 47 • C) and Alloy 3 (T m = 138 • C) are shown in Table 2 at three different α (V 0 = 1 mm/s). It is again of interest to note that each material is characterized by a characteristic τ C regardless of the normal stress. While the τ C values themselves are different depending on the alloy (∼ 55 MPa for Alloy 1 and 86 MPa for Alloy 3), τ C /µ 0 ratio was again found to be ≈ 0.05 in both the alloys. Although empirical in nature and tested only over a rather limited range of temperatures, these observations suggest a potential scaling relationship between τ C and µ 0 , similar to that between the theoretical strength and modulus [41].

Shear banding to homogeneous flow transition
The observation of constant τ C over a range of conditions, coupled with the fact that shear band plane coincides with the maximum shear direction, suggests a condition for banding onset based on maximum shear stress. This also suggests a transition in the flow mode to homogeneous-type when the shearing stress τ is < τ C . This transition was explored in our experiments by altering the material's shear flow stress using external strain rate and temperature as control parameters. As an example, Figure 6 shows in situ observations of the plastic flow in Wood's metal, where the artificial grid (obtained using PIV flow analysis) is shown superimposed on the raw images. The loading condition in Fig. 6(a) is identical to that in Fig. 3, except that V 0 = 0.01 mm/s and the correspondingγ ∼ 0.1 /s are two orders lower. Figure 6 It is important to note that the principal deformation process in Fig. 6 is one of shear deformation confined along a thin zone/plane OA. This, for instance, can be clearly observed from the deformation of square grids into elongated rhomboid shapes as material exits the plane OA (Fig. 6). This in turn suggests that the material's shear flow stress τ along this plane can be estimated using measured forces. In fact, replacing F 1,C and F 2,C terms in Eq. 1 with steady-state forces F 1 and F 2 , and φ with the shear plane orientation with respect to V 0 , reduces τ C calculation to τ for the steady-state homogeneous flow. This calculation shows that τ for the two cases shown in Fig. 6 are 52 MPa and 58 MPa, respectively. It is evident that these values are well below the critical τ C required for band nucleation.
The transition between the two flow modes and corresponding stress attributes are further elaborated in Fig. 7  It is also of interest to note that in the regionγ γ C , τ τ C , irrespective of the strain (Fig. 8). This should not be surprising given the weak strain-dependence noted for the present alloy (see Fig. S1). Therefore, the critical stress condition for flow transition in the present case reduces to a critical rate condition (in the absence of temperature effects). This equivalence can be also demonstrated from the alloy's constitutive description (τ = Kγ m at 298 K), whereγ =γ C = 10 /s results in a τ of 75 MPa, close to the measured τ C .

Temperature effects
The ambient temperature effects on the flow transition and band nucleation characteristics Results from in situ observations of the flow are plotted using T 0 vs.γ phase diagram, as shown in Fig. 9. As before, the blue •'s and red 's represent the homogeneous flow and shear banding modes, respectively. It is clear from the phase diagram that temperature has a remarkable effect on the critical rateγ C at which transition occurs. For example, lowering the temperature to −3 • C is seen to reduceγ C to 1 /s, insomuch by an order of magnitude compared to the room-temperature value. In fact, it was found that the critical rate at which flow transition occurs can be approximately described asγ C ∝ exp(−1/T 0 ), see black dashed line in Fig. 9. We will discuss the origin of this exponential scaling in Sec. 4. However, for present purposes, it is sufficient to note that the measured τ values in the vicinity of this boundary were all found to be close to τ C . This suggests that τ C is also insensitive to T 0 , at least in the investigated temperature range.
Note that in our study, deformation-induced plastic heating and associated temperature rise effects are minimized due to the low cutting speeds (characteristic Péclet numbers 1). That the temperature rise during cutting is small and no greater than 4-5 • C even at the highest V 0 = 10 mm/s has been established in our earlier study using thermocouple measurements [21]. Given that the maximum change in the flow stress (∼ 2 MPa) due to this temperature rise is sufficiently small and well within the experimental uncertainty, the plastic heating effects can be ignored in the present study.
Taken together with the observations in Figs. 7 and 8, this reaffirms that the exact strain, strain rate and temperature conditions that mark the flow transition are all governed by a common τ τ C condition.

Discussion
It is well-established that the condition for intense strain localization along a shear band is that, following the initiation of a thin band plane somewhere in the material, the next increment of strain-hardening is cancelled out by the accompanying material softening, mediated either by deformation-induced temperature rise [30,35] or other microstructural mechanisms [7,18]. The present study has focused on how such a thin plane is initiated in the first place.
Using in situ imaging of isolated shear bands, it is shown that shear band initiation occurs by nucleation of strain inhomogeneity at a stress concentration (tool tip in the present case, there exists a minimum shear stress τ C required for nucleating a band, with this stress being a physical characteristic of the material and proportional to the shear modulus. This is clearly brought out in our τ C measurements over a wide range of experimental conditions (Fig. 5), and also demonstrated across three alloys with different melting points (Table 2). That at stress levels < τ C , a different flow mode -homogeneous flow -without shear bands occurs is also unambiguously established using our in situ experiments (Fig. 6). The domains where homogeneous and shear banding flow modes operate are illustrated using phase diagrams ( Figs. 8 and 9).
The critical stress condition for the flow transition was demonstrated in Figs. 7 and 8 using direct stress measurements. Additional support for this condition from a physical standpoint is presented in Fig. 10, where the critical rate-temperature combinations marking the flow transition are plotted as log(γ) vs. 1/T 0 . It is seen that all the data fall on a straight line, with the deviations being within experimental uncertainty of accurately determining the criticalγ C for flow transition. In fact, it turns out that a generalized thermo-viscoplastic material with a flow stress dependence of the form τ ∝ {γ exp(Q/RT )} m [35] exhibits exactly this type of linear scaling between rate and temperature at a constant τ . In this picture, Q is the activation energy for plastic flow, R is the universal gas constant, and m as before is the rate sensitivity. For the data plotted in Fig. 10, the best straight line fit (black dashed line) is obtained at τ = 73 MPa and Q = 42 kJ/mol. Clearly, this τ is close to the τ C measured independently in our other experiments (Fig. 5). Equally importantly, Q is close to the activation energy for self-diffusion for all the constitutive elements [42]. This is suggestive of an underlying plastic flow mechanism where the rate-controlling step is vacancy diffusion to or from the climbing dislocations held up at discrete obstacles [43].
Based on foregoing observations and analysis, a plausible microscopic mechanism by which a thin shear band plane nucleates can be now discussed. Consider, for instance, dislocationmediated plastic flow connected with dislocation jumps across potential obstacles (e.g., precipitates, second-phase particles, forest dislocations), and let τ C be the shear stress required to set the dislocation in motion around the obstacles in the absence of any thermal contribution.
At temperatures above absolute zero, even if the applied stress is less than τ C , dislocation can overcome the obstacle with the help of thermal fluctuations, provided sufficient time is made available for it to make the jump. However, at high strain rates, the specimen's internal plastic rate can lag the external displacement rate, and when stress reaches a critical value, one can envision breakdown of piled-up dislocations along a plane as a sudden 'burst'. In this picture, shear band nucleation at a critical τ C is associated with strain transients resulting from sudden breakdown of dislocation barriers. If the temperature rise or structural changes accompanying the pile-up breakthrough are such that they overcome the strain/strain-rate hardening effect, strain localization results along a freshly nucleated band. While shear band initiation based on this type of dislocation pile-up 'avalanche' mechanism has been postulated before by several authors [44][45][46][47] based on theoretical considerations, we believe our study provides the first quantitative experimental support in terms of activation energy and nucleation stress. It is of interest to note that formation of microscopic slip bands is also known to be mediated by similar dislocation pile-up breakdown mechanisms [48].
It is noteworthy that the above microscopic description of shear band nucleation is con-sistent with our earlier observations that normal stresses play a little role in band nucleation, as expected for any dislocation-mediated process. Furthermore, the fact that τ C scales with µ 0 is in line with the classical rate-dependent plasticity theory where the athermal stress component is often taken to be proportional to the shear modulus [43,49]. Importantly, this description also provides a unified explanation for several important observations related to shear banding in metals. For example, factors that promote shear banding such as multi-phase or precipitation-hardened microstructures (as opposed to single-phase) [50], very low temperatures [51], high deformation rates and high levels of cold-work [2] -all of which restrict dislocation motion -can be understood from the above dislocation basis.
However, the precise mechanism of how dislocation slip that is confined to specific crystallographic planes can result in sample-scale bands that cut across several grains remains an open problem [52].
It is pertinent to also briefly discuss possible generalization of our results to other metals, beyond the model alloys investigated in this study. In this context, we compare our results with prior, albeit limited, data on band nucleation stresses. Using dynamic torsional experiments, Duffy's group reported τ C values for steels in the range of 350-1100 MPa, depending on the material's microstructure, composition and prior heat treatment [50]. The corresponding τ C /µ 0 ratios are between 0.01-0.02, substantially lower than observed in the present study. It appears therefore that, as with regular dislocation nucleation [53], the activation energy, and therefore, characteristic stresses required for band nucleation depend considerably on the type and nature of the obstacles/barriers themselves. This hypothesis clearly requires additional study. However, our current work has established the correlation between band dynamics and the corresponding force trace (Fig. 3) so that τ C measurements can be made across different material systems by merely observing the force traces, without need for detailed in situ imaging. Such measurements would be of practical value as they can be used in conjunction with the activation energies to predict the critical strain, strain rate and temperature conditions for the localized flow onset.
Lastly, the dislocations Burgers vector and the fact that dislocations do not 'run away' after breaking through the obstacles but are characterized by a finite (stress-dependent) velocity [44] introduce intrinsic length-and time-scales into the band nucleation problem.
In this regard, detailed examination of shear displacements and band propagation velocities during shear band nucleation, and their possible connection with dislocation dynamics at the microscale, will be likely valuable for placing shear banding within a wider dislocation physics framework.

Conclusions
The nucleation of isolated shear bands in metals has been studied using a special shear loading configuration and in situ imaging using high-speed photography. This enabled band nucleation to be decoupled from the subsequent strain-intensive growth phase. Based on synchronous imaging and force measurements, the existence of a critical shear stress (τ C ) required for band nucleation was established across three different alloy systems. The independence of τ C with regards to normal stresses, its constancy over a wide range of testing conditions, and scaling with the shear modulus (µ 0 ): τ C ≈ 0.05µ 0 , were also demonstrated.
It is shown that these results can be rationalized from the viewpoint that shear band nucleation is a mechanical instability of the crystal lattice arising due to sudden breakdown of dislocation barriers. A smooth, homogeneous flow mode without shear bands was reproduced at stress levels < τ C . Based on these observations, phase diagrams that delineate the strain rate, strain and temperature domains under which each of the flow modes (homogeneous vs. shear banding) occur were developed. It is shown that the observed boundaries demarcating the two flow modes are consistent with τ τ C condition.           Fig. 9. The points are seen to closely fall on a single straight line, indicating that the criticalγ c for flow transition scales as exp(−1/T 0 ). The best straight line fit to the data is shown as a black dashed line.  Table 1: Thermophysical properties of low melting-point alloys at 23

Shear band formation at α = 20 •
The two-stage mechanism of shear band formation shown in Fig. 3 of the main manuscript is common across different experimental parameters. Figure S2 shows single shear band formation in Wood's metal (Alloy 2) at α = 20 • , where 6 high-speed image frames and the corresponding time instances on the force trace are shown. All the essential features of banding dynamics -band nucleation at the tool tip (arrow, frames A), propagation towards the free surface to establish a well-defined band interface (frame C), and sliding along this interface (frames D-F ) -are replicated in this experiment. It is also of interest to note that the formation of interface OO coincides with the peak in F 1 , and sliding occurs under a dropping load. Nucleation stress calculations for the present case shows τ C = 77 MPa, consistent with the τ C values from other experiments (see Fig. 5, main manuscript).
As before, the band orientation φ (46 • ) is again closely aligned with the maximum shear direction, which is at 43 • with respect to V 0 direction.