Machining is among the most versatile material removal processes in the manufacturing industry. To better optimize the machining process, the knowledge of shear strains and shear strain rates within the primary shear zone (PSZ) during chip formation has been of great interest. The objective of this study is to study the strain and strain rate progression within the PSZ both in the chip flow direction and along the thickness direction during machining equal channel angular extrusion (ECAE) processed titanium (Ti). ECAE-processed ultrafine-grained Ti has been machined at cutting speeds of 0.1 and 0.5 m/s, and the shear strain and the shear strain rate have been determined using high speed imaging and digital image correlation (DIC). It is found that the chip morphology is saw-tooth at 0.1 m/s while continuous at 0.5 m/s. The cumulative shear strain and the incremental shear strain rate of the saw-tooth chip morphology can reach approximately 3.9 and 2.4 × 103 s−1, respectively, and those of the continuous chip morphology may be approximately 1.3 and 5.0 × 103 s−1, respectively. There is a distinct peak shift in the shear strain rate distribution during saw-tooth chip formation while there is a stable peak position of the strain rate distribution during continuous chip formation. The PSZ thickness during saw-tooth chip formation is more localized and smaller than that during continuous chip formation (28 versus 35 μm).

## Introduction

Machining is among the most versatile material removal processes in the manufacturing industry. The introduction of computer numeric controlled machining centers has led to significant improvements in the production rate, accuracy, and repeatability for industrial machining. As machining centers and cutting tools continue to advance, there is a need to further optimize machining process parameters. Process optimization is critical to reducing the overall machining cost and improving the production efficiency. This requires the use of analytic models and/or finite element modeling to predict optimum cutting conditions, and both approaches require knowledge of the constitutive behavior within the primary shear zone (PSZ).

Several constitutive material behavior models have been proposed to predict the flow stress of workpiece within the PSZ. Among these, the Johnson–Cook model has shown great success in predicting the flow stress during both quasi-static and dynamic deformation applications [15]. The Johnson–Cook model requires material specific constants that calibrate it for a specific loading condition, which are determined from stress–strain and stress–strain rate relations using standard mechanical property testing. However, standard mechanical property tests are limited to relatively low strains and strain rates compared to those that occur within the PSZ. In addition, current methods used to approximate the shear strain ($γ$) and the shear strain rate ($γ˙$) require knowledge of the PSZ thickness or characteristics of the chip morphology, which are often difficult to measure. Thus, if a constitutive model is to be calibrated for machining conditions, an approximation of shear strains and shear strain rates within the PSZ is required.

The objective of this study is to study the strain and strain rate progression within the PSZ both in the chip flow direction and along the thickness direction during machining equal channel angular extrusion (ECAE)-processed titanium (Ti). The shear strain and the shear strain rate are determined using digital image correlation (DIC), and for the first time, an accurate measurement of the PSZ thickness is made possible. While Ti is machined in this study, the method described herein is applicable during ductile machining of other materials. The resulting knowledge of this study can be used as a foundation to develop constitutive relations, which will allow for the development of advanced predictive models that better optimize cutting processes.

## Background

Various analytical models [6,7] as well as numerical simulations [8,9] have been developed to evaluate the strain and the strain rate during machining. However, the majority of analytical models [6,10] are only valid for continuous chip formation and do not account for chip segmentation. While numeric simulations are not limited by the chip morphology, they require a constitutive relation or damage model as a prior knowledge [2,11], which is usually elusive under machining conditions. Since any changes in cutting conditions will inevitably affect the cutting forces, tool wear, and surface integrity, there is an inherent need to quantify the variation of strains and strain rates within the PSZ when cutting conditions change.

The strain and the strain rate during machining have usually been estimated analytically by using Merchant's classical machining model as the foundation [6,10]. Merchant's model evaluates the strain and the strain rate within the PSZ during continuous chip formation using the cutting geometry and cutting conditions. More recently, several cutting and chip geometry-based approaches have been proposed to account for chip segmentation and further evaluate the strain and strain rate within the shear band (SB) and adjacent segments [1214]. Much like Merchant's model, these models have utilized the cutting geometry in addition to measurements of the chip geometry taken from postmortem metallographs. He et al. [12] measured individual chip segments from stainless steel chip metallographs and determined shear strains and shear strain rates as 10–20 and 107 s−1, respectively, by assuming the PSZ thickness to be between 0.01 and 1.0 μm. Similarly, Duan et al. [13] found the shear strains and shear strain rates as high as 30 and 105 s−1, respectively, during high speed machining of 30CrNi3MoV steel. Cotterell and Byrne [14] used high-speed imaging to approximate the sliding distance of individual saw-tooth segments [15] and further calculated the shear strain. The PSZ thickness was approximated to be 8 μm, and shear strains within the segment and inside the shear band were estimated to be 1–2 and 9, respectively. While these geometry-based approaches provided an approximation of the strain and strain rate during chip formation, there was a clear discrepancy between the estimated strains and strain rates using different geometry-based approaches. This was primarily due to the different approximation of the PSZ thickness, which was assumed to be between 0.01 and 8 μm [12,14]. Moreover, there are changes between the instantaneous chip morphology and the postmortem chip morphology, so extracting the chip geometry information from the postmortem chip morphology may also result in errors when approximating the strain and strain rate. Thus, further work is required to better estimate the strain and strain rate within the PSZ during the chip formation process.

To overcome the challenges associated with measurement errors when using geometry-based approaches, digital image-based optical approaches have been adopted to approximate the strain and strain rate during chip formation. These approaches include DIC [16] or particle image velocimetry (PIV) [1719], which track particle displacements on the workpiece surface through a sequence of images. The current image is correlated to a previous image in the sequence using the gray-level pixel intensities, which are used to maximize a correlation function [20]. The particle displacements that maximize the correlation function are used to calculate the strain and strain rate on the workpiece surface. Using this method, there are no errors due to measurement differences in the chip morphology or PSZ thickness. Pottier et al. [16] applied DIC to measure the principal strains and principal strain rates during plane strain sliding of Ti-6Al-4V and found the strain and strain rate to be as high as 3 and 4 × 103 s−1, respectively. Within the PSZ, the strain was found to be a maximum at the tool tip, while the strain rate was a maximum at the free surface (FS). Different from those assumed or predicted using Merchant's theory, the shear strain and the shear strain rate were not constant throughout the PSZ. Similarly, Lee et al. [17] used PIV to map the surface strain during a plane strain sliding process for commercially pure copper. The maximum shear strain and shear strain rate during continuous chip formation were found to be 4.5 and 300 s−1, respectively, which agree well with the analytic method using Merchant's theory. It should be noted that the same method was applied for commercially pure lead and 6061 aluminum alloy and considerable error was found when comparing the PIV results with those predicted using Merchant's theory, which was attributed to the presence of built up edge (BUE) [17]. Recently, Guo et al. [19] used PIV to study the influence of tool rake angle on the shear strain and shear strain rate distributions with the PSZ. Strains and strain rates as high as 3 and 12 s−1, respectively, were found when machining Ti-6Al-4V using a plane strain sliding speed of 1 mm/s.

Furthermore, the concept of DIC/PIV was applied to a machining setup to evaluate the machining-induced strain and/or strain rate. Efe et al. [18] utilized PIV to measure the surface strain during large strain extrusion machining of AZ31B magnesium alloy. In this approach, a constraint was placed at the free surface in order to hinder chip segmentation and allow for simultaneous cutting and extrusion. Shear strains and shear strain rates of approximately 1 and 100 s−1, respectively, were observed. This result was slightly lower than those obtained using Merchant's theory for continuous chip formation, which was most likely due to the change in the shear strain and shear strain rate distributions due to the presence of the additional constraint at the free surface. While these studies sufficiently quantified the strain and strain rate during machining, they were conducted using uncommon machining conditions such as plane strain sliding and relatively low cutting speeds.

Generally, the strain and strain rate measurements obtained from digital image-based approaches are lower than those calculated using analytical- and geometry-based approaches. This observation is most likely due to the assumptions made for the shear band thickness or the experimental setups used during imaging. Generally, two experimental setups have been used for imaging studies: plane strain sliding [16,17,19] or conventional turning [18]. In plane strain sliding operations, cutting speeds are achieved by linear translation as opposed to workpiece rotation. Due to the available length of workpieces, this significantly hinders the achievable cutting speeds compared to traditional single point turning operations. In addition, plane strain sliding is interrupted cutting in nature and only small lengths of material are removed during one cycle/stroke. As such, it may be difficult to achieve steady-state conditions similar to those that develop through several workpiece rotations during orthogonal turning. On the other hand, while conventional turning may provide higher achievable cutting speeds, it has been done under cutting conditions (such as machining under a free surface constraint [18]) that differ from traditional single point turning. Moreover, the knowledge on the shear strain and shear strain rate progression during conventional machining is still elusive.

## Experimental Methods

### Experimental Machining Setup

#### Cutting Conditions and Workpiece Preparation.

The microstructure of ECAE-processed ultrafine-grained metals can be refined down to the submicron level through extensive shearing [21,22]. Square Grade 4 Ti bars (CG 4, Service Steel Aerospace Corp., Seattle, WA) were ECAE-processed and cut as 25 mm diameter bars for orthogonal turning studies for their great mechanical, physical, and biocompatible properties [21,22]. In particular, the ECAE-processed Ti bars herein were rolled at 320 °C to approximately 6% in thickness between extrusion passes at an extrusion speed of 0.05 m/s while using route 4Bc (90 deg rotation about the long axis for four passes). Further details regarding the ECAE-processing conditions for the Ti bars used in this study are detailed elsewhere [21,23]. Orthogonal cutting experiments were designed and performed using a Whacheon HL-435 machine lathe (Whacheon USA, Long Beach, CA) using uncoated carbide cutting tools (NG3125R K313, Kennametal Inc., Latrobe, PA) with a fixed rake angle of 0 deg. A schematic of the experimental setup is shown in Fig. 1.

In particular, a disk profile (Fig. 1(a)) was machined 2 mm from the free end of the 25 mm diameter workpiece using grooving, thus the width of cut was fixed at 2 mm. An undeformed chip thickness (equivalent feedrate) was held constant at 0.1 mm/rev, while the cutting speed was varied from 0.1 to 0.5 m/s in order to observe changes in the chip formation characteristics within and around the PSZ as a function of cutting speed. It was difficult to achieve higher cutting speeds due to the high rotational speeds required when small diameter workpieces were machined for high speed imaging on the Whacheon lathe, thus the cutting speed was limited to 0.5 m/s for safety concerns. Prior work [23] utilized higher cutting speeds to examine changes in postmortem chip morphology; however, in this experiment, a different metal cutting setup and a different equipment were used to conduct in situ and DIC chip formation analysis, thus limiting the maximum cutting speed to 0.5 m/s. Different from previous translation-based or planning experimental setups [16,17], which involved interrupted cutting, in this study continuous cutting tool motion occurred along the cutting tool axis (pure translation along the radial Z-direction). Unlike previous translation-based or planning experimental setups [16,17], in this study, the workpiece was cut under realistic machining conditions while imaged at significantly higher cutting speeds.

Prior to imaging, the workpiece was first machined using a nominal cutting speed, feedrate, and depth of cut of 0.5 m/s, 0.025 mm/rev, and 0.1 mm, respectively. This set of cutting conditions was chosen to achieve acceptable surface finish and feed mark distance. Then, the workpiece surface was lightly sanded with a 1200 grit silicon carbide sand paper, which removed the feed marks left by the previous finish machining pass and introduced particle asperities that could be tracked using DIC. The surface was further cleaned in succession with both acetone and ethanol prior to imaging to remove any debris left during the finishing operation.

#### Imaging Setup.

As shown in Fig. 1, images of the chip formation process within and around the PSZ were taken using a high-speed imaging system [24], which was composed of a high speed camera, microscopic zoom lens, two light-emitting diode-based light sources, and xy adjustable stages. The high-speed camera (Photron SA-5, Photron USA, Inc., San Diego, CA) was mounted directly to the cross-slide of the Whacheon machine lathe so that it moved in unison with the cutting tool, allowing the tool to remain relatively stationary in each captured image. The Photron camera system had a 12 bit ADC CMOS sensor with a 1024 × 1024 pixel resolution. During the machining tests, a constant frame rate of 50,000 frames per second, which produced approximately 41,000 images in the sequence, was used to image the deformation inside the PSZ. Due to the scale of the PSZ and the chip thickness, a microscopic zoom lens (Navitar 12×, Navitar Inc., Rochester, NY) was utilized for magnification during imaging. The microscope lens was coupled with a 1× lens attachment and a 2× lens extender to obtain a final magnification of 15×. At this magnification rate, the field of view and spatial resolution were approximately 0.5 × 0.5 mm and 5 μm, respectively. The focal length of the lens was approximately 50 mm and fine focus was achieved by a manual adjustment on the lens. At high frame rates and high magnification rates, the cutting area had to be well illuminated in order to obtain a well-defined image. As such, two SugarCUBE light-emitting diode-based light sources (Nathaniel Electronics, Vergennes, VT) were used to illuminate the cutting area. In particular, one SugarCUBE was coupled with the Navitar lens system using coaxial illumination, and the other was used as an external light source.

### Digital Image Processing

#### Digital Image Correlation.

In this study, davisstrainmaster, a specialized DIC software package (LaVision Inc., Goettingen, Germany), was used to track the motion of chip flow as the workpiece moved through the PSZ and along the tool rake face. This software is preferred due to its robustness and versatility over other commercially available or matlab-based software. Herein, the subset and step sizes are 15 × 15 and 3 pixels, respectively. From the displacement field, other parameters such as velocity, shear strain, and shear strain-rate can be calculated. DIC is carried out using a differential correlation procedure where each image is compared with the previous image in the sequence. The incremental displacement field is then summed to obtain the cumulative displacement distribution. This method is advantageous over the relative-to-first correlation procedure due to the incremental correlation between successive images as opposed to comparing each image in the sequence to the first image (reference image), which is often difficult when large deformations occur. Using this correlation scheme, both the cumulative and incremental strain and strain rate distributions can be obtained. Due to the fact that the pixel displacement increases when the cutting speed increases, five sections of 50–150 frames were extracted from each image sequence for further processing. The cumulative distribution is used to show the maximum shear strain since it shows the total deformation contributions from the primary and secondary shear zones as well as the influence due to segmentation. However, the maximum shear strain rate is presented using the incremental distribution, which shows only the deformed regions for two sequential images rather than the total, accumulated deformation during chip formation. The maximum shear strain rate is obtained by dividing the incremental shear strain by the interframe time-step as stated elsewhere [25]. Once each distribution is determined, davisstrainmaster is used to plot the shear strain and shear strain rate distributions.

#### Cumulative Distribution Measurements.

Using the cumulative shear strain distribution, the maximum shear strain is determined for cutting speeds from 0.1 to 0.5 m/s along the center of the chip in the chip flow (perpendicular to the PSZ) direction. The chip center is assumed to represent average shear strain through the chip thickness. Measurements are taken using a Eulerian coordinate system where the locations remain constant throughout the image sequence. Figure 2(a) shows a schematic representation for the measurement locations at each cutting speed.

Point A represents a location within the chip, which shows the deformation after the workpiece exits the PSZ (as well as the secondary shear zone next to the tool rake face) and moves parallel to the tool rake face. In the case of the cumulative distribution, deformation is a maximum as it passes through point A and no further deformation may occur beyond this point. Point B represents a point within the PSZ, where the material undergoes large deformation as part of the workpiece flows as the chip. Point C represents the deformation within the workpiece as the incoming segment begins to deform. It is noted that the shear strain at point C is usually very small as relatively little deformation occurs within the bulk workpiece.

#### Incremental Distribution Measurements.

As with the cumulative shear strain distribution, the incremental shear strain rate distribution is determined for cutting speeds from 0.1 to 0.5 m/s along both the chip flow direction (perpendicular to the PSZ) and the transverse direction through the chip thickness (parallel to the PSZ). Figure 2(b) shows a schematic representation of the measurement locations for both directions. Along the chip flow direction, line ABC is drawn within a single segment and is perpendicular to the shear band of the previous segment. The definition of points A, B, and C is the same for the incremental distribution as for the cumulative distribution. In the case of continuous chip formation, where no pronounced segmentation occurs, line ABC is drawn perpendicular to the PSZ. Similarly, the incremental shear strain rate distribution is plotted in the transverse direction through the chip thickness. Point D represents the shear strain rate at the tool tip, point E represents the shear strain rate through the midsection of the PSZ, and point F represents the shear strain rate at the free surface. In this way, the shear strain rate occurring within the PSZ only can be determined. The shear strain rate for the formation of a single segment is analyzed rather than the formation of multiple segments as for the cumulative distribution. In addition, the incremental shear strain rate is used to characterize the PSZ geometry.

#### Displacement Uncertainty.

Estimation of the displacement uncertainty for image correlation during DIC has proven to be a difficult task. In general, the most critical factors that contribute to the accuracy of DIC algorithms are the pattern quality, particle, and subset size [16,26]. Typically, a pattern used for image correlation consists of a random distribution of speckles painted on the workpiece surface to produce high contrast between the foreground and the background. Speckle patterns can be produced by first applying a uniform base coat of white spray paint followed by sparsely painting a black spray paint. However, the intense shear strain, shear strain rates, and temperatures during machining make the application of a uniform base coat unpractical. Alternatively, it is possible to use particle asperities and the natural surface texture of the workpiece at low cutting speeds [27]. Unfortunately, poor image quality (low resolution, changes in light intensity, or camera noise) raises questions about the accuracy of DIC displacement fields when using the workpiece surface texture [16].

The quality of both speckle patterns and surface textures can be assessed by calculating the autocorrelation function of the surface or mean intensity gradient (MIG). Bornert et al. [27] numerically generated artificial speckle patterns of different speckle sizes then calculated the autocorrelation function using the method detailed elsewhere [28]. This offers an efficient method of qualitatively investigating the speckle intensity distribution used for DIC in addition to assessing the surface quality when a natural surface texture is employed. Several studies involving DIC/PIV have found that a natural surface texture provides a sufficient contrast [16,19,29]. In this study, the autocorrelation function for the natural surface texture is based on the machined surface of the ECAE-processed Ti. Figure 3(a) shows the texture of the machined surface after finishing, while Fig. 3(b) depicts the comparison between the calculated autocorrelation function and those presented in a previous study [27]. To ensure a good DIC correlation, the autocorrelation of the natural surface texture should fall between the fine and coarse speckle autocorrelation functions. As shown in Fig. 3(b), the experimental autocorrelation function of the ECAE-processed Ti machined surface shows that the natural surface texture is of sufficient quality for DIC.

In addition to the autocorrelation function, the MIG can be used to assess the quality of the natural surface texture or speckle pattern [26]. The MIG (δf) of the workpiece surface is given by
$δf=∑i=1W∑j=1H|∇f(xij)|/(W×H)$
(1)

where W and H are the image width and height (in pixels), respectively, and $∇f(xij)$ is the modulus of the local intensity gradient [26]. Pan et al. [26] used the MIG to evaluate the surface quality of different workpiece surfaces. The calculated MIGs ranged from 34.64 to 9.03 for workpiece surfaces with a fine speckle pattern to a polished workpiece surface, respectively. Pan et al. [26] further concluded that the higher the MIG, the lower the displacement error using DIC. In this study, the MIG of the ECAE-processed Ti surface shown in Fig. 3(a) is calculated to be 23.32. Thus, the experimentally calculated MIG indicates that the natural surface texture is sufficient for DIC.

Using the MIG, the standard deviation of measured displacements can be estimated, and Pan et al. [26] used the following equation to estimate the random error of measured displacements:
$std(u)=2σN×δf$
(2)

where σ is the standard deviation of measurement noises and N (=15 × 15 pixels) is the subset size. In the present study, σ is the measurement noise evaluated by taking successive motionless images and estimated to be less than 2 pixels [16,26]. Thus, the standard error of measured displacements is approximately std(u) = 0.0098 pixels = 0.0116 μm (δf = 23.32).

Furthermore, the displacement error and speckle quality can be evaluated by comparing the DIC output with the calculated displacements through rigid body rotations. The incremental displacements in the x-direction during turning are determined to be dx = 2.711704 pixels = 3.19024 μm and dx = 8.643557 pixels = 10.16889 μm based on the workpiece radius and cutting speeds (V = 0.1 m/s and V = 0.5 m/s, respectively). In order to compare with the DIC output, ten successive images are taken of an undeformed region on the workpiece surface, and the resultant displacement is extracted. The measured cumulative displacements are found to be dx = 27.10795 pixels = 31.8917 μm and dx = 85.9816 pixels = 101.1543 μm (ten images) for the cutting speeds V = 0.1 m/s and V = 0.5 m/s, respectively. Thus, the displacement error is found to be 0.0107 μm for V = 0.1 m/s and 0.0146 μm for V = 0.5 m/s, respectively. Therefore, displacement uncertainties found from rigid body rotations are comparable with the standard error found using Eq. (2).

## Experimental Results and Discussion

Figure 4 shows representative high-speed images produced during machining the ECAE-processed Ti. Figures 4(a)4(e) show the saw-tooth chip morphology using a cutting speed of 0.1 m/s, which has been commonly reported during machining of titanium and its alloys using a wide range of cutting speeds [2,3034]. Its chip morphology is considered aperiodic saw-tooth at a cutting speed of 0.1 m/s based on the chip surface topography and the chip formation images [35] even that the chip free surface looks like folds. However, as the cutting speed increases to 0.5 m/s, the saw-tooth features at the free surface are suppressed and a continuous chip forms as shown in Figs. 4(f)4(j). The transition in chip morphology as the cutting speed increases is due to changes in the constitutive behavior of workpiece materials. In addition to being used for image correlation, the high-speed images can also be used to measure the chip thickness, shear angle, and segmentation displacements [12,13]. This provides an advantage over metallurgical methods since high-speed images present instantaneous snapshots of the chip formation progression. Moreover, Figs. 4(a)4(e) can be used to delineate regions of the saw-tooth chip, which is important during DIC analysis. As shown in Figs. 4(a)4(e), each saw-tooth segment consists of two regions: the SB and the bulk segment (S). The shear band is on the trailing edge of each segment, where the majority of the deformation occurs (SB1, SB2, etc.). The bulk segment consists of the region of material between two sequential shear bands (S1, S2, etc.). If not specified, the images presented herein (Figs. 46) were taken at 0, 0.10, 0.30, 0.50, and 0.75 ms for the 0.1 m/s machining case and 0, 0.20, 0.25, 0.30, and 0.35 ms for the 0.5 m/s machining case.

As seen from the image sequence shown in Figs. 4(a)4(e), five stages of deformation can be identified. Stage 1, shown in Fig. 4(a), is characterized by compression of a wedge-shaped volume of material against the tool rake face. Upsetting occurs in stage 2 as shown in Fig. 4(b) where the free surface rises above the horizontal plane. Stage 3, shown in Fig. 4(c), is characterized by the initiation of instability in the form of adiabatic shear localization. Stage 4 (Fig. 4(d)) occurs when the segment begins to slide along the adjacent workpiece. Finally, Fig. 4(e) shows stage 5, which starts when the saw-tooth segment slides further up the tool rake face and upsetting of the next segment begins. It is assumed that no further primary shear deformation takes place once stage 5 occurs and the next cycle begins. While multiple stages of deformation can be established for the saw-tooth chip morphology, no pronounced segmentation occurs during continuous chip formation as illustrated by the image sequence in Figs. 4(f)4(j). Thus, the deformation has a uniform appearance throughout the image sequence, and individual stages as seen from the saw-tooth chip formation are not discernable. It is noted that some dark bands, almost parallel to the shear direction, are observed in Figs. 4(f)4(j), which are similar to those reported before [19,36]. While further work is required to pinpoint the source for these dark bands, it is attributed to inhomogeneous shearing due to the secondary shear zone in a continuous chip [37] or variations in the workpiece texture.

### Shear Strain Distribution

#### Saw-Tooth Chip Formation at 0.1 m/s.

Figure 5 shows the cumulative shear strain distribution for saw-tooth chip formation at the cutting speed of 0.1 m/s as well as the corresponding shear strain distribution along the chip flow direction ABC. Herein, the progression of S2 and SB2 are discussed since S1 and SB1 have already completed their deformation cycle. Figure 5(a) shows the shear strain distribution at stage 1 of chip formation, which is characterized by wedge-shaped compression. During stage 1, the shear strain within SB2 is not localized and the bulk segment (S2) is being deformed. Figure 5(b) illustrates the shear strain distribution during upsetting (stage 2), where the FS rises above the horizontal. During this stage, the shear strain begins to increase along SB2 and localize at the tool tip. This indicates that upsetting starts resulting in a change in the local shear angle in the vicinity of the tool tip, increasing the shear strain. Figure 5(c) shows the shear strain distribution just after the initiation of instability (stage 3) along SB2. Instability occurs along SB2 once the shear strain localization front propagates to the free surface and reaches a value of approximately 2.0 as measured at point B. This suggests that a critical shear strain is required for the formation of saw-tooth chips and is taken as 2.5 in this study. In addition, the shear strain distribution accompanying Fig. 5(c) shows two peaks, one of SB1 (25 μm along ABC) and the other of r SB2 (90 μm along ABC). It is noted that the measured shear strain of SB2 is twice that measured of SB1. This is because SB1 is midway in its formation when image correlation begins, while the current image sequence shows a complete cycle for the formation of SB2. Figure 5(d) shows the shear strain distribution once S2 begins to slide along the adjacent forming segment/workpiece. It is observed that while the majority of the deformation occurs within the PSZ, the relative sliding between SB2 and the adjacent forming segment/workpiece also contributes to the total shear strain within SB2. The strain increase contribution to the total shear strain is estimated as 1 based on the difference between the maximum strains in SB2 as seen from Figs. 5(c) and 5(d). Figure 5(e) shows that S2 slides along the workpiece as well as the tool rake face. After the segment S2 reaches this position, stage 1 initiates for the incoming segment (S3), and the cycle repeats. Consequently, the shear strain distributes into the bulk of the next segment (S3) as upsetting begins. This is the only deformation that the bulk segment undergoes during deformation. At this stage, the shear strain within SB2 is approximately 3.9 and the shear band ceases to undergo further primary shear deformation once the upsetting phase of the next segment (S3) begins.

#### Continuous Chip Formation at 0.5 m/s.

Figure 6 shows the cumulative shear strain distribution for continuous formation at the cutting speed of 0.5 m/s as well as the corresponding shear strain distribution along the chip flow direction ABC. The continuous chip morphology at 0.5 m/s is different from the saw-tooth chip morphology obtained at 0.1 m/s, which reflects the change in chip formation mechanisms. As a result, the shear strain distribution is considerably different from that shown in Fig. 5. Figure 6(a) shows the initial image in the sequence where the workpiece first enters the PSZ, where the shear strain changes dramatically. Figure 6(b) shows that the shear strain increases as the workpiece flows through the PSZ, which is assumed to be the zone where the majority of deformation occurs. As the workpiece exits the PSZ, the shear strain stops increasing at approximately 1.3 as measured at point A as shown in Fig. 6(c). In addition, Fig. 6(c) shows that the width of the PSZ remains constant and the shear strain increases gradually within this region. Figures 6(d) and 6(e) show the similar shear strain distributions. After the workpiece material enters the chip region, little further deformation occurs and the shear strain reaches an average value of approximately 1.3. There is some slight shear strain oscillation in the chip region. While the DIC images produced herein provide some insight into the chip formation mechanisms, some difficulties may arise in identifying key features during machining, in particular, at high cutting speeds and in the secondary shear zone. In order to improve the quality and accuracy of the correlation, further work is required to better acquire high speed images and delineate specific features of chip and workpiece surfaces, especially within the secondary shear zone (such as the dark layer at the chip-tool interface from Fig. 6(c)).

### Shear Strain Rate Distribution

#### Saw-Tooth Chip Formation at 0.1 m/s.

While the overall deformation is characterized using the cumulative shear strain, the deformation rate is evaluated using the incremental shear strain rate. Figure 7 shows the shear strain rate distribution for the five stages of deformation during the formation of a single saw-tooth segment. The shear strain rate is plotted along line ABC, which extends in the chip flow direction through the PSZ as shown in Fig. 2(b) and the insets of Fig. 7. Each shear strain rate distribution can be correlated to a specific event during formation of segments. In addition, using the shear band threshold, which is defined as two standard deviations from the mean shear strain rate, the upper and lower boundaries of the PSZ can be determined. In addition, it is assumed that the peak (maximum shear strain rate) is located along the center of the PSZ and separates the forming segment from the adjacent workpiece.

Figure 7(a) shows stage 1, the initial stage of deformation, where the incoming workpiece is compressed against the tool rake face, resulting in multiple peaks in the shear strain rate distribution. The overall width of the PSZ is larger in Fig. 7(a) than in Figs. 7(b)7(e). Two peaks are observed at 60 μm (point B) and approximately 75 μm along ABC. Figure 7(b) shows a shear strain rate distribution midway through upsetting (stage 2), where the shear strain localizes to a narrow region within the PSZ approximately half as wide as the overall width from Fig. 7(a). Compared with stage 1, the shear strain rate peak is significantly narrower. At the end of the upsetting stage, the shear strain rate reaches a maximum (2.4 × 103 s−1) and instability occurs in the form of catastrophic shear localization (stage 3). Figure 7(c) shows the point just after instability where the segment slides along the adjacent workpiece for a short distance and flattening of the shear strain rate peak between 55 and 65 μm along ABC is observed. This flattening of the peak occurs within the workpiece material. At this stage, the saw-tooth segment is distinguishable from the adjacent workpiece; the upper and lower boundaries of the PSZ can be established as shown by the shaded regions. If the shear strain rate falls below the shear band threshold, that portion of the shear strain rate distribution is considered to be within the bulk segment (S2 in Fig. 4) or relatively undeformed workpiece. From the shear strain rate distribution at stage 3, the upper and lower boundaries of the PSZ are found approximately 42 μm and 90 μm along ABC, respectively. The portion of deforming material within SB2 can be delineated from the portion being strained on the workpiece surface as illustrated by the shaded regions in Fig. 7(c). Figure 7(d) shows the shear strain rate distribution as the segment slides further against the adjacent workpiece (stage 4). A flattened peak at 50–60 μm within the workpiece is observed in this segment sliding stage. In addition, the portion of the PSZ within SB2 increases as it simultaneously moves up the tool rake face and slides up the workpiece surface. At this stage, the shear strain rate begins to change as the chip's flow speed transitions from the cutting velocity to the chip flow velocity. Finally, Fig. 7(e) shows the shear strain rate distribution for stage 5 as S2 continues to slide along the adjacent workpiece. Additional flattening of the shear strain rate peak is observed between about 40 and 50 μm along ABC, which is within S2. This new peak shape is in stark contrast to the previous two stages in which peak flattening is observed within the workpiece material side of the PSZ. Furthermore, the portion of PSZ within SB2 widens dramatically from the previous stage (Fig. 7(d)), while the workpiece portion of PSZ narrows significantly.

Figure 7(f) is simply the strain rate profiles from Figs. 7(a)7(e), plotted simultaneously. These plots show that the location of PSZ shifts in the chip flow direction as a saw-tooth segment forms during saw-tooth chip formation. There is a gradual shift of the shear strain rate peak at each stage of deformation. Consequently, while the upper and lower boundaries of the PSZ can be defined at a given time, the location of the PSZ is not fixed relative to the cutting tool and instead shifts in the direction of chip flow due to segment sliding. The full width half max (FWHM) of the shear strain rate peaks, which in this study is taken as the PSZ thickness, is shown in the inset of Fig. 7(f). From Fig. 7, it is apparent that the formation of a shear band occurs progressively and that the thickness of the PSZ is initially large and decreases as the shear strain rate (and shear strain) begins to localize. Once localization occurs, the PSZ thickness increases slightly to 28 μm (stage 3). The final thickness (stage 5) of the PSZ is approximately 25 μm, and the majority of PSZ is contained within SB2.

Additionally, Fig. 8 shows the shear strain rate distribution through the chip thickness (parallel to the PSZ). Each shear strain rate distribution shown in Fig. 8 corresponds to the same stage of deformation shown in Fig. 7 (for example, Figs. 7(a) and 8(a) corresponding to stage 1 of deformation). The dashed lines in Figs. 8(a)8(e) are introduced to approximate the slope of the shear strain rate in the shear propagation (Figs. 8(a) and 8(b)) and the shear flattening regions (Figs. 8(c)8(e)) by showing its overall trend. In general, the shear strain rate is a maximum at the tool tip and decreases toward the free surface. Figure 8(a) shows the shear strain rate distribution corresponding to stage 1 of deformation, where the bulk segment (S2) is being deformed. A second peak is observed at approximately 35 μm along DEF and is due to the propagation of localized shear strain front toward the free surface (FS in Fig. 4). As the deformation begins to localize along SB2, there is significant broadening of the shear strain rate peak at the tool tip as shown in Fig. 8(b). In addition, there are several smaller peaks similar to those shown in Fig. 8(a), which illustrate the propagation of localized front of shear strain rate toward the free surface along the PSZ. The waviness in the strain rate plots along ABC may be attributed to the fact that ABC is not drawn exactly along the center of the PSZ; nonetheless, the gradual decrease from peak to peak shows an average decreasing strain rate toward the free surface. Figures 8(c)8(d) demonstrate segment sliding-induced gradual flattening through the midsection and free surface regions. In Fig. 8(e), the peak near the tool tip narrows. Figure 8(f) shows the slope of the flattening region along the chip thickness direction, which is the region between about halfway through the chip thickness and the free surface. In general, several trends can be observed from the shear strain rate distributions as seen in Fig. 8. First, the shear strain rate is a maximum at the tool tip and gradually decreases toward the free surface. Prior to localization, the slope is generally negative. Once instability occurs, the slope of the flattening region changes to positive as the segment slides along the workpiece. Second, periodic broadening followed by narrowing of the shear strain rate peak in the vicinity of the tool tip is observed.

#### Continuous Chip Formation at 0.5 m/s.

Figure 9 illustrates the incremental shear strain rate distributions of the continuous chip morphology when machining at 0.5 m/s. In particular, Fig. 9(a) shows a representative shear strain rate distribution along the chip flow direction with a singular peak around 5000 s−1 measured at 0.2 ms in the image sequence. As seen from Fig. 9(b), such a peak does not shift during the chip formation process, and its value is almost constant and nearly double that observed for the saw-tooth chip morphology when machining at 0.1 m/s. Moreover, the PSZ thickness is measured from the FWHM during the progression of continuous chips. As shown in the inset of Fig. 9(b), the PSZ thickness is approximately 35 μm, which is slightly larger than that found for the saw-tooth chip morphology. The location of the peak shear strain rate along the chip flow direction can be assumed to be a constant and occurs at approximately 60 μm (point B) along ABC. Thus, contrary to saw-tooth chip formation, a characteristic feature of continuous chip formation is that the location of the PSZ remains constant.

Figure 9(c) shows a representative shear strain rate distribution along the thickness direction. Much like the shear strain rate of the saw-tooth chip morphology, the shear strain rate is a maximum at the tool tip and gradually decreases toward the free surface. Contrary to saw-tooth chip formation, the slope of the shear strain rate along the chip thickness direction (Fig. 9(d)) is always negative, indicating that the shear strain rate is always lowest at the free surface. Moreover, the length of the flattening region along the chip thickness direction (along DEF) only deviates by ±5 μm.

## Discussion

Recent advances in DIC technologies have allowed new insights to be gained on the mechanics of the metal cutting process. The transition in chip morphology from saw-tooth (0.1 m/s) to continuous (0.5 m/s) raises a perplexing question on the changes in dynamic material behavior between slow and moderate cutting speed regimes, which in large part is a function of shear strain and shear strain rate. New insights on the dynamic material behavior from DIC are discussed subsequently, followed by a comparison of the measured shear strain and strain rate with those estimated using analytic models.

### Analysis of Digital Image Correlation Results.

During the formation of a saw-tooth chip, the material within the shear localized region is weakened either through strain softening [2] or catastrophic thermoplastic shear [37]. The extremely large strain (approximately 3.9) coupled with titanium's (ECAE and commercially pure) low thermal conductivity makes it difficult to determine with certainty after what mechanism the material is weakened, allowing for the segment to slide along the plane defined by SB2. Segment sliding, as observed in Fig. 5(d), further increases the cumulative shear strain due to friction generated between the two sliding surfaces [38,39].

In a continuous chip, some shear strain oscillation is observed in the chip region (Figs. 6(a)6(e)) and may be due to changes in the surface texture or changes in the shear strain in the secondary shear zone due to sliding along the tool rake face. Komanduri et al. [37] suggested that the deformation within the secondary shear zone becomes more influential during continuous chip formation. The effect of the secondary shear zone is also evident by examination of Fig. 6, whereby the shear strain is highest near the tool rake face. The shear strain distribution observed for the continuous chip (Fig. 6(e)) is different from that observed for the saw-tooth chip morphology (Fig. 5(e)) and is attributed to the fact that during saw-tooth chip formation, intersegment sliding also contributes to the overall deformation within the shear bands (such as SB2 in Fig. 5(d)), whereas during continuous chip formation no segmentation occurs. This indicates that the workpiece is deformed uniformly within the entire forming chip as opposed to cyclically as for saw-tooth chip formation.

This work is of the first to identify characteristic shear strain-rate distributions and peak shapes, corresponding to either continuous chip formation or different stages of saw-tooth chip formation. The formation of multiple peaks in Fig. 7(a) at 60 μm (point B) and approximately 75 μm along ABC in addition to the relatively larger width of the PSZ indicates that deformation is not localized within the PSZ during the initial stages of saw-tooth chip formation. Instead, the bulk segment (S2 in Fig. 4) is being deformed, which can be considered analogous to uniform deformation for continuous chip formation. Narrowing of the shear strain-rate peak shown in Fig. 7(b), midway through upsetting (stage 2) shows the shear strain-rate localization prior to the instability. The localization of strain observed in Fig. 7(b) is identified by the formation of a singular, narrow strain rate peak, thus illustrating that upsetting happens before the onset of shear localization. Instability occurs in the form of catastrophic shear localization (stage 3) and segment sliding as shown in Fig. 7(c). A characteristic feature of segment sliding is the flattening of the shear strain-rate peak, as shown in Fig. 7(c) by the flattening of the shear strain rate peak between 55 and 65 μm along ABC, and Fig. 7(d) by the flattened peak at 50–60 μm along ABC. This linear portion of the shear strain-rate peak is characteristic of the segment sliding stage and is analogous to the shear strain distribution in laminar fluid flow between two parallel plates. Additionally, it is characteristic that the shear strain-rate peak is widened during the segment sliding stage (Figs. 7(b) and 7(c)).

### Comparison With Analytic Model Results.

The high speed camera images are used to measure the parameters required to analytically predict the shear strain and shear strain rate. In this study, measurements of the chip morphology in the Fig. 4 image sequences are used to calculate the shear strain and shear strain rate according to the models presented elsewhere for saw-tooth chips [1214] and for continuous chips [6,10]. While most of these models use postmortem chips to calculate the shear strain and shear strain rates, the use of high-speed imaging system provides the advantage of observing the chip formation process in situ. For comparison, both the calculated and the measured shear strains and shear strain rates are shown in Fig. 10. Figure 10(a) shows a comparison of the shear strains for S2 and within SB2. In the figure, shear band is abbreviated as sb, and segment is abbreviated as seg. In general, the shear strains measured using DIC agree well with those calculated using the analytical models. However, the model presented in He et al. [12] overestimates the shear strain as found using DIC, while the models of Duan et al. [13] and Cotterell and Byrne [14] underestimate it. Moreover, all analytical models slightly overestimate the shear strain within S2 compared to that found using DIC. Furthermore, the analytical models are used to calculate the shear strain rate in the shear band and the comparison is shown in Fig. 10(b). For all cutting speeds investigated in this study, the analytical models slightly underestimate the shear strain rate. It is worth noting that the analytical models require the PSZ thickness of to be measured for shear strain/strain rate calculations. This study uses the FWHM of the DIC measurements as the mean PSZ thickness. Compared with other reported values for strains and strain rates, which measure the PSZ thickness from chip metallographs, the DIC measurements report a smaller thickness value. This discrepancy may be responsible for the difference in shear strain rates. Overall, DIC yields a good estimation of shear strains and shear strain rates during chip formation.

## Conclusions and Future Work

Equal channel angular extrusion-processed ultrafine-grained titanium (Ti) was machined at cutting speeds of 0.1 and 0.5 m/s, and high speed imaging and DIC were applied to elucidate the shear strain and shear strain rate distributions within the PSZ and neighboring segments during machining. The DIC analysis was used to identify characteristic shear strain-rate distributions for the continuous chip formation or for different stages of saw-tooth chip formation, delineate different regions of the PSZ, and identify the cyclic formation of an unstable BUE (cyclic BUE) that affects the shear strain and shear strain rate distributions during saw-tooth chip formation.

Some main conclusions are listed as follows: (1) the chip morphology is saw-tooth at 0.1 m/s while continuous at 0.5 m/s; (2) the cumulative shear strain and incremental shear strain rate of the saw-tooth chip morphology can reach approximately 3.9 and 2.4 × 103 s−1, respectively, and the cumulative shear strain and incremental shear strain rate of the continuous chip morphology may be approximately 1.3 and 5.0 × 103 s−1, respectively; (3) five stages of saw-tooth chip formation are identified: wedge-shaped compression, segment upsetting, adiabatic shear localization, segment sliding along the adjacent workpiece, and segment sliding up the rake face with simultaneous upsetting of the next segment; (4) shear localization within the PSZ always occurs, regardless of the chip morphology or cutting conditions. There is a distinct peak shift in the shear strain rate distribution during saw-tooth chip formation, indicating that the chip formation in terms of strain and strain rate localizes within a thin region/band between adjacent segments; there is a stable peak position of the strain rate distribution during continuous chip formation, indicating that the chip formation occurs uniformly along the chip flow direction. The PSZ thickness during saw-tooth chip formation is more localized and smaller than that during continuous chip formation (28 versus 35 μm); and (5) the DIC results match well with model predictions if the appropriate PSZ thickness is used, demonstrating that the DIC is a powerful tool for the experimental study of chip formation process.

Some future work may include: (1) to apply high speed imaging and digital image correlation to study the chip formation process of other engineering materials, (2) to develop better high speed imaging facilities for the better imaging of high speed machining processes and elucidate the workpiece dynamic deformation behavior over a wider range of shear strain rates, (3) generally, the shear strain rate increases the propensity for saw-tooth chip formation [14,30,32]; however, in this study, the higher strain rate at 0.5 m/s has resulted in continuous chip formation, while the lower strain rate at 0.1 m/s has led to aperiodic saw-tooth chip formation. In order to determine the specific microstructural mechanisms responsible for the changes in the chip formation mechanism, further work is required, and (4) to determine the accuracy of the DIC method by comparing the estimated strains with those measured directly as well as theoretical analysis.

## Acknowledgment

The titanium bars were provided by Professor K. T. Hartwig of Texas A&M University.

## Funding Data

• National Science Foundation (Grant No. CMMI-1404926).

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