Applied Science, Faculty of
Mechanical Engineering, Department of
DSpace
UBCV
Fazlali, Mohammad Reza
Doctor of Philosophy - PhD
University of British Columbia
Advanced metals, such as titanium and nickel alloys, have been widely used in the production of high-value and high-performance components in various industries, including aerospace, biomedical, automotive, etc. Shear localization is the dominant chip formation mechanism in machining of these metallic components, limiting the cutting productivity due to high cutting temperature, rapid tool wear, and tool breakage. The machined surface integrity is also affected by shear localization with wavy topography of the finished surface and residual stress fluctuation. Therefore, it is of practical importance to investigate shear localization and its effects during machining processes. This thesis proposes a new analytical thermo-mechanical model for shear localization by assuming an extended primary deformation zone and complete localization of plastic deformation inside shear bands after their formations. Unlike the classical cutting mechanics, the extended primary deformation zone enables capturing the material behavior from homogeneous deformation, to instability onset, and finally, to localization. It will be shown that localization changes the contact mechanism between the tool and the chip, where both sticking and sliding lengths change with respect to time in each localization cycle. Subsequently, the thermal load on the tool’s rake face is affected by the shear band formation in front of the tool. The model predicts: 1) shear band’s stress, temperature, and strain rate inside the primary deformation zone; 2) shear band spacing (subsequently, cutting force frequency) and width, 3) shear band’s displacement, its distribution and chip morphology; 4) time-varying sticking-sliding contact lengths; 5) time-varying cutting forces; 6) normal stress and shear stress distributions on the tool’s rake face; and 7) transient temperature distribution at the tool-chip interface. It is experimentally shown that: 1) the produced shear bands roll on the cutting tool’s rake face, causing an intimate contact between the tool and high temperature shear band. 2) the large displacement at the shear band’s location (and subsequently, the chip morphology) can be predicted without any fracture assumption and criterion; 3) both sliding and sticking contact lengths change with respect to time in each localization cycle.
Analytical Modeling of Shear Localization in MachiningProcesses and Its Effects on the Tool-Chip InterfacebyMohammad Reza FazlaliA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Mechanical Engineering)The University of British Columbia(Vancouver)May 2023© Mohammad Reza Fazlali, 2023The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the thesis entitled:Analytical Modeling of Shear Localization in Machining Processes and ItsEffects on the Tool-Chip Interfacesubmitted by Mohammad Reza Fazlali in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in Mechanical Engineering.Examining Committee:Xiaoliang Jin, Associate Professor, Mechanical Engineering, UBCSupervisorGary Schajer, Professor, Mechanical Engineering, UBCSupervisory Committee MemberMatthias Militzer, Professor, Materials Engineering, UBCUniversity ExaminerAdam Clare, Professor, Mechanical Engineering, UBCUniversity ExaminerAdditional Supervisory Committee Members:Steve Cockcroft, Professor, Materials Engineering, UBCSupervisory Committee MemberYusuf Altintas, Professor, Mechanical Engineering, UBCSupervisory Committee MemberiiAbstractAdvanced metals, such as titanium and nickel alloys, have been widely used in the production ofhigh-value and high-performance components in various industries, including aerospace, biomedi-cal, automotive, etc. Shear localization is the dominant chip formation mechanism in machining ofthese metallic components, limiting the cutting productivity due to high cutting temperature, rapidtool wear, and tool breakage. The machined surface integrity is also affected by shear localiza-tion with wavy topography of the finished surface and residual stress fluctuation. Therefore, it isof practical importance to investigate shear localization and its effects during machining processes.This thesis proposes a new analytical thermo-mechanical model for shear localization by as-suming an extended primary deformation zone and complete localization of plastic deformationinside shear bands after their formations. Unlike the classical cutting mechanics, the extended pri-mary deformation zone enables capturing the material behavior from homogeneous deformation,to instability onset, and finally, to localization. It will be shown that localization changes the con-tact mechanism between the tool and the chip, where both sticking and sliding lengths change withrespect to time in each localization cycle. Subsequently, the thermal load on the tool’s rake faceis affected by the shear band formation in front of the tool. The model predicts: 1) shear band’sstress, temperature, and strain rate inside the primary deformation zone; 2) shear band spacing(subsequently, cutting force frequency) and width, 3) shear band’s displacement, its distributionand chip morphology; 4) time-varying sticking-sliding contact lengths; 5) time-varying cuttingiiiforces; 6) normal stress and shear stress distributions on the tool’s rake face; and 7) transient tem-perature distribution at the tool-chip interface. It is experimentally shown that: 1) the producedshear bands roll on the cutting tool’s rake face, causing an intimate contact between the tool andhigh temperature shear band. 2) the large displacement at the shear band’s location (and subse-quently, the chip morphology) can be predicted without any fracture assumption and criterion; 3)both sliding and sticking contact lengths change with respect to time in each localization cycle.ivLay SummaryAdvanced metals are widely used in aerospace, biomedical, and automotive industries because oftheir desired thermomechanical properties. During machining of these metals, however, the plasticdeformation is localized in very narrow regions, called adiabatic shear bands. The shear band for-mation in machining of these alloys decreases the final part’s quality and machining productivity.This thesis investigates the material deformation behavior in machining of high performancemetals, including modeling and experimental investigations. The workpiece temperature, the de-veloped cutting forces, the cutting tool’s temperature, and the frequency of their fluctuations arepredicted. The results can be used to select suitable machining parameters to increase the processperformance based on the metal being cut, the cutting tool’s life, and the finished surface quality.vPrefaceThe research presented in this PhD thesis is the original work performed by the author, Moham-mad Reza Fazlali, under the supervision of Dr. Xiaoliang Jin. The author was responsible for themajor areas of conceptualization, all experimental investigations, and drafting all manuscripts andpresentations. Dr. Jin supervised the entire research and provided feedback on the manuscripts andpresentations. Dr. Mauricio Ponga collaborated in the first two chapters of this thesis by providingconstructive feedback and by editing the manuscripts. All research was conducted at The Univer-sity of British Columbia, Vancouver, Canada.Chapter 2 is based on versions of two published manuscripts:• Fazlali M, Ponga M, Jin X. Analytical Modeling of Shear Localization in Orthogonal Cut-ting Processes. ASME Journal of Manufacturing Science and Engineering. 2022 Feb1;144(2).• Fazlali M, Ponga M, Jin X. Analysis of the Unique Mechanics of Shear Localization in MetalCutting Processes. International Manufacturing Science and Engineering Conference 2022Jun 27 (Vol. 85819, p. V002T05A054). American Society of Mechanical Engineers.In this chapter, all machining experiments, metallurgical sample preparations, scanning electronmicroscopy observations, and interpretations of the results were performed by the author, Mo-hammad Reza Fazlali. Moreover, the author developed the analytical thermomechanical model tovipredict the shear band’s stress and temperature evolution inside the primary shear zone by consid-ering a finite thickness for the primary deformation zone, and proposed a model for predicting thespacing between neighboring shear bands during orthogonal cutting processes. The author alsowrote the initial draft of the paper. Dr. Xiaoliang Jin (corresponding author) and Dr. MauricioPonga were involved in the supervision and paper editing.A version of Chapter 3 has been published in the Journal of Material Processing Technology(JMPT):• Fazlali M, Ponga M, Jin X. Predictive model of chip segmentation in machining of high-strength metallic alloys. Journal of Materials Processing Technology. 2022 Oct 1;308:117723.In this chapter, the author, Mohammad Reza Fazlali, performed all the machining experiments,metallurgical sample preparations of titanium and nickel alloys, and optical, laser, and scanningelectron microscopy observations. The results were also interpreted by the author with constructivefeedback of Dr. Xiaoliang Jin and Dr. Mauricio Ponga. The author also predicted the shear bands’displacement and its distribution as well as the chip morphology. The manuscript was written bythe author and edited by Dr. Xiaoliang Jin and Dr. Mauricio Ponga.The content of Chapter 4 is under review under the following title:• Fazlali M. and Jin X., Time-Varying Tool-Chip Contact in the Cutting Mechanics of ShearLocalization. Under Review.The author, Mohammad Reza Fazlali, identified the time-varying nature of the contact length atthe tool-chip interface via experimental observations. The theoretical modeling was performed bythe author to predict the time-varying contact lengths, the contact normal and shear stresses, thetransient temperature at the contact interface, and the contact force fluctuations. The author alsowrote the initial manuscript while Dr. Xiaoliang Jin edited it.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxviiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxviiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Shear Localization and Adiabatic Shear Bands . . . . . . . . . . . . . . . . . . . . 11.2 Which metallic alloys show shear localization during machining? . . . . . . . . . . 31.3 Practical Importance of Shear Localization in Machining Processes . . . . . . . . . 7viii1.4 Need for Analytical Modeling of Shear Localization . . . . . . . . . . . . . . . . . 81.5 Thesis Objectives and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Analytical Modeling of Shear Localization in Orthogonal Cutting Processes . . . . . 162.1 Introduction and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Experimental Investigations of Shear Localization in Cutting Ti-6Al-4V . . . . . . 192.3 Analytical Mechanics Model of Shear-Localization . . . . . . . . . . . . . . . . . 272.3.1 Governing equations and solution in homogeneous deformation . . . . . . 282.3.2 Localization in shear band formation . . . . . . . . . . . . . . . . . . . . . 322.3.3 Shear band spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 Prediction of Shear Band Displacement and Rolling Phenomenon . . . . . . . . . . 533.1 Introduction and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Experimental Investigations of Chip Segmentation in Ti-based and Ni-based Metal-lic Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2.1 Chip segmentation in Ti-based alloys . . . . . . . . . . . . . . . . . . . . . 573.2.2 Chip segmentation in Ni-based alloys . . . . . . . . . . . . . . . . . . . . 613.3 Effects of elastic unloading at the tool-chip interface . . . . . . . . . . . . . . . . . 673.4 Prediction of strain distribution and chip morphology . . . . . . . . . . . . . . . . 753.5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964 Time-Varying Tool-Chip Contact in Cutting Mechanics of Shear Localization . . . . 994.1 Introduction and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 994.2 Mechanisms of Shear Localization and Time-Varying Tool-Chip Contact . . . . . . 102ix4.3 Modeling of Cutting Forces and Time-Varying Contact Lengths . . . . . . . . . . . 1124.3.1 Sliding region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.3.2 Sticking region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.4 Temperature Distribution at the Tool-Chip Interface . . . . . . . . . . . . . . . . . 1264.5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314.5.1 Shear band’s analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314.5.2 Time-varying sticking and sliding contacts . . . . . . . . . . . . . . . . . . 1354.5.3 Time-varying cutting forces and stresses . . . . . . . . . . . . . . . . . . . 1474.5.4 Transient temperature at the tool-chip interface . . . . . . . . . . . . . . . 1544.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565 Conclusions and Future Research Directions . . . . . . . . . . . . . . . . . . . . . . 160Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165A Shear Band’s Failure Surface and Bifurcation . . . . . . . . . . . . . . . . . . . . . 176xList of TablesTable 2.1 Contact length s and displacement f of shear-localized chips for the cuttingspeeds 60 m·min−1 and 90 m·min−1. In each cutting speed, four measurementsat different locations as well as the mean values are represented. . . . . . . . . . 25Table 2.2 Shear band thickness 2δ and the thickness of the plastic deformation at thetool-chip interface p for the cutting speeds 60 m·min−1 and 90 m·min−1. Ineach cutting speed, the measurements for four different shear bands as well asthe mean values are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Table 2.3 Thermo-mechanical properties of Ti-6Al-4V alloy [82]. . . . . . . . . . . . . . 39Table 2.4 Experimentally measured chip thickness and the calculated shear angle for fivedifferent cutting speeds when cutting Ti-6Al-4V with uncut chip thickness d =120 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Table 2.5 Parameters of the Johnson-Cook flow law for Ti-6Al-4V alloy [82]. . . . . . . . 42Table 2.6 Initial and maximum strain rates for the cutting speeds 30, 60, and 90 m·min−1.The ratio γ˙ in/γ˙h is also represented. . . . . . . . . . . . . . . . . . . . . . . . . 43Table 2.7 Reference variables for nondimensionalization. . . . . . . . . . . . . . . . . . . 46Table 2.8 Comparison of the predicted temperatures in continuous and shear-localizedchip formation models for the cutting speeds 30, 60, and 90 m·min−1. . . . . . . 48xiTable 3.1 Calculation of the shear angle for annealed Inconel 718 and Waspaloy by ex-perimental measurements of chip thickness in different cutting speeds. . . . . . 81Table 3.2 Thermo-mechanical properties of Ti-6Al-4V [82, 112], Inconel 718 [6], andWaspaloy [24]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Table 3.3 Johnson-Cook parameters for Ti-6Al-4V [65, 82], aged Inconel 718 [70, 89],annealed Inconel 718 [89], and Waspaloy [93, 95]. . . . . . . . . . . . . . . . . 83Table 3.4 Summary of the effects of thermo-mechanical parameters, cutting condition,and tool geometry on the characteristic response of the workpiece material tochip segmentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Table 3.5 Strain rate magnitudes when cutting Ti-6Al-4V and Inconel 718 at the cuttingspeed vc = 60 m·min−1 in three different stages. . . . . . . . . . . . . . . . . . 92Table 3.6 Experimental measurements and theoretical predictions of strains and displace-ments when cutting Waspaloy and Ti-6Al-4V at different cutting speeds. . . . . 96Table 4.1 Calculation of the shear angle for Ti-6Al-4V by experimental measurements ofchip’s maximum thickness for two different cutting speeds of vc = 1 m·s−1 andvc = 1.5 m·s−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Table 4.2 Comparison between the experimental values of the shear band’s active lengthλ (t) and the prediction of Eq. (4.2) at the cutting speed vc = 1 m·s−1. Fivemeasurements at different places are performed for each λmax and λmin, and theaverage value is reported. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Table 4.3 Comparison of the total contact lengths predicted by the theoretical models ofLee and Shaffer [64], Kato et al. [52], Hahn [44], Rubenstein [99], and Sutter[115]. The contact length magnitudes for the cutting condition of vc = 0.65m·s−1, d = 200 µm, α = 0◦, and φ = 38◦ are also presented. . . . . . . . . . . 143xiiList of FiguresFigure 1.1 Adiabatic shear bands in (a) AISIS 1018 cold-rolled steel forming in the cornerof the impact crater with the impact speed of 100 m·s−1 [98], (b) an aluminumtarget impacted by a projectile with the impact speed of 353 m·s−1 (magni-fication times 6) [123]. The shear bands have long and narrow appearanceoccurring in the corner of the impact crater, causing failure of the target. . . . . 2Figure 1.2 (a) Shear band formation pattern shown in a transmission electron microscopymicro-graph of aluminum shocked to 0.9 GPa [40]. A cross-hatched defor-mation pattern occurs during the passage of the shock wave induced by thecompression impact. (b) Adiabatic shear band formed in a 7039 Aluminumcompression specimen at a strain rate of 8×103 s−1 (magnification times 20)[128]. The shear band extends from top-right corner to the bottom-left cornerof the specimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Figure 1.3 Intense narrow shear bands (between arrows) happen periodically betweenthick segments when cutting Ti-6Al-4V [55]. . . . . . . . . . . . . . . . . . . 4xiiiFigure 1.4 Some applications of advanced metals. (a) In jet engines, titanium alloys areused in the parts where high strength-to-weight ratio, corrosion and creep re-sistance, and toughness are required while nickel alloys are used where main-taining strength at elevated temperatures is needed. Adapted from [118]. (b)High strength steels are used in different parts of a car for their wear and fa-tigue resistance. Adapted from [48]. (c) Titanium alloys are widely used inbiomedical industry due to their high strength-to-weight ratio, corrosion resis-tance, and biocompatibility. Adapted from [116]. (d) Magnesium alloys areused in biomedical industry as they are biodegradable, making them excel-lent for the healing process of the human body. From top-left to bottom-right:Cardiovascular stents [31], MAGNEZIX screw [7], biodegradable orthopedicimplant [120], and wound-closing device [45]. (e) Cobalt alloys are employedin biomedical industry because of their excellent mechanical properties, highcorrosion and wear resistance, and low friction coefficient. Adapted from [7]. . 6Figure 1.5 (a) Scanning electron microscopy (SEM) micro-graph of tool wear on the rakeface when machining Ti-6Al-4V [55]. The maximum tool wear is located at thetool-tip due to high temperature and pressure produced by shear localization onthe tool-tip. (b) Tool-tip breakage when machining Ti-6Al-4V with tungstencarbide (WC-Co) inserts [5]. Shear localization causes high fluctuation of thecutting force acting on the small tool-chip contact length, resulting in rapidtool-tip breakage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Figure 1.6 Shear-localized chips and machined surfaces of AerMet100 steel at differentlocalization intensities [113]. Shear localization in the chip causes wavy ma-chined surface whose amplitude is proportional to the localization intensity. . . 9xivFigure 1.7 The spatial variation of the plastic deformation zone at the finished surfacewhen cutting Ti-6Al-4V at the cutting speed 250 m·min−1 [129]. The thicknessof plastic deformation zone at the machined surface fluctuates with the samefrequency of shear localization (SL). The minimum and maximum thicknessesare 2.5 µm and 5 µm, respectively. . . . . . . . . . . . . . . . . . . . . . . . . 10Figure 1.8 Thesis flowchart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Figure 2.1 (a) Employed experimental setup for the orthogonal cutting experiments. Theuncut chip thickness d was measured along the feed direction, while the widthof cut w was toward the center of the workpiece’s cross section. (b) Tool holderwith a small free length to avoid the effects of static deflection on the results.(c) Cutting tool insert with zero degree rake angle, zero degree inclination an-gle, and flat rake face. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 2.2 Schematics of the shear-localized chip formation stages. At stage (a), SB1forms in front of the cutting tool. At stage (b), new workpiece material pilesup in front of the cutting tool until SB2 occurs. . . . . . . . . . . . . . . . . . 23Figure 2.3 SEM micro-graph of a shear-localized chip obtained under the cutting speedvc = 90 m·min−1. Both shear band and chip segmentation are clear. Each shearband is located between two successive segments. . . . . . . . . . . . . . . . . 23Figure 2.4 SEM micro-graph of a shear-localized chip at the tool-chip interaction side.The cutting speed is 60 m·min−1. The rolling of the shear bands on the cuttingtool rake face is shown for two successive shear bands. . . . . . . . . . . . . . 24Figure 2.5 Comparison of the segment’s contact length at the tool-chip interface s and theshear band’s displacement at the free surface side f for two different segments,showing that they are equal in each segment (i.e., f1 = s1 and f2 = s2). Thecutting speed is 60 m·min−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24xvFigure 2.6 Connection between the shear band thickness 2δ and the thickness of the plas-tic deformation at the tool-chip interface p via the rolling process. The majorpart of the plastic deformation at the tool-chip interface is due to the shearband that has rolled on the cutting tool rather than the friction at the tool-chipinterface. The cutting speed is 90 m·min−1. . . . . . . . . . . . . . . . . . . . 25Figure 2.7 Micro-lines in the SEM micro-graph of a shear-localized chip under the cuttingspeed vc = 60 m·min−1. Two micro-lines are distinguished by red and greencolors, and the shear bands’ locations are also shown. The micro-lines showdiscontinuity (jump) at the shear bands’ locations while remain almost straightwithing the segments. It shows that the majority of plastic deformation hasbeen localized inside the shear bands with negligible plastic deformation insidethe segments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 2.8 Comparison of the plastic deformation inside and the close vicinity of a shearband using a metallurgically prepared sample and employing high magnifica-tion SEM micro-graph. The microstructure is almost intact outside of the shearband while heavily deformed in its inside. The cutting speed vc = 60 m·min−1. 27Figure 2.9 Schematic of the shear-localized chip formation mechanism in orthogonal cut-ting. The total deformation region AOB includes the shear deformation regionAOC and the upsetting region BOC. The formation of the new shear band in thedeformation region AOB is also shown in the inset as well as the correspondingthickness for each region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 2.10 Schematic of the simplified shear deformation region AOC along with x-y La-grangian coordinate system. The shear velocity v/2 is applied on the bordersof the region AOC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29xviFigure 2.11 Schematic representation of shear band formation at the plane OC. The nondi-mensional shear velocity v¯2 is applied on the shear band. The shear band actsas a boundary layer on which the segment slides rigidly. . . . . . . . . . . . . 35Figure 2.12 Homogeneous stress and temperature evolution for the cutting speed vc = 60m·min−1. The onset of shear localization is also shown. . . . . . . . . . . . . . 42Figure 2.13 Evolution of the Shear band’s half-width with respect to the cutting speed. Asthe cutting speed increases, the shear band’s width decreases asymptotically.The experimental measurements of the shear band’s width based on i) SEManalysis and ii) micro-marker analysis presented in Refs. [101, 105] are alsoshown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 2.14 Visually distinguishable border between the shear band with large shear de-formation and the surrounding segments with low-plastic deformation for thecutting speed vc = 105 m·min−1. . . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 2.15 Measurement of shear band’s width at different locations for the cutting speedvc = 90 m·min−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 2.16 Stress and temperature evolution inside the shear band for the cutting speedvc = 60 m·min−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 2.17 Comparison of the stress and temperature evolution inside the shear band be-tween the cutting speeds vc = 60 m·min−1 and vc = 90 m·min−1. . . . . . . . . 47Figure 2.18 Variation of the shear band spacing with respect to the cutting speed. The er-ror bars show the minimum and maximum experimental measurements of theshear band spacing for each cutting speed. The mean value of the experimentalmeasurements for each cutting speed as well as the fitted curve on the experi-mental data are also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Figure 2.19 Stress and temperature oscillations of shear bands in the deformation regionAOB for the cutting speed vc = 60 m·min−1. . . . . . . . . . . . . . . . . . . . 51xviiFigure 3.1 Details of the experimental investigations. . . . . . . . . . . . . . . . . . . . . 58Figure 3.2 Machined chips’ optical micro-graphs for (a) CP titanium under the cuttingspeed vc = 60 m·min−1, (b) CP titanium under the cutting speed vc = 90m·min−1, (c) Ti-6Al-4V under the cutting speed vc = 60 m·min−1, (d) Ti-6Al-4V under the cutting speed vc = 90 m·min−1. The grains are heavily distortedat the shear bands’ locations, showing large plastic deformation at the shearbands’ locations. The segments are more regular in Ti-6Al-4V than CP titanium. 59Figure 3.3 Laser micro-graph of the shear band formation when cutting: (a) Ti-6Al-4Vand (b) CP titanium under the cutting speed vc = 60 m·min−1. In Ti-6Al-4V,the shear bands are distinct from the segment (appear as white narrow regions)while in CP titanium, they retain their initial microstructure with heavy plasticdeformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 3.4 Chip formation mechanisms in Waspaloy at different cutting speeds: (a) atvc = 30 m·min−1, homogeneous deformation occurs, (b) the cutting speed vc =45 m·min−1 is a transition cutting speed where both continuous and shear-localized chips occur, (c) the cutting chip at vc = 90 m·min−1 is shear-localized. 62Figure 3.5 Shear band interaction with the cutting tool’s rake face when cutting annealedInconel 718 at the cutting speed 90 m·min−1. For the shear band O′A′C′D′, theedge O′A′ has been rolled on the rake face while the edge C′D′ is almost straight. 64Figure 3.6 Schematic presentation of chip segmentation in Ni-based alloys. When a newshear band occurs in front of the cutting tool (SB1), the previously formedshear band (SB2) rolls on the cutting tool through its thickness. This causesthe plastic region at the tool-chip interface to have a variable thickness. Thepossible behaviors of a single shear band are also shown in the sub-figures. . . 65xviiiFigure 3.7 High-magnification laser micro-graphs of a shear band when cutting annealedInconel 718 under vc = 200 m·min−1 at three different locations: (a) at the freesurface side, (b) at the middle of the chip, (c) at the tool-chip interface. Thedifference in the thickness of the shear band at different locations shows thatthe shear band is in direct contact with the cutting tool. . . . . . . . . . . . . . 67Figure 3.8 Schematic presentation of local deformation zone at the tool-chip interface,which is induced by the elastic deformation of the workpiece material due tothe contact with the cutting tool’s rake face. . . . . . . . . . . . . . . . . . . . 68Figure 3.9 Schematic evolution of stress and temperature of a potential shear band fromthe time when it enters the primary deformation region (boundary OA) until thetime when it exits the primary deformation region (boundary OB). Differentzones show different stages of the shear band formation. In zone I, the work-piece material experiences homogeneous deformation. In zone II, the criticalstrain γcr has reached and the stress and temperature variations are intense. Inzone III, the shear band becomes mature with slight change in its stress andtemperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 3.10 Schematic diagram of the shear-localized chip morphology and its correspond-ing parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 3.11 Laser micro-graph of the experimentally measured shear angles during cuttingof: (a) Waspaloy at the cutting speed vc = 120 m·min−1, (b) Ti-6Al-4V at thecutting speed vc = 90 m·min−1. The maximum chip thickness rmax, averagechip thickness ravg , and minimum chip thickness rmin are also shown in thefigure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Figure 3.12 Developed temperature (absolute) and stress in the shear band when cuttingInconel 718 (aged) and Ti-6Al-4V at the cutting speed vc = 60 m·min−1 anduncut chip thickness d = 120 µm. . . . . . . . . . . . . . . . . . . . . . . . . 85xixFigure 3.13 Variation of strain rate with respect to the cutting speed for aged Inconel 718and Ti-6Al-4V. The developed strain rate in Ti-6Al-4V is higher than Inconel718 for all cutting speeds, and they diverge as the cutting speed increases. . . . 86Figure 3.14 The measurement method of the shear bands’ widths at two different magni-fications for Waspaloy chips under the cutting speed vc = 120 m·min−1. Theborder between the shear band and the surrounding material is visually distin-guishable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 3.15 Comparison between the predicted and experimental values of shear band’shalf-width δ for Ti-6Al-4V and aged Inconel 718. The error bars show themaximum and minimum values of the measured widths for each cutting speed.The Ti-6Al-4V shear bands are thinner than aged Inconel 718 for the samecutting speed. In both cases, the shear band’s width asymptotically decreaseswith respect to the cutting speed. The SEM data of Ti-6Al-4V (set 2) is adoptedfrom Ref. [105]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Figure 3.16 The prediction of the shear band’s half-width of Waspaloy compared to theexperimental results. At low cutting speeds both homogeneous and inhomoge-neous deformation take place, causing the theory to lose its accuracy. At highcutting speeds (high than 70 m·min−1, which is called cut-off speed, the shearband’s half-width is predicted with high accuracy. . . . . . . . . . . . . . . . . 91Figure 3.17 Effect of elastic unloading at the tool-chip interface on the strain rate evolu-tion of the shear band at the cutting speed vc = 60 m·min−1 for (a) Ti-6Al-4Vand (b) Inconel 718 (aged). The elastic unloading at the tool-chip interfacecauses an overshoot in the strain rate of the shear band when the homogeneousdeformation changes to shear localization. . . . . . . . . . . . . . . . . . . . . 93xxFigure 3.18 Comparison between the strain rate distribution of Ti-6Al-4V and aged Inconel718 at the cutting speed 60 m·min−1. The maximum strain rate of Ti-6Al-4Vis∼ 5.5×105 s−1 while for aged Inconel 718 is∼ 1.6×105 s−1. The strain ratedistribution is wider for Inconel 718 compared to Ti-6Al-4V. . . . . . . . . . . 95Figure 4.1 High speed image sequence of shear localization in cutting of Ti-6Al-4V atvc = 0.65 m·s−1 and d = 200 µm without any image processing technique. Atfirst, the tool-chip contact length is under sliding condition due to the shearstress drop inside SB1. At t = 125 µs, new workpiece material starts pilingup (upsetting) in front of the cutting tool, and therefore, the sticking lengthstarts from the tool-tip. During the upsetting, the shear band SB1 rolls onthe cutting tool rake face causing an intimate contact between the shear bandSB1 and the cutting tool’s rake face. As the cutting tool advances further,the sticking length increases and the sliding length decreases until the nextshear band (SB2) occurs. Finally, the shear band SB2 initiates in front of thecutting tool and the contact length returns back to its initial state at which allthe contact length is of the sliding nature. . . . . . . . . . . . . . . . . . . . . 106Figure 4.2 Flow distribution map obtained from PIV analysis when cutting Ti-6Al-4Vunder the cutting speed vc = 0.65 m·s−1 and uncut chip thickness d = 200µm. The green arrows show the distribution of the velocity vectors within theworkpiece material and cutting chip. At first, the entire cutting chip slides onthe cutting tool’s rake face. As more material piles up in front of the cuttingtool, the sticking region (i.e., the region with zero velocity) increases as shownin (b), (c), (d), and (e). Finally, when enough material piles up in front of thecutting tool, a new shear band occurs and the contact length returns to its initialstate, where all the contact length is of the sliding type. . . . . . . . . . . . . . 108xxiFigure 4.3 Distribution map of the velocity in the chip flow direction. The increasinglength of the sticking region is clear in frames (b), (c), and (d) with dark bluecolor near the tool-tip. At first (frame (a)), the sticking length is negligiblewhile as the cycle progresses, the sticking length increase to 131.28 µm (frame(b)), to 182.10 (frame (c)), and finally to 223.91 µm (frame (d)). At the end(frame (e)), a new shear band occurs in front of the cutting tool and the stickinglength returns to its initial state. . . . . . . . . . . . . . . . . . . . . . . . . . 109Figure 4.4 The upsetting direction of the new workpiece material in front of the cuttingtool. Note that the upsetting direction is horizontal near the tool-chip interface,causing the stationary contact between the tool’s rake face and the workpiecematerial at the tool-chip interface. Note that everything above SB1 is beingpushed upward by the upsetting region. . . . . . . . . . . . . . . . . . . . . . 110Figure 4.5 Out-of-plane deformation (or “bulging”) of the material near the tool-chip in-terface due to the stationary condition between the cutting chip and the tool’srake face in the sticking region. The material sticks to the cutting tool but itneeds to accommodate the plastic deformation imposed by the advancement ofthe tool. Therefore, it deforms in out-of-plane direction the tool-chip interfacedue to the incompressibility of metallic alloys,. . . . . . . . . . . . . . . . . . 112Figure 4.6 The contact between the fresh workpiece material (which piles-up in front ofthe cutting tool) and the cutting tool’s rake face. At first (Fig. 4.6a), the contactlength is small and located at the tool-tip. As the cutting tool advances, thecontact length increases until it reaches its maximum value at which the nextshear band (SB2) initiates (Fig. 4.6d). This time-varying sticking contact be-tween the workpiece material and the rake face is accompanied by sliding ofthe previous shear band (SB1) and the remainder of the chip along the tool’srake face. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113xxiiFigure 4.7 The force system acting during shear-localized chip formation. The forces aredivided into two groups, namely, the forces on the sliding region (Fv1 and Fu1)and the forces on the sticking region (Fv2 and Fu2). . . . . . . . . . . . . . . . 114Figure 4.8 The normal stress distribution on the sliding and sticking regions (i.e., σv1(xˆ, t)and σv2(x, t), respectively). Two different Cartesian coordinate systems areemployed: (i) the Lagrangian coordinate system xˆ-yˆ and the Eulerian coor-dinate system x-y. The time-varying stress amplitude in the sliding region isσˆ0(t) while σˆ0(t)+σ0(t) in the sticking region. Note that there is a slope dis-continuity in the normal stress distribution on the rake face at the location ofthe shear band SB1 (i.e., the border between the sticking and sliding regions.) . 117Figure 4.9 The forces acting on the deformation region AOB: Fv2(t) and Fu2(t) are ap-plied by the cutting tool’s rake face in the sticking region, Fs1(t) and Fn1(t) areapplied by the shear band SB1, and Fs2(t) and Fn2(t) are applied by the freshworkpiece material in front of the cutting tool. . . . . . . . . . . . . . . . . . . 120Figure 4.10 The minimum sticking length lminst between the piled-up region and the cuttingtool’s rake face due to the cutting tool’s edge radius. The part of edge from thestagnation point to the horizontal axis determines lminst . . . . . . . . . . . . . . 122xxiiiFigure 4.11 The control volume with the spatial domain of 0 ≤ y ≤ b and 0 ≤ x ≤ lmaxtotcontaining the shear-localized chip. Heat conduction in both x and y directionsas well as heat advection in x direction are taken into account. The spatialdomain is discretized into Nx×Ny elements, where ∆x and ∆y are the spatialincrements in x and y directions, respectively. The five boundary conditionsare: two Convection boundary conditions with convection coefficient hair andtemperature θair at y = b and x = lmaxtot ; two Dirichlet boundary conditions withtemperature θh(t) at x = 0 and θsb(t) at y = 0 (0 ≤ x ≤ lst(t)); and one heatflux (Neumann) boundary condition at y = 0 (lst(t) < x ≤ ltot(t)). The FTCSscheme is employed to solve the governing heat equation using finite differencemethod as shown in the subset. . . . . . . . . . . . . . . . . . . . . . . . . . 127Figure 4.12 Optical micro-graph of the shear-localized chip at the cutting speed vc= 1.5m·s−1 and uncut chip thickness d = 120 µm. Narrow shear bands betweenthick segments are clear. The maximum chip thickness, which is employedto calculate the shear angle, is shown. The magnitude of shear band’s activelength λ (t) at the beginning and end of a cycle, i.e., λ (0) = b and λ (tcycle),respectively, are also illustrated. . . . . . . . . . . . . . . . . . . . . . . . . . 132Figure 4.13 Laser micro-graph of the ex situ sample for the cutting speed vc= 1.5 m·s−1 anduncut chip thickness d = 120 µm. The shear band has different morphologythan the segments and the interaction between the shear band and the tool rakeface via rolling is clear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133xxivFigure 4.14 The variation of the shear band’s active length for five successive shear bands atthe cutting speed vc = 1 m·s−1 and uncut chip thickness d = 120 µm. At first,the shear band’s active length is λmax = b. As it travels inside the deformationregion AOB, its active length decreases until it reach its minimum value λmin.After exiting the primary deformation region, the next shear band enter thedeformation region with the initial active length of λmax = b. The experimentalmeasurements for five neighboring shear bands from the optical micro-graphof the prepared chip are also shown. . . . . . . . . . . . . . . . . . . . . . . . 134Figure 4.15 High-speed image sequence of two segmentation cycles when cutting Ti-6Al-4V at the cutting speed vc = 0.65 m·s−1 and uncut chip thickness d = 200µm. Each row shows one segmentation cycle in three different stages: (i) theearly stage; (ii) a random internal stage, and (iii) the final stage. At the earlystage, the sliding contact length is at its maximum, while as time progresses,the sliding contact length decreases until it reaches its minimum value. At thesame time, the sticking length increases from its minimum value at the earlystage to its maximum at the final stage. . . . . . . . . . . . . . . . . . . . . . . 138xxvFigure 4.16 Comparisons of the predicted sliding and sticking contact lengths with the ex-perimental results for nine experimental cycles of segmentation. (a) At the be-ginning of each segment, the sliding length is at its maximum value of 674.34µm (theoretical prediction) and decreases linearly until it reaches the minimumvalue of 358.82. At the same time, the sticking length starts from the minimumvalue of 6.98 µm and increases linearly until it reaches the maximum value of158.98 µm. At this time, enough material has piled up in front of the cuttingtool for the next shear band’s onset. Then, the sliding length jumps back toits maximum value while the sticking length drops back to its minimum. (b)Variations in the measurements of lmaxsl , lminsl , and lmaxst and their comparisonswith the theoretical predictions. The theoretical prediction for lmaxsl is locatedat the top of the measurement bar while for lminsl and lmaxst , the predicted valuesare well close to the mean value of the measurements. . . . . . . . . . . . . . . 142Figure 4.17 Comparison of the predicted total contact length with the experimental mea-surements when cutting Ti-6Al-4V at the cutting speed vc = 0.65 m·s−1 anduncut chip thickness d = 200 µm. The total contact length starts at 681 µm inthe beginning of the segmentation cycle and decreases linearly up to 518 µm.Similar trend is observable in the experimental results. The constant contactlength predicted by Lee and Shaffer [64] is also presented. . . . . . . . . . . . 144xxviFigure 4.18 The effects of cutting speed vc and uncut chip thickness d on the fluctuationof sliding and sticking contact lengths. As the cutting speed increases from0.65 m·s−1 to 1 m·s−1 (the uncut chip thickness d = 120 µm), lmaxsl remainsconstant, lminsl decreases, and the segmentation frequency increases from 8.85kHz to 11.72 kHz. As the uncut chip thickness increases from 120 µm to200 µm (cutting speed is constant and set at 0.65 m·s−1), both lmaxsl and lminslincrease while the segmentation frequency drops significantly from 8.85 kHzto 3.34 kHz. For the sticking length, the increase in the cutting speed causeslarger lmaxst . By increasing the uncut chip thickness from 120 µm to 200 µm,lmaxst largely increases from 68.98 µm to 158.98 µm. The minimum stickinglength remains constant for all three cutting conditions. . . . . . . . . . . . . . 146Figure 4.19 Variation of the Shear force Fs1(t) in one cycle of segmentation for the cuttingspeed vc = 1 m·s−1 and uncut chip thickness d = 120 µm. The shear forcedrops 211.25 N to 63.77 N in 85 µs. . . . . . . . . . . . . . . . . . . . . . . . 148Figure 4.20 The fluctuations of the contact forces Fv1(t) and Fu1(t) in the sliding region,and Fv2(t) and Fu2(t) in the sticking region for five segmentation cycles atvc = 1 m·s−1 and d = 120 µm. In each cycle of segmentation, Fv1(t) dropsfrom 417.27 N to 125.97 N while Fv2(t) increases from 6.69 N to 230.24 N.The friction force Fu1(t) decreases from 166.91 N to 50.39 N while Fu2(t)increases from 7.93 N to 44.16 N. The opposite trends between the normal andfrictional forces in the sliding and sticking regions are clear. . . . . . . . . . . 149Figure 4.21 Fluctuations of total normal and frictional forces, i.e., Fv(t) and Fu(t), respec-tively, for vc = 1 m·s−1 and d = 120 µm. In each segmentation cycle, Fv(t)drops from 423.96 N to 356.21 N and Fu(t) decreases from 174.83 N to 94.55N. Comparing with Fig. 4.20, the fluctuation amplitude is less for total forcescompared to sliding and sticking forces. . . . . . . . . . . . . . . . . . . . . . 150xxviiFigure 4.22 Prediction of the normal stress distribution on the cutting tool’s rake face whencutting Ti-6Al-4V at vc = 1 m·s−1 and d = 120 µm. Three different framesshow the time-varying nature of the stress distribution from the early stage oflocalization when t = 10.06 µs to the end of localization where t = 85.32 µs.As time increases, the sticking length increases from 13.53 µm to 78.98 µ ,while the sliding length decreases from 380.57 µm to 241.81 µm. The normalstress amplitude at the tool tip drops from 13.58 GPa to 3.85 GPa. . . . . . . . 152Figure 4.23 Prediction of the shear stress distribution on the cutting tool’s rake face fromt = 10.06 µs to t = 85.32 µs. At t = 10.06 µs, the sticking length is 15.47µm with the yield stress of 864.15 MPa. In the sliding region, the shear stresschanges from 0 MPa to 396.14 MPa. At t = 42.23 µs, the sticking lengthincreases to 42.62 µm with the yield stress of 665.12 MPa. The shear stresschanges from 0 MPa to 304.90 MPa in the sliding region. Finally and at theend of the segmentation cycle (t = 85.32 µs), the sticking length is 78.98 µmwith the yield stress of 559.09 MPa while the shear stress changes from 0 MPato 256.30 MPa in the sliding region. . . . . . . . . . . . . . . . . . . . . . . . 154Figure 4.24 Transient temperature distribution at the tool-chip interface when cutting Ti-6Al-4V at the cutting speed vc = 1 m·s−1 and uncut chip thickness d = 120µm in one segmentation cycle. At first, the maximum temperature is 602.00K located at 2.67 µm away from the tool-tip. At t = 30 µs, the maximumtemperature increases to 1037.91 K. At t = 60 µs, the maximum temperatureincreases further to 1167.25 K from 2.67 µm to 56.20 µm. Finally, the finalposition of the maximum temperature is at 77.61 µm distance from the tool-tipwith the magnitude of 1233.26 K. . . . . . . . . . . . . . . . . . . . . . . . . 156xxviiiFigure A.1 Top view SEM image of the Ti-6Al-4V shear-localized chip at the cuttingspeed vc = 60 m·min−1 and uncut chip thickness d = 120 µm. The periodicshear bands as well as the segments are clear. . . . . . . . . . . . . . . . . . . 177Figure A.2 Top view SEM image of the shear band for the cutting speed vc = 60 m·min−1and uncut chip thickness d = 120 µm. Two different regions are clear: parabolicdimples and smooth sliding surfaces. . . . . . . . . . . . . . . . . . . . . . . . 178Figure A.3 Bifurcation in shear band at the cutting speed vc = 60 m·min−1 and uncut chipthickness d = 120 µm, where the shear band branches off near the tool-chipinterface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179xxixList of Symbolsα rake angle, degreeβ Taylor-Quinney coefficient, -χ thermal diffusivity, m2·s−1∆t time increment, µs∆u local deformation at the tool-chip interface in the shear angle’s direction, µm∆un thickness of the local deformation zone at the tool-chip interface, µm∆x spatial increment in the x-direction, µm∆y spatial increment in the y-direction, µmδ shear band’s half-width, µmδ f fluid’s boundary layer thickness, mmγ˙ equivalent plastic strain rate, 1/sΓ˙in nondimensional strain rate inside the shear band, -γ˙ in shear band’s strain rate, s−1γ˙h homogeneous strain rate, s−1xxxγ˙re f reference strain rate, s−1εn elastic strain normal to the rake face, -εt elastic strain tangential to the rake face, -η stagnation angle, radiansη1 factor for considering the shear zone thickness(Oxley’s model)η2 proportion of the heat conducted into the workpiece(Oxley’s model)γ equivalent plastic shear strainγ in shear band’s strain, -γh homogeneous strain, -γcr critical strain for the shear band formationσˆ0(t) normal stress amplitude in the sliding region, MPahˆ thickness of the plastic deformation at the tool-chip interface, µmλ (t) active length of the shear band, µmElocal elastic energy at the local deformation zone, Jµ friction coefficient, -ν Poisson’s ratio, -Ω coefficient capturing the effects of elastic modulus E and Poisson’s ratio ν , GPaxxxiφ shear angle, degreeφseg segment’s angle, degreeψ oscillation frequency of the cutting force, kHzρ mass density, kg·m−3σ0(t) normal stress change in the sticking region, MPaσn normal elastic stress at the tool-chip interface, MPaσv1(xˆ, t) normal stress distribution on the rake face in the sliding region, MPaσv2(x, t) normal stress distribution on the rake face in the sticking region, MPaτ(y, t) shear stress, MPaτ∗ reference shear stress, MPaτ in shear stress inside the shear band, MPaτ0(t) shear stress amplitude in the sliding region, MPaτh homogeneous shear stress, MPaτu1(xˆ, t) shear stress distribution in the sliding region, MPaθ(y, t) relative temperature, Kθ ∗ reference temperature, Kθ in temperature inside the shear band, Kθ out temperature outside the shear band, Kθh homogeneous temperature, Kxxxiiθair air temperature, Kϒ momentum diffusivity, m2·s−1ϑ effect of complex stress state in the deformation zone, -ξ (t) segmentation degree, -ζ distribution exponent, -a thermal softening coefficient, 1/KA, B Johnson-Cook Constants, MPab shear plane’s length, µmC strain rate sensitivity, -c heat capacity, J·(kg.K)−1CL longitudinal wave speed, m·s−1d uncut chip thickness, µmdd thickness of the total deformation region, µmds thickness of the upsetting region, µmdt thickness of the shear deformation region, µmE elastic modulus, GPaf displacement of the shear band at the free surface, µmFc resultant cutting force, NFu total friction force on the tool’s rake face, NxxxiiiFv total normal force on the tool’s rake face, NFn1(t) normal force on the shear band, NFn2(t) homogeneous normal force in the primary deformation zone, NFs1(t) shear force on the shear band, NFs2(t) homogeneous shear force in the primary deformation zone, NFu1(t) friction force on the tool’s rake face in the sliding region, NFu2(t) friction force on the tool’s rake face in the sticking region, NFv1(t) normal force on the tool’s rake face in the sliding region, NFv2(t) normal force on the tool’s rake face in the sticking region, Nhair air convection heat coefficient, W/(m2·K)hmax maximum thickness of the segmented chip, µmk thermal conductivity, W·(m.K)−1lsl(t) time-varying sliding contact length, µmlst(t) time-varying sticking contact length, µmlmaxst maximum sticking length, µmlminst minimum sticking length, µmltot(t) total contact length, µmM spindle speed, rev·min−1m thermal softening exponent, -xxxivN number of shear bands in a medium with limited thickness, -n work-hardening coefficient, -Nx number of elements in the x-direction, -Ny number of elements in the y-direction, -p thickness of the plastic deformation regionat the tool-chip interface, µmqsl(t) heat flux at the tool-chip interface in the sliding zone, W/m2R heat partition at the tool-chip interface, -r chip compression ratio, -r0 edge radius, µmravg. average chip thickness, µmrmax maximum thickness of the chip, µmrmin minimum thickness of the chip, µms contact length of the segment with the tool, µmT (y, t) absolute temperature, KtB traveling time in the deformation region AOB, µstC traveling time in the shear deformation region AOC, µstH heat diffusion characteristic time, µsTm melting temperature, KxxxvTr room temperature, KTs temperature at the shear plane(Oxley’s model), KtS shear band formation characteristic time, µstcycle time period of each segmentation cycle, µstcr critical time for shear band formation, µsv shear velocity, m·s−1v∗ reference shear velocity, m·s−1vc cutting speed, m·s−1vs Cutting tool’s velocity in the shear angle’s direction, m·s−1vch chip velocity, m·s−1w width of cut, mmPr Prandtl number, nondimensionalxxxviList of AbbreviationsAM additively manufacturedCEL coupled Eulerian-LagrangianFD finite differenceFE finite elementJC Johnson-CookPIV particle image velocimetrySEM scanning electron microscopySL shear localizationTi titaniumxxxviiAcknowledgmentsFirst, I would like to express my gratitude to my supervisor, Dr. Xiaoliang Jin, for his guidance,support and encouragement throughout my PhD journey. I want to thank the Manufacturing Au-tomation Laboratory (MAL) and Professor Yusuf Altintas for providing the experimental facilitiesI needed for my experiments. During my thesis, I had the privilege of working with and learningfrom bright minds. Dr. Ahmad Mohammadpanah is one of them whom I have learned a lot from.I would also like to thank all my friends at The University of British Columbia and AdvancedManufacturing Processes (AMP) laboratory for their support.My family’s support has made this journey possible, despite the distance between us. I wantto thank my parents Hossein and Maryam, sisters Elahe and Elham, brothers Ruholla and Ra-soul, brothers-in-law Payman and Saeed, sister-in-law Sepideh, and finally, my lovely nieces Yas,Samin, Ali, Zahra, Amirhossein, Hosna, and Reyhane for all their love and care. I want my dad toknow that he is my role model. Throughout my life, I have constantly seen him spreading kind-ness unconditionally, which I am still striving to emulate. I am grateful to have both him and mybeautiful mom in my life, and I love them from the bottom of my heart.Thank you all for being a part of my journey and for making this experience unforgettable.xxxviiiDedicationTo my dad, Hossein,my mom, Maryam,&my niece, YasxxxixChapter 1Introduction1.1 Shear Localization and Adiabatic Shear BandsThermoplastic deformation localization, or shear localization (SL), occurs where strong gradientsin the motions of a material body are imparted. Because of this strong motion gradient, the stabilityof the material’s constitutive response is crucial for the homogeneous and uniform deformation.Otherwise, deformation localization can happen. There is a particular interest in deformation local-ization where the dissipated heat traps adiabatically, causing a local thermal and/or microstructuralsoftening due to the strong motion gradient. In any case, the local softening mechanism causeslocalization of the dissipated plastic deformation in a narrow, nearly planar region of the materialcalled adiabatic shear band.Adiabatic shear bands occur in extreme loading conditions, including (1) ballistic impact, (2)fragmentation, (3) explosive loading (shock wave application), (4) dynamic blanking, (5) surfacerubbing, (6) impact erosion, (7) rolling, and (8) machining (see Refs. [27, 28, 126] for more appli-cations). For instance, in penetration mechanics, the goal is to design a target with the minimumcost and weight to protect against a projectile impact, or to design a projectile that can penetrate1the target by minimum energy and material. Fig. 1.1 shows the sectioned AISI 1018 cold-rolledsteel and aluminum targets impacted by projectiles with the velocities of 100 m·s−1 and 353 m·s−1,respectively. Based on Fig. 1.1, the shear band occurs in the corner of the impact crater due to theimpact of the projectile. The shear band has long and narrow appearance starting at a corner ofthe crater due to a stress concentration and extending deep into the specimen in the direction ofimpact. Fig. 1.1 shows that the shear band generation is the reason for the failure under dynamicimpact.116 H. C. ROGERS Figure 12. Transformed shear band at corner of impact crater in 1018 steel, cold rolled 67 percent and impacted at 100 m/s. Etched in nita!. 70 -30 BRASS, ANNEALED 1/8" PROJECTILE 269 FT/SEC Figure 13. Deformation of back side of an annealed 70-30 brass tar-get, .250 in. thick, showing general deformation and bulging. Target was mounted on a backing plate with a central .500 in. diameter hole. (a)128 R .L . Woodward FIGURE 6. Exarmples of the failure described in Fig. 5. (a) Plugging of a titanium alloy in which shear bands, although nucleating independently, appear to join up circumferentially to allow complete separation of the plug along shear bands. 8.5X. (b) Partial shear banding followed by a ductile fracture due to bending in an aluminium alloy. 6X. (c) Partial shear band failure followed by fracture by bending where anisotropy has influenced the fracture path. 6X. (b)Figure 1.1: Adiabatic shear bands in (a) AISIS 1018 cold-rolled steel forming in the cornerof the impact crater with the impact speed of 100 m·s−1 [98], (b) an aluminum targetimpacted by a projectile with the impact speed of 353 m·s−1 (magnification times 6)[123]. The shear bands have long and narrow appearance occurring in the corner of theimpact crater, causing failure of the target.SL plays a central role in shock failure too. Fig. 1.2 shows the heterogeneous shear defor-mation in the form of very fine shear bands in an aluminum specimen shocked to 0.9 GPa andcompares the observation with the shear band formation in a compression test of 7039 aluminumat a strain rate of 8×103 s−1. Based on Fig. 1.2a, a cross-hatched deformation pattern takes placeduring the passage of the shock wave induced by the compression impact. Shear band formation2in machining processes is in the form of narrow shear bands between thick segments happeningat a certain frequency. Fig. 1.3 shows the micro-graph of a Ti-6Al-4V shear-localized chip whereintense shear bands (between arrows) are formed between the segments. The term “localization”shows the extreme thinness of the produced shear bands compared to the thick segments. Theperiodic formation of the shear bands between the segments is also clear in Fig. 1.3.PROPERTIES OF PRECIPITATION HARDENED ALUMINUM (c) 14.7 GPa Shock (d) 9.0 GPa SHOCK Fig. 7c,d. Optical and TEM Metallographs of Shock-Deformed Aluminum. 93 (a)The high strain rate compression of 7039 aluminium 613 FIG. 5. (a) deformation cross observed at a strain of 0-24 (manification times 20). (b) Shear band formed at a strain of 0-26 (magnification times 20). (c) Cracking along the shear band at a strain of 0-35 (magnification times 20). All at a strain rate of 8 x 10 ~ s -~. (b)Figure 1.2: (a) Shear band formation pattern shown in a transmission electron microscopymicro-graph of aluminum shocked to 0.9 GPa [40]. A cross-hatched deformation pat-tern occurs during the passage of the shock wave induced by the compression impact.(b) Adiabatic shear band formed in a 7039 Aluminum compression specimen at a strainrate of 8×103 s−1 (magnification times 20) [128]. The shear band extends from top-right corner to the bottom-left corner of the specimen.1.2 Which metallic alloys show shear localization duringmachining?Advanced metallic alloys, including titanium, nickel, and iron alloys, are widely used for highvalue, high performance components in industries such as aerospace, automotive, and biomedi-cal. Titanium alloys, such as Ti-6Al-4V, Ti-5Al-2.55n, or Ti-13V-11Cr-3Al, show high strength-to-weight ratio, high toughness, superior corrosion and creep resistance, and biocompatibility3Figure 1.3: Intense narrow shear bands (between arrows) happen periodically between thicksegments when cutting Ti-6Al-4V [55].[118]. These characteristics make titanium alloys highly desirable in different industries, suchas aerospace (e.g. jet engine components as shown in Fig. 1.4a), gas turbine, rocket, nuclear,chemical vessels, and biomedical applications (e.g. joint as shown in Fig. 1.4c). On the otherhand, nickel alloys, such as Inconel 718 and Waspaloy, retain their strength when exposing to hightemperatures, making them ideal for use in the turbine sections of jet engines (see Fig. 1.4a). Forinstance, around 50% of a jet engine’s weight is composed of Inconel-718 [118].Magnesium alloys, whose Young’s modulus is close to that of human bones, are biodegradable,making them useful in facilitating the healing process after surgical procedures and in eliminatingthe need for a secondary surgery. These alloys are being considered for use in various medical ap-plications, such as cardiovascular stents and orthopedic implants, such as screws, plates, and otherfixing devices as shown in Fig. 1.4d. On the other hand, cobalt alloys are widely used for bearingsurfaces in artificial joint prosthetics due to their superior mechanical properties, high resistanceto corrosion and wear, and low friction coefficient. Compared to other metallic biomaterials, suchas stainless steels and titanium alloys, cobalt alloys exhibit exceptional wear resistance. For ex-ample, Co-28Cr-6Mo alloy is used in artificial hip joint ball and cup components, as well as in the4sliding components of artificial knee joints as shown in Fig. 1.4e. Regarding iron alloys, hardenedsteels are used in energy and automotive industries (e.g. gear boxes) due to their wear and fatigueresistant (see Fig. 1.4b) while stainless steels are used as biomaterials due to their low cost, goodmechanical and corrosion resistant properties, and biocompatibility [7].Machining of advanced metallic alloys is an essential process for achieving the final geometryand good surface quality. Adiabatic shear banding is the dominant type of plastic deformation inthe machining of these metallic components, such as titanium [66], nickel [59], iron [61], mag-nesium [104, 131], and cobalt alloys [23], under practical cutting conditions. In fact, machiningof almost all metallic alloys can cause adiabatic shear banding but for metals with moderate ther-momechanical properties, such as aluminum and mild steel, the cutting condition at which a shearband occurs does not lie in the practical cutting condition range employed in industry (up to 200-250 m·min−1 [4]). For Aluminum 7075, shear banding occurs at 459.6 m·min−1 (uncut chipthickness 100 µm) [135], which is not commonly used for machining of this aluminum alloy.In addition to the machining of the parts manufactured with well-established manufacturingmethods (for example, casting, forming, and rolling processes), recent investigations show thatshear localization (SL) is the dominant type of chip formation during machining of additively man-ufactured (AM) titanium-based and nickel-based parts, including the parts made with Direct En-ergy Deposition and Electron Beam Melting methods [3, 10, 69]. The generally expected thermo-mechanical properties, including high specific strength, low thermal conductivity, and maintainedproperties at elevated temperature, from the newly developed metallic alloys make SL the domi-nant chip formation mechanism in their machining as will be shown in Chapter 3. Therefore, it isimportant to understand the SL mechanism during machining processes in order to prevent its neg-ative effects, and increase the productivity and quality of the machined parts made of the currentlydeveloped metallic alloys as well as those to be developed in the future.5(a) (b)(c)(d)(e)Figure 1.4: Some applications of advanced metals. (a) In jet engines, titanium alloys areused in the parts where high strength-to-weight ratio, corrosion and creep resistance,and toughness are required while nickel alloys are used where maintaining strength atelevated temperatures is needed. Adapted from [118]. (b) High strength steels are usedin different parts of a car for their wear and fatigue resistance. Adapted from [48]. (c)Titanium alloys are widely used in biomedical industry due to their high strength-to-weight ratio, corrosion resistance, and biocompatibility. Adapted from [116]. (d) Mag-nesium alloys are used in biomedical industry as they are biodegradable, making themexcellent for the healing process of the human body. From top-left to bottom-right:Cardiovascular stents [31], MAGNEZIX screw [7], biodegradable orthopedic implant[120], and wound-closing device [45]. (e) Cobalt alloys are employed in biomedicalindustry because of their excellent mechanical properties, high corrosion and wear re-sistance, and low friction coefficient. Adapted from [7].61.3 Practical Importance of Shear Localization in MachiningProcessesShear localization has major effects on the cutting performance, productivity, and part quality,including high cutting tool temperature, rapid tool wear, tool-tip breakage, and cutting force fluc-tuation [90, 91, 106]. Komanduri [55] observed that the tool wear during cutting of Ti-6Al-4V isnot the conventional crater wear where the maximum crater depth occurs at a distance away fromthe tool-tip, but is a crater with the maximum wear depth at the tool-tip (see Fig. 1.5a). Theyattributed this type of tool wear to shear banding which causes the maximum temperature at thetool-chip interface occurs near the tool-tip. Employing high-resolution imaging, Saketi et al. [107]showed that the cutting edge experiences severe plastic deformation and macro cracking duringcutting of Ti-6Al-4V with uncoated carbide inserts, which may result in catastrophic failure ofthe edge. High tool-chip interface temperature and high chemical reactivity of Ti-6Al-4V werementioned as the cause of failure. Jadam et al. investigated the tool wear modes in machining ofTi-6Al-4V [50] and Inconel 718 [51] with different cutting tool materials. They identified adhesionas the most dominant tool wear mechanism, and severe notch wear was identified during cuttingof Inconel 718 with PCBN tools. Amin et al. [5] observed catastrophic tool wear and tool-tipbreakage when cutting Ti-6Al-4V at the cutting speeds beyond 0.67 m·s−1 as shown in Fig. 1.5b.They mentioned the low thermal conductivity of Ti-6Al-4V causes serrated chips (shear-localizedchips), which in turn leads to very poor machinability of titanium alloys. shear localization causeshigh fluctuation of the cutting force acting on a small tool-chip contact length.Recently, it has been shown that SL has a direct effect on the surface integrity of the final prod-uct. It causes micro-waves at the finished surface of the machined part which, in turn, increasesthe surface roughness [113]. Fig. 1.6 shows the chips and finished surfaces of high-strength alloysteel AerMet100 after machining including micro-waves, whose amplitude increases when shearlocalization in the chip increases. Oritz-de Zarate et al. [87] showed that the frequency of the7Rake FaceCutting Edge(a) (b)Figure 1.5: (a) Scanning electron microscopy (SEM) micro-graph of tool wear on the rakeface when machining Ti-6Al-4V [55]. The maximum tool wear is located at the tool-tipdue to high temperature and pressure produced by shear localization on the tool-tip. (b)Tool-tip breakage when machining Ti-6Al-4V with tungsten carbide (WC-Co) inserts[5]. Shear localization causes high fluctuation of the cutting force acting on the smalltool-chip contact length, resulting in rapid tool-tip breakage.periodic wavy topography of the finished surface is equal to the localization frequency.The residual stress at the finished surface also fluctuates with the same frequency as the shearband formation’s frequency [129]. Fig. 1.7 shows that the thickness of the plastic deformationzone at the finished surface varies spatially, which is an indication of the residual stress variationat the finished surface [129]. The variation frequency is equal to the segmentation frequency dueto shear localization. The mentioned major effects of SL on the cutting performance and the finalproduct’s quality as well as the fact that it is the dominant chip formation type during cutting ofadvanced metallic alloys necessitate the understanding of its mechanism in machining processes.1.4 Need for Analytical Modeling of Shear LocalizationThe experimental measurement of stress, temperature, and strain rate developed during shear-localized chip formation is extremely challenging due to the small thickness of the shear band and8120m·min−1Chip (top-view) Chip (thickness-view) Machined surface Surface roughness200m·min−1Figure 1.6: Shear-localized chips and machined surfaces of AerMet100 steel at different lo-calization intensities [113]. Shear localization in the chip causes wavy machined sur-face whose amplitude is proportional to the localization intensity.the high oscillation frequency. For instance, the SEM observations of the machined Ti-6Al-4Vchips show that for the cutting speed 60 m·min−1 and uncut chip thickness 100 µm, the shear bandwidth is around 4 µm [101], and the shear band spacing is around 75 µm, equivalent to 13 kHzfrequency of the cutting force oscillation [82]. The difficulties in the measurement of stress andtemperature in such a narrow band at oscillation frequency above 10 kHz make physical modelsand simulations to gain importance in order to understand the mechanism for process planning andoptimization.In the literature, different finite element (FE) analyses have been performed on SL during cut-ting processes. Molinari et al. [82] showed that a shear band initiates at the tool-tip and propagatestowards the free surface of the workpiece material. They also showed that as the cutting speedincreases, the shear band spacing (i.e., the shortest distance between two neighboring shear bands)increases until it reaches a limit. Childs et al. [17] proposed a new thermo-mechanical flow law in9of the refined layer are considered, and the specific size of the grainscannot be compared.The temperature and strain distribution along with the depth ofpeak (see Figs. 18(a)–18(c)) and valley (see Figs. 18(b)–18(d )) ofrefinement layer at different cutting speeds are shown in Fig. 19.Based on the dDRX theory (Eqs. (7)–(16)), the dDRX process isinduced by the combined effects of the temperature, strain, andstrain rate.From Fig. 19(a), the depth satisfying the dDRX critical tempera-ture condition is about 5 µm, which is similar to the thicknesses ofthe layer obtained in Fig. 18. However, the strains at the temperatureare different (see Fig. 19(b)), which show that the thickness dependson the position of the critical temperature line on the machinedsurface. Therefore, it can be concluded that temperature is themain determinant of the refinement layer thickness.Noting that there are two stages during the serrated chip forma-tion process and the state of the cutting force has a big differencein the two stages, resulting in inconsistent thermal–mechanicalloads on the machined surface (see Figs. 14(h) and 14(i)).Figure 20 shows the relationship between the cutting force fluctua-tion and the thickness of the surface refinement layer at 250 m/min,Fig. 18 Simulation and experimental results for physical characteristics in the machined surface atcutting speeds of 250 m/min and 500 m/minTable 6 Comparison of refinement layer characteristicsbetween simulation and experimental testsVc= 250 m/min Vc= 500 m/minPeak(µm)Valley(µm)Period(µm)Peak(µm)Valley(µm)Period(µm)Simulation 5.9 3 65 5.5 2.1 67.6Experimental 5 2.5 80 4.7 1.8 65Error 18.00% 20.00% 18.75% 17.00% 16.70% 4.00%101006-14 / Vol. 143, OCTOBER 2021 Transactions of the ASMEDownloaded from http://asmedigitalcollection.asme.org/manufacturingscience/article-pdf/143/10/101006/6687598/manu_143_10_101006.pdf by The University of British Columbia Library, Mohammadreza Fazlali on 17 February 2023Figure 1.7: The spatial variation of the plastic deformation zone at the finished surface whencutting Ti-6Al-4V at the cutting speed 250 m·min−1 [129]. The thickness of plasticdeformation zone at the machined surface fluctuates with the same frequency of SL.The minimum and maximum thicknesses are 2.5 µm and 5 µm, respectively.FE simulation of machining Ti-6Al-4V without the need of considering large strain softening. Thematerial failure and its subsequent partial recovery were implemented in the proposed flow law.The process of shear band formation in cutting processes is highly dynamic due to its rapid prop-agation and formation [34]. Therefore, mass scaling has been implemented in the FE simulationsto reduce the computational cost. Ducobu et al. [30] investigated the effect of mass scaling on thesimulation results when cutting Ti-6Al-4V, showing that mass scaling affects the chip morphologyand the simulated cutting force fluctuation extensively.The FE simulation of cutting processes de ends on a l rg set of parameters [30]. In most FEsimulations, the chip segmentation is heavily dependent on defining the fracture (damage) behav-ior of the workpiece material, which is generally calibrated by the exp rimental results [68]. The10process is that the fracture threshold (whether the critical strain or energy) value is modified untilthe chip morphology matches with the experimental observations, or the simulated cutting forceapproaches the experimentally measured value. The dependence of SL on the assumed damagethreshold is a common issue among different FE approaches. There are different approaches forassigning the damage threshold values for the workpiece material: (i) a single damage thresholdis allocated to the entire workpiece material, (ii) different damage thresholds are allocated to dif-ferent locations of the workpiece materials (i.e., uncut chip thickness, sacrifice layer, and final part[100]), or (iii) the Coupled Eulerian-Lagrangian (CEL) approach where a single fracture thresholdis allocated to the entire Eulerian domain of the workpiece [129], the damage threshold is a part ofthe simulation of shear banding machining.The problem with the calibration of the chip morphology with the experimental results and set-ting a threshold to match the chip morphology is that in this case, the saw-tooth-like appearance ofthe chip is due to the crack formation and propagation rather than catastrophic thermo-mechanicalinstability (i.e., shear band). Therefore, the developed stress and temperature inside the shear bandand its ductile behavior are under questions, since the formation of the crack is accompanied by theelement failure and potential deletion. This causes the underestimation of the developed stress andtemperature inside the adiabatic shear bands. As will be shown in Section 3.4, the displacement atthe shear bands’ locations (and subsequently, the chip morphology) can be predicted analyticallywithout any need for a fracture criterion. Therefore, the saw-tooth-like appearance of the cuttingchip is solely due to the displacement inside the shear band rather than crack propagation. Rulingout the fracture’s effect on the shear band’s displacement is due to the fact that a shear band itself isa type of material instability which can cause a drastic loss in load-bearing capacity of the material[126]. Therefore, FE simulations of SL in cutting processes must be improved further in terms ofspatial and temporal resolutions (which will cause even higher computational cost) in order to beable to capture such rapid changes in stress and temperature within a thin region of the shear band.11As an example, when cutting Ti-6Al-4V at the cutting speed 1 m·s−1, the shear band’s thicknessis around 3.6 µm and propagates from the cutting tool-tip to the free surface of the workpiecematerial with a propagation velocity at least an order of magnitude higher than the cutting speed[101, 104]. Sufficient number of elements is needed in such a thin shear band to capture the rapidvariations in stresses and temperature. Advanced re-meshing techniques have been investigated inthe theory of shear bands in the literature [27] where a single shear band is nucleated and prop-agated in a slab with simple velocity boundary conditions. Adopting these approaches in cuttingprocesses where a number of shear bands happen in front of the cutting tool (for example, around14 shear bands in only 1 cm cutting of Ti-6Al-4V at cutting speed 1 m·s−1 and uncut chip thickness120 µm [33]) needs to be investigated further.Analytical models, on the other hand, do not suffer from high computational cost and low spa-tial resolution during SL, although simplified assumptions are inevitable in their development. Inthe machining literature, there is a lack of analytical model that predicts the stress and tempera-ture of the shear band and subsequently, the cutting tool and workpiece by taking the shear bandformation into account. Burns et al. [13–15] analytically predicted the fluctuation in the cuttingforce during SL. However, the force and temperature fluctuations were assumed to be caused bythe elastic loading-unloading at the tool-chip interface rather than the catastrophic shear insidethe shear band. As the elastic energy stores at the tool-chip interface, the cutting force increasesand subsequently, the material’s plastic deformation at the shear plane increases due to the forceequilibrium condition between the forces on the cutting tool’s rake face and the forces on the shearplane in the absence of inertia. Afterwards, the thermal softening of the material at the shear planecauses a decrease in the stored elastic energy at the tool-chip interface due to the force equilibrium,causing the cutting force variation with respect to time. In this approach, the cutting mechanics ofSL is the same as that of the continuous chip formation where all the cutting action takes place at asingle shear plane (i.e., Merchant cutting model), the tool-chip contact length is constant, and the12entire material passing the shear plane is under plastic deformation [13–15]. As will be shown inSection 3.3, the stored energy at the tool-chip interface will be proved to have a minimal impacton SL and cannot be its main cause.1.5 Thesis Objectives and OutlineThe main goal of this research is to develop an analytical thermo-mechanical cutting model that isable to capture the effects of shear band formation in machining processes. The model can predictthe workpiece and cutting tool temperatures, cutting force oscillation’s amplitude and frequency,and tool-chip contact mechanism. The specific aims of this thesis are described in more detailsbelow.■ To develop a thermo-mechanical model for SL, the shear band formation mechanism inmachining processes must be well understood. Therefore, the first specific aim is to experi-mentally investigate the shear band formation in orthogonal cutting of Ti-6Al-4V (as one ofthe most common metallic alloys subjected to SL in machining). Based on the experimen-tal observations, an analytical thermo-mechanical model is developed to predict the shearband’s thickness, its temperature and stress, and strain rate in machining processes.■ Different metallic alloys show different tendency to SL. Therefore, shear banding in themachining of different metallic alloys that cover a wide range of thermomechanical proper-ties are investigated. Based on the observations, the second aim is to expand the analyticalmodel to predict the shear band displacement and chip morphology as well as to investigatethe effect of elastic energy storage at the tool-chip interface (which is assumed to be the mainreason of SL in the literature). The model also helps understand and quantify the effects of13different thermo-mechanical properties and cutting condition on the shear localization mech-anism.■ Having obtained a thermo-mechanical model for SL in machining processes, investigating itseffects on the tool-chip interface, including the contact mechanism, forces, and temperature,is the third aim of this research. To this end, high-speed image sequences of shear bandformation is employed to experimentally investigate the contact mechanism of SL. Basedon the experimental characterization, a complete cutting force model for SL is proposed topredict the contact forces at the tool-chip interface. Predicting the sticking-sliding contactlengths, the contact stresses are also predicted. Finally, based on the developed model, thetemperature at the tool-chip interface is also predicted.Fig. 1.8 shows the flowchart of this thesis in detail.14Experiments( Ti-6Al-4V)* Orthogonal cutting experiments* Metallurgical sample preparation* SEM characterizationExperiments(CP Ti, Ti-6Al-4V,Waspaloy, Inconel 718)* Orthogonal cutting experiments* Metallurgical sample preparation* Optical, laser, SEM characterizationExperiments* PIV Analysis of high-speed image sequences* Orthogonal cutting experiments* SEM observations* Rolling of shear band on tool* Localization Pattern * Thickness vs. cutting speed* Spacing vs. cutting speed Analytical modelling of shear bandsin orthogonal cutting processes:* Shear band’s stress*Shear band’s temperature*Shear band’s strain rate* Spacing * Shear band’s microstructure* Three types of chips in Ni alloys* Different rolling phenomena in Ti and Ni alloysAnalytical Modelling of:* Elastic energy release at the tool-chip interface* Shear band’s displacement and its distribution* Chip morphology* Time-varying sticking and sliding contact lengthsAnalytical Modelling of:* Time-varying sticking and sliding contact lengths* Contact forces at the tool-chip interface*Contact stresses at the tool-chip interface* Transient temperature at the tool-chip interfaceChapter 2 Chapter 3 Chapter 4Figure 1.8: Thesis flowchart.15Chapter 2Analytical Modeling of Shear Localizationin Orthogonal Cutting Processes2.1 Introduction and Literature ReviewShear band formation in front of the cutting tool is accompanied by high frequency oscillationof stress, temperature, and strain rate in the primary deformation zone. The experimental mea-surement of these variables are quite challenging due to the small thickness of the shear band andhigh oscillation frequency as described in Section 1.4. Therefore, the experimental studies in theliterature mainly focus on the investigation of the shear localization mechanism via in situ and exsitu measurements. In a series of experimental studies and with the aid of high-speed imaging,Komanduri et al. [55, 58, 60] showed that a wedge-shaped region of the workpiece material infront of the cutting tool is flattened on the rake face as the cutting tool advances, which cannot bejustified by the Merchant’s shear plane model [73, 74]. Based on high-speed imaging observations,Sagapuram et al. [102, 105] described the shear band as an infinitesimally thin interface and thematerial on its either side, i.e., the chip segments, as viscous half-spaces which rigidly slide withrespect to each other over the shear band. Employing high-speed imaging techniques, Sagapuram16et al. [103, 104] showed that formation of a shear band in front of the cutting tool consists oftwo stages. At first, the shear band nucleating at the tool-tip propagates toward the free surfaceof the workpiece material with a propagation speed at least an order of magnitude higher than thecutting speed. Then, the material at the shear band acts like a viscous fluid being sheared acrossits boundaries. Yadav and Sagapuram [131, 132] showed the cutting force fluctuation during SLwhen cutting a bismuth-based alloy commonly known as Wood’s metal. The reason for choos-ing Wood’s metal was to observe shear localization at a considerably low cutting speed (around6×10−4 m·s−1). They showed that there is a close correlation between the peaks and valleys ofthe cutting force evolution with respect to time and the shear band formation in front of the cuttingtool. These experimental observations show that there are unique characteristics of SL which mustbe taken into account for its physical modeling.In the literature, physical models are mainly focused on determining the shear band spacingand shear localization criteria, which represent the cutting condition (uncut chip thickness andcutting speed) corresponding to the onset of shear localization [94, 108, 136]. They are on/offcriteria indicating whether the assumed cutting condition leads to SL or not. The physical studyperformed by Burns and Davis [13, 14] is one of the few physical models trying to predict the stressand temperature oscillations as explained in detail in Section 1.4. In other dynamic deformationapplications, Wright and Walter [127] and Walter [121] employed numerical methods to model theshear band formation for rigid-perfectly plastic material in the unidirectional shearing of a slab.It was shown that the evolution of a shear band started from an early, homogeneous stage, to atransition stage with rapid changes in the stress, temperature, and strain rate, and finally a steady-state stage. In the steady-state stage, the strain rate takes a constant value while the temperatureand stress gradually change. Accordingly, Wright and Ravichandran [125] investigated the shearband’s width where the governing equations on the stress and temperature were assumed to beindependent of time. They showed that the structure of a shear band is canonical and dependent on17certain material properties and external loads. Due to the extreme thinness of shear bands, Glimmet al. [36, 37] divided the slab containing a shear band into a three-level structure including aquasi-static core, a surrounding thermal layer, and external flow, where the core acts as a disconti-nuity in the shear velocity. Olmstead et al. [86] and DiLellio and Olmstead [25, 26] further studiedthe effects of the elastic response of the material as well as the wave reflection from the boundaries.The spacing between neighboring shear bands is another important factor in the cutting pro-cess, as it determines the frequency of the cutting force oscillation in front of the cutting tool [82].Mechanics models have been developed to predict the shear band spacing based on the linear per-turbation analyses by Wright and Ockendon [124] and Molinari [78] in a unidirectional shearingof a slab, and momentum diffusion analysis by Grady and Kipp [39]. These theories have beendeveloped in the context of dynamics loading of materials, in particular, under impact and dy-namic deformation in torsional split−Hopkinson bar experiments. According to these theories,the shear band spacing ds decreases with the increase of the strain rate γ˙ at the rate of ds ∝ γ˙−34[39, 78, 124]. These theories, however, do not reflect the experimental observations of shear bandspacing in cutting processes. Optical and SEM examinations on the chip morphology show thatthe average spacing between shear bands increases as the cutting speed increases, and then asymp-totically approaches a limiting value. This trend has been experimentally observed when cuttingtitanium-based alloys [22, 114], high-strength steels [13, 19–21], and nickel-based alloys [21, 29].The similarity in the trend for different materials as well as the regularity of the spacing in theshear-localized chips provide a motivation for physical modeling of the shear band spacing in thecutting process.Based on the available literature, there is a lack of an analytical mechanics model that canexplain the mechanism of the complete shear band formation procedure (including onset and evo-lution of the shear bands) in the cutting process, with high time and spatial resolutions and low18computational cost. In this chapter, a physics-based mechanics model is proposed to predict thestress, temperature, and strain rate associated with the shear band formation based on the bound-ary layer approach. Different stages of shear band formation from the homogeneous stage, to theshear localization, and the chip segmentation are considered in the proposed model. Moreover, thespacing between shear bands in the cutting process is investigated and distinguished from otherdynamics deformation applications by taking the cutting mechanics into account. The predictionsof the shear band spacing and shear band width from the proposed model are compared with ex-perimental results.The remaining sections are organized as follows. Section 2.2 presents an experimental charac-terization of shear band formation in Ti-6Al-4V alloy. In particular, microscopic characterizationof the cutting chips is performed using SEM analysis. Next, in Section 2.3, the thermo-mechanicalboundary layer model for shear band formation is presented. The main results and comparison withthe experimental data is discussed in Section 2.4. Finally, the concluding remarks are discussed inSection 2.5.2.2 Experimental Investigations of Shear Localization inCutting Ti-6Al-4VOrthogonal cutting experiments were carried out on Ti-6Al-4V workpieces using the longitu-dinal turning operation mode of Mori Seiki NT 3150 DCG. The uncut chip thickness d wasalong the longitudinal axis of the workpiece (i.e., feed per revolution), and the width of cut wwas toward the center of the workpiece’s cross section as shown in Fig. 2.1a. The uncut chipthickness was set to d = 120 µm, and the cutting speed vc varied from 20 m·min−1 to 105m·min−1. In all experiments, the width of cut was w = 2 mm, which is more than 10 timesthe uncut chip thickness to ensure the plane strain condition. The longitudinal turning operationis common in the literature for orthogonal cutting experiments [67, 71]. Employing this oper-19ation, the cutting speed slightly varies along the cutting edge with the maximum difference of∆vmax = vmax− vmin = MπDmax−MπDmin = 2Mπw, where M is the spindle speed (with the unitof rev·min−1), Dmax is the maximum diameter (with the unit of m), and Dmin is the diameter of theworkpiece after cut (with the unit of m). For example, for the cutting speed of vc = 60 m·min−1and width of cut 1.5 mm, the maximum error in the cutting speed was 4.7% for the Ti-6Al-4Vcylindrical workpieces with 50.8 mm (2 in.) diameter employed in the experiments. Since themaximum error in the cutting speed along the cutting edge was quite small, the variation of thecutting speed along the cutting edge was neglected.The tool-holder’s length was set as minimum as possible (as shown in Fig. 2.1b), and the freelength of the workpiece material (after clamping) was approximately equal to its cross section’sdiameter to avoid vibrations and minimize the static deflection in the experiments. Uncoated car-bide inserts with zero degrees rake angle (α =0◦), 10 µm nose radius, and flat rake face (withoutany chip breaker) were employed as shown in Fig. 2.1c. In order to avoid any effect of tool wearon the experimental results, the tool edge was examined using an optical microscope after everytwo turning tests, and the insert was replaced when the flank wear length was beyond 15 µm ap-proximately. This was roughly equivalent to 2 mm cutting length of Ti-6Al-4V for each insert.After each experiment, the cutting chips were collected for metallurgical investigations. Theywere then mounted in resin such that the chip’s cross section was perpendicular to the mountcross section to avoid any diversion in the measured thicknesses and spacing of the shear bands.Cold mounting with high curing time (around 12 hours, suitable for delicate chips) was employedrather than hot mounting to avoid any rotation of the thin cutting chips inside the resin due to themounting pressure. Grindings of the mounted chips were performed using silicon carbide (SiC)grinding papers with the grinding sequence of 400, 600, 800, and 1200 (P-800, P-1200, P-2400,and P-4000) grit. After the grinding steps, the samples were polished with VerduTex™ (coarse to20medium polishing), MicroFloc™ (fine polishing), and Chemomet™ (ultra-fine polishing) polishingclothes in the order of 6 µm, 3 µm, 1 µm, and 0.05 µm. For polishing, monocrystalline diamondsuspensions were employed. The polished chip specimens were then chemically etched in a so-lution of 92 wt% water, 6 wt% nitric acid, and 2 wt% hydrofluoric acid (Kroll’s reagent). Theprepared samples were then observed by Zeiss Σigma™ field emission SEM employing the SE2,In-lens, and backscattered detectors. For measuring the shear band’s width, the micro-graphs withthe highest resolution of 0.06 µm (4910× magnification) were captured while for measuring theother dimensions of the chip including the shear band’s displacement and segment’s thickness, themicro-graphs with the typical resolution of 0.25 µm (1170×magnification) were employed. In thelatter case, the measured values were rounded to the closest integer. This sample preparation pro-cess enables the examination of the plastic deformation pattern at different locations of the cuttingchips, including the shear band (within the shear band’s width), in its vicinity, and at the tool-chipinterface with high spatial resolution.Fig. 2.2 shows the schematics of the shear-localized chip formation. At stage (a), shear lo-calization occurs at the shear band (SB1) extending from the tool-tip to the free surface of theworkpiece material. When SB1 occurs, the workpiece material loses its load-bearing capacity,causing the cutting force to drop rapidly. At stage (b), the new workpiece material piles up in frontof the cutting tool until the cutting force reaches a critical value for the next shear band to occur(SB2). This process repeats periodically in the shear-localized chip formation. Fig. 2.3 shows anSEM image of the chip morphology obtained at the cutting speed 90 m·min−1, which includes theshear bands and chip segmentation.During stage (b) where SB2 is forming, the previously formed shear band, i.e., SB1, rolls onthe tool rake face due to the piling up of the workpiece material. Fig. 2.2b shows the rolling andpiling-up processes during the upsetting stage. During this stage, the free surface side of the shear21WWidth of CutUncut Chip ThicknessClock-wise Rotation(a) Experimental setup.Free Length(b) Tool holder.Cutting Edge(c) Cutting tool insert.Figure 2.1: (a) Employed experimental setup for the orthogonal cutting experiments. Theuncut chip thickness d was measured along the feed direction, while the width of cutw was toward the center of the workpiece’s cross section. (b) Tool holder with a smallfree length to avoid the effects of static deflection on the results. (c) Cutting tool insertwith zero degree rake angle, zero degree inclination angle, and flat rake face.band experiences large deformation at the shear band’s location shown by f in Fig. 2.2b. Fig. 2.4shows the SEM image of the shear band’s rolling on the cutting tool rake face for the cutting speed60 m·min−1 in more details. From the SEM characterizations, it was found that the rolling of theshear band on the tool rake face extends all the way through the thickness of the segment. Thiscan be observed by comparing the shear band’s displacement at the free surface f and the contactlength of the segment with the rake face s in Fig. 2.5 for the cutting speed 60 m·min−1. Accordingto Fig. 2.5, the displacement f and the contact length s are very close for each segment. Table 2.1further compares the contact length s and the displacement f for two different cutting speeds,where si and fi (i is the segment’s number) show the measurements in different segments. In eachcutting speed, four measurements in different locations of the chip were performed to ensure theconsistency in the results. According to Table 2.1, the displacement f and contact length s are22CuttingToolWorkpieceSegment(a) Shear localization stageWorkpieceSegment(b) Upsetting stageFigure 2.2: Schematics of the shear-localized chip formation stages. At stage (a), SB1 formsin front of the cutting tool. At stage (b), new workpiece material piles up in front of thecutting tool until SB2 occurs.Figure 2.3: SEM micro-graph of a shear-localized chip obtained under the cutting speed vc =90 m·min−1. Both shear band and chip segmentation are clear. Each shear band islocated between two successive segments.almost equal with about 1% error, showing that the shear band has rolled on the tool rake face allthe way through the segment’s thickness. Because of the intimate contact between the cutting tooland the shear band which occurs during the rolling process, the heat produced within the shearband is directly transferred to the cutting tool, causing rapid tool wear during shear-localized chipformation.23Figure 2.4: SEM micro-graph of a shear-localized chip at the tool-chip interaction side. Thecutting speed is 60 m·min−1. The rolling of the shear bands on the cutting tool rakeface is shown for two successive shear bands.Figure 2.5: Comparison of the segment’s contact length at the tool-chip interface s and theshear band’s displacement at the free surface side f for two different segments, showingthat they are equal in each segment (i.e., f1 = s1 and f2 = s2). The cutting speed is 60m·min−1.Fig. 2.6 shows the thickness of the shear band away from the tool-chip interface (2δ ) andthe thickness of the plastic deformation region at the tool-chip interface (p) for the cutting speed90 m·min−1. Table 2.2 further compares 2δ and p for different chip segments. According toTable 2.2, the shear band’s thickness has not changed significantly with sliding at the tool-chipinterface, showing that when shear-localized chip formation occurs, the main part of the plasticdeformation at the tool-chip interface is caused by the shear band’s plastic deformation rather than24Table 2.1: Contact length s and displacement f of shear-localized chips for the cutting speeds60 m·min−1 and 90 m·min−1. In each cutting speed, four measurements at differentlocations as well as the mean values are represented.Cutting speed f1 s1 f2 s2 f3 s3 f4 s4 favg. savg.(m·min−1) (µm) (µm) (µm) (µm) (µm) (µm) (µm) (µm) (µm) (µm)60 85 85 97 97 88 88 85 86 88.75 8990 94 93 93 97 92 94 97 96 94 95the friction at the tool-chip interface. This is different from the tool-chip interaction in continuouschip formation.Figure 2.6: Connection between the shear band thickness 2δ and the thickness of the plasticdeformation at the tool-chip interface p via the rolling process. The major part of theplastic deformation at the tool-chip interface is due to the shear band that has rolled onthe cutting tool rather than the friction at the tool-chip interface. The cutting speed is90 m·min−1.Table 2.2: Shear band thickness 2δ and the thickness of the plastic deformation at the tool-chip interface p for the cutting speeds 60 m·min−1 and 90 m·min−1. In each cuttingspeed, the measurements for four different shear bands as well as the mean values areshown.Cutting speed 2δ1 p1 2δ2 p2 2δ3 p3 2δ4 p4 2δavg. pavg.(m·min−1) (µm) (µm) (µm) (µm) (µm) (µm) (µm) (µm) (µm) (µm)60 3 4 4 5 4 5 4 5 3.75 4.7590 3 4 3 3 3 4 3 3 3 3.5025Fig. 2.7 shows an SEM image of a shear-localized chip under the cutting speed vc = 60m·min−1. The micro-lines, two of which are distinguished by green and red colors, show therelative displacements’ distribution in the chip. In order to retain the micro-lines, no sample prepa-ration process is performed on the cutting chip. Each micro-line used to be a continuous straightline along the direction of the cutting velocity before cutting. After the cutting process, while themicro-lines are still approximately straight inside the segments, they jump at the shear bands’ lo-cations. This shows that the major part of the plastic deformation of the workpiece material hasbeen localized inside the shear bands while relatively negligible plastic deformation occurs withinthe segments (outside the shear bands). Fig. 2.8 further demonstrates the localization of the plasticdeformation within the shear band, where sample preparation has been performed on the chip. Theresults show that while intense plastic deformation has occurred inside the shear band, the materialoutside the shear band is almost free of plastic deformation even in the close proximity of the shearband. The intense plastic deformation inside the shear band can also be seen from the top-view ofthe chip (normal to the cutting width) as shown in Appendix A.Figure 2.7: Micro-lines in the SEM micro-graph of a shear-localized chip under the cuttingspeed vc = 60 m·min−1. Two micro-lines are distinguished by red and green colors,and the shear bands’ locations are also shown. The micro-lines show discontinuity(jump) at the shear bands’ locations while remain almost straight withing the segments.It shows that the majority of plastic deformation has been localized inside the shearbands with negligible plastic deformation inside the segments.26Figure 2.8: Comparison of the plastic deformation inside and the close vicinity of a shearband using a metallurgically prepared sample and employing high magnification SEMmicro-graph. The microstructure is almost intact outside of the shear band while heav-ily deformed in its inside. The cutting speed vc = 60 m·min−1.2.3 Analytical Mechanics Model of Shear-LocalizationFig. 2.9 shows the schematic of chip formation process with shear localization. In Fig. 2.9, theregion AOC is under plastic shear deformation while the region BOC piles up in front of the cut-ting tool. As the cutting tool advances, the shear stress of the workpiece material in the regionAOC is influenced by the hardening effects from strain and strain rate, as well as the thermal andmicrostructural softening effects. When the uncut chip thickness and cutting speed increase be-yond certain values, the softening mechanisms outweigh the hardening mechanisms at line OCunder the critical strain γcr. In this study, it is assumed that thermal softening is the only softeningmechanism in the workpiece material, and the microstructural softening mechanisms, includingdynamic recovery and/or dynamic recrystallization, are neglected.Near the cutting tool, the stress state is highly complex. However, the stress field quicklybecomes dominated by a shearing stress that acts at 45◦ from the workpiece’s free surface [85, 88].Thus, the sliplines OA, OC, and OB are curved as shown in the illustration. Some simplificationsare taken to characterize the geometry of the boundaries of the primary shear zone as follows: It is27CuttingToolOAWorkpieceSegmentABCODeformation ZonePreviousShear BandNewShear BandBCFigure 2.9: Schematic of the shear-localized chip formation mechanism in orthogonal cut-ting. The total deformation region AOB includes the shear deformation region AOCand the upsetting region BOC. The formation of the new shear band in the deformationregion AOB is also shown in the inset as well as the corresponding thickness for eachregion.assumed that the initial boundary OA is a straight line extending from the tool-tip to the free surfaceof the workpiece, and the shear angle φ does not change inside region AOC during the shear bandformation. Moreover, the critical strain γcr at which instability starts at line OC is obtained fromthe homogeneous solution of the governing equations on stress and temperature, correspondingto the material deformation in region AOC. This provides the initial conditions for the stress andtemperature evolution after shear localization. The details are presented in the following sections.2.3.1 Governing equations and solution in homogeneous deformationFig. 2.10 shows the simplified plastic deformation region AOC in more details, where the coordi-nate system x-y is a Lagrangian or material coordinate system moving along the chip velocity withthe material being cut. The shear velocity v is the projected component of the cutting velocity vc inx direction, i.e., v = vc cosαcos(φ−α) . The governing equations on the plastic deformation region AOC28CuttingToolOAWorkpieceSegmentABCODeformation ZonePreviousShear BandNewShear BandBCFigure 2.10: Schematic of the simplified shear deformation region AOC along with x-y La-grangian coordinate system. The shear velocity v/2 is applied on the borders of theregion AOC.are expressed asτ,y = 0, momentum; (2.1)ρcθ,t = kθ,yy+βτγ˙, energy; (2.2)τ = f (γ, γ˙,θ), flow law. (2.3)In Eqs. (2.1) to (2.3), τ(y, t) is the shear stress, and θ(y, t) = T (y, t)−Tr is the relative temperature,where T (y, t) is the absolute temperature and Tr is the room temperature. The comma subscriptindicates partial derivative with respect to the corresponding variable. For example, θ,t = ∂θ/∂ tand θ,yy = ∂ 2θ/∂y2. The material constants ρ,c, and k are mass density, heat capacity, and thermalconductivity of the workpiece, respectively. The variables γ and γ˙ are the plastic shear strain andthe plastic strain rate, respectively. The source term of the energy equation (βτγ˙) represents thefraction β of the plastic work which is converted into thermal energy. In this study, the Taylor-Quinney coefficient β is assumed to be equal to 0.9 [119]. Finally, the flow law relation in Eq. (2.3)shows the dependence of the flow stress on the strain, strain rate, and temperature. The Johnson-29Cook (JC) viscoplastic flow law [126] is used,τ =1√3[A+B(γ√3)n](γ˙γ˙re f)C[1− (aθ)m] , (2.4)where A, B, C, n, and a = 1/(Tm−Tr) are material constants, and γ˙re f is the reference strain rate.The coefficients√3 are inserted to calculate the shear stress from the Von-Mises equivalent stressand strain. In Eq. (2.4), the original strain rate term has been approximated by its power series asgC = eClng = 1+Clng+O(Clng)2, (2.5)where g = γ˙/γ˙re f . The higher-order term O(Clng)2 is usually very small for common strain ratesensitivities C. This form of the flow law provides more computational flexibility for the followinganalytical solutions of the governing equations.Before adiabatic shear band onset and because of the high rate of plastic deformation, thematerial inside the plastic deformation region AOC does not have enough time to conduct heat[39, 126], i.e., adiabatic assumption is applicable1. It is also assumed that the deformation ishomogeneous before the shear localization, and this assumption renders the temperature and stressinside the region AOC independent of y. Therefore, the homogeneous temperature θh before theonset of shear localization can be obtained as a function of homogeneous shear strain γh by directintegration of the energy equation (2.2), where the thermal diffusion term vanishes, and the shearstress is replaced by the flow law (2.4):θh(γh) =1a− 1aexp{− βa√3ρc(γ˙hγ˙re f)C [A+Bn+1(γh√3)n]γh}, (2.6)1This can be seen if one analyzes the characteristic time for adiabatic shear band and heat diffusion. In heatdiffusion, the characteristic time is given by tH = (2δ )2/χ , where χ is the thermal diffusivity of the material. On theother hand, the characteristic time for shear band formation is given by tS =CL/2δ , where CL is the longitudinal wavespeed in the material. For titanium (Ti), it is easy to see that tH ≫ tS justifying the adiabatic condition.30where γ˙h is the homogeneous strain rate of the plastic region AOC. In obtaining Eq. (2.6), the ap-proximation 1− (aθ)m = 1− aθ +O[1−m] has been employed in the thermal softening term ofEq. (2.4). Since the thermal softening coefficient m for many materials is equal or close to 1 [76],the error O[1−m] is small. The homogeneous shear stress τh is then obtained from Eq. (2.4) asτh (γh) =1√3[A+B(γh√3)n](γ˙hγ˙re f)C(1−aθh) . (2.7)Eqs. (2.6) and (2.7) are valid only before the onset of shear localization and shear band for-mation. As soon as the shear localization starts, the solutions are not valid anymore due to therapid variation and nonlinearity of the variables after shear band formation. In order to obtain thecritical strain corresponding to shear localization, the instability criterion proposed by Bai et al.[8] is employed. The criterion establishes that the instability occurs when the hardening effects arebalanced with the thermal softening effects in the material, i.e., the slope on the stress-strain plotbecomes zero. Mathematically, the criterion can be written as−βτh(∂τh/∂θh)ρc(∂τh/∂γh)= 1. (2.8)Inserting the flow law from Eq. (2.4) and τh from Eq. (2.7) into Eq. (2.8), the critical strain underwhich localization starts at line OC is obtained as:γncr−ρcnaβτh(1−aθh)γn−1cr +AB(√3)n = 0. (2.9)Note that in Eq. (2.9), θh and τh are also dependent on γcr and need to be inserted from Eqs. (2.6)and (2.7), respectively. Solving Eq. (2.9) for γcr, and inserting back to Eqs. (2.6) and (2.7), thetemperature θh(γcr) and stress τh(γcr) at the onset of instability are obtained and employed as the31initial conditions for the localization stage.2.3.2 Localization in shear band formationRecent experimental studies on the shear-localized chip formation show that the shear band prop-agates from the cutting tool-tip to the free surface of the workpiece material with a propagationspeed at least an order of magnitude higher than the cutting speed [132, 133]. This shows thatdifferent time scales due to different phenomena, e.g., elastic wave propagation and heat diffusion,can play a role in shear localization. Moreover, the shear band’s width is not prescribed beforehandand can be orders of magnitude smaller than the uncut chip thickness and plastic deformation re-gion’s thickness (i.e., different length scale). Nondimensionalization enables different phenomenawith different time and length scales to be included in the model, making the governing equationswell-scaled [126]. Here, by nondimensionalization, the analysis can be performed independentof the material parameters. Different parameters included in the analysis can be grouped into anondimensional variable to study their whole effects together. Therefore, the governing equations(2.1) to (2.3) are nondimensionalized by scaling the distance y with the thickness of the plasticdeformation region AOC, i.e., dt , and the time t with the reciprocal of the homogeneous strain rateγ˙h:y¯ =ydt, t¯ = t γ˙h. (2.10)The over-bar indicates that the parameter is in its nondimensional form. The dependent variablesare nondimensionalized asτ¯ =ττ∗, v¯ =vv∗, θ¯ =θθ ∗, Γ˙=γ˙γ˙h, (2.11)where τ∗, v∗ = γ˙hdt , and θ ∗ = τ∗/(ρc) are the reference shear stress, velocity, and temperature,respectively. Inserting Eqs. (2.10) and (2.11) into Eqs. (2.1) to (2.3), the nondimensional governing32equations are expressed asτ¯,y¯ = 0, momentum; (2.12)θ¯,t¯ = k¯θ¯,y¯y¯+β τ¯Γ˙, energy; (2.13)τ¯ = f (γ, Γ˙, θ¯), flow law, (2.14)where the nondimensional mass density ρ¯ and thermal conductivity k¯ areρ¯ =ρ(v∗)2τ∗, k¯ =kρcdtv∗. (2.15)For simplicity, the strain hardening term in Eq. (2.14) is neglected and replaced by its value at theonset of shear localization:κ0 =1√3[A+B(γcr√3)n]. (2.16)Therefore, the flow law in Eq. (2.4) is simplified asτ = κ0(γ˙γ˙re f)C[1− (aθ)m] . (2.17)Employing Eq. (2.11), the nondimensional form of the flow law (2.17) is expressed asτ¯ = τˆΓ˙C[1− (aˆθ¯)m] , (2.18)where τˆ = (κ0/τ∗)(γ˙h/γ˙re f )C and aˆ = aθ ∗.The momentum equation (2.1) including inertia term is τ,y = ρv,t . Employing the nondime-sionalization scheme of Eqs. (2.10) and (2.11), the nondimensional inertia equation can be writtenas τ¯,y¯ = ρ¯ v¯,t¯ . Inserting ρ = 4420 Kg/m3, v∗ = 1.38 m/s, and τ∗ = 782×106 N/m2 as typical val-ues for Ti-6Al-4V (see Section 2.4), the nondimensional mass density is equal to ρ¯ = 1.06×10−533while the nondimensional thermal conductivity k¯ = 8× 10−2 which is three orders of magnitudelarger than ρ¯ . Therefore, the inertia effect has been neglected in Eq. (2.1) and subsequently, in itsnondimensional form Eq. (2.12).According to the SEM characterization of the shear bands shown in Figs. 2.7 and 2.8, themajority of the plastic strain, and subsequently, the plastic strain rate Γ˙, has been localized insidethe shear bands. The segments OCB in Fig. 2.9 are assumed to be free of plastic deformationand therefore, their plastic strain rate must be zero, i.e., Γ˙ ≈ 0 within the segments. The energyequation (2.13) is thus separated intoθ¯ out,t¯ = k¯θ¯out,y¯y¯ , segments (region OCB); (2.19)θ¯ in,t¯ = k¯θ¯in,y¯y¯+β τ¯inΓ˙in. shear band (plane OC), (2.20)In Eqs. (2.19) and (2.20), θ¯ out(y¯, t¯) is the temperature outside the shear band (i.e., withing thesegment), while θ¯ in(y¯, t¯), τ¯ in(y¯, t¯), and Γ˙in are the temperature, stress, and strain rate developedinside the shear band, respectively. Since the plastic deformation is localized inside the shearband, the segment rigidly slides on it while the shear band acts as a boundary layer of the segment,as shown in Fig. 2.11. The change in the shear velocity across the shear band is∆v¯ = v¯ = 2δ¯ Γ˙in, (2.21)where δ¯ is the half-thickness of the shear band in nondimensional form, i.e., δ¯ = δ/dt . Based on34the energy equation (2.20), the change in the temperature gradient across the shear band is∆θ¯ out,y¯ = θ¯out,y¯ (+δ¯ , t¯)− θ¯ out,y¯ (−δ¯ , t¯)=∫ +δ¯−δ¯θ¯ in,y¯y¯(y¯, t¯)dy¯=1k¯ddt∫ +δ¯−δ¯θ¯ in(y¯, t¯)dy¯− βk¯∫ +δ¯−δ¯τ¯ in(y¯, t¯)Γ˙indy¯. (2.22)Assuming that the temperature distribution is symmetric in the close vicinity of the shear bandCuttingToolOAWorkpieceSegmentABCODeformation ZonePreviousShear BandNewShear BandBCFigure 2.11: Schematic representation of shear band formation at the plane OC. The nondi-mensional shear velocity v¯2 is applied on the shear band. The shear band acts as aboundary layer on which the segment slides rigidly.during the cutting process [132], i.e., θ¯ in(−δ¯ , t¯) = θ¯ in(+δ¯ , t¯), the first term of Eq. (2.22) is equalto zero. Therefore, using Eq. (2.21), the change in the temperature gradient across the shear bandbecomes∆θ¯ out,y¯ =−βk¯τ¯ in(t¯)v¯. (2.23)Since the shear band is shared between two segments (see Fig. 2.11), the boundary conditionfor each segment is written asθ¯ out,y¯ (0, t¯) =−β2k¯τ¯ in(t¯)v¯. (2.24)Therefore, the boundary condition at y¯ = 0 is dependent on the shear band’s internal stress τ¯ in(t¯),35which in turn, depends on the developed temperature inside the shear band. The next boundarycondition is obtained by assuming that the outer boundary of the segment (previously formed shearband in Fig. 2.11) is back to the homogeneous temperature:θ¯ out(d¯s, t¯) = θ¯h(t¯), (2.25)where d¯s is the nondimensional thickness of the segment, i.e., d¯s = ds/dt .The partial differential equation (2.19) is analytically solved by the Green’s Functions method[43]. Since the segment’s thickness is much larger than the shear band’s thickness, i.e., d¯s ≫ δ¯ ,the previous shear band is assumed to be in an infinite distance with respect to the current shearband, making Eq. (2.25) of negligible importance in the Green’s function solution. Therefore, thetemperature evolution within the segment is obtained asθ¯ out(y¯, t¯) =∫ ∞0θ¯ out(λ ,0)G(y¯, t¯;λ ,0)dλ−k¯∫ t¯0θ¯ out,λ (0,η)G(y¯, t¯;0,η)dη , (2.26)where G(y¯, t¯;λ ,η) is the Green’s function for the heat equation, which is expressed as [43]G(y¯, t¯;λ ,η) =exp{−(y¯−λ )24k¯(t¯−η)}+ exp{− (y¯+λ )24k¯(t¯−η)}2√π k¯(t¯−η). (2.27)In Eq. (2.26), the first term is due to the initial condition (homogeneous temperature) and thesecond term is from the boundary condition at the shear band’s location. Inserting the Green’sfunction (2.27), the boundary conditions (2.24), and the initial condition (2.6) into the solution36(2.26), the temperature evolution within the segment is obtained asθ¯ out(y¯, t¯) =θ¯h2√π k¯t¯∫ ∞0exp{−(y¯−λ )24k¯t¯}+ exp{−(y¯+λ )24k¯t¯}dλ+β v¯2√π k¯∫ t¯0τ¯ in(η)exp{ −y¯24k¯(t¯−η)}√t¯−η dη , (2.28)where θ¯h = θ¯h(γcr). Replacing the shear stress inside the band τ¯ in(η) in the second term by theshear band’s temperature and strain rate via the flow law (2.18), Eq. (2.28) is updated asθ¯(y¯, t¯) = θ¯h+β v¯τˆ2√π k¯(Γ˙in)C∫ t¯01− [aˆθ¯ in(η)]m√t¯−η · exp{ −y¯24k¯(t¯−η)}dη , (2.29)where θ¯ in(η) is the temperature evolution inside the shear band. Since the band is located at y¯= 0,the temperature developed within the band is obtained by setting y¯ = 0 in Eq. (2.29) asθ¯ in(t¯) = θ¯h+β v¯τˆ2√π k¯(Γ˙in)C∫ t¯01− [aˆθ¯ in(η)]m√t¯−η dη . (2.30)The strain rate Γ˙in in the second terms of Eqs. (2.29) and (2.30) is of considerable importance.According to Eq. (2.21), in order to obtain the strain rate inside the shear band, its thickness needsto be known. We employ the relation proposed for the half-width of a shear band by Wright andRavichandran [125] in dimensional form asδ =√2Ckβaτ inγ˙ in, (2.31)where the Taylor-Quinney coefficient β is added to the original equation to represent the portionof plastic deformation energy transferred to heat. Therefore, the maximum strain rate developed37during shear-localized chip formation is obtained by inserting Eq. (2.31) into ∆v = v = 2δ γ˙ in:γ˙ in =(βaτ in2Ck)(v2)2. (2.32)Eq. (2.32) shows that the maximum strain rate developed in the shear-localized chip formationvaries by the square of the shear velocity v, and subsequently the cutting velocity vc. The strainrate increases with the thermal softening rate a and the shear stress τ in, while changes inverselyproportional with the strain rate sensitivity C and thermal conductivity k. Therefore, the nondi-mensional strain rate Γ˙in in Eq. (2.30) is written as Γ˙in = γ˙ in/γ˙h based on Eq. (2.32). SolvingEqs. (2.18) and (2.30) simultaneously, the shear band’s temperature θ in(t) and stress τ in(t) can beobtained.2.3.3 Shear band spacingIn the torsional split−Hopkinson bar applications, shear bands occur simultaneously [39, 78, 124],where the linearized governing equations on the stress, strain rate, and temperature for a slabwith fairly large thickness are investigated. When instability occurs, a finite number of Fouriercomponents of the strain rate solution (i = 1,2,3, ..., I) grow exponentially with different growthrates. The wavelength with the maximum rate of growth (i = 1) is assumed to determine the shearband spacing [78, 124]. In the case of a medium with limited thickness, the number of shear bandsthat occur is obtained as [126]N =L2π[γ˙3h a2τC3kc]1/4, (2.33)where N should be rounded to the closest integer. The finite thickness L is equivalent to the thick-ness of the shear deformation region dt in the cutting process (see Figs. 2.9 and 2.10). The shearzone thickness dt is assumed to be a fraction of the shear plane length b, i.e., dt ≈ 0.15b− 0.2b[4, 110]. Therefore, dt ≈ 25 µm for the uncut chip thickness d = 120 µm and typical shear an-gle φ = 45◦. The shear angle φ = 45◦ is chosen because the segments have negligible plastic38deformation, making the uncut and cut chip thicknesses almost equal [35, 60]. This will be fur-ther discussed in Section 2.4. Therefore, for Ti-6Al-4V whose material properties are given inTable 2.3, in order for at least two shear bands to occur simultaneously in the plastic deformationregion, i.e., N = 2, the strain rate must be around γ˙h = 5.8× 109 s−1, which is equivalent to thecutting speed vc = 1725 m·min−1 (vc = 28.75 m·s−1). This cutting speed is much larger than thepractical cutting speeds employed in cutting of Ti-6Al-4V (which is usually less than 100 m·min−1[110]). Therefore, in the metal cutting processes with practical cutting speeds, shear bands formsuccessively rather than simultaneously, unlike other dynamic deformations’ applications. When ashear band forms in the shear deformation zone AOC, it is convected away with the chip velocity,and new material enters the shear deformation zone for the next shear band to occur.Table 2.3: Thermo-mechanical properties of Ti-6Al-4V alloy [82].Alloy k c ρ Tm TrW·(m.K)−1 J·(kg.K)−1 kg·m−3 K KTi-6Al-4V 7.2 560 4420 1930 293According to Fig. 2.9, the spacing between two successive shear bands, i.e., the distance be-tween the plane OC and the final boundary OB, isds = dd−dt . (2.34)The critical strain γcr at the plane OC is obtained by Eq. (2.9). Therefore, the critical time tcr,i.e., the time during which the material travels inside the plastic deformation region AOC, can beexpressed astcr =γcrγ˙h, (2.35)where γ˙h is the homogeneous strain rate and assumed to change with respect to the cutting speedsimilar to Eq. (2.32). Accordingly, the distance dt is considered to be proportional to the critical39time tcr as dt ∝ vchtcr, where vch = vc sinφcos(φ−α) is the chip velocity.For the continuous chip formation, Zorev [137] showed that the stress state in the plastic defor-mation region is complex and not simple shear. This is because the shear lines in the wedge-shapeddeformation region (region AOB in Fig. 2.9) converge to the tool-tip, and are not parallel with eachother [137]. The exact geometries of the deformation boundaries have not been predicted. There-fore, in order to account for the complex stress state in the plastic deformation zone, and the curvedgeometries of the deformation boundaries, similar to Zorev [137], Eq. (2.34) is updated by intro-ducing a calibration parameter ϑ asds = dd−ϑv f tcr. (2.36)In Eq. (2.36), dd and ϑ are assumed to be constant and calibrated by the experimental measure-ments of shear band spacing ds under two cutting speeds. The results are applicable to determinethe band spacing at other cutting speeds.2.4 Results and DiscussionsOrthogonal cutting experiments with the cutting speeds in the range of vc = 20−50 m·min−1 wereperformed on Ti-6Al-4V workpieces whose thermo-mechanical properties are given in Table 2.3.The shear angle was calibrated based on the measured chip thickness as [4]φ = tan−1(r cosα1− r sinα). (2.37)In Eq. (2.37), r = d/h is the chip compression ratio, where h is the chip thickness. Table 2.4 showsthe measured maximum thickness of the chip and the calculated shear angle for different cuttingspeeds. For each cutting speed, the maximum chip thickness was measured at five different loca-tions, and the average value was obtained. Based on the results shown in Table 2.4, the average40shear angle is φ ∼ 43◦.Table 2.4: Experimentally measured chip thickness and the calculated shear angle for fivedifferent cutting speeds when cutting Ti-6Al-4V with uncut chip thickness d = 120 µm.Cutting speed Chip thickness (µm) Shear angle(m·s−1) 1 2 3 4 5 avg. (deg.)21 124 128.3 127.1 130 138.1 129.5 42.8227 122 125.6 126.8 131.9 124.5 126.16 43.5736 115.1 121.3 113 125.2 115.3 117.97 45.4955 151 132.7 135.5 140.8 155 143 4077 120.2 116.7 129.3 124.2 134.6 125 43.8Average 128.33 43.13Initial conditions of shear band formation: The thickness of the shear deformation region dtis assumed to be 0.15 of its length, i.e., dt ≈ 0.15b, where b= d/sinφ [4]. Therefore, for the uncutchip thickness d = 120 µm, the thickness dt is 27 µm. This thickness is employed to estimate theinitial strain rate γ˙h in the homogeneous deformation stage of the shear band formation. Table 2.5shows the parameters of JC flow law for Ti-6Al-4V workpiece material. Based on Table 2.5, thethermal softening coefficient a = 1/(Tm− Tr) = 6.1× 10−4 K−1. Fig. 2.12 shows the homoge-neous temperature (Th = θh(γ)+ Tr) and homogeneous shear stress τh(γ) for vc = 60 m·min−1,which are obtained from Eqs. (2.6) and (2.7), respectively. Based on Eq. (2.9), instability occurswhen the strain in the plastic deformation region AOC reaches the critical strain γcr = 0.324. Thiscritical strain is small compared to the strain developed during continuous chip formation (∼ 2-3for the rake angle α=0◦ [106]), showing that localization starts quickly after the material in thedeformation region AOC experiences plastic deformation. According to Fig. 2.12, θh = 129.8 K(Th =422.8 K), and τh = 1.137 GPa are considered as the initial conditions to predict the stress andtemperature after the shear localization onset.Strain rate in shear localization and shear band width: Table 2.6 compares the simulated41Table 2.5: Parameters of the Johnson-Cook flow law for Ti-6Al-4V alloy [82].Alloy A B n C m γ˙re f(MPa) (MPa) (1/s)Ti6Al4V 782 498 0.28 0.028 1.0 10−50 0.1 0.2 0.3 0.4 0.5800900100011001200Shear Stressh0 0.1 0.2 0.3 0.4 0.5250300350400450500Absolute TemperaturehFigure 2.12: Homogeneous stress and temperature evolution for the cutting speed vc = 60m·min−1. The onset of shear localization is also shown.homogeneous strain rate γ˙h and the maximum strain rate γ˙ in (developing inside the shear band)from Eq. (2.32). According to Table 2.6, forming a shear band during the cutting process resultsin a significant increase in the strain rate from the homogeneous stage to the shear localizationstage (two orders of magnitude increase in the case of the cutting speed 90 m·min−1). The ratioof the maximum strain rate γ˙ in to the homogeneous strain rate γ˙h is also shown in Table 2.6. Thisratio shows that unlike the conventional continuous chip formation, the maximum strain rate doesnot change linearly with the cutting speed in the shear-localized chip formation. This is becausethe shear band width 2δ is not constant with respect to the cutting speed, which is clear from therelationship between the shear band width and the strain rate in Eq. (2.31).To verify the obtained values of the shear band widths and the strain rates, Fig. 2.13 comparesthe simulated shear band half-widths obtained from Eq. (2.31) with the experimental results from42Table 2.6: Initial and maximum strain rates for the cutting speeds 30, 60, and 90 m·min−1.The ratio γ˙ in/γ˙h is also represented.Cutting speed γ˙h γ˙ in Ratio(m·min−1) (s−1) (s−1) -30 2.5×104 1.2×105 4.860 5.0×104 4.8×105 9.690 7.5×104 1.1×106 14.5SEM observations of the machined chips. In all experiments, the shear band with large shear de-formation was visually apparent from the adjacent low-strain segments as shown in Fig. 2.14. Thethickness of the region with large shear deformation with obvious flow lines was considered as theshear band’s width. For each cutting speed, the measured shear band width is the average value ofat least four measurements at different locations of the chip, as shown in Fig. 2.15. The maximumdeviation of the measurements from the average value was 8%, 14%, 15%, and 12% for 30, 60,90, and 105 m·min−1, respectively.According to Fig. 2.13, the shear band’s width decreases asymptotically with respect to the cut-ting speed, and levels off at high cutting speeds. The experimental data presented by Sagapuramet al. [101, 105] is also included in Fig. 13 for cutting of Ti-6Al-4V with the uncut chip thicknessd=125 µm. They measured the shear band’s width based on two definitions: i) the region which isdistinctly different than the adjacent segment (similar to this study), and ii) the region over whichthe plastic strain falls to 50% of its maximum value. For the second definition, the strain profileacross the band was mapped by the micro-marker technique. According to their experiments, theystated that the shear band’s width decreases asymptotically with the cutting speed to a small valueless than 1 µm when cutting Ti-6Al-4V [105], which is in agreement with the presented model inthis study. Therefore, the change from homogeneous deformation with dt =25 µm to the localiza-tion in the shear band with less than 1 µm width is captured.4320 40 60 80 100 120Vc [m.min-1]024 [m]predictionexperiment (SEM)SEM [97]micro-marker [97]Figure 2.13: Evolution of the Shear band’s half-width with respect to the cutting speed. Asthe cutting speed increases, the shear band’s width decreases asymptotically. Theexperimental measurements of the shear band’s width based on i) SEM analysis andii) micro-marker analysis presented in Refs. [101, 105] are also shown.Figure 2.14: Visually distinguishable border between the shear band with large shear defor-mation and the surrounding segments with low-plastic deformation for the cuttingspeed vc = 105 m·min−1.Stress and temperature evolution in the shear bands: The cutting velocity vc = 60 m·min−1is assumed as the reference velocity for the nondimensionalization scheme, and Table 2.7 showsthe reference variables obtained from Eqs. (2.10) and (2.11). The numerical integration of theshear band’s temperature (Eq. (2.30)) is achieved by discretizing the independent variable t¯ intosufficiently small steps. In order to obtain the current value of θ¯ in(t¯i), the temperature in the44Figure 2.15: Measurement of shear band’s width at different locations for the cutting speedvc = 90 m·min−1.previous time step, i.e., θ¯ in(t¯i−1), is used to perform numerical integration. After obtaining thecurrent temperature θ¯ in(t¯i), the current stress can be obtained from Eq. (2.18) for each time step.Moreover, for the thermal sensitivity m = 1, which is applicable for many metals according tothe experimental data of the JC flow law [76], Eq. (2.30) can be analytically solved to obtain thedeveloped temperature in the shear band explicitly asθ¯ in(t¯) =β v¯τˆ(Γ˙in)C√t¯+√π k¯θ¯h√π k¯+ aˆβ v¯τˆ(Γ˙in)C√t¯. (2.38)Both Eqs. (2.30) and (2.38) provide closed-form expression of temperature inside the shear bandduring shear-localized chip formation. With the developed solutions of strain rate and temperature,the stress evolution in the shear band is obtained from Eq. (2.18), expressed asτ¯ in(t¯) = τˆ(Γ˙in)C[1− (aˆθ¯ in(t¯))m] . (2.39)Fig. 2.16 shows the simulated stress and temperature evolution of the shear band for the cut-ting speed of vc = 60 m·min−1. At first, the temperature increases rapidly, until it approaches45Table 2.7: Reference variables for nondimensionalization.v∗ γ˙h τ∗ θ ∗(m·s−1) (s−1) (MPa) K1.35 50984 782 315.9to a steady-state condition. This trend is in agreement with the numerical simulations of shearband formation [127], where the temperature increases gradually even at the steady-state stage asa characteristic of shear band behavior. As shown in Fig. 2.16, the temperature increases fromTh = 422.8 K (θh = 129.8 K) to T = 1235.9 K after 80 µs. The temperature at t = 70 µs isT = 1212 K, meaning that it has increased only 2% of its magnitude during the final 10 µs. Thisshows that the temperature profile has reached a quasi-steady-state value. Therefore, the tempera-ture T = 1235.8 K is considered as the steady-state temperature developed within the shear band.Accordingly, the shear stress drops from τh = 1137.5 MPa to τ in = 558.21 MPa at the steady-statestage due to the shear localization.0 20 40 60 80050010001500200005001000temperaturestressFigure 2.16: Stress and temperature evolution inside the shear band for the cutting speedvc = 60 m·min−1.Fig. 2.17 shows the effect of cutting speed on the developed stress and temperature in the shearband. According to Fig. 2.17, for the cutting speed vc = 90 m·min−1, the temperature increasesfrom T = 405.2 K to T = 1390.7 K. By comparing with the final temperature at the cutting speed46vc = 60 m·min−1, the steady-state temperature of the shear band increases 13% when the cuttingspeed increases from 60 to 90 m·min−1. The shear stress at vc = 90 m·min−1 drops from 1149.5MPa to 443.14 MPa. Therefore, for vc = 90 m·min−1, the stress drop is 11% greater compared tovc = 60 m·min−1, showing larger decrease of the stress at higher cutting speeds.0 20 40 60 800500100015002000Vc=90 m/minVc=60 m/min0 20 40 60 8005001000Vc=90 m/minVc=60 m/minFigure 2.17: Comparison of the stress and temperature evolution inside the shear band be-tween the cutting speeds vc = 60 m·min−1 and vc = 90 m·min−1.For continuous chip formation and based on the shear plane model, Oxley [88] predicted thetemperature rise in the primary shear zone asTs = Tr +η1(1−η2) Fs cosαρcwd cos(φ −α) , (2.40)where Ts is the temperature at the shear plane and Fs = τsbw is the shear force acting on the shearplane. In Eq. (2.40), the parameter η1 (0 < η1 < 1) is a factor to consider the fact that not all theplastic work has occurred on the shear plane (the primary shear zone has a thickness), and η2 is theproportion of the heat conducted into the workpiece material, which is obtained from the empiricalequations [88]:η2 = 0.5−0.35log(RT tanφ) 0.04≤ RT tanφ ≤ 10.0η2 = 0.3−0.15log(RT tanφ) RT tanφ > 10.0, (2.41)47where RT =ρ¯ c¯v¯d¯k¯ . Using the thermo-mechanical properties provided in Tables 2.3 and 2.5, and as-suming a typical value of η1 ≈ 0.7 [4], the temperature in the primary shear zone can be predicted.Table 2.8 compares the steady-state temperature predicted by the shear-localized chip formationmodel presented in this study and the temperature obtained by the continuous chip formation modelof Oxley [88]. According to Table 2.8, the shear-localized chip model predicts the temperatures41%, 74%, and 93% larger than those predicted by the continuous chip model for the cuttingspeeds 30, 60, 90 m·min−1, respectively. This shows the importance of shear band formation inthe developed temperature during cutting processes. It should be noted that in the Oxley’s model,all the plastic work is assumed to be converted into heat (β=1) [88]. In order to consider β=0.9, thesecond term of Eq. (2.40) should be multiplied by β . This gives the temperature values of 649.4K, 667.22 K, and 677.75 K for the cutting speeds 30, 60, 90 m·min−1, respectively.Table 2.8: Comparison of the predicted temperatures in continuous and shear-localized chipformation models for the cutting speeds 30, 60, and 90 m·min−1.Cutting speed Continuous model Shear-localized model(m·min−1) (K) (K)30 689 972.860 708.8 1235.990 720.5 1390.7Shear band spacing: Fig. 2.18 shows the experimental measurements and the simulated shearband spacing with respect to the cutting speed. Extensive SEM investigations of the cutting chipswere performed for 10 different cutting speeds in the range of 20-105 m·min−1. For each cuttingspeed, the minimum distance between two neighboring shear bands was measured for at least 13different locations. The error bars in Fig. 2.18 show the maximum and minimum values of themeasured spacing in each cutting speed. The average value of the experimental measurements aswell as the fitted curve are also shown in Fig. 2.18. It is shown that as the cutting speed increases,48the shear band spacing increases rapidly at first and then approaches to an asymptotic value.In the simulation of the shear band spacing, two cutting speeds of vc = 36 m·min−1 and vc = 48m·min−1 were used to calibrate dd (the total deformation region’s thickness, i.e., the thickness ofthe region AOB in Fig. 2.9) and ϑ in Eq. (2.36). For each cutting speed, the critical time tcr(Eq. (2.35)) and the average shear band spacing ds, which was measured experimentally, were in-serted into Eq. (2.36). The two equations were then solved simultaneously, where dd = 101 µmand ϑ = 8.4 were obtained. Based on Eq. (2.36), dt changes with respect to the cutting speed ratherthan being a constant. For the cutting speed vc = 36 m·min−1, dt is about 42 µm while 31 µm forvc = 48 m·min−1. As the cutting speed increases, dt decreases and converges to a value around25 µm. This is in agreement with Zorev’s cutting mechanics model [137], which shows that asthe cutting speed increases, the borders of the plastic deformation region dt rotate clockwise withfaster rotation of the initial boundary OA, until they reach their final location. Then, the thicknesswill be constant for the higher cutting speeds.20 40 60 80 100vc [m/min]20406080100Spacing [m]mean valuefitted curveerror barmodelFigure 2.18: Variation of the shear band spacing with respect to the cutting speed. The errorbars show the minimum and maximum experimental measurements of the shear bandspacing for each cutting speed. The mean value of the experimental measurements foreach cutting speed as well as the fitted curve on the experimental data are also shown.49Overall, the comparison shows that the proposed model is able to capture the asymptotic in-crease of the shear band spacing with respect to the cutting speed. This is in agreement with theexperiment, while is in contrast to the opposite trend in Refs. [79, 134, 136]. The error in the shearband spacing prediction is attributed to the assumption that dd and ϑ are constant in Eq. (2.36),while it is expected that they change with respect to the cutting speed [137].Stress and temperature oscillations in the deformation region: Having obtained the stressand temperature evolution within the shear bands as well as the shear band spacing, the stress andtemperature oscillations in the plastic deformation region are now simulated. Fig. 2.19 shows thetemperature and stress oscillations at the uncut chip thickness of d =120 µm and the cutting speedof vc = 60 m·min−1. The amplitude of oscillation is 580 MPa for the stress, and 813 K for thetemperature. The simulated oscillation frequency is ψ = vchds = 12.3 kHz, where vch is the chipvelocity obtained from the cutting velocity and the shear angle using vch = vsinφcos(φ−α) , and ds is thesimulated shear band spacing. The experimental oscillation frequency is ψexp = vchdsexp = 12.8 kHz,where dsexp is the average shear band spacing obtained from the SEM analyses.The formation of a shear band is accompanied by the rapid increase of the temperature asshown in Fig. 2.19. When a shear band occurs at plane OC shown in Fig. 2.2a, i.e., (SB1)OC,the temperature increases significantly. The rate of temperature rise decreases as the cutting tooladvances until SB1 reaches the final boundary of the deformation zone, i.e., (SB1)OB in Fig. 2.19.The temperature suddenly drops as the shear band leaves the deformation region and slides onthe rake face. At this stage, the next shear band has already formed in the deformation region,i.e., (SB2)OC, causing the temperature to rise again. This temperature oscillation in the primaryshear zone with large amplitude and high frequency may significantly affect the cutting tool’s life.A similar trend is seen for the stress oscillation, where the stress drops because of shear bandformation and increases again when new material enters the plastic deformation region.500 200 400 600 800t [ s]0200400600800100012001400T [K]0 200 400 600 800t [ s]0200400600800100012001400 [MPa]Figure 2.19: Stress and temperature oscillations of shear bands in the deformation regionAOB for the cutting speed vc = 60 m·min−1.2.5 ConclusionsA new analytical thermo-mechanical model for shear band formation in cutting processes was pre-sented in this section. The plastic deformation of the workpiece material inside the deformationregion was divided into homogeneous (pre-instability) and localization (post-instability) stages.The model predicts the stress, strain rate, and temperature inside the shear band before and afterlocalization. The predicted strain rate was validated with SEM observations of the shear band’sthickness, where the results show that the strain rate’s change due to shear localization is signifi-cant, and it increases nonlinearly with respect to the cutting speed. The nonlinear change in strainrate is because the shear band width decreases nonlinearly when the cutting speed increases. Thedeveloped temperature inside the shear band is validated with the existing temperature results ofthe cutting tool available in the literature. It is shown that the temperature of the tool rake face issignificantly affected by the shear band’s temperature due to the intimate contact between them.The proposed model is able to capture the band spacing during cutting processes by considering51a large deformation region in front of the cutting tool. The results, which are validated with ex-tensive SEM investigations, show an asymptotic increase in the shear band spacing with respect tothe cutting speed. The difference between the shear band formation in other dynamic deformationapplications and cutting processes is emphasized by showing that in cutting processes, shear bandsoccur subsequently rather than simultaneously. Moreover, SEM observations reveal different as-pects of shear localization in cutting processes. It is shown that the rolling of the shear band onthe rake face plays an important role in the shear-localized chip formation, especially, the cuttingtool’s temperature. The rolling extends all the way through the segment’s thickness and is equalto the displacement of the shear band at the free surface. The proposed analytical model revealsthe mechanism of the complete shear-localized chip formation process with emphasis on its dif-ferences with continuous chip formation, and quantitatively determines the stress and temperaturewith high computational efficiency. The model can be extended to consider the effects of cuttingfluids and cutting tool’s edge radius and more detailed experimental results can be obtained aboutthe stress and temperature evolution in the shear band during shear-localized chip formation byemploying a set of infrared detectors integrated with the cutting setup.52Chapter 3Prediction of Shear Band Displacement andRolling Phenomenon3.1 Introduction and Literature ReviewShear localization in machining of metallic alloys is influenced by a wide range of thermo-mechanicalproperties as well as the cutting condition [59, 61, 66, 104, 131]. However, it is still unclear howthey quantitatively determine the localization behavior and intensity during machining processes.The second question is about the role of elastic energy accumulated at the tool-chip interface inshear localization. In the machining literature, the elastic energy stored at the tool-chip interfaceis introduced as the main reason for the stress and temperature oscillations in front of the cuttingtool owing to its periodic accumulation and release. Based on the physical modeling of Burns andDavies [13, 14], and Burns et al. [15], when the cutting tool advances, the workpiece material atthe tool-chip interface is compressed locally. This compression is assumed to follow the simpleuni-axial stress-strain relation of σn = Eεn, where σn is the stress normal to the rake face, E is theelastic modulus, and εn is the elastic strain normal to the rake face, by neglecting the plane-straincondition of orthogonal cutting for simplicity. Due to the momentum balance between the forces53on the cutting tool’s rake face and the forces on the primary shear zone in the absence of inertia,the increase in the elastic deformation at the tool-chip interface causes the heat generation insidethe whole primary deformation zone to increase. The increased heat production, in turn, causesthe thermal softening of the material inside the primary shear zone, which subsequently unloadsthe elastic deformation at the tool-chip interface. Therefore, the shear localization was interpretedas the interaction between the primary deformation zone and the tool-chip interface inspired bythe study performed by Gray and Scott [41] on chemical oscillations and instabilities. In thesestudies, the catastrophic shear, and subsequently, the localization of the plastic deformation insidethe shear bands were neglected and assumed that all the material inside the primary deformationzone is experiencing thermal softening rather than the shear band itself.Due to the lack of a thermo-mechanical cutting model for shear localization, the developedstrain and displacement in the workpiece material (and subsequently, the chip morphology) hasnot been predicted in the literature. Many investigations have tried to obtain the strain and dis-placement of shear localization based on the experimental measurements on the chip morphologyby post-mortem investigations on the machined chips. Cotterell and Byrne [18] proposed a geo-metrical relation for the developed strain inside the produced shear bands when cutting Ti-6Al-4Vin a series of interrupted orthogonal cutting tests. They showed that the relations obtained basedon the assumption of continuous chip formation cannot capture the developed strain in chip seg-mentation. Dong et al. [29] performed experimental observations on the chip roots of Inconel 718by employing a quick-stop device. They proposed geometrical relations for obtaining the strainand strain rate of chip segmentation based on the assumption that the shear plane is converted intoa shear body when chip segmentation occurs. In these post-mortem investigations, the producedchips must be collected and their dimensions must be measured each time the cutting conditionchanges. In addition, since the rolling process has not been recognized, only the displacement ofthe shear band at the free surface is considered to be the total displacement of the shear band. In54FE simulations, the strains and displacements of chip segmentation are tuned with the aid of thefracture characteristics of the workpiece material due to the insufficient number of elements insidethe produced shear bands (as explained in Section 1.4). This posteriori calibration causes an un-derestimation of the temperature prediction inside the shear bands because of the element deletiondue to fracture (crack) propagation.This chapter presents three main contributions based on the surveyed literature mentionedabove. First, the formation of a shear band and its contact with the cutting tool via a process calledrolling is demonstrated for Ti-based and Ni-based alloys, and the differences are emphasized. Tothis end, extensive experimental observations are employed to study the shear band formation andchip segmentation in Ti-based and Ni-based metallic alloys. It is shown that while chip segmenta-tion is the only type of chip formation mechanism in Ti-based alloys in the cutting speed range of20 to 200 m·min−1, Ni-based alloys show three different types of chip formation within the samecutting speed range. Second, it is shown that the energy stored at the tool-chip interface, whichis believed to play a major role in chip segmentation (see Burns and Davies [13, 14], Burns et al.[15]), cannot be the reason for chip segmentation since it has a temporary effect on the behaviorof the workpiece material inside the primary deformation zone. It is shown that the elastic unload-ing at the tool-chip interface causes an overshoot in the shear band’s strain rate which disappearsquickly after shear band formation. Third, the developed strain, displacement, and chip mor-phology of different metallic alloys are predicted without post-mortem measurement of the shearband’s morphology. The results show good agreement with the experimental measurements withthe maximum error of 10% in the displacement prediction. It is shown that the chip segmentationcan be predicted accurately without employing fracture criteria. This outcome proves that unlikethe crack formation, the continuity of the material preserves in the chip segmentation. Moreover,it is shown that the rolling process has to be considered to accurately predict the developed strainand displacement inside the shear band. Neglecting the rolling effect results in an underestimation55of the strain and displacement in the shear band by a factor of two.The remaining sections are organized as follows. Section 3.2 presents and compares the exper-imental characterizations of shear band formation in Ti-6Al-4V, commercially pure (CP) titanium,Inconel 718, and Waspaloy. The machined chips’ optical, laser, and SEM micro-graphs are pre-sented and investigated. Based on these observations, different aspects of chip segmentation arediscussed. Next, the theoretical aspects of chip segmentation, including the elastic energy releaseat the tool-chip interface (Section 3.3) and the strain, strain rate, and displacement predictions(Section 3.4) are presented. The results are presented and discussed in Section 3.5. Finally, thesummary of the presented work and the concluding remarks are provided in Section 3.6.3.2 Experimental Investigations of Chip Segmentation inTi-based and Ni-based Metallic AlloysOrthogonal cutting experiments were performed on commercially pure (CP) titanium (grade 2 tita-nium), Ti-6Al-4V, Inconel 718 (in annealed condition), and the Ni-based alloy Waspaloy, using thelongitudinal turning operation mode of Mori Seiki NT 3150 DCG (see Section 2.2 for the detailsof the experimental setup). These workpiece materials were chosen for their propensity to shear-localization while they cover a wide range of thermo-mechanical properties. For Ti-6Al-4V, CPtitanium, and Inconel 718, cylindrical workpieces with the diameter of 50.8 mm (2 in.) were em-ployed while for Waspaloy, the workpieces’ diameter was 120 mm. Depending on the workpiecematerial, the cutting speed vc was varied from 20 m·min−1 to 200 m·min−1, while the uncut chipthickness d was set to 120 µm to investigate the effects of the cutting speed on the experimentalresults. To avoid the tool wear on the experimental results (based on Section 2.2), the cutting insertwas replaced after 2.3 mm cutting length for CP titanium, 2 mm for Ti-6Al-4V, 1.3 mm for Inconel718, and 1 mm for Waspaloy.56After mounting and grinding (as described in Section 2.2), colloidal silica suspensions wereused for polishing the CP titanium and Ti-6Al-4V samples, whereas, polycrystalline diamond slur-ries were employed for the Waspaloy and Inconel 718 samples in order to maximize the removalrate. Soon after the final polishing step, the specimens were chemically etched in a solution of92 wt% water, 6 wt% nitric acid, and 2 wt% hydrofluoric acid for CP titanium and Ti-6Al-4V,and 5 grams cupric chloride (CuCl2), 100 ml hydrochloric acid, and 100 ml ethanol for Waspaloyand Inconel 718. The Ti-based samples were immersed in the etchant, while the Ni-based sampleswere scrubbed by a cotton swab soaked with the etchant due to their high corrosion resistance. Theetching time varied from a few seconds to a minute depending on the material and its correspond-ing etchant.To examine the chip morphology and shear band structure, optical images were captured byNikon Epishot 300 Inverted optical microscope, while Olympus Lext 3000 confocal scanning lasermicroscope and Σigma™ field emission scanning electron microscope (SEM) were used to captureimages with higher magnifications. The results based on the experimental observations include theshear band’s structure, the plastic deformation pattern, and the grain distribution in the machinedchips. Fig. 3.1 shows the details of the experimental investigations.3.2.1 Chip segmentation in Ti-based alloysFig. 3.2 shows the optical micro-graphs of the CP titanium and Ti-6Al-4V machined chips at thecutting speeds vc = 60 m·min−1 and vc = 90 m·min−1, respectively. There is a high density of thegrain boundaries in CP titanium at the shear bands’ locations observable in Figs. 3.2a and 3.2b,showing that the grains are heavily distorted at the shear bands’ locations. On the other hand, thelarge grains inside the segments shows that the plastic deformation inside the segment is small.Since the plastic deformation is localized inside the shear band, the segment slides on the shear57Cold MountingGrindingTitanium Alloys Nickel Alloys* Inconel 718* Waspaloy* Ti-6Al-4V* Pure TitaniumEtchingEtchingStored under VacuumSEM Optical MicroscopeLaser MicroscopeOrthogonal Cutting Experiment* Longitudinal turning* Cutting speed: 20-200 m/min* Uncut chip thickness: 120 μm * High curing time (12 hours) Epofix resin * Chips perpendecular to the mount’s cross-section* Silicon Carbide (SiC) papers* Sequence: 400, 600, 800, 1200 grit* Platen Speed: 150-300 rpm Polishing* Colloidal silica suspensions* 6 μm, 3 μm, 1 μm, 0.05 μmPolishing* Polycrystalline dymond slurries* 6 μm, 3 μm, and 1 μm* Kroll’s reagent* Immersed in etachant* Etching time: 30 - 45 sec.* Kalling’s reagent (Inconel 718) * Modified Kalling’s (Waspaloy) * Scrubbed by cotton swab* Etching time: 5 - 15 sec.* To avoid hydride and oxide formationSputter Coating* To facilitate electron discharge* Iridium coating* 2-4 nm coating thickness* Zeiss field emission SEM* Detectors: SE2, In-lens, and backscattered* High-res. images to measure: - shear band’s width - plastic deformation at the tool-chip interface - rolling phenomenon * Olympus Lext 3000 confocal SLM* High-res. images to capture: - grain size and distribution - shear band’s microstructure - segment’s microstructure - rolling phenomenon* Nikon Epishot 300 Inverted OM* Optical images to measure: - chip’s morphology - shear band’s spacing - rolling phenomenon Figure 3.1: Details of the experimental investigations.58band similar to a rigid body, causing the saw-tooth-like appearance of the cutting chip. In all thecutting chips, the shear bands are narrow bands located periodically between the segments whilefor the both cutting speeds, the shear bands occur more regularly in Ti-6Al-4V compared to CPtitanium.(a) CP titanium, vc = 60 m·min−1.Large Density of Grain BoundariesShear Band(b) CP titanium, vc = 90 m·min−1Shear Band Spacing(c) Ti-6Al-4V, vc = 60 m·min−1Shear BandDifferent Morphology Compared to the Segments(d) Ti-6Al-4V, vc = 90 m·min−1Figure 3.2: Machined chips’ optical micro-graphs for (a) CP titanium under the cutting speedvc = 60 m·min−1, (b) CP titanium under the cutting speed vc = 90 m·min−1, (c) Ti-6Al-4V under the cutting speed vc = 60 m·min−1, (d) Ti-6Al-4V under the cuttingspeed vc = 90 m·min−1. The grains are heavily distorted at the shear bands’ locations,showing large plastic deformation at the shear bands’ locations. The segments are moreregular in Ti-6Al-4V than CP titanium.In both CP titanium and Ti-6Al-4V, the shear band spacing (i.e., the shortest distance between59two successive shear bands or, in other words, the segments’ thickness as shown in Fig. 3.2c)increases as the cutting speed increases from 60 m·min−1 to 90 m·min−1. For CP titanium, theaverage shear band spacing (the average of measurements in at least 10 different locations) was 51µm for vc = 60 m·min−1 while 58 µm for vc = 90 m·min−1. For Ti-6Al-4V, the shear band spac-ing was measured as 72 µm for vc = 60 m·min−1 while 80 µm for vc = 90 m·min−1. The spacingbetween the shear bands is generally smaller in CP titanium than Ti-6Al-4V for the same cuttingcondition. This observation can be related to the difference between the yield strengths of theseworkpiece materials, their thermo-mechanical properties, and the friction at the tool-chip interface.For instance, CP titanium has lower yield strength compared to Ti-6Al-4V (around 480 MPa yieldstrength of CP titanium compared to around 800 MPa yield strength of Ti-6Al-4V [109]). Accord-ing to the Zorev’s cutting model, the higher strength of the material causes the plastic deformationregion in front of the cutting tool to be larger. Since the shear band spacing is determined by thearea of the plastic deformation region ( see Section 2.3.3), the alloy with higher strength has largershear band spacing. Subsequently, the segmentation frequency can be obtained as ψ = vch/ds,where vch = vc sinφ/cos(φ −α) is the chip velocity, and ds is the shear band spacing. For thecutting speed vc = 60 m·min−1, the rake angle α = 0◦, and the typical shear angle φ = 43◦ (seeChapter 2), ψ = 19.6 kHz for CP titanium, while ψ = 13.9 kHz for Ti-6Al-4V.According to Fig. 3.2, the morphology of the shear bands formed during cutting of Ti-6Al-4Vand CP titanium is different. To further investigate these differences, Fig. 3.3 shows two lasermicro-graphs for Ti-6Al-4V and CP titanium under the cutting speed vc = 60 m·min−1. As de-picted in Fig. 3.3, the shear bands formed during the cutting of Ti-6Al-4V are distinct from thesegments (the shear bands have a white appearance), while in CP titanium, the shear bands re-tain their appearance (similar to the segments) although the grains are heavily deformed (the largedensity of the grain boundaries in the shear bands’ locations is clear in Fig. 3.3b). In other dy-namic deformation applications, including torsional split-Hopkinson bar, the different appearance60of the Ti-6Al-4V shear bands is attributed to different microstructural mechanisms active at ele-vated temperatures during the large deformations in Ti-6Al-4V, including recrystallization [96, 97]and phase transformation [75], without any consensus on the reason.(a) Ti-6Al-4V (b) CP titaniumFigure 3.3: Laser micro-graph of the shear band formation when cutting: (a) Ti-6Al-4V and(b) CP titanium under the cutting speed vc = 60 m·min−1. In Ti-6Al-4V, the shear bandsare distinct from the segment (appear as white narrow regions) while in CP titanium,they retain their initial microstructure with heavy plastic deformation.3.2.2 Chip segmentation in Ni-based alloysDifferent chip formation mechanisms at different cutting speeds were observed during the cuttingof the Ni-based alloys (both Waspaloy and annealed Inconel 718). Fig. 3.4 shows the SEM micro-graphs of the machined chips produced during cutting of Waspaloy at the uncut chip thicknessd = 120 µm and three different cutting speeds of vc = 30 m·min−1, 45 m·min−1, and 90 m·min−1,respectively. At the lowest cutting speed (vc = 30 m·min−1), the chip formation is continuous,which indicates that the material is being cut homogeneously at the entire length of the chip with-out shear localization. This type of chip formation is similar to the chips formed when cutting mildsteel (AISI 1045) up to the cutting speed of 400 m·min−1 (for the uncut chip thickness of 250 µm)[108]. According to the cutting mechanics of continuous chip formation proposed by Oxley and61Shaw [88], the plastic deformation zone in front of the cutting tool (i.e., the primary deformationzone) extends from the tool-tip to the free surface of the workpiece material. In the produced con-tinuous chips, the chip thickness is constant with constant cutting forces, and the temperature inthe primary deformation zone reaches a steady-state value rather than time-varying.(a) vc = 30 m·min−1 (b) vc = 45 m·min−1(c) vc = 90 m·min−1Figure 3.4: Chip formation mechanisms in Waspaloy at different cutting speeds: (a) at vc =30 m·min−1, homogeneous deformation occurs, (b) the cutting speed vc = 45 m·min−1is a transition cutting speed where both continuous and shear-localized chips occur, (c)the cutting chip at vc = 90 m·min−1 is shear-localized.By increasing the cutting speed to vc = 45 m·min−1, the machined Waspaloy chip shows inho-mogeneity in the plastic deformation in some parts of the chip. A certain part of the chip where62both continuous and segmented sections (i.e., both homogeneous deformation and inhomogeneousdeformation) exist is shown in Fig. 3.4b. The occurrence of the mixed chip formation mecha-nism in one cutting experiment shows that the cutting speed vc = 45 m·min−1 lies in the transitionzone between the continuous chip formation and the shear-localized chip formation. It should benoted that the transition from the continuous chip formation to the shear-localized chip formationis smooth, meaning that there is a range of cutting speeds where both chip formation mechanismsare observable (the transition starts from a cutting speed lower than vc = 45 m·min−1 and endsat a cutting speed higher than vc = 45 m·min−1). Determining the accurate cutting speeds underwhich the transition starts and ends needs multiple cutting experiments in a limited range of cuttingspeeds, which is not the focus of this thesis and shall be studied in a future work.As the cutting speed increases further, the inhomogeneity in the plastic deformation was ob-served in all the cutting chips with no evidence of the continuous chip formation (Fig. 3.4c).Fig. 3.4c shows the machined chip under the cutting speed vc = 90 m·min−1, where localization ofthe plastic deformation inside the shear bands is evident via the saw-tooth-like appearance of thechip. Similar behaviors (i.e., continuous chip formation, transition zone, and shear-localization)were observed in Inconel 718 (annealed). However, in annealed Inconel 718, the transition zonewas observed at a cutting speed higher than Waspaloy (the approximate cutting speed was vc = 60m·min−1). Compared to the cutting experiments on titanium alloys, where shear localization wasobserved in all the cutting speeds (from vc = 20 m·min−1 to vc = 120 m·min−1), Ni-based alloys(Waspaloy and Inconel 718) are less prone to shear-localization.Using high-magnification SEM images of the prepared Ti-6Al-4V samples, the regions wherethe shear bands roll on the cutting tool’s rake face at the tool-chip interface were shown in Chap-ter 2. Then, it was shown that the thickness of plastic deformation region at the tool-chip interfaceis almost equal to the shear bands’ width. Therefore, it was concluded that most of the plastic63deformation at the tool-chip interface belongs to the shear band, which has been rolled on the rakeface, rather than the sliding friction at the tool-chip interface. Fig. 3.5 shows the optical imageof the shear band structure for annealed Inconel 718 at the cutting speed vc = 90 m·min−1. As itis evident in the micro-graph, during cutting of the nickel alloys where the produced shear bandsare thicker than the titanium shear bands (cf. Figs. 3.3 and 3.5), the experimental results show adifferent rolling mechanism. According to Fig. 3.5, the rolling occurs from only one edge of theshear band (the edge O′A′) while the other edge remains almost straight (the edge C′D′). Whilethis causes the shear band to be in direct contact with the cutting tool, the thickness of the plasticregion at the tool-chip interface varies correspondingly.Figure 3.5: Shear band interaction with the cutting tool’s rake face when cutting annealedInconel 718 at the cutting speed 90 m·min−1. For the shear band O′A′C′D′, the edgeO′A′ has been rolled on the rake face while the edge C′D′ is almost straight.There are two important aspects in the shear band formation during cutting processes. First,how a new shear band forms in front of the cutting tool, and second, how the previously formedshear bands evolve when a new shear band is forming. Fig. 3.6 shows the schematic of shear-localization in the orthogonal cutting processes where the cutting force oscillates with respect totime due to the formation of shear bands in front of the cutting tool. According to Fig. 3.6, bypiling up of the new workpiece material in front of the cutting tool, the cutting force reaches a crit-64ical value (i.e., a maximum value) for a new shear band (called SB1 in Fig. 3.6) to occur. Duringthe piling up process, the previously formed shear band, i.e., SB2 in Fig. 3.6, rolls on the cuttingtool from its bottom edge and therefore, it is in direct contact with the cutting tool. According toFig. 3.6, the shear band SB2 is thick enough so that when the cutting tool moves forward, the bot-tom edge of the shear band rolls on the rake face while the top edge remains approximately straightwithout any significant rolling. In this way, the shear band can accommodate the significantly highshear strain imposed by the cutting tool while at the same time, it satisfies the compatibility condi-tion at the tool-chip interface, that the workpiece material must take the shape of the flat rake faceof the cutting tool. In Fig. 3.6, A′C′D′O′ shows the points corresponding to ACDO after the rollingphenomenon.With RollingWorkpiece(a) (b)CuttingToolUpsettingWithout RollingCuttingToolUpestting RegionFigure 3.6: Schematic presentation of chip segmentation in Ni-based alloys. When a newshear band occurs in front of the cutting tool (SB1), the previously formed shear band(SB2) rolls on the cutting tool through its thickness. This causes the plastic region atthe tool-chip interface to have a variable thickness. The possible behaviors of a singleshear band are also shown in the sub-figures.Under the shear loading, an isolated shear band would tend to freely deform in the direction ofshear velocity if no geometrical constraints were imposed by the cutting tool (Fig. 3.6a). However,in the cutting processes: (i) the upsetting region is pushing the previously formed shear band (see65Fig. 3.6), (ii) the friction at the tool-chip interface (even if it is small compared to the continuouschip formation) holds one side of the shear band while the other side (at the free surface side) isfreely moving along the chip velocity, (iii) the shear band must take the flat shape at the tool-chipinterface due to the geometrical constraint of the cutting tool’s rake face. Therefore, the shear bandtakes the form shown in Fig. 3.6b rather than the shape in the simple shear of Fig. 3.6a. Thesekinematic differences cause the shear band to be in contact with the cutting tool over the entirecontact area of the tool-chip interface. Subsequently, since the shear band’s temperature is highdue to localization, the generated heat inside the band is conducted to the tool’s rake face, causinghigh cutting tool temperature and rapid tool wear. Moreover, based on Fig. 3.6, the rolling causesthe total displacement of the shear band to be ( f + s) rather than f since the shear band’s bottomand top edges move in different directions as shown in Fig. 3.6b.The partial rolling in Ni-based alloys (i.e., the rolling from only one edge of the shear band)causes the shear band’s thickness to vary at its two ends (i.e., the free surface end and the end atthe tool-chip interaction). Fig. 3.7 shows the high magnification laser micro-graphs of annealedInconel 718 at the cutting speed vc = 200 m·min−1 in three different regions of one shear band,including the free surface side (Fig. 3.7a), middle section of the chip (Fig. 3.7b), and the tool-chipinterface area (Fig. 3.7c). It is clear that the thickness of the shear band at the free surface side(Fig. 3.7a) is not constant and reduces as it approaches the free surface. Considering Fig. 3.7b, thethickness is almost constant in the middle of the shear band while in Fig. 3.7c, the thickness in-creases as it approaches the tool-chip interface. This thickness variation shows that the remainingthickness of the shear band near the free surface is actually in direct contact with the cutting toolat the tool-chip interface and proves the schematics of Fig. 3.6.66(a) . (b)(c)Figure 3.7: High-magnification laser micro-graphs of a shear band when cutting annealedInconel 718 under vc = 200 m·min−1 at three different locations: (a) at the free surfaceside, (b) at the middle of the chip, (c) at the tool-chip interface. The difference in thethickness of the shear band at different locations shows that the shear band is in directcontact with the cutting tool.3.3 Effects of elastic unloading at the tool-chip interfaceFig. 3.8 shows the local deformation zone with the thickness of ∆un at the tool-chip interface. Thecutting tool is in contact with the chip with the contact area of ltot ×w, where ltot is the total tool-chip contact length. As the tool advances, it exerts a normal compression force to the chip over thelocal deformation area whose thickness is ∆un (see Fig. 3.8). Therefore, an elastic stress called σnis applied perpendicular to the rake face due to this local compression. Note that in the following,67the subscripts n and t denote the normal and tangential directions with respect to the cutting tool’srake face, respectively, as shown in Fig. 3.8. The local compression stress σn is written based onthe plane-strain condition of the orthogonal cutting process asσn =E(1+ν)(1−2ν) [(1−ν)εn+νεt ] , (3.1)where E is the workpiece’s elastic modulus, ν is its Poisson’s ratio, εn is the elastic strain normalto the rake face, and εt is the elastic strain tangential to the rake face. Taking the time derivative ofEq. (3.1), the changing rate of the normal stress σn at the tool-chip interface isdσndt=E(1+ν)(1−2ν)[(1−ν)dεndt+νdεtdt]. (3.2)WorkpieceSegmentFigure 3.8: Schematic presentation of local deformation zone at the tool-chip interface,which is induced by the elastic deformation of the workpiece material due to the con-tact with the cutting tool’s rake face.The elastic strain normal to the rake face (i.e., εn) is assumed to be equal to εn = ∆un/ravg.,68where ravg. = d/sinφ is the average thickness of the chip. Therefore, the changing rate of theelastic strain εn isdεndt=1ravg.· d(∆un)dt=sinφ cos(φ −α)d· d(∆u)dt, (3.3)where ∆u is the projection of ∆un on the plane OC where the shear band’s center is located (seeFig. 3.8). According to Oxley and Shaw [88], the strain rate in the tangential direction (i.e., therate of εt) isdεtdt=vchhˆ, (3.4)where hˆ is the thickness of the plastic deformation at the tool-chip interface. As discussed inSection 2.2, the shear localization process includes two different stages. At the first stage, theworkpiece material is piled up in front of the cutting tool while the previously formed shear bandrolls on the cutting tool’s rake face. At the second stage, a new shear band occurs in front ofthe cutting tool, causing the chip to slide (with the velocity of vch) along the rake face. Duringthe first stage (i.e. upsetting stage), there is almost no relative motion between the segment beingformed and the rake face (i.e., vch ≈ 0). Sliding with the velocity of vch occurs at the second stagewhere a new shear band initiates in front of the cutting tool. Therefore, at the first stage where theworkpiece material is compressed at the tool-chip interface, there is a local quasi-static conditionat the tool-chip interface. Hou and Komanduri [47] and Komanduri [55] employed high-speedimaging techniques to observe the motion of the workpiece material during shear-localization.Their experimental observations show the quasi-static condition during the piling-up stage (i.e.,the first stage) based on which there is no motion in the chip velocity direction (i.e., vch = 0).Accordingly, the rate of change of normal stress presented in Eq. (3.2) is simplified asdσndt=(1−ν)E(1+ν)(1−2ν) ·dεndt=Ωdεndt, (3.5)where the parameter Ω= (1−ν)E(1+ν)(1−2ν) is defined for conciseness to take the effects of elastic mod-69ulus E and the Poisson’s ratio ν into account. Inserting Eq. (3.3) into Eq. (3.5), the rate of changeof normal stress is written asdσndt=Ωsinφ cos(φ −α)d· d(∆u)dt. (3.6)The term d(∆u)/dt in Eq. (3.6) shows the compression rate at the tool-chip interface due to thedifference between the cutting tool’s velocity vs and the shearing velocity of the workpice materialv both in the direction of the shear angle φ (i.e., along the plane OC). Therefore, Eq. (3.6) is writtenasdσndt=Ωsinφ cos(φ −α)d(vs− v) . (3.7)Initially, when the cutting tool starts contacting the cutting chip, vs > 0 and v= 0, causing a suddenincrease in the elastic energy of the material at the tool-chip interface, i.e., dσ/dt > 0. Afterwards,when the material is being cut homogeneously, vs = v and therefore, there is no change in theelastic energy of the workpiece material at the tool-chip interface, i.e., dσ/dt = 0. However, whena shear band occurs in front of the cutting tool, there is a sudden increase in the plastic strain rateand therefore, vs < v, causing the workpiece material at the tool-chip interface to unload elastically,i.e., dσ/dt < 0.Neglecting the small inertia terms, the momentum balance requires that σnltotwcos(φ −α) =τwd/sinφ . Therefore, the evolution equation of the shear stress on the shear band is obtainedbased on Eq. (3.7) asdτdt=ltotΩd2sin2φ cos2(φ −α)(vs− v) . (3.8)The cutting tool velocity is obtained as vs = v = γ˙hdt at the homogeneous deformation stage.Since the cutting tool velocity vs is always constant with respect to time (i.e., rigid motion ofthe cutting tool), vs = γ˙hdt remains valid irrespective of the deformation stage of the workpiece70material. Focusing on the localization stage (when shear band occurs in front of the cutting tool),the shearing velocity of the workpiece material is obtained as v= 2γ˙ inδ , where γ˙ in is the strain rateinside the shear band and 2δ is the shear band’s width. Therefore, from the homogeneous stage tothe localization stage, the shearing deformation’s thickness drops from dt to 2δ , and the strain rateincreases from γ˙h to γ in:dτdt=ltotΩd2sin2φ cos2(φ −α)(γ˙hdt −2γ˙ inδ), (3.9)where vs and v are replaced by γ˙hdt and 2γ˙ inδ , respectively. Note the shear band’s strain rateγ˙ in and its half-width δ (which are related via ∆v = v = 2δ γ˙ in) can be obtained from Eqs. (2.31)and (2.32), respectively.To interpret Eq. (3.9), the shear stress variation at different stages of shear band formation andevolution is needed. Fig. 3.9 shows the schematic of the simplified behavior based on the analyti-cal results of Chapter 2 when the workpiece material (the potential shear band) enters the primarydeformation region AOB from the boundary OA until when it exits the primary deformation regionAOB from the boundary OB (see Fig. 2.9). The shear band’s stress and temperature variationsconsist of three different stages: homogeneous (zone I), transient (zone II), and steady-state (zoneIII) as shown in Fig. 3.9. The material is under homogeneous deformation in the zone I accordingto Fig. 3.9. This is equivalent to the deformation from OA to OC in Fig. 2.9 (where localization hasnot occurred in the primary deformation region yet). The homogeneous stress and temperature ofthe workpiece material can be obtained from the homogeneous solution of the governing equations(see Section 2.3.1). As soon as the strain reaches the critical strain γcr (which is obtained fromEq. (2.9)), the material cannot respond to the applied dynamic loading homogeneously anymore.Therefore, instability initiates in the material behavior, causing a partial loss in its load-bearingcapacity as shown in Fig. 3.9 where there is a rapid variation in stress and temperature evolution71in the transient zone (zone II). However, this rapid variation quickly gives way to slow asymptoticbehavior according to which the shear band becomes mature (quasi-steady-state stage or zone IIIin Fig. 3.9).Figure 3.9: Schematic evolution of stress and temperature of a potential shear band from thetime when it enters the primary deformation region (boundary OA) until the time whenit exits the primary deformation region (boundary OB). Different zones show differentstages of the shear band formation. In zone I, the workpiece material experiences ho-mogeneous deformation. In zone II, the critical strain γcr has reached and the stress andtemperature variations are intense. In zone III, the shear band becomes mature withslight change in its stress and temperature.Based on the formation and evolution of the shear band during the cutting process explained inFig. 3.9, the behavior of the shear band is affected by the elastic response of the material in zoneII. Employing the nondimensionalization scheme proposed in Eqs. (2.10) and (2.11), Eq. (3.9) iswritten in nondimensional form asdτ¯dt¯= E¯(1−2δ¯ Γ˙in) , (3.10)72where E¯ is the nondimensional elastic modulus at the tool-chip interface obtained asE¯ =ltotdtd2τ∗Ωsin2φ cos2(φ −α). (3.11)The nondimensional elastic response Eq. (3.10), along with the nondimensional energy Eq. (2.13)and the nondimenional flow law Eq. (2.18), determine the shear band response under the elasticunloading at the tool-chip interaction. Solving this system of equations analytically is challeng-ing since there is a sudden change in the thickness of the material under plastic deformation fromhomogeneous dt to localized 2δ , and rapid increase in the strain rate from homogeneous γ˙h tolocalized γ˙ in. Therefore, based on the assumption that the heat conduction at the early time ofthe stress collapse is negligible, i.e., adiabatic condition at the early time of the stress collapse[54, 127], the general features of the elastic response of the workpiece are explained below.When instability starts at the plane OC, the homogeneous deformation gives way to shearlocalization. Considering the first increment right after the instability onset, Eq. (3.10) is presentedin the incremental form as∆τ¯ = E¯(∆t¯−2δ¯∆γ) , (3.12)where ∆ is the increment operator. Since the heat conduction at the early time of the stress collapseis not significant, the energy Eq. (2.13) can be written in its incremental form as∆θ¯ = β τˆΓ˙C[1− (aˆθ¯)m]∆γ. (3.13)Since the strain rate sensitivity C is very small, and the thermal softening coefficient m is close to 1for many metallic alloys (see Ref. [76] for common strain rate sensitivities and thermal softeningcoefficients of metallic alloys), the incremental form of the flow law Eq. (2.18) at the early stage73of localization is approximated as∆τ¯ =−aˆτˆΓ˙C∆θ¯ . (3.14)Inserting Eq. (3.13) into Eq. (3.14):∆τ¯ =−β aˆτˆ2Γ˙2C [1− (aˆθ¯)m]∆γ. (3.15)Right after localization onset, the term (aˆθ¯)m is much smaller than 1. For example, for Ti-6Al-4V,under the cutting speed vc = 60 m·min−1, θ¯ = 0.41 and aˆ= 0.19 (see Section 2.4), resulting in theterm (aˆθ¯)m of 0.078. Therefore, Eq. (3.15) is written as∆τ¯ ≈−β aˆτˆ2Γ˙2C∆γ. (3.16)Inserting Eq. (3.16) into the incremental form of the elastic unloading at the tool-chip interface,i.e., Eq. (3.12), the incremental form of the strain rate right after the instability onset is obtained as∆γ∆t¯=v¯2δ¯ − (β aˆτˆ2/E¯) . (3.17)Eq. (3.17) is interpreted as follows: when the workpiece is considered as rigid-viscoplastic materialat the tool-chip interface, the nondimensional elastic modulus E¯ can be assumed to be infinite.Therefore, the second term in the denominator is equal to zero and the strain rate increment iswritten as (∆γ)/(∆t¯) = v¯/(2δ¯ ). This is the same as the strain rate obtained without considering thelocal deformation zone at the tool-chip interface (see Section 2.3.2). However, the second term inthe denominator (which captures the local deformation zone’s effect) always takes a positive value.Subsequently, the denominator is smaller than 2δ¯ and the strain rate is larger than v¯/(2δ¯ ). Thiscauses that in zone II, the instantaneous strain rate becomes larger than the strain rate in zone III.In order to understand this behavior, the elastic energy accumulated at the local deformation zone74of the tool-chip interface is written (in dimensional form) asElocal =∫ ∆u012σεdu =σ2∆u2E. (3.18)This energy is stored at the tool-chip interface due to the local compression. When the suddenunloading because of the shear band formation occurs, this energy is released with the rate ofσ(∆u)σ˙/(2E). Based on the momentum equation σ ltotw = τwd/sinφ , the rate of energy releaseis expressed asE˙local = (dltot sinφ)2(∆u)ττ˙. (3.19)In zone II where the stress drops significantly, E˙local is a large negative value, showing the con-siderable energy release. Based on the energy conservation Elocal +Etool = Ediss (where Etool isthe energy imposed by the cutting tool and Ediss is the energy dissipated in the shear deformationregion), both elastic energy and energy applied by the cutting tool causes large strain rate in zoneII. However, in zone III, the stress change is almost zero, and therefore, the strain rate is deter-mined solely by the input energy of the cutting tool. In conclusion, Eq. (3.17) captures the effectof elastic unloading on the shear band’s strain rate while Eq. (3.19) demonstrates that this effect istemporary and specific to zone II.3.4 Prediction of strain distribution and chip morphologyFig. 3.10 shows different parameters describing the produced shear-localized chip during chip seg-mentation. In Fig. 3.10, f is the displacement at the free surface side of the shear band, s is theinteraction length of the shear band with the cutting tool due to the rolling process, ds is the shearband spacing, rmax is the maximum thickness of the chip, rmin is the minimum thickness of thechip, b is the deformation zone’s length, and φseg is the segment’s angle.Interpreting the shear band’s strain rate integrated with time as the shear band’s strain γ in, it75CuttingToolASegmentBCFigure 3.10: Schematic diagram of the shear-localized chip morphology and its correspond-ing parameters.can be written asγ in =∫ tB0γ˙ indt, (3.20)where the integration limit starts from 0 (i.e., when the material enters the upsetting region AOBfrom the boundary OA) and end at the time tB (i.e., when the shear band exits the boundary OB,see Fig. 2.9). Therefore, tB = dd/vch. The developed strain in Eq. (3.20) is divided toγ in =∫ tC0γ˙hdt+∫ tBtCγ˙ indt = γcr +∫ tBtCγ˙ indt≃∫ tBtCγ˙ indt, (3.21)where γcr can be obtained from Eq. (2.9). Note that since the critical strain γcr is quite smallcompared to the shear band strain, it has been neglected in Eq. (3.21). Assuming that the strainrate is constant after localization, the strain in Eq. (3.21) can be approximated asγ in = γ˙ in(tB− tC) = γ˙ in · dd−dtvch = γ˙in · dsvch, (3.22)76where (tB− tC) is the time during which the shear band is under loading in the primary deformationregion. When a shear band forms, its bottom part rolls on the cutting tool as shown in Figs. 3.5and 3.7. Consequently, the total displacement accumulated inside the shear band is actually ( f +s)because the two edges of the shear band are moving in different directions as shown in Fig. 3.6.Since the shear band’s rolling on the rake face is equal to the displacement at the free surface (i.e.,s≃ f ) as shown in Fig. 3.6, the displacement f is half of the total displacement accumulated insidethe shear band, i.e., f = γ inδ (note that the shear band’s width is 2δ ). In Eq. (3.22), the strainrate inside the shear band (i.e., γ˙ in) is significantly high, therefore, a small error in prediction ofthe time during which the shear band is in the primary deformation zone (i.e., tB− tC) can cause alarge error compared to the experiments.According to the plasticity theory, the unloading in the plastic region is time-history dependentand the problems involving unloading must be solved incrementally in order to capture the historydependency [53]. The reason is that for a unique strain value, more than one stress value can existif unloading has occurred. However, in Fig. 3.9, the unloading is not because of the load removal,but it is caused by the shear instability. In other words, the unloading is not accompanied by nega-tive strain increment which is the reason of history dependency. Therefore, for each strain value, aunique stress value exists, making Eq. (3.22) valid to use.According to Fig. 3.9, in the steady state region (i.e., zone III), the temperature increases veryslowly, i.e., θ¯,t¯ ≃ 0, causing the temperature increase term in the energy Eq. (2.13) to be negligible.Therefore, Eq. (2.13) in the zone III is approximated ask¯θ¯,y¯y¯+β τ¯Γ˙≃ 0. (3.23)Eq. (3.23) states that in the steady-state stage of the shear band formation, a balance between heat77conduction out of the shear band and heat production inside the shear band is achieved. In orderto obtain the spatial distribution of the strain, it is assumed that the strain rate is dependent onthe location, i.e., Γ˙ = 𭟋(y). Therefore, from the nondimensional JC flow law (Eq. (2.18)), thetemperature distribution in the vicinity of the shear band, i.e., θ¯(y¯) is obtained asθ¯(y¯) =1aˆ[1− τ¯τˆ𭟋(y¯)−c]. (3.24)Note that although the temperature and strain rate are dependent on the location y¯, the stress isconstantly distributed inside the shear band because the inertia effect is negligible, i.e., τ¯y¯ = 0.Taking the second derivative of the temperature, i.e., θ¯ ′′(y¯) from Eq. (3.24) and inserting the resultinto Eq. (3.23), one can finally obtain𭟋′′(y¯)− (C+1)𭟋′(y¯)2𭟋(y¯)+β aˆτˆk¯𭟋(y¯)C+2 = 0. (3.25)In order to solve Eq. (3.25) analytically, the approximations 1+C = 1+O(C) and 2+C = 2+O(C) were made. Since the strain rate sensitivity C is small for many metallic alloys [76], theseapproximations are reasonable. Therefore, Eq. (3.25) is updated as𭟋′′(y¯)− 𭟋′(y¯)2𭟋(y¯)+β aˆτˆk¯𭟋(y¯)2 = 0. (3.26)In order to solve Eq. (3.25), two boundary conditions are needed. One boundary condition is𭟋(0) = Γ˙in, where Γ˙in is the strain rate at the center of the shear band, which can be obtained fromEq. (2.32) (Note that Eq. (2.32) is in dimensional form). Since the maximum strain rate happensat the center of the shear band and the strain rate distribution is continuous inside the band, thesecond boundary condition is 𭟋′(0) = 0. Solving the ordinary differential Eq. (3.26) with the78stated boundary conditions, the strain rate distribution is obtained asΓ˙(y¯) =𭟋(y¯) = Γ˙insech2√ aˆmΓ˙inβ τˆ2Ck¯y¯= Γ˙insech2( y¯δ¯), (3.27)where, the shear band’s half-width from Eq. (2.31) has been inserted into the result (in its nondi-mensional form). Making the results dimensional, the strain rate distribution is written as γ˙(y) =Γ˙(y¯)γ˙h. The strain distribution is obtained based on Eqs. (3.22) and (3.27) asγ(y) = γcr + γ˙(y)∆t ≃ γ˙(y)∆t. (3.28)Having obtained the displacement at the shear band’s location, i.e., f , the degree of segmentation(ξ ) is defined asξ =rmax− rminrmax=fb=fdsinφ . (3.29)Finally, the angle between the shear band and the free surface side of the segment (i.e., φseg inFig. 3.10) is obtained from the law of sines (the subfigure in Fig. 3.10) asφseg = sin−1(fscos(α+φ)) =π2− (α+φ). (3.30)3.5 Results and DiscussionsTable 3.1 shows the experimentally measured thicknesses of the shear-localized chips when cut-ting annealed Inconel 718 and Waspaloy. The maximum chip thickness rmax (see Fig. 3.10) wasmeasured at least in five different segments, and the average value was obtained for each cut-ting condition as presented in Table 3.1. The averaged values were then employed to obtain theshear angle via φ = tan−1( r cosα1−r sinα ) where r = d/(rmax)avg is the chip compression ratio. It shouldbe noted that the maximum chip thickness rmax is used to obtain the shear angle rather than the79minimum chip thickness rmin or the average chip thickness ravg = (rmax + rmin)/2 because: (i) itprovides the shear angle corresponding to the boundary OA (see Fig. 3.10). This shear angle isassumed to be constant in the primary deformation zone (see Section 2.3). (ii) It is independent ofthe displacement developed inside the shear band. Unlike the maximum chip thickness rmax, theminimum chip thickness rmin and the average chip thickness ravg are both dependent on the dis-placement inside the shear band. Since the displacement inside the shear band can be quite large,the minimum chip thickness rmin can be very small and can even approach zero in high cuttingspeeds (in this case, the chip becomes discontinuous). Therefore, obtaining the shear angle basedon the minimum or average thicknesses may result in the shear angles much larger than 45◦ (incase of discontinuous chip formation, it approaches 90◦). As an example, when cutting Ti-6Al-4Vunder the cutting speed of 90 m·min−1, the mean values of the average and minimum chip thick-nesses were measured as 115.13 µm and 73.95 µm, which result in the shear angles of 46.19◦and 58.37◦, respectively. Komanduri [55] discussed about the physical importance of shear anglein cutting mechanics and how large shear angles cannot be justified in case of chip segmentation.Therefore, in this study, the maximum thickness of the shear-localized chip was used for obtainingthe shear angle of the primary deformation zone. Adopting the maximum chip thickness to obtainthe shear angle, the average shear angles of annealed Inconel 718 and Waspaloy are φ = 27.9◦ andφ = 29.6◦, respectively, as shown in Table 3.1. For Ti-6Al-4V and age-hardened Inconel 718, theshear angle was obtained as φ = 43.1◦ (Section 2.4) and φ = 37.4◦ [59], respectively.The main purposes of finding the shear angle in chip segmentation is to obtain the componentof the cutting velocity (and the cutting force) in the shear band’s direction (i.e., finding the shearvelocity v and the shear force Fs as shown in Fig. 3.6). The shear velocity is subsequently employedin the analysis to obtain the homogeneous strain rate applied on the shear band. Miscalculatingthe shear angle causes the shear velocity (and shear force) to be overestimated (if the shear angleis less than the experimental results) or underestimated (if the predicted value is larger than the80Table 3.1: Calculation of the shear angle for annealed Inconel 718 and Waspaloy by experi-mental measurements of chip thickness in different cutting speeds.Workpiece Cutting speed Chip thickness (µm) Shear angleMaterial (m·min−1) 1 2 3 4 5 avg. (deg.)Inconel 718120 222 220 205 218 223 217 28.9200 244 235 238 238 231 237 26.8(annealed) Average 227 27.9Waspaloy120 221 237 215 217 233 224 28.1200 220 197 194 189 200 200 31.0Average 212 29.6experimental results). Fortunately, the formation of distinct shear bands during chip segmentationenables the experimental measurement of the shear bands’ orientation for validating the predictedshear angle. Fig. 3.11 shows the laser micro-graphs of the experimentally measured shear an-gles when cutting Waspaloy at the cutting speed vc = 120 m·min−1 and Ti-6Al-4V at the cuttingspeed vc = 90 m·min−1, respectively. The average value of the shear angle (for at least seven mea-surements in different locations) was obtained as 30.43◦ for Waspaloy and 41.53◦ for Ti-6Al-4V.Comparing these values with the predicted values of 29.6◦ for Waspaloy and 43.1◦ for Ti-6Al-4Vshows that the calculated shear angle based on the maximum chip thickness rmax provides accurateresults. It should be noted that while the measurement of the shear angle based on the shear band’sorientation requires metallurgical sample preparation to reveal the shear band’s microstructure, theprediction of the shear angle based on the chip thickness can be performed on the cutting chipwithout any sample preparation, which justifies the shear angle’s prediction based on the measuredchip’s thickness.The thermo-mechaincal properties for Ti-6Al-4V, Inconel 718, and Waspaloy are shown in Ta-ble 3.2, while Table 3.3 represents the JC parameters for Ti-6Al-4V, age-hardened Inconel 718,annealed Inconel 718, and Waspaloy used for the simulations. Note that the thermo-mechanicalproperties and the JC parameters of Ti-6Al-4V is the same as Table 2.3 and Table 2.5 in Sec-81 CuttingTool(a) Waspaloy CuttingTool(b) Ti-6Al-4VFigure 3.11: Laser micro-graph of the experimentally measured shear angles during cuttingof: (a) Waspaloy at the cutting speed vc = 120 m·min−1, (b) Ti-6Al-4V at the cuttingspeed vc = 90 m·min−1. The maximum chip thickness rmax, average chip thicknessravg , and minimum chip thickness rmin are also shown in the figure.tion 2.4, which have repeated here for the sake of comparison with other alloys. The JC parametersare extracted from the studies in the literature that are performed on the metal cutting processes.Regarding Inconel 718, both types of heat treatments (annealed and age-hardened) are commonlyused in industry and therefore, their JC parameters are well-established in the literature. Ozel et al.[89] compared different JC parameters proposed for Inconel 718 in the literature and chose the setof parameters proposed by Lorentzon et al. [70] for their simulations. Liu et al. [68] compareddifferent JC parameters proposed for Ti-6Al-4V in the literature and chose the one proposed byLee and Lin [65] for their simulations. For CP titanium, some studies have tried to modify theJC flow law model (more specifically, its strain hardening and thermal softening terms) in order tobe able to capture the dynamic deformation behavior of CP titanium by JC model [84, 111]. Thisrequires modification of the derived equations in this thesis for the proposed modified JC models,82which is out of the scope of this thesis. Therefore, Ti-6Al-4V is considered as the representativeof titanium alloys in this section. The experimental results of the age-hardened Inconel 718 are de-rived from the experimental investigations performed by Sagapuram and Viswanathan [101, 102]and Sagapuram et al. [105].Table 3.2: Thermo-mechanical properties of Ti-6Al-4V [82, 112], Inconel 718 [6], and Was-paloy [24].Alloyk¯ c¯ ρ¯ Tm E νW/(m.K) J/(kg.K) kg/m3 K GPa −Ti-6Al-4V 7.2 560 4420 1930 114 0.34Inconel 718 11.54 451 8199 1570 206 0.33Waspaloy 11.7 520 8200 1633 211 0.33Table 3.3: Johnson-Cook parameters for Ti-6Al-4V [65, 82], aged Inconel 718 [70, 89], an-nealed Inconel 718 [89], and Waspaloy [93, 95].Alloy A B n C m Γ˙re f(MPa) (MPa) - - - (1/s)Ti-6Al-4V 782 498 0.28 0.028 1.0 10−5Inconel 7181241 622 0.6522 0.0134 1.3 1(aged)Inconel 718450 1700 0.65 0.017 1.3 10−3(annealed)Waspaloy 750 330 1.05 0.01 1.5 1The shear deformation region’s thickness (dt in Fig. 2.9) is usually assumed to be 0.15b−0.2bwhere b = d/sinφ [4]. Therefore, for the uncut chip thickness d = 120 µm and the obtainedshear angles, dt becomes 27 µm, 30 µm, and 38 µm, and 36 µm for Ti-6Al-4V, aged Inconel718, annealed Inconel 718, and Waspaloy, respectively. Based on the melting temperatures pro-vided in Table 3.2, the thermal softening coefficient a= 1/(Tm−Tr) is obtained as 6.1×10−4 K−1,7.83×10−4 K−1, and 7.46×10−4 K−1 for Ti-6Al-4V, Inconel 718, and Waspaloy, respectively.83Employing Eq. (2.9), for the cutting speed vc = 60 m·min−1, the critical strain γcr is obtainedas γcr = 0.324 for Ti-6Al-4V and γcr = 0.727 for Inconel 718 (aged), respectively. Substituting thecritical strain γcr into Eqs. (2.6) and (2.7), the homogeneous temperature and stress at the onset oflocalization can be obtained as θh = 129.8 K (T = 422.8 K) and τh = 1.137 GPa for Ti-6Al-4V,and θh = 196.56 K (T = 489.56 K) and τh = 1.152 GPa for Inconel 718. These values are consid-ered as the initial conditions for the shear localization analysis. Fig. 3.12 compares the developedstress and temperature inside the shear band between aged Inconel 718 and Ti-6Al-4V under thecutting speed vc = 60 m·min−1 and uncut chip thickness d = 120 µm obtained from Eqs. (2.18)and (2.30), respectively. According to Fig. 3.12, when the shear band starts to occur, the temper-ature (stress) rapidly increases (decreases) due to the shear instability. However, soon after theshear band onset, the stress and temperature asymptotically approach a quasi-steady state wheretheir variations become significantly small compared to the onset of localization.A shear band during the cutting process is under loading until the next shear band occurs infront of the cutting tool. Therefore, the loading time of a shear band (i.e., the time during which theshear band is in the primary deformation zone) can be obtained as ds/vch. The shear band spacingfor Inconel 718 was measured around ds =110 µm while ds =80 µm for Ti-6Al-4V. Therefore, theloading time is obtained as 157 µs for Inconel 718 and 85 µs for Ti-6Al-4V, respectively. There-fore, both temperature and stress diagrams are truncated at the time 157 µs for Inconel 718 and atthe time 85 µs for Ti-6Al-4V. According to Fig. 3.12, the final absolute temperature developed inInconel 718 (aged) is T =1149 K while T =1275 K for Ti-6Al-4V. Subsequently, the final shearstress is obtained as τ = 558 MPa for Inconel 718 (aged) and τ = 545 MPa for Ti-6Al-4V.Fig. 3.13 shows the developed strain rate inside the shear band for Inconel 718 (aged) andTi-6Al-4V with the cutting speed. First, unlike continuous chip formation where the strain rate840 50 100 150t [ s]050010001500T [K]Inconel 718 (aged)Ti6Al4V(a) .0 50 100 150t [ s]050010001500 [MPa]Inconel 718 (aged)Ti6Al4V(b)Figure 3.12: Developed temperature (absolute) and stress in the shear band when cuttingInconel 718 (aged) and Ti-6Al-4V at the cutting speed vc = 60 m·min−1 and uncutchip thickness d = 120 µm.increases linearly with the cutting speed [88], the shear band’s strain rate increases nonlinearlywith the cutting speed in shear-localized chip formation. Second, the developed strain rate in Ti-6Al-4V is larger than the one in Inconel 718 for all cutting speeds, showing that Ti-6Al-4V isunder extreme loading conditions for the same cutting speed and uncut chip thickness. Therefore,Ti-6Al-4V is more prone to localization. For example, at the cutting speed vc = 60 m·min−1, thestrain rate in Ti-6Al-4V is around 0.5×106 s−1 while 0.17×106 s−1 in Inconel 718. Third, as thecutting speed increases, the maximum strain rate in Ti-6Al-4V and Inconel 718 diverges, where theincreasing trend for Ti-6Al-4V is more than Inconel 718. The difference in the strain rate increasesfrom 0.33×106 s−1 at the cutting speed vc = 60 m·min−1 to 1.02×106 s−1 at the cutting speedvc = 100 m·min−1.Since the experimental measurements of the shear band’s stress, temperature, and strain rateand their instantaneous changes during cutting are challenging as explained in Chapter 1, one wayto indirectly prove the predicted stress, temperature, and strain rate of the shear band during cut-ting processes is to compare the predicted shear band’s width with the experimental measurementsperformed after sample preparation and by high magnification imaging. If the shear band’s widthcan be predicted accurately, the strain rate can be obtained via γ˙ in = v/(2δ ), which in turn, can8520 40 60 80 100Vc [m.min-1]00.511.52max [s-1]106Inconel 718 (aged)Ti6Al4VFigure 3.13: Variation of strain rate with respect to the cutting speed for aged Inconel 718and Ti-6Al-4V. The developed strain rate in Ti-6Al-4V is higher than Inconel 718 forall cutting speeds, and they diverge as the cutting speed increases.validate the prediction in Fig. 3.13. Moreover, according to Eqs. (2.18) and (2.30), the developedtemperature and stress inside the shear band depend on the prediction of the shear band’s width,and the shear band acts as a heat source with variable power and the thickness of 2δ .For Inconel 718 and Waspaloy, due to the large thickness of the shear bands compared to Ti-based alloys, high magnification laser micro-graphs and SEM images can be used to measure theshear bands’ width. However, for titanium alloys, since the thickness of the shear bands is muchsmaller, high magnification SEM images are employed to measure the shear band’s thickness accu-rately. Fig. 3.14 shows the laser micro-graph of the developed shear band when cutting Waspaloyat two different magnifications. In all measurements, there was always a distinct boundary betweenthe shear band and the surrounding materials, as shown in Fig. 3.14b. The shear band’s width wasmeasured as the minimum distance between two borders of the band. Note that, in order to avoidthe rolling phenomenon’s effect explained in Section 3.2 (see Fig. 3.7), the shear bands’ width wasmeasured in the middle of the chip (not at the tool-chip interface or the free surface side), wherethe thickness of the shear band is constant. At least five different measurements for each cuttingspeed were performed, and the average values were employed.86(a) (b)Figure 3.14: The measurement method of the shear bands’ widths at two different magnifi-cations for Waspaloy chips under the cutting speed vc = 120 m·min−1. The borderbetween the shear band and the surrounding material is visually distinguishable.Fig. 3.15 shows the predictions of the shear band’s half-width for aged Inconel 718 and Ti-6Al-4V, and their comparisons with the experimental results. It can be seen that the predictionscan capture the shear band’s half-width with high accuracy for both aged Inconel 718 and Ti-6Al-4V. In general, the shear bands are thicker in Inconel 718 compared to Ti-6Al-4V for the samecutting speed. For instance, at the cutting speed vc = 60 m·min−1, the shear band half-width isδ = 4.27 µm for Inconel 718 while δ = 1.33 µm for Ti-6Al-4V (experiments: 3.8 µm for Inconel718 while 1.8 µm for Ti-6Al-4V). According to Eq. (2.31), high thermal conductivity, high strainrate sensitivity, and low thermal sensitivity cause thicker shear bands during cutting processes.Moreover, higher shear angle φ causes a thinner primary shear deformation region and, therefore,thinner shear bands. Based on Tables 3.1 to 3.3, Ti-6Al-4V has the lowest thermal conductivity,highest strain rate sensitivity, lowest thermal sensitivity, and highest shear angle. These intrinsicproperties of Ti-6Al-4V cause narrower shear bands, and subsequently, more intense localized de-formation during cutting of Ti-6Al-4V. Although the cutting tool’s rake angle was constant in thisstudy (α = 0◦), the increase in the cutting tool’s rake angle α causes thicker shear bands due to the87lower strain rate applied to the shear deformation zone via γ˙h = vc cosαdt cos(φ−α) . Table 3.4 summarizesthe effects of different thermo-mechanical parameters of the workpiece material as well as the cut-ting conditions on the chip segmentation during machining processes based on the proposed modeland the simulation results in this study.20 40 60 80 100 120Vc [m.min-1]0510 [m]Shear Band Half-WidthInconel 718 (aged)Inconel 718 SEMTi6Al4VTi6Al4V SEM (set 1)Ti6Al4V SEM (set 2)Figure 3.15: Comparison between the predicted and experimental values of shear band’s half-width δ for Ti-6Al-4V and aged Inconel 718. The error bars show the maximum andminimum values of the measured widths for each cutting speed. The Ti-6Al-4V shearbands are thinner than aged Inconel 718 for the same cutting speed. In both cases, theshear band’s width asymptotically decreases with respect to the cutting speed. TheSEM data of Ti-6Al-4V (set 2) is adopted from Ref. [105].88Table 3.4: Summary of the effects of thermo-mechanical parameters, cutting condition, andtool geometry on the characteristic response of the workpiece material to chip segmen-tation.Parameter DescriptionThermal softening (a)Larger thermal softening coefficient causes:• more propensity to localization (smaller critical strain γcrbased on Eq. (2.9)).• higher strain rate based on Eq. (2.32).• larger overshoot in the strain rate due to elastic loadingthe tool-chip interface based on Eq. (3.17).• more segmentation of the chip, i.e., larger strain and dis-placement based on Eq. (3.22).Strain rate hardening (C)Larger strain rate hardening causes:• less propensity to localization (larger critical strain γcrbased on Eq. (2.9)).• thicker shear band based on Eq. (2.31).• smaller strain rate based on Eq. (2.32).• less segmentation of the chip based on Eq. (3.22).Strain hardening (n)Larger stain hardening coefficient leads to:• less propensity to localization because of larger criticalstrain γcr based on Eq. (2.9).• Larger overshoot during elastic unloading at the tool-chip interface because of its effect on the parameter τˆ inEq. (3.17).Thermal conductivity (k)Higher thermal conductivity causes:• less propensity to localization based on Eq. (2.9).• thicker shear band based on Eq. (2.31).• smaller strain rate based on Eq. (2.32).• less segmentation of the chip, i.e., smaller strain and dis-placement based on Eq. (3.22).89Cutting speed (vc)Increase in cutting speed vc leads to:• thinner shear bands based on Eq. (2.31).• higher strain rate based on Eq. (2.32).• larger overshoot in the strain rate based on Eq. (3.17).• more segmentation of the chip based on Eq. (3.22).• higher propensity to localization based on Eq. (2.9).Rake angle (α)Increase in the cutting tool’s rake angle leads to:• lower strain rate γ˙ in based on Eq. (2.32) since the shearvelocity projected on the shear band (v) decreases.• thicker shear band due to the lower strain rate applied tothe shear deformation zone (Eq. (2.31)).• smaller overshoot in the strain rate based on Eq. (3.17).• smaller strain and displacement (less segmentation) basedon Eq. (3.22).At all considered cutting speeds, shear-localized chip formation was observed for Ti-6Al-4Vand aged Inconel 718. However, as explained in Section 3.2, for Waspaloy and annealed Inconel718, different chip formation mechanisms were observed in different cutting speed ranges. For thelow cutting speeds, the machined chips are of continuous type. Then, the chip consists of bothhomogeneous and inhomogeneous deformation for a range of cutting speeds, which depends onthe workpiece material. At this cutting speed range, the deformation process cannot be capturedby the proposed theory as the fundamental assumption of this study is the periodical occurrenceof shear-localization in front of the cutting tool. The inhomogeneous deformation is the dominantdeformation mechanism for higher cutting speeds, but the shear band’s thickness is considerablylarger than the shear band’s thickness of Ti-6Al-4V and aged Inconel 718 cutting chips. For in-stance, the average shear band’s half-width was measured as 22 µm for Waspaloy versus 1.45 µmfor Ti-6Al-4V at the cutting speed vc = 90 m·min−1.90Fig. 3.16 shows the predicted half-width of the Waspaloy shear bands with the experimentalresults. According to Fig. 3.16, at the low cutting speeds, the theory loses its accuracy while athigh cutting speeds, it can predict the shear band’s width with high accuracy. The loss of accuracyis attributed to the fact that in order to see shear localization as a dominant chip formation mech-anism, the cutting speed needs to be higher than the cut-off value corresponding to the activationof only inhomogeneous deformation. When both homogeneous and inhomogeneous deformationshappen (see Fig. 3.4b), the chip formation mechanism needs different theoretical assumptions sincethe fundamental assumption of periodical occurrence of shear-localization in front of the cuttingtool is violated. In Fig. 3.16, the cut-off cutting speed was experimentally estimated to be vc =70m·min−1, where irregular chip formation gives way to the dominant localization.50 100 150 200Vc [m.min-1]020406080100 [m]Shear Band Half-Width (Waspaloy)predictionexperiment (SEM)Figure 3.16: The prediction of the shear band’s half-width of Waspaloy compared to the ex-perimental results. At low cutting speeds both homogeneous and inhomogeneousdeformation take place, causing the theory to lose its accuracy. At high cutting speeds(high than 70 m·min−1, which is called cut-off speed, the shear band’s half-width ispredicted with high accuracy.Fig. 3.17 presents the effect of elastic deformation at th tool-chip interface on the shear band’sstrain rate evolution for Ti-6Al-4V and aged Inconel 718 at the cutting speed vc = 60 m·min−1.According to Fig. 3.17, at the homogeneous stage, the strain rate is 0.56×105 s−1 for Ti-6Al-4Vwhile 0.52×105 s−1 for Inconel 718. This is equivalent to the traveling of the workpiece material91from the initial boundary OA to the plane OC in Fig. 2.9. During this stage, the local elastic energyat the tool chip interface does not change since vs = v, i.e., dσ/dt = 0. As soon as the localiza-tion starts at the plane OC, the elastic energy release occurs at the tool-chip interface, causing asudden change in the strain rate to 7.55×105 s−1 for Ti-6Al-4V and to 1.65×105 s−1 for Inconel718. A portion of this strain rate jump is because of localization of the plastic deformation insidethe shear band (the deformation region’s thickness changes from dt to 2δ ), while another portionis because of the elastic energy release at the tool-chip interface. Soon after the formation of theshear band, the effect of elastic energy at the tool-chip interface disappears and the strain rate takesits steady-state value, i.e., 4.98×105 s−1 for Ti-6Al-4V and to 1.54×105 s−1 for Inconel 718. Ta-ble 3.5 summarizes the strain rate magnitudes when cutting Ti-6Al-4V and Inconel 718 in the threedifferent stages. It should be noted that the normal stress at the tool-chip interface may enter theplastic region due to the intense compressive pressure existed at the tool-chip interface. However,assuming elastic response at the tool-chip interface provides an upper-bound for the overshoot inthe strain rate evolution of the shear band since for a certain plastic strain, the stored energy islarger for the fully elastic response compared to the elastic-plastic response. Fig. 3.17 shows thateven for the fully elastic behavior at the tool-chip interface, the stored energy causes only a tran-sient change in the shear band’s strain rate which quickly disappears.Table 3.5: Strain rate magnitudes when cutting Ti-6Al-4V and Inconel 718 at the cuttingspeed vc = 60 m·min−1 in three different stages.Strain RateMaterial Homogeneous Transient Steady-state×105[s−1] ×105[s−1] ×105[s−1]Ti-6Al-4V 0.56 7.55 4.98Inconel 718 0.52 1.65 1.54Regarding the local elastic deformation at the tool-chip interface, two more factors are note-worthy. First, the strain rate’s jump due to the elastic energy release in Ti-6Al-4V is larger than the920 1 2 3 4 5t [ s]02468max [s-1] 105 Ti-6Al-4V(a)0 1 2 3 4 5t [ s]00.511.52max [s-1] 105 Inconel 718(b)Figure 3.17: Effect of elastic unloading at the tool-chip interface on the strain rate evolutionof the shear band at the cutting speed vc = 60 m·min−1 for (a) Ti-6Al-4V and (b) In-conel 718 (aged). The elastic unloading at the tool-chip interface causes an overshootin the strain rate of the shear band when the homogeneous deformation changes toshear localization.jump in the strain rate of Inconel 718 as shown in Fig. 3.17. The nondimensional elastic modulusE¯ is 12.28 for Ti-6Al-4V while 13.39 for Inconel 718 based on Eq. (3.11). The constant (β aˆτˆ2/E¯)in the denominator of Eq. (3.17) is therefore 0.035 for Ti-6Al-4V and 0.021 for Inconel 718. Thisshows that the elastic deformation at the tool-chip interface is comparable when cutting Ti-6Al-4Vand Inconel 718. On the other hand, the term (2δ¯ − β aˆτˆ2/E¯) in the denominator of Eq. (3.17)is much smaller for Ti-6Al-4V than for Inconel 718 due to the smaller shear band’s width in Ti-6Al-4V. Therefore, the strain rate jump is larger in Ti-6Al-4V compared to Inconel 718. This isreasonable noting that since the shear band’s width is larger in Inconel 718 compared to Ti-6Al-4V,the same amount of elastic energy released at the tool-chip interface causes smaller jump in thestrain rate of Inconel 718. Second, according to the classical cutting mechanics, when estimatingthe strain rate during the cutting process, the material being cut affects the prediction of the shearangle through the chip compression ratio r [88]. Therefore, in the classical cutting mechanics the-ory, for a certain uncut chip thickness and the cutting tool geometry, the cutting chips are collected93and their thicknesses are measured to predict the shear angle similar to Table 3.1. Then, the plasticdeformation’s thickness is considered as 0.15-0.2b, where b is the shear plane’s length. Similarly,the homogeneous strain rate in Table 3.5 is quite close for Ti-6Al-4V and Inconel 718. However, assoon as shear localization occurs, the material dependency of the strain rate increases significantlyas shown in Table 3.5. This dependency is because the thickness of the shear band is not onlydependent on the cutting condition but also on the thermo-mechanical properties of the workpiecematerial (see Eq. (2.31)) which differs from Ti-6Al-4V to Inconel 718 (see Table 3.2).Fig. 3.18 shows the simulated strain rate distribution inside the primary shear zone AOB (seeFig. 2.9) when cutting Ti-6Al-4V and Inconel 718 (aged) under the cutting speed vc = 60 m·min−1using Eq. (3.27). Considering Ti-6Al-4V, the localization of the plastic strain rate inside the shearband is clear in Fig. 3.18a, where there is an extreme change in the strain rate magnitude in a verysmall length. On a larger scale (Fig. 3.18b), the distribution of the strain rate inside the band isclear. The symmetric distribution of the strain rate (and subsequently, the displacement and tem-perature) in Fig. 3.18b agrees with the experimental results of Yadav and Sagapuram [132]. Basedon Fig. 3.18, by the formation of the shear band, the upsetting region (see Fig. 3.6) slides similarto a rigid body on the shear band (because of the zero plastic strain rate shown in Fig. 3.18) whilethe band is under intense plastic deformation. The maximum strain rate of Ti-6Al-4V occurs at thecenter of the band with the magnitude of 5.5×105 s−1.Fig. 3.18 also compares the strain rate distributions of aged Inconel 718 with Ti-6Al-4V at thecutting speed vc = 60 m·min−1. The maximum strain rate of Inconel 718 is 1.6×105 s−1 comparedto 5.5×105 s−1 of Ti-6Al-4V. However, the strain rate profile of Inconel 718 is wider than Ti-6Al-4V as shown in details in Fig. 3.18b. This causes that the heat source strength of Inconel 718 to becomparable with that of Ti-6Al-4V for the same cutting speed, which is reflected in the predictedtemperature of Fig. 3.12a.94-20 0 20 40 60 80y [ m]0246 [s-1] 105Inconel 718 (aged)Ti6Al4V(a) .-10 -5 0 5 10y [ m]0246 [s-1] 105Inconel 718 (aged)Ti6Al4V(b)Figure 3.18: Comparison between the strain rate distribution of Ti-6Al-4V and aged Inconel718 at the cutting speed 60 m·min−1. The maximum strain rate of Ti-6Al-4V is ∼5.5×105 s−1 while for aged Inconel 718 is∼ 1.6×105 s−1. The strain rate distributionis wider for Inconel 718 compared to Ti-6Al-4V.Based on Fig. 3.18, the strain rate distribution of the shear band in the primary deformationzone at the steady-state stage is similar to the fluid motion near solid boundaries. If we considerthe strain rate profile similar to the fluid velocity profile between two solid boundaries, the max-imum strain rate occurs at the center of the band which is at the furthest possible distance withrespect to the solid boundaries. Based on the fluid boundary layer theory, the boundary layerthickness (flow velocity thickness) is δ f ∝√ϒ, where ϒ is the momentum diffusivity (kinematicviscosity) of the fluid. For a shear band, the shear band’s thickness is proportional to its thermalconductivity, i.e., δ ∝√k, based on Eq. (2.31). Therefore, two different types of diffusion mech-anisms, i.e., momentum diffusion for the fluid boundary layer and energy (heat) diffusion for theshear band boundary layer, govern on the flow localization. The ratio of the momentum diffusivity(ϒ) to thermal diffusivity (χ = k/(ρc)) is called Prandtl dimensionless number Pr = ϒ/χ . Formetals, the Prandtl number is very small (in the range of 0.01 to 0.001), meaning that the energydiffusion dominates the momentum diffusion. Therefore, unlike fluid boundary layer (which is the95response of the fluid to large velocity gradient), the metallic alloys respond to the large gradient ofvelocity (and, subsequently, strain rate as shown in Fig. 3.18) by energy diffusion mechanisms dueto the very small value of Pr.Table 3.6 presents the experimental measurements and theoretical predictions of strains anddisplacements during cutting of Waspaloy and Ti-6Al-4V at different cutting speeds. The shearband’s width 2δ was obtained from Eq. (2.31), while Eq. (2.32) was employed to predict the shearband’s strain rate γ˙ in. For Waspaloy, the average spacing of shear band at the cutting speed vc = 120m·min−1 was obtained as 96 µm based on Section 2.3.3. Therefore, the time during which a shearband is under loading in the upsetting region (i.e., the region AOB in Fig. 2.9) is ∆t = 84.87 µs.Therefore, the total strain of the shear band is obtained as γ in = 6.59 by employing Eq. (3.22).This causes the total displacement of the shear band to be ( f + s) = γ in× 2δ = 169.74 µm, andsubsequently, the fracture length at the free surface is f = 84.87 µm. Therefore, f is predictedwith around 3% error. The same procedure was adopted in obtaining the strain and displacementof Ti-6Al-4V with 10% error.Table 3.6: Experimental measurements and theoretical predictions of strains and displace-ments when cutting Waspaloy and Ti-6Al-4V at different cutting speeds.Experiment TheoryWorkpiece vc 2δ γave fave 2δ γ˙ in ∆t γ f[m·min−1] [µm] - [µm] [µm] [s−1] [µs] - [µm]Waspaloy 120 31 6 95 29.6 7.76×104 84.87 6.59 97.47Ti-6Al-4V 60 4 53 112 2.74 5.60×105 45.07 45 123.483.6 ConclusionsIn this chapter, a theoretical model along with new experimental evidences are presented for thechip segmentation and shear band formation during cutting of Ti-based and Ni-based metallic96alloys. The main conclusions are given as follows.1. Experimental evidences show that while shear localization is the only type of chip formationduring cutting of Ti-based alloys in the cutting range of 20 to 200 m·min−1, three differenttypes of chip formation occur during cutting of Ni-based alloys in the same cutting speedrange. At low cutting speed (30 m·min−1 for Waspaloy and 40 m·min−1 for annealed In-conel 718), the chip was continuous, while at the mid-range cutting speeds (45 m·min−1for Waspaloy and 60 m·min−1 for annealed Inconel 718), the produced chip contained bothcontinuous and segmented chip morphologies, causing a chaotic behavior during the chipformation mechanism. However, at high cutting speeds, shear-localization dominated, andchip segmentation occurred.2. Based on the experiments, the produced shear bands during cutting of Ti-based alloys roll onthe cutting tool’s rake face by their entire thickness. On the other hand, the produced shearbands in Ni-based alloys were thicker than Ti-based alloys, causing the shear band to rollonly from one edge while the other edge remained straight. The partial rolling of the shearband in Ni-based alloys caused the plastic deformation thickness at the tool chip interface tovary linearly from zero to the thickness of the band.3. The effects of the stored energy at the tool-chip interface on the chip segmentation andshear band’s behavior were investigated. It is shown that due to the local deformation of theworkpiece material, a certain amount of elastic energy is stored at the tool-chip interface assoon as the cutting chip comes into contact with the cutting tool. During the homogeneouscutting stage, this elastic stored energy remains constant since the cutting tool velocity andthe shear velocity of the material are equal. Moreover, as soon as shear localization happensand a shear band initiates in front of the cutting tool, the stored elastic energy is suddenlyreleased, causing a jump in the shear band’s strain rate. However, the jump is temporary,and the elastic energy quickly dissipates, causing the shear band to take its final steady-state97value of strain rate. Due to the transient effect of the stored elastic energy on the shearband’s strain rate, it is unlikely that is the cause of the shear localization as believed in someliterature studies [13–15].4. The displacement during chip segmentation, the developed strain, and the degree of segmen-tation were predicted without the post-mortem measurement of the produced shear bands. Itwas shown that the shear bands could tolerate considerable strain and, subsequently, largedisplacements without recurring to any fracture criterion. Therefore, the developed displace-ment during chip segmentation is due to shear band formation rather than crack nucleationand propagation. This finding can help future FE simulations of chip segmentation to avoidcompensating the low predicted displacement (due to the low mesh density and restrictivespatial resolution) by crack propagation criteria. Including the effect of cutting fluid and cut-ting tool’s edge radius can be a reasonable extension of the proposed model. Finally, sincethe model can capture the effects of different thermo-mechanical parameters, cutting condi-tions, and rake angle on the chip segmentation, it can serve as the primary tool to investigatedifferent methods of controlling shear-localization during cutting processes.98Chapter 4Time-Varying Tool-Chip Contact in CuttingMechanics of Shear Localization4.1 Introduction and Literature ReviewFrom the Merchant’s shear plane model [73, 74] to the Zorev’s [137] and Oxley’s [88] cuttingmodels with finite thickness primary deformation zone, the secondary deformation zone is alwaysdefined as a zone with constant length where the chip locally deforms and moves along the tool’srake face. Over a certain length of the contact, the chip sticks to the rake face due to the highnormal stress, which is called the sticking region. At the sticking region, the friction stress is ap-proximately equal to the yield shear stress of the workpiece material, and the chip moves over thestuck material in the chip velocity direction. As soon as the normal stress drops enough, the stick-ing stops, and the material at the tool-chip interface starts sliding along the rake face. At a certainpoint on the rake face, the chip loses its contact with the cutting tool and becomes separated fromit. The contact forces in the secondary deformation zone are always in equilibrium with the cuttingforces in the primary deformation zone. Since the cutting forces in the primary deformation zoneare always constant (due to the homogeneous cutting of the workpiece material in continuous chip99formation, i.e., steady-state condition), the contact forces and subsequently, the contact lengthsare also constant. This chapter shows that the steady-state contact forces, and subsequently, theconstant contact lengths are not applicable in the case of shear localization.In the literature, no difference in the tool-chip contact mechanism has been identified betweenshear localization and continuous chip formation. The sticking-sliding contact lengths are alwaysconstant, and the contact forces are steady-state and in equilibrium with the forces in the primarydeformation zone [13–15]. However, as investigated in Chapters 2 and 3, the shear force in theprimary deformation zone varies rapidly due to the drop in the load-bearing capacity of the shearband. The question arises as to what are the effects of the drop in the shear force tolerated bythe shear band in the primary deformation zone on the contact forces and lengths. This chapteranswers this question based on both experimental investigations and analytical modeling. In thischapter, the effects of shear localization on the tool-chip interface (secondary deformation zone)are investigated. If the contact forces on the tool-chip interface are predicted, the complete forcesystem in the case of shear localization can be simulated. The prediction of contact forces on thetool-chip interface along with the prediction of sticking-sliding contact lengths result in simulatingthe normal and shear contact stresses at the tool chip interface. The results can then be employedto predict the cutting temperature distribution at the tool-chip interface.From temperature prediction point of view, different thermal models have been proposed topredict the developed temperature at the tool-chip interface during cutting processes. In a seriesof studies, Komanduri and Hou [56, 57] proposed an analytical thermal model for predicting thesteady-state temperature distribution inside the workpiece material in the case of continuous chipformation by assuming different heat sources on the shear plane and tool-chip interface. Moufki etal. [83] proposed an explicit equation for the steady-state temperature distribution at the tool-chipinterface by considering an average friction coefficient and neglecting the heat conduction in the100chip velocity direction (i.e., one-dimensional heat equation). Budak et al. [12] and Barzegar andOzlu [9] investigated the effect of the tertiary deformation zone on the steady-state temperature atthe tool-chip interface in continuous chip formation. Finite difference (FD) numerical scheme isused in the case of interrupted machining, such as milling, with the same assumption of continu-ous chip formation [63]. Islam et al. [49] employed FD method to propose a three-dimensionalthermal model for the case of interrupted cutting for continuous chip formation. In the literature,the theoretical thermal models are dominantly allocated to the case of continuous chip formation.However, based on the described shear localization mechanism, there is a need for a thermal modelto predict the temperature distribution, its maximum value, and its corresponding location at thetool-chip interface in the case of shear localization. Unlike continuous chip formation, the heatproduction rate inside the shear bands rapidly varies with respect to time in shear localization(Chapter 2). Moreover, taking the rolling of the shear band on the tool’s rake face into account(which cannot be justified by shear plane models) is another theoretical challenges in proposing athermal model for shear localization. As illustrated in Chapters 2 and 3, the rolling phenomenoncauses an intimate contact between the shear band and the tool’s rake face which can largely affectthe temperature at the tool’s rake face due to the accumulated temperature inside the shear bands.This chapter presents the cutting mechanics of shear localization by introducing the time-varying nature of the contact at the tool-chip interface. First, it will be experimentally shownthat the sticking and sliding contact lengths change with respect to time in each localization cycle,and therefore, they have a cyclic nature. Second, the normal and frictional forces on the tool’srake face in the sliding region are modeled based on the force equilibrium conditions between theforces on the shear band, and those on the tool’s rake face. It will be shown that since the shearforce on the shear band drops due to the loss in its load-bearing capacity, the sliding contact lengthdecreases with respect to time in each localization cycle (i.e., the shear band is the reason for timevarying sliding length). Third, the force system in the sticking region is modeled by considering101the upsetting region in front of the cutting tool, showing an increase in the sticking length withrespect to time in each localization cycle. The time-varying sticking length explains the high nor-mal pressure at the tool-tip. Fourth, the proposed cutting model is then employed to predict thetransient temperature at the tool-chip interface.The remaining sections of this chapter are organized as follows. Section 4.2 presents the ex-perimental observations of shear localization during cutting processes using high-speed image se-quences and SEM investigations. Next, in Section 4.3, for each sliding and sticking regions, theshear and normal forces on the shear bands and normal and frictional forces on the cutting tool’srake face are predicted. Then, the time-varying contact length for each sticking and sliding regionis obtained. Assuming a general normal stress distribution on the cutting tool’s rake face, the stressdistribution on the cutting tool’s rake face is discussed and predicted. Having obtained the com-plete force prediction acting on the cutting chip, Section 4.4 presents the prediction of the transienttemperate at the tool-chip interface considering the time-varying sticking-sliding contact lengths,time-varying heat production at the tool-chip interface, and the shear band’s rolling phenomenon.The results are presented and discussed in Section 4.5. Finally, the summary of the presented workand the conclusion are provided in Section 4.6.4.2 Mechanisms of Shear Localization and Time-VaryingTool-Chip ContactBoth ex situ and in situ experiments were employed to investigate the contact mechanism duringshear localization. The ex situ orthogonal cutting experiments were performed on the Ti-6Al-4V,Waspaloy, and Inconel 718 (in annealed condition) cylindrical workpieces with uncoated carbideinserts (zero degree rake angle, α = 0◦, and 10 µm edge radius, r0 = 10 µm) as in Section 2.2.Both virgin and metallurgically prepared chips (see Section 3.2) were investigated to identify dif-102ferent aspects of shear localization. For the ex situ experiments, Nikon Epishot 300 Inverted opticalmicroscope (with the magnifications of 50×, 100×, 200×) provided ≈ 0.3 µm per pixel resolu-tion (for the magnification of 200×), and Olympus Lext 3000 confocal scanning laser microscope(with the magnifications of 500× and 1000×) provided ≈ 0.12 µm per pixel resolution.The in-situ chip formation analyses are based on the cutting experiments performed by Saga-puram et al. [104] with high-speed imaging. In order to prevent the out-of-plane flow (satisfyingplane-strain condition), a glass plate was used to constraint the side surface of the workpiece ma-terial. The inter-frame interval of imaging is 25 µs and ten frames in each localization cycle iscaptured in average. Employing the produced high-speed images, Sagapuram et al. [103, 104]focused on the experimental study of a single shear band’s propagation and formation in front ofthe cutting tool. They concluded that a weak zone occurs in front of the cutting tool starting fromthe tool-tip to the free surface of the workpiece material and then, the shear band happens at theweak zone by large plastic deformation localized in its inside. On the other hand, this chapterfocuses on the interaction between the shear band and the tool rake face after its formation, in-cluding shear band bending, rolling, and sliding. The high-speed imaging results (483 frames intotal) are employed to investigate the behavior of the shear bands after they form, and their effectson the cutting mechanics, including the contact lengths. The high speed images were taken byCMOS camera equipped with an optical microscope to provide a total resolution of 1.4 µm perpixel. In all cases (whether ex situ or in situ), the measurements were rounded to the closest integer.Fig. 4.1 shows the high-speed image sequence of shear localization during orthogonal cuttingof Ti-6Al-4V at the cutting speed vc = 0.65 m·s−1, uncut chip thickness d = 200 µm, rake angleα = 0◦, and edge radius r0 = 10 µm with markers added to qualitatively describe the shear lo-calization process,. The process description starts from the time instant when the shear band SB1occurs in front of the cutting tool as shown in Fig. 4.1a. At this time instant, the shear stress inside103the shear band SB1 drops due to the material instability, and therefore, by assuming momentumequilibrium condition between the forces acting on the cutting chip [4], the normal force on thecutting tool’s rake face drops subsequently. Due to the decrease in the normal force acting on thetool’s rake face, the entire tool-chip contact length is under sliding condition as can be seen fromFig. 4.1a to Fig. 4.1b where the entire cutting chip moves in the chip velocity direction (in thevch direction). After the formation of SB1, a fresh workpiece material starts piling up in front ofthe cutting tool for the next shear band to occur (upsetting stage). At first, the initial contact ofthe fresh workpiece material and the cutting tool is at the tool-tip. Although the 1.4 µm per pixelresolution of the in situ imaging is not adequate to focus on the tool-tip, the wedge-like shape ofthe fresh workpiece material in front of the cutting tool due to the oblique orientation of the shearband with the shear angle φ shows that the early contact of the shear band and the cutting tool startsfrom the tool-tip (see Figs. 4.1a and 4.1b). During upsetting, the previous shear band rolls on thecutting tool’s rake face as illustrated in Fig. 4.1b. The rolling process in titanium and nickel alloyshave been experimentally shown and discussed in Chapters 2 and 3 for Ti-6Al-4V, Inconel 718,and Waspaloy (See also the laser micro-graph of Fig. 4.13 in Section 4.5). Therefore, the upsettingof the workpiece material and the rolling of the shear band SB1 in front of the cutting tool are thetwo main characteristics of this stage which happen simultaneously.With further advancement of the cutting tool, more material piles up in its front, and subse-quently, the sticking contact length increases further as shown in Fig. 4.1c. Fig. 4.1c shows aframe of chip formation 275 µs after the SB1 onset. At this stage, the sticking length has increasedfurther, and the shear band has rolled on the cutting tool’s rake face more. The amount of thematerial piling up in front of the cutting tool has also clearly increased in Fig. 4.1c. By further ad-vancement of the cutting tool, enough material piles up in its front for the next shear band to occuras shown in Fig. 4.1d. In Fig. 4.1d, the new shear band SB2 is about to occur in front of the cuttingtool. At this stage, the previous shear band (SB1) is at the exit border of the primary deformation104zone, and the new shear band SB2 is close to initiate in front of the cutting tool. As soon as SB2initiates, its shear stress drops, and subsequently, the normal force on the cutting tool’s rake facedrops because of the momentum equilibrium condition of the cutting chip. Accordingly, the con-tact length returns back to the sliding nature, and the cutting chip slides on the cutting tool’s rakeface again with a new cycle starting to happen. This process repeats each time a shear band occursin front of the cutting tool, causing the cyclic formation of the shear bands and the time-varyingnature of the tool-chip contact.To describe the above process in more detail, particle image velocimetry (PIV) analysis wasperformed to map the material flow during the cutting process both qualitatively and quantitatively.PIV is a flow visualization technique where the motion of a medium (mostly fluids) is analyzedby tracking the instantaneous location of seeded particles via the captured high-speed images [1].The motion of the seeded particles is then employed to calculate the full-field velocity distributionin the medium. PIVlab, which is a GUI-based open source tool [117], is employed to perform thePIV analyses on 484 captured frames of the shear localization process. Each PIV analysis con-sists of three different steps: image pre-processing, image evaluation, and post-processing. In thepre-processing phase, the contrast of the high-speed images was enhanced by adopting ContrastLimited Adaptive Histogram Equalization (CALHE) to improve the probability of detecting thevalid experimental vectors [117]. In the image evaluation phase, the Fast Fourier Transform (FFT)approach, which has been implemented in PIVlab, was used to obtain the correlation matrix inthe frequency domain with low computational cost. Finally, in the post-processing phase, the lowcontrast filter is used to filter out the low contrast background signals.105Sliding(a) t = 0 µsSlidingStickingRollingUpsetting(b) t = 125 µsSlidingStickingRollingUpsetting(c) t = 275 µsSlidingRollingStickingPotentialUpsetting(d) t = 400 µsSliding(e) t = 475 µsFigure 4.1: High speed image sequence of shear localization in cutting of Ti-6Al-4V at vc =0.65 m·s−1 and d = 200 µm without any image processing technique. At first, the tool-chip contact length is under sliding condition due to the shear stress drop inside SB1.At t = 125 µs, new workpiece material starts piling up (upsetting) in front of the cuttingtool, and therefore, the sticking length starts from the tool-tip. During the upsetting, theshear band SB1 rolls on the cutting tool rake face causing an intimate contact betweenthe shear band SB1 and the cutting tool’s rake face. As the cutting tool advances further,the sticking length increases and the sliding length decreases until the next shear band(SB2) occurs. Finally, the shear band SB2 initiates in front of the cutting tool and thecontact length returns back to its initial state at which all the contact length is of thesliding nature.Fig. 4.2 represents the flow motion pattern by PIV analysis for cutting of Ti-6Al-4V in onecycle of segmentation at the cutting speed vc = 0.65 m·s−1 and uncut chip thickness d = 200 µm(same segmentation cycle as Fig. 4.1 but with higher magnification to focus on the sticking zone).106The velocity vectors, whose lengths are proportional to the magnitude of velocity and directionsshow the direction of velocity, are clear in Fig. 4.2. At t = 0 µs, a shear band has formed in frontof the cutting tool (this can be observed by large displacement at the free-surface-side of the shearband), and therefore, almost all the plastic motion is localized inside the band. Therefore, theworkpiece material approaches the cutting tool with the cutting velocity vc, followed by shearingacross the shear band with shear velocity v, and then, moves along the cutting tool’s rake face withthe chip velocity vch. At t = 225 µs, the upsetting process is clear where the material at the tool-chip interface in the sticking zone is under stationary condition (i.e., zero chip velocity, vch = 0).This can be observed by the velocity vectors with almost negligible lengths in the sticking zone.In the upsetting region and far from the tool’s rake face, the velocity vector shows the upsettingdirection in front of the cutting tool. As the cutting tool moves forward (Fig. 4.2c), more materialhas piled up inside the upsetting region, and the sticking length (the length with zero chip velocity)has increased clearly. By further advancement of the cutting tool (Fig. 4.2d and Fig. 4.2e), theupsetting region has enlarged even more with further increase of the sticking length. Finally, whenenough material accumulates in the upsetting stage, a new shear band occurs in front of the cuttingtool and the tool-chip contact returns back to the sliding condition as clearly shown in Fig. 4.2f.At this moment, the sticking length is in its minimum value and the entire segment moves in thedirection of chip velocity vch same as the beginning of the segmentation cycle. Although the ve-locity vectors are almost negligible at the tool-chip interface in the sticking region, they produce aclockwise vorticity (spin) according to Fig. 4.2d and Fig. 4.2e. This clockwise vorticity shows therolling direction of the material (and subsequently, the shear band) on the cutting tool’s rake faceat the tool-chip interface.Fig. 4.3 shows the quantitative velocity distribution map in the chip flow direction (i.e., in thedirection of vch) in five different frames. This is the same segmentation cycle as Figs. 4.1 and 4.2.In Fig. 4.3, the images’ backgrounds as well as the cutting tool are masked (shown by red surfaces)107(a) t = 0 µs (b) t = 225 µs (c) t = 375 µs(d) t = 425 µs (e) t = 450 µs (f) t = 575 µsFigure 4.2: Flow distribution map obtained from PIV analysis when cutting Ti-6Al-4V underthe cutting speed vc = 0.65 m·s−1 and uncut chip thickness d = 200 µm. The greenarrows show the distribution of the velocity vectors within the workpiece material andcutting chip. At first, the entire cutting chip slides on the cutting tool’s rake face. Asmore material piles up in front of the cutting tool, the sticking region (i.e., the regionwith zero velocity) increases as shown in (b), (c), (d), and (e). Finally, when enoughmaterial piles up in front of the cutting tool, a new shear band occurs and the contactlength returns to its initial state, where all the contact length is of the sliding type.in order to avoid any noise signal effects on the results. At Fig. 4.3a, the cutting velocity vc = 0.65m·s−1 transforms to the chip velocity vch = 0.523 m·s−1 across the shear band. In Fig. 4.3b, thezero velocity region is clearly observable by the dark blue color near the tooltip with the lengthof 131.28 µm. As the cutting tool moves forward (Fig. 4.3c), the sticking length (the region withzero chip velocity at the tool-chip interface) increases to the value of 182.10 µm, showing a clearincrease in the sticking length. In Fig. 4.3d, the sticking length has increased to 223.91 µm, andfinally, the sticking length drops (showing a new shear band has occurred in front of the cutting108tool), and the whole chip slides in the chip velocity’s direction (see Fig. 4.3e). The describedprocess with quantitative results shows a clear change in the sticking contact length during shearlocalization. 00.10.20.30.40.5 [m/s](a) t = 0 µs 00.10.20.30.40.5 [m/s](b) t = 225 µs 00.10.20.30.40.5 [m/s](c) t = 375 µs 00.10.20.30.40.5 [m/s](d) t = 425 µs 00.10.20.30.40.5 [m/s](e) t = 575 µsFigure 4.3: Distribution map of the velocity in the chip flow direction. The increasing lengthof the sticking region is clear in frames (b), (c), and (d) with dark blue color nearthe tool-tip. At first (frame (a)), the sticking length is negligible while as the cycleprogresses, the sticking length increase to 131.28 µm (frame (b)), to 182.10 (frame(c)), and finally to 223.91 µm (frame (d)). At the end (frame (e)), a new shear bandoccurs in front of the cutting tool and the sticking length returns to its initial state.From Fig. 4.3, it is clear that while the material piles up in front of the cutting tool in the upset-ting stage (Figs. 4.3b to 4.3c), the previously formed segments move in the chip velocity direction.This is possible only if the piled-up region pushes the previously formed chip upward during the109upsetting stage. Another observation based on Figs. 4.2 and 4.3 is as follows. From both exper-imental and theoretical evidences, Chapters 2 and 3 showed that during shear localization, shearbands roll on the cutting tool’s rake face. They explained the mechanism of rolling in differentmetallic alloys and showed that shear band rolls all the way through the segment’s thickness, andtherefore, it is the shear band which is in the direct contact with the cutting tool rather than thesegment itself. Returning back to Fig. 4.2, the connection between the upsetting and shear bandrolling is now clear. The upsetting of the material in front of the cutting tool is accompanied bythe rolling of the shear band on the cutting tool’s rake face because the upsetting region pushes theshear band SB1 in the upsetting direction as shown in Fig. 4.4 which schematically shows the up-setting process and rolling of the shear band on the cutting tool. In Fig. 4.4, the red arrows near thetool-chip interface show horizontal upsetting direction. Note that since the contact lengths changewith respect to time as illustrated previously, they are considered to be functions of time as shownin Fig. 4.4. Therefore, in Fig. 4.4, lsl(t) and lst(t) represent the time-varying sliding and stickinglengths, respectively.CuttingToolWorkpieceSegmentFigure 4.4: The upsetting direction of the new workpiece material in front of the cutting tool.Note that the upsetting direction is horizontal near the tool-chip interface, causing thestationary contact between the tool’s rake face and the workpiece material at the tool-chip interface. Note that everything above SB1 is being pushed upward by the upsettingregion.Fig. 4.5 shows the SEM images of the Ti-6Al-4V, Waspaloy, and Inconel 718 (in annealed110condition) samples obtained from the ex situ experiments. According to Fig. 4.5, the stationarycondition between the tool and the material in the time-varying sticking zone causes local out-of-plane deformation (henceforth, is called “bulging”) at the tool-chip interface due to the incom-pressibility of metallic alloys [16]. Note that since the ex situ orthogonal cutting experiments wereperformed by the turning operation, bulging occurs at the free surface (where plane stress conditionapplies) while the plane strain condition governs in most section along the width of cut. Accordingto Fig. 4.5, bulging occurs during shear localization irrespective of the material being cut. Thisshows that bulging is due to shear localization, and therefore, the process described in Figs. 4.1to 4.3 applies to all the metallic alloys showing shear localization during cutting processes.111(a) Ti-6Al-4V, vc = 1.33 m·s−1, d = 120 µm (b) Waspaloy, vc = 1.43 m·s−1, d = 120 µm(c) Inconel 718, vc = 1.5 m·s−1, d = 120 µmFigure 4.5: Out-of-plane deformation (or “bulging”) of the material near the tool-chip inter-face due to the stationary condition between the cutting chip and the tool’s rake face inthe sticking region. The material sticks to the cutting tool but it needs to accommodatethe plastic deformation imposed by the advancement of the tool. Therefore, it deformsin out-of-plane direction the tool-chip interface due to the incompressibility of metallicalloys,.4.3 Modeling of Cutting Forces and Time-Varying ContactLengthsThe cutting mechanics model for predicting the cutting forces and time-varying tool-chip contactlengths due to shear localization is proposed in this section based on the shear localization mech-anism described in Section 4.2 and summarized schematically in Fig. 4.6. According to Fig. 4.6,the sticking and sliding lengths change with respect to time as the cutting tool moves forward. In112the beginning, the sticking length is in its minimum value and increases with respect to time untilit reaches its maximum value at the end of the segmentation cycle. In the meantime, the slidinglength starts from its maximum value at the beginning of the segmentation cycle and decreasesuntil it reaches its minimum value at the end of the segmentation cycle. This is due to the shearforce drop on the shear band SB1 as will be explained in Section 4.3.1.CuttingToolWorkpieceSegment(a) Initial contactCuttingToolWorkpieceSegment(b) Early contactCuttingToolWorkpieceSegment(c) Late contactCuttingToolWorkpieceSegment(d) Final contactFigure 4.6: The contact between the fresh workpiece material (which piles-up in front of thecutting tool) and the cutting tool’s rake face. At first (Fig. 4.6a), the contact lengthis small and located at the tool-tip. As the cutting tool advances, the contact lengthincreases until it reaches its maximum value at which the next shear band (SB2) initiates(Fig. 4.6d). This time-varying sticking contact between the workpiece material and therake face is accompanied by sliding of the previous shear band (SB1) and the remainderof the chip along the tool’s rake face.Fig. 4.7 shows the schematic of force generation associated with shear-localized chip forma-tion. The forces on the rake face are divided into two different regions: (i) sliding region and(ii) sticking region. The reason for this division is that the formation of shear band SB1 causesa discontinuity in the load-bearing capacity of the chip over the tool-chip contact length as wellas the separation of the upsetting region (below SB1) from the remainder of the cutting chip. InSections 4.3.1 and 4.3.2, each region is discussed in detail.113CuttingToolWorkpieceFigure 4.7: The force system acting during shear-localized chip formation. The forces aredivided into two groups, namely, the forces on the sliding region (Fv1 and Fu1) and theforces on the sticking region (Fv2 and Fu2).4.3.1 Sliding regionThe analysis starts by obtaining the shear force Fs1(t) (which is the main cause of time-varyingsticking-sliding contact lengths) acting on the shear band SB1 shown in Fig. 4.7. The shear forceFs1(t) is expressed asFs1(t) = w∫ λ (t)0τ in(t)dλ = τ in(t)λ (t)w, (4.1)where τ in(t) is the time-varying stress of the shear band SB1, λ (t) is the time-varying active lengthof the shear band SB1 (see Fig. 4.7), and w is the width of cut. Note that the qualitative evolution ofτ in(t) with respect to time is shown in Fig. 3.9, while Section 2.3 presents its analytical prediction.In Eq. (4.1), the shear stress on the shear band, i.e., τ in(t), is assumed to be distributed evenly overthe shear band’s length [126].As far as the shear band SB1 is under loading (i.e., in the upsetting region AOB), its lengthtolerating the shear force Fs1(t) changes from b to λ (t) (see Fig. 4.7). This change in length (whichis defined as the shear band’s active length) is due to the large displacement f at the free surface side114of the shear band. Note that this large displacement is not due to the crack propagation as shown inChapter 4. In fact, the shear band itself can causes significantly large plastic displacement withoutany fracture presumption. The active length of the shear band tolerating the shear force Fs1(t) isλ (t) = b− f (t) = b[1−ξ (t)]. (4.2)In Eq. (4.2), ξ (t) = f (t)/b is the segmentation degree of the chip, where f (t) is the displacementlength shown in Fig. 4.7. The displacement length f (t) was analytically obtained in Section 3.4.Through experimental observations, it was shown that f (t) is equal to the sticking length lst(t) dueto the mass continuity. Eq. (4.2) shows that as the shear band SB1 moves along the cutting tool’srake face, its active length decreases. In its extreme case where the active length becomes zero(i.e., λ (t) = 0), the chip becomes discontinuous.Based on this description, the shear force on the shear band SB1 (i.e., Fs1(t)) shows that asthe shear band travels inside the deformation region, it loses its load-bearing capability. FromEq. (4.1) point of view, this can be deduced from the decrease in both τ in(t) and λ (t) with respectto time. In the sliding region and in the absence of inertia, the normal and shear forces on theshear band SB1 (Fn1 and Fs1, respectively) are in equilibrium with the normal and friction forceson the cutting tool in the sliding region (Fu1 and Fv1, respectively) as shown in Fig. 4.7. AssumingCoulomb’s friction law in the sliding region, the tangential force on the rake face in the slidingregion is Fu1(t) = µFv1(t) where µ is the nominal friction coefficient [81]. The friction coefficientµ is assumed to be constant and independent of temperature [17, 81, 89]. Projecting the actingforces on the directions parallel and perpendicular to the shear band SB1 (Fig. 4.7) and satisfyingthe force equilibrium conditions in those directions, the normal force on the rake face, i.e., Fv1(t),115and the normal force on the shear band SB1, i.e., Fn1(t), are obtained, respectively, asFv1(t) =Fs1(t)cos(φ −α)−µ sin(φ −α) , (4.3)Fn1(t) = Fu1(t)cos(φ −α)+Fv1(t)sin(φ −α)= [µ cos(φ −α)+ sin(φ −α)]Fv1(t). (4.4)In Eq. (4.3), the shear force Fs1(t) is predicted from Eq. (4.1). Note that in Eqs. (4.3) and (4.4), theshear angle is constant and is the angle of the plane where the shear band occurs, which can be ob-tained based on the maximum thickness of the cutting chips (Section 3.5). In obtaining Eqs. (4.3)and (4.4), the effect of inertia has been neglected similar to the other studies on the cutting me-chanics, such as Refs. [13–15]. The discussion on effect of inertia via dimensional analysis waspresented in Section 2.3.2.In order to obtain the contact stresses from the predicted contact forces, the distributions of thenormal and shear stresses on the cutting tool’s rake face (i.e., the stress distributions because ofFv1(t) and Fu1(t)) as well as the stress distribution on SB1 as a result of Fn1(t) must be known inadvance. Likewise, prediction of the sliding contact length lsl(t) depends on the stress distributionon the rake face (the contact lengths and stresses are interdependent). The constant normal stressdistribution assumption on the primary deformation zone (i.e., constant stress distribution becauseof Fn1(t)) is customary in the machining literature [4, 110] and hence, is adopted in this study.However, the normal and shear stress distributions on the cutting tool’s rake face are assumed tohave a general form as shown in Fig. 4.8. In this way, different profiles of normal stress distributionon the rake face can be assumed and generated. Fig. 4.8 shows the normal stress distribution onboth sliding and sticking regions. Two different Cartesian coordinate systems are adopted: (i)the Lagrangian coordinate system xˆ-yˆ, which moves with the boundary between the sticking andsliding regions, is used for deriving the stress equations governing on the sliding region; (ii) the116Eulerian coordinate system x-y, with its origin set at the tool-tip, is employed for obtaining thegoverning stress equations in the sticking region. The transformation equations of xˆ= x− lst(t) andyˆ = y exist between the two coordinate systems. Based on Fig. 4.8, the normal stress distributionin the sliding region, i.e., σv1(xˆ, t), takes the general form ofWorkpieceFigure 4.8: The normal stress distribution on the sliding and sticking regions (i.e., σv1(xˆ, t)and σv2(x, t), respectively). Two different Cartesian coordinate systems are employed:(i) the Lagrangian coordinate system xˆ-yˆ and the Eulerian coordinate system x-y. Thetime-varying stress amplitude in the sliding region is σˆ0(t) while σˆ0(t)+σ0(t) in thesticking region. Note that there is a slope discontinuity in the normal stress distributionon the rake face at the location of the shear band SB1 (i.e., the border between thesticking and sliding regions.)σv1(xˆ, t) = σˆ0(t)(1− xˆlsl(t))ζ0≤ xˆ≤ lsl(t), (4.5)where σˆ0(t) is the time-varying amplitude of the normal stress in the sliding zone, and ζ is thedistribution exponent controlling the stress decay when xˆ increases. For 0 ≤ ζ ≤ 1, there is asmooth decay at the stress profile while for larger ζ , the normal stress is accumulated close toxˆ = 0. Eq. (4.5) provides the normal stress distribution with the sliding length lsl(t) according tothe fact that the normal stress is in its minimum value at the top boundary (i.e., when xˆ = lsl(t))and reaches its maximum value at the lower boundary (i.e., when xˆ = 0) as shown in Fig. 4.8. The117normal stress σv1(xˆ, t) is written in terms of Fv1(t) asw∫ lsl(t)0σv1(xˆ, t)dxˆ = Fv1(t). (4.6)Inserting Eq. (4.5) into Eq. (4.6), and integrating the result:wσˆ0(t)∫ lsl(t)0(1− xˆlsl(t))ζdxˆ =wlsl(t)1+ζσˆ0(t) = Fv1(t). (4.7)In Eq. (4.7), the term lsl(t)σˆ0(t)1+ζ is the area under the normal stress distribution in the sliding region(see Fig. 4.8). Therefore, the stress amplitude σˆ0(t) is written in terms of Fv1(t) based on Eq. (4.7)asσˆ0(t) =(1+ζ )Fv1(t)wlsl(t). (4.8)Note that through Eqs. (4.1), (4.3) and (4.8), the amplitude σˆ0(t) is related to the shear band’sstress τ in(t). Since the shear band’s stress τ in(t) drops in each cycle of the chip segmentation asthe shear band evolves (see Fig. 3.9), the amplitude σˆ0(t) also drops as the shear band occurs inthe upsetting region. According to Eq. (4.8), the amplitude σˆ0(t) is obtained explicitly in terms ofthe known constants and variables based on the process condition except the sliding length lsl(t)which still needs to be obtained. Taking moments around xˆ = 0, at any point xˆ on the rake face,the normal force per unit width is σˆv1dxˆ, and its moment about xˆ = 0 is xˆσˆv1dxˆ. Hence, the totalmoment of Fv1 about xˆ = 0 isσˆ0(t)∫ lsl(t)0xˆ(1− xˆlsl(t))ζdxˆ =l2sl(t)ζ 2+3ζ +2σˆ0(t). (4.9)This moment must be balanced by the moment of the normal stress on the shear band, i.e., the118moment of Fn1(t) about xˆ = 0, expressed asl2sl(t)ζ 2+3ζ +2σˆ0(t) =Fn1(t)λ (t)2w(4.10)Therefore, Eqs. (4.8) and (4.10) provide two explicit equations for obtaining lsl(t) and σˆ0(t) in thesliding region. Omitting σˆ0(t) from Eqs. (4.8) and (4.10), the time-varying sliding contact lengthcan be obtained in terms of the forces aslsl(t) =(ζ 2+3ζ +2)λ (t)Fn1(t)(1+ζ )Fv1(t). (4.11)From Eq. (4.11), it can be seen that the distribution exponent ζ affects the sliding contact length. Itmeans that the assumption of the normal stress profile at the rake face changes the sliding contactlength lsl(t), which is reasonable since different distribution profiles generate different momentsaround xˆ = 0. Having obtained the normal stress distribution, the shear stress distribution alongthe sliding region, i.e., τu1(xˆ, t) is obtained by employing the Coulomb’s friction law asτu1(xˆ, t) = τ0(t)(1− xˆlsl(t))ζ0≤ xˆ≤ lsl(t), (4.12)where τ0(t) = µσ0(t). In Eq. (4.12), the sliding length lsl(t) is obtained from Eq. (4.11).4.3.2 Sticking regionFig. 4.9 shows the free body diagram of the sticking region during shear-localized chip formation.Three different sets of forces are acting on the material piling up in front of the cutting tool: (i)reaction forces Fs1 and Fn1 which are being applied by the sliding region through the shear bandSB1 as shown in Fig. 4.7, (ii) interaction forces Fu2 and Fv2 at the tool-chip interface which arerespectively parallel and perpendicular to the cutting tool’s rake face, and (iii) shear and normal119forces at the plane OA, i.e., Fs2 and Fn2, applied by the workpiece material.Figure 4.9: The forces acting on the deformation region AOB: Fv2(t) and Fu2(t) are appliedby the cutting tool’s rake face in the sticking region, Fs1(t) and Fn1(t) are applied bythe shear band SB1, and Fs2(t) and Fn2(t) are applied by the fresh workpiece materialin front of the cutting tool.The analysis of the cutting forces and stresses in the sticking zone is more complicated thanthe sliding zone because: (i) the contact length in the sticking zone (i.e., lst(t)) cannot be obtainedfrom moment equilibrium around the cutting tool’s edge anymore. This is because the materialin the very neighborhood of the tool’s rake face is rotating (or in another words, the shear bandis rolling) on the cutting tool’s rake face. This rotation is local and limited to the region near thecutting tool’s rake face; (ii) in addition to the shear and normal forces on the plane OA (i.e., Fs2(t)and Fn2(t), respectively) and the tangential and normal forces on the cutting tool rake face in thesticking zone (i.e., Fu2(t) and Fv2(t), respectively), two additional forces (i.e., Fs1(t) and Fn1(t))are being applied by the previously formed shear band, i.e., SB1, in the directions parallel andperpendicular to the top border of the piling up region, which must be also taken into account (seeFig. 4.9).To address the first issue (i.e., predicting the time-varying contact length lst(t)), the experi-mental observations presented in Section 4.2 as well as the shear band spacing (ds) prediction in120Section 2.3.3 are employed. According to Fig. 4.1a, the sticking contact area is significantly smallat the initial contact of the cutting tool with the new workpiece material (in the beginning of theupsetting stage) defined as lminst . As the cutting tool advances further, the workpiece material startspiling up in its front (simultaneously, the shear band rolls on the cutting tool’s rake face) withthe speed of vch until the sticking contact length reaches its maximum value lmaxst . After that, anew shear band initiates in front of the cutting tool, and the sticking length return to its minimumvalue lminst . Based on this description of the process, the time-varying sticking contact length in oneperiod of segmentation can be expressed aslst(t) = lminst + vcht. (4.13)The sticking contact length lst(t) ideally starts from zero (when the cutting tool’s edge radiusis zero). However, in reality, the cutting tool always has an edge radius, defined as r0, whichmakes the sticking contact length start from a non-zero value. Therefore, the term lminst is addedto Eq. (4.13) in order to take the edge radius effect on the contact length into account. Fig. 4.10demonstrates the relation between the minimum sticking length lminst and the edge radius of thecutting tool r0. As the workpiece material moves towards the cutting tool’s edge, the materialabove the stagnation point O′ is converted into the machined chip while the material below O′ endsup at the finished surface of the machined workpiece. Based on Fig. 4.10, lminst is written aslminst = r0(π2−η), (4.14)where η is the stagnation angle, i.e., the angle between the stagnation point O′ and the point wherethe edge reaches the flank face [42] as shown Fig. 4.10. Reminding that the shear band spacingds is the smallest distance between two subsequent shear bands, the maximum sticking length lmaxstcan be written as (see Fig. 4.9)121SegmentFigure 4.10: The minimum sticking length lminst between the piled-up region and the cuttingtool’s rake face due to the cutting tool’s edge radius. The part of edge from the stag-nation point to the horizontal axis determines lminst .lmaxst =dscos(φ −α) . (4.15)According to Eq. (4.13), in each cycle of shear band formation, the sticking length lst(t) startsfrom lminst and increases with the speed of vch until it reaches lmaxst . In the meantime, the slidinglength lsl(t) starts from its maximum value and decreases until it reaches its minimum value at theend of the cycle based on Eq. (4.11). Then, the cycle repeats again. Having obtained the time-varying sliding and sticking lengths respectively from Eqs. (4.11) and (4.13), the total interactionlength of the shear-localized chip with the cutting tool, i.e., ltot, is defined asltot(t) = lsl(t)+ lst(t). (4.16)In continuous chip formation, different methods are employed for obtaining the total contact lengthbetween the cutting chip and the cutting tool. It is obtained based on the equilibrium between thecutting forces on the shear plane and the forces at the tool-chip interface [4] or using slip-line fieldtheory [64]. In all these cases, the contact length between the cutting chip and the cutting toolis always constant. However, Eqs. (4.11), (4.13) and (4.16) show that in the shear-localized chipformation, the sliding, sticking, and total contact lengths (i.e., lsl(t), lst, and ltot(t), respectively)122are all time-dependent, which is in accordance with the high-speed imaging results of Section 3.2.After obtaining the time-varying contact length of the sticking region, i.e., lst(t), the contactforces and stresses are investigated. For the sake of explanation, the process for a special case inwhich the shear angle φ is equal to the rake angle α (i.e., φ = α) is analyzed first. This causesthe frictional force on the rake face Fu2(t) to be in the direction of the normal forces Fn1(t) andFn2(t) while the normal force on the rake face Fv2(t) to be in the direction of the shear forcesFs1(t) and Fs2(t) (see Fig. 4.9). At the early stage of process when the shear band SB1 has justoccurred, the localization has just started and the shear stress at the shear band τ in(t) starts fromthe homogeneous stress τh(t). Note that τh(t) is the homogeneous shear stress due to homoge-neous deformation of the material which can be obtained from Eq. (2.7) and is observable in zone1 of Fig. 3.9. Moreover, at the starting point, the displacement at the shear band is negligible, i.e.,f (t)≈ 0, and therefore, λ (t) = b. Hence, according to Eq. (4.1), the shear force on the shear bandSB1 is Fs1(t) = τ in(t)λ (t)w ≈ τh(t)bw at the initial contact. Meanwhile, at the early stage of thecontact, the border OA is under homogeneous deformation since no new shear band has occurredyet. Therefore, the shear force on the plane OA is Fs2(t) = τh(t)bw due to the homogeneous defor-mation. Because of the force equilibrium in the plane OA direction, Fv2(t) = Fs2(t)−Fs1(t) ≈ 0,meaning that the normal force in the sticking region is close to zero in the beginning of the upset-ting stage. However, the shear force on the shear band SB1, i.e., Fs1(t), quickly starts decreasingdue to the material instability in the shear band as shown schematically in Fig. 3.9 while the shearforce Fs2(t) is still in the homogeneous state and approximately constant (because no new shearband has occurred yet). Therefore, the normal force Fv2(t) = Fs2(t)−Fs1(t) increases rapidly asthe cutting tool moves forward. The increase in Fv2(t) continues so far as the shear force Fs1(t)decreases and the sticking length lst(t) increases (i.e., until the end of the segmentation cycle).Similar trend exists for the frictional force Fu2(t) = Fn2(t)−Fn1(t). At the early stage of the stick-ing contact, Fn2(t)≈ Fn1(t) and therefore, Fu2(t)≈ 0 (see Fig. 4.9). However, soon after the stress123drops inside the shear band SB1, the normal force Fn1(t) drops based on Eq. (4.4) while Fn2(t)is almost constant (due to homogeneous deformation). Therefore, Fu2(t) increases rapidly as thesegmentation cycle continues further.Considering the general case in which the shear angle φ is not necessarily equal to the rakeangle α , the force equilibrium in the direction of plane OA is (see Fig. 4.9)[Fs2(t)−Fs1(t)]−Fv2(t)cos(φ −α)+Fu2(t)sin(φ −α) = 0. (4.17)Rearranging Eq. (4.17) and writing Fv2(t) with respect to the known forces Fs2(t), Fs1(t), andFu2(t), it is obtained asFv2(t) = [Fs2(t)−Fs1(t)]sec(φ −α)+Fu2(t) tan(φ −α). (4.18)Satisfying the force equilibrium in the direction normal to the plane OA and the shear band SB1,the normal force Fn2(t) is obtained asFn2(t) = Fn1(t)+Fv2(t)sin(φ −α)+Fu2(t)cos(φ −α). (4.19)Having obtained the sticking contact length and the forces acting on the piling-up material, thenormal and shear stresses in the sticking zone is obtained. The normal stress on the sticking region,i.e., σv2(x, t), is assumed to have the general form ofσv2(x, t) = σ0(t)(1− xlst(t))ζ+ σˆ0(t) 0≤ x≤ lst(t), (4.20)where σ0(t) is the time-varying change of the normal stress amplitude in the sticking region. Ac-cording to Eq. (4.20) and as shown in Fig. 4.8, in the sticking zone, the normal stress σv2(x, t) does124not start from zero at x = lst(t) but from σˆ0(t) which is the amplitude of the normal stress in thesliding region. This is because of the fact that the shear band SB1 is still capable of tolerating theshear stress with the time-varying magnitude of τ in(t). By force equilibrium at the shear band’slocation (i.e., at x = lst(t)), the normal stress on the top and bottom boundaries of the shear bandSB1 must be equal. Therefore, according to Eq. (4.5), the normal stress in the sliding zone startsfrom zero at xˆ = lsl(t) and reaches σˆ0(t) at xˆ = 0 (i.e., the shear band’s location). In the stickingzone, the normal stress starts from σˆ0(t) (at the shear band’s location) at x = lst(t) and reachesσ0(t)+ σˆ0(t) at the tooltip (x= 0) as shown in Fig. 4.8. The normal force at the tool-chip interfacein the sticking zone is written asw∫ lst(t)lminσv2(x, t)dx = Fv2(t). (4.21)Inserting Eq. (4.20) into Eq. (4.21), the stress amplitude σ0(t) is obtained asσ0(t) =lst(t)ζ (1+ζ )(lst(t)− lminst )1+ζ[Fv2(t)w− σˆ0(t)[lst(t)− lminst]](4.22)In the early stage of contact, the sticking length lst(t) is close to the minimum length lminst (seeEq. (4.13)). Therefore, the denominator term in Eq. (4.22), i.e., lst(t)− lminst , is considerably small,causing the normal stress amplitude σ0(t) to take a large value. Therefore, the cutting tool is undersignificant pressure at the beginning of each cycle. When multiple segmentation cycles happen, thetool-tip is periodically under a large stress fluctuation, causing tool fatigue and finally, breakage,which justifies the experimental observations of different studies in the literature, including Refs.[50, 51, 55, 107].1254.4 Temperature Distribution at the Tool-Chip InterfaceBased on the proposed cutting model in Section 4.3, the time-varying nature of the contact lengthsas well as the cutting forces cause the temperature at the tool-chip interface varies in each cycleof localization unlike the steady-state condition in the continuous chip formation. As will be seenin Section 4.5, these changes are rapid and therefore, significant in the temperature distributionand evolution. Therefore, the developed cutting model in Section 4.3 is employed to predict thetemperature at the tool-chip interface where the two-dimensional transient heat equation governingon the shear-localized chip is expressed asχ[∂ 2θ∂x2+∂ 2θ∂y2]=∂θ∂ t+ vch∂θ∂x. (4.23)In Eq. (4.23), vch is the advection velocity (here, chip velocity), and χ = kρcp is thermal diffusivitywhere k is thermal conductivity, ρ is mass density, and cp is heat capacity. Eq. (4.23) governson the control volume containing the shear localized chip in the spatial domain of 0 ≤ y ≤ b and0≤ x≤ lmaxtot where lmaxtot = lmaxst + lmaxsl as shown in Fig. 4.11. Based on Eq. (4.23), the heat conduc-tion in both chip velocity and shear directions (i.e., x and y directions, respectively) are taken intoaccount. Moreover, the term vch∂θ/∂x considers the cutting chip’s movement (and subsequently,the heat advection) in the chip velocity direction with respect to the xy coordinate system.In the case of dry cutting, the control volume is in direct contact with air at y= b and at x= lmaxtotboundaries as shown in Fig. 4.11. Hence, the convection boundary conditions at x= lmaxtot and y= bboundaries are written, respectively, as−k∂θ∂y= hair [θair−θ(x,b, t)] , convection at y = b, (4.24)−k∂θ∂x= hair [θair−θ(lmaxtot ,y, t)] , convection at x = lmaxtot , (4.25)126AdvectionTool-ChipInterfaceConvection: ,Convection:,Dirichlet:Figure 4.11: The control volume with the spatial domain of 0 ≤ y ≤ b and 0 ≤ x ≤ lmaxtotcontaining the shear-localized chip. Heat conduction in both x and y directions aswell as heat advection in x direction are taken into account. The spatial domain isdiscretized into Nx×Ny elements, where ∆x and ∆y are the spatial increments in xand y directions, respectively. The five boundary conditions are: two Convectionboundary conditions with convection coefficient hair and temperature θair at y = band x = lmaxtot ; two Dirichlet boundary conditions with temperature θh(t) at x = 0 andθsb(t) at y = 0 (0 ≤ x ≤ lst(t)); and one heat flux (Neumann) boundary condition aty = 0 (lst(t)< x≤ ltot(t)). The FTCS scheme is employed to solve the governing heatequation using finite difference method as shown in the subset.where hair and θair are the air convection heat coefficient and air temperature, respectively. Atthe left boundary (i.e., x = 0 where the plane OA is located), the workpiece material is beingcut homogeneously. Hence, the homogeneous temperature can be obtained by the homogeneoussolution of the energy and momentum equations presented in Section 2.3.1. Therefore, Dirichletboundary condition is considered at x = 0 asθ(0,y, t) = θh(t), Dirichlet at x = 0, (4.26)where, θh(t) is the homogeneous temperature which can be obtained from Eq. (2.6).Based on the cutting force model presented in Section 4.3, the tool-chip interface (i.e., the127boundary y = 0 in Fig. 4.11) is divided into two different regions: (i) sticking region where 0 ≤x≤ lst(t) and (ii) sliding region where lst(t)< x≤ ltot(t). In the sticking region, the material upsetsin front of the cutting tool, and the chip velocity is zero (vch). Therefore, due to the shear bandrolling on the cutting tool, the temperature in the sticking zone is derived from Dirichlet boundarycondition:θ(x,0, t) = θ in(t) if 0≤ x≤ lst(t). (4.27)The time-varying shear band’s temperature θ in(t) is obtained from Eq. (2.30). In the sliding regionand due to the friction, the heat production at the tool-chip interface is qsl(t) = τu1(t)vch, whereτu1(t) is the time-varying shear stress in the sliding region that can be obtained from Eq. (4.12).The produced heat is partitioned between the tool and chip with the partition coefficient of R(0 < R < 1). Therefore, the heat flux boundary condition at the tool-chip interface in the slidingregion is∂θ∂y∣∣∣∣y=0=−Rτu1(t)vchkif lst(t)≤ x≤ lmaxtot . (4.28)Finally, the initial temperature of the domain is equal to the fresh workpiece temperature enteringthe primary deformation zone.The governing heat equation of Eq. (4.23) along with the boundary conditions presented inEqs. (4.24) to (4.28) and the initial condition are solved using the FD Method. The domain is dis-cretized into Nx elements (Nx+1 grid points) in the x-direction and Ny elements (Ny+1 grid points)in the y-direction as illustrated in Fig. 4.11. Therefore, each element has the width of ∆x= lmaxtotNxandheight of ∆y = bNy . Here, the total simulation time is considered to be the time required for eachcycle of segmentation, i.e., tcycle = dsvch . Dividing the time span into Nt number of steps, the timeincrement is ∆t = tcycleNt .By employing the explicit FTCS (forward time, centered space) scheme on the computational128grid shown in Fig. 4.11, the finite difference (FD) form of the governing heat equation (Eq. (4.23))is obtained asθ n+1i, j = θni, j +Dx[θ ni−1, j−2θ ni, j +θ ni+1, j]+Dy[θ ni, j−1−2θ ni, j +θ ni, j+1]−Cx2[θ ni+1, j−θ ni−1, j], (4.29)where the subscripts i and j represent the spatial increments in x and y directions, respectively,while the superscript n is the increment in time as shown in Fig. 4.11. In Eq. (4.29), Dx =χ∆t∆x2 , Dy =χ∆t∆y2 , and Cx =vch∆t∆x . Application of von Neumann’s stability criterion [46] shows that Eq. (4.29) isstable whenD2x +D2y ≤12& C2x ≤ Dx. (4.30)The truncation error of Eq. (4.29) is in the order of O(∆x2)+O(∆y2)+O (∆t), showing thatthe spatial and temporal errors have different orders (first-order temporal error while second-orderspatial error).Eq. (4.29) must be modified for the four cases of y = 0, y = b, x = 0, and x = lmaxtot to satisfythe boundary conditions presented in Eqs. (4.24) to (4.28). For y = b and x = lmaxtot , Eq. (4.29),respectively, becomesθ n+1i,Ny+1 = θni,Ny+1+Dx[θ ni−1,Ny+1−2θ ni,Ny+1+θ ni+1,Ny+1]+2Dy[θ ni,Ny −θ ni,Ny+1−∆yhairk(θair−θ ni,Ny+1)]−Cx2[θ ni+1,Ny+1−θ ni−1,Ny+1]: for y = b (i.e., j = Ny+1), (4.31)129θ n+1Nx+1, j = θnNx+1, j +2Dx[θ nNx, j−θ nNx+1, j−∆xhairk(θair−θ nNx+1, j)]+Dy[θ nNx+1, j−1−2θ nNx+1, j +θ nNx+1, j+1]+Cx∆xhairk[θair−θ nNx+1, j]: for x = lmaxtot (i.e., i = Nx+1). (4.32)Moreover, the Dirichlet boundary condition of Eq. (4.26) in its FD form isθ n1, j = θnh , (4.33)where θ nh = θh(tn).Based on Eq. (4.27), the governing FD equation at the tool-chip interface (y= 0) in the stickingzone isθ ni,1 = (θin)n. (4.34)In the sliding zone (y= 0, and lst(t)≤ x≤ lmaxtot ), the FD form of Eq. (4.28) is obtained by two-pointcentral difference approximation asθ n+1i,1 = θni,1+Dx[θ ni−1,1−2θ ni,1+θ ni+1,1]+2Dy[θ ni,2−θ ni,1+RT qsl(t)k∆y]−Cx2[θ ni+1,1−θ ni−1,1]. (4.35)As a final remark, the effect of time-varying sticking and sliding contact lengths has beencaptured by the time-varying boundary condition at the tool-chip interface based on the time-varying sticking and sliding lengths.1304.5 Results and DiscussionsThe thermo-mechanical properties and the JC parameters of Ti-6Al-4V, which are commonly usedin the literature, were presented in Tables 3.2 and 3.3, respectively. Fig. 4.12 shows the opticalmicro-graph of the prepared shear-localized chip at the cutting speed vc= 1.5 m·s−1 and uncutchip thickness d = 120 µm from the ex situ experiments. Five cycles of complete segmentationare shown in Fig. 4.12, where narrow shear bands between the wide segments are clear. The shearangle φ was calculated based on measuring the maximum thickness of the shear-localized chips,i.e., h, shown in Fig. 4.12 according to Section 3.5. Table 4.1 presents the measured thicknessesof the produced Ti-6Al-4V chips under the cutting speeds vc = 1 m·s−1 and vc = 1.5 m·s−1 at fivedifferent segments for each cutting speed. To obtain the shear angle for each cutting speed, theaverage value of the five measurements is then introduced into φ = tan−1( r cosα1−r sinα ), where r = d/his the chip compression ratio [4]. The shear angle for the cutting speeds vc = 1 m·s−1 and vc = 1.5m·s−1 are φ = 40.12◦ and φ = 39.84◦, respectively. Fig. 4.13 shows the laser micro-graph of thesame sample focused at the tool-chip interface. The interaction between the shear band and thetool-chip interface via rolling is clear.Table 4.1: Calculation of the shear angle for Ti-6Al-4V by experimental measurements ofchip’s maximum thickness for two different cutting speeds of vc = 1 m·s−1 and vc = 1.5m·s−1.Cutting speed Chip thickness (µm) Shear angle(m·s−1) 1 2 3 4 5 avg. (deg.)1 145 143 141 143 140 142.4 401.5 147 147 144 140 141 143.8 394.5.1 Shear band’s analysisShear band’s temperature and stress: At first, the homogeneous temperature θh(t) and stressτh(t) are obtained from Eqs. (2.6) and (2.7), respectively. As shown in Section 2.4, the critical131Figure 4.12: Optical micro-graph of the shear-localized chip at the cutting speed vc= 1.5m·s−1 and uncut chip thickness d = 120 µm. Narrow shear bands between thick seg-ments are clear. The maximum chip thickness, which is employed to calculate theshear angle, is shown. The magnitude of shear band’s active length λ (t) at the begin-ning and end of a cycle, i.e., λ (0) = b and λ (tcycle), respectively, are also illustrated.strain under which the instability initiates is γcr = 0.324 for the cutting speed vc = 1 m·min−1 anduncut chip thickness d=120 µm. Before γcr = 0.324, the workpiece material’s behavior followsthe homogeneous deformation while after γcr = 0.324, the homogeneous assumption is not validanymore because of the material instability onset. Therefore, according to the numerical resultsof Section 2.4, the stress and absolute temperature (Th(γ) = θh(γ)+Tr) at the onset of instabilityare 1137 MPa and 422.8 K (θ = 129.8 K), respectively, which were shown in Fig. 2.12. Afterthe instability onset, the workpiece material’s behavior does not follow the homogeneous solutionanymore. Therefore, the localization of the plastic deformation inside the shear bands must betaken into account. The strain rate inside the shear band, i.e., γ˙ in, is obtained from Eq. (2.32).Inserting this strain rate into Eq. (2.30), the shear band’s temperature θ in(t) and subsequently, theshear band stress τ in(t) are obtained. The final stress and absolute temperature of the shear bandare τ in =558.21 MPa and T in =1235.8 K, respectively, as shown in Fig. 2.16.Shear band’s active length: As the shear band travels inside the upsetting zone AOB, its active132RollingShear BandSegmentTool-Chip InterfaceFigure 4.13: Laser micro-graph of the ex situ sample for the cutting speed vc= 1.5 m·s−1 anduncut chip thickness d = 120 µm. The shear band has different morphology than thesegments and the interaction between the shear band and the tool rake face via rollingis clear.length tolerating the shear force Fs1(t) changes from λmax = λ (0) = b to λmin = λ (tcycle) due tothe large displacement inside the shear band and its rolling on the cutting tool’s rake face as shownin Fig. 4.7. The maximum and minimum active lengths, i.e., λmax and λmin, respectively, can bemeasured from the high-resolution optical and laser micro-graphs of the prepared shear-localizedchips as illustrated in Figs. 4.12 and 4.13. Fig. 4.14 compares the predicted values of λ (t) obtainedfrom Eq. (4.2) with the ex situ experimental measurements for five segmentation cycles (i.e., fivesuccessive shear bands) at the cutting speed vc = 1 m·s−1 and uncut chip thickness d = 120 µm. InFig. 4.14, linear variation between the experimental values of λmax and λmin is assumed. Accord-ing to Fig. 4.14, at the shear band’s onset, the active length is equal to the shear plane’s length b(i.e., length of OA in Fig. 4.7). As long as the shear band is under loading (i.e., inside the upsettingregion AOB), its active length decreases according to Eq. (4.2). As soon as the shear band exits theregion AOB, the next shear band starts tolerating the loading. Therefore, the active length returnsto λmax = b.1330 100 200 300 400 500t ( s)100120140160180200(t) (m)Active Length OscillationTheoryExperimentFigure 4.14: The variation of the shear band’s active length for five successive shear bandsat the cutting speed vc = 1 m·s−1 and uncut chip thickness d = 120 µm. At first, theshear band’s active length is λmax = b. As it travels inside the deformation regionAOB, its active length decreases until it reach its minimum value λmin. After exitingthe primary deformation region, the next shear band enter the deformation regionwith the initial active length of λmax = b. The experimental measurements for fiveneighboring shear bands from the optical micro-graph of the prepared chip are alsoshown.Table 4.2 compares the predicted and experimental values of the shear band’s active length λ (t)for the same cutting condition, i.e., cutting speed vc = 1 m·s−1 and uncut chip thickness d = 120µm. It can be seen that Eq. (4.2) predicts λmax and λmin with prediction error of 3.00% and 0.76%,respectively. Although the error is lower for λmin, it can be seen from Fig. 4.14 and Table 4.2 thatthe variation of λmin is more than that of λmax in the experimental results. In fact, the standarddeviation of measured values for λmin is 11 µm while 3 µm for λmax. This is because during shearband formation, the strain rate inside the shear band is significantly high. For instance, in the caseof vc = 1 m·s−1, the shear band’s strain rate is about 4.56× 105 s−1 (see Eq. (2.32)). Therefore,even a small difference in the time during which the shear bands are inside the primary deforma-tion region causes a large displacement deviation.134Table 4.2: Comparison between the experimental values of the shear band’s active length λ (t)and the prediction of Eq. (4.2) at the cutting speed vc = 1 m·s−1. Five measurements atdifferent places are performed for each λmax and λmin, and the average value is reported.ExperimentSimulation Error(1) (2) (3) (4) (5) Averageλmax 191 196 190 194 188 192 186 3.00%[µm]λmin 106 102 118 130 110 113 114 0.76%[µm]4.5.2 Time-varying sticking and sliding contactsFig. 4.15 is a high-speed image sequence of two consecutive segmentation cycles, i.e., two con-secutive shear bands with the contact lengths data added. The cutting conditions are: vc = 0.65m·s−1, d = 200 µm, α = 0◦, r0 = 10 µm, and φ = 38◦. For each segmentation cycle, three differentframes are chosen in Fig. 4.15 so that: (i) the first frame approximately shows the beginning of thesegmentation cycle when the shear band just occurs in front of the cutting tool, i.e., lsl(t0) = lmaxsland lst(t0) = lminst ; (ii) the second frame shows an inter-frame in which neither lsl(t) nor lst(t) hasreached its maximum/minimum value; and (iii) the final frame showing the time just before thenew shear band initiation in front of the cutting tool (while the previous shear band is about to exitthe upsetting zone). Therefore, in this frame, lsl(t2) = lminsl and lst(t2) = lmaxst as shown in Fig. 4.15c.Hence, the first and final frames in each row show the start and end of each segmentation cycle. Itshould be noted that the shear band’s onset, propagation, and formation during cutting processeshappen with high propagation speed. For example, the propagation speed of a shear band from thetool-tip to the free surface of the workpiece material is around 8 m·s−1 when cutting Ti-6Al-4V atthe cutting speed of 0.65 m·s−1 [103]. Therefore, the time resolution provided by the high-speedimage sequence (ten frame in each cycle) may not be enough to capture all aspects of shear lo-calization in cutting processes. Therefore, the first and last frames for each segmentation cyclepresented in Fig. 4.15 are approximate since the exact time at which the shear band occurs in front135of the cutting tool or exits the deformation region may not be captured by any of the frames. InFig. 4.15, it has been tried to choose the closest frames to the start and end of each segmentationcycle.At the starting point of segment 1 shown in Fig. 4.15a, all the contact length is under slidingcondition (see Section 4.2) where the sliding contact length is measured as lsl(t0) = lmaxsl = 616µm. Here, the measurement method of the contact lengths is noteworthy. As mentioned in Sec-tion 4.2, the chip segmentation frames are captured by a high resolution camera coupled with anoptical microscope to provide the resolution 1.4 µm per pixel. Moreover, in order to measure thecontact length by visual inspection, the contact zone is magnified by image processing toolbox ofMATLAB software. This provides a sufficient accuracy to determine the point at which the chipleaves the cutting tool’s rake face. In all these frames, the gap between the chip and the cuttingtool’s rake face always shows a “V-shape” separation as is clear in all frames of Fig. 4.15, which isdue to the chip curling happening during the cutting process. The tip of this ”V-shape” gap is thepoint at which the chip starts to become separated from the tool’s rake face. Therefore, the distancebetween the cutting tool’s tip to the separation point is measured as the total contact length, i.e,ltot(t). In Fig. 4.15a where t0 = 0 µs, the sticking contact length is at its minimum value, i.e., lminst .It should be noted that according to Fig. 4.10, the minimum sticking length (which happens at thebeginning of the cycle, i.e., Fig. 4.15a) is a portion of the cutting tool edge’s perimeter obtainedfrom Eq. (4.14). However, since the resolution of the high-speed sequence is not enough to mea-sure that minimum sticking length, it is assumed that lminst ≈ 0 at the beginning of the segmentation.Fig. 4.15b shows a middle frame at t1 = 200 µs where the shear band SB1 has rolled on thecutting tool’s rake face, and the material has piled up in front of the cutting tool based on thedescription in Section 4.2. The sticking contact length along which the material is in stationarycondition with respect to the cutting tool’s rake face is the distance from the tool-tip to the active136shear band rolling on the cutting tool’s rake face (i.e., SB1). It is lst(t1) = 93 µm in this frame.Therefore, from t0 = 0 to t1 = 200 µs, the sticking length increases from 0 µm to 93 µm while thesliding length decreases from 616 µm to 428 µm as shown in Figs. 4.15a and 4.15b. Fig. 4.15cshows the sticking and sliding contact lengths at t2 = 300 µs where more material has piled up infront of the cutting tool. Based on Fig. 4.15c, a new shear band SB2 is about to happen in frontof the cutting tool starting from the tool-tip. However, since it has not occurred yet, the cuttingforces have not dropped yet, and therefore, the last segment is still in the sticking condition ratherthan sliding. At this stage, the sticking length takes its maximum value of lst(t2) = lmaxst = 131µm while the sliding length has been decreased to lsl(t2) = lminsl = 365 µm. Similar behavior isobservable in the next segmentation cycle (i.e., segment 2) shown in the second row of Fig. 4.15.For segment 2 (the second row of Fig. 4.15), the maximum sliding length is lmaxsl = 578 µm asshown in Fig. 4.15d, while in the intermediate frame (Fig. 4.15e), the sliding length decreases tolsl(t1) = 438 µm, and the sticking length increases to lst(t1) = 91 µm. The segmentation cycleends at the sticking length of lmaxst = 162 µm, and the sliding length of lminsl = 384 µm as shown inFig. 4.15f. In all these frames, the decrease in the sliding contact length lsl(t) and increase in thesticking contact length lsl(t) as the time progresses during each segmentation cycle is clear. Thisshows the time-varying nature of the contact length in shear-localization chip formation, which isin contrast to the continuous chip formation where the contact length is constant during the cuttingprocess given a certain cutting condition.137(a) t0 = 0 (b) t1 = 200 µs (c) t2 = 300 µs(d) t0 = 0 (e) t1 = 200 µs (f) t2 = 300 µsFigure 4.15: High-speed image sequence of two segmentation cycles when cutting Ti-6Al-4V at the cutting speed vc = 0.65 m·s−1 and uncut chip thickness d = 200 µm. Eachrow shows one segmentation cycle in three different stages: (i) the early stage; (ii)a random internal stage, and (iii) the final stage. At the early stage, the sliding con-tact length is at its maximum, while as time progresses, the sliding contact lengthdecreases until it reaches its minimum value. At the same time, the sticking lengthincreases from its minimum value at the early stage to its maximum at the final stage.At the cutting condition chosen for the experimental observation of Fig. 4.15 (vc = 0.65 m·s−1,d = 200 µm, α = 0◦, r0 = 10 µm, and φ = 38◦), the homogeneous strain rate, temperature, andstress are obtained as γ˙h = 16927.86 s−1, θh = 130.56 K, and τh = 1106.33 MPa with the same pro-cedure described earlier in this section. The segmentation cycle is tcycle = ds/vch = 152/0.5078 =299.33 µs, where the shear band spacing ds can be obtained from the theoretical or the experi-138mental observations (Section 2.3.3). Accordingly, the shear band’s final stress and temperature areobtained from Eq. (2.4) and Eq. (2.30) as τ in = 503.40 MPa and θ in = 1282.49 K, respectively.Based on Eq. (4.2), the active length of the shear band, i.e., λ (t), decreases from the initial value of324.85 µm to the final value of 172.85 µm in each segmentation cycle. In order to predict the min-imum sticking length lminst from Eq. (4.14), the stagnation angle η is needed. The stagnation angleη ranges from 40◦ to 65◦ for round cutting edge based on the literature survey performed by Guo[42] and chosen to be equal to η = 50◦ (η =5π18) in this study. Based on Molinari et al. [82] andMiguelez et al. [77] for the orthogonal cutting simulation of Ti-6Al-4V, the friction coefficient atthe tool-chip interface is assumed to be µ = 0.4 for the carbide tool. Finally, in order to obtain thesliding contact length, the distribution exponent ζ is needed in advance (see Eqs. (4.5) and (4.20)).The distribution exponent ζ is sometimes assumed to be zero (i.e., uniform stress distribution) forthe sake of simplicity [4, 110]. However, the procedure to obtain the actual distribution exponent isto experimentally or numerically obtain the normal stress distribution at the tool-chip interface andthen, fit a curve to the normal stress distribution. The exponent of the fitted curve is then assumedto be the distribution exponent. Adopting the same procedure, Molinari et al. [80, 81] considereddifferent values for ζ and adopted the distribution exponent of 0.23, causing a mild decay in thestress profile as mentioned in Section 4.3.1.Fig. 4.16a compares the predicted values of the sticking and sliding contact lengths with theexperimental measurements for nine different segmentation cycles. Note that the first two seg-mentation cycles shown in Fig. 4.16a are those two cycles illustrated and shown in Fig. 4.15. Thesame measurement approach was adopted for seven more segmentation cycles using the high-speed image sequence. For each segmentation cycle, three different frames were chosen to showthe beginning of the segmentation cycle, the intermediate time instance, and the end of the seg-mentation cycle. Hence, in total, 27 different frames (54 distinct measurements of lsl(t) and lst(t))are shown in Fig. 4.16a. For each measurement, the separation point between the cutting tool and139the segmented chip is considered to be the final point on the tool-chip contact length as explainedpreviously. As mentioned previously, the active shear band (i.e., the shear band under plastic defor-mation) is the place where the sticking length separates from the sliding length, making it possibleto measure the sticking and sliding lengths from the high-speed imaging sequences. Using the im-age processing toolbox of MATLAB software, for each 27 frames, the sticking and sliding lengthswere experimentally measured. In Fig. 4.16a, the predicted sliding length lsl(t) is obtained fromEq. (4.11), while the sticking length lst(t) is predicted via Eqs. (4.13) to (4.15).As shown in Fig. 4.16a, the predicted sliding length lsl(t) starts from the maximum value oflmaxsl = 674.34 µm and decreases linearly until it reaches its minimum value of lminsl = 358.82 µm.This decrease in the sliding contact length happens in one segmentation cycle with the time periodof tcycle = 299.3 µs. During the same period of time, the predicted sticking length lst(t) increasesfrom the minimum value of lminst = 6.98 µm to the maximum value of lmaxst = 158.98 µm. Atthe end of the cycle, the sliding length lsl(t) jumps back to the maximum value of 674.34 µmand the sticking length lst(t) drops back to its minimum value of 6.98 µm, and the cycle repeatsagain. Comparing the theoretical and experimental results of Fig. 4.16a, it can be observed thatthe described trend is in accordance with the experimental results. The intermediate frame in eachsegmentation cycle helps to see if the variation between the maximum and minimum values islinear as predicted by the theoretical results or not. For both sticking and sliding lengths, the vari-ation between the minimum and maximum values follow a linear trend with respect to time. If thetime at which the internal frame is captured was in the middle of the initial time and final time ofthe cycle, the linearly of the contact length variation could be judged better. However, as can beseen in Fig. 4.16a, some of the internal frames are close to the end frames. This is because of thedifficulties in the measurement of the sticking length lst(t) in the frames close to the starting pointof the cycle due to its small magnitude. As explained earlier, the minimum value of the stickinglength lminst is assumed to be zero in the experimental results, which is reflected in Fig. 4.16a.140In Fig. 4.16a, the experimentally measured values of lmaxsl , lminsl , and lmaxst have some variationsin different segmentation cycles. Therefore, Fig. 4.16b shows the variations in the measured val-ues of lmaxsl , lminsl , and lmaxst for the nine segmentation cycles and compares them with the theoreticalpredictions. For each of the lmaxsl , lminsl , and lmaxst , the average values of the measurements are alsoshown by solid circles in Fig. 4.16b. The maximum measured value of lmaxsl is 674 µm while itsminimum measured value is 578 µm. From Fig. 4.16b, it is clear that the predicted lmaxsl , which isshown by a diamond sign, is at the upper border of the experimental bar, showing that the measuredmaximum sliding lengths are mostly less than the predicted value. This can be attributed to the em-ployed JC parameters (since the homogeneous stress τh(γ), which is the value of the shear band’sstress at the beginning of the cycle, is dependent on the JC parameters based on Eq. (2.7)), theinitial active length of the shear band λ (0), the friction coefficient µ , or the distribution exponentζ . For lminsl shown in Fig. 4.16b, the maximum measured value is 440 µm, the minimum measuredvalue is 294 µm, and the average is 378 µm. It can be seen that the predicted value lies well inthe middle of the experimentally measured values and very close to the average value. Same holdstrue for lmaxst with better accuracy in its prediction where the maximum, minimum, and averagemeasured values are 232 µm, 114 µm, and 163 µm, respectively. Note that lminst is not presentedin Fig. 4.16b due to the difficulty in measuring the early contact of the workpiece material with thecutting tool’s edge by high-speed imaging.Contact in Shear-localized vs. in continuous chip formation: Having obtained the time-varying values of the sticking and sliding contact lengths, it is beneficial to compare the resultswith those obtained from the theories developed for continuous chip formation. Table 4.3 presentsthe total contact lengths predicted by some of the theoretical models developed for continuous chipformation in the literature. Employing the slip-line field theory, Lee and Shaffer [64] predicted thecontact length in terms of uncut chip thickness d, shear angle φ , and rake angle α as shown in1410 1000 2000 3000t ( s)0200400600800Contact Length (m)StickingSlidingSticking-Exp.Sliding-Exp.(a)lslmax lslmin lstmax100200300400500600700Contact Length ( m)Measurement Variationslslmax-Exp.lslmin-Exp.lstmax-Exp.lslmax-Theorylslmin-Theorylstmax-Theory(b)Figure 4.16: Comparisons of the predicted sliding and sticking contact lengths with the ex-perimental results for nine experimental cycles of segmentation. (a) At the beginningof each segment, the sliding length is at its maximum value of 674.34 µm (theoret-ical prediction) and decreases linearly until it reaches the minimum value of 358.82.At the same time, the sticking length starts from the minimum value of 6.98 µm andincreases linearly until it reaches the maximum value of 158.98 µm. At this time,enough material has piled up in front of the cutting tool for the next shear band’sonset. Then, the sliding length jumps back to its maximum value while the stickinglength drops back to its minimum. (b) Variations in the measurements of lmaxsl , lminsl ,and lmaxst and their comparisons with the theoretical predictions. The theoretical pre-diction for lmaxsl is located at the top of the measurement bar while for lminsl and lmaxst ,the predicted values are well close to the mean value of the measurements.Table 4.3. Satisfying the equilibrium conditions between the forces on the cutting tool’s rakeface and those on the Merchant’s shear plane, Hahn [44] presented an explicit equation to obtainthe total contact length. Later, Rubenstein [99] generalized the equation by assuming a generalstress distribution on the cutting tool’s rake face. This theory was expanded further to consider theedge radius by Molinari et al. [80] and to take the inclination angle into account (during obliquecutting) by Budak and Ozlu [11]. Experimental approach is another method to predict the tool-chipcontact length in the literature for continuous chip formation. Kato et al. [52] proposed a simpleapproximation based on which the contact length is almost twice the uncut chip thickness. Sutter[115] proposed another experimental model based on the shear plan’s length b and uncut chip142thickness d to predict the contact length. In the literature, many other theoretical and numericalmodels have been proposed to predict the contact length. An extensive investigation on differenttheoretical models can be found in Refs. [32, 80, 81]. The numerical value of each analyticalmodel for d = 200 µm, α = 0◦, and φ = 38◦ is also presented in Table 4.3.Table 4.3: Comparison of the total contact lengths predicted by the theoretical models of Leeand Shaffer [64], Kato et al. [52], Hahn [44], Rubenstein [99], and Sutter [115]. Thecontact length magnitudes for the cutting condition of vc = 0.65 m·s−1, d = 200 µm,α = 0◦, and φ = 38◦ are also presented.Method Total Contact Length [µm] Prediction [µm]Lee and Shaffer ltot =d√2sinφ cos(45◦+φ −α) 589.37Kato et al. ltot = 2d 400Hahn ltot =d sin(φ +ϑ −α)sinφ cosϑ302.40Rubenstein ltot =ζ +22d sin(φ +ϑ −α)sinφ cosϑ337.17Sutter ltot = 1.92b−0.09d 605.72d: uncut chip thickness, φ : shear angle, α: rake angle, ϑ : friction angle at the rake face.ζ : distribution exponent.Fig. 4.17 compares the predicted total contact length ltot(t) from Eq. (4.16) with the experi-mental measurements using high-speed image sequence. For the experimental measurements, thetime-varying sticking and sliding contact lengths are added at each instance of time in order to ob-tain the total contact length’s variation with respect to time. According to Fig. 4.17, ltot(t) followsthe same trend as lsl(t) in which the total contact length is the maximum at the beginning of thecycle (i.e., 681 µm) while linearly decreases with respect to time until it reaches its minimum atthe end of the cycle (i.e., 518 µm). Although the sticking length lst(t) has an increasing trend ineach segmentation cycle, the decreasing rate of the sliding length lsl(t) is larger, causing the totallength to decrease with respect to time during each segmentation cycle. Similar trend has been ob-143served in the experimental results as shown in Fig. 4.17. The contact length predicted by Lee andShaffer [64] is also presented in Fig. 4.17. It can be seen that all the theories presented in Table 4.3,including Lee and Shaffer [64], predict a constant contact length while only two predicted values(those obtained from Lee and Shaffer [64] and Sutter [115]) lie within the experimental variationof the total contact length. This shows that the mechanism of shear localization, which is differentin nature than continuous chip formation, must be taken into account for the accurate prediction ofthe contact lengths.0 1000 2000 3000t ( s)500600700800l tot(t) (m)PredictionExperimentLee and ShafferFigure 4.17: Comparison of the predicted total contact length with the experimental measure-ments when cutting Ti-6Al-4V at the cutting speed vc = 0.65 m·s−1 and uncut chipthickness d = 200 µm. The total contact length starts at 681 µm in the beginning ofthe segmentation cycle and decreases linearly up to 518 µm. Similar trend is observ-able in the experimental results. The constant contact length predicted by Lee andShaffer [64] is also presented.Effects of cutting condition on the contact length: Fig. 4.18 shows the effects of cuttingspeed vc and uncut chip thickness d on the variations of sliding and sticking contact lengths. Thecutting condition with the cutting speed vc = 0.65 m·s−1 and uncut chip thickness d = 120 µm(referred as Condition 1) is considered as the reference. The effect of cutting speed vc is investi-gated by increasing it to vc = 1 m·s−1 while the uncut chip thickness keeps constant at d = 120144µm (referred as Condition 2). Then, the effect of uncut chip thickness is studied by keeping thecutting speed at vc = 0.65 m·s−1 while increasing the uncut chip thickness to d = 200 µm (referredas Condition 3). According to Fig. 4.18a, at Condition 1, the sliding length fluctuates between themaximum value of lmaxsl = 394.44 µm and the minimum value of lminsl = 263.01 µm (therefore, thefluctuation amplitude is 131.43 µm), while the fluctuation frequency is 8.85 kHz. When the cuttingspeed increases to vc = 1 m·s−1 (i.e., Condition 2), the sliding length fluctuates between 394.44µm and 241.81 µm (i.e., the fluctuation amplitude of 152.63 µm) with the fluctuation frequencyof 11.72 kHz. It is clear that the maximum sliding length lmaxsl is constant for both conditions andtherefore, insensitive to the cutting speed. This is because according to Eq. (4.11), lmaxsl dependson the distribution exponent ζ , the initial active length of the shear band λ (0), and the force ratioFn1(0)/Fv1(0). The distribution exponent ζ is constant, and the force ratio Fn1(t)/Fv1(t) is depen-dent on the shear angle φ and rake angle α according to Eq. (4.4). Since the shear angle averagedfor different cutting speeds is employed (see Table 4.1), the force ratio Fn1(t)/Fv1(t) does notchange with respect to the cutting speed. Note that the shear angle’s change is less than a degreewhen the cutting speed increases by 0.5 m·s−1 according to Table 4.1, making the constant shearangle assumption valid for the analysis. Although the active length λ (t) is dependent on the cuttingspeed through the segmentation parameter ξ (t) in Eq. (4.2), its initial value λ (0) is independentof the cutting speed and equals to the shear plane length b according to Eq. (4.2). Therefore, themaximum contact length lmaxsl does not change with respect to the cutting speed.As the cutting speed increases from Condition 1 to Condition 2, the displacement inside theshear band increases because the shear band’s strain rate is higher for higher cutting speed (seeEq. (3.22) in Section 3.4). Therefore, the larger shear band’s displacement causes smaller lminsl forCondition 2 as shown in Fig. 4.18a. The increase in the cutting speed also increases the frequencyof fluctuation from 8..85 kHz (Condition 1) to 11.72 kHz (Condition 2). Therefore, for Condition1, the loading time of each shear band (i.e., the time during which one shear band is under loading)1450 200 400 600 800 1000 1200t ( s)0200400600Contact Length (m)Sliding60 m/min, 120 m39 m/min, 120 m39 m/min, 200 m(a)0 200 400 600 800 1000 1200t ( s)050100150200250Contact Length (m)Sticking60 m/min, 120 m39 m/min, 120 m39 m/min, 200 m(b)Figure 4.18: The effects of cutting speed vc and uncut chip thickness d on the fluctuation ofsliding and sticking contact lengths. As the cutting speed increases from 0.65 m·s−1 to1 m·s−1 (the uncut chip thickness d = 120 µm), lmaxsl remains constant, lminsl decreases,and the segmentation frequency increases from 8.85 kHz to 11.72 kHz. As the uncutchip thickness increases from 120 µm to 200 µm (cutting speed is constant and setat 0.65 m·s−1), both lmaxsl and lminsl increase while the segmentation frequency dropssignificantly from 8.85 kHz to 3.34 kHz. For the sticking length, the increase in thecutting speed causes larger lmaxst . By increasing the uncut chip thickness from 120µm to 200 µm, lmaxst largely increases from 68.98 µm to 158.98 µm. The minimumsticking length remains constant for all three cutting conditions.is larger. Similar trend can be observed in the sticking length fluctuation shown in Fig. 4.18b.When the cutting speed increases from Condition 1 to Condition 2, the maximum sticking lengthlmaxst increases from 68.98 µm (Condition 1) to 78.99 µm (Condition 2). Note that the minimumsticking length lminst depends on the geometry of the cutting tool according to Eq. (4.14), and there-fore, remains constant with respect to the cutting speed as shown in Fig. 4.18b.According to Fig. 4.18a, the effect of uncut chip thickness on the contact length fluctuationis more pronounced than that of the cutting speed. By increasing the uncut chip thickness fromd = 120 µm to d = 200 µm (i.e., Condition 1 to Condition 3), the maximum sliding length lmaxslincreases from 394.44 µm to 674.34 µm, while the minimum sliding length increases from 263.01µm to 358.82 µm. Therefore, the fluctuation amplitude significantly changes from 131.43 µm to146315.52 µm. Moreover, the segmentation frequency drops from 8.85 kHz to 3.34 kHz as the uncutchip thickness increases from 120 µm to 200 µm. Accordingly, the time during which each shearband is under loading (i.e., segmentation period) increases from 113 µs to 299 µs by increasingthe uncut chip thickness. In general, it can be concluded that increase in the uncut chip thicknesscauses large drop in the segmentation frequency, more loading time for each shear band (the shearband is under loading for longer period of time), and therefore, larger amplitude of fluctuation inthe contact length (and subsequently the cutting forces).4.5.3 Time-varying cutting forces and stressesFig. 4.19 shows the variation of shear force Fs1(t) in one cycle of segmentation for the cuttingspeed vc = 1 m·s−1 and uncut chip thickness d = 120 µm obtained from Eq. (4.1). According toFig. 4.19, at the beginning of the cycle, the shear force on the shear band SB1 is 211.25 N, and itdrops to 63.77 N (i.e., 69.81% drop in its initial value) at the end of the cycle. Having obtained theshear force Fs1(t) acting on the shear band, the normal force on the rake face in the sliding region,i.e., Fv1(t) is obtained from Eq. (4.3), and the friction force on the rake face in the sliding region,i.e., Fu1(t), is obtained from the Coulomb’s friction law of Fu1(t) = µFv1(t). Fig. 4.20a shows thenormal and frictional force fluctuations on the cutting tool’s rake face in the sliding region for thecutting speed vc = 1 m·s−1 and uncut chip thickness d = 120 µm for five segmentation cycles. Ac-cording to Fig. 4.20a, at each segmentation cycle, Fv1(t) decreases from 417.27 N to 125.97 N dueto the shear band’s evolution inside the deformation region. When the shear band travels along thechip velocity direction, it finally exits the primary deformation region and the cutting force jumpsback to the maximum value of 417.27 N. Afterwards, the next shear band initiates in front of thecutting tool and Fv1(t) decreases with the trend shown in Fig. 4.20a again. This process repeats ineach cycle of segmentation and causes the cyclic characteristic of force fluctuation on the cuttingtool’s rake face in the sliding region. According to Fig. 4.20a, the friction force Fu1(t) follows the147same trend as Fv1(t) due to using the Coulomb’s friction law and decreases from 166.91 N to 50.39N in each cycle of segmentation.0 20 40 60 80 100t ( s)50100150200250F s1 (N)Shear Force on SB1End of CycleBeginning of CycleFigure 4.19: Variation of the Shear force Fs1(t) in one cycle of segmentation for the cuttingspeed vc = 1 m·s−1 and uncut chip thickness d = 120 µm. The shear force drops211.25 N to 63.77 N in 85 µs.Fig. 4.20b shows the fluctuations of the normal and frictional forces on the tool’s rake face inthe sticking region, i.e., Fv2(t) and Fu2(t), respectively. In the sticking region, Fv2(t) is obtainedfrom Eq. (4.18) while Fu2(t) is obtained from Fu2(t) = τ in(t)lst(t)w as explained in Section 4.3.2.Due to the rolling of the shear band on the cutting tool’s rake face in the sticking region, the yieldstress of the workpiece material in the sticking zone is equal to the shear band’s stress τ in(t). Infact, this is the reason why the workpiece material is always under yielding condition in the stick-ing region irrespective of the sticking contact length. According to Fig. 4.20b, Fv2(t) increase from6.69 N (at the beginning of the cycle) to 230.24 N (at the end of the cycle) showing an oppositetrend compared to Fv1(t) in Fig. 4.20a. This is because in each cycle of segmentation, the stickinglength increases and more material piles up in front of the cutting tool, causing Fv2(t) to increasewhile Fv1(t) drops due to the drop in the shear band’s force Fs1(t). This piling up (increase in the148sticking length) causes increase in the contact force in the sticking length as shown in Fig. 4.20b.At the same time, the friction force Fu2(t) increases from 7.93 N to 44.16 N as shown in Fig. 4.20b.0 100 200 300 400 500t ( s)0100200300400500Force (N)SlidingFv1(t) Fu1(t)(a)0 100 200 300 400 500t ( s)050100150200250Contact Force (N)StickingFv2(t) Fu2(t)(b)Figure 4.20: The fluctuations of the contact forces Fv1(t) and Fu1(t) in the sliding region, andFv2(t) and Fu2(t) in the sticking region for five segmentation cycles at vc = 1 m·s−1and d = 120 µm. In each cycle of segmentation, Fv1(t) drops from 417.27 N to 125.97N while Fv2(t) increases from 6.69 N to 230.24 N. The friction force Fu1(t) decreasesfrom 166.91 N to 50.39 N while Fu2(t) increases from 7.93 N to 44.16 N. The oppositetrends between the normal and frictional forces in the sliding and sticking regions areclear.The prediction of the cutting force variation in each sliding and sticking zone (shown inFig. 4.20b) is related to the distributions of the normal and shear stresses on the rake face andsubsequently, the distribution of the cutting tool’s temperature on the rake face. Also, the cuttingtool’s wear is heavily dependent on the stress distributions on the tool’s rake face. Therefore, it isimportant to obtain the normal and frictional forces separately for each sliding and sticking regionas shown in Fig. 4.20. However, it is also beneficial to predict the total normal and frictional forcesexerted on the cutting tool based on the obtained results of Fig. 4.20. Fig. 4.21a shows the fluctua-tion of the total normal force on the cutting tool’s rake face obtained from Fv(t) = Fv1(t)+Fv2(t).According to Fig. 4.21a, Fv(t) starts from the maximum value of 423.96 N and decreases to theminimum value of 356.21 N at the end of the segmentation cycle. Therefore, the fluctuation am-149plitude is 67.75 N. Comparing this amplitude to the fluctuation amplitude of 291.3 N in slidingforce and that of the sticking force, i.e., 223.55 N, it can be seen that the force fluctuation ampli-tude is smaller for the total normal force Fv(t) compared to individual sticking and sliding forcecomponents. This smaller fluctuation amplitude is because in each cycle of segmentation, and atthe beginning of segmentation, the sliding force Fv1(t) decreases while the sticking force Fv2(t)increases, both with asymptotic trend as shown in Fig. 4.20. Similar trend exists for the frictionalforce Fu(t) shown in Fig. 4.21b where it drops from the maximum value of 174.83 N to the mini-mum value of 94.55 N.0 100 200 300 400 500t ( s)340360380400420440F v (N)Total Normal Force(a)0 100 200 300 400 500t ( s)80100120140160180F u (N)Total Friction Force(b)Figure 4.21: Fluctuations of total normal and frictional forces, i.e., Fv(t) and Fu(t), respec-tively, for vc = 1 m·s−1 and d = 120 µm. In each segmentation cycle, Fv(t) drops from423.96 N to 356.21 N and Fu(t) decreases from 174.83 N to 94.55 N. Comparing withFig. 4.20, the fluctuation amplitude is less for total forces compared to sliding andsticking forces.Normal stress at the tool-chip interface: Fig. 4.22 shows the normal stress distribution onthe cutting tool’s rake face in three different frames from the beginning (t = 10.06 µs) to the endof the segmentation cycle (t = 85.32 µs) for vc = 1 m·s−1 and d = 120 µm. The origin is chosento be at the bottom-right corner of Fig. 4.22 so that the rake face is located at the right-hand sidein accordance with Figs. 4.7 and 4.8 where the cutting tool cuts from right to left. In Fig. 4.22,150the normal stress amplitude is obtained from Eq. (4.8) in the sliding region (i.e., σˆ0(t)), and fromEq. (4.22) in the sticking region (i.e., σ0(t)). Moreover, the distribution of the normal stress is de-rived from Eq. (4.5) in the sliding region (i.e., σv1(xˆ, t)) and from Eq. (4.20) in the sticking region(i.e., σv2(x, t)). In Fig. 4.22a, σv1(xˆ, t) starts from zero at 392.02 µm and finishes at 16 µm withthe magnitude of σˆ0(t) = 990.04 MPa. Note that since the distribution exponent is ζ = 0.23, thenormal stress takes a rather mild distribution with a sudden drop to zero at the end of the contactlength as shown in Fig. 4.22a. The border between the sliding and sticking regions is the placewhere the shear band is located as shown in Fig. 4.22a. Below the shear band (i.e., in the stickingzone) where the material has piled up in front of the cutting tool, the sticking length is considerablysmall (lst(t) = 16 µm), causing a large magnitude of the normal stress at the tool-tip. The stressamplitude at the sticking region is σ0 =13054 MPa, and therefore, the total stress amplitude at thetool-tip is σ0+ σˆ0 = 14044 MPa (see Fig. 4.8).By the advancement of the cutting tool, the sticking length increases to 42.62 µm while thesliding length decreases to 318.90 µm at t = 42.23 µs as shown in Fig. 4.22b. According toFig. 4.22b, the stress amplitude at the sliding region becomes σˆ0(t) = 762.26 MPa while in thesticking region, it is σ0(t) = 5115.7 MPa. Therefore, the total normal stress at the tool tip be-comes σ0 + σˆ0 = 5878 MPa. At the end of the segmentation cycle (t = 85.32 µs), the stickinglength is at its maximum of 78.98 µm while the sliding length is at its minimum of 241.81 µm.The stress amplitude on the sliding region is σˆ0(t) = 640.75 MPa while in the sticking length, itis σ0(t) = 3212.8 MPa. Interestingly, although the sliding contact length is decreasing from thebeginning to the end of the cycle, the normal stress decreases rather than increasing. This decreaseis because of the fact that the normal force on the sliding zone has dropped due to the shear bandformation. Therefore, the normal stress amplitude is determined by the competition between thedecrease in the cutting force and decrease in the contact length as in Eq. (4.8). Based on Fig. 4.22,it can be seen that the tool-tip is under large normal stress fluctuation in each cycle of segmentation.151050001000015000 (MPa)0100200300400500Contact Length (m)t=10.06 sRake FaceSlidingSticking Shear Band’sLocation(a)0200040006000 (MPa)0100200300400500Contact Length (m)t=42.23 sRake FaceSlidingSticking Shear Band’sLocation(b)01000200030004000 (MPa)0100200300400500Contact Length (m)t=85.32 sRake FaceSlidingStickingShear Band’sLocation(c)Figure 4.22: Prediction of the normal stress distribution on the cutting tool’s rake face whencutting Ti-6Al-4V at vc = 1 m·s−1 and d = 120 µm. Three different frames show thetime-varying nature of the stress distribution from the early stage of localization whent = 10.06 µs to the end of localization where t = 85.32 µs. As time increases, thesticking length increases from 13.53 µm to 78.98 µ , while the sliding length decreasesfrom 380.57 µm to 241.81 µm. The normal stress amplitude at the tool tip drops from13.58 GPa to 3.85 GPa.shear stress at the tool-chip interface: Fig. 4.23 shows the shear stress distribution on thecutting tool’s rake face in both sliding and sticking regions at vc = 1 m·s−1 and d = 120 µm. Att = 10.06 µs, the sticking length is considerably small (15.47 µm), and the material’s yield stressis 864.15 MPa. In the sliding region, the shear stress is zero at 371.92 µm and increases to 396.14152MPa at 15.47 µm. At the border between the sticking and sliding lengths, there is a sudden changein the shear stress due to the existence of the shear band at the border. At t = 42.23 µs, the stickinglength has increased to 42.62 µm and the material is yielded at 665.12 MPa. In the sliding length,the shear stress starts from zero at 361.51 µm and increases to 304.90 MPa at 42.62 µm, which isthe border between the sticking and sliding lengths. Finally, at the end of the segmentation cyclewhen t = 85.32 µs, the sticking length increases to 78.98 µm with the material yield stress of559.09 MPa. In the sliding region, the shear stress reaches 256.30 MPa at the border between thesticking and sliding lengths.An important note about the shear stress distribution shown in Fig. 4.23 is that the material’syield stress in the sticking zone decreases from 864.15 MPa at t = 10.06 µs to 665.12 MPa att = 42.23 µs, and finally, reaches 559.09 MPa at t = 85.32 µs. The reason for this time-varyingyield stress at the sticking region is the rolling of the shear band on the cutting tool’s rake face.When the shear band forms in front of the cutting tool, its shear flow stress drops from 864.15MPa at t = 10.06 µs to 559.09 MPa at t = 85.32 µs with the asymptotic trend shown in Fig. 2.16.Simultaneously, when it rolls on the cutting tool’s rake face, the yield stress at the tool-chip inter-face in the sticking zone follows the same trend, causing a time-varying yield stress at the stickingregion as shown in Fig. 4.23.15302004006008001000 (MPa)0100200300400500Contact Length (m)t=10.06 sRake FaceSlidingStickingShear Band’sLocation(a)0200400600800 (MPa)0100200300400500Contact Length (m)t=42.23 sRake FaceSlidingStickingShear Band’sLocation(b)0200400600 (MPa)0100200300400500Contact Length (m)t=85.32 sRake FaceSlidingStickingShear Band’sLocation(c)Figure 4.23: Prediction of the shear stress distribution on the cutting tool’s rake face fromt = 10.06 µs to t = 85.32 µs. At t = 10.06 µs, the sticking length is 15.47 µm withthe yield stress of 864.15 MPa. In the sliding region, the shear stress changes from 0MPa to 396.14 MPa. At t = 42.23 µs, the sticking length increases to 42.62 µm withthe yield stress of 665.12 MPa. The shear stress changes from 0 MPa to 304.90 MPain the sliding region. Finally and at the end of the segmentation cycle (t = 85.32 µs),the sticking length is 78.98 µm with the yield stress of 559.09 MPa while the shearstress changes from 0 MPa to 256.30 MPa in the sliding region.4.5.4 Transient temperature at the tool-chip interfaceIn order to obtain the temperature distribution at the tool-chip interface using FD method, the spa-tial domain of 0≤ y≤ 186 µm and 0≤ x≤ 401.4 µm is discretized into Nx = Ny = 150 elements154in x and y directions. Therefore, each FD element has the width of ∆x = 2.68 µm and the heightof ∆y = 1.24 µm. For the segmentation cycle of tcycle = 85.32 µs, the time increment is obtainedas ∆t =tcycleNt= 1.14×10−7 s by assuming 603 time steps (i.e., Nt = 603). The employed spatialand time increments satisfy both stability criteria presented in Eq. (4.30). The initial temperatureis assumed to be equal to the room temperature, i.e. 293 K, and the Dirichlet boundary conditionis obtained from the homogeneous solution of the temperature. The air convection coefficient isequal to hair = 20 W/(m2·K) [62]. The heat partition between the chip and the tool is usually as-sumed to be 0.5 in the absence of the tool-chip contact analysis considering surface asperities [81](see Ref. [2] for reviewing the heat partition values adopted in some of the literature studies). Onthe other hand, it generally depends on the thermal conductivity of the tool and workpiece as wellas the cutting speed [17]. For the orthogonal cutting of Ti-6Al-4V with uncoated carbide insert,the experimental equation of R = 0.707−0.229vch+0.035v2ch has been suggested [72] and there-fore, employed here to consider the effect of cutting speed on the heat partition. Therefore, for thecutting speed vc = 1 m·s−1 (vch = 0.8439 m·s−1), the heat partition is R = 0.54.Fig. 4.24 represents the transient temperature distribution on the cutting tool’s rake face fromthe beginning to the end of one segmentation cycle. At the beginning of the segmentation cyclewhen t = 1 µs, the maximum temperature is 602.00 K located at the distance x = 2.67 µm fromthe cutting tool-tip as shown in Fig. 4.24a. As the cutting tool moves forward, the maximum tem-perature increases and its distribution expands along the cutting tool’s rake face in the chip velocitydirection. At t = 30 µs, The maximum temperature is 1037.91 K located from x = 2.67 µm tox = 32.11 µm with the distribution shown in Fig. 4.24b. At t = 60 µs, the maximum temperatureat the tool-chip interface increases to 1167.25 K located from 2.67 µm to 56.20 µm. Finally, atthe end of the segmentation cycle, i.e., t = 85 µs, the maximum temperature reaches 1233.26 Kfrom x = 2.67 µm to x = 77.61 µm from the tool-tip. Fig. 4.24 demonstrates the transient natureof the temperature evolution at the cutting tool’s rake face from the beginning to the end of the155segmentation cycle with the amplitude of temperature fluctuation equal to 631.26 K. This largeamplitude of temperature fluctuation is harmful for the cutting tool and elevates the cutting toolwear [92].t=1 s020y ( m)0100200300400Tool-Chip Interface (m)3003504004505005506001030(a)t=30 s020y ( m)0100200300400Tool-Chip Interface (m)30040050060070080090010001030(b)t=60 s0y ( m)0100200300400Tool-Chip Interface (m)400600800100020 1030(c)t=85 s0y ( m)0100200300400Tool-Chip Interface (m)4006008001000120020 1030(d)Figure 4.24: Transient temperature distribution at the tool-chip interface when cutting Ti-6Al-4V at the cutting speed vc = 1 m·s−1 and uncut chip thickness d = 120 µm inone segmentation cycle. At first, the maximum temperature is 602.00 K located at2.67 µm away from the tool-tip. At t = 30 µs, the maximum temperature increasesto 1037.91 K. At t = 60 µs, the maximum temperature increases further to 1167.25 Kfrom 2.67 µm to 56.20 µm. Finally, the final position of the maximum temperature isat 77.61 µm distance from the tool-tip with the magnitude of 1233.26 K.4.6 ConclusionsThis chapter presents the cutting mechanics of shear localization, and proposes an analytical cut-ting model including the shear band formation, its rolling on the cutting tool’s rake face, andtime-varying nature of sticking-sliding contact lengths. The main conclusions of this study aregiven as follows.1. Employing the high-speed image sequence of shear localization during orthogonal cutting ofTi-6Al-4V, it was shown that when a shear band occurs in front of the cutting tool, the tool-chip contact length is dominated by sliding nature with negligible sticking length at the tool-156tip. As the cutting tool moves forward, new workpiece material starts upsetting in its front,the formed shear band starts rolling on the cutting tool’s rake face starting from the tool-tip, and the sticking length starts increasing due to the upsetting phenomenon. By furtheradvancement of the cutting tool, more material piles up in its front, and the sticking lengthincreases subsequently. This increase continues until the cutting forces are high enough forthe next shear band to occur. As soon as the new shear band occurs in front of the cuttingtool, the normal and tangential forces on the cutting tool’s rake face drop and therefore, theentire contact length returns back to the sliding condition. This process repeats each time ashear band occurs in front of the cutting tool, causing the cyclic nature of shear localization.2. The velocity magnitude at the tool-chip interface in the sticking region is almost zero. It isshown the length across which there is a zero velocity magnitude at the tool-chip interfaceincreases with respect to time in each segmentation cycle, showing the time-varying nature ofthe sticking contact length at the tool-chip interface. The small magnitude velocity vectorsat the tool-chip interface in the sticking length produce a vorticity (a curl) which showsthe upsetting direction (the direction of the shear band’s rolling) on the cutting tool’s rakeface. Moreover, it is shown that during shear localization and due to the stationary conditionbetween the cutting chip and the cutting tool at the sticking zone, there is an out-of-planedeformation (bulging) in the cutting chip which was captured by SEM imaging of Ti-6Al-4V, Inconel 718, and Waspaloy.3. Based on the experimental observations and analytical analyses on the produced shear bandsin cutting, a new analytical cutting mechanics model is developed for the shear localization.The model takes the shear band’s thermo-mechanical behavior, its rolling on the tool’s rakeface, and the time-varying nature of the contact length into account. To this end, the forcegeneration is divided into two different regions, namely, sliding and sticking regions, con-nected to each other by the equilibrium condition at the shear band’s location. In the sliding157region, the Coulomb’s friction law determines the relation between the tangential and nor-mal forces at the tool-chip interface while they are under the equilibrium condition with theshear and normal forces on the shear band. The shear force on the shear band drops in eachsegmentation cycle due to the drop in the shear stress of the shear band and decrease in itsactive length. This decrease in the shear force of the shear band, in turn, causes a decreasein the normal and tangential forces at the tool-chip interface in the sliding region, causing amonotonous decrease in the sliding length in each segmentation cycle.4. At the sticking region, the sticking length increases with the rate of chip velocity vch fromthe minimum value determined by tool’s edge radius until it reaches its maximum valuedetermined by the shear band spacing. The free body diagram of the upsetting region isemployed to obtain the normal and tangential forces at the tool-chip interface in the stickingregion. The effect of the sliding forces is taken into account by the forces acting on theshear band. Based on the calculated normal and tangential forces in the sticking region, thenormal and tangential stress are predicted. Regarding the shear stress, due to the rolling ofthe formed shear band on the tool’s rake face, it is equal to the time-varying shear stressinside the shear band.5. Based on the proposed cutting model, the transient temperature at the tool-chip interface ispredicted using the finite difference (FD) technique. To this end, the boundary condition atthe tool-chip interface is obtained from the predicted shear stress in the sticking and slidinglengths. It is shown that the temperature at the tool rake face fluctuates due to the cyclicnature of shear localization, which may cause the thermal fatigue of the cutting tool. In eachcycle of shear localization, the location of the maximum temperature at the tool-chip inter-face moves along the tool’s rake face and is not at a constant location unlike continuous chipformation. The magnitude of the maximum temperature occurs at the end of each segmenta-tion cycle and is considerably high. For instance, the maximum temperature reaches 1495.4158K in cutting Ti-6Al-4V at vc = 1 m·s−1.6. Finally, the predicted results are presented in detail and compared with the experimentalresults showing a good agreement in predicting the active length of the shear band, segmen-tation frequency, the time-varying sticking length, the time-varying sliding length, and thetotal contact length. The effect of the cutting speed and uncut chip thickness on the resultsare also presented.159Chapter 5Conclusions and Future ResearchDirectionsShear localization widespread in the machining of high performance metallic alloys results in lowmachinability, rapid tool wear and low quality of the finished surface. This thesis studied the shearlocalization mechanism in the orthogonal cutting of metallic alloys from analytical and experi-mental points of view. It developed an analytical thermo-mechanical cutting model by consideringshear localization to predict the stress, temperature, strain rate, and displacement developed inthe primary deformation zone as well as the contact forces, lengths, stresses, and temperature atthe tool-chip interface (secondary deformation zone). The model was supported by experimentalinvestigations revealing different aspects of shear localization in cutting processes, including therolling mechanism, lack of plastic deformation in the segments, shear localization intensity in dif-ferent metallic alloys, and time-varying contact lengths.First, it was experimentally shown that the shear localization process includes upsetting of thefresh workpiece material in front of the cutting tool, and subsequently, rolling of the shear bandon the tool’s rake face. Therefore, the shear band’s simulation in the primary deformation zone160was performed by assuming an extended primary deformation region and localization of plasticdeformation inside the shear band. The former is necessary for the modeling of shear localizationsince the softening and hardening mechanisms (whose relative magnitude is the reason of mate-rial instability) develop gradually in the workpiece material, making the shear plane assumptionproposed in the literature oversimplified for the case of shear localization. The latter is proved bythe micro-graphs of micro-markers on the chip as well as the shear band’s microstructure, showingthat the plastic deformation outside the shear band is negligible. Therefore, the governing equa-tions (energy, inertia, and flow law equations) were divided into two separate groups applicable oninside and outside the shear band. The main difference between these two sets is the large heat pro-duction inside the band compared to the negligible heat production in its outside (i.e., segments).They were solved analytically using the Green’s function method. The results were analytical pre-diction of the stress, temperature, and strain rate inside the primary deformation region by coupledalgebraic and integral equations. The shear band spacing (and subsequently, the cutting force fluc-tuation frequency) was also predicted, and it was emphasized that only one shear band happens inthe primary deformation zone under practical cutting conditions employed in industry. The shearbands happen successively rather than simultaneously in machining unlike other dynamic defor-mation applications, such as impact projectile, shock waves, and split-Hopkinson bar applications.Second, the proposed model was further developed to predict the shear band’s displacement,its distribution, and the localization degree without any post-mortem measurement of the producedshear bands. It was shown that in order to predict the shear band’s displacement, there is no needfor crack assumption as the shear band itself is a type of material failure (different than crack for-mation). In order to predict the shear band’s displacement, the time during which it is inside theprimary deformation zone, its strain rate, and its rolling on the tool’s rake face must be taken intoaccount. The model was employed to study the effects of elastic energy storage at the tool-chip in-terface. It was shown that it only causes a jump in the shear band’s strain rate when changing from161homogeneous to localized deformation, and therefore, cannot be the reason for shear localization.Third, experimental observations showed different aspects of shear localization in titanium(more specifically, Ti-6Al-4V and CP titanium) and nickel (Waspaloy and Inconel 718) alloys.While titanium alloys show shear-localized chip in all the employed cutting speeds, nickel alloysproduce continuous, chaotic, and shear-localized chip in different cutting speed ranges, showingless susceptibility of nickel alloys than titanium alloys to shear localization. The developed ana-lytical model predicts the effect of different thermo-mechanical properties and cutting conditionson shear localization as shown in Table 3.4. Moreover , in nickel alloys (whose shear bands arethicker than those of titanium alloys), the shear band rolls on the tool’s rake face only from oneedge resulting in the thickness variation of the plastic deformation at the tool-chip interface.Fourth, using high-speed imaging results and image processing techniques, it was shown thatboth sticking and sliding lengths change with respect to time in each cycle of localization. In thebeginning of the localization cycle, the sticking length is at its minimum value (at the tool-tip)while the sliding length is at its maximum value. As time passes, more material piles up in frontof the cutting tool, and therefore, the sticking length increases while the sliding length decreasesdue to the shear force drop at the shear band. At the end of the localization cycle, the stickinglength is maximum while the sliding length is minimum. As the shear band leaves the primarydeformation zone, the process starts again causing periodic change of sticking and sliding lengthsduring machining of high performance metallic alloys.As a consequence of time-varying contact lengths, both normal and friction forces at the tool-chip interface vary with respect to time. In each localization cycle, the normal and friction forcesincrease in the sticking region while decrease in the sliding region. Explicit analytical equationswere proposed for the contact forces in each region in Chapter 4. However, in both sticking andsliding regions, the normal and friction stress decrease in each localization cycle. Regarding the162sticking region, the normal stress decreases because the sticking contact length increases (althoughthe normal force decreases). The shear stress decreases because it corresponds to the shear stressinside the shear band which has rolled on the tool’s rake face. Considering the sliding region, bothdecrease in the contact forces and increase in the sliding length are in favor of decrease in thecontact stresses. Chapter 4 discusses the contact forces and stresses in detail.Finally, the temperature at the tool-chip interface is not constant but varies with respect to time.From the beginning to the end of the localization cycle, the amplitude of the temperature increaseis large. For example, in cutting of Ti-6Al-4V at the cutting speed and uncut chip thickness, themaximum temperature at the tool-chip interface increases from to. The temperature distributionalso predicted in Chapter 4. It was shown that the location of maximum temperature is not constantbut moves in the chip velocity direction. This is different than the steady-state temperature at thetool-chip interface in the case of continuous chip formation.Future possible directions of this thesis as as follows:1. The presented model can be extended to capture the spatial fluctuation of the residual stressat the machined surface when machining high performance metals. First, the time-varyingploughing force, which occurs due to the tool’s edge radius, needs to be modeled by con-sidering shear localization. If the tool is worn, the friction due to the flank face also needsto be taken into account. Then, the predicted time-varying normal and shear stresses in theprimary deformation region in this thesis and the ploughing and friction stresses can be as-sumed as moving mechanical loads applying on the machined surface in the cutting speeddirection. The predicted temperature in the primary deformation zone can also be consideredas a moving heat source in the cutting speed direction to investigate its effect on the residualstress at the machined surface. Then, a theoretical solution based on the theory of contactmechanics can be developed to capture the residual stress distribution and its fluctuation.1632. Considering the coolant’s effect on the stress and temperature simulation in the primary andsecondary deformation zones is another extension of this thesis. The shear band formationfrequency, its width (and subsequently, strain rate), temperature, and stress can be affectedby the application of coolant. The shear band’s hardness will probably increases as thecooling rate increases significantly. The boundary conditions of the developed model arealso changed due to the effects of coolant.3. The extension of the model to oblique cutting, and subsequently, milling operations can befollowed. In homogeneous cutting, the transformation from orthogonal cutting to obliquecutting is performed by some geometrical equations that take the inclination angle into ac-count. 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Metal cutting mechanics. → pages 40, 49, 50, 99, 177175Appendix AShear Band’s Failure Surface andBifurcationThe experimental results provided so far are normal to the chip’s cross section to obtain infor-mation about the plastic deformation pattern of the shear band and segments, their shear band’smicrostructure, and tool-chip interface. However, the top-view (normal to the cutting width) pro-vides information about the failure mechanism inside the shear band during machining. Moreover,during investigation of some metallurgically prepared samples, irregularities were observed in themorphology of few shear bands morphology near the tool-chip interface, which is discussed in thisappendix.Shear band failure surface: Fig. A.1 shows the top-view SEM image of the Ti-6Al-4V shear-localized chip at the cutting speed vc = 60 m·min−1 and uncut chip thickness d = 120 µm. Thesegments and shear bands happening periodically are clear in Fig. Fig. A.1. Based on Fig. Fig. A.1,the segments slide on the shear bands, causing the shear bands to experience large plastic strain.Figure Fig. A.2 shows the fracture surface of the shear band in more detail. The fracture surfaceof the shear band exhibits parabolic dimples and smooth surfaces. The elongated dimple struc-176Figure A.1: Top view SEM image of the Ti-6Al-4V shear-localized chip at the cutting speedvc = 60 m·min−1 and uncut chip thickness d = 120 µm. The periodic shear bands aswell as the segments are clear.ture shows that considerable plastic deformation occurs inside the shear bands, and the fractureis ductile. Noting that the material inside the primary deformation zone is under shear as wellas hydrostatic compression (due to the mechanics of cutting [137]), forming of elongated dimplestructure (which is known to be the result of void nucleation [38]) under the compression imposedby the cutting tool is noteworthy. The smooth surfaces (which are the main parts of the fracturesurface) are probably due to the mutual rubbing of the shear band’s surfaces.Bifurcation in shear-localized chip formation: During the investigation of the prepared sam-ples in different cutting speeds, it was observed that in some of the produced shear bands, thetwo thin shear bands branch off from the main shear band at a certain distance from the cuttingtool’s rake face, as shown in Fig. Fig. A.3. Figure Fig. A.3 shows a shear band formation near thetool-chip interface under the cutting speed of vc = 60 m·min−1 and uncut chip thickness of d =120 µm. The shear band has been branched into two different shear bands with less thickness ata distance away from the cutting tool’s rake face. The total thickness of the new shear bands isalmost equal to the initial shear band’s thickness, and the two branches join each other again at thetool-chip interface.177Figure A.2: Top view SEM image of the shear band for the cutting speed vc = 60 m·min−1and uncut chip thickness d = 120 µm. Two different regions are clear: parabolicdimples and smooth sliding surfaces.Bifurcation has been observed in shear bands during the explosion of cylinders made of Ti-6Al-4V [130], and AISI 4340 steel [122]. It has been justified by the spiral trajectory of the shear band[122]. In their cutting experiments, Komanduri et al. [55] observed that the shear band initiatesat the tool-tip in the direction of the cutting velocity (i.e., the horizontal direction in orthogonalcutting) and gradually curves up until it reaches the free surface (i.e., curve slip-lines [88]). There-fore, it can be concluded that the spiral trajectory of the shear band in cutting is near the tool-tipwhere the shear band has a curved path to reach the free surface, making it prone to bifurcationnear the cutting tool’s tip.178Figure A.3: Bifurcation in shear band at the cutting speed vc = 60 m·min−1 and uncut chipthickness d = 120 µm, where the shear band branches off near the tool-chip interface.179
Thesis/Dissertation
eng
Mechanical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Jin, Xiaoliang
Analytical modeling of shear localization in machining processes and its effects on the tool-chip interface
Text