Shear banding is a type of plastic flow instability with often adverse implications for cutting and deformation processing of metals. Here, we study the mechanics of plastic flow evolution within single shear bands in Ti- and Ni-based alloy systems. The local shear band displacement profiles are quantitatively mapped at high resolution using a special micromarker technique. The results show that shear bands, once nucleated, evolve by a universal viscous sliding mechanism that is independent of microstructural details. The evolution of local deformation around the band is accurately captured by a momentum diffusion equation based on a Bingham-type flow rule. The predicted band viscosity is very small, compared to those of liquid metals. A plausible explanation for this small viscosity and fluid-like behavior at the band, based on phonon drag, is presented.
Introduction and Background
Shear banding is an oft-occurring phenomenon in large-strain plastic deformation of ductile metals, where an essentially smooth and continuous deformation pattern abruptly gives way to a highly localized flow, restricted to thin planar (shear) bands, with regions outside undergoing negligible deformation. The intense local deformation within the bands often leads to fracture and therefore poses adverse implications for metal shaping processes that rely on large-strain deformation. The work of Zener in the 1940s [1,2] is widely credited with drawing attention to shear banding in metals processing, its problematic aspects, and the birth of research into related phenomena. However, observations of the localized heterogeneous flows (e.g., Lüders bands, “heat lines”) in themselves date much earlier to Tresca and others [3,4]. Subsequent research has established key materials–mechanics aspects promoting shear banding, namely low workability or low strain-hardening capacity (e.g., hexagonal close-packed metals) , low thermal diffusivity [6,7], high strain rates [8,9], and ultra-low temperatures [10,11].
Shear banding is a major challenge in machining of advanced structural metals, e.g., high-strength steels, Ti- and Ni-based superalloys, pervasive in the discrete products manufacturing sector including aerospace, biomedical, and energy applications. The unit material action intrinsic to all cutting processes, e.g., turning, milling, drilling, is shown in Fig. 1. This action involves a hard wedge-shaped tool creating a new surface by removing material in the form of a chip. In the absence of shear banding, a chip of uniform thickness (tc) forms via intense plastic shear that is typically confined to a narrow zone, as shown in Fig. 1(a). In contrast, under shear banding conditions, the periodic localization of plastic flow results in a “saw-tooth” chip morphology. Such a chip is typically characterized by periodic bands of intense localized strains, well-separated by broad segments where the material has undergone little deformation. The challenges arising from shear banding in cutting are twofold. First, periodic band formation induces force oscillations [12,13] that affect machining stability and tool life, while also imposing limitations on the cutting speeds. Second, it has been known that the occurrence of unsteady modes of plastic flow such as segmentation and shear banding  can adversely impact surface finish and microstructure homogeneity of the machined surface [15,16]. In contrast to these undesirable consequences, it is also possible to envision benefits arising from shear banding. For instance, extrapolating from Zener's observations [1,2], cutting may be achieved with reduced forces under shear band flow conditions.
From cutting processes standpoint, there is hence a critical need for understanding how shear bands initiate and evolve, and the mechanics of the transition from uniform to localized flow. Such understanding is of value for not only devising strategies to suppress shear band formation but also for using shear band instability phenomena to our advantage where possible. Additional motivation comes from fundamental interest in understanding the stability of flows in large-strain plasticity—the onset of shear banding being one kind of plastic instability—important for modeling material behavior in deformation processing.
The phenomenon of shear banding has attracted the attention of the manufacturing community following pioneering investigations into cutting behavior of Ti and steels in the 50 s by Shaw and Merchant and coworkers [17,18]. The subsequent works by Recht , Lemaire and Backofen , Komanduri and Schroeder , and Komanduri and Hou  have highlighted the role of temperature rise during machining, and analyzed the onset of shear band instability based on the competition between strain and strain-rate hardening on one hand, and deformation-induced thermal softening on the other. Over the last two decades, finite element simulations have been extensively used to predict shear band formation and overall chip morphology over a range of cutting process parameters. Most of these studies are based on Johnson–Cook type material models, typically with an additional ductile failure criterion based on damage [22–24]. More recently, phenomenological models taking into account dynamic recrystallization processes and associated flow softening mechanisms have been also incorporated into finite element modeling framework [25,26].
While these studies have all focused on the onset of shear band instability, aspects related to the shear band growth following instability onset have received far less attention and thus remained elusive. Systematic measurements of deformation and temperature fields in the vicinity of growing shear bands are extremely few. Preliminary studies using in situ high-speed imaging of cutting have shown that shear bands develop in two distinct nucleation and evolution phases . These measurements have suggested that most of the strain imposition within shear bands occurs during the second evolution phase. On the other hand, the primary role of the nucleation phase is to create a “weak” interface (or plane) over which blocks of material on either side can then undergo relative plastic sliding. Given that most of the localization and concurrent stress relaxation (which causes unloading on the cutting tool) occurs during the latter phase, it is critical to look beyond the nucleation phase and at the ensuing evolution phase. Detailed characterization and improved understanding of the shear band evolution phase will also lead to material-agnostic approaches to controlling shear band flows in processing, as suggested by earlier studies [27,28].
In this study, we expand on our previous observations of shear banding with specific focus on the shear band evolution phase. We use a high-resolution micromarker technique to map local material deformation characteristics around shear bands. Using direct shear band displacement profiles, we deduce that shear bands are intrinsically characterized by a viscous-like flow behavior. We use this fluid mechanics connection to analyze shear band flow evolution using a rate-dependent Bingham fluid model. The model is able to capture the measured shear band displacement profiles in different alloy systems across a range of deformation rates, thus suggesting a common viscous sliding mechanism for the shear band evolution. Importantly, the model also demonstrates that the shear-banded region flows with a very small viscosity, a few mPa·s, typical of liquid metals. We show that the phonon drag is at the origin of viscous flow and small shear band viscosity.
Plane-strain cutting experiments were carried out in three material systems: Ti-6Al-4V alloy (mill annealed condition, hardness: 346 HV), Ni-base superalloy Inconel 718 (age-hardened condition, 458 HV), and commercially pure (CP) Ti (grade 2, annealed condition, 215 HV). These materials were chosen for their propensity to shear band, and, importantly, for their relevance in key industrial applications. Dry cutting experiments (without lubricant application) were carried out on a lathe by feeding the cutting tool edge radially into rotating thin disk-shaped samples. Tungsten carbide tools were used (edge radius, 10 μm) at fixed rake angle and undeformed chip thickness t0 = 125 μm. The width of cuts was ∼ 3 mm, and cutting speed V0 was varied from 0.1 m/s to 12.5 m/s.
Early experiments showed that Ti-6Al-4V chips exhibited saw-tooth chip morphology over the entire range of cutting speeds investigated. At lower V0 (<0.5 m/s) conditions, fracture between low-strain segments was prevalent, while higher speeds resulted in continuous shear banded chips without fracture. With Inconel 718 and CP Ti, a transition in the chip morphology from uniform (Fig. 1(a)) to shear-banded type (Fig. 1(a)) was observed with increasing V0. This transition occurred roughly at 1 m/s in Inconel 718 and 2 m/s in CP Ti. Cutting conditions that resulted in only shear banded chips (without fracture) are focused in this study.
Local material deformation in the vicinity of shear bands was quantified using a high-resolution micromarker technique. In this method, finely spaced concentric markers (∼5 μm spacing)—initially inscribed on the disk workpiece side surface—were tracked in the machined chip to compute local displacements around shear bands. A pointed carbide indenter was used to inscribe these markers on the workpiece. Figure 2(a) shows a typical marker pattern produced by this method. Plane-strain condition at the marked surface during cutting deformation was ensured by constraining this surface using an identical disk sample. The micromarker application to track local material displacements is analogous to the classical viscoplasticity techniques that have been used to map plastic flow fields in machining and deformation processes .
Microstructure of the shear-banded chip specimens was analyzed using optical and electron microscopy techniques. Chip specimens were polished and chemically etched at room temperature for observations under scanning electron microscope (SEM, FEI XL-40) to reveal the flow pattern around shear bands. The chemical etchants used were: Kroll's reagent (92 ml water, 6 ml nitric acid and 2 ml hydrofluoric acid) for Ti-6Al-4V and CP Ti; and Kalling's reagent (5 g copper(II) chloride, 100 ml hydrochloric acid and 100 ml ethanol) for Inconel 718. High-resolution characterization of shear band microstructure was also done using bright-field transmission electron microscopy (TEM) and related diffraction analysis using an FEI Tecnai microscope. Samples for TEM of shear bands were prepared using the focused-ion beam (FIB) lift-out method in a FIB-SEM Dual Beam FEI Nova 200.
The marker method enabled high-resolution measurements of local displacement fields in the vicinity of shear bands in two alloy systems (Ti-6Al-4V and Inconel 718), and illustrated key common features of the localized plastic flow, including their rate dependence. These measurements have also enabled a momentum diffusion-based model for studying localized band displacements fields, and established a common viscous sliding mechanism for the shear band evolution phase.
Shear Band Deformation: Phenomenology.
Figure 2(b) shows an SEM image of a shear-banded Ti-6Al-4V chip produced at V0 = 2 m/s. The typical serrated, saw-tooth morphology of the chip is immediately evident. The micromarkers, which appear as light striations on the chip, were initially marked on the disk workpiece side surface such that they are roughly parallel to the free surface or V0 direction (cf. Fig. 1). Thus, markers in this case are equivalent to streaklines commonly used to visualize flow in fluid mechanics, where local fluid motion is tracked by the injection of colored dyes. Several important observations pertaining to the localized shear band flow can be made. First, it can be seen that markers remain almost parallel (except near the chip rake face in contact with the tool) in the chip segments A and B, indicating nearly homogeneous straining within these segments. However, in between the segments, markers are seen to undergo sharp displacements over a very thin interface—the shear band. The displaced positions of markers on either side of the band, for example labeled 1, 2, and 3 for three markers (Fig. 2(b)), illustrate large displacements at the band. It must be noted that these displacements are simply a result of extreme plastic shear over a thin interface, and not because of fracture. Electron microscopy observations have been used to confirm that fracture did not occur under cutting conditions presented here. For example, see Fig. 3 that shows SEM image of shear banded region from etched chip cross section. A highly sheared morphology at the sliding interface (shear band) between low-strain segments is seen without any evidence of fracture.
Second, the displacements along the length of the interface are found to be nearly uniform at ∼ 100 μm, indicating that the band must have developed predominantly via simultaneous sliding (between segments A and B) across the entire interface. This picture of shear band formation is consistent with our earlier high-speed photography observations of shear-banded chip formation [16,27], which revealed that most of the localized straining takes place during the second shear band sliding (or evolution) phase, following band nucleation. It should be noted that this picture of band formation is a departure from other phenomenological descriptions of shear bands as analogous to moving cracks. In these latter viewpoints, the plastic dissipation is viewed as occurring predominantly around the propagating shear band tip [30,31] (as opposed to across the entire band length or interface). Also, the fact that localized strain profiles develop via gross plastic sliding between adjacent segments implies that the extent of localization can be suppressed by geometrically constraining the sliding phase, despite the onset of shear band instability.
Finally, it may be noted that the curved marker profiles near the sliding interface (Fig. 2(b)) are highly reminiscent of the classical fluid boundary layer profiles that develop, for example, in fluid flow past a rigid interface . In this latter case, fluid motion in the boundary layer near the interface is retarded due to viscous drag. Given the extreme local deformation and associated heating within shear bands, similar viscous-like effects may be expected at the shear band sliding interface. Marker observations around shear bands in Inconel 718 chips revealed essentially identical boundary layer-like band displacement profiles as in Ti-6Al-4V.
Shear Band Displacement Profiles.
The measured shear band displacement profiles for Ti-6Al-4V and Inconel 718 across a range of V0 conditions are summarized in Fig. 4. Figure 4(a) is a schematic of the marker profile (curved line) and reference axes for the displacement plots shown in Figs. 4(b) and 4(c). Here, y = 0 corresponds to the interface across which the segments slide at a velocity VS. This sliding velocity VS is simply the V0 component resolved along the band interface , and can be given by V0/cosϕ (for rake angle ), where ϕ is the shear band angle with respect to V0 (see Fig. 1(b)). The displacements, U(y), parallel to the interface were measured from SEM images of the markers. For consistency, only those markers passing roughly through the center of the band length were considered for the displacement data. Figures 4(b) and 4(c) show the displacement data plotted as a function of normalized y for Ti-6Al-4V and Inconel 718, respectively, at different V0. In these plots, the displacement values were normalized with respect to the maximum displacement Umax on either side of the band, see Fig. 4(a). This normalization enables direct comparison of displacement profiles at different V0 and across different materials.
From the figure, it can be seen that the displacement profiles are characterized by a sigmoidal shape and remarkably similar in both the materials. More importantly, the effect of V0 (or equivalently, VS) seems to be universal. At increased sliding velocity, the displacements are increasingly confined close to the interface. This type of rate dependence is exactly similar to that exhibited by viscous fluids, wherein the boundary layer thickness decreases with the sliding velocity because of the limited (momentum) diffusion times at higher velocities . This suggests that the plastic flow during the shear band sliding phase can be analyzed using fluid flow principles [32,34]. The similarity of displacement profiles in different metal systems also indicates that the shear band flow could be described using a general continuum model without need for precise microstructure details.
The marker method can be also used for estimating the strain and strain-rate profiles of shear bands. Since a band forms along the direction of maximum shear, the local gradient (slope) of the displacement profile (such as in Fig. 4) gives the local shear strain. In Ti-6Al-4V (Fig. 4(b)), the strain at the interface (y = 0) is estimated to be 20 for V0 = 0.5 m/s and as large as 50 for V0 of 5 m/s. Based on the assumption of steady-state sliding across the interface (i.e., at constant VS), the strain rate at the interface is estimated as 106/s for the V0 range investigated. It is well known that metals exhibit a rate-dependent viscous-type plastic behavior under these extreme strain-rate conditions [35–37]. Similar measurements for Inconel 718 showed that the interface shear strain (i.e., at band center) increases from 10 at V0 of 1 m/s to ∼70 at 5 m/s. Such extreme strains must definitely arise from some kind of a fluid-like viscous flow mechanism. Guided by these observations, we analyze the material displacements across the band using a simple viscous slider model. Details of the model are presented next.
Viscous Slider Model.
where is the complementary error function. The availability of experimental data for the shear band displacement profiles (at the end of the sliding process), as in Fig. 4, enables direct validation of our model for the band sliding phase. Note that in Eq. (3), the only adjustable parameter is the kinematic viscosity, ν. The other parameters such as the sliding velocity and the sliding time are known from experiments.
The model comparison with the experimental data is summarized in Fig. 5. In Fig. 5(a), the raw displacement data for Ti-6Al-4V and Inconel 718 at different V0 (or VS), as obtained from the markers (Figs. 4(b) and 4(c)), are used to fit Eq. (3). Unlike a conventional least-squares approach, the marker data are fitted to Eq. (3) by appropriately scaling the y, which provides a corresponding value for ν. It can be seen that all the data from both the material systems roughly collapse onto a single “curve” that is coincident with the theoretical displacement curve (Eq. (3)). It is seen that the model agreement with the data is slightly better at lower velocities. A small but systematic deviation is observed at higher V0 (curves marked by and ×), likely due to the finite rate dependence of μ (or ν) (e.g., shear thinning behavior) above a critical V0. Note that an assumption in our model is that μ is constant, independent of V0. However, the fact that the model agrees well with the data from two different alloy systems and over a range of deformation suggests the validity of the viscous sliding assumption. Importantly, this also points to a common mechanism for the shear band evolution phase that is governed by (plastic) momentum diffusion and largely insensitive to the microstructure-level details.
Another important observation from the model is the shear band viscosity, obtained from fitting the experimental data to Eq. (3). The dynamic viscosity μ = ρν, thus obtained, is shown as a function of V0 in Fig. 5(b) for both the materials. It is seen that μ is in the millipascal second (mPa·s) range for both materials. It may be noted that the viscosity in mPa·s range is almost at the bottom end of the entire viscosity spectrum for materials that ranges over 25 orders of magnitude: from ∼ 10–4 Pa·s for liquid nitrogen to 1021 Pa·s estimated for Earth's crust. Interestingly, the liquid metal viscosity for Ti and Ni-base alloys at their respective melting points also falls exactly in the mPa·s range [38,39]. A few plausible mechanisms underlying this fluid-like shear band sliding behavior are discussed in Sec. 4. Additionally, it is seen that μ decreases with V0. This reduction in μ with V0 suggests shear thinning behavior at the band interface, and likely explains the deviation in the displacement fits in Fig. 5.
The marker-based measurements of shear band displacement fields under plane-strain cutting have provided insights into localized plastic flow during shear band evolution. In particular, the observations have highlighted several common features of shear band evolution in different metal systems. These include accumulation of intense localized strains by relative plastic sliding between two adjacent material blocks along a thin, weak interface (Fig. 2); fluid-like boundary layer structure of the shear band; and thinning of this (plastic) boundary layer with increasing deformation rate or sliding velocity (Fig. 4). A simple viscous slider model, proposed in light of these observations, has captured experimental shear band deformation profiles and their rate dependence in two different material systems.
where y is the perpendicular distance from the band center in the direction of heat flow (e.g., see Fig. 4(a)), t is the sliding time, κ is the thermal diffusivity, and C is the heat capacity. This solution predicts maximum temperature rise is at the band center, which monotonically increases with the sliding time. If we take Ti-6Al-4V (κ = 3 × 10–6 m2/s, C = 600 J/kg/K) as an example, the total sliding time tf is ∼ 70 μs at V0 = 1 m/s (VS = 1.46 m/s). Further assuming that all plastic dissipation is converted into heat, Q can be given by τV S, where τ is the shear band flow stress as before. The cutting force measurements showed that τ is ∼ 400 MPa at V0 = 1 m/s. Using these values, it is seen that the temperature rise () at the band center is only ∼ 300 °C. Using Loewen–Shaw's method  of estimating cutting temperatures, it can be further shown that the shear zone temperature prior to band nucleation is no more than 300 °C. This indicates that the maximum shear band temperature would not have been more than 600 °C. Even at higher cutting velocities (5 m/s), the maximum band temperatures were found to be at most 850 °C. These band temperatures are well below the melting temperatures. In fact, they are even below the phase transformation temperature of 995 ± 15 °C in Ti-6Al-4V.
The absence of phase transformation within shear bands, supporting the above conclusion, has been also confirmed in our study via transmission electron microscopy of the band microstructure. Figure 6 shows bright-field TEM images of shear band microstructures in Ti-6Al-4V (V0 = 1 m/s) and CP Ti (grade 2, V0 = 3 m/s), with the corresponding diffraction patterns shown below. In both cases, the shear band microstructure is seen to be highly refined—undoubtedly, a consequence of extreme plastic shear within the band. The grain size within the band is 20–50 nm in Ti-6Al-4V and ∼ 100–200 nm in CP Ti. An analysis of diffraction patterns showed that the band microstructure is predominantly α phase. No evidence for the β phase was found in both the materials. This observation, particularly in Ti-6Al-4V, is surprising given that this alloy in as-received condition is characterized by a two-phase (α and β) structure. This, therefore, suggests dissolution of the β into the α phase upon extreme deformation. Similar observations pertaining to dissolution of second-phase particles into the matrix (e.g., cementite dissolution into ferrite in steels) under extreme deformation at shear bands and sliding surfaces have been also reported elsewhere [42,43]. TEM investigation of two more shear bands confirmed the repeatability of the β phase disappearance. The absence of β (or of acicular α or which usually form upon rapid cooling from the β phase) indicates that phase transformation has not occurred, consistent with our above temperature calculations.
While the possibility of microscopic melting (for example, at grain/phase boundaries) cannot be entirely ruled out, our band temperature estimates and electron microscopy observations conclusively suggest that the band temperatures are well below the melting point. In view of these considerations, the local Bingham fluid-like behavior at the band cannot be attributed to melting. Other alternative microscopic mechanisms that result in Bingham-type flow processes in a solid crystalline state are lattice/grain boundary diffusion and damping of dislocations due to phonon interactions . In lattice diffusion, the viscosity term (μ) is given by where d is the grain size, k is the Boltzmann constant, T is the temperature (in K), A is a constant (approximately 10–40), Ω is the atomic volume, and Dv is the lattice (vacancy) diffusion coefficient. Taking d to be roughly 100 nm (Fig. 6), T to be 600–800 °C, and using standard values for rest of the parameters (k, Ω, and Dv) , it can be easily seen that the viscosity μ will be about 1012 Pa·s, which is several orders of magnitude greater than the model-predicted values in the range of 10–3 Pa·s. A similar analysis of viscosity assuming grain boundary diffusion again yields μ (∼ 105 Pa·s) that is much larger than the observed value. This shows that lattice and grain boundary diffusion mechanisms do not play a significant role in accommodating shear band plastic flow, and suggests that the low band viscosity should inevitably arise from glide-controlled plasticity.
As noted before, under very large strain rates exceeding 104/s, metal plasticity is characterized by a linear rate-dependent flow. In this large strain-rate regime, the drag on mobile dislocation experienced as a result of its interaction with phonons is considered to be the microscopic mechanism underlying the purely rate-dependent flow . The effective phonon drag force F (per unit length of dislocation) is given as F = Bv, where v is the dislocation velocity and B is the (phonon) drag coefficient. Now, using Orowan's equation relating the strain rate to dislocation velocity and mobile dislocation density (ρm), the macroscopic viscosity term can be written as , where b is the Burgers vector. Taking the theoretical estimate for B (10–4–10–5 Pa·s [45,46]) and b to be 3 × 10–10 m, it can be seen that sufficiently low shear band viscosities of a few m Pa·s require that ρm within the band to be about 1015–1017/m2. These densities, while much larger than those in typical cold-worked metals, are not unrealistic given the extreme deformation within shear bands. Therefore, of all mechanisms, phonon dislocation damping seems to be most plausible mechanism underlying viscous shear band flow.
A study has been made of the shear banding phenomenon in cutting of Ti-6Al-4V, Inconel 718, and CP Ti alloy systems. Local shear band displacement profiles measured using a micromarker technique have revealed that shear band, once nucleated, evolves by a common sliding mechanism characterized by viscous flow. A slider model with a Bingham-type rheology for the shear band material, proposed for shear band evolution, has reproduced the experimental displacement profiles well. Interestingly, the shear band viscosities predicted by the model are very small (few mPa·s), and comparable to those of liquid metals at their melting point. Possible microscopic mechanisms that contribute to this fluid-like flow at the band are analyzed. It is shown that phonon drag on dislocation motion is the most likely origin of viscous shear banding.
Authors acknowledge the support from the Department of Industrial and Systems Engineering at Texas A&M University and Texas A&M Engineering Experiment Station (TEES).
US Army Research Office (Award W911NF-15-1-0591).
National Science Foundation (Grant No. DMR 1610094).
US Department of Energy EERE program (Award DE-EE0007868).