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FINITE ELEMENT SIMULATION OF CHIP SEGMENTATION IN MACHINING A Ti 6Al4V ALLOY By KAREEM SYED Bachelor of Engineering Osmania University Hyderabad, India June, 2001 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE December, 2004 ii FINITE ELEMENT SIMULATION OF CHIP SEGMENTATION IN MACHINING A Ti 6Al4V ALLOY Thesis approved: Dr. Ranga Komanduri Thesis Adviser Dr. Hong Bing Lu Dr. Samit Roy Dr. Gordon Emslie Dean of the Graduate College iii ACKNOWLEDGEMENT I wish to express my sincere thanks to my advisor, Dr.Ranga Komanduri, for his intelligent supervision, constructive guidance, financial support, inspiration and motivation. His wise advices made it possible for me to quickly grow not only technically but also in all aspects during the course of my M.S study. I truly appreciate the encouragement and technical guidance from my committee members: Dr. Samit Roy and Dr. Hongbing Lu. I would also like to thank Dr. Bo Wang for his guidance in deriving the equations for Recht’s catastrophic shear failure criterion. This project has been funded by a grant from the division of Design, Manufacturing and Industrial Innovation (DMII) of the National Science Foundation (NSF). I would like to thank Dr. George Hazelrigg for his interest and support of this work. I would like to thank Chief Technical Officer Dr. Troy Marusich of Third Wave Systems, Inc. and Support Engineers Mr. Christopher Brand and Mr. Deepak Agarwal for their technical support in using AdvantEdge™ software. I wish to express my sincere gratitude to Mr. Dhananjay Joshi and Mr. Parag Konde for their contribution in discussing, deriving and implementing the equations of Recht’s catastrophic shear failure criterion into the user subroutine code (UMAT). I would also like to extend my gratitude to all other members of our research group for their support and friendship. iv I would like to thank the Department of Mechanical and Aerospace Engineering for providing me with the opportunity to pursue M.S at Oklahoma State University. I am ever thankful to my parents who were always a constant source of my inspiration and encouragement at all times. Finally, I would like to extend my gratitude to my brother and sisters for their inspiration and love. v TABLE OF CONTENTS Chapter Page 1. Introduction………………………………………………………....…….........1 1.1 State Of Art: Finite Element Simulation of the Cutting Process………………………………………………………...1 1.2 Historical Developments…………………………………........3 1.3 Principles of Metal Cutting…………………………………....5 1.4 Principles of Finite Element Modeling………………………..6 1.5 Metallurgical Aspects of Titanium alloys……………………..7 1.6 Machinability aspects of Ti 6Al4V………………………….10 1.7 Mechanism of ShearLocalized Chip Formation in Ti 6Al4V……………………………………………………..11 1.8 Thesis Outline………………………………………………...13 2. Literature Review………………………..………………………………….....16 2.1 Numerical Study of ShearLocalized Chip Formation………..16 2.2 ShearLocalized Chip Formation in Ti 6Al4V: Experiments...25 2.3 ShearLocalized Chip Formation in Ti 6Al4VFEM Simulations…………………………………………………….36 3. Finite Element Formulation of Metal Cutting .................................................47 3.1 Introduction………..……………………………….……….....47 3.2 Finite Element Formulation …………………………………..48 3.3 Friction formulation along toolchip interface ………………. 57 3.4 Formulation of Contact Conditions …………………………..58 3.5 Adaptive remeshing …………………………………………..59 3.6 Chip formation………………………………………………...61 4. Problem Statement………………………………………………………….......63 5. ShearLocalized Chip Formation in Ti 6Al4V machining…………………..65 5.1 ShearLocalized Chip in Ti 6Al4V …………………………...65 5.2 Mechanism of ShearLocalized Chip formation………… ……66 5.3 Criterion for Thermo Plastic Shear Instability…………………68 vi 5.4 Metallurgical Aspects of ShearLocalized Chip Formation…..70 6. FEM Simulation of Chip Segmentation in Ti 6Al4V ………………………..72 6.1 Introduction………..………………………………………....72 6.2 Material Constitutive Model ………………………………...73 6.3 Formulation of Recht’s Catastrophic Shear Failure Criterion 75 6.4 Stress Update Algorithm …………………………………….80 7. Results and Discussions………..………………………………………….........84 7.1 Process Model and Material Properties……………………... 84 7.2 Simulation Approach and Cutting Conditions………………..88 7.3 Observations of Chip Formation Process…………………….90 7.4 Temperature and Equivalent Plastic Strain distribution in the chip……………………………………………………94 7.5 Effects of Cutting Speed and Feed rate……………………....97 7.6 Effects of Rake angle………………………………………..111 7.7 Effects of Coefficient of friction…………………………….116 7.8 Validation of Simulation Results…………………………….119 7.9 Discussion…………………………………………………...126 8. Conclusions and Future Work ………………………………………………..128 8.1 Conclusions…………………………………………………..128 8.2 Future Work..………………………………………………...131 References.......………………………………………………………………...133 vii LIST OF TABLES Table No. Page No. 7.1 Physical properties of Ti 6Al4V…..………………………………………..86 7.2 JohnsonCook material properties for Ti 6Al4V…………………………...87 7.3 Cutting conditions used for FEM simulations……………………………....89 7.4 Tool dimensions used in FEM simulations………………………………....90 7.5 Adaptive mesh options input to the FEM software…………………………90 7.6 Finite element simulation results and experimental data of cutting and thrust force with percent deviation……………………………………..120 7.7 Finite element results and experimental data for cutting force with percent deviation………………………………………………………122 viii LIST OF FIGURES Figure No. Page 1.1 Orthogonal metal cutting model ……………………………………………….....6 1.2 Microstructure of Ti 6Al4V alloy showing primary αphase and transformed βgrains……………………………………………………………....9 2.1 Stressstrain diagram for a combination of strain hardening, thermal softening and failure softening……………………………………………………………. 22 2.2 Adiabatic shearlocalized chip in a Ti 6Al4V alloy, obtained by orthogonal cutting at the velocity 1.2 m/s…………………………………………………... 28 2.3 Chip morphology as a function of cutting speed and feed in the orthogonal cutting of a Ti 6Al4V alloy…………………………………………………….. 29 2.4 ShearLocalized chip formed in machining a Ti 6Al4V alloy at cutting speeds of 60 and 120 m/min using feed rate of 0.127 mm/rev………………………..... 31 2.5 Roughness of workpiece material (Ti 6Al4V) as a function of cutting length at different cutting speeds……………………………………………….. 32 2.6 Voids and cracks in an adiabatic shear band of a Ti 6Al4V alloy specimen deformed at 700°C and 2000 s1 strain rate…………………………...33 2.7 Temperature dependence of flow stress in a Ti 6Al4V alloy at very low strain rates of 103 s1 ………………………………………………………34 2.8 Pseudotemperature variation with drill depth showing a linear increase ……...35 2.9 Serrated chips obtained in machining a Ti 6Al4V alloy at cutting speeds (a) 80 m/min and (b) 140 m/min……………………………….. 36 2.10 Chip formation process showing shear localization in finite element simulation of a Ti 6Al4V alloy at cutting speed of 5 m/s and feed 0.3 mm using 8° rake angle tool………………………………………... 37 ix 2.11 Toolworkpiece mesh system used in finite element simulation of orthogonal metal cutting a Ti 6Al4V alloy……………………………….. 39 2.12 Serrated chip formation with cutting length in finite element simulation of machining a Ti 6Al4V alloy at cutting speed 30 m/min, feed 0.25 mm and 20° rake angle……………………………………………... 39 2.13 Serrated chip formed in finite element simulation of orthogonal metal cutting of a Ti 6Al4V alloy at cutting speed 30 m/min, feed 0.25 mm and 20° rake angle…………………………………………….. 40 2.14 Initial geometry and workpiecetool mesh used in finite element simulation of orthogonal metal cutting of a Ti 6Al4V alloy…………………41 2.15 Segmented chip formation process in finite element simulation of orthogonal metal cutting of a Ti 6Al4V alloy at 10 m/s cutting speed and 0.5 mm feed using a 3° rake angle tool …………………………. 42 2.16 Segmented chip formation process in finite element simulation of orthogonal metal cutting of a Ti 6Al4V alloy at 50 m/s cutting speed and 0.04 mm cutting depth using a 10° rake angle tool ……………….43 2.17 Equivalent plastic strain distribution in serrated chip simulated at cutting speeds of 1.2, 120 and 600 m/min in a Ti 6Al4V alloy……………...45 3.1 Contacting surfaces in a mesh showing (a) predictor configuration and (b) kinematically compatible configuration…………………………….. 59 3.2 Sixnodes and three quadrature points shown in a typical sixnoded triangular element used in finite element mesh of workpiece and tool………60 3.3 Element shape in a finite element mesh (a) before adaptive remeshing and (b) after adaptive remeshing……………………………………………..60 5.1 Description of shearlocalized chip formed due to adiabatic shear localization…………………………………………………………………...67 5.2 Xray diffraction spectra for chip and uncut material in a Ti6Al4V alloy….. 71 6.1 Model used for determination of temperature gradient with strain x in catastrophic shear zone…………………………………………………... 77 6.2 Variation of Recht’s criterion value (R) with temperature for Ti 6Al4V alloy at different strain rates and strain of 3…….............................................79 6.3 Variation of Recht’s criterion value (R) with temperature for Ti 6Al4V alloy at different strain rates and strain of 4……............................................80 7.1 Workpiecetool system used for FEM simulations showing initial mesh….. 86 7.2 True stressstrain curves of Ti 6Al4V alloy based on JohnsonCook material model in the temperature range of 200 to 16000 C at high strain rate of 1x 104 s1…………………………………………………….. 87 7.3 The variation of True stress as a function of temperature for Ti 6Al4V in the different strain range of 0.005 to 4.0 at high strain rate of 1x 104 s1 88 7.4 Various stages of shearlocalized chip formation in machining simulation of Ti 6Al4V conducted at 30 m/min cutting speed and 0.5 mm depth of cut for 15° rake angle showing equivalent plastic strain localization in narrow shear bands…………………………..93 7.5 Temperature distribution in shearlocalized chip formation obtained from FEM simulation of machining Ti 6Al4V at 30 m/min cutting speed and 0.5 depth of cut for 15° rake angle ……….…………...95 7.6 Equivalent plastic strain distribution in the shearlocalized chip formation from FEM simulation of machining Ti 6Al4V at a cutting speed of 30m/min and a depth of cut of 0.5 mm for a 15° rake angle…….96 7.7 Temperature and equivalent plastic strain contour plots in shearlocalized chips formed in machining simulations of Ti 6Al4V for a depth of cut of 0.25 mm at different cutting speeds varying from 10 to100 m/min using 0° rake angle tool…………………………….100 7.8 Temperature and equivalent plastic strain contour plots in shearlocalized chips formed in machining simulations of Ti 6Al4V for a depth of cut of 0.5 mm at different cutting speeds varying xi from 10 to100 m/min using 0° rake angle tool……………………………104 7.9 Effect of cutting speed on (a) the shear zone temperature and (b) the rake face temperature for FEM simulations of Ti 6Al4V at cutting speeds from 10 to 100 m/min for two different depths of cut of 0.25 and 0.5 mm………………………………………………...104 7.10 Effect of cutting speed on (a) equivalent plastic strain and (b) plastic strain rate for FEM simulations of Ti 6Al4V at cutting speeds varying from 10 to 100 m/min for two different depths of cut of 0.25 and 0.5mm……………………………………………..…..105 7.11 Cutting and thrust force plots with time for finite element simulations of Ti 6Al4V for a cutting speed of 30 m/min and depth of cut of 0.5 mm using different rake angles of 15°, 0°, 15°, 30° and 45°………………….107 7.12 Temperature and equivalent plastic strain contour plots in the chip formed in machining simulation of Ti 6Al4V for a depth of cut of 0.25 mm and a cutting speed of 5 m/min using a 0° rake angle tool……..108 7.13 Temperature and equivalent plastic strain contour plots in the chip formed in machining simulation of Ti 6Al4V for a depth of cut of 0.5 mm and a cutting speed of 5 m/min using a 0° rake angle tool………109 7.14 Effect of cutting speed on (a) average cutting force and (b) average thrust force for FEM simulations of Ti 6Al4V at cutting speeds from 10 to 100 m/min for two different depths of cut of 0.25 and 0.5 mm, respectively…………………………………………………110 7.15 Effect of cutting speed on (a) average power consumption and (b) number of segments for FEM simulations of Ti 6Al4V at cutting speeds from 10 to 100 m/min for two different depths of cut of 0.25 and 0.5 mm, respectively………………………………….111 7.16 Temperature and equivalent plastic strain contour plots in chip xii segmentation in machining simulations of Ti 6Al4V for a depth of cut of 0.5 mm at a cutting speed of 30 m/min using different rake angles of 15°, 0°, 15°, 30° and 45…………………………………...114 7.17 Effect of rake angle on (a) shear zone and rake face temperatures (b) equivalent plastic strain in the shear zone (c) plastic strain rate and (d) cutting forces for FEM simulations of Ti 6Al4V at cutting speed 30 m/min and depth of cut of 0.5 mm……………………………. 115 7.18 Temperature and equivalent plastic strain contour plots in chip segmentation in machining simulations of Ti 6Al4V for a depth of cut of 0.5 mm and cutting speed of 30 m/min using different friction coefficients (0.3, 0.5, 0.7, 0.9)…………………………………....118 7.19 Effect of friction coefficient on (a) shear zone temperature and (b) rake face temperature for FEM simulations of Ti 6Al4V at a cutting speed of 30 m/min and depth of cut of 0.5 mm……………………118 7.20 Effect of friction coefficient on (a) equivalent plastic strain and (b) average cutting forces for FEM simulations of Ti6Al4V at a cutting speed 30 m/min and a depth of cut of 0.5 mm…………………..118 7.21 FEM and experimental results of cutting and thrust force compared for feed rates 0.02, 0.05, 0.075, 0.1mm/rev………………………………. 121 7.22 Results of cutting forces obtained from FEM simulations compared with experimental data for feed rates 0.127 and 0.35mm/rev…...122 7.23 Results of chip comparison from FEM simulations and experimental data for 180m/min cutting speed and feed rates of 0.04, 0.06, 0.08, and 0.1mm/rev………………………………………………………..124 7.24 Results of chip morphology from FEM simulation conducted at 1.2 m/s and 0.5mm/rev feed rate compared with that of experimental data.125 1 CHAPTER 1 INTRODUCTION 1.1 State of Art: Finite Element Simulation of the Cutting Process Improvements in manufacturing technologies require better modeling and simulation of metal cutting processes. Theoretical and experimental investigations of metal cutting have been extensively carried out using various techniques. On the other hand, complicated mechanisms usually associated in metal cutting, such as interfacial friction, heat generated due to friction, large strains in the cutting region and high strain rates, have somewhat limited the theoretical modeling of chip formation. So, many researches are focusing on computer modeling and simulation of metal cutting process to solve many complicated problems arising in the development of new technologies. One of the stateofart efforts in manufacturing engineering is the finite element simulation of the metal cutting process. These simulations would greatly enhance our understanding of the metal cutting process and in reducing the number of trial and error experiments, which is used traditionally for tool design, process selection, machinability evaluation, chip formation and chip breakage investigations. According to a comprehensive survey conducted by the CIRP Working Group on Modeling of Machining Operations during 19961997 [1], among the 55 major research groups active in modeling, 43% were active in empirical modeling, 32% in analytical modeling, and 18% in numerical modeling in 2 which finite element modeling techniques are used as the dominant tool. More attention to the finite element method has been paid in the past decade in respect to its capability of numerical modeling of different types of metal cutting problems. Advantage of finite element method is the entire process can be simulated using a computer. Compared to empirical and analytical methods, finite element methods used in the analysis of chip formation have advantages in several respects, namely, (1) Material properties can be handled as a function of strain, strain rate, and temperature. (2) Interaction between the chip and the tool can be modeled as sticking and sliding. (3) Nonlinear geometric boundaries, such as the free surface of the chip can be represented and used. (4) In addition to the global variables such as, the cutting force, thrust force and chip geometry, local variables, such as stresses, temperature distributions, etc., can also be obtained. Finite element method has been used to simulate machining operations since the early 1970s [45]. With the development of faster processors and larger memory, model limitations and computational difficulty have been overcome to a large extent. In addition, more commercial FE codes are being developed for cutting simulations, including ABAQUS , AdvantEdge , DEFORM 2D , LS DYNA , FORGE 2D , MARC , FLUENT and ALGOR . Significant progress has been made in this field such as: 3 (1) Lagrangian approach is used to simulate the cutting process including incipient chip formation. (2) Segmented chip formation is modeled to simulate highspeed machining. (3) 3D simulation is performed to analyze oblique cutting. (4) A diversity of cutting tools and workmaterials is used in the simulation of cutting process. 1.2 Historical Developments The earliest finite element chip formation studies simulated the loading of tool against a preformed chip avoiding the problems of modeling large flows [46]. Small strain elasticplastic analysis demonstrated the development of plastic yielding along the primary shear plane as the tool was displaced against the chip. This work had a number of limitations, making it only of historical interest. The limitations of this initial work were removed by Shirakashi and Usui [3], who developed an iterative way of changing the shape of the preformed chip until the generated plastic flow was consistent with the assumed shape. They also included realistic chip/tool friction conditions and material flow stress variations with strain, strain rate and temperature measured from high strain rate Hopkinson bar tests. The procedure of loading a tool against an already formed chip greatly reduced computing capacity requirements. The justification of the method was that it gave good agreement with experiments but it did not follow the actual path by which a chip should be formed. Rigid–plastic modeling however, does not require the actual loading path to be followed. Iwata et al. [4] developed steady state rigidplastic modeling (within a eulerian 4 framework) adjusting an initially assumed flow field to bring it into agreement with the computed field. They included friction, work hardening, and a chip fracture criterion. Experiments at low cutting speeds supported their predictions. The mid1980s saw the first nonsteady chip formation analyses, following the development of a chip from first contact of a cutting edge with the workpiece as in machining. Updated Lagrangian elasticplastic analysis was used, and different chip/ work separation criteria at the cutting edge were developed. Strenkowski and Carol [5] used a strainbased separation criterion. At that time, neither a realistic friction model nor coupling of elasticplastic to thermal analysis was included. The 1990s have seen the development of nonsteady state analysis, from transient to discontinuous chip formation, the first threedimensional analyses, and the introduction of adaptive meshing techniques particularly to cope with the flow around the cutting edge of a tool. A simple form of remeshing at the cutting edge, instead of the geometrical crack, was introduced to accommodate the separation of chip from the work. Both rigidplastic and elasticplastic adaptive remeshing softwares have been developed and are being applied for chip formation simulations [14, 15]. Marusich and Oritz [14] developed a twodimensional finite element code for the simulation of orthogonal cutting that includes sophisticated adaptive remeshing, thermal effects, a criterion for brittle fracture and tool stiffness. They seem to be more effective than Arbitrary LagrangianEulerian (ALE) methods in which the mesh is neither fixed in space nor in the workpiece. Thus, the 1970s to the 1990s has seen the development and testing of finite element techniques for chip formation processes and during this period, many researchers have concentrated 5 more on the development of the new methods in the finite element simulations of metal cutting [2]. 1.3 Principles of Metal Cutting Metal cutting is classified as the secondary process by which material is removed to transform a raw material to a part with certain shape, size, dimensional tolerance and surface finish. The theory of machining is concerned with the various features of the cutting process including the forces, strain and strain rates, temperatures, and wear of cutting tools. All metal cutting operations, such as turning, drilling, boring, milling, grinding, reaming and other metal removal processes produce chips in a similar fashion. Therefore, analysis of chip formation can give better understanding to the mechanics of machining processes. Metal cutting involves concentrated shear along a rather distinct shear plane. As metal approaches the shear plane, it does not deform until the shear plane is reached. It then undergoes a substantial amount of simple shear as it crosses the thin shear zone. There is essentially no further plastic flow as the chip proceeds up the face of the tool. The tool is a single point tool that is characterized by the rake angle α and the clearance angle θ as shown in Fig. 1.1. When the rake face of the tool is in the clockwise position from the workpiece then the rake angle is considered positive and if it is counter clockwise then it is considered negative [6]. A small clearance angle is generally used to keep the tool from damaging the finished surface of the workpiece. The rake face of the tool is the surface over which the chip flows and has a contact length lc, which is the chiptool contact interaction. The prescribed velocity V, known as the cutting speed is in the feed direction. 6 Fig. 1.1 Orthogonal metal cutting model [6] The localized straining in the workpiece enforced by the tool causes plastic deformation of the undeformed chip t, which proceeds to become deformed chip thickness tc. Large forces are generated during the cutting process. The cutting force Fc acts in the direction of cutting velocity and the thrust force Ft acts normal to the cutting velocity in the direction perpendicular to workpiece. A knowledge of the basic force relationships and the associated geometry occurring in the cutting process of metal cutting is a necessity if the solution of engineering problems arising in that field is to be handled by FEM simulations. 1.4 Principles of Finite Element Modeling Finite element analysis is an approximate numerical analysis tool to study the behavior of a continuum or a system to an external influence, such as stress, heat and pressure. This involves generation of a mathematical formulation of the physical process followed by a numerical solution of the mathematics model. Basic concept of finite element method involves division of a given domain into a set of simple subdomains, called, finite elements accompanied with polynomial approximations of solution over 7 each element in terms of nodal values and applying the calculated finite solutions to the whole geometry to solve the problem. The advantage of finite element method is, it provides approximate solutions to complex problems that are difficult to solve analytically. Finite element analysis involves three stages of activity, namely, preprocessing, processing and postprocessing. Preprocessing involves the preparation of data, such as nodal coordinates, connectivity, boundary conditions, and loading and material information. The processing stage involves stiffness generation, stiffness modification, and solution of equations, resulting in the evaluation of nodal variables. Other derived quantities, such as gradients or stresses may be evaluated at this stage. The postprocessing stage deals with the presentation of results. Typically, the deformed configuration, mode shapes, temperature and stress distribution are compared and displayed at this stage. A complete finite element analysis is a logical interaction of these three stages. 1.5 Metallurgical Aspects of Titanium alloys Titanium alloys have been extensively studied over the past few decades due to their important technological applications. Their high strength, low density, corrosion resistance, good formability, weldability, and good metallurgical stability prompted the use of these alloys in a wide variety of applications ranging from aircraft engine and structural components to biomedical implants. Among titanium alloys, Ti 6Al4V accounts for the largest share of the present market and hence studied in depth. Ti 6Al4V is currently used in a wide range of low and high temperature applications, such as blades and other components for turbines in aircraft engine applications, steam turbine blades, 8 marine components, structural forgings and biomedical implants [7]. Despite the increased usage and production of titanium alloys, they are expensive when compared to many other metals because of the complexity of the extraction process, difficulty of melting, and problems associated with fabrication and machining. In order to improve processing as well as to lay ground to new titanium alloys, it is important to understand the deformation mechanisms and microstructural evolutions associated with these deformations. Titanium alloys are usually divided into four main groups according to their basic metallurgical characteristics: αalloys, near αalloys, αβ alloys, and βalloys [8]. αalloys: This group contains α stabilizers, sometimes in combination with neutral elements, and hence have an αphase microstructure. One such singlephase αalloy, Ti5Al2.5Sn, is still available commercially and is the only one of its type to survive besides commercially pure titanium. The alloy has excellent tensile properties and creep stability at room temperature and elevated temperatures upto 300°C. αalloys are used chiefly for corrosion resistance and cryogenic applications. Near αalloys: These alloys are highly αstabilized and contain only limited quantities of βstabilizing elements. They are characterized by a microstructure consisting of αphase containing only small quantities of βphase. Ti8Al1MoIV and Ti6Al5Zr 0.5Mo0.25Si are examples of near αalloys. They behave more like αalloys and are capable of operating at greater temperatures of between 400°C and 520°C. αβ alloys: This group of alloys contains additions of α and βstabilizers and possess microstructures consisting of mixtures of α and βphases. Ti 6Al4V and Ti4Al 2Sn4Mo0.5Si are its most common alloys. They can be heattreated to high strength 9 levels and hence are used chiefly for high strength applications at elevated temperatures of between 350°C and 400°C. βalloys: These alloys contain significant quantities of βstabilizers and are characterized by high hardenability, improved forgeability, and cold formability as well as high density. Basically, these alloys offer an ambient temperature strength equivalent to that of αβ alloys but their elevated temperature properties are inferior to those of the αβ alloys. Ti 6Al4V, an alloy introduced in 1954, comes as close to being a generalpurpose grade as possible in titanium. In fact, it is considered as the workhorse of titanium alloys and is available in all product forms. In Ti 6Al4V, both α and βphases are Al solid solutions in Ti. Fig 1.2 Microstructure of Ti 6Al4V alloy showing primary αphase and transformed β grains [44] Various impurity atoms, such as O, C, N, and H are usually present. βphase may be stabilized at room temperature by adding βstabilizing elements such as V, Fe and Mn. At room temperature, stabilized βphase contains more V than nominal 4%. Above 10 527°C, α transforms to βphase, while above 980°C the whole microstructure is composed of equiaxed β grains. The flow stress of this alloy is strongly dependent on temperature and deformation rate [10]. At temperatures above 527°C, the flow stress decreases sharply with temperature while the strain rate sensitivity increases. The flow mechanisms and kinetics are different in these two phases. This renders to a large number of deformation mechanisms responsible for the macroscopic behavior of the alloy. Their identification is important for understanding the mechanical response of this alloy. 1.6 Machinability aspects of Ti 6Al4V Machinability is defined as the ease or the difficulty with which a material can be machined under a given set of cutting conditions including cutting speed, feed, and depth of cut. It is mainly assessed during the cutting operation by measuring component forces, chip morphology, surface finish generated and tool life. The machinability of Ti 6Al4V and other titanium alloys has not kept pace with advances in manufacturing processes due to their several inherent properties [7, 8]: 1) Its high chemical reactivity with almost all tool materials results in rapid wear of the tool at high cutting speeds. Also the tendency to weld to the cutting tool during machining increases leading to chipping and premature failure. 2) Its low thermal conductivity increases the temperature at the tool/workpiece interface, which affects the toollife adversely. 3) Its high melting temperature and high temperature strength further impairs its machinability. 11 4) Besides high cutting temperatures, high mechanical pressure and high dynamic loads in machining of titanium alloys result in rapid tool wear. 5) Its low modulus of elasticity can cause slender workpieces to deflect more than comparable pieces of steel. This can create problems of chatter, tool contact and holding tolerances. Despite the increased usage and production of this alloy, it is expensive when compared to many other metals and alloys because of the complexity of the extraction process, difficulty of melting and problems during machining and fabrication. Serious vibrations are often encountered during machining that impose limit on the material removal rate, and consequently, productivity. Also this alloy is spectacular to machine because of sparks generated at high speeds. Most tool materials used for machining this alloy wear rapidly even at moderate cutting speeds. Titanium alloys are generally difficult to machine at cutting speeds over 30 m/min with HSS steel tools and over 60 m/min with cemented tungsten carbide tools. Other types of tool materials, including ceramic, diamond and cubic boron nitride are highly reactive with titanium alloys and consequently not used in the machining of these alloys. These problems can be minimized by employing very rigid machines, using proper cutting tools and setups, using low cutting speeds, maintaining high feed rates, minimizing cutting pressures, providing copious coolant flow and designing special tools or nonconventional cutting methods. 1.7 Mechanism of ShearLocalized Chip Formation in Ti 6Al4V Titanium and other aerospace structural alloys are extremely difficult to machine at high cutting speeds due to limitations associated with its several inherent properties. It 12 has been observed that in metal cutting the thermomechanical behavior at the workpiece/tool interface significantly influences the chip morphology, which in turn affects the tool life. In order to increase tool life and productivity in machining these alloys it is necessary to study the mechanism of chip formation and its effect on machinability and tool life. Depending on the type of workmaterial, its metallurgical conditions and the cutting conditions used, three types of chip formation are commonly encountered in metal cutting process. They are the continuous chip, shearlocalized chip and discontinuous chip. Traditionally most of the investigations on metal cutting have focused on the continuous chip formation because continuous chip is an ideal chip for analysis as it is relatively stable and many conditions can be simplified. However, long continuous chips are not preferred in machining because, in practice they interfere with the process and may cause unpredictable damage on machined surface and tool. Shearlocalized chips are found in the case of most difficulttomachine materials with poor thermal properties. This type of chip on the other hand is easier to break and considered as a relatively ideal chip to dispose off when the machining process is automated. In the case of machining Titanium alloys, chip is segmented and the strain in it is not uniformly distributed but is confined mainly to narrow bands between the segments. Whereas in continuous chip formation, the deformation is largely uniform. The sequence of events leading to segmented chip formation when machining Ti 6Al4V was described by Komanduri and Von Turkovich [11] based on a detailed study of video sequence of low speed machining experiments conducted inside the scanning electron microscope, high speed movie films of the chip formation process at higher speeds and the micrographs of midsections of the chips. The mechanism of chip formation when 13 machining Ti 6Al4V was found to be different from the continuous chip formation [26]. There are two stages involved in this process. One stage involves plastic instability and strain localization in a narrow band in the primary shear zone leading to catastrophic shear failure along the shear surface. The other stage involves gradual buildup of segment with negligible deformation by the flattening of the wedgeshaped work material ahead of the advancing tool. Generally, adiabatic shearing caused by thermomechanical instability is held responsible for this process. This nature of instability, frequently referred to as adiabatic shear, was originally expressed by Recht [12] as one in which the rate of thermal softening exceeds the rate of strain hardening i.e., the slope of the shear stressshear strain curve becomes zero. Also, alternative theories have been formulated based on damage models and crack formation processes which is probably applicable at low speeds. Finite element simulations of machining a Ti 6Al4V alloy, allows study of chip formation and the mechanism of chip segmentation in detail. Such simulations have shown that it is indeed possible to form strongly segmented chips by the described process without the necessity of crack formation. 1.8 Thesis outline A brief description of each chapter in this study is given in the following: Chapter 1 gives a brief introduction of finite element simulation of the cutting process, its historical developments and principles, metallurgical aspects and machinability issues of Ti 6Al4V alloy and mechanism of shearlocalized chip formation in Ti 6Al4V. 14 Chapter 2 presents literature review on numerical analysis of chip segmentation, experimental work on Ti 6Al4V alloy and finite element simulations of shearlocalized chip formation in machining a Ti 6Al4V alloy. Chapter 3 contains finite element formulations of metal cutting mechanics, friction along the toolchip interface, contact conditions, adaptive remeshing and chip formation. Chapter 4 gives a brief description of machinability issues of Ti 6Al4V alloy and outlines the rationale and motivation behind this work along with the objectives and research approach. Chapter 5 deals with the mechanism of shearlocalized chip formation and explains the criterion used in this study to simulate it. This chapter also illustrates the metallurgical aspects of Ti 6Al4V alloy that influence this type of chip formation in machining. Chapter 6 gives a description of material model used to represent deformation behavior of Ti 6Al4V alloy under high strains, strain rates, and temperatures. This chapter also derives the equations of Recht’s catastrophic shear failure criterion and stress update algorithm used in the userdefined subroutine code (UMAT). Chapter7 presents the physical properties of Ti 6Al4V alloy, parameters of JohnsonCook material model, simulation approach and cutting conditions used. This chapter also provides chip formation process, temperature and plastic strain distribution in the chip, and comparison of finite element simulations with experimental results reported in the literature. Finally, it discusses the effect of cutting speed, feed rate, rake 15 angles and coefficient of friction on cutting forces, temperatures, strains and chip segmentation. Chapter 8 draws conclusions on the work done in this study and presents proposed scope for future work. 16 CHAPTER 2 LITERATURE REVIEW 2.1 Numerical Study of ShearLocalized Chip Formation Over the past ten years or so, numerical study of the machining processes has been the subject of intense research in which various aspects of shearlocalized chip formation, discontinuous chip formation and algorithms for element separation have been addressed. Various criteria for Shearlocalized chip formation, such as effective strain criterion, maximum principal stress criterion, maximum shear stress criterion, catastrophic shear failure criterion and so on have been utilized by many researchers for the chip formation process using FEM. According to Xie et al. [13], most shearlocalized chips are formed by flow (shear) localization during the chip deformation. Some bands of intense shear dividing the chip into segments occur in metal cutting process. Accordingly, this band is a very thin layer with extremely concentrated shear strains, which may cause chip to become easily separated and broken. They developed a mechanistic model to predict quantitatively the critical cutting conditions for a shearlocalized chip formation. This was done by establishing a relationship between the flow localization parameter and related governing cutting conditions, i.e. cutting speed and feed rate. Flow localization parameter β is defined as 17 ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ + + − ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ + ⎟⎠ ⎞ ⎜⎝ ⎛ ∂ ∂ = − + Vf K Vf K n Vf K c n T m γ γ γ ρ τ γ β 1 1 1 1 1.328 0.664 1 1 1.328 0.9 3 (2.1) where m is strain rate sensitivity parameter, n is strain hardening exponent, K1 is thermal diffusivity of work material, γ is shear strain and T is temperature. The governing cutting process parameters, cutting speed V and feed rate f are associated with possible onset of shear localization. The flow localization parameter, β is used to rank the tendency for strain concentration within a material. A certain critical value of β must exceed to reach at the onset of strain localization. The above equation is used to predict the flow localization parameter, β, for given cutting conditions and material properties. The value of β increases as either the cutting speed V increases or the feed rate f increases. Usually, β at which shear banding is possible is determined experimentally and for Ti 6Al4V the value is 4.41. The formation of shearlocalized chip involves several material, mechanical, and thermophysical properties including density, specific heat, strain hardening exponent, thermal diffusivity, strain rate sensitivity and conductivity. Basically, as a cutting condition (Vf) reaches the critical value at which shearlocalized chips are formed, the plastic deformation rate becomes high and toolworkpiece friction becomes more severe, increasing the rate of heat generation. The adiabatic or quasiadiabatic condition may be reached due to high accumulation of heat. In this case, temperature can be very high locally in some places of the workpiece, resulting in further thermal softening. This further thermal softening reduces strainhardening capacity so the 18 instability takes place in a narrow band of the chip. Thus, β may be used as material property to judge and predict shear localization. Marusich et al. [14] implemented a fracture model that allows for arbitrary crack initiation and propagation in the regime of shearlocalized chips. They presented a model of highspeed machining using a lagrangian code to simulate large unconstrained plastic flow with continuous adaptive meshing and remeshing as principal tools for sidestepping the difficulties associated with deformation induced element distortion. Accordingly, when slip induced transgranular cleavage is the dominant mechanism, fracture of mild steel can be described in terms of a critical stress criterion. The critical stress σf appears to be relatively independent of temperature and strain rate and can be inferred from toughness KIC through the small scale yield relationship σf = l KIC 2π (2.2) The critical distance l correlates with the spacing of the grain boundary carbides. Under mixedmode conditions, such as expected in machining, the crack might kink or follow a curved path as it grows. To predict the crack trajectory under conditions of brittle fracture, they adopted maximum hoop stress criterion, according to which crack propagates along the angle θ from the crack face at which hoop stress σθθ attains a relative maximum. Combining maximum hoop stress along the angle θ θ max σθθ and critical stress criterion, the criteria for mixedmode crack growth is given by θ max σθθ (l, θ)= σf (2.3) The critical crack tip opening displacement (CTOD) criterion for mode I crack propagation can be recast as the attainment of critical value εp f of the effective plastic strain at a distance l directly ahead of the crack tip. The criterion can be expressed as 19 θ max εp(l, θ)= εp f (2.4) with the understanding that crack propagates at an angle θ for which the criterion is met. The critical effective plastic strain can be estimated as εp f ≈2.48e1.5p/σ (2.5) where p =σkk/3 is the hydrostatic pressure. Ceretti et al. [15] implemented the Cockroft and Latham damage criterion for material fracture [16]. According to the numerical model, material fracture is simulated by deleting the mesh elements that have been subjected to high deformation and stress. Accordingly, the damage criterion is evaluated by the Equation 2.6: Ci = ε σ ε σ σ d f ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ∫ * 0 (2.6) where Ci is the critical damage value given by uniaxial tensile test, εf the strain at the breaking condition, σ* is the maximum stress. The criterion predicts the damage when the critical value Ci is exceeded. The damage is evaluated for each element of the workpiece. Element deletion occurs when damage value is reached. To simulate the shearlocalized chip formation they used commercial software DEFORM2D and customized it with new algorithms incorporating the damage criterion. Hashemi et al. [17] developed a fracture algorithm for simulating chip segmentation and separation during orthogonal cutting process. This criterion evaluates the principal stress at each node in each computational cycle. If the magnitude of principal stress exceeds a predetermined value, which can be taken as material fracture strength, a crack is assumed to initiate and propagate along the direction normal to the stress vector. 20 Obikawa and Usui [18] proposed an effective strain based criterion in their FEM simulation of serrated chip formation in cutting Ti 6Al4V. They postulated that when the effective plastic strain at specific node reaches the preset critical value, this node is then separated indicating crack initiation and propagation. According to them, in the machining of titanium alloys, serrated chips are produced due to ductile cracks propagating from the chip free surface. The following fracture criterion was applied for the crack initiation and propagation: c ε p >ε (2.7) where ⎟⎠ ⎞ ⎜⎝ − + ⎛ + ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = − 293 273 0.09 exp 37.8 , 0 100 max 0.075 ln . ε ε σ ii θ p c where max [,] is a function giving the maximum value in the bracket. This criterion is based on equivalent plastic strain. This predetermined critical strain value εc is a function of strain, strain rate . p ε , hydrostatic pressure σ ii , and temperature θ. Rice [19] simulated shearlocalized chip formation using critical shear strain criterion. They developed the failure criterion from the studies of mechanism of chip segmentation using photographs of segmented chips taken at various stages during machining. Accordingly, they proposed that chip transforms from continuous to segmental when nominal shear strain in the primary shear zone reaches critical value. Iwata et al. [20] proposed the following stressstrain based criterion in their FEM analysis of orthogonal cutting as: ( ∫ ε ε f 0 +b1σm+ b2) dε = b3 (2.8) 21 where the constants b1, b2 and b3 are given as functions of metallurgical properties However, the authors claimed that obtaining these constants needs complicated experiments that requires high pressure. Shivpuri et al. [21] implemented the deformation energybased criterion proposed originally by Cockroft and Latham [16] to simulate shearlocalized chip formation of Ti 6Al4V using FEM. This criterion was based on critical damage value given by: Ci= ε σ ε σ σ d f ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ∫ * 0 (2.9) The Ci value is a workpiece material constant and does not depend on the workmaterial or the tool material. The Ci value is a function of temperature and microstructure of the point of interest and can be expressed as Ci = f (T, m), where m denotes the microstructure. When the temperature is below the βtransformation temperature, the microstructure is presented by α+β, otherwise microstructure is β. The critical damage value is evaluated for each element of the workpiece. Element deletion occurs when damage value is reached. Thus the crack is initiated and propagated. In the early 1970’s Nakayama [22] found that sawtooth chips were produced when highly cold worked (60 percent reduction) 40/60 (Zn/Cu) brass was turned under orthogonal conditions. Based on that he proposed a theory of sawtooth chip formation. Accordingly, shear crack initiates at the free surface of the chip in the primary shear zone and runs from the surface downward along shear plane towards the tool tip. With further advance of the tool, the chip glides outward like a friction slider along the cracked surface until the next crack forms and a new cycle begins. A shear crack will initiate at the free surface of the chip where the crack arresting normal stress is zero and proceeds 22 downward along the shear plane toward the tool tip. Initially the crack will be continuous across the width of the chip but will become discontinuous as higher crack arresting normal stresses are encountered. Owen and Vaz Jr. [23], applied computational techniques to investigate highspeed machining. They focused on the simulation of Ti 6Al4V machining involving material failure due to adiabatic strain localization and addressed important issues, such as adaptive mesh refinement and material failure. According to them, material failure under adiabatic strain localization conditions results from the accumulation of large plastic deformation and microscopic damage. They employed a twoparameter model to describe material failure, i.e. a failure indicator (or fracture criterion), I, and an energy release factor, Wr. Accordingly, the former indicates failure onset and the latter defines the amount of energy released during the softening process as shown in the Fig. 2.1. Fig. 2.1 Stressstrain diagram for a combination of strain hardening, thermal softening and failure softening [23] The failure softening path, BC, is governed by the softening modulus, Et, as Et = ∂ε ∂ Y T =  r Y cr e W T h 2 , 2 (2.10) 23 where he is the characteristic length of the particular element, TY is the true stress of the material at yield point and TY,cr is the critical yield stress of the material where failure initiates. They adopted a material failure criterion based on the uncoupled integration of Lemaitre’s ductile model. Lemaitre postulated that damage progression is governed by void growth represented by the damage evolution equation s r Y Y D D ⎟ ⎠ ⎜ ⎞ ⎝ ⎛ − − = ∂ = − ∂ 1 . . . ψ γ γ (2.11) where .D is the damage variable, r and s are damage parameters, Ψ is the dissipation potential and Y is the damage strain energy release rate. They also incorporated a stress update algorithm which is called as operatorsplit algorithm to which the computation of the fracture indicator and failure softening were added. The methodology consists of decomposing the problem into elastic and plastic components, in which the former assumes that deformation is entirely elastic whereas the latter solves a highly nonlinear system of equations comprising the constitutive relations, evolution laws and plastic consistency using the elastic predictor stage as the initial condition. In order to overcome the problem of severe element distortion when using lagrangian formulations in their simulations, they incorporated error estimator based on the principle that the adaptive procedure should not only capture the progression of the plastic deformation but also provide refined meshes at regions of possible material failure. Samiatin and Rao [24] developed another model for shearlocalization, which incorporates a simple heat transfer analysis, and material properties, such as the strainhardening rate, the temperature dependence of flow stress, and the strain rate sensitivity of the flow stress to establish the tendency towards the localized flow. Using their 24 literature data, they found the nonuniform flow in metal cutting to be imminent when the ratio of the normalized flowsoftening rate to the strain rate sensitivity is equal to or greater than 5. Recht in 1964 [12] developed a classical model of catastrophic shear instability for shear localization in metals under dynamic plastic conditions. Accordingly, catastrophic shear will occur at a plastically deforming location within a material when the slope of the true stressstrain function becomes zero. He formulated a simple criterion for catastrophic slip in the primary shear zone based on the thermophysical response of the work material under the conditions of cutting. The catastrophic slip can be written as: where τ, ε and θ refer to the shear stress, shear strain, and temperature, respectively. Material will shear catastrophically when this ratio lies between zero and one; catastrophic shear will be imminent when the ratio equals to one. Komanduri and Hou [25] developed a thermal model for the thermoplastic shear instability in the machining of a Ti6Al4V alloy. It is based on the analysis of the shearlocalized chip formation process due to various heat sources (primary, preheating and image) in the shear band. They determined the temperature in the shear band analytically using the Jaeger’s classical stationary and moving heat source methods and used Recht’s catastrophic shear instability model to determine the onset of shear localization. Accordingly, the shear stress in the shear band is calculated at the shear band temperature and compared with the shear strength of the bulk material at the preheating temperature. According to Recht, if the shear stress in the shear band is less than or equal to the shear (2.12) 25 strength of the bulk material, then shear localization is imminent. The cutting speed at which this occurs is taken as the critical speed for the onset of shear localization. The effect of depth of cut on the critical speed is determined and found that lower the depth of cut, higher the critical speed for onset of shear localization. The cutting speed for the onset of the shear localization in the machining of Ti 6Al4V was found to be extremely low, ~0.42 m/min for a depth of cut of 0.2 mm. A bestfit relationship between the critical cutting speed for shear localization and depth of cut was developed using the analytical data and is given by 1.0054 0 0.082667 V = × a − cri (2.13) 2.2 ShearLocalized Chip Formation in Ti 6Al4V: Experiments Numerous studies on the machining of titanium alloys (analysis of chip formation and cutting forces) have been carried out in a range of cutting velocities lower than 5 m/s, by Komanduri and von Turkovich [11], Komanduri [26], Narutaki and Murakoshi [47], Larbi [50], Bayoumi and Xie [39] and Diack [49]. In the work of Hoffmiester et al. [48] and Molinari et al. [28], larger velocities were considered in the range of 20 to 100 m/s. These studies illustrated several unique features associated with the machining of these alloys, including the following [26]: (1) The role of poor thermal properties of titanium alloys which interact with the physical properties in controlling the nature of plastic deformation (i.e. strain localization) in the primary zone is illustrated. (2) Periodic gross inhomogeneous deformation occurs in the primary zone. 26 (3) Instability in the chip formation process results in the segmented or cyclic chip. (4) Oscillations in the cutting and thrust components of force cause chatter and the need to have a rigid toolworkmachine tool system. (5) High toolchip interface temperatures and high chemical reactivity of titanium in machining with almost any tool material are responsible for the rapid tool wear. (6) The low modulus of elasticity which decreases rapidly, even at moderate temperatures, causes undue deflections of the workpiece, especially when machining slender parts, and inaccuracies in the finished part. Chip formation studies were conducted at various machining speeds from an extremely low speed of 0.127 mm/min to a moderately high speed of 5.1 m/s. The low speed experiments were conducted inside a scanning electron microscope and the cutting process was recorded on a video tape. Chip formation studies at higher cutting speeds were conducted on a lathe with the aid of a HiCam highspeed movie camera (camera speed up to 8000 frames/s) using the technique developed earlier by Komanduri and Brown [19]. The sequence of events leading to cyclic chip formation when machining titanium was described by Komanduri and von Turkovich [27] based on a detailed study of videotapes of low speed machining experiments inside the scanning electron microscope, highspeed movie films of the chip formation process at higher speeds and the micrographs of the mid section of the chips. Accordingly, they described two stages in the chip formation process. One stage involves plastic instability and strain localization in a narrow band in the primary shear zone leading to catastrophic shear failure along a shear surface. The other stage involves gradual buildup of the segment with negligible 27 deformation by the flattening of the wedgeshaped work material ahead of the advancing tool. Molinari et al. [28] carried out experimental analysis of shear localization and chip segmentation in Ti 6Al4V at cutting speeds in a range from 0.01 to 73 m/s. To cover a wide range of cutting speeds, two different devices were used. The low cutting speeds (0.01 to 1.0 m/s) were obtained with a universal highspeed testing machine. The second arrangement was constituted by an airgun setup for speeds from 1.0 to 73 m/s. Accordingly, to avoid fracture of the tools at high impact velocities, small depths of cut were used. All tests were carried out with carbide tools with a rake angle 0° and the tools were square shaped without chipbreaker. The collected chips were embedded into a resin and the lateral section polished and etched for observation in the optical microscope. The following conclusions were drawn from their experimental studies: (1) Chip segmentation was observed to be related to adiabatic shear banding. (2) Adiabatic shear bands are the manifestation of a thermomechanical instability resulting in the concentration of large shear deformations in narrow layers. (3) For velocities lower than 1.2 m/s, chip serration is related to the development of deformed shear bands as shown in Fig. 2.2, which are the manifestation of thermomechanical instability. However, at low values of cutting velocities, the instability process is weak and the localization is not as sharp as for higher velocities. (4) The patterning of adiabatic shear bands were observed experimentally by measurements of shear band width and chip segment width. Patterning is seen to be strongly dependent on cutting velocity V. 28 Fig. 2.2 Adiabatic shearlocalized chip in a Ti 6Al4V alloy obtained by orthogonal cutting at a cutting speed of 1.2 m/s [28] Barry et al. [29] conducted orthogonal cutting tests to investigate the mechanism of chip formation and to assess the influences of such on acoustic emission (AE) for a Ti 6Al4V alloy of 330HV with an uncoated P10/P20 carbide tool. Here, AE refers to the transient displacement of the surface of a body, of the order of 1012 m, due to the propagation of high frequency elastic stress waves. The surface displacement is detected by the AE sensor and typically output as a proportional voltage. Accordingly, orthogonal cutting tests were undertaken using a Ti 6Al4V alloy of 330 HV with uncoated P10/P20 carbide tool with a rake angle of 6° and a clearance angle of 12°. The workpiece used was in the form of a 25 mm diameter disc, 1.1mm in width and was held on a mandrel during cutting. The tests were performed on a Daewoo PUMA 43A CNC lathe under a constant surface speed control. Acoustic emission signals were captured with a Kistler wide band piezoelectric AE sensor and the signal amplified and conditioned using a Kistler Piezotron unit. Fig. 2.3 shows the influence of cutting speed and feed on chip morphology in the orthogonal cutting of Ti 6Al4V. They classified all chips obtained for different cutting conditions in the range 20100 μm feed and 0.253 m/s cutting speed as either aperiodic sawtooth or periodic saw tooth. Accordingly, it was seen that with low values of cutting 29 speed and undeformed chip thickness (e.g., 20 μm), aperiodic sawtooth chips were produced. Increase in either or both of these parameters resulted in a transition from aperiodic to periodic sawtooth chip formation. They observed occurrence of welding between chip and tool during machining and with degree of welding increasing with cutting speed. Fig. 2.3 Chip morphology as a function of cutting speed and feed in orthogonal cutting of a Ti 6Al4V alloy [29] In machining of Ti 6Al4V, Barry et al. [29] observed that catastrophic failure occurs not only within the primary shear zone, but also within the weld formed between the chip and the tool rake face and the fracture of such welds appears to be the dominant source of acoustic emission in machining Ti 6Al4V with cutting speeds greater than 0.5 m/s. This is precisely the mechanism proposed by Komanduri and von Turkovich [11] (See Fig.5.1 in chapter 5). Xie and Bayoumi [13] conducted orthogonal cutting tests and various metallurgical analysis techniques were used to examine the chip formation process and the role of shear instability. Cutting speeds were varied from 0.5 to 8.0 m/s and the feed 30 rates from 0.03 to 0.5 mm/rev. The following are their conclusions from the experimental study: (1) The results from metallurgical studies showed no diffusiontype phase transformation in the machined chips, while the Xray diffraction tests identified some nondiffusional phase transformation from βphase into α phase during chip formation. (2) Intensive shear takes place in a narrow zone rather than in a plane as is often assumed by some investigators in the analysis of orthogonal machining process. (3) For each work material there exists a critical value (Vf) of chip load at which shear localized chips were observed. The cutting conditions also influence the shear banding in a way that the shear banding frequency increases with an increase in feed rate or a decrease in cutting speed. Shivpuri et al. [21] conducted experiments on Ti 6Al4V using a CNC turning center at cutting speeds of 60, 120 and 240 m/min, feed rates of 0.127 and 0.35 mm/rev and depth of cut of 2.54 mm. They collected the deformed chips for different cutting speeds and observed the chip morphology under a microscope. The cutting forces were measured with a Kistler dynamometer (Type 9121). A general carbide tool was used with a rake angle of 15° and a relief angle of 6°. They showed that (Fig. 2.4) as the cutting speed is increased, the chip fracture observed at lower cutting speeds gradually reduces, and the flow localization and strain between the serrated chip gradually increases resulting in changing chip morphology (discontinuous chip becoming continuous but serrated). They observed that the crack, which determines the serrated chip during cutting, always occurs in the primary shear zone on the tool tip side and at low cutting 31 speeds, the crack propagates to the tool tip because the temperature in the chip being formed is much lower than the βtransus temperature. Whereas at high cutting speeds, the crack propagates to the free surface as the temperature in the secondary shear zone at the tool face is much higher than that in the free surface. This temperature is above βtransus Fig. 2.4 Shearlocalized chip formed in machining a Ti 6Al4V alloy at cutting speeds of 60 and 120 m/min and feed rate of 0.127 mm/rev [21] temperature and causes microstructural changes resulting in the rise of ductility in the shear zones. The chip just formed during cutting process connects to the workpiece forming the serrated chip morphology. Ribeiro et al. [30] carried out turning tests on Ti 6Al4V with conventional uncoated carbide tools for cutting speeds of 55, 70, 90 and 110 m/min, using 0.1 mm/rev feed rate and 0.5 mm depth of cut. The objective of this work was to optimize the cutting speed for best finish in the machining of titanium alloy. Fig. 2.5 shows the variation of roughness with the length of cut at different cutting speeds. They found 90 m/min to be the optimum cutting speed for best finish. 32 Fig. 2.5 Roughness of workpiece material (Ti 6Al4V) as a function of cutting length at different cutting speeds [30] Lee and Lin [31] investigated the high temperature deformation behavior of Ti 6Al4V alloy by conducting mechanical tests using compression split Hopkinson bar under high strain rate of 2×103 s1 and temperatures varying from 7001100°C in the intervals of 100°C. Fracture features of the specimens after the mechanical tests were observed using optical and scanning electron microscopy. They showed that extensive localized shearing dominates the fracture behavior of this material and adiabatic shear bands run across the specimen. Another important observation made relating to the shear band is the formation and coalescence of voids in an adiabatic shear band, which might lead to variation in mechanical properties of the material. Fig. 2.6 shows a typical array of coalesced voids in a welldeveloped shear band. Initially, the voids are observed to be spherical, but when the diameters reach the thickness of the shear bands, the voids coalesce and their extension along the shear band results in elongated cavities and smoothsided cracks. 33 Fig. 2.6 Voids and cracks in an adiabatic shear band of a Ti6Al4V alloy specimen deformed at 700°C and 2×103 s1 strain rate [31] In each specimen, a transformed adiabatic shear band appears in the microstructure, indicating that a catastrophic localized shear occurred during deformation. Failure analysis of the specimens indicated that adiabatic shear bands are the sites where the fracture of the material occurs, and that the thickness and microhardness of the adiabatic shear bands vary completely with temperature. Picu et al. [10] investigated plastic deformation of Ti 6Al4V alloy under low and moderate strain rates and at various temperatures. Mechanical testing was performed in the temperature range 6501340 K and at strain rates from 103 to 10 s1. A discontinuity in the flow stress versus temperature curve was reported. Fig. 2.7 shows temperature dependence of the yield stress at low strain rates (103 s1) and comparison with the published results. The curve shows discontinuity at temperature T~ 800 K. At temperatures above 800 K, the flow stress sharply decreases with temperature. The discontinuity observed in the flow stresstemperature curve suggests that additional deformation mechanisms become active at that temperature and this leads to dramatic reduction in strain hardening. Texture has a significant effect on the flow stress at 34 temperatures below the discontinuity, while higher temperatures rapidly decrease the texture sensitivity. At temperatures in the 11001350 K range, phase transformation from α to β becomes the impetus behind the mechanical behavior. Also associated with this Fig. 2.7 Temperature dependence of flow stress in a Ti 6Al4V alloy at very low strain rates of 103 s1 [10] transformation is a pronounced variation in the strain rate sensitivity (m), with the sensitivity being higher in the 100% β material. Reissig et al. [32] investigated different machining processes, such as drilling, shot peening and electrochemical drilling. According to them, it is usually very difficult or often impossible to measure directly the surface temperatures introduced by machining processes, such as deep hole drilling. Therefore, they presented a postmortemmethod which allows the determination of maximum temperatures during machining by measuring the local vanadium concentration in a Ti 6Al4V alloy. They proposed a method to determine a local pseudo temperature to determine in regions at a distance as small as 50 nm from the surface. In a Ti 6Al4V alloy, vanadium is used as a βstabilizer and highest vanadium concentration is found in the βphase. By using the method they 35 proposed, in drilling maximum pseudo temperature occurs at the end of the drilled hole and is very close to the βtransus temperature of Ti 6Al4V alloy. Initially the temperature was about 490 K and increased linearly by 5 K/mm between the start of the drill hole and the end. They accounted for this increase to be due to insufficient coolant Fig. 2.8 Pseudotemperature variation with drill depth showing a linear increase [32] supply. The pseudotemperature measurements in shot peening showed that the maximum value was 1106 K in a surface region of about 50 nm thickness and the electro chemical drilling showed no significant increase in pseudo temperature. Lacalle et al. [33] conducted milling tests on Ti 6Al4V alloy to study the tool influence of the tool geometry and coating as well as the influence of cutting conditions on the productivity of the milling process. They used cutting speeds between 11 and 14 m/min, feed between 0.04 and 0.15 mm/tooth, helix angles of 30°, 45°, 60°, number of teeth of 3, 4, 6 and uncoated cemented carbide milling cutters as cutting tools. Based on their studies they reached the following conclusions: 1) Cutting speed has a crucial role in the tool roughening. Serrated chips are found for all the cutting speeds. Fig.2.9 shows serrated chips for 80 m/min and 140 m/min. 36 (a) (b) Fig. 2.9 Serrated chips obtained in machining a Ti 6Al4V alloy at cutting speeds (a) 80 m/min and (b) 140 m/min [33] 2) For high feed values, thick chips are generated and they result in increasing cutting loads, increasing chip deformation and separation. 3) As far as usage of coating materials for HSS steel tool in the machining of titanium alloys, it is found that flank wear can be delayed with TiCN coatings. 2.3 ShearLocalized Chip Formation in Ti 6Al4VFEM Simulations Xie et al. [13] presented a quasistatic finite element model of chip formation and shear banding in orthogonal metal cutting of Ti 6Al4V using a commercial FEA code (NIKE2D™). The updated Lagrangian formulation for plane strain conditions is used in this investigation. The tiebreak sideline was used to separate the newly formed chip from the workpiece surface. The effective plastic strain is used as the material failure criterion and a strainhardening thermalsoftening model for the flow stress is used for shearlocalized chip simulation. A series of finite element simulated machining tests with different tool rake angles ranging from –16° to 20° were carried out at a cutting speed of 37 5 m/s and a feed of 0.3 mm to study the effect of rake angle on shear band angle and cutting forces. Fig. 2.10 (a) and (b) shows machining process modeled with a negative rake angle 8°. The simulated results show that the effective plastic strain is within a narrow area along the shear zone angle, and the field of the maximum shear stress matches with the area of the primary and secondary shear zones. They reported that the finite element model predicts the detailed deformation in front of the tool tip and the initiation of the shear band. (a) (b) Fig. 2.10 Chip formation process showing shear localization in finite element simulation of a Ti 6Al4V alloy at cutting speed of 5m/s and feed 0.3mm using 8° rake angle tool [13] However, from their investigation, it can be observed that they have not used a reliable material model to represent strainhardening and thermalsoftening behavior of Ti 6Al4V during metal cutting process as they mentioned in their study that they arbitrarily composed the thermalsoftening part of the material behavior by making the stress from strainhardening part to thermalsoftening part to decrease by 50%. Xie et al. [13] also mentioned that, they used a debond (node separation) criterion based on critical 38 effective plastic strain for chip formation. It can be observed from their investigation that they have used an arbitrary value of 0.5 for critical effective plastic strain. Also from the Fig. 2.10 it can be noticed that, they have just shown the onset of chip segmentation and did not show complete chip formation process. Obikawa and Usui [18] developed a finite element model for the computational machining of titanium alloy Ti 6Al4V. Fig.2.11 shows the cutting model they used in the finite element analysis. A cemented carbide tool (FGHI) and a titanium alloy (Ti 6Al4V) (ABCDEF) were modeled with four node (linear) quadratic isoparametric elements. The tool was assumed to be rigid and the finite elements in it were used only for temperature calculations. The cutting speed used was 30 m/min, the undeformed chip thickness was 0.25 mm and the tool rake angle was 20°. The parallelogram ABCF was part of the workpiece to be removed as chip. Twodimensional elastic plastic analysis was formulated in updated Lagrangian form and procedures needed for metal cutting were developed for unsteady state heat conduction and material nonlinearities. The friction on the rake face and the complicated flow stress characteristic of the titanium alloy at high strain rates and high temperatures were also considered. Fig.2.12 shows the serrated chip formation process in case of the titanium alloy that was obtained by the FEM simulation. In this investigation, to simulate serrated chip formation they used a geometrical criterion based on fracture strain and defined boundary conditions in such a way that crack propagation occurs in the predetermined path. Hence these simulations lack actual physical phenomenon and mechanism of chip segmentation. 39 Fig. 2.11 Toolworkpiece mesh system used in finite element simulation of orthogonal metal cutting a Ti 6Al4V alloy [18] Fig. 2.12 Serrated chip formation with cutting length in finite element simulation of machining a Ti 6Al4V alloy at cutting speed 30 m/min, feed 0.25 mm and 20° rake angle [18] Maekawa et al. [34] used an iterative convergence method (ICM) to simulate metal cutting process. The ICM uses flow lines which consist of trajectories of particles or a series of finite elements. The chip is supposed to be preformed on the surface of the work material and to be stress free. Calculation proceeds by incrementally displacing the workpiece towards the tool so that a load develops between the chip and the tool. A plastic state develops in the chip deformation zone and it is checked with the assumed 40 chip shape and automatically altered and the calculation is repeated. However, this method simulated only continuous chip. Later, to simulate serrated and discontinuous chips, a sophisticated methodology was developed for approximation of such discontinuity. A failure strain criterion was introduced into the ICM methodology, so that crack initiates at the tool side within the highly deformed workpiece and propagates towards the free surface side, resulting in the periodic segmentation of the chip. Applying this methodology, Ti 6Al4V alloy machining simulations were carried out at 30 m/min cutting speed and 0.25 mm/rev feed rate using 20° rake angle tool. Fig. 2.13 shows the predicted serrated chip shape in titanium alloy machining simulation. They concluded that serration arises in the chip from a small fracture strain of the alloy and not due to adiabatic shear instability. But then, as stated above, they introduced a failure strain criterion so that crack initiates at the tool side within the highly deformed workpiece and propagates towards the free surface side. And as their simulation approach is based on geometric criterion with preformed chip shape, this work does not represent actual mechanism of chip segmentation observed in machining. Fig. 2.13 Serrated chip formed in finite element simulation of orthogonal metal cutting of a Ti 6Al4V alloy at cutting speed 30 m/min, feed 0.25 mm and 20° rake angle [34] Owen and Vaz Jr. [23], simulated machining of Ti 6Al4V alloy in which, they addressed such issues as evaluation of the mesh refinement procedure, strain localization 41 process, and material failure process. The geometry and the initial mesh they used for a rake angle of 3° are depicted in Fig.2.14. An enhanced oneGauss point element was used in the simulation. The simulations employed an error estimator based on uncoupled integration of Lemaitre’s damage model. Tests were undertaken to evaluate the effect of Fig. 2.14 Initial geometry and workpiecetool mesh used in finite element simulation of orthogonal machining of Ti 6Al4V alloy [23] the cutting speed (from 5 to 20 m/s) and rake angle (from 9° to 9°) and to assess the capacity of the remeshing procedure to describe the process evolution. They assumed fracture strain as the governing parameter of material failure in highspeed machining and assessed two indicators of failure, namely, equivalent plastic strain and a fracture strain based on Lemaitre’s damage model. The chip breakage process for a fracture strain based on the equivalent plastic strain was illustrated in Fig 2.15 which shows the elements undergoing a failure softening and fracture propagation. This criterion assumes that fracture initiates when effective plastic strain is equal to failure strain which is greater than or equal to 1.0. 42 However, from the Fig.2.15, it can be observed that only onset of chip segmentation process is shown in this investigation and it does not represent the exact physical phenomenon and mechanism of chip segmentation. (a) (b) (c) Fig. 2.15 Segmented chip formation process in machining simulation of a Ti 6Al4V alloy at 10 m/s cutting speed and 0.5 mm feed using a 3° rake angle tool [23] Baker et al. [35] developed a twodimensional finite element model to simulate highspeed machining of Ti 6Al4V using the commercial software, ABAQUS, together with a special mesh generator programmed in C++. In ABAQUS, they used standard program system, which allows the definition of complex contact conditions, leaves many possibilities to define material behavior, and can be customized in many regards by including userdefined subroutines. Preprocessor they used for automatic remeshing was written in C++ using standard class libraries to ensure that elements never become too distorted and refined mesh is created in the shear zone. They assumed that chip segmentation is caused solely by adiabatic shear band formation and that no material failure or cracking occurs in the shear zone. They used fournoded quadrilateral elements which converge better than triangular elements. The number of elements they used in the simulations varied with the number of segments. About 5000 elements and 7000 nodes were used at the beginning of the simulation and 10000 elements and 12000 nodes near 43 the end. The element edge was about 7 μm in the shear zone and Fig. 2.16 shows the finite element meshes at different stages of the cutting process with segmented chip formation. Fig. 2.16 Segmented chip formation process in finite element simulation of orthogonal metal cutting of a Ti 6Al4V alloy at 50 m/s cutting speed and 0.04 mm cutting depth using a 10° rake angle tool [35] Baker et al. [35] also investigated the influence of the elastic modulus and cutting speed on chip segmentation. They reported that elastic modulus affects the degree of segmentation. They also studied the influence of thermal conductivity on chip segmentation and showed that the degree of segmentation decreases with increasing thermal conductivity. In this investigation, Baker et al. [35] used a pure deformation process to simulate metal cutting process without node separation. The material that overlaps with the tool as 44 the tool advances is removed using remeshing technique. This phenomenon however does not represent actual physical process of chip formation in machining. They also mentioned that they have found the strainhardening part of plastic flow curves experimentally and determined the thermal softening part arbitrarily to facilitate the adiabatic shear bands formation which is unrealistic. In their simulations, it can be observed that high speeds are used, no friction is assumed between tool and the workpiece and heat flow into the tool is neglected. Also, Fig.2.16 does not represent actual mechanism of chip segmentation. Sandstrom and Hodowany [36] modeled the highspeed orthogonal machining of Ti 6Al4V using the commercial FEM code, Mach2D™ at a cutting speed of 10.16 m/s. Simulation results included chip segmentation, dynamic cutting forces, unconstrained plastic flow of material during chip formation, and thermomechanical environments of the workpiece and the cutting tool. They reported good agreement of the simulated cutting force to the experimental data. However, it can be observed that their investigation lack low speed simulations. Shivpuri et al. [21] used a commercial finite element software (DEFORM 2D™) which is a lagrangian implicit code designed for metal forming processes, to simulate the orthogonal machining of Ti 6Al4V. They modeled workpiece as a rigidviscoplastic material owing to large plastic deformations taking place in the primary and the secondary deformation zones during the machining process. Furthermore, high mesh density was defined around the tool tip and excessively deformed workpiece mesh was automatically remeshed as needed during simulation. They modeled the tool as rigid (or elastic) material so that stresses in the tool body can be predicted. Dynamic flow stress 45 model was used to represent material behavior dependence on strain, strain rate, and temperature. Ductile fracture criterion was used to simulate crack initiation and propagation for chip segmentation. Simulations were conducted at cutting speeds of 1.2, 120 and 600 m/min and a feed rate of 0.127 mm/rev. Fig.2.17 shows the plastic strain distribution that presents the initiation of shear plane and the formation of shear zone. Based on simulations, they proposed that chip segmentation during cutting Ti 6Al4V alloy is caused by flow localization within the primary deformation zone and upsetting Fig. 2.17 Equivalent plastic strain distribution in serrated chip simulated at cutting speeds of 1.2, 120 and 600 m/min in a Ti 6Al4V alloy [21] deformation zone by moving tool rake face on the segment to be formed ahead of it. The flow localization induces fracture that separates segment from the workpiece matrix. Flow localization causes crack in the primary deformation zone while the secondary deformation zone controls the chip morphology (segmented or discontinuous). Discontinuous chip was formed at low speed (1.2 m/min) and segmented chips at higher cutting speeds (120 and 600m/min). At low cutting speeds, crack was observed to initiate 46 at the tool tip and propagates to the free surface of the chip while at higher cutting speeds, crack propagates from free surface towards the tool tip. In their investigation, Shivpuri et al. [21] used maximum tensile stress criterion for ductile fracture to simulate serrated chip formation while maximum shear stress criterion accurately models serrated chip formation. Also, it can be observed from Fig.2.17, that chip morphology does not look like a serrated chip. Although most of the researchers reported that their simulation results were in good agreement with the corresponding experimental data and their material model accurately predicts material deformation behavior in simulations, there is no consensus on which criterion represents the best for shearlocalized chip formation in orthogonal metal cutting simulation of Ti 6Al4V alloy. Some of the major drawbacks of the finite element simulations of shearlocalized chip formation in Ti 6Al4V alloy found in the literature include: 1) The methodology proposed by many researchers using noncommercial FEM codes makes it difficult for end users. 2) Few researchers used extensive computer time and engineering effort, which makes their technique not economical to use. 3) The lack of reliable material data under specific process conditions such as, strain, strain rate and temperature that must be used as inputs to any material flow simulation program results in difficulty in applying to practical processes. 4) Most of the finite element simulations of Ti 6Al4V alloy, lack proper experimental validations (especially chip morphology). 5) Lack of low speed machining simulations (< 10 m/min). 47 CHAPTER 3 FINITE ELEMENT FORMULATION OF METAL CUTTING 3.1 Introduction Improvements in manufacturing processes require better modeling and simulation techniques of metal cutting. The process involves very complicated mechanisms such as interfacial frictional behavior, extremely high temperatures, complex stress state between tool chip interface, high strain rates, different types of chip formation, work hardening and thermal softening. Unfortunately, these complicated mechanisms associated have limited the performance of cutting process modeling and in recent years researchers in the metal cutting field are paying more and more attention to the finite element method due to its capability of numerically modeling different metal cutting problems. The advantages of using finite element method to study machining can be summarized as follows: 1) Material properties can be handled as a function of strain, strain rate, and temperature. 2) The toolchip interaction can be modeled as sticking and sliding. 3) Nonlinear geometric boundaries can be represented and used, such as free surface of chip. 48 Although the ideas of finite element analysis may date back much further, it was after further treatment of plane elasticity problem by Clough in 1960, that researches began to recognize the efficacy of finite element method in the engineering field. The advent and continuous improvements of digital computers have made finite element analysis a useful analytical tool which has been applied very efficiently in almost every area of engineering field. One of the most important reasons that finite element analysis is so widely used is that it can be routinely used. There are a definite set of several basic and distinct steps used in the FEM simulations: 1) Discretization of the continuum. 2) Selection of the interpolation function. 3) Determination of the element properties. 4) Assembly of the element properties in order to obtain the system equations. 5) Determination of the constraints and other boundary conditions. 6) Solution of the system equations. 7) Computation of the derived variables. Inspite of the success of the finite element method in solving a very large number of complex problems, there are still many areas where more work needs to be done. Some examples are the handling of problems involving material failures and the modeling of nonlinear material behavior. 3.2 Finite Element Formulation The basic idea of using the finite element method is to seek a solution to the momentum equation [37]: 49 . ij , j i i σ + b = ρυ (i, j = 1,2,3), (3.1) where σij,j is the Cauchy stress tensor, ρ is the current mass density, bi is the body force, vi is the particle velocity in Cartesian coordinates, . i υ is the acceleration, and j indicates partial differentiation with respect to xj. Equation 3.1 satisfies the trajectory boundary condition: σij nj =Ti (3.2) where nj is the unit normal to the boundary and Ti is the surface traction on the plane with a unit normal nj. In this study, displacement based finite element method, which is the most widely used formulation in engineering among many finite element analysis techniques available, is applied to model the machining process. The displacement boundary condition that Equation 3.1 should satisfy at time t is given by: ( ) i i u t = u (3.3) For elasticplastic deformations, the strain rate tensor is usually decomposed into elastic and plastic parts such that ( ), 2 1 , , . ε ij = vi j + v j i . . . i p ij e ε ij =ε +ε (3.4) From Hook’s law, we have kl o ijkl e ij ε e C 1 σ . = − ; ijkl ik jl ij kl e v Gv C Gδ δ δ δ 1 2 2 2 − = + (3.5) where, Ce ijkl is the elasticity tensor, G and v are the shear modulus and poisons ratio, respectively, and ij oσ is the corotational strain given by 50 ik ij ik jk ij o ij σ =σ −σ ω +σ ω . (3.6) with ik ω being the material spin. Using the von Mises flow rule, ij p .ε is obtained as ij ij p s τ ε γ 2 . = (3.7) where sij is the deviatoric stress, 2 1 . . . 2 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = p ij p ij γ ε ε is the effective strain rate and ( )2 1 2 1 ij ij τ = s s is the effective shear stress. The effective strain rate in Equation 3.7 is determined from the yield criterion of the rateindependent materials. For von Mises materials, yield criterion is given by ( ) 0 2 f = J −τ γ = y , (3.8) where J2 is the second invariant of sij, from the consistency condition one can obtain the relation J h s sij ij o 2 . 2 γ = , γ τ ∂ = ∂ h , (3.9) where h is the hardening/softening modulus. Combining Equations 3.2 to 3.5, 3.7 and 3.9 we obtain, ij ijkl ij o C .ε σ = , (3.10) where Cijkl is the elasticplastic incremental tensor and is given by ijkl p ijkl e ijkl C = C − C , ( ) ijkl ij kl p s s G h G C + = 2 2 τ (3.11) For ratedependent materials, Equations 3.4 and 3.5 yield 51 ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = − ijkl kl kl e ij o C s τ σ ε γ 2 . . (3.12) These equations are implemented in all finite element programs. The solutions of engineering problems require the application of three conditions: 1) Equilibrium of forces. 2) Compatibility of deformations. 3) Constitutive relationship (material behavior). These three conditions are used to generate the system of equations with stresses or displacements as unknowns. The former approach is called the force method and latter approach is called the displacement method. In general among many finite element techniques available displacementbased finite element method is most widely used formulation in engineering. The displacement method generates finite element equations of the form [38]: [K]{u}( ) {F} e = (3.13) where [K] is the global stiffness matrix, {F} is the vector of all applied loads (known variables), and {u}(e) is the nodal displacement vector (unknown variables). It should be noted that applied load is not necessarily the force; it may be stress, displacement rate, etc. Solving the above equation yields the nodal displacement vector {u}(e). Then, the element strains {ε} can be determined by the straindisplacement relationship and stress {σ} can be calculated from the constitutive relationship [38] as: { } [ ]{ }( ) { } [ ][ ]{ }(e) [ ]{ }(e) e M B u S u B u = = = σ ε (3.14) 52 where [B] is the straindisplacement matrix which usually has all constants, [M] is the material property matrix, and [S] = [M][B] is usually called the element stress matrix. The following coupled system of ordinary differential equations are used: [M] u F u,u,T {P(u,b,t,T )} . int .. = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟⎠ ⎞ ⎜⎝ + ⎛ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ (3.15) where {Fint} is the internal nodal force vector and {P} is the external nodal force vector which can be a function of nodal displacement u, body force per unit volume b, time t and nodal temperature T. The element stiffness [K](e) can be calculated as: [ ]( ) ([ ]( ) ) [ ](e)[ ](e) e T D K e B e M B dD e = ∫ (3.16) where [B](e) is the straindisplacement matrix, [M](e) is the material matrix and De is the element volume. The element contribution to the internal nodal force vector {Fint}(e) can be obtained as: {F }( ) ([B]( ) ) { }dD i T D e e e = ∫ τ int (3.17) Then the global stiffness matrix [K] and internal load vector {Fint} can be constructed as: [ ] [ ]( ) { } { }(e) e e e F F K K Σ Σ = = int int (3.18) It is necessary to point out that for metal cutting simulations, the matrices [B](e) and [M](e) are no longer constant as metal cutting analysis is a nonlinear, large deformation analysis. Therefore [B](e) and [M](e) need to be evaluated at each step of the finite element calculation. For quasistatic analysis of metal cutting process, the finite element equations (Equation 3.13) are simplified by eliminating the inertial effects, and thus have the form: 53 ( ) F u,u,T {P u,b,t,T } . int = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟⎠ ⎞ ⎜⎝ ⎛ (3.19) To obtain the solution at time step tn+1, the finite element equations are first linearized about the configuration at tn as: [ ] { } { } { }n n n K u P F1 int 0 Δ = − + (3.20) where [K]n is the stiffness matrix based on the configuration at tn and {Δu}° is the increment in displacement where the superscript denotes iteration number. {Δu}° can be solved from Equation 3.20 and displacement is updated by: { } { } 0{ }0 1 u u s u n n = + Δ + (3.21) where s° is a parameter between 0 and 1 which is determined by line search scheme. Finally, the equilibrium iterations can be performed by solving the following equations: [ ] { } { } +1 { int } +1 { } +1 Δ = − = n i n i n i i K P F Q (3.22) where {Q}i n+1 is the residual. And the convergence is determined by examining both the displacement norm { } d i u u ≤ε Δ max , (3.23) and energy norm: ({ } ) { } ({ } ) { } e n T n i T i u Q u Q ≤ε Δ Δ + + 1 0 0 1 , (3.24) In the Equations 3.23 and 3.24, umax is the maximum displacement norm above all of the n steps including the current iteration, εd and εe are the tolerances that are typically 102 to 54 103 or smaller and are usually adjustable for different problems. If the convergence is not attained and solution is divergent, the solution is obtained by: { } { } i { }i n i n i u +1 = u +1 + s Δu (3.25) where si is the parameter same as s° at the ith iteration. Iterations will then continue and if the solutions are determined to be divergent, or convergence fails to occur within an assigned number of iterations the stiffness matrix is [K] is reformed using the current estimation of the geometry before continuing the equilibrium iteration. The commercial software AdvantEdge™ is used in this study to simulate the metal cutting process. The finite element formulation used in this software is a Lagrangian formulation. The central difference rule is used to integrate the equations of motion. AdvantEdge™ uses a lumped mass formulation for efficiency, which produces a diagonal mass matrix that renders the solution of the conservation of the momentum equation given by [37]: ij j i i b u .. , σ + ρ = ρ (3.26) This equation shows the inertial forces and internal forces, due to the current state of stress, are in balance with the applied boundary loads. The weak form of the above equation based on the principle of virtual work becomes: v v b dV v ui dV B i i i B i ij j .. ∫ σ , + ρ = ∫ρ (3.27) solving and rearranging the above equation gives: ∫ + ∫ = ∫ Ω+ ∫ ∂ B j i i B ij i ij B i i j B i ρv u dV v σ dV vσ n d v ρb dV , .. (3.28) Finite element discretization of the above equation provides: 55 N N u dV N dV N d N b dV B a i B ij a i B b ib a j B ∫ a + ∫ = ∫ Ω + ∫ ∂ ρ σ τ ρ , .. (3.29) It can be represented in matrix form as: ext n n n Ma R R 1 int +1 +1 + + = (3.30) where 0 0 0 M N N dV a b B ab = ∫ρ (3.31) is the mass matrix and 0 0 0 0 = ∫ + ∫ Ω ∂ R b N dV N d a B a i B i ext ia τ (3.32) is external force array and , 0 int 0 R P N dV a j B ia = ∫ ij (3.33) is the internal force array. In the above expressions, Na, a=1,….,numnp are the shape functions, repeated indices imply summation and a (,) represents partial differentiation with respect to the corresponding spatial coordinate, and Pij is the first PiolaKirchoff stress tensor analogous to nominal or engineering stress. Using the Newmark family of algorithms for temporal integration of the discretized weak form, second order accurate explicit analysis can be achieved through the central difference scheme. Since plasticity results in softening of stiffness matrix, it suffices to look to the bounding case of linear elasticity, for which the generalized eigen problem is given by: ( − ) = 0 l ij j l ij K λ M d (3.34) Accurate computation of the system eigen values is essential due to changes in mesh geometry (eigen values) arising from deformation. Since the largest system eigen value is 56 bounded by the largest element eigen value, it suffices to compute the largest eigen value of each element to determine the critical time step for the mesh. Heat generation and transfer are handled via the Second Law of Thermodynamics. A discretized weak form of the law is given by: . n 1 n n T = T + ΔtT + (3.35) 1 . +1 +1 + + = n n n CT KT Q (3.36) A lumped capacitance matrix is used to eliminate the need for solving any equations. CT + KT = Q . (3.37) where T is the array of nodal temperatures. 0 C c N N dV a b B ab t = ∫ ρ (3.38) is the heat capacity matrix and K D N N dV a i b j B ab ij , , 0 = ∫ (3.39) is the conductivity matrix and = ∫ + ∫ B q a B a a Q sN dV hN dS t τ (3.40) is the heat source array with h, having an appropriate value for the chip or tool. In machining applications, the main sources of heat are plastic deformation in the shear zone and frictional sliding in the toolworkpiece interface. The rate of heat supply due to the first is estimated as: P s W . = β (3.41) 57 where P W . is the plastic power per unit deformed volume and TaylorQuinney coefficient β is of the order of 0.9. The rate at which the heat generated at the frictional contact, on the other hand is given by: h = −t ⋅ v (3.42) where t is the contact traction and v is the jump in velocity across the contact. A staggered procedure is adopted to couple mechanical and thermal equations. Geometrically identical meshes for the mechanical and thermal models are used. Mechanical and thermal computations are staggered assuming constant temperature during the mechanical step and constant heat generation during the thermal step. A mechanical step is taken first based on current distribution of temperatures, and the heat generated is computed from plastic working and frictional heat generation. The heat thus computed is transferred to the thermal mesh and the temperatures are recomputed by recourse to the forwardEuler algorithm. The resulting temperatures are transferred to the mechanical mesh and are incorporated into the thermalsoftening model which completes one time step cycle. 3.3 Friction formulation along toolchip interface Friction along the toolchip contact during the cutting process is a very complex phenomenon. Friction influences chip formation, builtup edge formation, cutting temperature and tool wear. Therefore it is necessary to understand the friction mechanism across the tool face and around the edge of the tool, in order to develop accurate models 58 for cutting forces and temperature. In the AdvantEdge™ software, the friction model incorporated is coulomb friction model and it is represented by: τ = μσ (3.43) where τ is the frictional shear stress and σ is the normal stress to the surface. Usually the friction coefficient μ is assumed to be constant for a given interface. In the metal cutting, the cutting pressure at the toolchip interface will become several times the yield stress of the workpiece material. In this extreme case, the real contact between the tool and workpiece is so nearly complete in the sticking region that sliding occurs only beyond this region. Therefore the frictional force becomes that required to shear the weaker of the two materials across the whole interface. This force is almost independent of the normal force, but is directly proportional to the apparent area of contact. 3.4 Formulation of Contact Conditions In metal cutting simulations, meshonmesh contact occurs between the workpiece and the tool and this contact is formulated in AdvantEdge™ software using predictorcorrector method of PRONTO2D™ explicit dynamics code. The two contacting surfaces are designated as master and slave surfaces (Fig. 3.1). Assuming that no contact has occurred, nodal accelerations from the outofbalance forces are calculated and nodal positions, velocities, and accelerations are predicted by predictor algorithm. A resulting predictor configuration shows penetration of the master surface into the slave surface. The contact conditions are designated by an auxiliary consecutive numbering of the nodes on the contacting surfaces. The penetration distances for all nodes on the slave surface are then calculated. The contact force required to prevent penetration is equal to 59 the force required to keep the master surface remain stationary on predictor configuration. Tangential force exerted by the master surface on the slave node cannot exceed the maximum frictional resistance. This condition should be satisfied to prevent unwanted penetration. A balanced masterslave approach in which surfaces alternately act as master and slave is also employed; however, rigid surfaces are always treated as master surfaces. Fig. 3.1 Contacting surfaces in a mesh showing (a) predictor configuration and (b) kinematically compatible configuration [14] 3.5 Adaptive remeshing In AdvantEdge™ finite element formulation incorporates a sixnoded quadratic triangular element with three corner and three midside nodes providing quadratic interpolation of displacements within the element as shown in Fig. 3.2. The element is integrated with threepoint quadrature interior to the element. At the integration points, the constitutive response of the material is computed and consequently linear pressure distribution is provided within the element. 60 Fig. 3.2 Sixnodes and three quadrature points shown in a typical sixnoded triangular element used in the finite element mesh of the workpiece and tool [14] During metal cutting, workpiece material flows around the cutting tool edge. In this process, at the tool vicinity elements get distorted and the accuracy is lost. To alleviate element distortion, finite element mesh is updated periodically, refining large elements, remeshing distorted elements and coarsening smaller elements. For instance, if an element needs refinement, the diagonal of the element is split, a midside node becomes new corner node and new midside nodes are added to both elements as shown in Fig.3.3. (a) (b) Fig. 3.3 Element shape in a finite element mesh (a) before and (b) after adaptive remeshing [14] For adaptive remeshing, an adaptation criterion based on the equidistribution of plastic power is used. In this approach, elements with plastic power content exceeding a 61 prescribed tolerance TOL, are targeted for refinement. The criterion can be represented as [14]: ∫ Ω Ω W d e h p . >TOL (3.44) here Ωh e denotes the domain of the element e and the plastic power for an element is given by . . W p =σ ε p (3.45) if, despite this continuous remeshing, elements arise with unacceptable aspect ratios, the mesh is subjected to Laplacian smoothing. Besides sidestepping the problem of element distortion, adaptive remeshing provides a means of simultaneously resolving multiple scales in the solution. Transport of data, such as displacement and temperature from old mesh to new mesh after remeshing is done by interpolation technique. 3.6 Chip formation Different numerical techniques for modeling chip separation exist and they can be divided into two categories geometrical and physical. The geometrical model is usually based on the tied slideline interface which debonds when certain criterion is fulfilled. This criterion may be a certain level of stress, strain or simply when the cutting edge is close enough to the front nodes. On the other hand, physical model is based on the physical behavior of the material, such as plastic deformation, crack initiation and crack propagation without predetermining its path. AdvantEdge™ simulation software uses critical stress intensity factor, KIC, as a fracture criterion for brittle materials and crack tip opening displacement (CTOD) for ductile fracture. Brittle fracture, such as the one that occurs below the transition temperature, proceeds by cleavage. In particular, conditions for brittle fracture are found to be consistent with the attainment of a critical opening stress σf at a critical distance l. 62 The critical stress σf is found to be relatively independent of temperature and strain rate and can be inferred from the toughness KIC through the small scale yielding relation [14] l KIC f π σ 2 = (3.46) The crack trajectory under conditions of brittle fracture is predicted using maximum hoop stress criterion according to which crack propagates along the angle θ from the crack face at which the hoop stress σ θθ attains a relative maximum. Void growth and coalescence are known to be principal mechanisms of ductile fracture. The rate of growth of voids is accelerated by the blunting of the crack tip, which has the effect of raising the hydrostatic stress at the location of the void. The crack tip opening displacement (CTOD) criterion for ductile fracture can be recast as the attainment of critical effective plastic strain εp f of the effective plastic strain at a distance l ahead of the crack tip [14]. The criterion can be represented as: ( ) ε θ = θ max p l, εp f (3.47) where θ is the angle at which the crack propagates when the criterion is met. The critical effective plastic strain (εp f) can be estimated as εp f σ p e 1.5 2.48 − ≈ (3.48) where p= σ kk/3 is the hydrostatic pressure. 63 CHAPTER 4 PROBLEM STATEMENT As mentioned in the literature review, many researchers in the past have developed analytical models to explain the theory of chip segmentation. Many theories based on adiabatic shear failure, damage model and crack initiation and propagation have been proposed for segmented chip formation in Ti 6Al4V. But most of the models suffer from a lack of adequate acceptable methodologies for application under a wide range of cutting conditions. In addition, very little work has been done on finite element simulations of machining Ti6Al4V alloy (especially at very low speeds). Most of these simulations have limited experimental validation under few selected cutting conditions and tool geometries. This is an area deserving more study because a reliable finite element model of chip segmentation needs a realistic material model to describe material behavior accurately under high temperatures, high strains, and high strain rates and a proper failure criterion to represent the mechanism of chip segmentation. Thus an attempt has been made in this study to deal with the above mentioned aspects. The specific objectives of this study are: 1) To study the mechanism of chip formation in machining Ti 6Al4V using finite element simulations. 64 2) To formulate a reliable material model, such as the JohnsonCook model into the user defined material code (UMAT) so that it can represent material flow stress with respect to strain, strain rate, and temperature at different machining conditions. 3) To derive equations for Recht’s catastrophic shear failure criterion and incorporate it into UMAT code as a failure criterion for shearlocalized chip formation in Ti 6Al4V. 4) To validate the finite element simulations with experimental results obtained from the literature by comparing the cutting forces, rake face temperature and chip morphology. 5) To predict the critical cutting speed for the onset of chip segmentation. 6) To conduct finite element simulations using commercial finite element software, AdvantEdge™. 7) To study the effect of different machining conditions, such as cutting speed, feed, rake angle and coefficient of friction on cutting forces, rake face temperature, shear zone temperature, equivalent plastic strain, frequency of chip segmentation, shear band width, and power consumption. 65 CHAPTER 5 SHEAR LOCALIZATION IN Ti 6Al4V MACHINING 5.1 ShearLocalized Chip in Ti 6Al4V Titanium alloys are one of the most attractive materials because of their high specific strength maintained at high temperatures, excellent fracture resistance and corrosion resistance. They are much sought after materials for aerospace applications. Although Ti 6Al4V comprises about 45% to 60% of titanium products in practical use, it is considered as difficulttomachine material because of its unfavorable thermal properties. Generally, two types of chip formation are observed in machining Ti 6Al4V based on the cutting speed. They are the continuous chip and the segmented chip. Segmented chip is also called shearlocalized chip because of intense plastic deformation in the narrow band between the chip segments and negligible deformation within the segment. According to Komanduri and Hou [25], for titanium alloys shear localization occurs even at very low cutting speeds (<0.5 m/min) and continues over the entire conventional cuttingspeed range. Consequently, continuous chip can be observed only at very low speeds (below 0.5 m/min). The chips formed in Ti 6Al4V machining looks like saw tooth with chip being inhomogeneous and shows two regions, shear band with very large shear strains between the chip segments and a trapezoidal shaped segments body 66 with relatively small deformation. Once the shear localization is initiated, the chip topside looks like saw tooth with each tooth corresponding to a segment. Many researchers have done extensive studies on the mechanism of chip formation when machining titanium alloys since the early 1950s using several techniques, such as highspeed photography, insitu machining inside a scanning electron microscope and metallurgical analysis of chips generated in machining. Based on these studies many models were formulated to describe the mechanism of segmented chip formation in titanium alloys. In this study, Recht’s thermoplastic shear instability criterion is used to simulate shearlocalized chip formation. 5.2 Mechanism of ShearLocalized Chip formation In machining Titanium alloys, the chips formed are segmented and strain is not uniformly distributed but is confined to narrow bands between the segments. Whereas, in the continuous chip formation, the deformation is largely uniform. The sequence of events leading to shearlocalized chip formation in Ti 6Al4V was described by Komanduri and Turkovich [11] based on the detailed chip formation studies of video tapes of low speeds experiments inside the scanning electron microscope, highspeed movie films at higher cutting speeds and the micrographs of midsection of chips. They observed the mechanism of chip formation to be invariant with respect to cutting speed. The mechanism of shearlocalized chip formation can be explained based on the Fig. 5.1. The process can be divided into two basic stages. The first stage involves shear instability and strain localization in a narrow band in the primary shear zone ahead of the tool. This narrow band originates from the tool tip almost parallel to the cutting velocity vector and 67 gradually curves with the concave surface upwards until it meets the free surface. The shear failure of the chip appears as a crack on the outside while it is a heavily deformed band inside. Fig. 5.1 Description of shearlocalized chip formed due to adiabatic shear localization [26] The second stage involves upsetting of an inclined wedge of work material by the advancing tool, with negligible deformation, forming a chip segment. During upsetting of the chip segment in the primary shear zone ahead of the tool tip, intense shear takes place at approximately 45° to the cutting direction. This occurs not between the tool face and chip but between the last segment formed and the one just forming. The initial contact of the tool face with the segment being formed is very less and it gradually increases as the upsetting progresses. There is almost no relative motion between the bottom surface of the chip segment being formed and the tool face until the end of the upsetting stage of the 68 segmentation process. The gradual upsetting process slowly pushes the segment formed previously upwards. The velocity of the chip along the rake face is same as the upsetting chip until the shear is initiated and progresses rapidly. Once this occurs, it pushes the chip segment being formed faster parallel to the shear surface. This will then push the previous segment formed rather rapidly. Thus the chip velocity along the rake face fluctuates cyclically. This mechanism of segmented chip formation occurs almost at all speeds. However, as the cutting speed increases, intense shear in the narrow shear band occurs so rapidly that contact area between any two segments decreases to a stage that individual segments of the chip get separated. 5.3 Criterion for Thermo Plastic Shear Instability Recht in 1964 [12] developed a criterion for the prediction of catastrophic shear instability in metals under dynamic plastic conditions. Accordingly, shearlocalized chip formation can be attributed to dynamic plastic behavior of the material which is influenced by internally generated temperature gradients. These gradients are function of thermophysical properties of the workmaterial as well as strain rate and shear strength. Catastrophic shear will occur at a plastically deformed region within a material when the slope of the true stresstrue strain curve becomes zero, i.e. when local rate of change of temperature has a negative effect on the strength which is equal to or greater than the positive effect of strain hardening. The criterion for catastrophic shear failure can be represented as: 69 0 ≤ 1.0 ∂ − ∂ ∂ ∂ ≤ ε θ θ τ ε τ d d (5.1) where τ, ε, θ refer to shear stress, shear strain, and temperature, respectively. Accordingly, material will shear catastrophically when this ratio lies between zero and one; catastrophic shear will be imminent when this ratio equals one. No catastrophic shear will occur when this ratio is greater than one. High positive values above one indicate that strain hardening is predominant and shear deformation will distribute throughout the material, in which case material will strain harden more than it will thermal soften. Negative values indicate that material will actually become stronger with an increase in temperature and that shear deformation will distribute. Thus thermoplastic shear instability (frequently referred to as adiabatic shear) is a major contributor to chip segmentation. Just so long as material can withstand shear stress by virtue of its shear strength, it will remain thermoplastically stable. As stress increases, material strains and if rate of change of strength matches the rate of change of stress, material will undergo stable deformation. When instability occurs, the applied stress will be borne by the remaining (diminishing) strength of the material. And inertial reactions associated with, accelerate instability, leading to catastrophic shear failure. Recht thus provided the first explanation for segmented chip formation in machining. Since the fundamental contributions of Recht, much work has been done on the adiabatic shear band formation, often in connection with the applications, such as armor penetration and explosive fragmentation. 70 5.4 Metallurgical Aspects of ShearLocalized Chip Formation In metal cutting, the temperatures generated in the primary and secondary deformation zones can be high enough to cause several changes in the workpiece material, such as thermal softening, phase transformation, and even grain shape and size changes. These changes to a certain extent can effect the prediction of optimal cutting conditions for a given machining operation. Adiabatic shear banding is a phenomenon observed in machining titanium alloys. The formation of shearlocalized chip in titanium alloys is also accompanied by certain metallurgical changes. Komanduri [26] noted that during segmented chip formation in titanium alloys, there may be a transition from the lowtemperature hexagonal close packed (HCP) structure to the body centered cubic (BCC) structure with a corresponding increase in the number of available slip systems. Accordingly, this phenomenon further localizes the shear strain. This transition in the crystal structure results primarily from the increase in temperature and this increase can cause further increase in plastic deformation. Ti 6Al4V is an alloy with α+β structure which consists of lamellar α structure and intergranular β structure. Bayoumi and Xie [39] conducted metallurgical studies using a scanning electron microscope (SEM) and an Xray diffraction studies of both uncut and machined Ti6Al4V chips. Fig.5.2 shows Xray diffraction spectra for Ti 6Al 4V alloy before and after cutting. They compared the spectrum of the chip with uncut material and found that the peaks corresponding to βphase structure disappearing after cutting, indicating that a nondiffusional phase transformation took place in the process of shear band formation. When the temperature in the chip reaches the βtransus 71 temperature of Ti 6Al4V (which is 980°C) during the cutting process and cools back from this high temperature, the temperature change along with tremendous cutting pressure would produce the lamellar α structure from βphase by nucleation and growth. Accordingly, they concluded that there is only grain change during the phase transformation and no chemical change. Fig. 5.2 Xray diffraction spectra for chip and uncut material in a Ti 6Al4V alloy [39] From the microstructural examination of the chip and uncut Ti 6Al4V, they concluded that as both are similar in appearance, the shearlocalized chips may be caused by material flow in shear bands due to adiabatic shear instability. 72 CHAPTER 6 FEM SIMULATION OF CHIP SEGMENTATION IN Ti 6Al4V 6 .1 Introduction One of the stateofart efforts in manufacturing engineering is the finite element simulation of the metal cutting process. These simulations would have great value in increasing our understanding of the metal cutting process and reducing the number of trial and error experiments, which traditionally was the approach used for tool design, process selection, machinability evaluations, chip formation and chip breakage investigations. Compared to empirical and analytical methods, finite element methods used in the analysis of chip formation has advantages in several respects, such as developing material models that can handle material properties as a function of strain, strain rate and temperature. The toolchip interaction can be modeled as sticking and sliding; and nonlinear geometric boundaries such as the free surface of the chip can be represented and used. In addition to the global variables, such as cutting force, thrust force and chip geometry; the local variables such as stress, temperature distributions, can also be obtained. As mentioned in the literature review, very limited work has been done on FEM simulation of titanium alloys. Qualitative analysis of such simulations seems to suffer with lack of accurate material model to describe the material behavior of titanium alloys 73 under high temperature, high strain and high strain rates. Also, most of the simulations lack appropriate failure criterion to simulate shearlocalized chip formation. Further, most of the simulations were performed for highspeed machining while in titanium alloys segmented chip is found to occur even at very low speeds. In this study, JohnsonCook material model is used to represent material behavior as a function of strain, strain rate, and temperature; Recht’s catastrophic shear failure criterion is used for chip segmentation and a commercial, generalpurpose FE software, AdvantEdge is used to simulate orthogonal metal cutting of Ti 6Al4V under various machining conditions. AdvantEdge machining modeling software is a twodimensional central difference explicit finite element code using a Lagrangian mesh. The material model of the software accounts for elasticplastic strains, strain rates and temperature. It has an isotropic power law for strain hardening and as the material flow properties are temperature dependent it also accounts for thermal softening. A staggered method for coupled transient mechanical and heat transfer analyses is utilized. A sixnode quadratic triangular element with three quadrature points is used. This software also features adaptive remeshing option to alleviate mesh distortion. 6.2 Material Constitutive Model The flow stress or instantaneous yield strength at which work material starts to deform plastically is mostly influenced by temperature, strain, and strain rate. The constitutive model proposed by Johnson and Cook describes the flow stress of a material as a product of strain, strain rate, and temperature that are individually determined by the following equation: 74 σ = [A+B(ε)n][1+Cln( . ε * )][1+(T*)m] (6.1) where σ is the effective stress, ε is the effective plastic strain, . ε * is the normalized effective plastic strain rate, n is the work hardening exponent, m is the thermal softening exponent and A, B, C and m are constants [9]. It may be noted that formulation of JohnsonCook model is empirically based. The expression in the first set of brackets gives the stress as a function of strain for . ε * =1.0 and T*=0.The expressions in the second and third sets of brackets represent the effects of strain rate and temperature, respectively. The parameter A is in fact the initial yield strength of the material at room temperature at a strain rate of 1 s1. The nondimensional parameter T* is defined as T* = (T Troom)/(Tmelt Troom) (6.2) where T is the current temperature, Troom is the ambient temperature, and Tmelt is the melting temperature. Temperature term in this model reduces the flow stress to zero at the melting temperature of the workmaterial, leaving the constitutive model with no temperature effects. The nondimensional normalized effective plastic strain rate . ε * is the ratio of the effective plastic strain rate ε p to the reference strain rate ε° (usually equal to 1.0). In general, the parameters A, B, C, n and m are fitted to the data obtained by several mechanical tests conducted at low strains and strain rates as well as split Hopkinson pressure bar (SHPB) tests and ballistic impact tests. JohnsonCook model provides good fit for strain hardening behavior of the metals. It is numerically robust and can be easily used in finite element simulation models. 75 6.3 Formulation of Recht’s Catastrophic Shear Failure Criterion Perhaps the first criterion for shear localization was developed by Recht [12]. According to this criterion, shear instability was predicted to occur when the rate of strain hardening of the material is balanced by thermal softening. This indicates that when an increase in shear strain is associated with a decrease in shear stress, the strain hardening slope would become negative: ≤ 0 γ τ d d with γ τ γ τ γ τ d dT d T d ∂ + ∂ ∂ = ∂ (6.3) The criterion R can be written as follows: γ τ γ τ d dT T R ∂ − ∂ ∂ ∂ = and 0≤ R ≤ 1 (6.4) where τ ,γ and T represent shear stress, shear strain and temperature respectively. When R is equal to zero, instability starts and when R is equal to one catastrophic shear failure occurs. The numerator and denominator terms of R are derived from JohnsonCook flow stress equation: ( ( ) ) ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − − − ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = + + m r r o n T T T T A B 1 C ln 1 . . ε σ ε ε (6.5) 76 where A, B, C, n, m are constants. T is the current temperature, Tr is the ambient temperature, Tm is the melt temperature. ε is the strain, . ε is the effective plastic strain rate and . ε o is the reference plastic strain rate. The shear stress τ and the shear strain γ can be expressed in terms of strain and strain rate as follows: 3 τ = σ and γ =ε 3 (6.6) Substituting the Equation 6.6 into Equation 6.5, we get ⎥⎦ ⎤ ⎢⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − − − ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ = + m r r n T T T T A B 1 C ln 1 3 3 1 . 0 . γ τ γ γ (6.7) The partial differentiation of Equation 6.7 with respect to the shear strainγ gives the strain hardening term γ τ ∂ ∂ as ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − − − ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ = ∂ ∂ − m m r r n T T T T C nB 1 ln 1 3 3 . 0 1 . γ γ γ γ τ (6.8) while the partial differentiation of Equation 6.8 with respect to temperature T gives the thermal softening term ∂T ∂τ as ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − − − − ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ = + ∂ ∂ m m r r r n T T T T T T m A B C T . 0 . 1 ln 3 3 1 γ τ γ γ (6.9) 77 The second term in the denominator of Recht’s criterion dγ dT is the rate of change of temperature with strain in the catastrophic shear zone. Recht developed a model [12] to determine the temperature gradient with strain. According to this model, the zone in Fig. 6.1 with unit area A and unit thickness T is assumed to be the weakest zone within the length L of the specimen. Catastrophic shear is achieved by applying a constant rate of average strain L x . high enough to produce catastrophic slip in this zone and when this is achieved, this shear zone will remain thin and it can be seen that plastic deformation is confined to this region. Since this zone is very thin, it is assumed to be a plane of uniform heat generation. Fig. 6.1 Model used for determination of temperature gradient with strain in catastrophic shear zone [12] Thus the heat generation rate over the unit area A is given by .γ τ W L q = (6.10) where q is the heat generation rate per unit area, τ is the shear strength in the weak zone, L is the specimen length, .γ is the average shear strain rate equal to L x . and W is the work equivalent of heat. Using Carslaw and Jaeger’s solution for the temperature on a plane in 78 an infinite medium at constant heat generation, the instantaneous temperature TA on unit area A is given as C t W L T y A πκρ τ γ . = (6.11) where y τ is the initial shear yield strength, t is the time, κ is the thermal conductivity, C is the specific heat and ρ is the specific weight. Differentiating the above equation with respect to time t, gives dt W Ct L dT y A πκρ τ γ 1 2 1 . = (6.12) But, for constant strain rate, y γ =γ t +γ . and .γ γ = dt d where γ is the unit shear strain and y γ is the initial yield strain. Substituting these two in Equation 6.12, we get, ( )y A y W C L d dT πκρ γ γ τ γ γ − = . 2 1 (6.13) Substituting Equations 6.8, 6.9 and 6.13 in Equation 6.4, will give the following equation which can be used to formulate the Recht’s criterion into the UMAT code. ( )⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − − − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − − − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ = − y y m m r r r n m m r r n W C L T T T T T T m A B T T nB T T R πκρ γ γ γ τ γ γ . 1 2 1 3 3 1 1 3 3 (6.14) For each integration point of the element of workpiece mesh, Recht criterion is evaluated. When Recht criterion is satisfied by all the integration points of the element, 79 the stress state of all these integration points is made zero. The code of the element is stored in a temporary file. All these coded (listed) elements are then deleted and then the border of the workpiece is extracted and smoothed. This smoothing operation reduces the loss of volume in the workpiece determined by e
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Title  Finite Element Simulation of Chip Segmentation in Machining a Ti 6al4v Alloy 
Date  20041201 
Author  Syed, Kareem 
Keywords  Ti6Al4V, ShearLocalized Chip, Orthogonal Cutting, Metal Cutting, Segmented Chip, Aerospace Alloys 
Department  Mechanical Engineering 
Document Type  
Full Text Type  Open Access 
Abstract  Ti 6Al4V, a titanium alloy introduced in 1954, is considered the workhorse amongst the titanium alloys and is available in all product forms. It is extensively used in aerospace industry because of its excellent strengthtoweight ratio maintained at elevated temperatures, fracture resistance characteristics and exceptional corrosion resistance. The machinability of this alloy is generally considered to be poor owing to its several inherent properties. It is very reactive chemically and therefore has a tendency to weld to the cutting tool during machining. Its low thermal conductivity increases the temperature at the tool/workpiece interface, which affects the tool life adversely. Additionally, its high strength maintained at elevated temperature, low modulus of elasticity further impairs its machinability. In order to overcome the machinability issues associated with machining Ti 6Al4V, an attempt has been made in this study to observe the effect of machining conditions on the chip formation, rake face and shear zone temperatures and cutting forces. To simulate orthogonal metal cutting of Ti 6Al4V a commercial, generalpurpose FE code (AdvantEdge) has been used. AdvantEdge has the facility to incorporate userdefined material subroutine (UMAT). Using this, JohnsonCook material model and Recht's catastrophic shear failure criterion are incorporated into the UMAT subroutine code. Finite element simulations are conducted for a range of cutting speeds from 10 m/min to 100 m/min using two different depths of cut of 0.25 and 0.5 mm, for different rake angles from 15� to 45� using depth of cut of 0.5 mm and cutting speed of 30 m/min and for different values of coefficient of friction ranging from 0.3 to 0.9 using depth of cut of 0.5 mm and cutting speed of 30 m/min. Results of the simulations are compared with the experimental data and are found to be in close agreement. Mechanism of chip formation studied from simulations closely matched with that proposed in the literature. Effect of cutting speed, depth of cut, rake angles and coefficient of friction on cutting forces, temperature, strains and chip morphology is studied. Finally, cutting speed for the onset on chip segmentation is found for two different depths of cut. 
Note  Thesis 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  FINITE ELEMENT SIMULATION OF CHIP SEGMENTATION IN MACHINING A Ti 6Al4V ALLOY By KAREEM SYED Bachelor of Engineering Osmania University Hyderabad, India June, 2001 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE December, 2004 ii FINITE ELEMENT SIMULATION OF CHIP SEGMENTATION IN MACHINING A Ti 6Al4V ALLOY Thesis approved: Dr. Ranga Komanduri Thesis Adviser Dr. Hong Bing Lu Dr. Samit Roy Dr. Gordon Emslie Dean of the Graduate College iii ACKNOWLEDGEMENT I wish to express my sincere thanks to my advisor, Dr.Ranga Komanduri, for his intelligent supervision, constructive guidance, financial support, inspiration and motivation. His wise advices made it possible for me to quickly grow not only technically but also in all aspects during the course of my M.S study. I truly appreciate the encouragement and technical guidance from my committee members: Dr. Samit Roy and Dr. Hongbing Lu. I would also like to thank Dr. Bo Wang for his guidance in deriving the equations for Recht’s catastrophic shear failure criterion. This project has been funded by a grant from the division of Design, Manufacturing and Industrial Innovation (DMII) of the National Science Foundation (NSF). I would like to thank Dr. George Hazelrigg for his interest and support of this work. I would like to thank Chief Technical Officer Dr. Troy Marusich of Third Wave Systems, Inc. and Support Engineers Mr. Christopher Brand and Mr. Deepak Agarwal for their technical support in using AdvantEdge™ software. I wish to express my sincere gratitude to Mr. Dhananjay Joshi and Mr. Parag Konde for their contribution in discussing, deriving and implementing the equations of Recht’s catastrophic shear failure criterion into the user subroutine code (UMAT). I would also like to extend my gratitude to all other members of our research group for their support and friendship. iv I would like to thank the Department of Mechanical and Aerospace Engineering for providing me with the opportunity to pursue M.S at Oklahoma State University. I am ever thankful to my parents who were always a constant source of my inspiration and encouragement at all times. Finally, I would like to extend my gratitude to my brother and sisters for their inspiration and love. v TABLE OF CONTENTS Chapter Page 1. Introduction………………………………………………………....…….........1 1.1 State Of Art: Finite Element Simulation of the Cutting Process………………………………………………………...1 1.2 Historical Developments…………………………………........3 1.3 Principles of Metal Cutting…………………………………....5 1.4 Principles of Finite Element Modeling………………………..6 1.5 Metallurgical Aspects of Titanium alloys……………………..7 1.6 Machinability aspects of Ti 6Al4V………………………….10 1.7 Mechanism of ShearLocalized Chip Formation in Ti 6Al4V……………………………………………………..11 1.8 Thesis Outline………………………………………………...13 2. Literature Review………………………..………………………………….....16 2.1 Numerical Study of ShearLocalized Chip Formation………..16 2.2 ShearLocalized Chip Formation in Ti 6Al4V: Experiments...25 2.3 ShearLocalized Chip Formation in Ti 6Al4VFEM Simulations…………………………………………………….36 3. Finite Element Formulation of Metal Cutting .................................................47 3.1 Introduction………..……………………………….……….....47 3.2 Finite Element Formulation …………………………………..48 3.3 Friction formulation along toolchip interface ………………. 57 3.4 Formulation of Contact Conditions …………………………..58 3.5 Adaptive remeshing …………………………………………..59 3.6 Chip formation………………………………………………...61 4. Problem Statement………………………………………………………….......63 5. ShearLocalized Chip Formation in Ti 6Al4V machining…………………..65 5.1 ShearLocalized Chip in Ti 6Al4V …………………………...65 5.2 Mechanism of ShearLocalized Chip formation………… ……66 5.3 Criterion for Thermo Plastic Shear Instability…………………68 vi 5.4 Metallurgical Aspects of ShearLocalized Chip Formation…..70 6. FEM Simulation of Chip Segmentation in Ti 6Al4V ………………………..72 6.1 Introduction………..………………………………………....72 6.2 Material Constitutive Model ………………………………...73 6.3 Formulation of Recht’s Catastrophic Shear Failure Criterion 75 6.4 Stress Update Algorithm …………………………………….80 7. Results and Discussions………..………………………………………….........84 7.1 Process Model and Material Properties……………………... 84 7.2 Simulation Approach and Cutting Conditions………………..88 7.3 Observations of Chip Formation Process…………………….90 7.4 Temperature and Equivalent Plastic Strain distribution in the chip……………………………………………………94 7.5 Effects of Cutting Speed and Feed rate……………………....97 7.6 Effects of Rake angle………………………………………..111 7.7 Effects of Coefficient of friction…………………………….116 7.8 Validation of Simulation Results…………………………….119 7.9 Discussion…………………………………………………...126 8. Conclusions and Future Work ………………………………………………..128 8.1 Conclusions…………………………………………………..128 8.2 Future Work..………………………………………………...131 References.......………………………………………………………………...133 vii LIST OF TABLES Table No. Page No. 7.1 Physical properties of Ti 6Al4V…..………………………………………..86 7.2 JohnsonCook material properties for Ti 6Al4V…………………………...87 7.3 Cutting conditions used for FEM simulations……………………………....89 7.4 Tool dimensions used in FEM simulations………………………………....90 7.5 Adaptive mesh options input to the FEM software…………………………90 7.6 Finite element simulation results and experimental data of cutting and thrust force with percent deviation……………………………………..120 7.7 Finite element results and experimental data for cutting force with percent deviation………………………………………………………122 viii LIST OF FIGURES Figure No. Page 1.1 Orthogonal metal cutting model ……………………………………………….....6 1.2 Microstructure of Ti 6Al4V alloy showing primary αphase and transformed βgrains……………………………………………………………....9 2.1 Stressstrain diagram for a combination of strain hardening, thermal softening and failure softening……………………………………………………………. 22 2.2 Adiabatic shearlocalized chip in a Ti 6Al4V alloy, obtained by orthogonal cutting at the velocity 1.2 m/s…………………………………………………... 28 2.3 Chip morphology as a function of cutting speed and feed in the orthogonal cutting of a Ti 6Al4V alloy…………………………………………………….. 29 2.4 ShearLocalized chip formed in machining a Ti 6Al4V alloy at cutting speeds of 60 and 120 m/min using feed rate of 0.127 mm/rev………………………..... 31 2.5 Roughness of workpiece material (Ti 6Al4V) as a function of cutting length at different cutting speeds……………………………………………….. 32 2.6 Voids and cracks in an adiabatic shear band of a Ti 6Al4V alloy specimen deformed at 700°C and 2000 s1 strain rate…………………………...33 2.7 Temperature dependence of flow stress in a Ti 6Al4V alloy at very low strain rates of 103 s1 ………………………………………………………34 2.8 Pseudotemperature variation with drill depth showing a linear increase ……...35 2.9 Serrated chips obtained in machining a Ti 6Al4V alloy at cutting speeds (a) 80 m/min and (b) 140 m/min……………………………….. 36 2.10 Chip formation process showing shear localization in finite element simulation of a Ti 6Al4V alloy at cutting speed of 5 m/s and feed 0.3 mm using 8° rake angle tool………………………………………... 37 ix 2.11 Toolworkpiece mesh system used in finite element simulation of orthogonal metal cutting a Ti 6Al4V alloy……………………………….. 39 2.12 Serrated chip formation with cutting length in finite element simulation of machining a Ti 6Al4V alloy at cutting speed 30 m/min, feed 0.25 mm and 20° rake angle……………………………………………... 39 2.13 Serrated chip formed in finite element simulation of orthogonal metal cutting of a Ti 6Al4V alloy at cutting speed 30 m/min, feed 0.25 mm and 20° rake angle…………………………………………….. 40 2.14 Initial geometry and workpiecetool mesh used in finite element simulation of orthogonal metal cutting of a Ti 6Al4V alloy…………………41 2.15 Segmented chip formation process in finite element simulation of orthogonal metal cutting of a Ti 6Al4V alloy at 10 m/s cutting speed and 0.5 mm feed using a 3° rake angle tool …………………………. 42 2.16 Segmented chip formation process in finite element simulation of orthogonal metal cutting of a Ti 6Al4V alloy at 50 m/s cutting speed and 0.04 mm cutting depth using a 10° rake angle tool ……………….43 2.17 Equivalent plastic strain distribution in serrated chip simulated at cutting speeds of 1.2, 120 and 600 m/min in a Ti 6Al4V alloy……………...45 3.1 Contacting surfaces in a mesh showing (a) predictor configuration and (b) kinematically compatible configuration…………………………….. 59 3.2 Sixnodes and three quadrature points shown in a typical sixnoded triangular element used in finite element mesh of workpiece and tool………60 3.3 Element shape in a finite element mesh (a) before adaptive remeshing and (b) after adaptive remeshing……………………………………………..60 5.1 Description of shearlocalized chip formed due to adiabatic shear localization…………………………………………………………………...67 5.2 Xray diffraction spectra for chip and uncut material in a Ti6Al4V alloy….. 71 6.1 Model used for determination of temperature gradient with strain x in catastrophic shear zone…………………………………………………... 77 6.2 Variation of Recht’s criterion value (R) with temperature for Ti 6Al4V alloy at different strain rates and strain of 3…….............................................79 6.3 Variation of Recht’s criterion value (R) with temperature for Ti 6Al4V alloy at different strain rates and strain of 4……............................................80 7.1 Workpiecetool system used for FEM simulations showing initial mesh….. 86 7.2 True stressstrain curves of Ti 6Al4V alloy based on JohnsonCook material model in the temperature range of 200 to 16000 C at high strain rate of 1x 104 s1…………………………………………………….. 87 7.3 The variation of True stress as a function of temperature for Ti 6Al4V in the different strain range of 0.005 to 4.0 at high strain rate of 1x 104 s1 88 7.4 Various stages of shearlocalized chip formation in machining simulation of Ti 6Al4V conducted at 30 m/min cutting speed and 0.5 mm depth of cut for 15° rake angle showing equivalent plastic strain localization in narrow shear bands…………………………..93 7.5 Temperature distribution in shearlocalized chip formation obtained from FEM simulation of machining Ti 6Al4V at 30 m/min cutting speed and 0.5 depth of cut for 15° rake angle ……….…………...95 7.6 Equivalent plastic strain distribution in the shearlocalized chip formation from FEM simulation of machining Ti 6Al4V at a cutting speed of 30m/min and a depth of cut of 0.5 mm for a 15° rake angle…….96 7.7 Temperature and equivalent plastic strain contour plots in shearlocalized chips formed in machining simulations of Ti 6Al4V for a depth of cut of 0.25 mm at different cutting speeds varying from 10 to100 m/min using 0° rake angle tool…………………………….100 7.8 Temperature and equivalent plastic strain contour plots in shearlocalized chips formed in machining simulations of Ti 6Al4V for a depth of cut of 0.5 mm at different cutting speeds varying xi from 10 to100 m/min using 0° rake angle tool……………………………104 7.9 Effect of cutting speed on (a) the shear zone temperature and (b) the rake face temperature for FEM simulations of Ti 6Al4V at cutting speeds from 10 to 100 m/min for two different depths of cut of 0.25 and 0.5 mm………………………………………………...104 7.10 Effect of cutting speed on (a) equivalent plastic strain and (b) plastic strain rate for FEM simulations of Ti 6Al4V at cutting speeds varying from 10 to 100 m/min for two different depths of cut of 0.25 and 0.5mm……………………………………………..…..105 7.11 Cutting and thrust force plots with time for finite element simulations of Ti 6Al4V for a cutting speed of 30 m/min and depth of cut of 0.5 mm using different rake angles of 15°, 0°, 15°, 30° and 45°………………….107 7.12 Temperature and equivalent plastic strain contour plots in the chip formed in machining simulation of Ti 6Al4V for a depth of cut of 0.25 mm and a cutting speed of 5 m/min using a 0° rake angle tool……..108 7.13 Temperature and equivalent plastic strain contour plots in the chip formed in machining simulation of Ti 6Al4V for a depth of cut of 0.5 mm and a cutting speed of 5 m/min using a 0° rake angle tool………109 7.14 Effect of cutting speed on (a) average cutting force and (b) average thrust force for FEM simulations of Ti 6Al4V at cutting speeds from 10 to 100 m/min for two different depths of cut of 0.25 and 0.5 mm, respectively…………………………………………………110 7.15 Effect of cutting speed on (a) average power consumption and (b) number of segments for FEM simulations of Ti 6Al4V at cutting speeds from 10 to 100 m/min for two different depths of cut of 0.25 and 0.5 mm, respectively………………………………….111 7.16 Temperature and equivalent plastic strain contour plots in chip xii segmentation in machining simulations of Ti 6Al4V for a depth of cut of 0.5 mm at a cutting speed of 30 m/min using different rake angles of 15°, 0°, 15°, 30° and 45…………………………………...114 7.17 Effect of rake angle on (a) shear zone and rake face temperatures (b) equivalent plastic strain in the shear zone (c) plastic strain rate and (d) cutting forces for FEM simulations of Ti 6Al4V at cutting speed 30 m/min and depth of cut of 0.5 mm……………………………. 115 7.18 Temperature and equivalent plastic strain contour plots in chip segmentation in machining simulations of Ti 6Al4V for a depth of cut of 0.5 mm and cutting speed of 30 m/min using different friction coefficients (0.3, 0.5, 0.7, 0.9)…………………………………....118 7.19 Effect of friction coefficient on (a) shear zone temperature and (b) rake face temperature for FEM simulations of Ti 6Al4V at a cutting speed of 30 m/min and depth of cut of 0.5 mm……………………118 7.20 Effect of friction coefficient on (a) equivalent plastic strain and (b) average cutting forces for FEM simulations of Ti6Al4V at a cutting speed 30 m/min and a depth of cut of 0.5 mm…………………..118 7.21 FEM and experimental results of cutting and thrust force compared for feed rates 0.02, 0.05, 0.075, 0.1mm/rev………………………………. 121 7.22 Results of cutting forces obtained from FEM simulations compared with experimental data for feed rates 0.127 and 0.35mm/rev…...122 7.23 Results of chip comparison from FEM simulations and experimental data for 180m/min cutting speed and feed rates of 0.04, 0.06, 0.08, and 0.1mm/rev………………………………………………………..124 7.24 Results of chip morphology from FEM simulation conducted at 1.2 m/s and 0.5mm/rev feed rate compared with that of experimental data.125 1 CHAPTER 1 INTRODUCTION 1.1 State of Art: Finite Element Simulation of the Cutting Process Improvements in manufacturing technologies require better modeling and simulation of metal cutting processes. Theoretical and experimental investigations of metal cutting have been extensively carried out using various techniques. On the other hand, complicated mechanisms usually associated in metal cutting, such as interfacial friction, heat generated due to friction, large strains in the cutting region and high strain rates, have somewhat limited the theoretical modeling of chip formation. So, many researches are focusing on computer modeling and simulation of metal cutting process to solve many complicated problems arising in the development of new technologies. One of the stateofart efforts in manufacturing engineering is the finite element simulation of the metal cutting process. These simulations would greatly enhance our understanding of the metal cutting process and in reducing the number of trial and error experiments, which is used traditionally for tool design, process selection, machinability evaluation, chip formation and chip breakage investigations. According to a comprehensive survey conducted by the CIRP Working Group on Modeling of Machining Operations during 19961997 [1], among the 55 major research groups active in modeling, 43% were active in empirical modeling, 32% in analytical modeling, and 18% in numerical modeling in 2 which finite element modeling techniques are used as the dominant tool. More attention to the finite element method has been paid in the past decade in respect to its capability of numerical modeling of different types of metal cutting problems. Advantage of finite element method is the entire process can be simulated using a computer. Compared to empirical and analytical methods, finite element methods used in the analysis of chip formation have advantages in several respects, namely, (1) Material properties can be handled as a function of strain, strain rate, and temperature. (2) Interaction between the chip and the tool can be modeled as sticking and sliding. (3) Nonlinear geometric boundaries, such as the free surface of the chip can be represented and used. (4) In addition to the global variables such as, the cutting force, thrust force and chip geometry, local variables, such as stresses, temperature distributions, etc., can also be obtained. Finite element method has been used to simulate machining operations since the early 1970s [45]. With the development of faster processors and larger memory, model limitations and computational difficulty have been overcome to a large extent. In addition, more commercial FE codes are being developed for cutting simulations, including ABAQUS , AdvantEdge , DEFORM 2D , LS DYNA , FORGE 2D , MARC , FLUENT and ALGOR . Significant progress has been made in this field such as: 3 (1) Lagrangian approach is used to simulate the cutting process including incipient chip formation. (2) Segmented chip formation is modeled to simulate highspeed machining. (3) 3D simulation is performed to analyze oblique cutting. (4) A diversity of cutting tools and workmaterials is used in the simulation of cutting process. 1.2 Historical Developments The earliest finite element chip formation studies simulated the loading of tool against a preformed chip avoiding the problems of modeling large flows [46]. Small strain elasticplastic analysis demonstrated the development of plastic yielding along the primary shear plane as the tool was displaced against the chip. This work had a number of limitations, making it only of historical interest. The limitations of this initial work were removed by Shirakashi and Usui [3], who developed an iterative way of changing the shape of the preformed chip until the generated plastic flow was consistent with the assumed shape. They also included realistic chip/tool friction conditions and material flow stress variations with strain, strain rate and temperature measured from high strain rate Hopkinson bar tests. The procedure of loading a tool against an already formed chip greatly reduced computing capacity requirements. The justification of the method was that it gave good agreement with experiments but it did not follow the actual path by which a chip should be formed. Rigid–plastic modeling however, does not require the actual loading path to be followed. Iwata et al. [4] developed steady state rigidplastic modeling (within a eulerian 4 framework) adjusting an initially assumed flow field to bring it into agreement with the computed field. They included friction, work hardening, and a chip fracture criterion. Experiments at low cutting speeds supported their predictions. The mid1980s saw the first nonsteady chip formation analyses, following the development of a chip from first contact of a cutting edge with the workpiece as in machining. Updated Lagrangian elasticplastic analysis was used, and different chip/ work separation criteria at the cutting edge were developed. Strenkowski and Carol [5] used a strainbased separation criterion. At that time, neither a realistic friction model nor coupling of elasticplastic to thermal analysis was included. The 1990s have seen the development of nonsteady state analysis, from transient to discontinuous chip formation, the first threedimensional analyses, and the introduction of adaptive meshing techniques particularly to cope with the flow around the cutting edge of a tool. A simple form of remeshing at the cutting edge, instead of the geometrical crack, was introduced to accommodate the separation of chip from the work. Both rigidplastic and elasticplastic adaptive remeshing softwares have been developed and are being applied for chip formation simulations [14, 15]. Marusich and Oritz [14] developed a twodimensional finite element code for the simulation of orthogonal cutting that includes sophisticated adaptive remeshing, thermal effects, a criterion for brittle fracture and tool stiffness. They seem to be more effective than Arbitrary LagrangianEulerian (ALE) methods in which the mesh is neither fixed in space nor in the workpiece. Thus, the 1970s to the 1990s has seen the development and testing of finite element techniques for chip formation processes and during this period, many researchers have concentrated 5 more on the development of the new methods in the finite element simulations of metal cutting [2]. 1.3 Principles of Metal Cutting Metal cutting is classified as the secondary process by which material is removed to transform a raw material to a part with certain shape, size, dimensional tolerance and surface finish. The theory of machining is concerned with the various features of the cutting process including the forces, strain and strain rates, temperatures, and wear of cutting tools. All metal cutting operations, such as turning, drilling, boring, milling, grinding, reaming and other metal removal processes produce chips in a similar fashion. Therefore, analysis of chip formation can give better understanding to the mechanics of machining processes. Metal cutting involves concentrated shear along a rather distinct shear plane. As metal approaches the shear plane, it does not deform until the shear plane is reached. It then undergoes a substantial amount of simple shear as it crosses the thin shear zone. There is essentially no further plastic flow as the chip proceeds up the face of the tool. The tool is a single point tool that is characterized by the rake angle α and the clearance angle θ as shown in Fig. 1.1. When the rake face of the tool is in the clockwise position from the workpiece then the rake angle is considered positive and if it is counter clockwise then it is considered negative [6]. A small clearance angle is generally used to keep the tool from damaging the finished surface of the workpiece. The rake face of the tool is the surface over which the chip flows and has a contact length lc, which is the chiptool contact interaction. The prescribed velocity V, known as the cutting speed is in the feed direction. 6 Fig. 1.1 Orthogonal metal cutting model [6] The localized straining in the workpiece enforced by the tool causes plastic deformation of the undeformed chip t, which proceeds to become deformed chip thickness tc. Large forces are generated during the cutting process. The cutting force Fc acts in the direction of cutting velocity and the thrust force Ft acts normal to the cutting velocity in the direction perpendicular to workpiece. A knowledge of the basic force relationships and the associated geometry occurring in the cutting process of metal cutting is a necessity if the solution of engineering problems arising in that field is to be handled by FEM simulations. 1.4 Principles of Finite Element Modeling Finite element analysis is an approximate numerical analysis tool to study the behavior of a continuum or a system to an external influence, such as stress, heat and pressure. This involves generation of a mathematical formulation of the physical process followed by a numerical solution of the mathematics model. Basic concept of finite element method involves division of a given domain into a set of simple subdomains, called, finite elements accompanied with polynomial approximations of solution over 7 each element in terms of nodal values and applying the calculated finite solutions to the whole geometry to solve the problem. The advantage of finite element method is, it provides approximate solutions to complex problems that are difficult to solve analytically. Finite element analysis involves three stages of activity, namely, preprocessing, processing and postprocessing. Preprocessing involves the preparation of data, such as nodal coordinates, connectivity, boundary conditions, and loading and material information. The processing stage involves stiffness generation, stiffness modification, and solution of equations, resulting in the evaluation of nodal variables. Other derived quantities, such as gradients or stresses may be evaluated at this stage. The postprocessing stage deals with the presentation of results. Typically, the deformed configuration, mode shapes, temperature and stress distribution are compared and displayed at this stage. A complete finite element analysis is a logical interaction of these three stages. 1.5 Metallurgical Aspects of Titanium alloys Titanium alloys have been extensively studied over the past few decades due to their important technological applications. Their high strength, low density, corrosion resistance, good formability, weldability, and good metallurgical stability prompted the use of these alloys in a wide variety of applications ranging from aircraft engine and structural components to biomedical implants. Among titanium alloys, Ti 6Al4V accounts for the largest share of the present market and hence studied in depth. Ti 6Al4V is currently used in a wide range of low and high temperature applications, such as blades and other components for turbines in aircraft engine applications, steam turbine blades, 8 marine components, structural forgings and biomedical implants [7]. Despite the increased usage and production of titanium alloys, they are expensive when compared to many other metals because of the complexity of the extraction process, difficulty of melting, and problems associated with fabrication and machining. In order to improve processing as well as to lay ground to new titanium alloys, it is important to understand the deformation mechanisms and microstructural evolutions associated with these deformations. Titanium alloys are usually divided into four main groups according to their basic metallurgical characteristics: αalloys, near αalloys, αβ alloys, and βalloys [8]. αalloys: This group contains α stabilizers, sometimes in combination with neutral elements, and hence have an αphase microstructure. One such singlephase αalloy, Ti5Al2.5Sn, is still available commercially and is the only one of its type to survive besides commercially pure titanium. The alloy has excellent tensile properties and creep stability at room temperature and elevated temperatures upto 300°C. αalloys are used chiefly for corrosion resistance and cryogenic applications. Near αalloys: These alloys are highly αstabilized and contain only limited quantities of βstabilizing elements. They are characterized by a microstructure consisting of αphase containing only small quantities of βphase. Ti8Al1MoIV and Ti6Al5Zr 0.5Mo0.25Si are examples of near αalloys. They behave more like αalloys and are capable of operating at greater temperatures of between 400°C and 520°C. αβ alloys: This group of alloys contains additions of α and βstabilizers and possess microstructures consisting of mixtures of α and βphases. Ti 6Al4V and Ti4Al 2Sn4Mo0.5Si are its most common alloys. They can be heattreated to high strength 9 levels and hence are used chiefly for high strength applications at elevated temperatures of between 350°C and 400°C. βalloys: These alloys contain significant quantities of βstabilizers and are characterized by high hardenability, improved forgeability, and cold formability as well as high density. Basically, these alloys offer an ambient temperature strength equivalent to that of αβ alloys but their elevated temperature properties are inferior to those of the αβ alloys. Ti 6Al4V, an alloy introduced in 1954, comes as close to being a generalpurpose grade as possible in titanium. In fact, it is considered as the workhorse of titanium alloys and is available in all product forms. In Ti 6Al4V, both α and βphases are Al solid solutions in Ti. Fig 1.2 Microstructure of Ti 6Al4V alloy showing primary αphase and transformed β grains [44] Various impurity atoms, such as O, C, N, and H are usually present. βphase may be stabilized at room temperature by adding βstabilizing elements such as V, Fe and Mn. At room temperature, stabilized βphase contains more V than nominal 4%. Above 10 527°C, α transforms to βphase, while above 980°C the whole microstructure is composed of equiaxed β grains. The flow stress of this alloy is strongly dependent on temperature and deformation rate [10]. At temperatures above 527°C, the flow stress decreases sharply with temperature while the strain rate sensitivity increases. The flow mechanisms and kinetics are different in these two phases. This renders to a large number of deformation mechanisms responsible for the macroscopic behavior of the alloy. Their identification is important for understanding the mechanical response of this alloy. 1.6 Machinability aspects of Ti 6Al4V Machinability is defined as the ease or the difficulty with which a material can be machined under a given set of cutting conditions including cutting speed, feed, and depth of cut. It is mainly assessed during the cutting operation by measuring component forces, chip morphology, surface finish generated and tool life. The machinability of Ti 6Al4V and other titanium alloys has not kept pace with advances in manufacturing processes due to their several inherent properties [7, 8]: 1) Its high chemical reactivity with almost all tool materials results in rapid wear of the tool at high cutting speeds. Also the tendency to weld to the cutting tool during machining increases leading to chipping and premature failure. 2) Its low thermal conductivity increases the temperature at the tool/workpiece interface, which affects the toollife adversely. 3) Its high melting temperature and high temperature strength further impairs its machinability. 11 4) Besides high cutting temperatures, high mechanical pressure and high dynamic loads in machining of titanium alloys result in rapid tool wear. 5) Its low modulus of elasticity can cause slender workpieces to deflect more than comparable pieces of steel. This can create problems of chatter, tool contact and holding tolerances. Despite the increased usage and production of this alloy, it is expensive when compared to many other metals and alloys because of the complexity of the extraction process, difficulty of melting and problems during machining and fabrication. Serious vibrations are often encountered during machining that impose limit on the material removal rate, and consequently, productivity. Also this alloy is spectacular to machine because of sparks generated at high speeds. Most tool materials used for machining this alloy wear rapidly even at moderate cutting speeds. Titanium alloys are generally difficult to machine at cutting speeds over 30 m/min with HSS steel tools and over 60 m/min with cemented tungsten carbide tools. Other types of tool materials, including ceramic, diamond and cubic boron nitride are highly reactive with titanium alloys and consequently not used in the machining of these alloys. These problems can be minimized by employing very rigid machines, using proper cutting tools and setups, using low cutting speeds, maintaining high feed rates, minimizing cutting pressures, providing copious coolant flow and designing special tools or nonconventional cutting methods. 1.7 Mechanism of ShearLocalized Chip Formation in Ti 6Al4V Titanium and other aerospace structural alloys are extremely difficult to machine at high cutting speeds due to limitations associated with its several inherent properties. It 12 has been observed that in metal cutting the thermomechanical behavior at the workpiece/tool interface significantly influences the chip morphology, which in turn affects the tool life. In order to increase tool life and productivity in machining these alloys it is necessary to study the mechanism of chip formation and its effect on machinability and tool life. Depending on the type of workmaterial, its metallurgical conditions and the cutting conditions used, three types of chip formation are commonly encountered in metal cutting process. They are the continuous chip, shearlocalized chip and discontinuous chip. Traditionally most of the investigations on metal cutting have focused on the continuous chip formation because continuous chip is an ideal chip for analysis as it is relatively stable and many conditions can be simplified. However, long continuous chips are not preferred in machining because, in practice they interfere with the process and may cause unpredictable damage on machined surface and tool. Shearlocalized chips are found in the case of most difficulttomachine materials with poor thermal properties. This type of chip on the other hand is easier to break and considered as a relatively ideal chip to dispose off when the machining process is automated. In the case of machining Titanium alloys, chip is segmented and the strain in it is not uniformly distributed but is confined mainly to narrow bands between the segments. Whereas in continuous chip formation, the deformation is largely uniform. The sequence of events leading to segmented chip formation when machining Ti 6Al4V was described by Komanduri and Von Turkovich [11] based on a detailed study of video sequence of low speed machining experiments conducted inside the scanning electron microscope, high speed movie films of the chip formation process at higher speeds and the micrographs of midsections of the chips. The mechanism of chip formation when 13 machining Ti 6Al4V was found to be different from the continuous chip formation [26]. There are two stages involved in this process. One stage involves plastic instability and strain localization in a narrow band in the primary shear zone leading to catastrophic shear failure along the shear surface. The other stage involves gradual buildup of segment with negligible deformation by the flattening of the wedgeshaped work material ahead of the advancing tool. Generally, adiabatic shearing caused by thermomechanical instability is held responsible for this process. This nature of instability, frequently referred to as adiabatic shear, was originally expressed by Recht [12] as one in which the rate of thermal softening exceeds the rate of strain hardening i.e., the slope of the shear stressshear strain curve becomes zero. Also, alternative theories have been formulated based on damage models and crack formation processes which is probably applicable at low speeds. Finite element simulations of machining a Ti 6Al4V alloy, allows study of chip formation and the mechanism of chip segmentation in detail. Such simulations have shown that it is indeed possible to form strongly segmented chips by the described process without the necessity of crack formation. 1.8 Thesis outline A brief description of each chapter in this study is given in the following: Chapter 1 gives a brief introduction of finite element simulation of the cutting process, its historical developments and principles, metallurgical aspects and machinability issues of Ti 6Al4V alloy and mechanism of shearlocalized chip formation in Ti 6Al4V. 14 Chapter 2 presents literature review on numerical analysis of chip segmentation, experimental work on Ti 6Al4V alloy and finite element simulations of shearlocalized chip formation in machining a Ti 6Al4V alloy. Chapter 3 contains finite element formulations of metal cutting mechanics, friction along the toolchip interface, contact conditions, adaptive remeshing and chip formation. Chapter 4 gives a brief description of machinability issues of Ti 6Al4V alloy and outlines the rationale and motivation behind this work along with the objectives and research approach. Chapter 5 deals with the mechanism of shearlocalized chip formation and explains the criterion used in this study to simulate it. This chapter also illustrates the metallurgical aspects of Ti 6Al4V alloy that influence this type of chip formation in machining. Chapter 6 gives a description of material model used to represent deformation behavior of Ti 6Al4V alloy under high strains, strain rates, and temperatures. This chapter also derives the equations of Recht’s catastrophic shear failure criterion and stress update algorithm used in the userdefined subroutine code (UMAT). Chapter7 presents the physical properties of Ti 6Al4V alloy, parameters of JohnsonCook material model, simulation approach and cutting conditions used. This chapter also provides chip formation process, temperature and plastic strain distribution in the chip, and comparison of finite element simulations with experimental results reported in the literature. Finally, it discusses the effect of cutting speed, feed rate, rake 15 angles and coefficient of friction on cutting forces, temperatures, strains and chip segmentation. Chapter 8 draws conclusions on the work done in this study and presents proposed scope for future work. 16 CHAPTER 2 LITERATURE REVIEW 2.1 Numerical Study of ShearLocalized Chip Formation Over the past ten years or so, numerical study of the machining processes has been the subject of intense research in which various aspects of shearlocalized chip formation, discontinuous chip formation and algorithms for element separation have been addressed. Various criteria for Shearlocalized chip formation, such as effective strain criterion, maximum principal stress criterion, maximum shear stress criterion, catastrophic shear failure criterion and so on have been utilized by many researchers for the chip formation process using FEM. According to Xie et al. [13], most shearlocalized chips are formed by flow (shear) localization during the chip deformation. Some bands of intense shear dividing the chip into segments occur in metal cutting process. Accordingly, this band is a very thin layer with extremely concentrated shear strains, which may cause chip to become easily separated and broken. They developed a mechanistic model to predict quantitatively the critical cutting conditions for a shearlocalized chip formation. This was done by establishing a relationship between the flow localization parameter and related governing cutting conditions, i.e. cutting speed and feed rate. Flow localization parameter β is defined as 17 ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ + + − ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ + ⎟⎠ ⎞ ⎜⎝ ⎛ ∂ ∂ = − + Vf K Vf K n Vf K c n T m γ γ γ ρ τ γ β 1 1 1 1 1.328 0.664 1 1 1.328 0.9 3 (2.1) where m is strain rate sensitivity parameter, n is strain hardening exponent, K1 is thermal diffusivity of work material, γ is shear strain and T is temperature. The governing cutting process parameters, cutting speed V and feed rate f are associated with possible onset of shear localization. The flow localization parameter, β is used to rank the tendency for strain concentration within a material. A certain critical value of β must exceed to reach at the onset of strain localization. The above equation is used to predict the flow localization parameter, β, for given cutting conditions and material properties. The value of β increases as either the cutting speed V increases or the feed rate f increases. Usually, β at which shear banding is possible is determined experimentally and for Ti 6Al4V the value is 4.41. The formation of shearlocalized chip involves several material, mechanical, and thermophysical properties including density, specific heat, strain hardening exponent, thermal diffusivity, strain rate sensitivity and conductivity. Basically, as a cutting condition (Vf) reaches the critical value at which shearlocalized chips are formed, the plastic deformation rate becomes high and toolworkpiece friction becomes more severe, increasing the rate of heat generation. The adiabatic or quasiadiabatic condition may be reached due to high accumulation of heat. In this case, temperature can be very high locally in some places of the workpiece, resulting in further thermal softening. This further thermal softening reduces strainhardening capacity so the 18 instability takes place in a narrow band of the chip. Thus, β may be used as material property to judge and predict shear localization. Marusich et al. [14] implemented a fracture model that allows for arbitrary crack initiation and propagation in the regime of shearlocalized chips. They presented a model of highspeed machining using a lagrangian code to simulate large unconstrained plastic flow with continuous adaptive meshing and remeshing as principal tools for sidestepping the difficulties associated with deformation induced element distortion. Accordingly, when slip induced transgranular cleavage is the dominant mechanism, fracture of mild steel can be described in terms of a critical stress criterion. The critical stress σf appears to be relatively independent of temperature and strain rate and can be inferred from toughness KIC through the small scale yield relationship σf = l KIC 2π (2.2) The critical distance l correlates with the spacing of the grain boundary carbides. Under mixedmode conditions, such as expected in machining, the crack might kink or follow a curved path as it grows. To predict the crack trajectory under conditions of brittle fracture, they adopted maximum hoop stress criterion, according to which crack propagates along the angle θ from the crack face at which hoop stress σθθ attains a relative maximum. Combining maximum hoop stress along the angle θ θ max σθθ and critical stress criterion, the criteria for mixedmode crack growth is given by θ max σθθ (l, θ)= σf (2.3) The critical crack tip opening displacement (CTOD) criterion for mode I crack propagation can be recast as the attainment of critical value εp f of the effective plastic strain at a distance l directly ahead of the crack tip. The criterion can be expressed as 19 θ max εp(l, θ)= εp f (2.4) with the understanding that crack propagates at an angle θ for which the criterion is met. The critical effective plastic strain can be estimated as εp f ≈2.48e1.5p/σ (2.5) where p =σkk/3 is the hydrostatic pressure. Ceretti et al. [15] implemented the Cockroft and Latham damage criterion for material fracture [16]. According to the numerical model, material fracture is simulated by deleting the mesh elements that have been subjected to high deformation and stress. Accordingly, the damage criterion is evaluated by the Equation 2.6: Ci = ε σ ε σ σ d f ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ∫ * 0 (2.6) where Ci is the critical damage value given by uniaxial tensile test, εf the strain at the breaking condition, σ* is the maximum stress. The criterion predicts the damage when the critical value Ci is exceeded. The damage is evaluated for each element of the workpiece. Element deletion occurs when damage value is reached. To simulate the shearlocalized chip formation they used commercial software DEFORM2D and customized it with new algorithms incorporating the damage criterion. Hashemi et al. [17] developed a fracture algorithm for simulating chip segmentation and separation during orthogonal cutting process. This criterion evaluates the principal stress at each node in each computational cycle. If the magnitude of principal stress exceeds a predetermined value, which can be taken as material fracture strength, a crack is assumed to initiate and propagate along the direction normal to the stress vector. 20 Obikawa and Usui [18] proposed an effective strain based criterion in their FEM simulation of serrated chip formation in cutting Ti 6Al4V. They postulated that when the effective plastic strain at specific node reaches the preset critical value, this node is then separated indicating crack initiation and propagation. According to them, in the machining of titanium alloys, serrated chips are produced due to ductile cracks propagating from the chip free surface. The following fracture criterion was applied for the crack initiation and propagation: c ε p >ε (2.7) where ⎟⎠ ⎞ ⎜⎝ − + ⎛ + ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = − 293 273 0.09 exp 37.8 , 0 100 max 0.075 ln . ε ε σ ii θ p c where max [,] is a function giving the maximum value in the bracket. This criterion is based on equivalent plastic strain. This predetermined critical strain value εc is a function of strain, strain rate . p ε , hydrostatic pressure σ ii , and temperature θ. Rice [19] simulated shearlocalized chip formation using critical shear strain criterion. They developed the failure criterion from the studies of mechanism of chip segmentation using photographs of segmented chips taken at various stages during machining. Accordingly, they proposed that chip transforms from continuous to segmental when nominal shear strain in the primary shear zone reaches critical value. Iwata et al. [20] proposed the following stressstrain based criterion in their FEM analysis of orthogonal cutting as: ( ∫ ε ε f 0 +b1σm+ b2) dε = b3 (2.8) 21 where the constants b1, b2 and b3 are given as functions of metallurgical properties However, the authors claimed that obtaining these constants needs complicated experiments that requires high pressure. Shivpuri et al. [21] implemented the deformation energybased criterion proposed originally by Cockroft and Latham [16] to simulate shearlocalized chip formation of Ti 6Al4V using FEM. This criterion was based on critical damage value given by: Ci= ε σ ε σ σ d f ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ∫ * 0 (2.9) The Ci value is a workpiece material constant and does not depend on the workmaterial or the tool material. The Ci value is a function of temperature and microstructure of the point of interest and can be expressed as Ci = f (T, m), where m denotes the microstructure. When the temperature is below the βtransformation temperature, the microstructure is presented by α+β, otherwise microstructure is β. The critical damage value is evaluated for each element of the workpiece. Element deletion occurs when damage value is reached. Thus the crack is initiated and propagated. In the early 1970’s Nakayama [22] found that sawtooth chips were produced when highly cold worked (60 percent reduction) 40/60 (Zn/Cu) brass was turned under orthogonal conditions. Based on that he proposed a theory of sawtooth chip formation. Accordingly, shear crack initiates at the free surface of the chip in the primary shear zone and runs from the surface downward along shear plane towards the tool tip. With further advance of the tool, the chip glides outward like a friction slider along the cracked surface until the next crack forms and a new cycle begins. A shear crack will initiate at the free surface of the chip where the crack arresting normal stress is zero and proceeds 22 downward along the shear plane toward the tool tip. Initially the crack will be continuous across the width of the chip but will become discontinuous as higher crack arresting normal stresses are encountered. Owen and Vaz Jr. [23], applied computational techniques to investigate highspeed machining. They focused on the simulation of Ti 6Al4V machining involving material failure due to adiabatic strain localization and addressed important issues, such as adaptive mesh refinement and material failure. According to them, material failure under adiabatic strain localization conditions results from the accumulation of large plastic deformation and microscopic damage. They employed a twoparameter model to describe material failure, i.e. a failure indicator (or fracture criterion), I, and an energy release factor, Wr. Accordingly, the former indicates failure onset and the latter defines the amount of energy released during the softening process as shown in the Fig. 2.1. Fig. 2.1 Stressstrain diagram for a combination of strain hardening, thermal softening and failure softening [23] The failure softening path, BC, is governed by the softening modulus, Et, as Et = ∂ε ∂ Y T =  r Y cr e W T h 2 , 2 (2.10) 23 where he is the characteristic length of the particular element, TY is the true stress of the material at yield point and TY,cr is the critical yield stress of the material where failure initiates. They adopted a material failure criterion based on the uncoupled integration of Lemaitre’s ductile model. Lemaitre postulated that damage progression is governed by void growth represented by the damage evolution equation s r Y Y D D ⎟ ⎠ ⎜ ⎞ ⎝ ⎛ − − = ∂ = − ∂ 1 . . . ψ γ γ (2.11) where .D is the damage variable, r and s are damage parameters, Ψ is the dissipation potential and Y is the damage strain energy release rate. They also incorporated a stress update algorithm which is called as operatorsplit algorithm to which the computation of the fracture indicator and failure softening were added. The methodology consists of decomposing the problem into elastic and plastic components, in which the former assumes that deformation is entirely elastic whereas the latter solves a highly nonlinear system of equations comprising the constitutive relations, evolution laws and plastic consistency using the elastic predictor stage as the initial condition. In order to overcome the problem of severe element distortion when using lagrangian formulations in their simulations, they incorporated error estimator based on the principle that the adaptive procedure should not only capture the progression of the plastic deformation but also provide refined meshes at regions of possible material failure. Samiatin and Rao [24] developed another model for shearlocalization, which incorporates a simple heat transfer analysis, and material properties, such as the strainhardening rate, the temperature dependence of flow stress, and the strain rate sensitivity of the flow stress to establish the tendency towards the localized flow. Using their 24 literature data, they found the nonuniform flow in metal cutting to be imminent when the ratio of the normalized flowsoftening rate to the strain rate sensitivity is equal to or greater than 5. Recht in 1964 [12] developed a classical model of catastrophic shear instability for shear localization in metals under dynamic plastic conditions. Accordingly, catastrophic shear will occur at a plastically deforming location within a material when the slope of the true stressstrain function becomes zero. He formulated a simple criterion for catastrophic slip in the primary shear zone based on the thermophysical response of the work material under the conditions of cutting. The catastrophic slip can be written as: where τ, ε and θ refer to the shear stress, shear strain, and temperature, respectively. Material will shear catastrophically when this ratio lies between zero and one; catastrophic shear will be imminent when the ratio equals to one. Komanduri and Hou [25] developed a thermal model for the thermoplastic shear instability in the machining of a Ti6Al4V alloy. It is based on the analysis of the shearlocalized chip formation process due to various heat sources (primary, preheating and image) in the shear band. They determined the temperature in the shear band analytically using the Jaeger’s classical stationary and moving heat source methods and used Recht’s catastrophic shear instability model to determine the onset of shear localization. Accordingly, the shear stress in the shear band is calculated at the shear band temperature and compared with the shear strength of the bulk material at the preheating temperature. According to Recht, if the shear stress in the shear band is less than or equal to the shear (2.12) 25 strength of the bulk material, then shear localization is imminent. The cutting speed at which this occurs is taken as the critical speed for the onset of shear localization. The effect of depth of cut on the critical speed is determined and found that lower the depth of cut, higher the critical speed for onset of shear localization. The cutting speed for the onset of the shear localization in the machining of Ti 6Al4V was found to be extremely low, ~0.42 m/min for a depth of cut of 0.2 mm. A bestfit relationship between the critical cutting speed for shear localization and depth of cut was developed using the analytical data and is given by 1.0054 0 0.082667 V = × a − cri (2.13) 2.2 ShearLocalized Chip Formation in Ti 6Al4V: Experiments Numerous studies on the machining of titanium alloys (analysis of chip formation and cutting forces) have been carried out in a range of cutting velocities lower than 5 m/s, by Komanduri and von Turkovich [11], Komanduri [26], Narutaki and Murakoshi [47], Larbi [50], Bayoumi and Xie [39] and Diack [49]. In the work of Hoffmiester et al. [48] and Molinari et al. [28], larger velocities were considered in the range of 20 to 100 m/s. These studies illustrated several unique features associated with the machining of these alloys, including the following [26]: (1) The role of poor thermal properties of titanium alloys which interact with the physical properties in controlling the nature of plastic deformation (i.e. strain localization) in the primary zone is illustrated. (2) Periodic gross inhomogeneous deformation occurs in the primary zone. 26 (3) Instability in the chip formation process results in the segmented or cyclic chip. (4) Oscillations in the cutting and thrust components of force cause chatter and the need to have a rigid toolworkmachine tool system. (5) High toolchip interface temperatures and high chemical reactivity of titanium in machining with almost any tool material are responsible for the rapid tool wear. (6) The low modulus of elasticity which decreases rapidly, even at moderate temperatures, causes undue deflections of the workpiece, especially when machining slender parts, and inaccuracies in the finished part. Chip formation studies were conducted at various machining speeds from an extremely low speed of 0.127 mm/min to a moderately high speed of 5.1 m/s. The low speed experiments were conducted inside a scanning electron microscope and the cutting process was recorded on a video tape. Chip formation studies at higher cutting speeds were conducted on a lathe with the aid of a HiCam highspeed movie camera (camera speed up to 8000 frames/s) using the technique developed earlier by Komanduri and Brown [19]. The sequence of events leading to cyclic chip formation when machining titanium was described by Komanduri and von Turkovich [27] based on a detailed study of videotapes of low speed machining experiments inside the scanning electron microscope, highspeed movie films of the chip formation process at higher speeds and the micrographs of the mid section of the chips. Accordingly, they described two stages in the chip formation process. One stage involves plastic instability and strain localization in a narrow band in the primary shear zone leading to catastrophic shear failure along a shear surface. The other stage involves gradual buildup of the segment with negligible 27 deformation by the flattening of the wedgeshaped work material ahead of the advancing tool. Molinari et al. [28] carried out experimental analysis of shear localization and chip segmentation in Ti 6Al4V at cutting speeds in a range from 0.01 to 73 m/s. To cover a wide range of cutting speeds, two different devices were used. The low cutting speeds (0.01 to 1.0 m/s) were obtained with a universal highspeed testing machine. The second arrangement was constituted by an airgun setup for speeds from 1.0 to 73 m/s. Accordingly, to avoid fracture of the tools at high impact velocities, small depths of cut were used. All tests were carried out with carbide tools with a rake angle 0° and the tools were square shaped without chipbreaker. The collected chips were embedded into a resin and the lateral section polished and etched for observation in the optical microscope. The following conclusions were drawn from their experimental studies: (1) Chip segmentation was observed to be related to adiabatic shear banding. (2) Adiabatic shear bands are the manifestation of a thermomechanical instability resulting in the concentration of large shear deformations in narrow layers. (3) For velocities lower than 1.2 m/s, chip serration is related to the development of deformed shear bands as shown in Fig. 2.2, which are the manifestation of thermomechanical instability. However, at low values of cutting velocities, the instability process is weak and the localization is not as sharp as for higher velocities. (4) The patterning of adiabatic shear bands were observed experimentally by measurements of shear band width and chip segment width. Patterning is seen to be strongly dependent on cutting velocity V. 28 Fig. 2.2 Adiabatic shearlocalized chip in a Ti 6Al4V alloy obtained by orthogonal cutting at a cutting speed of 1.2 m/s [28] Barry et al. [29] conducted orthogonal cutting tests to investigate the mechanism of chip formation and to assess the influences of such on acoustic emission (AE) for a Ti 6Al4V alloy of 330HV with an uncoated P10/P20 carbide tool. Here, AE refers to the transient displacement of the surface of a body, of the order of 1012 m, due to the propagation of high frequency elastic stress waves. The surface displacement is detected by the AE sensor and typically output as a proportional voltage. Accordingly, orthogonal cutting tests were undertaken using a Ti 6Al4V alloy of 330 HV with uncoated P10/P20 carbide tool with a rake angle of 6° and a clearance angle of 12°. The workpiece used was in the form of a 25 mm diameter disc, 1.1mm in width and was held on a mandrel during cutting. The tests were performed on a Daewoo PUMA 43A CNC lathe under a constant surface speed control. Acoustic emission signals were captured with a Kistler wide band piezoelectric AE sensor and the signal amplified and conditioned using a Kistler Piezotron unit. Fig. 2.3 shows the influence of cutting speed and feed on chip morphology in the orthogonal cutting of Ti 6Al4V. They classified all chips obtained for different cutting conditions in the range 20100 μm feed and 0.253 m/s cutting speed as either aperiodic sawtooth or periodic saw tooth. Accordingly, it was seen that with low values of cutting 29 speed and undeformed chip thickness (e.g., 20 μm), aperiodic sawtooth chips were produced. Increase in either or both of these parameters resulted in a transition from aperiodic to periodic sawtooth chip formation. They observed occurrence of welding between chip and tool during machining and with degree of welding increasing with cutting speed. Fig. 2.3 Chip morphology as a function of cutting speed and feed in orthogonal cutting of a Ti 6Al4V alloy [29] In machining of Ti 6Al4V, Barry et al. [29] observed that catastrophic failure occurs not only within the primary shear zone, but also within the weld formed between the chip and the tool rake face and the fracture of such welds appears to be the dominant source of acoustic emission in machining Ti 6Al4V with cutting speeds greater than 0.5 m/s. This is precisely the mechanism proposed by Komanduri and von Turkovich [11] (See Fig.5.1 in chapter 5). Xie and Bayoumi [13] conducted orthogonal cutting tests and various metallurgical analysis techniques were used to examine the chip formation process and the role of shear instability. Cutting speeds were varied from 0.5 to 8.0 m/s and the feed 30 rates from 0.03 to 0.5 mm/rev. The following are their conclusions from the experimental study: (1) The results from metallurgical studies showed no diffusiontype phase transformation in the machined chips, while the Xray diffraction tests identified some nondiffusional phase transformation from βphase into α phase during chip formation. (2) Intensive shear takes place in a narrow zone rather than in a plane as is often assumed by some investigators in the analysis of orthogonal machining process. (3) For each work material there exists a critical value (Vf) of chip load at which shear localized chips were observed. The cutting conditions also influence the shear banding in a way that the shear banding frequency increases with an increase in feed rate or a decrease in cutting speed. Shivpuri et al. [21] conducted experiments on Ti 6Al4V using a CNC turning center at cutting speeds of 60, 120 and 240 m/min, feed rates of 0.127 and 0.35 mm/rev and depth of cut of 2.54 mm. They collected the deformed chips for different cutting speeds and observed the chip morphology under a microscope. The cutting forces were measured with a Kistler dynamometer (Type 9121). A general carbide tool was used with a rake angle of 15° and a relief angle of 6°. They showed that (Fig. 2.4) as the cutting speed is increased, the chip fracture observed at lower cutting speeds gradually reduces, and the flow localization and strain between the serrated chip gradually increases resulting in changing chip morphology (discontinuous chip becoming continuous but serrated). They observed that the crack, which determines the serrated chip during cutting, always occurs in the primary shear zone on the tool tip side and at low cutting 31 speeds, the crack propagates to the tool tip because the temperature in the chip being formed is much lower than the βtransus temperature. Whereas at high cutting speeds, the crack propagates to the free surface as the temperature in the secondary shear zone at the tool face is much higher than that in the free surface. This temperature is above βtransus Fig. 2.4 Shearlocalized chip formed in machining a Ti 6Al4V alloy at cutting speeds of 60 and 120 m/min and feed rate of 0.127 mm/rev [21] temperature and causes microstructural changes resulting in the rise of ductility in the shear zones. The chip just formed during cutting process connects to the workpiece forming the serrated chip morphology. Ribeiro et al. [30] carried out turning tests on Ti 6Al4V with conventional uncoated carbide tools for cutting speeds of 55, 70, 90 and 110 m/min, using 0.1 mm/rev feed rate and 0.5 mm depth of cut. The objective of this work was to optimize the cutting speed for best finish in the machining of titanium alloy. Fig. 2.5 shows the variation of roughness with the length of cut at different cutting speeds. They found 90 m/min to be the optimum cutting speed for best finish. 32 Fig. 2.5 Roughness of workpiece material (Ti 6Al4V) as a function of cutting length at different cutting speeds [30] Lee and Lin [31] investigated the high temperature deformation behavior of Ti 6Al4V alloy by conducting mechanical tests using compression split Hopkinson bar under high strain rate of 2×103 s1 and temperatures varying from 7001100°C in the intervals of 100°C. Fracture features of the specimens after the mechanical tests were observed using optical and scanning electron microscopy. They showed that extensive localized shearing dominates the fracture behavior of this material and adiabatic shear bands run across the specimen. Another important observation made relating to the shear band is the formation and coalescence of voids in an adiabatic shear band, which might lead to variation in mechanical properties of the material. Fig. 2.6 shows a typical array of coalesced voids in a welldeveloped shear band. Initially, the voids are observed to be spherical, but when the diameters reach the thickness of the shear bands, the voids coalesce and their extension along the shear band results in elongated cavities and smoothsided cracks. 33 Fig. 2.6 Voids and cracks in an adiabatic shear band of a Ti6Al4V alloy specimen deformed at 700°C and 2×103 s1 strain rate [31] In each specimen, a transformed adiabatic shear band appears in the microstructure, indicating that a catastrophic localized shear occurred during deformation. Failure analysis of the specimens indicated that adiabatic shear bands are the sites where the fracture of the material occurs, and that the thickness and microhardness of the adiabatic shear bands vary completely with temperature. Picu et al. [10] investigated plastic deformation of Ti 6Al4V alloy under low and moderate strain rates and at various temperatures. Mechanical testing was performed in the temperature range 6501340 K and at strain rates from 103 to 10 s1. A discontinuity in the flow stress versus temperature curve was reported. Fig. 2.7 shows temperature dependence of the yield stress at low strain rates (103 s1) and comparison with the published results. The curve shows discontinuity at temperature T~ 800 K. At temperatures above 800 K, the flow stress sharply decreases with temperature. The discontinuity observed in the flow stresstemperature curve suggests that additional deformation mechanisms become active at that temperature and this leads to dramatic reduction in strain hardening. Texture has a significant effect on the flow stress at 34 temperatures below the discontinuity, while higher temperatures rapidly decrease the texture sensitivity. At temperatures in the 11001350 K range, phase transformation from α to β becomes the impetus behind the mechanical behavior. Also associated with this Fig. 2.7 Temperature dependence of flow stress in a Ti 6Al4V alloy at very low strain rates of 103 s1 [10] transformation is a pronounced variation in the strain rate sensitivity (m), with the sensitivity being higher in the 100% β material. Reissig et al. [32] investigated different machining processes, such as drilling, shot peening and electrochemical drilling. According to them, it is usually very difficult or often impossible to measure directly the surface temperatures introduced by machining processes, such as deep hole drilling. Therefore, they presented a postmortemmethod which allows the determination of maximum temperatures during machining by measuring the local vanadium concentration in a Ti 6Al4V alloy. They proposed a method to determine a local pseudo temperature to determine in regions at a distance as small as 50 nm from the surface. In a Ti 6Al4V alloy, vanadium is used as a βstabilizer and highest vanadium concentration is found in the βphase. By using the method they 35 proposed, in drilling maximum pseudo temperature occurs at the end of the drilled hole and is very close to the βtransus temperature of Ti 6Al4V alloy. Initially the temperature was about 490 K and increased linearly by 5 K/mm between the start of the drill hole and the end. They accounted for this increase to be due to insufficient coolant Fig. 2.8 Pseudotemperature variation with drill depth showing a linear increase [32] supply. The pseudotemperature measurements in shot peening showed that the maximum value was 1106 K in a surface region of about 50 nm thickness and the electro chemical drilling showed no significant increase in pseudo temperature. Lacalle et al. [33] conducted milling tests on Ti 6Al4V alloy to study the tool influence of the tool geometry and coating as well as the influence of cutting conditions on the productivity of the milling process. They used cutting speeds between 11 and 14 m/min, feed between 0.04 and 0.15 mm/tooth, helix angles of 30°, 45°, 60°, number of teeth of 3, 4, 6 and uncoated cemented carbide milling cutters as cutting tools. Based on their studies they reached the following conclusions: 1) Cutting speed has a crucial role in the tool roughening. Serrated chips are found for all the cutting speeds. Fig.2.9 shows serrated chips for 80 m/min and 140 m/min. 36 (a) (b) Fig. 2.9 Serrated chips obtained in machining a Ti 6Al4V alloy at cutting speeds (a) 80 m/min and (b) 140 m/min [33] 2) For high feed values, thick chips are generated and they result in increasing cutting loads, increasing chip deformation and separation. 3) As far as usage of coating materials for HSS steel tool in the machining of titanium alloys, it is found that flank wear can be delayed with TiCN coatings. 2.3 ShearLocalized Chip Formation in Ti 6Al4VFEM Simulations Xie et al. [13] presented a quasistatic finite element model of chip formation and shear banding in orthogonal metal cutting of Ti 6Al4V using a commercial FEA code (NIKE2D™). The updated Lagrangian formulation for plane strain conditions is used in this investigation. The tiebreak sideline was used to separate the newly formed chip from the workpiece surface. The effective plastic strain is used as the material failure criterion and a strainhardening thermalsoftening model for the flow stress is used for shearlocalized chip simulation. A series of finite element simulated machining tests with different tool rake angles ranging from –16° to 20° were carried out at a cutting speed of 37 5 m/s and a feed of 0.3 mm to study the effect of rake angle on shear band angle and cutting forces. Fig. 2.10 (a) and (b) shows machining process modeled with a negative rake angle 8°. The simulated results show that the effective plastic strain is within a narrow area along the shear zone angle, and the field of the maximum shear stress matches with the area of the primary and secondary shear zones. They reported that the finite element model predicts the detailed deformation in front of the tool tip and the initiation of the shear band. (a) (b) Fig. 2.10 Chip formation process showing shear localization in finite element simulation of a Ti 6Al4V alloy at cutting speed of 5m/s and feed 0.3mm using 8° rake angle tool [13] However, from their investigation, it can be observed that they have not used a reliable material model to represent strainhardening and thermalsoftening behavior of Ti 6Al4V during metal cutting process as they mentioned in their study that they arbitrarily composed the thermalsoftening part of the material behavior by making the stress from strainhardening part to thermalsoftening part to decrease by 50%. Xie et al. [13] also mentioned that, they used a debond (node separation) criterion based on critical 38 effective plastic strain for chip formation. It can be observed from their investigation that they have used an arbitrary value of 0.5 for critical effective plastic strain. Also from the Fig. 2.10 it can be noticed that, they have just shown the onset of chip segmentation and did not show complete chip formation process. Obikawa and Usui [18] developed a finite element model for the computational machining of titanium alloy Ti 6Al4V. Fig.2.11 shows the cutting model they used in the finite element analysis. A cemented carbide tool (FGHI) and a titanium alloy (Ti 6Al4V) (ABCDEF) were modeled with four node (linear) quadratic isoparametric elements. The tool was assumed to be rigid and the finite elements in it were used only for temperature calculations. The cutting speed used was 30 m/min, the undeformed chip thickness was 0.25 mm and the tool rake angle was 20°. The parallelogram ABCF was part of the workpiece to be removed as chip. Twodimensional elastic plastic analysis was formulated in updated Lagrangian form and procedures needed for metal cutting were developed for unsteady state heat conduction and material nonlinearities. The friction on the rake face and the complicated flow stress characteristic of the titanium alloy at high strain rates and high temperatures were also considered. Fig.2.12 shows the serrated chip formation process in case of the titanium alloy that was obtained by the FEM simulation. In this investigation, to simulate serrated chip formation they used a geometrical criterion based on fracture strain and defined boundary conditions in such a way that crack propagation occurs in the predetermined path. Hence these simulations lack actual physical phenomenon and mechanism of chip segmentation. 39 Fig. 2.11 Toolworkpiece mesh system used in finite element simulation of orthogonal metal cutting a Ti 6Al4V alloy [18] Fig. 2.12 Serrated chip formation with cutting length in finite element simulation of machining a Ti 6Al4V alloy at cutting speed 30 m/min, feed 0.25 mm and 20° rake angle [18] Maekawa et al. [34] used an iterative convergence method (ICM) to simulate metal cutting process. The ICM uses flow lines which consist of trajectories of particles or a series of finite elements. The chip is supposed to be preformed on the surface of the work material and to be stress free. Calculation proceeds by incrementally displacing the workpiece towards the tool so that a load develops between the chip and the tool. A plastic state develops in the chip deformation zone and it is checked with the assumed 40 chip shape and automatically altered and the calculation is repeated. However, this method simulated only continuous chip. Later, to simulate serrated and discontinuous chips, a sophisticated methodology was developed for approximation of such discontinuity. A failure strain criterion was introduced into the ICM methodology, so that crack initiates at the tool side within the highly deformed workpiece and propagates towards the free surface side, resulting in the periodic segmentation of the chip. Applying this methodology, Ti 6Al4V alloy machining simulations were carried out at 30 m/min cutting speed and 0.25 mm/rev feed rate using 20° rake angle tool. Fig. 2.13 shows the predicted serrated chip shape in titanium alloy machining simulation. They concluded that serration arises in the chip from a small fracture strain of the alloy and not due to adiabatic shear instability. But then, as stated above, they introduced a failure strain criterion so that crack initiates at the tool side within the highly deformed workpiece and propagates towards the free surface side. And as their simulation approach is based on geometric criterion with preformed chip shape, this work does not represent actual mechanism of chip segmentation observed in machining. Fig. 2.13 Serrated chip formed in finite element simulation of orthogonal metal cutting of a Ti 6Al4V alloy at cutting speed 30 m/min, feed 0.25 mm and 20° rake angle [34] Owen and Vaz Jr. [23], simulated machining of Ti 6Al4V alloy in which, they addressed such issues as evaluation of the mesh refinement procedure, strain localization 41 process, and material failure process. The geometry and the initial mesh they used for a rake angle of 3° are depicted in Fig.2.14. An enhanced oneGauss point element was used in the simulation. The simulations employed an error estimator based on uncoupled integration of Lemaitre’s damage model. Tests were undertaken to evaluate the effect of Fig. 2.14 Initial geometry and workpiecetool mesh used in finite element simulation of orthogonal machining of Ti 6Al4V alloy [23] the cutting speed (from 5 to 20 m/s) and rake angle (from 9° to 9°) and to assess the capacity of the remeshing procedure to describe the process evolution. They assumed fracture strain as the governing parameter of material failure in highspeed machining and assessed two indicators of failure, namely, equivalent plastic strain and a fracture strain based on Lemaitre’s damage model. The chip breakage process for a fracture strain based on the equivalent plastic strain was illustrated in Fig 2.15 which shows the elements undergoing a failure softening and fracture propagation. This criterion assumes that fracture initiates when effective plastic strain is equal to failure strain which is greater than or equal to 1.0. 42 However, from the Fig.2.15, it can be observed that only onset of chip segmentation process is shown in this investigation and it does not represent the exact physical phenomenon and mechanism of chip segmentation. (a) (b) (c) Fig. 2.15 Segmented chip formation process in machining simulation of a Ti 6Al4V alloy at 10 m/s cutting speed and 0.5 mm feed using a 3° rake angle tool [23] Baker et al. [35] developed a twodimensional finite element model to simulate highspeed machining of Ti 6Al4V using the commercial software, ABAQUS, together with a special mesh generator programmed in C++. In ABAQUS, they used standard program system, which allows the definition of complex contact conditions, leaves many possibilities to define material behavior, and can be customized in many regards by including userdefined subroutines. Preprocessor they used for automatic remeshing was written in C++ using standard class libraries to ensure that elements never become too distorted and refined mesh is created in the shear zone. They assumed that chip segmentation is caused solely by adiabatic shear band formation and that no material failure or cracking occurs in the shear zone. They used fournoded quadrilateral elements which converge better than triangular elements. The number of elements they used in the simulations varied with the number of segments. About 5000 elements and 7000 nodes were used at the beginning of the simulation and 10000 elements and 12000 nodes near 43 the end. The element edge was about 7 μm in the shear zone and Fig. 2.16 shows the finite element meshes at different stages of the cutting process with segmented chip formation. Fig. 2.16 Segmented chip formation process in finite element simulation of orthogonal metal cutting of a Ti 6Al4V alloy at 50 m/s cutting speed and 0.04 mm cutting depth using a 10° rake angle tool [35] Baker et al. [35] also investigated the influence of the elastic modulus and cutting speed on chip segmentation. They reported that elastic modulus affects the degree of segmentation. They also studied the influence of thermal conductivity on chip segmentation and showed that the degree of segmentation decreases with increasing thermal conductivity. In this investigation, Baker et al. [35] used a pure deformation process to simulate metal cutting process without node separation. The material that overlaps with the tool as 44 the tool advances is removed using remeshing technique. This phenomenon however does not represent actual physical process of chip formation in machining. They also mentioned that they have found the strainhardening part of plastic flow curves experimentally and determined the thermal softening part arbitrarily to facilitate the adiabatic shear bands formation which is unrealistic. In their simulations, it can be observed that high speeds are used, no friction is assumed between tool and the workpiece and heat flow into the tool is neglected. Also, Fig.2.16 does not represent actual mechanism of chip segmentation. Sandstrom and Hodowany [36] modeled the highspeed orthogonal machining of Ti 6Al4V using the commercial FEM code, Mach2D™ at a cutting speed of 10.16 m/s. Simulation results included chip segmentation, dynamic cutting forces, unconstrained plastic flow of material during chip formation, and thermomechanical environments of the workpiece and the cutting tool. They reported good agreement of the simulated cutting force to the experimental data. However, it can be observed that their investigation lack low speed simulations. Shivpuri et al. [21] used a commercial finite element software (DEFORM 2D™) which is a lagrangian implicit code designed for metal forming processes, to simulate the orthogonal machining of Ti 6Al4V. They modeled workpiece as a rigidviscoplastic material owing to large plastic deformations taking place in the primary and the secondary deformation zones during the machining process. Furthermore, high mesh density was defined around the tool tip and excessively deformed workpiece mesh was automatically remeshed as needed during simulation. They modeled the tool as rigid (or elastic) material so that stresses in the tool body can be predicted. Dynamic flow stress 45 model was used to represent material behavior dependence on strain, strain rate, and temperature. Ductile fracture criterion was used to simulate crack initiation and propagation for chip segmentation. Simulations were conducted at cutting speeds of 1.2, 120 and 600 m/min and a feed rate of 0.127 mm/rev. Fig.2.17 shows the plastic strain distribution that presents the initiation of shear plane and the formation of shear zone. Based on simulations, they proposed that chip segmentation during cutting Ti 6Al4V alloy is caused by flow localization within the primary deformation zone and upsetting Fig. 2.17 Equivalent plastic strain distribution in serrated chip simulated at cutting speeds of 1.2, 120 and 600 m/min in a Ti 6Al4V alloy [21] deformation zone by moving tool rake face on the segment to be formed ahead of it. The flow localization induces fracture that separates segment from the workpiece matrix. Flow localization causes crack in the primary deformation zone while the secondary deformation zone controls the chip morphology (segmented or discontinuous). Discontinuous chip was formed at low speed (1.2 m/min) and segmented chips at higher cutting speeds (120 and 600m/min). At low cutting speeds, crack was observed to initiate 46 at the tool tip and propagates to the free surface of the chip while at higher cutting speeds, crack propagates from free surface towards the tool tip. In their investigation, Shivpuri et al. [21] used maximum tensile stress criterion for ductile fracture to simulate serrated chip formation while maximum shear stress criterion accurately models serrated chip formation. Also, it can be observed from Fig.2.17, that chip morphology does not look like a serrated chip. Although most of the researchers reported that their simulation results were in good agreement with the corresponding experimental data and their material model accurately predicts material deformation behavior in simulations, there is no consensus on which criterion represents the best for shearlocalized chip formation in orthogonal metal cutting simulation of Ti 6Al4V alloy. Some of the major drawbacks of the finite element simulations of shearlocalized chip formation in Ti 6Al4V alloy found in the literature include: 1) The methodology proposed by many researchers using noncommercial FEM codes makes it difficult for end users. 2) Few researchers used extensive computer time and engineering effort, which makes their technique not economical to use. 3) The lack of reliable material data under specific process conditions such as, strain, strain rate and temperature that must be used as inputs to any material flow simulation program results in difficulty in applying to practical processes. 4) Most of the finite element simulations of Ti 6Al4V alloy, lack proper experimental validations (especially chip morphology). 5) Lack of low speed machining simulations (< 10 m/min). 47 CHAPTER 3 FINITE ELEMENT FORMULATION OF METAL CUTTING 3.1 Introduction Improvements in manufacturing processes require better modeling and simulation techniques of metal cutting. The process involves very complicated mechanisms such as interfacial frictional behavior, extremely high temperatures, complex stress state between tool chip interface, high strain rates, different types of chip formation, work hardening and thermal softening. Unfortunately, these complicated mechanisms associated have limited the performance of cutting process modeling and in recent years researchers in the metal cutting field are paying more and more attention to the finite element method due to its capability of numerically modeling different metal cutting problems. The advantages of using finite element method to study machining can be summarized as follows: 1) Material properties can be handled as a function of strain, strain rate, and temperature. 2) The toolchip interaction can be modeled as sticking and sliding. 3) Nonlinear geometric boundaries can be represented and used, such as free surface of chip. 48 Although the ideas of finite element analysis may date back much further, it was after further treatment of plane elasticity problem by Clough in 1960, that researches began to recognize the efficacy of finite element method in the engineering field. The advent and continuous improvements of digital computers have made finite element analysis a useful analytical tool which has been applied very efficiently in almost every area of engineering field. One of the most important reasons that finite element analysis is so widely used is that it can be routinely used. There are a definite set of several basic and distinct steps used in the FEM simulations: 1) Discretization of the continuum. 2) Selection of the interpolation function. 3) Determination of the element properties. 4) Assembly of the element properties in order to obtain the system equations. 5) Determination of the constraints and other boundary conditions. 6) Solution of the system equations. 7) Computation of the derived variables. Inspite of the success of the finite element method in solving a very large number of complex problems, there are still many areas where more work needs to be done. Some examples are the handling of problems involving material failures and the modeling of nonlinear material behavior. 3.2 Finite Element Formulation The basic idea of using the finite element method is to seek a solution to the momentum equation [37]: 49 . ij , j i i σ + b = ρυ (i, j = 1,2,3), (3.1) where σij,j is the Cauchy stress tensor, ρ is the current mass density, bi is the body force, vi is the particle velocity in Cartesian coordinates, . i υ is the acceleration, and j indicates partial differentiation with respect to xj. Equation 3.1 satisfies the trajectory boundary condition: σij nj =Ti (3.2) where nj is the unit normal to the boundary and Ti is the surface traction on the plane with a unit normal nj. In this study, displacement based finite element method, which is the most widely used formulation in engineering among many finite element analysis techniques available, is applied to model the machining process. The displacement boundary condition that Equation 3.1 should satisfy at time t is given by: ( ) i i u t = u (3.3) For elasticplastic deformations, the strain rate tensor is usually decomposed into elastic and plastic parts such that ( ), 2 1 , , . ε ij = vi j + v j i . . . i p ij e ε ij =ε +ε (3.4) From Hook’s law, we have kl o ijkl e ij ε e C 1 σ . = − ; ijkl ik jl ij kl e v Gv C Gδ δ δ δ 1 2 2 2 − = + (3.5) where, Ce ijkl is the elasticity tensor, G and v are the shear modulus and poisons ratio, respectively, and ij oσ is the corotational strain given by 50 ik ij ik jk ij o ij σ =σ −σ ω +σ ω . (3.6) with ik ω being the material spin. Using the von Mises flow rule, ij p .ε is obtained as ij ij p s τ ε γ 2 . = (3.7) where sij is the deviatoric stress, 2 1 . . . 2 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = p ij p ij γ ε ε is the effective strain rate and ( )2 1 2 1 ij ij τ = s s is the effective shear stress. The effective strain rate in Equation 3.7 is determined from the yield criterion of the rateindependent materials. For von Mises materials, yield criterion is given by ( ) 0 2 f = J −τ γ = y , (3.8) where J2 is the second invariant of sij, from the consistency condition one can obtain the relation J h s sij ij o 2 . 2 γ = , γ τ ∂ = ∂ h , (3.9) where h is the hardening/softening modulus. Combining Equations 3.2 to 3.5, 3.7 and 3.9 we obtain, ij ijkl ij o C .ε σ = , (3.10) where Cijkl is the elasticplastic incremental tensor and is given by ijkl p ijkl e ijkl C = C − C , ( ) ijkl ij kl p s s G h G C + = 2 2 τ (3.11) For ratedependent materials, Equations 3.4 and 3.5 yield 51 ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = − ijkl kl kl e ij o C s τ σ ε γ 2 . . (3.12) These equations are implemented in all finite element programs. The solutions of engineering problems require the application of three conditions: 1) Equilibrium of forces. 2) Compatibility of deformations. 3) Constitutive relationship (material behavior). These three conditions are used to generate the system of equations with stresses or displacements as unknowns. The former approach is called the force method and latter approach is called the displacement method. In general among many finite element techniques available displacementbased finite element method is most widely used formulation in engineering. The displacement method generates finite element equations of the form [38]: [K]{u}( ) {F} e = (3.13) where [K] is the global stiffness matrix, {F} is the vector of all applied loads (known variables), and {u}(e) is the nodal displacement vector (unknown variables). It should be noted that applied load is not necessarily the force; it may be stress, displacement rate, etc. Solving the above equation yields the nodal displacement vector {u}(e). Then, the element strains {ε} can be determined by the straindisplacement relationship and stress {σ} can be calculated from the constitutive relationship [38] as: { } [ ]{ }( ) { } [ ][ ]{ }(e) [ ]{ }(e) e M B u S u B u = = = σ ε (3.14) 52 where [B] is the straindisplacement matrix which usually has all constants, [M] is the material property matrix, and [S] = [M][B] is usually called the element stress matrix. The following coupled system of ordinary differential equations are used: [M] u F u,u,T {P(u,b,t,T )} . int .. = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟⎠ ⎞ ⎜⎝ + ⎛ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ (3.15) where {Fint} is the internal nodal force vector and {P} is the external nodal force vector which can be a function of nodal displacement u, body force per unit volume b, time t and nodal temperature T. The element stiffness [K](e) can be calculated as: [ ]( ) ([ ]( ) ) [ ](e)[ ](e) e T D K e B e M B dD e = ∫ (3.16) where [B](e) is the straindisplacement matrix, [M](e) is the material matrix and De is the element volume. The element contribution to the internal nodal force vector {Fint}(e) can be obtained as: {F }( ) ([B]( ) ) { }dD i T D e e e = ∫ τ int (3.17) Then the global stiffness matrix [K] and internal load vector {Fint} can be constructed as: [ ] [ ]( ) { } { }(e) e e e F F K K Σ Σ = = int int (3.18) It is necessary to point out that for metal cutting simulations, the matrices [B](e) and [M](e) are no longer constant as metal cutting analysis is a nonlinear, large deformation analysis. Therefore [B](e) and [M](e) need to be evaluated at each step of the finite element calculation. For quasistatic analysis of metal cutting process, the finite element equations (Equation 3.13) are simplified by eliminating the inertial effects, and thus have the form: 53 ( ) F u,u,T {P u,b,t,T } . int = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟⎠ ⎞ ⎜⎝ ⎛ (3.19) To obtain the solution at time step tn+1, the finite element equations are first linearized about the configuration at tn as: [ ] { } { } { }n n n K u P F1 int 0 Δ = − + (3.20) where [K]n is the stiffness matrix based on the configuration at tn and {Δu}° is the increment in displacement where the superscript denotes iteration number. {Δu}° can be solved from Equation 3.20 and displacement is updated by: { } { } 0{ }0 1 u u s u n n = + Δ + (3.21) where s° is a parameter between 0 and 1 which is determined by line search scheme. Finally, the equilibrium iterations can be performed by solving the following equations: [ ] { } { } +1 { int } +1 { } +1 Δ = − = n i n i n i i K P F Q (3.22) where {Q}i n+1 is the residual. And the convergence is determined by examining both the displacement norm { } d i u u ≤ε Δ max , (3.23) and energy norm: ({ } ) { } ({ } ) { } e n T n i T i u Q u Q ≤ε Δ Δ + + 1 0 0 1 , (3.24) In the Equations 3.23 and 3.24, umax is the maximum displacement norm above all of the n steps including the current iteration, εd and εe are the tolerances that are typically 102 to 54 103 or smaller and are usually adjustable for different problems. If the convergence is not attained and solution is divergent, the solution is obtained by: { } { } i { }i n i n i u +1 = u +1 + s Δu (3.25) where si is the parameter same as s° at the ith iteration. Iterations will then continue and if the solutions are determined to be divergent, or convergence fails to occur within an assigned number of iterations the stiffness matrix is [K] is reformed using the current estimation of the geometry before continuing the equilibrium iteration. The commercial software AdvantEdge™ is used in this study to simulate the metal cutting process. The finite element formulation used in this software is a Lagrangian formulation. The central difference rule is used to integrate the equations of motion. AdvantEdge™ uses a lumped mass formulation for efficiency, which produces a diagonal mass matrix that renders the solution of the conservation of the momentum equation given by [37]: ij j i i b u .. , σ + ρ = ρ (3.26) This equation shows the inertial forces and internal forces, due to the current state of stress, are in balance with the applied boundary loads. The weak form of the above equation based on the principle of virtual work becomes: v v b dV v ui dV B i i i B i ij j .. ∫ σ , + ρ = ∫ρ (3.27) solving and rearranging the above equation gives: ∫ + ∫ = ∫ Ω+ ∫ ∂ B j i i B ij i ij B i i j B i ρv u dV v σ dV vσ n d v ρb dV , .. (3.28) Finite element discretization of the above equation provides: 55 N N u dV N dV N d N b dV B a i B ij a i B b ib a j B ∫ a + ∫ = ∫ Ω + ∫ ∂ ρ σ τ ρ , .. (3.29) It can be represented in matrix form as: ext n n n Ma R R 1 int +1 +1 + + = (3.30) where 0 0 0 M N N dV a b B ab = ∫ρ (3.31) is the mass matrix and 0 0 0 0 = ∫ + ∫ Ω ∂ R b N dV N d a B a i B i ext ia τ (3.32) is external force array and , 0 int 0 R P N dV a j B ia = ∫ ij (3.33) is the internal force array. In the above expressions, Na, a=1,….,numnp are the shape functions, repeated indices imply summation and a (,) represents partial differentiation with respect to the corresponding spatial coordinate, and Pij is the first PiolaKirchoff stress tensor analogous to nominal or engineering stress. Using the Newmark family of algorithms for temporal integration of the discretized weak form, second order accurate explicit analysis can be achieved through the central difference scheme. Since plasticity results in softening of stiffness matrix, it suffices to look to the bounding case of linear elasticity, for which the generalized eigen problem is given by: ( − ) = 0 l ij j l ij K λ M d (3.34) Accurate computation of the system eigen values is essential due to changes in mesh geometry (eigen values) arising from deformation. Since the largest system eigen value is 56 bounded by the largest element eigen value, it suffices to compute the largest eigen value of each element to determine the critical time step for the mesh. Heat generation and transfer are handled via the Second Law of Thermodynamics. A discretized weak form of the law is given by: . n 1 n n T = T + ΔtT + (3.35) 1 . +1 +1 + + = n n n CT KT Q (3.36) A lumped capacitance matrix is used to eliminate the need for solving any equations. CT + KT = Q . (3.37) where T is the array of nodal temperatures. 0 C c N N dV a b B ab t = ∫ ρ (3.38) is the heat capacity matrix and K D N N dV a i b j B ab ij , , 0 = ∫ (3.39) is the conductivity matrix and = ∫ + ∫ B q a B a a Q sN dV hN dS t τ (3.40) is the heat source array with h, having an appropriate value for the chip or tool. In machining applications, the main sources of heat are plastic deformation in the shear zone and frictional sliding in the toolworkpiece interface. The rate of heat supply due to the first is estimated as: P s W . = β (3.41) 57 where P W . is the plastic power per unit deformed volume and TaylorQuinney coefficient β is of the order of 0.9. The rate at which the heat generated at the frictional contact, on the other hand is given by: h = −t ⋅ v (3.42) where t is the contact traction and v is the jump in velocity across the contact. A staggered procedure is adopted to couple mechanical and thermal equations. Geometrically identical meshes for the mechanical and thermal models are used. Mechanical and thermal computations are staggered assuming constant temperature during the mechanical step and constant heat generation during the thermal step. A mechanical step is taken first based on current distribution of temperatures, and the heat generated is computed from plastic working and frictional heat generation. The heat thus computed is transferred to the thermal mesh and the temperatures are recomputed by recourse to the forwardEuler algorithm. The resulting temperatures are transferred to the mechanical mesh and are incorporated into the thermalsoftening model which completes one time step cycle. 3.3 Friction formulation along toolchip interface Friction along the toolchip contact during the cutting process is a very complex phenomenon. Friction influences chip formation, builtup edge formation, cutting temperature and tool wear. Therefore it is necessary to understand the friction mechanism across the tool face and around the edge of the tool, in order to develop accurate models 58 for cutting forces and temperature. In the AdvantEdge™ software, the friction model incorporated is coulomb friction model and it is represented by: τ = μσ (3.43) where τ is the frictional shear stress and σ is the normal stress to the surface. Usually the friction coefficient μ is assumed to be constant for a given interface. In the metal cutting, the cutting pressure at the toolchip interface will become several times the yield stress of the workpiece material. In this extreme case, the real contact between the tool and workpiece is so nearly complete in the sticking region that sliding occurs only beyond this region. Therefore the frictional force becomes that required to shear the weaker of the two materials across the whole interface. This force is almost independent of the normal force, but is directly proportional to the apparent area of contact. 3.4 Formulation of Contact Conditions In metal cutting simulations, meshonmesh contact occurs between the workpiece and the tool and this contact is formulated in AdvantEdge™ software using predictorcorrector method of PRONTO2D™ explicit dynamics code. The two contacting surfaces are designated as master and slave surfaces (Fig. 3.1). Assuming that no contact has occurred, nodal accelerations from the outofbalance forces are calculated and nodal positions, velocities, and accelerations are predicted by predictor algorithm. A resulting predictor configuration shows penetration of the master surface into the slave surface. The contact conditions are designated by an auxiliary consecutive numbering of the nodes on the contacting surfaces. The penetration distances for all nodes on the slave surface are then calculated. The contact force required to prevent penetration is equal to 59 the force required to keep the master surface remain stationary on predictor configuration. Tangential force exerted by the master surface on the slave node cannot exceed the maximum frictional resistance. This condition should be satisfied to prevent unwanted penetration. A balanced masterslave approach in which surfaces alternately act as master and slave is also employed; however, rigid surfaces are always treated as master surfaces. Fig. 3.1 Contacting surfaces in a mesh showing (a) predictor configuration and (b) kinematically compatible configuration [14] 3.5 Adaptive remeshing In AdvantEdge™ finite element formulation incorporates a sixnoded quadratic triangular element with three corner and three midside nodes providing quadratic interpolation of displacements within the element as shown in Fig. 3.2. The element is integrated with threepoint quadrature interior to the element. At the integration points, the constitutive response of the material is computed and consequently linear pressure distribution is provided within the element. 60 Fig. 3.2 Sixnodes and three quadrature points shown in a typical sixnoded triangular element used in the finite element mesh of the workpiece and tool [14] During metal cutting, workpiece material flows around the cutting tool edge. In this process, at the tool vicinity elements get distorted and the accuracy is lost. To alleviate element distortion, finite element mesh is updated periodically, refining large elements, remeshing distorted elements and coarsening smaller elements. For instance, if an element needs refinement, the diagonal of the element is split, a midside node becomes new corner node and new midside nodes are added to both elements as shown in Fig.3.3. (a) (b) Fig. 3.3 Element shape in a finite element mesh (a) before and (b) after adaptive remeshing [14] For adaptive remeshing, an adaptation criterion based on the equidistribution of plastic power is used. In this approach, elements with plastic power content exceeding a 61 prescribed tolerance TOL, are targeted for refinement. The criterion can be represented as [14]: ∫ Ω Ω W d e h p . >TOL (3.44) here Ωh e denotes the domain of the element e and the plastic power for an element is given by . . W p =σ ε p (3.45) if, despite this continuous remeshing, elements arise with unacceptable aspect ratios, the mesh is subjected to Laplacian smoothing. Besides sidestepping the problem of element distortion, adaptive remeshing provides a means of simultaneously resolving multiple scales in the solution. Transport of data, such as displacement and temperature from old mesh to new mesh after remeshing is done by interpolation technique. 3.6 Chip formation Different numerical techniques for modeling chip separation exist and they can be divided into two categories geometrical and physical. The geometrical model is usually based on the tied slideline interface which debonds when certain criterion is fulfilled. This criterion may be a certain level of stress, strain or simply when the cutting edge is close enough to the front nodes. On the other hand, physical model is based on the physical behavior of the material, such as plastic deformation, crack initiation and crack propagation without predetermining its path. AdvantEdge™ simulation software uses critical stress intensity factor, KIC, as a fracture criterion for brittle materials and crack tip opening displacement (CTOD) for ductile fracture. Brittle fracture, such as the one that occurs below the transition temperature, proceeds by cleavage. In particular, conditions for brittle fracture are found to be consistent with the attainment of a critical opening stress σf at a critical distance l. 62 The critical stress σf is found to be relatively independent of temperature and strain rate and can be inferred from the toughness KIC through the small scale yielding relation [14] l KIC f π σ 2 = (3.46) The crack trajectory under conditions of brittle fracture is predicted using maximum hoop stress criterion according to which crack propagates along the angle θ from the crack face at which the hoop stress σ θθ attains a relative maximum. Void growth and coalescence are known to be principal mechanisms of ductile fracture. The rate of growth of voids is accelerated by the blunting of the crack tip, which has the effect of raising the hydrostatic stress at the location of the void. The crack tip opening displacement (CTOD) criterion for ductile fracture can be recast as the attainment of critical effective plastic strain εp f of the effective plastic strain at a distance l ahead of the crack tip [14]. The criterion can be represented as: ( ) ε θ = θ max p l, εp f (3.47) where θ is the angle at which the crack propagates when the criterion is met. The critical effective plastic strain (εp f) can be estimated as εp f σ p e 1.5 2.48 − ≈ (3.48) where p= σ kk/3 is the hydrostatic pressure. 63 CHAPTER 4 PROBLEM STATEMENT As mentioned in the literature review, many researchers in the past have developed analytical models to explain the theory of chip segmentation. Many theories based on adiabatic shear failure, damage model and crack initiation and propagation have been proposed for segmented chip formation in Ti 6Al4V. But most of the models suffer from a lack of adequate acceptable methodologies for application under a wide range of cutting conditions. In addition, very little work has been done on finite element simulations of machining Ti6Al4V alloy (especially at very low speeds). Most of these simulations have limited experimental validation under few selected cutting conditions and tool geometries. This is an area deserving more study because a reliable finite element model of chip segmentation needs a realistic material model to describe material behavior accurately under high temperatures, high strains, and high strain rates and a proper failure criterion to represent the mechanism of chip segmentation. Thus an attempt has been made in this study to deal with the above mentioned aspects. The specific objectives of this study are: 1) To study the mechanism of chip formation in machining Ti 6Al4V using finite element simulations. 64 2) To formulate a reliable material model, such as the JohnsonCook model into the user defined material code (UMAT) so that it can represent material flow stress with respect to strain, strain rate, and temperature at different machining conditions. 3) To derive equations for Recht’s catastrophic shear failure criterion and incorporate it into UMAT code as a failure criterion for shearlocalized chip formation in Ti 6Al4V. 4) To validate the finite element simulations with experimental results obtained from the literature by comparing the cutting forces, rake face temperature and chip morphology. 5) To predict the critical cutting speed for the onset of chip segmentation. 6) To conduct finite element simulations using commercial finite element software, AdvantEdge™. 7) To study the effect of different machining conditions, such as cutting speed, feed, rake angle and coefficient of friction on cutting forces, rake face temperature, shear zone temperature, equivalent plastic strain, frequency of chip segmentation, shear band width, and power consumption. 65 CHAPTER 5 SHEAR LOCALIZATION IN Ti 6Al4V MACHINING 5.1 ShearLocalized Chip in Ti 6Al4V Titanium alloys are one of the most attractive materials because of their high specific strength maintained at high temperatures, excellent fracture resistance and corrosion resistance. They are much sought after materials for aerospace applications. Although Ti 6Al4V comprises about 45% to 60% of titanium products in practical use, it is considered as difficulttomachine material because of its unfavorable thermal properties. Generally, two types of chip formation are observed in machining Ti 6Al4V based on the cutting speed. They are the continuous chip and the segmented chip. Segmented chip is also called shearlocalized chip because of intense plastic deformation in the narrow band between the chip segments and negligible deformation within the segment. According to Komanduri and Hou [25], for titanium alloys shear localization occurs even at very low cutting speeds (<0.5 m/min) and continues over the entire conventional cuttingspeed range. Consequently, continuous chip can be observed only at very low speeds (below 0.5 m/min). The chips formed in Ti 6Al4V machining looks like saw tooth with chip being inhomogeneous and shows two regions, shear band with very large shear strains between the chip segments and a trapezoidal shaped segments body 66 with relatively small deformation. Once the shear localization is initiated, the chip topside looks like saw tooth with each tooth corresponding to a segment. Many researchers have done extensive studies on the mechanism of chip formation when machining titanium alloys since the early 1950s using several techniques, such as highspeed photography, insitu machining inside a scanning electron microscope and metallurgical analysis of chips generated in machining. Based on these studies many models were formulated to describe the mechanism of segmented chip formation in titanium alloys. In this study, Recht’s thermoplastic shear instability criterion is used to simulate shearlocalized chip formation. 5.2 Mechanism of ShearLocalized Chip formation In machining Titanium alloys, the chips formed are segmented and strain is not uniformly distributed but is confined to narrow bands between the segments. Whereas, in the continuous chip formation, the deformation is largely uniform. The sequence of events leading to shearlocalized chip formation in Ti 6Al4V was described by Komanduri and Turkovich [11] based on the detailed chip formation studies of video tapes of low speeds experiments inside the scanning electron microscope, highspeed movie films at higher cutting speeds and the micrographs of midsection of chips. They observed the mechanism of chip formation to be invariant with respect to cutting speed. The mechanism of shearlocalized chip formation can be explained based on the Fig. 5.1. The process can be divided into two basic stages. The first stage involves shear instability and strain localization in a narrow band in the primary shear zone ahead of the tool. This narrow band originates from the tool tip almost parallel to the cutting velocity vector and 67 gradually curves with the concave surface upwards until it meets the free surface. The shear failure of the chip appears as a crack on the outside while it is a heavily deformed band inside. Fig. 5.1 Description of shearlocalized chip formed due to adiabatic shear localization [26] The second stage involves upsetting of an inclined wedge of work material by the advancing tool, with negligible deformation, forming a chip segment. During upsetting of the chip segment in the primary shear zone ahead of the tool tip, intense shear takes place at approximately 45° to the cutting direction. This occurs not between the tool face and chip but between the last segment formed and the one just forming. The initial contact of the tool face with the segment being formed is very less and it gradually increases as the upsetting progresses. There is almost no relative motion between the bottom surface of the chip segment being formed and the tool face until the end of the upsetting stage of the 68 segmentation process. The gradual upsetting process slowly pushes the segment formed previously upwards. The velocity of the chip along the rake face is same as the upsetting chip until the shear is initiated and progresses rapidly. Once this occurs, it pushes the chip segment being formed faster parallel to the shear surface. This will then push the previous segment formed rather rapidly. Thus the chip velocity along the rake face fluctuates cyclically. This mechanism of segmented chip formation occurs almost at all speeds. However, as the cutting speed increases, intense shear in the narrow shear band occurs so rapidly that contact area between any two segments decreases to a stage that individual segments of the chip get separated. 5.3 Criterion for Thermo Plastic Shear Instability Recht in 1964 [12] developed a criterion for the prediction of catastrophic shear instability in metals under dynamic plastic conditions. Accordingly, shearlocalized chip formation can be attributed to dynamic plastic behavior of the material which is influenced by internally generated temperature gradients. These gradients are function of thermophysical properties of the workmaterial as well as strain rate and shear strength. Catastrophic shear will occur at a plastically deformed region within a material when the slope of the true stresstrue strain curve becomes zero, i.e. when local rate of change of temperature has a negative effect on the strength which is equal to or greater than the positive effect of strain hardening. The criterion for catastrophic shear failure can be represented as: 69 0 ≤ 1.0 ∂ − ∂ ∂ ∂ ≤ ε θ θ τ ε τ d d (5.1) where τ, ε, θ refer to shear stress, shear strain, and temperature, respectively. Accordingly, material will shear catastrophically when this ratio lies between zero and one; catastrophic shear will be imminent when this ratio equals one. No catastrophic shear will occur when this ratio is greater than one. High positive values above one indicate that strain hardening is predominant and shear deformation will distribute throughout the material, in which case material will strain harden more than it will thermal soften. Negative values indicate that material will actually become stronger with an increase in temperature and that shear deformation will distribute. Thus thermoplastic shear instability (frequently referred to as adiabatic shear) is a major contributor to chip segmentation. Just so long as material can withstand shear stress by virtue of its shear strength, it will remain thermoplastically stable. As stress increases, material strains and if rate of change of strength matches the rate of change of stress, material will undergo stable deformation. When instability occurs, the applied stress will be borne by the remaining (diminishing) strength of the material. And inertial reactions associated with, accelerate instability, leading to catastrophic shear failure. Recht thus provided the first explanation for segmented chip formation in machining. Since the fundamental contributions of Recht, much work has been done on the adiabatic shear band formation, often in connection with the applications, such as armor penetration and explosive fragmentation. 70 5.4 Metallurgical Aspects of ShearLocalized Chip Formation In metal cutting, the temperatures generated in the primary and secondary deformation zones can be high enough to cause several changes in the workpiece material, such as thermal softening, phase transformation, and even grain shape and size changes. These changes to a certain extent can effect the prediction of optimal cutting conditions for a given machining operation. Adiabatic shear banding is a phenomenon observed in machining titanium alloys. The formation of shearlocalized chip in titanium alloys is also accompanied by certain metallurgical changes. Komanduri [26] noted that during segmented chip formation in titanium alloys, there may be a transition from the lowtemperature hexagonal close packed (HCP) structure to the body centered cubic (BCC) structure with a corresponding increase in the number of available slip systems. Accordingly, this phenomenon further localizes the shear strain. This transition in the crystal structure results primarily from the increase in temperature and this increase can cause further increase in plastic deformation. Ti 6Al4V is an alloy with α+β structure which consists of lamellar α structure and intergranular β structure. Bayoumi and Xie [39] conducted metallurgical studies using a scanning electron microscope (SEM) and an Xray diffraction studies of both uncut and machined Ti6Al4V chips. Fig.5.2 shows Xray diffraction spectra for Ti 6Al 4V alloy before and after cutting. They compared the spectrum of the chip with uncut material and found that the peaks corresponding to βphase structure disappearing after cutting, indicating that a nondiffusional phase transformation took place in the process of shear band formation. When the temperature in the chip reaches the βtransus 71 temperature of Ti 6Al4V (which is 980°C) during the cutting process and cools back from this high temperature, the temperature change along with tremendous cutting pressure would produce the lamellar α structure from βphase by nucleation and growth. Accordingly, they concluded that there is only grain change during the phase transformation and no chemical change. Fig. 5.2 Xray diffraction spectra for chip and uncut material in a Ti 6Al4V alloy [39] From the microstructural examination of the chip and uncut Ti 6Al4V, they concluded that as both are similar in appearance, the shearlocalized chips may be caused by material flow in shear bands due to adiabatic shear instability. 72 CHAPTER 6 FEM SIMULATION OF CHIP SEGMENTATION IN Ti 6Al4V 6 .1 Introduction One of the stateofart efforts in manufacturing engineering is the finite element simulation of the metal cutting process. These simulations would have great value in increasing our understanding of the metal cutting process and reducing the number of trial and error experiments, which traditionally was the approach used for tool design, process selection, machinability evaluations, chip formation and chip breakage investigations. Compared to empirical and analytical methods, finite element methods used in the analysis of chip formation has advantages in several respects, such as developing material models that can handle material properties as a function of strain, strain rate and temperature. The toolchip interaction can be modeled as sticking and sliding; and nonlinear geometric boundaries such as the free surface of the chip can be represented and used. In addition to the global variables, such as cutting force, thrust force and chip geometry; the local variables such as stress, temperature distributions, can also be obtained. As mentioned in the literature review, very limited work has been done on FEM simulation of titanium alloys. Qualitative analysis of such simulations seems to suffer with lack of accurate material model to describe the material behavior of titanium alloys 73 under high temperature, high strain and high strain rates. Also, most of the simulations lack appropriate failure criterion to simulate shearlocalized chip formation. Further, most of the simulations were performed for highspeed machining while in titanium alloys segmented chip is found to occur even at very low speeds. In this study, JohnsonCook material model is used to represent material behavior as a function of strain, strain rate, and temperature; Recht’s catastrophic shear failure criterion is used for chip segmentation and a commercial, generalpurpose FE software, AdvantEdge is used to simulate orthogonal metal cutting of Ti 6Al4V under various machining conditions. AdvantEdge machining modeling software is a twodimensional central difference explicit finite element code using a Lagrangian mesh. The material model of the software accounts for elasticplastic strains, strain rates and temperature. It has an isotropic power law for strain hardening and as the material flow properties are temperature dependent it also accounts for thermal softening. A staggered method for coupled transient mechanical and heat transfer analyses is utilized. A sixnode quadratic triangular element with three quadrature points is used. This software also features adaptive remeshing option to alleviate mesh distortion. 6.2 Material Constitutive Model The flow stress or instantaneous yield strength at which work material starts to deform plastically is mostly influenced by temperature, strain, and strain rate. The constitutive model proposed by Johnson and Cook describes the flow stress of a material as a product of strain, strain rate, and temperature that are individually determined by the following equation: 74 σ = [A+B(ε)n][1+Cln( . ε * )][1+(T*)m] (6.1) where σ is the effective stress, ε is the effective plastic strain, . ε * is the normalized effective plastic strain rate, n is the work hardening exponent, m is the thermal softening exponent and A, B, C and m are constants [9]. It may be noted that formulation of JohnsonCook model is empirically based. The expression in the first set of brackets gives the stress as a function of strain for . ε * =1.0 and T*=0.The expressions in the second and third sets of brackets represent the effects of strain rate and temperature, respectively. The parameter A is in fact the initial yield strength of the material at room temperature at a strain rate of 1 s1. The nondimensional parameter T* is defined as T* = (T Troom)/(Tmelt Troom) (6.2) where T is the current temperature, Troom is the ambient temperature, and Tmelt is the melting temperature. Temperature term in this model reduces the flow stress to zero at the melting temperature of the workmaterial, leaving the constitutive model with no temperature effects. The nondimensional normalized effective plastic strain rate . ε * is the ratio of the effective plastic strain rate ε p to the reference strain rate ε° (usually equal to 1.0). In general, the parameters A, B, C, n and m are fitted to the data obtained by several mechanical tests conducted at low strains and strain rates as well as split Hopkinson pressure bar (SHPB) tests and ballistic impact tests. JohnsonCook model provides good fit for strain hardening behavior of the metals. It is numerically robust and can be easily used in finite element simulation models. 75 6.3 Formulation of Recht’s Catastrophic Shear Failure Criterion Perhaps the first criterion for shear localization was developed by Recht [12]. According to this criterion, shear instability was predicted to occur when the rate of strain hardening of the material is balanced by thermal softening. This indicates that when an increase in shear strain is associated with a decrease in shear stress, the strain hardening slope would become negative: ≤ 0 γ τ d d with γ τ γ τ γ τ d dT d T d ∂ + ∂ ∂ = ∂ (6.3) The criterion R can be written as follows: γ τ γ τ d dT T R ∂ − ∂ ∂ ∂ = and 0≤ R ≤ 1 (6.4) where τ ,γ and T represent shear stress, shear strain and temperature respectively. When R is equal to zero, instability starts and when R is equal to one catastrophic shear failure occurs. The numerator and denominator terms of R are derived from JohnsonCook flow stress equation: ( ( ) ) ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − − − ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = + + m r r o n T T T T A B 1 C ln 1 . . ε σ ε ε (6.5) 76 where A, B, C, n, m are constants. T is the current temperature, Tr is the ambient temperature, Tm is the melt temperature. ε is the strain, . ε is the effective plastic strain rate and . ε o is the reference plastic strain rate. The shear stress τ and the shear strain γ can be expressed in terms of strain and strain rate as follows: 3 τ = σ and γ =ε 3 (6.6) Substituting the Equation 6.6 into Equation 6.5, we get ⎥⎦ ⎤ ⎢⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − − − ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ = + m r r n T T T T A B 1 C ln 1 3 3 1 . 0 . γ τ γ γ (6.7) The partial differentiation of Equation 6.7 with respect to the shear strainγ gives the strain hardening term γ τ ∂ ∂ as ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − − − ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ = ∂ ∂ − m m r r n T T T T C nB 1 ln 1 3 3 . 0 1 . γ γ γ γ τ (6.8) while the partial differentiation of Equation 6.8 with respect to temperature T gives the thermal softening term ∂T ∂τ as ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − − − − ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ = + ∂ ∂ m m r r r n T T T T T T m A B C T . 0 . 1 ln 3 3 1 γ τ γ γ (6.9) 77 The second term in the denominator of Recht’s criterion dγ dT is the rate of change of temperature with strain in the catastrophic shear zone. Recht developed a model [12] to determine the temperature gradient with strain. According to this model, the zone in Fig. 6.1 with unit area A and unit thickness T is assumed to be the weakest zone within the length L of the specimen. Catastrophic shear is achieved by applying a constant rate of average strain L x . high enough to produce catastrophic slip in this zone and when this is achieved, this shear zone will remain thin and it can be seen that plastic deformation is confined to this region. Since this zone is very thin, it is assumed to be a plane of uniform heat generation. Fig. 6.1 Model used for determination of temperature gradient with strain in catastrophic shear zone [12] Thus the heat generation rate over the unit area A is given by .γ τ W L q = (6.10) where q is the heat generation rate per unit area, τ is the shear strength in the weak zone, L is the specimen length, .γ is the average shear strain rate equal to L x . and W is the work equivalent of heat. Using Carslaw and Jaeger’s solution for the temperature on a plane in 78 an infinite medium at constant heat generation, the instantaneous temperature TA on unit area A is given as C t W L T y A πκρ τ γ . = (6.11) where y τ is the initial shear yield strength, t is the time, κ is the thermal conductivity, C is the specific heat and ρ is the specific weight. Differentiating the above equation with respect to time t, gives dt W Ct L dT y A πκρ τ γ 1 2 1 . = (6.12) But, for constant strain rate, y γ =γ t +γ . and .γ γ = dt d where γ is the unit shear strain and y γ is the initial yield strain. Substituting these two in Equation 6.12, we get, ( )y A y W C L d dT πκρ γ γ τ γ γ − = . 2 1 (6.13) Substituting Equations 6.8, 6.9 and 6.13 in Equation 6.4, will give the following equation which can be used to formulate the Recht’s criterion into the UMAT code. ( )⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − − − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − − − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ = − y y m m r r r n m m r r n W C L T T T T T T m A B T T nB T T R πκρ γ γ γ τ γ γ . 1 2 1 3 3 1 1 3 3 (6.14) For each integration point of the element of workpiece mesh, Recht criterion is evaluated. When Recht criterion is satisfied by all the integration points of the element, 79 the stress state of all these integration points is made zero. The code of the element is stored in a temporary file. All these coded (listed) elements are then deleted and then the border of the workpiece is extracted and smoothed. This smoothing operation reduces the loss of volume in the workpiece determined by e 



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