Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
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Geometric flow control of shear bands by suppression of viscous sliding

Dinakar Sagapuram

Dinakar Sagapuram

Center for Materials Processing and Tribology, Purdue University, West Lafayette, IN, USA

[email protected]

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,
Koushik Viswanathan

Koushik Viswanathan

Center for Materials Processing and Tribology, Purdue University, West Lafayette, IN, USA

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,
Anirban Mahato

Anirban Mahato

Center for Materials Processing and Tribology, Purdue University, West Lafayette, IN, USA

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Narayan K. Sundaram

Narayan K. Sundaram

Department of Civil Engineering, Indian Institute of Science, Bangalore, India

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Rachid M'Saoubi

Rachid M'Saoubi

Seco Tools (UK) Ltd., Springfield Business Park, Alcester, Warwickshire, UK

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Kevin P. Trumble

Kevin P. Trumble

Center for Materials Processing and Tribology, Purdue University, West Lafayette, IN, USA

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and
Srinivasan Chandrasekar

Srinivasan Chandrasekar

Center for Materials Processing and Tribology, Purdue University, West Lafayette, IN, USA

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    Abstract

    Shear banding is a plastic flow instability with highly undesirable consequences for metals processing. While band characteristics have been well studied, general methods to control shear bands are presently lacking. Here, we use high-speed imaging and micro-marker analysis of flow in cutting to reveal the common fundamental mechanism underlying shear banding in metals. The flow unfolds in two distinct phases: an initiation phase followed by a viscous sliding phase in which most of the straining occurs. We show that the second sliding phase is well described by a simple model of two identical fluids being sheared across their interface. The equivalent shear band viscosity computed by fitting the model to experimental displacement profiles is very close in value to typical liquid metal viscosities. The observation of similar displacement profiles across different metals shows that specific microstructure details do not affect the second phase. This also suggests that the principal role of the initiation phase is to generate a weak interface that is susceptible to localized deformation. Importantly, by constraining the sliding phase, we demonstrate a material-agnostic method—passive geometric flow control—that effects complete band suppression in systems which otherwise fail via shear banding.

    1. Introduction

    Mesoscale flow modes in plasticity, such as shear banding [1,2], kinking [3] and sinuous flow [4,5], have a marked effect on the mechanical response of metals. In particular, shear bands have long been recognized as a common mode of non-homogeneous flow in condensed matter systems, ranging from metals [1,2,6] to geomaterials (rocks, soils) [7] and complex fluids [8]. Experimental observations of shear bands date back to more than a century ago [9,10]. Perhaps the most common manifestation of shear band flow is in large-strain deformation of ductile metals, wherein shape changes often occur via localized flow confined to thin planar bands, rather than in a smooth and continuous manner. Shear bands have also elicited broad interest because of their ubiquity and the range of length scales over which they occur (10 nm–100 km).

    Several phenomenological origins for shear bands have been identified to date [11]. The pioneering investigations of high strain-rate metal deformation highlighted the role of catastrophic adiabatic shear, driven by the competition between strain hardening and flow-stress softening due to deformation induced heating [1,12,13]. This mechanism is particularly pronounced in metals having low thermal diffusivity (e.g. Ti and Ni alloys), and under cryogenic conditions where the deformation induced heating is enhanced due to the very small specific heat capacity of the metal [14]. Interestingly, this type of catastrophic shear appears to have been recorded in steels as early as 1928 [13], as highlighted in a recent documentation of milestones in adiabatic shear banding literature [15]. In deformation of Mg alloys, the large differences in the magnitude of critical stresses needed for activating different slip modes result in localization of the flow in ‘soft’ areas, favourably oriented for easy basal slip [16]. And in highly worked metals such as cold-rolled Cu or brass, shear bands are triggered because of exhaustion of strain-hardening capacity that causes the flow to localize easily at material heterogeneities. Machine-system stiffness effects can also play a crucial role, as illustrated in the observations of serrated flow in Al at liquid He temperatures [17]. Despite these micromechanical explanations for banding, the key steps involved in the evolution of localized flow have remained obscure. This is because of the small timescales in which shear bands develop, which hinder direct observations.

    From a practical standpoint, shear banding is a recurrent issue in downstream material removal stages of manufacturing, with adverse consequences for process efficiency and product surface quality, e.g. finish, microstructure and strength homogeneity [18,19]. Equally importantly, as shear banding is a precursor to fracture, it imposes severe limitations in deformation processing of metals [20], especially in hexagonal close packed (hcp) systems such as Mg and Ti.

    Thus, the ability to control shear bands is of vital interest in engineering problems. Control methods based on introducing microstructural constraints [21] offer one possible solution, and have had some success in suppressing bands at the nanoscale in metallic glasses [22]. However, such microstructure modifications are generally not possible in applications where functional properties take precedence in materials selection. Methods to suppress bands based on macroscopic geometric considerations are the more attractive possibility, with potential applicability across the spectrum of material systems and processes.

    Our prior work [23,24] suggested the possibility of a material-agnostic method for controlling shear bands in cutting of metals. In this study, we build on these preliminary observations, and present a fundamental basis and general method of controlling shear bands at the meso- and macro-scales. Knowing the precise flow mechanism during band evolution is the key to development of this method. We deduce this mechanism experimentally using in situ imaging and high-resolution marker techniques. Our experiments show that shear band flow is a two-phase process—the first phase involving the initiation of a local weak zone, followed by a second viscous-like sliding phase that accounts for the large localized strains. This distinction suggests a purely geometric method—passive geometric flow control (PGFC)—to prevent localized flow, by suppression of the second phase. We apply PGFC to demonstrate band suppression in machining, and in forming of large metal samples that would otherwise fail by shear banding.

    2. Experimental procedure

    To understand how shear bands develop in large-strain deformation of metals, we initiated a study where two-dimensional plane strain cutting was used as a framework to impose controlled simple shear. A combination of in situ and ex situ techniques were used to quantitatively characterize the evolution of localized flow within shear bands at high spatial and temporal resolution. Cutting involves removal of a layer of material of predefined thickness (t0) continuously from a workpiece surface in the form of a ‘chip’ by a sharp wedge-shaped tool, as shown in figure 1a. When the flow is smooth (laminar/homogeneous), a continuous chip of uniform thickness (tc) results from simple shear imposed within a confined deformation zone (red shaded area, figure 1a). Under such conditions, the chip is subjected to uniform straining, with the shear strain (γ, typically in the range of 2–10) depending on the ratio tc/t0 and the rake angle α. The strain rate (γ˙) in the deformation zone depends on the cutting velocity (V0) to first order as γ˙=γV0/Δ, where Δ is the deformation zone thickness (typically 100 μm). In the presence of a shear band instability, however, the cutting is characterized by cyclic localization of the flow. This shear band flow manifests as a characteristic ‘saw-tooth’ chip pattern, with thin bands of severe strain localization separated by large segments that are much less strained (figure 1b). Examples of shear bands produced in cutting of Ti-6Al-4V alloy, Ni-based superalloy Inconel 718 and commercially pure (CP) Ti are given in figure 2. In all cases, the chip microstructure shows thin bands of intense shear, separated by mildly deformed grain structures indicative of low strains. The shear bands are macroscopic in length scale, extending across the chip width and thickness, and spanning many grains. This type of flow localization is to be distinguished from grain-level inhomogeneity of plastic flow, as in microscale slip bands [6].

    Figure 1.

    Figure 1. Schematic of plane strain cutting process: (a) continuous chip formation with smooth (laminar/homogeneous) flow, with deformation zone in red; (b) shear band flow with ‘saw-tooth’chip formation; and (c) experimental set-up for in situ imaging of the flow and deformation. The important cutting process parameters are labelled: V0, α, t0 and tc. In the shear banding case, tc is taken to be the maximum thickness of the chip. (Online version in colour.)

    Figure 2.

    Figure 2. Optical micrographs showing microstructure characteristics of typical shear banded chips in machining of (a) Ti-6Al-4V (V0= 1 m s−1), (b) Inconel 718 (V0=3 m s−1) and (c) CP Ti (V0=2 m s−1); α=0° and t0=125 μm. In all cases, notice how the material is highly deformed in the flow localized bands, while the broad segments in between the bands are only lightly deformed. (Online version in colour.)

    (a) Experimental details

    Shear band evolution was studied in Ti alloy Ti-6Al-4V (initial material condition: annealed, Vickers hardness: 346 HV), Ni-based superalloy Inconel 718 (age-hardened, 458 HV), commercially pure (CP) Ti (grade 2, annealed, 215 HV), Mg alloy AZ31B (annealed, 55 HV) and cold-worked 70/30 (single-phase) brass. These systems were chosen based on their pronounced susceptibility to shear banding and diversity in mechanical response. The brass was cold worked by rolling at ambient temperature to impose an effective initial strain of 2.1. Shear banding behaviour was studied at V0 of 10−4 m s−1 to 12.5 m s−1, corresponding to nominal strain rates of 1–105 s−1. Cutting was conducted using high-speed steel or tungsten carbide cutting tools (edge radius, 10 μm) with α=0°. The undeformed chip thickness (t0) was set at 125 μm for Ti-6Al-4V, Inconel 718 and CP Ti; 250 μm for Mg AZ31B; and 600 μm for cold-worked brass. The cutting width in all cases was much greater than t0 (10–25 times), thereby ensuring plane strain.

    The chips produced from Ti-6Al-4V, Mg AZ31B and cold-worked brass exhibited banding over the entire range of V0. In Mg AZ31B, fracture along the shear bands resulted in discrete chip particles at all speeds. By contrast, fracture in Ti-6Al-4V was observed only at low V0 conditions, while V0>0.5 m s−1 resulted in shear band flow without fracture, as in figure 2a. In the case of Inconel 718 and CP Ti, a transition from continuous to shear band flow was observed upon increasing V0. The critical V0 for this transition was around 1 m s−1 for Inconel 718 and 2 m s−1 for CP Ti.

    The deformation zone was observed in situ to capture details of the shear band development for V0 of 10−4–1 m s−1. These experiments were conducted on rectangular samples (35×25×2.5 mm) in a linear cutting configuration, where the sample was moved with respect to the cutting tool. A schematic of the experimental set-up for the in situ observations is shown in figure 1c. To prevent out-of-plane flow, the workpiece side surface was constrained by the use of a glass plate. Most of the imaging experiments were performed using a high-resolution CMOS camera (pco dimax), coupled to an optical microscope (Nikon Optishot), at a resolution of 1.4 μm per pixel. Quantitative characterization and mapping of the localized flow in terms of streaklines,1 velocity, displacements and strain rate was done using particle image velocimetry (PIV). PIV is an image correlation technique commonly used in fluid mechanics for flow visualization and analysis, wherein instantaneous locations of tracer particles seeded into the fluid are tracked to obtain full-field velocity data [25]. In the present adaptation of PIV to study metal plasticity, roughness features, deliberately introduced onto the workpiece side surface using abrasives, played the role of markers.

    Cutting experiments at high V0 (more than 1 m s−1) were conducted on disc shaped samples (3 mm width×50–150 mm diameter) in a rotary configuration on a lathe, where the cutting tool was fed radially into the disc at the rate of t0 per revolution. Plane strain conditions were again ensured by keeping the workpiece width (3 mm) ≫t0 and by sandwiching the workpiece disc between two other identical discs. Shear band flow in this high-speed cutting was analysed using a micro-marker method (ex situ). This method involves inscribing finely spaced (approx. 5 μm distance) parallel concentric markers on the disc side surface prior to cutting using a pointed scratching tool and tracking the resulting marker profiles in the chip. From the displacements, strain and strain rate data were obtained. This method of flow visualization and computation of deformation parameters is analogous to flow-line analyses used previously in studies of indentation [26] and severe sliding contacts [27]. Unlike the in situ imaging, which is limited by loss of spatial resolution at high deformation rates, the marker method is not limited by V0. In this study, the application of marker method has been used to map displacement profiles at speeds up to V0=12.5 m s−1.

    (b) Microstructure characterization

    Microstructure characterization of the localized flow was performed using optical and electron microscopy techniques. This provides information complementary to the in situ imaging and marker-based techniques. The flow inhomogeneity (or lack thereof), shear band morphology (e.g. thickness), grain size and diffraction patterns were analysed. Scanning electron microscopy (SEM) and electron backscattered diffraction (EBSD) were done in an FEI XL-40 scanning electron microscope. EBSD specimens were prepared by mechanical polishing with silicon carbide and diamond (1 μm) abrasives, followed by colloidal silica (20 nm). In the EBSD analysis, only material points having a confidence index (CI) ≥0.1 were considered, with points of CI<0.1 shown in black. Transmission electron microscopy (TEM) was done in FEI Tecnai microscope at 200 kV. TEM specimens for site-specific examination of the shear band microstructure in Ti alloy samples were prepared by using the focused ion beam (FIB) lift-out method in the FIB/SEM Dual Beam FEI Nova 200. TEM specimens from Mg AZ31B samples were prepared by electropolishing using 1% perchloric acid in ethanol at 50 V and −20°C. Metallography of chip specimens was performed using standard polishing and chemical etching procedures. Etchants used were 100 ml ethanol, 10 ml water, 6 g picric acid and 5 ml acetic acid for Mg AZ31B; 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.

    3. Results

    The experimental study has provided direct details of the phenomenology of shear band development common to various metal systems, and enabled high-resolution characterization of the localized flow fields. The observations show that the complex shear band flow can be decoupled into separate initiation and sliding phases, with the latter phase showing all the characteristics of a friction slider. This decoupling has led to a geometric method to control shear bands without the need for modifying the material microstructure.

    (a) Shear band development

    The in situ imaging of cutting of cold-worked brass, illustrated in figure 3, has enabled us to capture key features of shear band dynamics. Shear band flow is seen to be a two-phase process, involving distinct initiation and sliding phases. The initiation phase, shown in frames ac, is depicted using the strain-rate field that clearly highlights the shear band. In this field, the band is coincidental with high strain rate and appears as a bright yellow/orange line against the dark background. The band initiates near the wedge tip (O) and propagates towards the free surface at a speed of 2.5 mm s−1, which is an order of magnitude higher than V0. Higher magnification images of the instantaneous location of the shear band front (marked by crosses) are shown as insets. In frame c, the band has just reached the surface at O. Subsequent to this initiation phase, the second sliding phase ensues (frames df), in which the whole chip slides along the interface OO′ at a nearly constant sliding velocity VS=0.15 mm s−1, that is comparable to V0. Insets to frames df clearly show the development of a shear offset at the free surface (arrows) due to this sliding. While the initiation phase acts to ‘weaken’ the material, causing it to flow locally, it is during the sliding phase that the band is subjected to intense plastic strains. This is evident from the streaklines (yellow lines), shown superimposed on the raw images in frames df, that reveal sharp displacements at the interface OO′. It should be noted that the shear displacements at different locations along the band length are essentially equal, indicating uniform sliding across the interface.

    Figure 3.

    Figure 3. Shear band development in cutting of cold-worked brass. The six frames from a high-speed image sequence reveal the two phases of the shear band flow: initiation (frames ac, first phase) and sliding (frames df, second phase). In frames ac, the strain-rate field in the region of shear band activity is shown superimposed onto the raw images. The shear band, concurrent here with the line of high strain rate, initiates at the tool tip (O) and propagates towards the free surface, eventually reaching it at O′ in frame c. The regions adjacent to the band undergo negligible deformation. Insets to frames ac give the instantaneous location of the band front, as marked by crosses. The second sliding phase is shown in frames df using superimposed streaklines. Uniform sliding along the weak interface OO′, established in the first initiation phase, can be inferred from the streakline profiles. This sliding results in displacements (shear offsets) at the free surface (see at arrows in frames e and f). Insets to frames df show the development of these shear offsets. V0=0.1 mm s−1. (Online version in colour.)

    That this two-phase picture for shear banding is common to different material systems across deformation rates was also deduced from in situ experiments with Ti-6Al-4V. Figure 4 shows a high-speed image sequence of one shear band cycle in cutting of Ti-6Al-4V at V0 of 0.66 m s−1. Again, the initiation phase is quite fast, in this case occurring between just two frames (b and c) imaged at an inter-frame interval of 25 μs. This gives a lower bound of 8 m s−1 for the band front velocity. The weak interface OO′ established by the initiation phase is evident in frame c. The second sliding phase, shown in frames df, occurs at a much lower velocity, VS=0.78 m s−1, which is of the order of V0 as in the brass (figure 3). These observations indicate that the band front velocity during the initiation phase is a material-dependent quantity while the VS is determined by V0. A closer examination in fact shows that VS is just the component of V0 resolved along the interface, given by VScos(α)V0/cos(ϕ), where ϕ is the included angle between V0 and OO′ (figure 4). The similarity in the phenomenology of the shear band flow in brass and Ti-6Al-4V is surprising, given their completely different crystal structures and deformation mechanisms. These details of the two-phase shear band flow were also reproduced in finite-element (FE) simulations of cutting of Ti-6Al-4V. Details are presented in appendix A.

    Figure 4.

    Figure 4. High-speed image sequence showing two-phase shear band development in cutting of Ti-6Al-4V, involving initiation and sliding phases. The first band initiation phase is quite fast and occurs between just two frames (b and c). Frames df show the second sliding phase where the chip slides at a constant velocity, VS=0.78 m s−1, along the weak interface OO′ established by the initiation phase (see frame c) . V0=0.66 m s−1, inter-frame interval=25 μs.

    Ex situ observations of shear band profiles using the marker method have enabled quantification of the second sliding phase of band formation. Figure 5a shows a scanning electron microscope (SEM) image of the shear banded Ti-6Al-4V chip (V0=1 m s−1), where markers are seen as light striations on the chip. Physically, these markers are equivalent to streaklines, as in frames df in figure 3. It is evident that the markers experience sharp displacements at the shear band as a result of extensive sliding between segments A and B. One such marker profile is highlighted in yellow in the figure. It is important to note that the displacements across the band result purely from highly localized shear over a thin layer between the segments, and not from fracture, as clearly seen from the continuity of the markers across the band. The shear displacements (120 μm) along the length of the band are again uniform, similar to brass (figure 3). The marker profiles observed across the band are reminiscent of classical fluid boundary layers [28], where regions immediately adjacent to a sliding interface are retarded as a result of viscous drag.

    Figure 5.

    Figure 5. Shear band flow profile and microstructure in Ti-6Al-4V. (a) SEM image showing marker profiles across the shear band with one of them highlighted in yellow. The markers show sharp displacements and significant curving in the vicinity of the band due to sliding between segments A and B along a localized weak interface. The marker profiles are reminiscent of boundary layers in shearing fluids. (b) EBSD inverse pole figuremap showing shear band (black region) running diagonally across the image. Microstructure of the band is unresolvable due to highly concentrated plastic flow. By contrast, mildly distorted grain structures can be seen in the small-strain regions immediately surrounding the band. Colour code corresponding to crystallographic orientations of the grains is shown on top-left. (c) Bright-field TEM image from the band showing nanocrystalline structure with individual grains as small as 20 nm (arrows). V0= 1 m s−1. (Online version in colour.)

    The fact that sliding occurs over a highly localized, well-defined layer is also evident from microstructure observations. Figure 5b shows an EBSD scan of the chip, where the shear band is seen as a thin black zone approximately 4 μm in thickness. The band microstructure is not fully resolvable with EBSD because of highly concentrated flow. Bright-field TEM observations of the band, shown in figure 5c, reveal a nanocrystalline structure where grains as small as 20 nm (arrows) are visible. Immediately adjoining the band are segments subjected to relatively small strains; consequently the original grain structure is largely preserved in these areas (figure 5b).

    The sharp transition in the microstructure between the shear band and segment can be also seen from the TEM image collage shown in figure 6. This collage is of a perpendicular cross section of the band, with frames 1 and 4 corresponding to the band centre and segment, respectively, and microstructure transition occurring between frames 2 and 3. The structural changes, with increasing distance away from the band centre, can also be inferred from the diffraction patterns. The ring-like diffraction patterns from within the band (frames 1 and 2) reflect its highly refined, nanoscale structure, while spotted patterns from outside the band (frames 3 and 4) are indicative of a coarse-grained structure. The thickness of the nanostructured layer is found to be quite similar to that of the black zone in the EBSD image (figure 5b).

    Figure 6.

    Figure 6. Bright-field TEM image collage of a perpendicular cross-sectional spanning across the shear band in Ti-6Al-4V chip (V0=1 m s−1). Frame 1 corresponds to the shear band center, with frames 2–4 taken from adjacent locations at increasing increasing distance away from the band centre towards the segment. The diffraction patterns, taken from the imaged area in each frame, are shown on top. The microstructure transition from nanocrystalline to microcrystalline type across the band occurs between frames 2 and 3, as revealed by the change from a ring-like diffraction pattern (frame 2) to a spotted pattern (frame 3).

    The severity and spatial extent of localized flow in the sliding phase are determined by V0. The shear band displacement profiles, as measured using the marker method, are summarized in figure 7. Figure 7a is a schematic of the sliding phase of shear band flow along with a typical marker profile (red curved line). In this figure, y=0 is a stationary interface across which two adjacent segments slide at velocity VS. The displacements (U(y)) parallel to this interface are measured. Figure 7bd, shows the measured displacement profiles for Ti-6Al-4V, Inconel 718 and CP Ti at different V0 (equivalently VS). The normalized displacement profiles can be seen to be qualitatively similar in all the metals. Additionally, the effect of V0 on the flow is similar across all three metals: the deformation becomes increasingly localized at the interface with increasing velocity. This suggests that the flow during the sliding phase can be analysed using the principle of momentum balance in viscous fluid flows [28]. Equally importantly, the similarity of displacement fields in different metal systems indicates the possibility of describing the band flow using a general continuum model without need for precise microstructure details.

    Figure 7.

    Figure 7. Shear band displacement profile measurements using marker method. (a) Schematic of sliding phase of the shear band flow showing marker profile and reference axes for the plots. The interface (y=0) over which the segments slide is stationary. The shear displacements, U(y), are measured parallel to this interface, with Umax being the maximum sliding distance.The measured shear displacements, normalized with respect to Umax, for Ti-6Al-4V, Inconel 718 and CP Ti are shown in (bd), respectively, at different V0 conditions. Note that in all the three materials, the deformation becomes increasingly localized at the interface with increasing V0 (or VS). (Online version in colour.)

    An evaluation of the deformation conditions under which the localized flow takes place shows that band strains in cutting are among the highest reported for shear bands in the literature. Because a shear band forms along the direction of maximum shear, the local shear strain can be measured from the local gradient of the displacement profile (such as in figure 7). In Ti-6Al-4V (figure 7b), the strain at the interface (y=0) is estimated to be 15 for V0=0.5 m s−1 and as large as 56 for V0=5 m s−1. The interface strains in Inconel 718 and CP Ti are in the range of 8–73 and 12–80, respectively, over the V0 range shown in figure 7. Similar to that in Ti-6Al-4V, the larger strains occur at higher V0. Given these extremely large strains, microstructure refinement in the band down to the nanoscale (figures 5c and 6) is to be expected. Assuming constant VS in the sliding phase, the strain rate at the interface for all three metals is estimated to be between 105 s−1 and 107 s−1 for the V0 range of interest.

    It is noteworthy that these large strains are accommodated within the band without fracture (as illustrated in figures 2 and 5), which can perhaps be explained by viscous flow during the sliding phase. The viscous-like deformation may arise because of the nanocrystalline structure of the band, where rate-dependent mechanisms such as grain boundary sliding contribute to the flow [29], or due to phonon drag on dislocations that becomes prominent at high strain rates [30]. Indeed, viscous flow has also been noted elsewhere in a variety of sliding contacts typically characterized by large strains and strain rates [31,32].

    (b) Viscous slider model

    Despite the material-specific details of the first initiation phase of shear band flow, the marker profiles (figure 7) have established common features of the second sliding phase across material systems. In order to explain the observed displacement profiles, the band is modelled as an infinitesimally thin planar interface over which two material segments (e.g. A and B in figure 5a) slide. The interface itself is formed in the initiation phase and the two sliding segments are assumed to be Newtonian viscous half-spaces (figure 7a), with the same kinematic viscosity ν and density ρ.

    At t=0, the segments are stationary with respect to each other. This instant can be taken to correspond to the end of the initiation phase. For 0<t<tf, a constant sliding velocity VS/2 is imposed, equal and opposite on both the segments and remote from the interface (y=0). For this configuration, the final displacement profile across the interface (band), U(y,tf), is given by [28]

    UUmax=[2ξ2erfc(ξ)+erf(ξ)+2ξπexp(ξ2)],3.1
    where ξ=y/4νtf is a dimensionless variable, and erf(ξ) and erfc(ξ) are the error function and complementary error function, respectively. In this model, the kinematic viscosity ν is the only adjustable parameter, with the sliding time tf given in terms of Umax (figure 7a) and VS by tf=2Umax/VS.

    The raw experimental displacement data for Ti-6Al-4V, Inconel 718 and CP Ti from the markers (figure 7) is used to fit equation (3.1) and obtain a value for ν at different V0. Unlike a conventional least-squares approach, equation (3.1) is fitted to the marker data by scaling the independent variable y, which provides a corresponding value for ν.

    As y in equation (3.1) appears in the combination ξ=y/4νtf, the raw displacement data are scaled as a function of the dimensionless variable ξ. The scaled data for all three material systems, shown in figure 8a, appears to collapse onto a single master curve. The displacement profile predicted by the slider model (equation (3.1)) is shown as solid black curve in the figure. The validity of the model is evident from the close match between model curve (black) and experiment profiles for all three materials. More importantly, this points to a general viscous sliding mechanism for the sliding phase that is insensitive to precise microstructure details. The agreement is best at the lowest sliding velocities (corresponding to V0 of 0.5 m s−1 for Ti-6Al-4V, 1 m s−1 for Inconel, and 4 m s−1 for CP Ti). However, a small but systematic deviation is observed at higher V0 (curves marked by crosses and circles) for all three systems. This is likely due to the occurrence of shear thinning rheology in the band at high strain rates.

    Figure 8.

    Figure 8. Viscous sliding model describing the second sliding phase of shear band formation. (a) Plot of marker displacements for Ti-6Al-4V, Inconel 718 and CP Ti at different V0. The normalized displacement data (U/Umax), rescaled using a fitted viscosity ν, are shown as a function of ξ=y/4νtf, where tf is given by 2Umax/VS. The displacements for all the three metals are seen to collapse onto a common curve, coinciding with the model prediction which is shown as the solid black curve (equation (3.1)). The raw displacement data from experiments used to generate the plot are provided in figure 7. (b) Plot showing dependence of dynamic viscosity μ=ρν (from model) on V0. For all the three materials, the μ values fall in the viscosity range for liquid metals (mPa⋅s), and decrease with increasing V0. (Online version in colour.)

    The dynamic viscosity μ=ρν, obtained from the fit ν values, is shown as a function of V0 in figure 8b. The μ values are in the millipascal second (mPa⋅s) range: 0.2–2 mPa⋅s for Ti-6Al-4V, 0.35–11.25 mPa⋅s for Inconel 718 and 4.95–30.9 mPa⋅s for CP Ti. It may be noted that these values fall exactly in the viscosity range for metals in their liquid state [33]. The μ for all materials can be also seen to slightly decrease with increasing V0, which is consistent with the deviation of the experimental displacement profiles from the model prediction at high V0.

    (c) Passive geometric flow control of shear bands

    The occurrence of unconstrained viscous sliding in the second phase of shear band flow suggests a powerful general method—PGFC—to suppress banding. This method is based on inhibiting the sliding phase by application of macroscopic constraints to the flow, thereby reducing or even eliminating flow localization at the meso- and macro-scales.

    In the cutting configuration, PGFC involves imposing a separate constraint directly across from the cutting tool edge at the free surface as shown schematically in figure 9. This constraint physically prevents the sliding phase from developing, even though the initiation phase could still occur due to intrinsic material instability. Furthermore, any geometric flow softening by load-bearing area reduction due to sliding along the band is simultaneously reduced by the presence of the constraint. The extent of sliding (Umax) along the band in the second phase is determined by the parameter λ=tc/t0, where tc is the maximum thickness of the flow localized chip in free cutting. PGFC experiments with different constraint levels showed that when λ is below a critical value, it is possible to fully suppress shear banding. For other values of λ, varying degrees of flow localization were observed.

    Figure 9.

    Figure 9. Schematic of PGFC in cutting using application of a geometric constraint. The degree of constraint on the flow is set by the parameterλ=tc/t0. The laminar flow under conditions where shear banding is suppressed is illustrated by the smooth (dark red) streaklines, with the deformation zone shaded in light red. (Online version in colour.)

    Figure 10 demonstrates the effect of geometric confinement in PGFC on shear banding in cutting of Mg AZ31B. This alloy is particularly known for exhibiting intense flow localization even at small strains [16]. As seen in figure 10, free cutting of Mg AZ31B results in the formation of discrete chip particles because of severe shear banding and consequent fracture. Figure 10b is an optical micrograph of a discrete Mg AZ31B chip, where fracture between segments is evident. The preferential dark etching contrast of the segment margins (arrows, figure 10b) shows that fracture is preceded by flow localization. The tendency for fracture during the second sliding phase in Mg is most probably a result of its low ductility due to limited slip activity. Upon using the PGFC, however, the discrete chips are transformed into a long continuous ribbon chip (figure 10a). That shear banding is suppressed can be seen from the optical micrograph of the PGFC chip, shown in figure 10c. The homogeneous, dynamically recrystallized microstructure confirms the occurrence of uniform plastic flow. It is found that a critical level of constraint, λ≤0.7, is required to fully suppress banding in this Mg alloy.

    Figure 10.

    Figure 10. Suppression of shear banding in cutting of Mg AZ31B by PGFC. (a) Comparison of free cutting and constrained cutting (PGFC): free cutting results in discrete chip particles due to severe shear banding and fracture, whereas constrained cutting eliminates shear banding, resulting in a continuous ribbon chip. (b) Optical micrograph of discrete chips from free cutting showing fracture between individual segments. The dark etching contrast at the segment margins (arrows) suggests the occurrence of localized flow before fracture. (c) Homogeneous and fully dynamically recrystallized microstructure of the continuous ribbon produced by PGFC with λ=0.7. V0=1 m s−1. (Online version in colour.)

    The Mg chip produced by PGFC (figure 10c) has a much smaller grain size (2.3 μm) than the starting workpiece (approx. 15 μm). This large refinement in the grain size is a direct consequence of the severe plastic deformation in PGFC, in combination with the deformation induced heating in the plastic zone. Based on assumption of simple shear along a thin plane (red shaded area, figure 9), the shear strain in the PGFC for λ=0.7 is estimated to be 2. The temperature rise in the zone is determined by V0 (deformation rate) and is estimated as 125–225°C for V0 in range of 0.25–4 m s−1 [23]. By varying the deformation temperature using V0, the extent of dynamic recrystallization can be controlled to achieve microstructures finer than in figure 10c. As an example, figure 11 shows a TEM image from a ribbon created by PGFC at a lower V0 of 0.25 m s−1. The microstructure is composed of ultrafine equiaxed grains, of approximately 500 nm in size. This type of ultrafine microstructure in Mg alloys is difficult, if not impossible, to achieve using conventional deformation processes (e.g. rolling, extrusion) because of shear banding. Additionally, the confinement of flow by PGFC increases the hydrostatic pressure in the deformation zone, which is useful for suppressing or delaying fracture in deformation processing [20].

    Figure 11.

    Figure 11. TEM image showing ultrafine-grained microstructure of Mg AZ31B ribbon, enabled by the PGFC method. Well-defined, equiaxed grains of approximately 500 nm in diameter can be clearly seen from the image. PGFC conditions: λ=0.7 and V0=0.25 m s−1.

    Suppression of shear banding by PGFC has also been successfully achieved in the Ti alloy systems, Ti-6Al-4V and CP Ti (figure 12). In Ti-6Al-4V, PGFC with λ=0.6 results in the formation of continuous long ribbons with smooth surfaces (figure 12a(i)). The ribbon shows a uniformly sheared flow-line type microstructure (figure 12a(ii)), typical of homogeneous flow in cutting [32]; this is to be contrasted with the non-uniform structure of the shear banded chip in figure 2a, produced under same cutting conditions but without PGFC. Some small-scale heterogeneity in the PGFC ribbon microstructure of figure 12a(ii) is likely a result of the initiation phase still being operative. Similar band suppression by PGFC has been achieved in CP Ti with λ=1.4. Figure 12b(i) shows a coiled ribbon formed by cutting CP Ti with PGFC. The resulting homogeneous microstructure is shown on (ii) (cf. shear banded chip in figure 2c). Additional experiments showed that PGFC is also capable of suppressing the fracture in (free) cutting of Ti-6Al-4V at low V0 (less than 0.5 m s−1), resulting in continuous ribbons similar to that in figure 12a.

    Figure 12.

    Figure 12. Application of PGFC to suppress shear banding in Ti alloy systems. (a) Long Ti-6Al-4V ribbons with smooth surfaces (i), exhibiting a uniformly sheared microstructure (ii) produced by PGFC at λ=0.6 and V0= 1 m s−1. Fine band-like features in the microstructure are likely due to the presence of the initiation phase. (b) Coiled CP Ti ribbon (i) produced at λ=1.4 and V0=2 m s−1. Optical micrograph on (ii) shows homogeneous microstructure of the chip, devoid of shear bands. The shear banded chips that result from free cutting (no constraint), under same process conditions are shown in figure 2a,c. (Online version in colour.)

    Because the sliding phase of the shear banding flow is common to different materials, and is largely insensitive to the microstructure, PGFC is expected to be of broad applicability for a range of material systems, beyond the hcp Mg and Ti alloys demonstrated here. In fact, PGFC was found to be successful in eliminating banding in cutting of Inconel 718, electrical sheet steels (Fe–Si alloys) and cold-worked brass—again based on the principle of suppressing the sliding phase. It is worth pointing out here that while suppression of shear bands by PGFC has been demonstrated in cutting, its application to other deformation processes can be also envisioned. For example, in rolling, it is feasible to impose geometric constraints at the sheet surface to restrict shear band flow. This can be done say by bonding a ductile metal foil onto the sheet or by sandwiching the block being rolled between metal sheets.

    4. Discussion

    The experimental observations of plane strain cutting have provided a detailed description of shear band flow development and its key attributes in large-strain deformation of metals. Shear band flow is found to be a two-phase process, involving distinct initiation and sliding phases, at both small (figure 3) and large strain rates (figure 4). The first initiation phase develops quite rapidly, at a speed an order of magnitude greater than the cutting speed. Its primary role is to establish a locally weak interface in the material, along which flow localizes in the ensuing sliding phase. The strain imposed in the initiation phase is quite small, constituting a small fraction (5–10%) of the total strain in the band. The mechanism of the initiation phase, as well as its quantitative details (e.g. shear band angle, band front velocity) is very likely determined by the nature of the workpiece material and its initial microstructure/texture.

    (a) Sliding phase of shear band flow

    The second phase of shear band development is characterized by sliding along the weak interface established by the initiation phase. It is during this sliding phase that intense localized (shear) strains, typically greater than 10, are developed in the band. The magnitude of straining increases with increasing V0, with shear strain values reaching up to 80 at high V0. The strain rates in the band are also very large in this sliding phase, of the order of 107 s−1 at higher V0 of 5–10 m s−1. The deformation conditions of strain and strain rate are quite uniform across the length of the band, as deduced from the essentially equal shear offset distances at different locations along the band length (figures 3 and 5a). This uniformity implies that the large strains and strain rates occur over macroscopic regions. Incidentally, the shear band strains measured here are among the highest reported over macroscopic volumes. Prior reports of similar large strains in shear bands have mostly been based on the assessment of deformation over small volumes, typically on the length scale of microstructural features [34].

    The large strains are generally accommodated in the band without fracture, as seen from the continuity of the marker profiles across the band (figure 5a). It is only in semi-brittle materials like Mg (figure 10b), or at low cutting speeds (in Ti-6Al-4V), that fracture is observed along the bands. While the experimental observations from the different metal systems have formed the basis for the two-phase model of band flow, this model is also supported by FE simulations (figure 13).

    Figure 13.

    Figure 13. FE simulation showing the evolution of effective strain-rate field during shear band flow in cutting of Ti-6Al-4V (V0=5 m s−1), with background streaklines and velocity vectors. Frames 1–2 show the first (band initiation) phase, while frames 3–6 show the second (sliding) phase. Note the intensely local nature of the deformation, indicated by the thin zone of very high strain rate. The blue dot is a material point that lies in the shear band, and approximately at its centre. The greendots represent neighbouring material points lying initially on a horizontal line with the blue dot, selected to highlight shearing action within the band. The vectors show velocity distribution in a frame of reference co-moving with the blue dot. The dominant motion is clearly one of shear parallel to the band, in opposite directions on either side of the band. (Online version in colour.)

    The boundary-layer like displacement profiles in the vicinity of the band (figures 5a and 7) have led us to model the second sliding phase of shear band flow as shearing of two viscous fluids (figure 7a). In this model, two identical materials—part of the workpiece and a chip segment—are sheared across a common weak interface established in the initiation phase. The equivalent shear band viscosity, computed by fitting the viscous sliding model to the measured displacement profiles, is very close to typical viscosity values for liquid metals (figure 8b). This correspondence is hardly fortuitous since viscosity values for metals span over 20 orders of magnitude depending on factors such as liquid/solid fraction, grain size and temperature [35].

    While the precise microstructural origin for this fluidity is still unclear, a few possible sources are suggested by various observations: viscous flow in shear bands can arise from microstructure refinement down to the nanometre length scale, where rate-dependent mechanisms such as grain boundary sliding play an important role [29]. This type of structure refinement has been observed in the present experiments (e.g. figures 5c and 6) and also well documented elsewhere [11]. Other possible origins for band fluidity are phonon drag on dislocations [30], which becomes increasingly important at high strain rates; structural changes due to intense plastic straining [1,21]; and, perhaps, even local melting in some instances [36]. But irrespective of the origin of the fluidity, the agreement between the model and experiments affirms viscous flow in the shear bands during the sliding phase.

    The micro-marker measurements have established the striking similarity of displacement fields of shear bands across metal systems with different crystal structures and material response. Coupled with the observations of similar flow profiles in systems as diverse as rock deformation [37], ballistic impact (see fig. 3 in [38]) and torsion of metals (figs. 8, 11 and 12 in [39]), and cutting of plastics [40] and glasses (figs. 4 and 13 in [41]), this points to a universal mechanism for band formation across length scales and materials. The two-phase flow observations in fact suggest that such commonalities can arise if sliding across a weak interface is key to the evolution of all shear bands. One can imagine a situation in which the first (nucleation) phase arises from interplay of varied and complex processes involving material/thermo-mechanical/microstructural phenomena [1,2,12,14,16,17]. However, irrespective of the initiation mechanism, once a weakened interface is generated it must inevitably be followed by a sliding phase up and until fracture intervenes. This is also corroborated by our experiments, where the bulk of the deformation is observed to occur in the viscous sliding phase; and the development of the sliding phase is negligibly influenced by the initial workpiece microstructure, and hence largely material-independent.

    (b) Geometric flow control

    Delineation of the initiation phase from the strain-intensive sliding phase of shear banding immediately suggests possibilities for effecting its control. Suppressing the initiation phase, which is material dependent and whose phenomenological time and length scales are small, is likely difficult or impossible without recourse to specialized microstructure/texture design. This design option is also precluded in practice by the fact that materials selection is usually dictated by functional considerations (and properties). By contrast, modification of the sliding phase is a promising alternative option for band control, because it does not warrant material modifications. Furthermore, as nearly all of the localized straining occurs in this phase, controlling the sliding reduces the degree of flow localization. That this option is indeed feasible has been demonstrated in our experiments employing PGFC.

    The fundamental basis of PGFC is that the sliding phase can be limited and even suppressed by suitable application of a constraint in the deformation zone (figure 9). The efficacy of PGFC in suppressing shear bands has been demonstrated unambiguously in the experiments not only across a range of material systems, but also at the high cutting speeds typical of industrial practice (up to 2 m s−1). In fact, PGFC has been found to be successful for band suppression studied at even higher speeds (e.g. 5–10 m s−1). It is worth noting here again that the mere application of PGFC does not in itself assure suppression of shear bands; rather, for PGFC to be effective, the constraint on the flow has to be sufficiently high (i.e. small λ) so as to fully suppress the shear offset occurring in the sliding.

    The use of a small λ with PGFC has another key benefit: increasing the level of hydrostatic pressure in the deformation zone, a condition quite beneficial for suppressing fracture [23]. It is perhaps for this reason that PGFC, with λ=0.7, is effective in suppressing the shear band flow and fracture altogether in cutting of Mg alloy AZ31B (figure 10). As a consequence, the discrete chip particles arising from shear banding in conventional machining of this alloy are transformed into continuous ribbon chips with a fine- or even an ultrafine-grained microstructure (figures 10c and 11) when PGFC is employed during the cutting. The creation of such fine-grained microstructures in thin sheet and foil forms—desirable for their superior strength and formability properties—in Mg alloys, and more generally in hcp alloys, has long been a challenge. This is because large thickness reductions by multi-stage rolling in these alloys are limited by the inhomogeneous flow and fracture unless annealing steps are employed between rolling stages.

    The implications of shear band control/suppression by PGFC for processing of metals are wide-ranging. In cutting and machining, shear banding causes force and tool-chip contact area fluctuations which adversely impact tool wear and surface finish. Furthermore, as the deformation state of the machined surface is usually a replica of the chip deformation state, shear band flow can result in heterogeneous straining and non-uniform microstructure/properties on the machined surface [42]. The implementation of PGFC, even in the finish machining stage, can therefore be beneficial for product quality and machining performance. Besides its utility in machining, PGFC is of obvious benefit in metal deformation processes (e.g. rolling) where flow localization is a limiting factor. The use of PGFC also opens up other routes for production of sheet and foil, such as by hybrid extrusion–machining, as demonstrated in figures 10 and 12. Finally, the ability to vary the degree of shear banding by applying constraint levels less severe than those needed for full suppression offers an opportunity to explore shear band dynamics systematically using PGFC.

    5. Conclusion

    The development of shear bands in large-strain deformation of metals has been studied at high spatial and temporal resolution using a combination of in and ex situ observational techniques. Plane strain cutting is used as a framework to impose large simple shear at strain rates of 1–105 s−1, in metal systems such as Ti, Mg and Ni. These were selected based on their pronounced susceptibility to shear banding, varied shear band origins and diverse deformation response.

    Shear band flow is found to unfold in two phases: an initiation phase which creates a localized weak zone, followed by a viscous sliding phase across this zone that results in large localized strains. While the initiation phase is material-specific in details, the second sliding phase is found to be largely material-independent. The characterization of the bands has revealed key phenomenological details of how these two phases develop, and provided direct measurements of the displacement and deformation fields. The localized flow in shear bands is characterized by very large shear strains, up to as much as 80, and strain rates up to 107 s−1, that prevail over macroscopic regions. The shear band strains in machining are uniform across the length of the bands and are among the largest recorded in macroscopic shear bands. Most of the strain in the bands is imposed in the second sliding phase, which is found to be well described by a model of viscous shearing across a fluid interface. The equivalent viscosity for this interface in all of the metals is very close to that of the liquid metals, in the mPa⋅s range. The study has also established the striking similarity of displacement fields of shear bands in very different material systems, suggesting a universal mechanism for band formation across length scales and materials.

    The picture of two-phase flow for shear band development has enabled a general method—PGFC—for controlling shear bands. This method involves only macroscopic modification of the flow geometry in the deformation zone, and not any changes to the workpiece material microstructure which are inherently limited or infeasible in practice. The application of PGFC to suppress shear banding in highly band susceptible Mg and Ti alloys is demonstrated. The benefits of PGFC are wide-ranging, including improvements in product quality and process performance in machining, and enhancements in deformation processing capability for creating thin metal sheet and foil.

    Data accessibility

    The data supporting this paper are available as electronic supplementary material.

    Authors' contributions

    D.S., A.M. and R.M. planned and conducted the experiments pertaining to shear band flow development; K.V. and N.S. were responsible for the modelling; D.S., S.C. and K.T. were responsible for the geometric flow control design; and all authors contributed to data analysis, formulation of conclusions and preparation of the manuscript.

    Competing interests

    We declare we have no competing interests.

    Funding

    The authors would like to acknowledge support from US DOE grant no. DE-EE0005762 (via Third Wave Systems), US Army Research Office Award W911NF-15-1-0591, and NSF grant no. CMMI 1562470 and DMR 1610094.

    Appendix A. Finite-element analysis

    The two-stage sliding model observed in experiments is also seen on performing FE simulations of cutting of Ti-6Al-4V. These FE simulations of shear banding are carried out using the ABAQUS/EXPLICIT dynamics solver in a Lagrangian framework. While FE simulations of shear banding in machining are routine [43], as is the FE procedure used in the present work, interrogating the resulting strain-rate and velocity fields in particular ways can be very helpful in revealing features of shear band flow.

    (a) Procedure

    Fully coupled thermal/stress analysis is performed to capture the thermoplastic behavior of the workpiece. Ti-6Al-4V, one of the most widely studied alloys in the context of shear banding, is selected as the prototypical example. The cutting tool is modelled as an isothermal (T=300 K) analytical rigid body. The workpiece specimen is modelled using 22130 four-noded, plane-strain, reduced-integration elements with displacement and temperature degrees of freedom at the nodes. The element size in the cutting zone is 3.8 μm with an aspect ratio of 0.7. The FEA mesh is generated using a custom mesh-maker.

    The tool is kept fixed, while the rectangular workpiece is slid against it by applying a velocity boundary condition V0=5 m s−1 to nodes on one vertical face of the specimen. The t0 is kept fixed at 125 μm. To successfully perform the simulation in a Lagrangian framework, a geometric chip separation criterion is used [44], with a separation layer thickness of 1.2 μm. The use of a separation layer in metal cutting has also been viewed as an indispensable feature to model the formation of new surfaces by ductile failure near the tool tip [45].

    Ti-6Al-4V is modelled using Johnson–Cook hardening [46] within a metal plasticity framework. This well-known phenomenological hardness model allows for the dependence of yield stress on strain, strain rate and temperature. All material properties, elastic, thermal and plastic, are considered to be temperature dependent [47,48]. A Taylor–Quinney coefficient of 0.9 is used in the simulation to model heat generation due to plasticity. Ductile failure is modelled for the workpiece, with Wierzbicki-type [49] stress-triaxiality dependent damage initiation, and linear softening of damage evolution. To eliminate mesh dependence, the maximum plastic displacement criterion is used, with half-element width (2 μm) as the displacement to failure. Element deletion is not used. Contact between the tool and workpiece is enforced using kinematic constraints. A capped-Coulomb friction model is used at the interface, with friction coefficient μ=0.15 and maximum interfacial shear stress τmax of 400 MPa. The total physical cutting time simulated is about 118 μs.

    (b) Results

    Figure 13 shows the evolution of the effective strain-rate field during shear band flow in cutting of Ti-6Al-4V. In frames 1 and 2, representing the band initiation phase, a zone of very high strain rate develops, starting from near the tool tip and growing towards the free surface. During this phase, the shear displacement in the band is quite limited, as indicated by the fact that the line of material points (dots in the figure) is not displaced significantly in a direction parallel to the band. However, in frames 3–6, representing the sliding phase, it is clearly seen that this line of material points is sheared intensely, forming a step. The blue dot represents a material point that lies approximately in the middle of the shear band and the green dots on either side of it displace in opposite directions parallel to the band. It is also evident that the serration on the free surface of the chip essentially develops during the sliding phase (frames 3–6). The superimposed quiver plots in the figure depict the instantaneous velocity field in a frame of reference that co-moves with the band, and also clearly highlight that the principal motion is one of sliding parallel to the band. Lastly, it is also seen that the amplitude of the shear, indicated by the magnitude of the kink formed in each streakline, is quite uniform through the thickness of the chip.

    It should be noted that principal characteristics of the shear band flow (shear band angle, spacing between bands) are not affected significantly by the use of different Johnson–Cook material parameters for Ti-6Al-4V [43] or different values of maximum shear stress in the friction model. However, the shear amplitude parallel to the band does vary to some extent.

    While these simulations are not meant to be used for quantitative, one-to-one comparison with experiments, they clearly show that the two-stage mechanism is characteristic of shear band flows. They also corroborate the idea that once flow localization is initiated by any mechanism (in this case, by damage-induced softening), the flow subsequently shows characteristics of a frictional slider.

    Footnotes

    1 A streakline is the locus of all particles that have passed earlier through a particular spatial point.

    Published by the Royal Society. All rights reserved.

    References