Sinuous flow in metals

Significance

It is counterintuitive, yet well known, that cutting a soft metal, vis-à-vis a hardened one, involves significantly larger forces, with the formation of a thick chip. Using in situ imaging we show that this phenomenon results from a hitherto unidentified flow mode in metals, called sinuous flow due to its repeatedly folded nature, that resembles irreversible flows in geological rocks and some complex fluids. We also demonstrate how sinuous flow can be suppressed, by simply applying common marking ink remote from the cutting interface—the forces are reduced significantly and the thick chip is eliminated. Besides explaining some important decades-old phenomena in metal cutting, our work has broad implications for many natural and industrial cutting processes.

Abstract

Annealed metals are surprisingly difficult to cut, involving high forces and an unusually thick “chip.” This anomaly has long been explained, based on ex situ observations, using a model of smooth plastic flow with uniform shear to describe material removal by chip formation. Here we show that this phenomenon is actually the result of a fundamentally different collective deformation mode—sinuous flow. Using in situ imaging, we find that chip formation occurs via large-amplitude folding, triggered by surface undulations of a characteristic size. The resulting fold patterns resemble those observed in geophysics and complex fluids. Our observations establish sinuous flow as another mesoscopic deformation mode, alongside mechanisms such as kinking and shear banding. Additionally, by suppressing the triggering surface undulations, sinuous flow can be eliminated, resulting in a drastic reduction of cutting forces. We demonstrate this suppression quite simply by the application of common marking ink on the free surface of the workpiece material before the cutting. Alternatively, prehardening a thin surface layer of the workpiece material shows similar results. Besides obvious implications to industrial machining and surface generation processes, our results also help unify a number of disparate observations in the cutting of metals, including the so-called Rehbinder effect.

A typical cutting process involves removal of material in the form of a continuous chip in plane–strain (2D) (Fig. S1). When the material being cut (workpiece) is in an initially annealed state, an unusually thick chip results and the forces involved in the process are very large. This difficulty in cutting, well known in industrial practice (1, 2), has hitherto eluded fundamental explanation. At the mesoscale ($\sim 100$ μm), the structure of the chip is assumed homogeneous, resulting from laminar plastic flow (3, 4). Using such a framework, augmented by ex situ observations (5), the high forces are attributed to the thick chip developed in the process, without an explanation of the cause of such anomalous chip formation. In this work, we revisit this long-held hypothesis using in situ observations and analysis, in the process unearthing a collective mode of plastic deformation.

Fig. S1.

Schematic of idealized plane–strain cutting showing chip formation by smooth laminar flow, with simple shear. The deformation zone is highlighted in blue. Initial chip thickness $h_0$ is measured from free surface to the surface of material separation. The workpiece material undergoes plastic shape transformation to form a chip with final thickness $h_c$. The velocity of bulk material flow against the tool is $V_0$.

Our model system consists of an annealed oxygen-free high-conductivity (OFHC) copper workpiece (grain size $\sim 500$ μm) cut by a hard steel wedge (tool) at a velocity of $V_0=0.42$ mm/s. Temperature effects are negligible at this low velocity. Part of the workpiece material undergoes plastic (irreversible) shape transformation to form the chip (Fig. S1). The tool face is fixed to be normal to $V_0$, whereas the cutting depth (initial chip thickness) is maintained at $h_0=50$ μm. This model system mimics many natural (6) and industrial cutting (2) processes. The flow of metal against the tool face is observed in situ and photographed using a high-speed camera. The images are postprocessed using particle image velocimetry (PIV) to obtain a comprehensive record of velocity, strain rate, and strain field histories. This enables quantitative characterization of material flow past the tool edge. Additional experimental details are provided in Materials and Methods.

Much to our surprise, the flow responsible for the shape transformation of the workpiece material into the chip, as revealed by the streakline pattern, bore little resemblance to any reported in classical plasticity. Fig. 1A, derived from a high-speed image sequence, shows streaklines that reveal a highly unsteady, sinuous flow with significant vorticity. The streaklines are extensively folded over in the chip, with peak-to-peak amplitudes in a single fold being as much as two-thirds of the chip thickness. Small surface protuberances, which form in the compressive field just ahead of the tool face, appear to trigger the folding; one such bump is bounded by two arrows (1 and 2) in Fig. 1. These arrows demarcate pinning points which are central to fold growth. The entire chip thus forms by repeated folding of the incoming material, i.e., sinuous flow—a collective plastic deformation mode, in the same genre as kinking (7) and shear banding (8, 9). In addition to Cu, sinuous flow was observed in other systems such as α-brass and commercially pure aluminum, showing that it is a truly mesoscopic mode, independent of the material’s crystal structure. Note that sinuous flow is not to be confused with the transition between laminar and rotational dislocation motion (10), which occurs at a much smaller scale.

Fig. 1.

Sinuous flow mode of deformation in annealed copper. (A) Streakline pattern illustrating the underlying folding. Points 1 and 2 (red arrows) are pinning locations—the material between them forms a surface bulge while continuously rotating due to differential displacements. This is seen by the inclination of the fold peak with respect to the horizontal. (B) Optical micrograph of the chip. The back surface of the chip shows regular mushroom-like corrugations, separated by gaps. These are often misinterpreted as surface cracks. (C) Strain distribution in the chip, with mean $\langle \mathit{ϵ}\rangle =5.62$. The highly inhomogeneous strain distribution inside the chip is clearly reflective of multiple folds during deformation. The figures in A–C correspond to different instants of time.

The occurrence of sinuous flow cannot be inferred purely from postmortem structural observations in the chip or force measurements. As an illustration, an optical micrograph of the removed chip is shown in Fig. 1B. The surface of the chip shows repeated mushroom-like formations, with gaps in between. This structure has been (erroneously) described as resulting from homogeneous flow, supplemented by cracking on the chip free surface (1). In situ analysis reveals that the strain field in the chip is actually highly nonhomogeneous (Fig. 1C). Interestingly, this inhomogeneity is not reflected in the measured forces; no detectable oscillations occur at the fold frequency of around 1.7 Hz (Fig. S2).

Fig. S2.

Energies and force in cutting annealed copper. A comparison between the cutting energy computed from force ($E_{force}$) measurements and PIV ($E_{PIV}$) analysis is shown. $E_{PIV}=E_{sin}+E_{sub}$, is obtained by integrating the stress along pathlines in the PIV flow field. $E_{sin}$ and $E_{sub}$ are the energies dissipated in the chip and the subsurface, respectively. The specific energy $U_{sin}$ for sinuous flow (see main text) is $E_{sin}$ per unit volume. The cutting force $F_C$ is also shown. Interestingly, no fluctuations are observed in the force trace at the fold formation frequency $\sim 1.7$ Hz.

The kinematics of fold nucleation and development is clear from the motion of the initial bump and its boundaries (compare Fig. 1A). The set of four frames in Fig. 2 shows the evolution of such a bump into a fold. The two points $P_1$ and $P_2$, bounding the initial bump, move along with the material in each frame. A white dotted line—the bump axis—joins the peaks of adjacent streaklines, indicating the orientation of the impending fold. The color scheme depicts the underlying strain rate field, from PIV calculations. The development of multiple such folds in sinuous flow, along with superimposed streaklines and strain field, is also shown in Movie S1.

Fig. 2.

Sequence of images with superimposed streaklines showing development of folds. The underlying strain rate field (with color bar inset) captures regions of local deformation. (A) $P_1$ and $P_2$ (marked by white arrows) are material locations that delimit the initial bump. These provide the pinning points for the surface bulge in B. As shear occurs closer to the tool face (C), this bulge rotates and is stretched diagonally, before finally forming an impending fold in D. Local shear is evident from the underlying strain rate field as well as the change in orientation of the bump axis (dashed white line) between B, C, and D.

In Fig. 2A, $P_1$ and $P_2$, likely grain boundaries, delimit the initial bump and act as the local pinning points. They force the bump to deform plastically, resulting in a pronounced bulge on the free surface in Fig. 2B. The underlying strain rate field reflects this deformation in the two local colored zones surrounding the initial bump (Fig. 2 A and B). The bump axis is nearly parallel in both frames. Simultaneously with surface bulging, the workpiece material is also constantly forced against the vertical tool face. This constraint imparts a vertical velocity to each material point. The bulge in Fig. 2B is hence sheared, causing the axis to rotate in a counterclockwise direction (Fig. 2C). The magnitude of shear increases as the material nears the tool face; see the strain rate field in Fig. 2C. The bulge is amplified while also reducing its original width (Fig. 2D), with the material between $P_1$ and $P_2$ now constituting a single impending fold. Folding is complete once the original bump axis is rotated by nearly $90°$, at which time another bulge is initiated ahead of the tool face and the process repeats.

The chip hence comprises a series of folds, developed one after another in the manner above. Corresponding folds in the streakline pattern provide quantitative geometric fold characteristics as well as variations along the chip thickness. The results of this analysis are summarized in Fig. 3 (see SI Materials and Methods for details).

Fig. 3.

Geometric parameters characterizing the observed folds. (A) Two adjacent streaklines (blue, red) are shown, demarcating a single fold. P is the fold peak (curve maximum), $M_1,M_2$ are fold troughs (curve minima), M is the midpoint of $M_1{M}_2$. $P^{\prime }$ is the maximum of the second streakline, the axial line $P{P}^{\prime }$ and the line $PM$ subtend angles ϕ and θ with $M_1{M}_2$. The fold amplitude A and width W are the lengths of $PM$ and $M_1{M}_2$, respectively. (B) Scatter plot of ϕ and θ. Data shown for the first ($○$), second ($\mathrm{◁}$), and third (□) streaklines from the free surface. Marker color indicates fold width W: red (cyan) corresponding to large (small) W. The values fall around the $45°$ line, deviation from which implies nonuniform streakline spacing. All of the wide folds undergo large shear whereas smaller ones remain upright $\left(\theta ,\varphi \sim 90°\right)$ (C) Histogram of fold widths W. The mean width (dashed line) $W_m\simeq 50$ μm and one SD (dotted lines) are also shown. The mean wavelength of the folds is $\sim 200$ μm at the point of formation.

Inhomogeneous shear in the material can easily be seen in a plot of ϕ vs. θ, as shown in Fig. 3B. For symmetrically sheared folds, maxima of adjacent streaklines are expected to lie on the line $PM$, corresponding to the $45°$ line (dashed) in Fig. 3B. However, local shear results in varying distance between adjacent streaklines, as indicated in Fig. 3A. Additionally, both ϕ- and θ-values are clustered near $0°$ and $180°$, which indicate large shear. Geometrically, this brings fold peaks P closer to the extrapolated minima line $M_1{M}_2$. Most wide folds undergo large shear, whereas a minor fraction (small folds) remain upright ($\theta ,\varphi \simeq 90°$) and form over existing larger folds.

The distribution for W, including the mean $W_m\simeq 50$ μm and one SD, is shown in Fig. 3C. The wider folds ($W\ge 150$ μm) occur near the beginning of the streaklines, getting progressively narrower as material flows past the tool. Subsequently, the small folds, constituting $\sim 10\text{\%}$ of the total, are developed. The mean and maximum fold widths are smaller than the initial grain size in the material ($\sim 500$ μm). The average fold wavelength is $\sim$ 200 μm at the point of formation.

The immediate consequence of this sinuous flow mechanism is that the resulting chip is quite thick—its final thickness $h_c$ being 14× the initial thickness $h_0$. However, this significant thickening is not a priori indicative of the actual unsteady folding phenomenon, for such a shape change can also be envisaged in the framework of ideal smooth laminar flow (Fig. S1). Characteristically, however, the sinuous flow also produces a highly nonuniform strain field in the chip, fluctuating between 4 and 8, that reflects the underlying fold pattern (Fig. 1C). It is interesting to note that the representative (volume-weighted) strain for sinuous flow, $\mathit{ϵ}\sim 5.6$, is actually much lower than for an equivalent shape change by laminar flow, where $\mathit{ϵ}\sim 8.1$. This latter value is obtained if only the chip thickness ratio ($h_c/h_0$) is considered, without accounting for the actual flow process (2).

Like the strain, the specific energy U (energy per unit volume) for chip formation, i.e., shape transformation, is also significantly smaller for the sinuous flow. By the usual integration of stress and strain along path lines in the sinuous flow field, $U_{sin}$ is obtained as 2.9 J/mm³ (SI Materials and Methods and Fig. S2). In comparison, the corresponding value for an equivalent laminar flow (with $\mathit{ϵ}=8.1$) is $U_{lam}=4.2$ J/mm³, which is 45% greater than $U_{sin}$.

Based on the strain and specific energy, the shape transformation into a chip is thus much more efficiently achieved by sinuous flow than by laminar flow. This is counterintuitive because, at first sight, the highly folded, sinuous flow appears quite inefficient, involving extensive redundant deformation. But, because selection of collective deformation modes is in general governed by their relative stability, the material’s preference for sinuous flow is likely the result of a flow instability in smooth laminar flow. Whereas there are other instances where large-strain plastic deformation occurs via nonhomogeneous modes, e.g., shear banding, kinking, and buckling, we are not aware of any prior observations of large shape changes being effected by the sinuous flow mode demonstrated here.

Our observations have uncovered a previously unidentified mesoscopic flow mode in the plastic deformation of ductile metals, in the process also explaining a longstanding problem in cutting. The mechanism of fold formation appears to be strongly tied in with the large grain size and ductility common to such annealed metals and is driven primarily by the ability of the material to undergo large plastic deformation. Microscopically, each grain roughly constitutes a single fold, consistent with both the formation mechanism (Fig. 2) and fold width distributions (Fig. 3C). In this respect, it seems to have a similar origin as the folding mechanism reported in sliding wear (11), although the resulting flow here is significantly different—there is actual material removal, and folds of very large amplitude occur. Fold patterns observed in the sinuous flow show remarkable resemblance with other well-studied folding phenomena in geophysics (12), thin films (13), fluid flow (14, 15), and non-Newtonian plastics (16).

The mechanism of fold development suggests that the difficulty in cutting annealed metals can be resolved if the sinuous flow mode is suppressed or eliminated altogether. Prestraining a thin surface layer (thickness $\sim h_0$) of the annealed material to strains $\mathit{ϵ}\ge 1$ causes refinement of the grain size and a reduction in the ductility. Doing so removes the two main triggers for sinuous flow, viz. bulge formation and the establishment of pinning points. Thus, prestraining should suppress the sinuous flow, a fact confirmed experimentally (Fig. 4A). The streakline pattern in the figure shows that the folding is eliminated, with an almost $70\text{\%}$ reduction in both the cutting force and strain. Additionally, the flow is completely laminar, with a sharp, well-defined deformation zone and thin chip, as in the classical plasticity model. This surface prestraining can be accomplished by a suitable surface deformation process (17, 18).

Fig. 4.

Suppression of sinuous flow. (A) Strain rate field with superimposed streaklines when cutting hardened copper. A sharply defined narrow shear zone is seen, as assumed in conventional plasticity models. The flow is laminar with insignificant bump formation ahead of the tool–chip interface. (B) Comparison of cutting forces (force in the direction of $V_0$) for different surface conditions. An annealed Cu sample (Inset, brown) is surface-coated over half its length with marking ink (Inset, blue). The cutting force in the uncoated region is very large (brown). A drastic reduction ($>50\text{\%}$) is seen when cutting the material coated with ink (blue), due to sinuous flow suppression. Cutting a Cu sample with a thin prestrained (hardened) surface layer (Inset, red) results in low cutting force (red), reflective of laminar flow. The application of ink on the free surface of such a hardened layer shows no measurable effect on forces, consistent with the explanation based on sinuous flow.

In light of the above observations, an intriguing possibility now presents itself—can sinuous flow be suppressed also by the simple application of a thin coating to the workpiece surface? Presumably, such a coating could be expected to modify the surface mechanical state (ductility, stiffness), thereby preventing the triggers for sinuous flow. To test this hypothesis, a layer of ink (Dykem) was painted onto the free surface of the annealed workpiece (Fig. 4B, Inset, and Fig. S1), before the cutting. Dykem ink, consisting of colored pigments in an alcohol (propanol + diacetone alcohol) medium, is commonly used to mark metals. Note that the surface of ink application is above the surface along which material separation occurs (Fig. S1), and away from the tool–chip contact. Interestingly, the bump formation ahead of the tool was much reduced by this ink application, with concomitant suppression of the folding and significantly lower forces resulting. Fig. 4B shows the large force drop observed when cutting an annealed Cu sample with the ink coated along half its length. When cutting the annealed workpiece region with no surface application (brown), sinuous flow occurs as expected, and the cutting force is very large. The force decreases by more than $50\text{\%}$ when cutting over the workpiece region with the ink coating (blue), owing to sinuous flow suppression. This effect of the ink is very similar to that due to the prestrained surface layer, also shown for comparison (red). Application of a variety of other substances such as resins and nail polish, and even marking the surface with a sketch pen, was found to suppress the sinuous flow to various degrees. Such surface layer applications, however, did not have any noticeable influence on the forces and flow when cutting prestrained Cu, where the flow is intrinsically laminar. We hope to further explore this suppression hypothesis via additional controlled experiments and modeling.

These results also offer a coherent picture of the Rehbinder effect (19) in the cutting of metals. This phenomenon concerns the small reduction ($<10\text{\%}$) in cutting forces upon application of a suitable volatile fluid (e.g., CCl₄) on the workpiece free surface (20, 21). The effect has traditionally been attributed to either “microcracks” on the workpiece surface promoting a physicochemical effect (21, 22) or a fundamental change in the dislocation structure near the surface (20). Besides the speculative nature of these explanations, the reports of the force reductions have been inconsistent (23, 24). Sinuous flow provides a natural explanation for this effect—any surface application, including volatile CCl₄, will modify the surface mechanical state of the workpiece and inhibit initial bump formation ahead of the tool. Consequently, the folding will be diminished, resulting in lower forces. The inconsistent force reductions observed are hence most certainly due to the large variability in the initial state (annealed, partially/fully hardened) of the workpiece arising from the specific preparation procedures.

Two important and contrasting implications of our observations involve the utility and suppression of sinuous flow. The highly “redundant” nature of the flow pattern suggests its use in designing new materials for energy absorption applications. This can be viewed as being complementary to materials with a tendency to form shear bands, such as metallic glasses (25). Chip structures, similar to those in Fig. 1B, have been reported in other metallic systems such as low carbon steels and pure iron (1, 5, 21), and microtomy of metals (26). This, together with our in situ observations in Cu, brass, and Al, points to the widespread occurrence of sinuous flow in annealed metals. Hence, this flow mode will also play an important role in micromachining (27) and surface deformation (17, 18) processes. On the other hand, as we have shown, a suitable surface treatment can suppress sinuous flow, thereby enabling easier processing of annealed metals. The large reduction in forces translates directly into an equivalent energy reduction. Furthermore, the reduced forces and energy dissipation should have favorable consequences for industrial machining, e.g., avoiding chatter-vibration instability across a broader range of process conditions, improved component surface quality, and enhanced tool life.

SI Materials and Methods

Energy Estimates.

Shape transformation by laminar flow.

It is interesting that despite the highly sinuous flow, the cutting process still results in a chip of essentially constant thickness. Based on the chip thickness (pre- and postcutting) alone, the chip shape transformation can be envisioned as occurring by smooth laminar flow with a sharp deformation zone (shear plane). For this idealized shape transformation, the shear strain γ in the chip can be estimated as $\gamma =\lambda +1/\lambda$, where λ is the ratio $h_c/h_0$ (Fig. S1). Based on Fig. 1 (main text), λ is 14. This gives $\gamma \sim 14.1$ and the von Mises strain $\mathit{ϵ}\left(=\gamma /\sqrt{3}\right)$ as 8.1.

Once ε is known, the specific energy (energy per unit volume) for the shape transformation can be estimated for the laminar flow as $U_{lam}={\displaystyle \int \sigma d\mathit{ϵ}}$. Using the constitutive model for Cu (28), we get for the stress σ

$$\sigma =\left(A+B{\mathit{ϵ}}^n\right)\left(1+C\ln \frac{\dot{\mathit{ϵ}}}{{\dot{\mathit{ϵ}}}_0}\right)\left[1-{\left(\frac{T-T_0}{T_m-T_0}\right)}^m\right],$$ [S1]

where ε and ${\dot{\mathit{ϵ}}}_0$ are the experimental and reference strain rates (s⁻¹), $T,T_m,T_0$ are temperature of work material, its melting temperature, and the room temperature, respectively. Coefficient A is the yield strength (MPa), B is the hardening modulus (MPa), C is the strain rate sensitivity coefficient, n is the hardening coefficient, and m is the thermal softening coefficient.

At low cutting speed, as in the present case ($V_0=0.42$ mm/s), then $\dot{\mathit{ϵ}}\simeq {\dot{\mathit{ϵ}}}_0,T\simeq T_0$ and the flow rule simplifies to

$$\sigma =A+B{\mathit{ϵ}}^n.$$ [S2]

For annealed OFHC copper, $A=90$ MPa, $B=292$ MPa, $n=0.31$. Specific energy $U_{lam}$ is obtained as $U_{lam}={\displaystyle {\int }_0^{{\mathit{ϵ}}_0}\sigma d\mathit{ϵ}={\left[A\mathit{ϵ}+\left(B{\mathit{ϵ}}^{n+1}/n+1\right)\right]}_0^{{\mathit{ϵ}}_0}}$. ${\mathit{ϵ}}_0=8.1$ is obtained from the chip thickness data. This gives for the specific energy $U_{lam}=4.18$ J/mm³.

Shape transformation by sinuous flow.

For the sinuous flow case, energy estimates are obtained by evaluating ${\displaystyle \int \sigma d\mathit{ϵ}}$ along pathlines in the high-speed images. Strain data, obtained from PIV analysis, are used for this purpose. The incremental strain $\mathrm{\Delta }{\mathit{ϵ}}_{\left(ij\right)}$ at any pixel location i at frame j is used to estimate the entire strain field, i.e., ${\mathit{ϵ}}_{\left(ij\right)}={\displaystyle {\sum }_{k=1}^j\mathrm{\Delta }{\mathit{ϵ}}_{\left(ik\right)}}$. Note that $i,j,k$ here are not tensor indices, but correspond to scalar strain values (effective strain) at pixel location i at frames j or k. Neglecting temperature and strain rate effects, and using the constitutive law for copper (see Eqs. S1 and S2 above), the flow stress and specific energies are estimated as

$${\sigma }_{\left(ij\right)}=\left(A+B{\mathit{ϵ}}_{\left(ij\right)}^n\right),$$ [S3]

$$U_{\left(ij\right)}={\displaystyle \sum _{k=1}^j\left(A+B{\mathit{ϵ}}_{\left(ik\right)}^n\right)\mathrm{\Delta }{\mathit{ϵ}}_{\left(ik\right)}}.$$ [S4]

The energy dissipated in a region is obtained as the volume average of $U_{\left(ij\right)}$ at a particular instant of time (frame j). The results are shown in Fig. S2. The values obtained using PIV ($E_{sin}$) for energy into the chip, when added to the energy imparted to the substrate, i.e., $E_{piv}=E_{sub}+E_{sin}$, match the values obtained from force measurements ($E_{force}$).

The same method is used to obtain the average strain in the chip. The ratio of the specific energies obtained for laminar and sinuous flows is relatively insensitive to the constitutive model used in the computations.

Numerical Details of Fold Analysis.

The method of analysis of the developed folds closely follows the principles used in the study of geological folding phenomena (29). The folding is captured by streakline data, generated from PIV calculations. A streakline is a 2D plane curve, parameterized by time. The x and y axes are oriented along the horizontal and vertical directions, respectively. Each streakline is characterized by its (spatially varying) curvature, which yields information about individual folds. To determine the curvature, the following scheme is used:

• i)The numerical representation of each streakline consists of $\left(x_i,y_i\right)$ pairs at a final time $t_{\text{inf}}$—each point on the streakline is introduced in the flow at some earlier time $t_i<t_{\text{inf}}$.
• ii)The unit tangent vector ${\widehat{t}}_i=\left(u_i,v_i\right)$ at each location i is determined using finite (central) differences

$${\widehat{t}}_i=\left(u_i,v_i\right)=\frac{1}{2}\frac{dt}{ds}\left(x_{i+1}-x_{i-1},y_{i+1}-y_{i-1}\right),$$ [S5]

• where $ds/dt=1/2\sqrt{{\left(x_{i+1}-x_{i-1}\right)}^2+{\left(y_{i+1}-y_{i-1}\right)}^2}$ is the speed along the curve, with arc length s.

• iii)The curvature measures the rate of change of ${\widehat{t}}_i$. To reduce numerical errors in continued finite difference calculations, the angle ϕ is determined from

$${\varphi }_i=\text{arctan}\left(v_i/u_i\right),$$ [S6]

• and this yields the curvature ${\kappa }_i\equiv d{\varphi }_i/ds\approx \left(dt/ds\right)\left(1/2\right)\left({\varphi }_{i+1}-{\varphi }_{i-1}\right)$.

Points of highest positive (negative) curvature are defined to be maximum (minimum) points. Inflection points correspond to locations of zero curvature. A single fold is defined to be the section of the curve bounded by two adjacent minima.

The characteristics of a single fold are defined as shown in Fig. 3 in the main text. These definitions are somewhat different from those used in structural geology, and are motivated by numerical difficulties in accurately determining inflection points on the curves. The fold width W is the length of the line joining the two extremities $M_1$ and $M_2$ of a single fold. The amplitude of the fold A is defined as the length of the line joining the maximum point P to the midpoint M of $M_1{M}_2$ and θ denotes the angle between the lines $PM$ and $M{M}_2$. The axial line $P{P}^{\prime }$ joins the maximum point P on one streakline to the corresponding maximum point $P^{\prime }$ on the next streakline. This axial line makes an angle ϕ with $M{M}_2$. Also, θ and ϕ are always measured with respect to the increasing s direction and hence lie in the range $\left[0°,180°\right]$. $A,W,\varphi$, and θ were determined for each fold in each streakline.

Materials and Methods

The workpiece is made of 99.99% OFHC copper Alloy 101 (McMaster-Carr). The samples were annealed in air at 750 °C for 4 h and oven cooled to room temperature. The average grain size was $\sim$ 500 μm after annealing. The workpiece surfaces were prepared by milling and then polishing to a sufficient depth, so as to remove any oxide layer present. The side face used for imaging was polished using 600 grit (SiC paper) to introduce enough surface features for PIV analysis. The final workpiece dimensions were 50 mm × 25 mm × 3 mm. The cutting tool was made of high-speed steel (Mo-Max M42 cobalt). The cutting was done using Mobil 1 lubricant. The speed $V_0$ and initial depth $h_0$ were fixed at 0.42 mm/s and 50 μm, respectively. Side flow of the workpiece was constrained by the use of a glass plate. All experiments were performed at room temperature ($T=300\hspace{0.5em}\text{K}$, $T\ll 0.5T_m$, where $T_m$ is the workpiece melting temperature). The region of interest, illuminated by white light, was imaged using a high-resolution complementary metal-oxide semiconductor (CMOS) camera (pco dimax), coupled to an optical microscope (Nikon Optihphot), with a resolution of 1.4 μm per pixel and a frame rate of 100 frames per second. A piezoelectric dynamometer (Kistler 9272) was used to measure the cutting forces, and the data were sampled at 1 kHz.