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UBC Theses and Dissertations

An integrated model for force prediction in peripheral milling operations Zou, Guiping 2011

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 AN INTEGRATED MODEL FOR FORCE PREDICTION IN PERIPHERAL MILLING OPERATIONS   by  Guiping Zou  B.A.Sc Dalian University of Technology, 1985 M.A.Sc Dalian University of Technology, 1988 Ph.D Dalian University of Technology, 1995   A THESIS SUBMITTED IN PARTIAL FULLFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  in  The Faculty of Graduate Studies  (Mechanical Engineering)      THE UNIVERSITY OF BRITISH COLUMBIA (VANCOUVER)     January 2011    © Guiping Zou, 2011  ii Abstract  This thesis is primarily concerned with the modeling of peripheral milling operations and, in particular, with cutting force prediction. The models developed incorporate novel approaches to deal with the difficulties described regarding edge geometry and chip thickness. The major contributions are outlined below. The first step in the analysis has been the development of models for small chip thickness. The model for cutting with a large cutting edge radius or chamfer (compared to chip thickness) is formulated as an Upper Bound where the primary and secondary zones are defined from a previous slip-line field model. This has the advantage of providing both force and moment equilibrium together with a realistic rake face stress situation. The second major contribution of the thesis lies in the formulation of a new model of oblique cutting and the direct application of this to the milling process. The new upper bound model that incorporates force equilibrium parallel to the cutting edge is proposed for oblique cutting operations. The energy approach is framed in terms of the normal shear angle and two fundamental variables that characterize the energy requirements of the oblique cutting process. Since the process of surface generation requires the analysis of entry and exit phenomenon, the attention has been directed towards simple slip-line field models of ploughing with single and double cutting edges. The analysis includes the influence of ploughing length on the ploughing/cutting transition. Finally the thesis presents the modeling of milling forces using the oblique cutting method as outlined and adding to the ploughing forces calculated from the new model. In addition, the model incorporates  iii suitable material constitutive equations so that the influence of strain, strain rate, and temperature effects can be accounted for as chip thickness varies.  The Upper Bound model at this stage is also able to incorporate the influence of surface slope and the kinematics of the process. The models developed have been tested with a variety of end mills on several work materials. These tests clearly demonstrate the need to account for the secondary factors not normally included in the modeling process.                              iv Preface A version of chapter 3 has been published. Zou, G. P., Yellowley, I., and Seethaler, R. J., 2009. A new approach to the modeling of oblique cutting processes, Int. J. of Machine Tools and Manufacture, 49, 701-707. I am responsible for the formulations and algorithms, the development of software, the numerical computations, the draft of the paper and further modifications. Prof. Yellowley is responsible for proposing the methodology, checking formulations, writing paper, replying to the reviewers and finalizing the paper. Prof. Seethaler is responsible for deriving the formulations related to the parameter identifications, and modifying the paper, also. A version of chapter 4 has been published. Zou, G. P., Yellwoley, I., and Seethaler, R. J., 2010. The extension of a simple predictive model for orthogonal cutting to include flow below the cutting edge, Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, special issue for 36 MATADOR, submitted. Zou, G. P., Yellowley, I., and Seethaler, R. J., 2010. Extension of a simple predictive model for orthogonal cutting to include below the cutting edge, 36 MATADOR Conf., Manchester, UK. I conducted research on deriving the formulations, compiling Maple optimization program, numerical computations, and writing the draft. Prof. Yellowley proposed the methodology, modified paper and replied to the reviewer. Prof. Seethaler checked formulations, modified the paper and presented to 36 MATADOR.     v Table of Contents Abstract.............................................................................................................................. ii Preface............................................................................................................................... iv Table of Contents .............................................................................................................. v List of Tables .................................................................................................................... ix List of Figures.................................................................................................................... x Nomenclature ............................................................................................................... xviii Acknowledgements ....................................................................................................... xxv 1         Introduction............................................................................................................ 1 1.1 Overview..................................................................................................................... 1 1.2 Objective and Scope of Thesis .................................................................................. 6 1.3 Outline......................................................................................................................... 8 2         Literature Survey................................................................................................. 10 2.1 Introduction.............................................................................................................. 10 2.2 Basic Mechanics of Metal Cutting.......................................................................... 10 2.2.1 Shear Plane Models .................................................................................... 11 2.2.2 Slip Line Field ............................................................................................ 13 2.2.3 Upper Bound Method ................................................................................. 21 2.2.4 Material Separation..................................................................................... 23 2.2.5 Finite Element Modeling of Machining ..................................................... 24 2.3 Oblique Cutting Models .......................................................................................... 26 2.4 Analysis of Practical Milling Tools......................................................................... 30 2.4.1 Modeling of Milling Forces........................................................................ 33 2.4.2 Modeling of Ploughing Forces ................................................................... 37 2.5 Conclusions............................................................................................................... 44 3         A New Approach to the Modeling of Oblique Cutting Processes.................... 46 3.1 Introduction.............................................................................................................. 46  vi 3.2 Analysis ..................................................................................................................... 46 3.3 Special Cases ............................................................................................................ 53 3.3.1 Zero SLIP ................................................................................................... 53 3.3.2 SLIP Approaching Unity ............................................................................ 54 3.4 Relationship between the Major Parameters ........................................................ 55 3.4.1 SLIP and RATIO ........................................................................................ 55 3.4.2 SLIP and Angle of Chip Flow on the Rake Face........................................ 56 3.5 Model Calibration from Measurement of Cutting Forces ................................... 58 3.6 Comparison with Existing Experimental Tests..................................................... 60 3.7 Conclusions............................................................................................................... 64 4         The Modeling of Tool Edge Ploughing Forces in Milling Operations ............ 66 4.1 Introduction.............................................................................................................. 66 4.2 Predictive Model for Orthogonal Cutting to Include Flow below the Cutting Edge...................................................................................................................... 67 4.2.1 Upper-Bound Formulation.......................................................................... 67 4.2.2 Optimization Procedure.............................................................................. 71 4.2.3 Force Prediction.......................................................................................... 72 4.2.4 Adapting the Predictive Model for Radiused Edge Tools .......................... 75 4.2.5 Case Studies of Predictive Cutting Model.................................................. 76 4.2.5.1 Optimized Upper Bound Geometries ................................................ 76 4.2.5.2 Validating the Proposed Model with Experimental Data .................. 78 4.2.6 Parameter Studies of Predictive Cutting Model ......................................... 81 4.2.6.1 Influence of Chamfer Angle on Cutting Forces................................. 81 4.2.6.2 Influence of Chamfered Land Length on Cutting Forces .................. 82 4.3 Ploughing Models..................................................................................................... 83 4.3.1 Single Edge Ploughing Model .................................................................... 83 4.3.2 Double Edge Ploughing Model .................................................................. 85 4.3.3 Ploughing Model with Rounded Edge Tool ............................................... 88 4.3.4 Ploughing Model Applications ................................................................... 89 4.3.4.1 Effect of Attack Angle....................................................................... 89 4.3.4.2 Edge Radius Effects........................................................................... 90 4.4 Transition of Ploughing to Cutting Processes ....................................................... 90  vii 4.4.1 Effect of Sliding Distance on the Single Edge Ploughing Model............... 91 4.4.2 Effect of Sliding Distance on Double Edge Ploughing Model ................... 93 4.4.3 Discussion on the Entry and Exit Milling Processes .................................. 94 4.4.4 Validating Entry and Exit with Worn Tool Milling Operations ................. 95 4.5 Conclusions............................................................................................................... 97 5         A New Predictive Model for Peripheral Milling Operations ........................... 99 5.1 Introduction.............................................................................................................. 99 5.2 The Development of a Methodology to Estimate Milling Forces....................... 100 5.3 The Estimation of Oblique Cutting Forces.......................................................... 102 5.3.1 Deformation Energy Dissipation on the Shear Plane ............................... 102 5.3.2  Friction Energy Dissipation at the Tool-chip Interface ............................ 104 5.4 The Influence of Work Hardening on the Average Normal Stress in the Shear Plane................................................................................................................... 106 5.5 The Estimation of Edge Force Components ........................................................ 109 5.6 The Effect of Work Surface Slope........................................................................ 110 5.6.1 Surface Slope Formations......................................................................... 110 5.6.2 Frictionless Orthogonal Cutting ............................................................... 112 5.6.3 Oblique Cutting Model............................................................................. 113 5.7 Force Prediction Based on the Shear Plane Forces ............................................ 114 5.8 Kinematics of Helical End Milling Tools ............................................................. 116 5.8.1 Coordinate System and Velocity Relationship ......................................... 117 5.8.2 Helical Tool Velocity Profile.................................................................... 119 5.8.3 Continuous Shear...................................................................................... 123 5.8.4 Special Case: Frictionless ......................................................................... 124 5.9 Cutting Force Prediction for a General Helical Tooth Milling Cutter ............. 127 5.10 Conclusions............................................................................................................. 131 6         Experimental Validation of the Predictive Model .......................................... 133 6.1 Introduction............................................................................................................ 133 6.2 Experimental Setup ............................................................................................... 133 6.3 Edge Radius Measurement ................................................................................... 135 6.4 Identification of the Constitutive Parameters ..................................................... 139  viii 6.5 Detailed Validation of the Proposed Model......................................................... 148 6.5.1 Validating Simulation Results for Aluminum Alloy ................................ 151 6.5.2 Validating Simulation Results for Titanium Alloy with 3/8” SGS Cutter 160 6.5.3 Validating Simulation Results for Titanium Alloy with 3/16” SGS Cutter....   .................................................................................................................. 168 6.5.4 Validating Simulation Results for Steel ................................................... 180 6.6 Conclusions............................................................................................................. 184 7         Conclusions......................................................................................................... 185 7.1 Dissertation Overview ........................................................................................... 185 7.2 Major Contributions.............................................................................................. 186 Bibliography .................................................................................................................. 188 Appendices..................................................................................................................... 200 Appendix A: Experimental Tests ................................................................................... 200 Appendix B: Dynamometer Calibration ....................................................................... 202                       ix List of Tables  Table 3.1 SLIP and RATIO from experimental data (Brown and Armarego (1964))................... 62 Table 3.2 SLIP and RATIO from experimental data (Pal and Koenigsberger (1968)) ................. 62 Table 3.3 Average values of RATIO and SLIP extracted from experimental data ....................... 62 Table 6.1 The range of cutting tests undertaken.......................................................................... 134 Table 6.2 Cutter information ....................................................................................................... 136 Table 6.3 The cutting operations and measured edge radii ......................................................... 136 Table 6.4 Workpiece material properties .................................................................................... 143 Table 6.5 The measured forces at the midpoint of central cuts ................................................... 144 Table 6.6 Identified J-C constitutive parameters for aluminum alloy, titanium alloy and steel.. 144 Table AA.1  Milling tests for aluminum alloy ............................................................................ 200 Table AA.2  Milling tests for titanium alloy with 2 teeth 3/8” end mill cutter ........................... 200 Table AA.3  Milling tests for titanium alloy with 2 teeth 3/16” end mill cutter ......................... 201 Table AA.4  Milling tests for AISI 4140 steel ............................................................................ 201 Table AB.1  Identified parameters used for charge amplifiers (Kistler Model 5004)................. 202                       x List of Figures  Figure 1.1 Edge radius of milling tools ........................................................................................... 3 Figure 1.2 Milling parameters ......................................................................................................... 3 Figure 1.3 Schematic showing 3 phases of milling ......................................................................... 4 Figure 1.4 Chip load and force relationship .................................................................................... 4 Figure 1.5 Milling operations .......................................................................................................... 5 Figure 2.1 Single shear plane model ............................................................................................. 12 Figure 2.2 Lee and Shaffer’s slip-line field solution ..................................................................... 14 Figure 2.3 Oxley’s thin zone model .............................................................................................. 15 Figure 2.4 Kudo’s slip line field.................................................................................................... 16 Figure 2.5 Oxley’s parallel zone model ........................................................................................ 17 Figure 2.6 Dewhurst model ........................................................................................................... 19 Figure 2.7 Geometry of the basic model ....................................................................................... 20 Figure 2.8 Equilibrium of chip ...................................................................................................... 21 Figure 2.9 Force in oblique cutting with single edge tool ............................................................. 28 Figure 2.10 Face milling geometry ............................................................................................... 31 Figure 2.11 End milling geometry................................................................................................. 31 Figure 2.12 Milling types .............................................................................................................. 32 Figure 2.13 Milling forces............................................................................................................. 34 Figure 2.14 Scheme of forces acting on the face and flank surface of the tool............................. 37 Figure 2.15 Endres’ dual-mechanism approach ............................................................................ 39 Figure 2.16 The division of slip-line field (Fang 2003) ................................................................ 40 Figure 2.17 Ren and Altintas’s model (2000) ............................................................................... 40 Figure 2.18 Schematic of Waldorf’s ploughing force model (1998)............................................. 41 Figure 2.19 Manjunathaiah and Endres’s geometric model of edge radius tool (2000)................ 43 Figure 3.1 The oblique cutting process ......................................................................................... 48 Figure 3.2 Power and derivative with different assumptions regarding the formulation of rake face force ....................................................................................................... 52 Figure 3.3 Normal shear angle versus inclination angle (RATIO=1) ........................................... 55 Figure 3.4 The relationship between SLIP and RATIO with varying inclination and normal shear angles ................................................................................................... 56  xi Figure 3.5 SLIP as a function of normalized flow angle............................................................... 58 Figure 3.6 The influence of normal rake angle on chip flow angle............................................... 61 Figure 3.7 Comparison of the predicted chip flow angle with experimental results (Moufki et al. (2000)) ................................................................................................ 63 Figure 3.8 Comparison of the predicted chip flow angle with experimental results (Pal and Koenigsberger (1968))................................................................................. 63 Figure 3.9 Comparison of the predicted chip flow angle at various normal rake angles with experimental data from several investigators .................................................... 64 Figure 4.1 Proposed slip-line field for orthogonal cutting chamfer tools...................................... 67 Figure 4.2 Hodograph of the proposed slip-line field ................................................................... 68 Figure 4.3 Moment equilibrium of chip ........................................................................................ 69 Figure 4.4 Force equilibrium on the tool-rake face ....................................................................... 73 Figure 4.5 Force equilibrium for chamfered tool .......................................................................... 74 Figure 4.6 Upper bound field for radiused edge............................................................................ 75 Figure 4.7 Upper bound solutions for the case of γ1=-700, γ2=00, βs=400...................................... 77 Figure 4.8 Upper bound solutions for different chamfer angles.................................................... 77 Figure 4.9 Calculation of normal stress on chamfer using two extreme cases of field geometry .................................................................................................................... 77 Figure 4.10 Comparison of predicted cutting forces with experimental results............................ 78 Figure 4.11 Variation of cutting and thrust forces versus uncut chip thickness............................ 80 Figure 4.12 Change in force ratio as uncut chip thickness increases ............................................ 80 Figure 4.13 Cutting force Fc versus chamfer angle for the rake angle of -100, 00 and 100 .............................................................................................................................. 81 Figure 4.14 Thrust force FT versus chamfer angle for the rake angle of -100, 00 and 100 .............................................................................................................................. 82 Figure 4.15 Variation of cutting force FC and thrust force FT with respect to chamfer length ......................................................................................................................... 83 Figure 4.16 Slip-line field for single edge ploughing model......................................................... 84 Figure 4.17 Double edge slip-line field......................................................................................... 85 Figure 4.18 Hodograph.................................................................................................................. 86 Figure 4.19 Round edge tool ploughing geometry ........................................................................ 88 Figure 4.20 Specific cutting power versus attack angle ................................................................ 89 Figure 4.21 Specific ploughing power with respect to ploughing depth....................................... 90 Figure 4.22 Specific power versus sliding distance for attack angle λs=350 ................................. 91  xii Figure 4.23 Specific power versus sliding distance for attack angle λs=450 ................................. 92 Figure 4.24 Specific power versus sliding distance for attack angle λs=550 ................................. 92 Figure 4.25 Slip-line field variation with sliding distance for attack angle λs=350 ....................... 93 Figure 4.26 Slip-line field variation with attack angle for sliding distance lslide=2h0 .................... 93 Figure 4.27 Specific power versus sliding distance for γ1=550, γ2=800 and LED=0.5 h0................ 93 Figure 4.28 Slip-line field with different sliding distance for ploughing angle γ1=550, γ2=800......................................................................................................................... 94 Figure 4.29 Simulation of micromilling processes........................................................................ 97 Figure 5.1 Diagram of predictive model ..................................................................................... 102 Figure 5.2 Coordinate system of oblique cutting ........................................................................ 105 Figure 5.3 Normal shear plane for oblique cutting...................................................................... 106 Figure 5.4 Slope variation in milling........................................................................................... 111 Figure 5.5 Orthogonal model with surface slope ........................................................................ 112 Figure 5.6 Shear angle relationship with RATIO........................................................................ 113 Figure 5.7 Force diagram for the shear plane.............................................................................. 116 Figure 5.8 Geometry of helical tool system ................................................................................ 117 Figure 5.9 Basic geometry of helical pitch.................................................................................. 120 Figure 5.10 Tangential velocity of the segment .......................................................................... 121 Figure 5.11 Plan view of chip...................................................................................................... 123 Figure 5.12 Power change versus depth of cut a and helical angle i ........................................... 126 Figure 5.13 Power change versus tool radius R and helical angle i ............................................ 126 Figure 5.14 Geometry of helical tooth ........................................................................................ 128 Figure 6.1 Mounted cutter samples ............................................................................................. 135 Figure 6.2 SEM image for Cutter A ............................................................................................ 137 Figure 6.3 SEM image for Cutter B ............................................................................................ 137 Figure 6.4 SEM image for Cutter C ............................................................................................ 137 Figure 6.5 SEM image for Cutter D ............................................................................................ 138 Figure 6.6 SEM image for Cutter E............................................................................................. 138 Figure 6.7 Forces acting on the rake face and shear plane .......................................................... 141 Figure 6.8 Centerline cutting force comparison as feedrate changes for central cutting aluminum alloy (cutter A)........................................................................................ 145  xiii Figure 6.9 Centerline cutting force comparison as feedrate changes for central cutting titanium alloy (cutter B)........................................................................................... 145 Figure 6.10 Centerline cutting force as feedrate changes for slotting of steel (cutter D) ............................................................................................................................. 146 Figure 6.11 Predicted shear angle versus uncut chip thickness for central cutting aluminum alloy (cutter A)........................................................................................ 146 Figure 6.12 Predicted shear angle versus uncut chip thickness for central cutting titanium alloy (cutter B)........................................................................................... 147 Figure 6.13 Predicted shear angle versus uncut chip thickness for slotting steel (cutter D) ............................................................................................................................. 147 Figure 6.14 Velocity profile for cutting aluminum alloy (Cutter A)........................................... 149 Figure 6.15 Velocity profile for cutting titanium alloy (Cutter B and C).................................... 149 Figure 6.16 Velocity profile for cutting titanium alloy (Cutter E) .............................................. 150 Figure 6.17 Velocity profile for cutting steel (Cutter D)............................................................. 150 Figure 6.18 Measured and predicted forces Fx and Fy for up milling of aluminum alloy (Vf=3.95 in/min, Cutter A).............................................................................. 151 Figure 6.19 Measured and predicted forces Fx and Fy for up milling of aluminum alloy (Vf=7.9 in/min, Cutter A)................................................................................ 152 Figure 6.20 Measured and predicted force Fx and Fy for down milling of aluminum alloy (Vf=1.28 in/min, Cutter A).............................................................................. 152 Figure 6.21 Measured and predicted force Fx and Fy for down milling of aluminum alloy (Vf=2.56 in/min, Cutter A).............................................................................. 153 Figure 6.22 Measured and predicted force Fx and Fy for down milling of aluminum alloy (Vf=3.95 in/min, Cutter A).............................................................................. 153 Figure 6.23 Measured and predicted force Fx and Fy for down milling of aluminum alloy (Vf=6.75 in/min, Cutter A).............................................................................. 154 Figure 6.24 Measured and predicted forces Fx and Fy for central cutting aluminum alloy (Vf=1.28 in/min, Cutter A).............................................................................. 154 Figure 6.25 Measured and predicted forces Fx and Fy for central cutting aluminum alloy (Vf=2.56 in/min, Cutter A).............................................................................. 155  xiv Figure 6.26 Measured and predicted forces Fx and Fy for central cutting aluminum alloy (Vf=4.4 in/min, Cutter A)................................................................................ 155 Figure 6.27 Measured and predicted forces Fx and Fy for central cutting aluminum alloy (Vf=6.75 in/min, Cutter A).............................................................................. 156 Figure 6.28 Measured and predicted forces Fx and Fy for central cutting aluminum alloy (Vf=7.9 in/min, Cutter A)................................................................................ 156 Figure 6.29 Force comparison with and without considering ploughing force on central cutting aluminum alloy (Vf=1.28 in/min, Cutter A) .................................... 157 Figure 6.30 Force comparison with and without considering ploughing force on central cutting aluminum alloy (Vf=2.56 in/min, Cutter A) .................................... 157 Figure 6.31 Force comparison with and without considering slope effect on central cutting aluminum alloy (Vf=7.9 in/min, Cutter A) .................................................. 158 Figure 6.32 Force comparison with and without considering slope effect on up milling aluminum alloy (Vf=7.9 in/min, Cutter A).................................................. 158 Figure 6.33 Measured and predicted forces Fx and Fy for slotting (center jump of depth of cut) aluminum alloy at different feeding velocities (Cutter A).................. 159 Figure 6.34 Measured and predicted forces Fx and Fy for down milling of titanium alloy (Vf=0.75 in/min, Cutter C).............................................................................. 160 Figure 6.35 Measured and predicted forces Fx and Fy for down milling of titanium alloy (Vf=1.5 in/min, Cutter C)................................................................................ 160 Figure 6.36 Measured and predicted forces Fx and Fy for down milling of titanium alloy (Vf=2.2 in/min, Cutter C)................................................................................ 161 Figure 6.37 Measured and predicted forces Fx and Fy for up milling of titanium alloy (Vf=0.75 in/min, Cutter C)....................................................................................... 161 Figure 6.38 Measured and predicted forces Fx and Fy for up milling of titanium alloy (Vf=1.5 in/min, Cutter C)......................................................................................... 162 Figure 6.39 Measured and predicted forces Fx and Fy for up milling of titanium alloy (Vf=4.4 in/min, Cutter C)......................................................................................... 162 Figure 6.40 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=0.75 in/min, Cutter B)....................................................................................... 163 Figure 6.41 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=1.5 in/min, Cutter B)......................................................................................... 163  xv Figure 6.42 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=2.2 in/min, Cutter B)......................................................................................... 164 Figure 6.43 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=3.95 in/min, Cutter B)....................................................................................... 164 Figure 6.44 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=4.4 in/min, Cutter B)......................................................................................... 165 Figure 6.45 Force comparison with and without considering ploughing force for central cutting titanium alloy (Vf=0.75 in/min, Cutter B)........................................ 166 Figure 6.46 Force comparison with and without considering ploughing force for up milling of titanium alloy (Vf=0.75 in/min, Cutter C)............................................... 166 Figure 6.47 Force comparison with and without considering ploughing force for down milling of titanium alloy (Vf=0.75 in/min, Cutter C)..................................... 167 Figure 6.48 Force comparison with and without considering slope effect for up milling of titanium alloy (Vf=4.4 in/min, Cutter C)................................................. 167 Figure 6.49 Force comparison with and without considering slope effect for central cutting titanium alloy (Vf=4.4 in/min, Cutter B) ..................................................... 168 Figure 6.50 Measured and predicted forces Fx and Fy for down milling of titanium alloy (Vf=0.75 in/min, Cutter E) .............................................................................. 169 Figure 6.51 Measured and predicted forces Fx and Fy for down milling of titanium alloy (Vf=1.5 in/min, Cutter E) ................................................................................ 169 Figure 6.52 Measured and predicted forces Fx and Fy for down milling of titanium alloy (Vf=2.2 in/min, Cutter E) ................................................................................ 170 Figure 6.53 Measured and predicted forces Fx and Fy for down milling of titanium alloy (Vf=2.56 in/min, Cutter E) .............................................................................. 170 Figure 6.54  Measured and predicted forces Fx and Fy for up milling of titanium alloy (Vf=1.5 in/min, Cutter E) ......................................................................................... 171 Figure 6.55  Measured and predicted forces Fx and Fy for up milling of titanium alloy (Vf=1.5 in/min, Cutter E) ......................................................................................... 171 Figure 6.56  Measured and predicted forces Fx and Fy for up milling of titanium alloy (Vf=2.2 in/min, Cutter E) ......................................................................................... 172 Figure 6.57  Measured and predicted forces Fx and Fy for up milling of titanium alloy (Vf=2.56 in/min, Cutter E) ....................................................................................... 172  xvi Figure 6.58 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=0.75 in/min, Cutter E) ....................................................................................... 173 Figure 6.59 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=1.5 in/min, Cutter E) ......................................................................................... 173 Figure 6.60 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=2.2 in/min, Cutter E) ......................................................................................... 174 Figure 6.61 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=2.56 in/min, Cutter E) ....................................................................................... 174 Figure 6.62 Comparison of the predicted forces with and without considering ploughing force for down milling of titanium alloy (Vf=0.75 in/min, Cutter E)................................................................................................................... 175 Figure 6.63 Comparison of the predicted forces with and without considering ploughing force for up milling of titanium alloy (Vf=0.75 in/min, Cutter E).............................................................................................................................. 176 Figure 6.64 Comparison of the predicted forces with and without considering ploughing force for central cutting titanium alloy (Vf=0.75 in/min, Cutter E).............................................................................................................................. 176 Figure 6.65 Comparison of the predicted forces with and without considering slope effect for down milling of titanium alloy (Vf=2.56 in/min, Cutter E) ..................... 177 Figure 6.66 Comparison of the predicted forces with and without considering slope effect for up milling of titanium alloy (Vf=2.56 in/min, Cutter E) .......................... 177 Figure 6.67 Comparison of the predicted forces with and without considering slope effect for central cutting titanium alloy (Vf=2.56 in/min, Cutter E) ........................ 178 Figure 6.68 Comparison of the effect of velocity profile for down milling of titanium alloy (Vf=2.56 in/min, Cutter E) .............................................................................. 178 Figure 6.69 Comparison of the effect of velocity profile for up milling of titanium alloy (Vf=2.56 in/min, Cutter E) .............................................................................. 179 Figure 6.70 Comparison of the effect of velocity profile for central cutting titanium alloy (Vf=2.56 in/min, Cutter E) .............................................................................. 179 Figure 6.71 Measured and predicted forces Fx and Fy for slotting of steel (Vf=0.75 in/min, Cutter D)...................................................................................................... 180  xvii Figure 6.72 Measured and predicted forces Fx and Fy for slotting of steel (Vf=1.28 in/min, Cutter D)...................................................................................................... 181 Figure 6.73 Measured and predicted forces Fx and Fy for slotting of steel (Vf=1.5 in/min, Cutter D)...................................................................................................... 181 Figure 6.74 Measured and predicted forces Fx and Fy for slotting of steel (Vf=2.2 in/min, Cutter D)...................................................................................................... 182 Figure 6.75 Measured and predicted forces Fx and Fy for slotting of steel (Vf=2.56 in/min, Cutter D)...................................................................................................... 182 Figure 6.76 Measured and predicted forces Fx and Fy for slotting of steel (Vf=3.95 in/min, Cutter D)...................................................................................................... 183 Figure 6.77 Comparison of force prediction with and without considering edge force for slotting of steel (Vf=0.75 in/min, Cutter D) ....................................................... 183 Figure AA.1 Geometry of central cutting titanium alloy by 3/16” end mill cutter ..................... 201 Figure AB.1 Calibration of the dynamometer in x direction....................................................... 203 Figure AB.2 Calibration of dynamometer in y direction............................................................. 203 Figure AB.3 Calibration of dynamometer in z direction............................................................. 204                       xviii Nomenclature  a Axial depth of cut a2 Penetration depth A Material constant of Johnson-Cook model Ac Plastic contact area on the rake face As Area of shear plane e sA  Elemental area of shear plane B Material constant of Johnson-Cook model Cm Material constant of Johnson-Cook model d Radial depth of cut D Tool diameter ds1 Elemental length along the slip-line α ds2 Elemental length along the slip-line β E Young’s modulus FC Principal cutting force CF ′  Cutting force for orthogonal cutting CEF  Edge force component in the principal cutting direction e CF  Elemental principal cutting force C,elasticF  Elastic force component in the cutting direction CEF'  Edge force component in the cutting direction for orthogonal cutting CHF'  Cutting force component in orthogonal cutting for high negative rake cutting CPF  Ploughing force component in the principal cutting direction CPF'  Ploughing force component in the cutting direction for orthogonal cutting CSF  Sharp tool cutting force Ff Resultant friction force acting on the rake face Ff,elastic Elastic friction force acting on the tool-chip interface Fft Friction force component parallel to the cutting edge on the rake face  xix Ffr Friction force component normal to the cutting edge on the rake face FN Normal force acting on elastic region of rake face FR Lateral force Fr Radial force component in milling REF  Edge force component in the lateral direction e RF  Elemental lateral force RPF  Ploughing force component in the lateral direction Fs Resultant shear force Fsr Shear force component normal to the cutting edge in the shear plane Fst Shear force component parallel to the cutting edge in the shear plane e SF  Elemental resultant shear force Ft Tangential force component in milling FT Thrust force TF ′  Thrust force for orthogonal cutting TEF  Edge force component in the transverse direction e TF  Elemental transverse cutting force TEF'  Edge force component in the transverse direction for orthogonal cutting T,elasticF  Elastic contact force component along the vertical direction THF'  Thrust force component in orthogonal cutting for high negative rake cutting TPF  Ploughing force component in the transverse direction TPF'  Ploughing force component in the transverse direction for orthogonal cutting TSF  Sharp tool thrust force Fx Milling force component in the x direction Fy Milling force component in the y direction Fz Milling force component in the z direction Gc Fracture toughness h Instantaneous uncut chip thickness in milling operations  xx h0 Uncut chip thickness h1 Chip thickness hav Average chip thickness in milling operations hcr Critical uncut chip thickness hm Minimum chip thickness i Inclination angle k Shear flow stress ek  Elemental shear yield stress K Thermal diffusivity N ABk  Shear yield stress in normal shear plane Kcf Normal force coefficient on the clearance face Kf Rake face force coefficient L Fan radius of ploughing model; Height of primary deformation zone for chamfer model l Length of cut LC Chip-tool contact length lc Chip length Lchmf Chamfer land length LE Elastic contact length Le Equivalent tool cutting length lN Normal shear plane length LP Plastic contact length ls Shear plane length lslide Sliding distance m Material constant of Johnson-Cook model n Material constant of Johnson-Cook model N Normal force on the rake face for orthogonal cutting Ne Number of segment for helical tool NEF Normal force acting on chamfer plane neq Equivalent work hardening term Nf Number of teeth  xxi Ns Normal force acting on the shear plane p Hydrostatic pressure P Total power consuming fP  Rate of energy dissipation by friction force shearP  Continuous volume shear power e shearP  Element continuous volume shear power sP  Shear plane plasticity energy dissipation rate PT Total power consumption for chamfer tool r Chip thickness ratio;  Nose radius r1 Ratio of tangential to radial cutting force r2 Ratio of tangential to radial parasitic force re Tool edge radius R Tool radius; Height of secondary deformation zone for chamfer model RPM Spindle speed [rev/min] Ri Inside chip radius Ro Outside chip radius Rtool Tool resultant cutting force S Specific heat St Feed per tooth T Temperature Tl Temperature at the lower boundary of shear zone Tu Temperature at the upper boundary of shear zone ur Chip flow velocity in the  radial direction ut Chip flow velocity in the tangential direction ux Chip flow velocity in the x direction uy Chip flow velocity in the y direction uz Chip flow velocity in the z direction ux’ Chip flow velocity in the x’ direction uy’ Chip flow velocity in the y’ direction uz’ Chip flow velocity in the z’ direction  xxii V0 Cutting velocity VC Chip velocity on the rake face e CV  Elemental chip velocity on the rake face Vcn Component of chip velocity normal to the cutting edge on the rake face e cnV  Elemental component of chip velocity normal to the cutting edge on the rake face Vct Component of chip velocity parallel to the cutting edge on the rake face e ctV  Elemental component of chip velocity parallel to the cutting edge on the rake face Vf Table feeding velocity Vi Interference volume Vs Shear velocity on the shear plane e SV  Elemental shear velocity on the shear plane N SV  Shear velocity component along normal shear plane Vsn Component of shear velocity normal to the cutting edge in the shear plane e snV  Elemental component of shear velocity normal to the cutting edge in the shear plane Vst Component of shear velocity parallel to the cutting edge in the shear plane e stV  Elemental component of shear velocity parallel to the cutting edge in the shear plane w Width of cut x Fraction of the cutting power transformed to heat; Elastic sliding distance relative to the plastic point B on the rake face for chamfer model; Global coordinate direction x x’ Local coordinate direction x’ y Global coordinate direction y y’ Local coordinate direction y’ z Distance between the plastic point B and FN for chamfer model; Global coordinate direction z  xxiii z’ Local coordinate direction z’ ze Height of the element zh Height of linear tangential velocity equal to constant tangential velocity β Friction angle βA Average friction angle βN Normal friction angle βs Elastic friction angle δ Fan angle of ploughing model δs Surface slope e sδ  Elemental surface slope ∆T Increased temperature ε Strain ε&  Strain rate 0ε&  Reference strain rate N ABε  Normal shear plane strain along middle shear zone N ABε&  Normal shear plane strain rate along middle shear zone pε&  Equivalent strain rate rθθθrr ε,ε,ε &&&  Tension and shear strain rates φ Shear angle φN Normal shear angle φN0 Normal shear angle when δs=0 γ Rake angle γ1 Chamfer angle; Rake angle for single edge ploughing model γ2 Chamfer tool rake angle γave Equivalent rake angle γe Effective rake angle γN Normal rake angle γoblique Oblique cutting strain  xxiv obliqueγ&  Strain rate of oblique cutting γv Velocity rake angle ηc Chip flow angle ηs Shear flow angle e sη  Elemental shear flow angle λ Runout angle λs Attack angle µ Coefficient of friction µcf Friction coefficient on the clearance face θ Angle between the resultant force and the shear plane θN Angle between the resultant force and the normal shear plane ρ Material density; Magnitude of runout; Ploughing bulge angle σ Yield stress; Normal stress distribution on the tool rake face N ABσ  Yield stress in normal shear plane σB Normal force at point B for chamfer tool model σN,GF Normal force acting at the plastic region GF for chamfer tool model σy Yield stress Ω Spindle speed ψ Tool rotation angle; Approach angle ψ1 Central cut entry angle ψ2 Central cut exit angle ψe Effective approach angle ψen Entry angle ψex Exit angle ψL Helix pitch rotation angle ψs Immersion angle      xxv Acknowledgements  I take this opportunity to express my sincere gratitude to my supervisors, Dr. Ian Yellowley and Dr. Rudolf Seethaler, for their expert guidance, valuable suggestions and advice throughout the course of this thesis. I also thank them for their financial support during the years of my doctoral program. A heart-felt thanks to Prof. Yellowley for his kindness and care throughout the doctoral program, although he is retired, he has still spent a great deal of time guiding the direction of my research. I am very grateful to Dr. Rudolf Seethaler for his help in the design and implementation of the experimental program which was conducted to verify the modeling process. I would like to thank technicians Markus Fengler and Erik Wilson for their help in testing. I would like to thank my wife Ping Lu and son Jie Zou for their sacrifices, endless inspiration and patience without which this work would not have been completed.                  1 1  Introduction  1.1 Overview  Milling is a commonly used machining operation in the aerospace, automobile and die-mold industries. The milling operation removes material by feeding a workpiece past a rotating single or multiple teeth cutter; the process is versatile and allows large amounts of material to be removed quickly. Competition in the manufacturing market requires the use of efficient and cost-effective planning systems and the optimization of process plans; both require the availability of good physical models of the machining processes. Many of the current machining models, whether for force, power, surface finish or tool life are expressed in empirical form and the required constants must be determined experimentally. These models are useful and necessary; however, the resulting equations and parameters are often restricted to a particular operation and condition tested. There is clearly considerable scope to improve the accuracy and validity of the process models on the basis of the fundamental physics and mechanics. The need for quantitative machining performance information has been recognized by many researchers and by industry (Armarego et al. 1995; Armarego 2000; Chiou et al. 2005; Wang et al. 2004; and Li and Shin 2006). Due to the wide variety of machining operations and numerous factors influencing each operation, the development of models for the prediction of machining performance presents a formidable task. In milling, as in other machining operations, the work material goes through a severe deformation process, at very high deformation strain (up to 4), strain rate (up to 106 1/s), and temperatures (up to 1000 0C). A major challenge in describing the milling 1  Introduction  2 forces analytically is the computation of the flow stress in the primary shear zone. The experimental methods commonly used to determine flow stress data through tensile and/or compression tests are not able to represent deformation behavior in these ranges of strain, strain rate, and temperature (Jaspers and Dautzenberg 2002). Jaspers and Dautzenberg (2002) conducted experiments using a Split Hopkinson’s Pressure Bar (SHPB) test to obtain the Johnson-Cook and Zerilli-Armstrong constitutive equations for AISI 1045 steel and 6082-T6 aluminum alloy. Sartkulvanich et al. (2004) proposed an approach to estimate the flow stress data using an orthogonal slot milling test. Finding approaches to describe the general complex milling operations with an accurate and reliable constitutive model is still a challenge. The milling process usually involves chip thicknesses that are small and variable; such chip thicknesses are often too small to promote cutting and instead result in either sliding or ploughing (Liu et. al. 2004). It is well known that the cutting tool is not perfectly sharp, and that there is always an edge radius at the cutting edge of a “sharp” tool; in addition, tool wear is a natural and inevitable process for all cutting tools that further degrades the cutting edge.  The edge is thought to play a significant role in the cutting forces, especially in milling where the radius of the edge relative to the chip thickness will be much higher.  The major difference between micro and conventional end milling operations, (from the perspective of this work), is that as the size of a tool decreases, the sharpness of the tool cannot be improved proportionally due to limitations in the tool fabrication processes and the need to provide some degree of strength at the tool edge. Figure 1.1 shows the edge radius of the milling tools, which can exert considerable influence on the cutting forces. Macro-chip formulation models are based on 1  Introduction  3 the assumption that the cutting tool completely removes the surface of the work piece (i.e., no material flows under the tool or is displaced sideward). On the other hand, the micro milling chip thickness is of the same order as tool edge radius and the same assumption cannot be made. h0 γ re  Figure 1.1 Edge radius of milling tools  The instantaneous chip thickness in milling can be approximated as ψψ sin)( tSh =                                                            (1.1) where ψ is the angle of immersion (shown in Figure 1.2). workpiece Ω St uncut chip thicknessh feed/per tooth ψ Feeding Direction  Figure 1.2 Milling parameters 1  Introduction  4 However, in reality the cutting process only proceeds once the elastic deflection and initial ploughing process have been completed, (see Figure 1.3 below which depicts the 3 stages). h0  h0  h0 (a) Ploughing  (b) Cutting initiated   ( c) Cutting  Figure 1.3 Schematic showing 3 phases of milling  Liu et al. (2004, 2006, and 2007) experimentally examined chip formulation and micro-cutting forces. They concluded that sudden change in thrust force corresponded to the shift from ploughing to cutting and hence occurred at what is termed the critical chip thickness. This sudden change in thrust forces was explained by a shifting from a ploughing/sliding process to normal cutting as shown in Figure 1.4. The same pattern of change in force and physical explanation has been reported in the milling of titanium alloy with worn tools by Yellowley et al. (1992). F o rc e p er  U n it  W id th  ( N /m m )  Figure 1.4 Chip load and force relationship 1  Introduction  5  In examining the peripheral milling processes one must differentiate between the up and down milling modes, (as shown in Figure 1.5). One expects that the entry and exit effects will be quite different between the down milling and up milling processes and need to be modeled with different fields. In addition, the work surface slope and the uncut chip thickness continuously vary in the milling processes somewhat resembling wave removing in dynamic cutting. The slope is a function of the cutter diameter and the feedrate in milling operations and is positive for up-milling and negative for down- milling. The effect of slope on the milling operations has been examined by previous authors (Pandey and Shan 1972; Altintas 1986). Pandy and Shan (1972) examined the slope change due to the advancement of shear plane, while Altintas (1986) only considered the chip thickness variation, more research is needed to study the effect of the total surface slope on the forces in milling operations as well as the relationship between the shear angle and the surface slope.   St Feed Direction Ω (a) Up milling Ω Feed Direction  (b) Down milling Figure 1.5 Milling operations  1  Introduction  6 1.2 Objective and Scope of Thesis Milling forces for a particular tool-workpiece pair are related to a variety of parameters, such as, width of cut, chip thickness, tool edge radius, constrained kinematics of the chip, cutting velocity, and milling type. Such parameters are difficult to incorporate in a conventional mechanistic modeling approach. This thesis intends to estimate the expected contribution of these influences so as to extend the applicability of a conventional modeling approach. Specific areas of concentration are stated as follows: 1) Practical milling cutters have significant edge radius and wear during their active life. Since the milling chip thicknesses are very small and variable, surface integrity and cutting forces are greatly influenced by the process at the tool edge. Traditionally, the majority of the modeling effort has concentrated on the deformation of the chip material. More recently a strong interest in understanding the physics of the process occurring at the cutting edge and below the tool has emerged. In addition, it is important to study all the milling force generation processes by considering the influences of the entry and exit processes. 2) Helical end milling tools have non-straight cutting edges and non-planar rake faces. Non-straight cutting edges cause the kinematic constraint in the cutting process. A kinematically admissible velocity field for the analysis of helical tools needs to be investigated to estimate the size of this influence. 3) Since helical milling cutters have non-planar rake faces and complex tool edge geometry, it is not possible to experimentally determine the shear angle from the chip geometry. This requires the development of an oblique cutting model that obviates the necessity for accurate shear angle data during calibration. 1  Introduction  7 4) Unlike the normal oblique cutting processes, the surface inclination to the cutting direction and the uncut chip thickness continuously vary in the milling processes (either increasing in up milling or decreasing in down milling). The usual shear plane force models show no dependence on the surface slope. In order to estimate the milling forces with sufficient accuracy, the influence of surface slope on the equilibrium value of shear angle needs to be investigated and included in a realistic milling model. 5) In practice no tool is perfectly sharp and the cutting edge always has a finite radius with corresponding forces acting at the cutting edge. In addition, due to wear, possible deformation of the cutting edge and perhaps the surface generation, additional forces will exist in this region. Because these forces are not considered to contribute to the chip removal process they are collectively referred to as the edge force. It is difficult to measure the edge force directly, especially in milling where the chip thickness is variable. The author investigates the design of a series of experimental tests and the development of a realistic methodology to identify the material properties and the characteristics of the primary shear zone based on the measured milling forces that avoid as  much as possible the influence of edge forces and slope on the forces used for calibration. (The changes expected from these are incorporated in the final force equations). 6) The thesis attempts to develop a simple predictive model for milling operations that includes the considerations for the complex milling cutter edge geometries, the influence of tool edge radius, kinematics and the surface slope. It also incorporates realistic material models to cope with varying values of strain, strain 1  Introduction  8 rate and temperature. The model considers both the conventional chip formation aspects of the process as well as subsurface flow. 1.3 Outline The chapters in this thesis are arranged as follows. Chapter 2 provides a review of metal cutting theory and milling operations. Basic orthogonal and oblique metal cutting models based on a single shear plane, slip-line fields, upper bound methods, and FEM approaches are introduced. The characteristics of milling operations, chip thickness models, edge forces and ploughing effects are presented. Chapter 3 presents a new upper-bound model that incorporates force equilibrium parallel to the cutting edge for oblique cutting operations. The energy approach is framed in terms of the normal shear angle and two new fundamental variables that characterize the energy requirements of the oblique cutting process. SLIP is a kinematic variable and is defined as the ratio of the shear velocity imparted to the chip on the shear plane parallel to the cutting edge, to the incoming velocity in the same direction. RATIO is a force based variable and is defined as the ratio of the friction force on the rake face to the resultant shear force in the shear plane. Calibration of the model for either real time identification purposes or for process planning/optimization requires experimental force data but no shear angle data making it very suitable for the analysis of cutting operations with non-straight cutting edges. Chapter 4 describes the extension of a simple predictive model for orthogonal cutting to accommodate flow under the cutting edge. The model utilizes an upper bound approach combined with primary and secondary boundaries that guarantee both force and moment equilibrium of the chip. Next, single and double edge ploughing models are 1  Introduction  9 developed to study the ploughing process, ploughing/cutting transition and minimum chip thickness. The issues of entry and exit of milling operations are discussed, also. Chapter 5 presents a predictive force model for peripheral milling operations based on the integration of the newly developed oblique cutting model and the incorporation of edge effects together with the adoption of a realistic material model, surface slope and admissible velocity profile. Chapter 6 presents the validation processes of the proposed model. The cutter edge radius is measured by scanning electron microscope (SEM). The predictive results are compared with experimental results for various workpiece materials, tool diameters, feeding velocities, and milling types. Chapter 7 points out the development and contribution of the present works; in particular recommends the importance of the second order effects in peripheral milling simulations.   10 2  Literature Survey  2.1 Introduction The milling operation is a metal cutting process that uses a rotating cutter that normally has multiple teeth. The process is usually oblique and a varying chip thickness is presented to the cutting edges. Chip formation in milling is the central issue that must be addressed in order to progress to a consideration of the practical variables of cutting force, tool wear and tool life, chatter, and tool breakage. The analysis of cutting forces generated during machining has been a popular research topic and from a practical point of view, force measurements have been shown to be valuable in the monitoring of cutting processes. This chapter provides a brief review of the previous research related to the modeling of machining operations. First, the fundamental physics and mechanics of metal cutting are discussed. Second, previous research covering the 3D cutting models, (especially the equivalent orthogonal models), is presented. Third, the modeling of practical milling operations is discussed. Special attention is given to the characteristics of milling operations, chip kinematics, cutting and edge force components and the incorporation of the ploughing effect within the analysis of milling processes. 2.2 Basic Mechanics of Metal Cutting The purpose of metal cutting process is to remove material and in doing so to create a new surface. The majority of modeling effort has concentrated on the deformation of the chip material and ignores the secondary factors. Researchers in this field have attempted to develop theories of the cutting process that can predict important 2  Literature Survey  11 cutting parameters without the need for empirical formulation. The following sections briefly review previous works based upon shear plane models, slip-line fields, simple upper bound models, and FEM analysis. 2.2.1 Shear Plane Models  The most widely used model is that proposed by Ernst and Merchant (1941) and Merchant (1944, 1945). The model describes shearing of incoming material as it passes through the primary shear zone. Earlier work by Piispanen (1937), considered that the shear process of metal is similar to cutting a deck of stacked cards; the cards being inclined at the effective shear plane angle. As the cards approach the tool, they are forced to slide over each other due to resistive force provided by the tool.  Merchant (1944) was the first to attempt to analyze the process and attempt to obtain a relationship between the shear angle and other variables. He began the metal cutting analysis by making certain assumptions, as follows 1) The tool is perfectly sharp and there is no contact along the clearance face; 2) The shear surface is a plane extending upward from the cutting edge to the free surface; 3) The chip is in a state of quasi-static equilibrium; 4) There is a constant coefficient of friction on the tool rake face; 5) The shear and normal stresses along the shear plane and tool are uniform. The forces acting between the chip and tool acting on the tool face, the shear plane and in the direction of relative work/tool velocity are shown in a free body diagram in Figure 2.1. The following relations follow from the assumption of equilibrium 2  Literature Survey  12  (a) Free body diagram   (b) Velocity diagram Figure 2.1 Single shear plane model  φφ φφ sincos sincos ⋅+⋅= ⋅−⋅= CTs TCs FFN FFF                                         (2.1) where β is the friction angle and µ the coefficient of friction given by ( ) N F FFN FFF f TC TCf = ⋅−⋅= ⋅+⋅= = − µ γγ γγ µβ sincos cossin tan 1                                          (2.2) Assuming that there is only sliding friction on the rake face and that there is a thin primary shear deformation area, then the power consuming force and the thrust force can be formulated as follows ( ) ( ) ( ) ( )     −+⋅ − =       −+⋅ − = γβφφ γβ γβφφ γβ cossin sin cossin cos 0 0 wkhF wkhF T C                                    (2.3) where w is the width of cut and h0 is the uncut chip thickness. The shear angle may be found experimentally by considering the material volume continuity 2  Literature Survey  13       ⋅− ⋅ = − γ γφ sin1 cos tan 1 r r                                                 (2.4) where r is the cutting ratio, or chip thickness ratio    ( )γφ φ − == cos sin 1 0 h h r                                                   (2.5) and h1 is the chip thickness. The velocity diagram is shown in Figure 2.1b. The cutting velocity, V0, is the velocity of the tool relative to the workpiece and is parallel to Fc. The chip velocity, Vc, is the velocity of the chip relative to the workpiece and is directed along the tool face. The shear velocity Vs is the velocity of chip relative to the workpiece; this component is directed along the shear plane.  ( ) ( ) 0 0 cos cos cos sin VV VV S C γφ γ γφ φ − = − =                                                 (2.6) 2.2.2 Slip Line Field Lee and Shaffer (1951) were the first researchers to apply engineering plasticity to solve the metal cutting problem; their slip line field is shown in Figure 2.2. The Lee and Shaffer model makes the following assumptions: 1) The work material is rigid-perfectly plastic; 2) The field is in a state of plain strain; 3) The cutting zone approximates a constant stress field; 4) No elastic contact is apparent, (hence the final boundary is stress free). 2  Literature Survey  14 Because the state of stress in ABC is uniform, a single Mohr circle diagram was used for its representation as shown in Figure 2.3. Since the maximum stress that can be withstood is k=τy, the radius of the circle is k. Further, since there is no force across AC then the Mohr circle passes through the origin. There are two possible circles satisfying these conditions, one in which all normal stress components are tensile and the other in which are compressive; the latter is the appropriate one. Based on the Mohr circle, the Lee and Shaffer solution gives the following expression for the shear angle 4 piγβφ =−+                                                             (2.7)  (a) Slip-line field R495 ,09 Shear Stress Normal Stress σB τB k β2β 2(φ−γ) p=k  (b) Mohr circle diagram Figure 2.2 Lee and Shaffer’s slip-line field solution  Oxley (1961, 1963) applied a simplified slip-line field to a thin zone model in metal cutting as shown in Figure 2.3. The deformation zone was assumed to be bounded by straight and parallel slip lines at an angle of φ to the direction of motion, and the point A to be a negligible distance from the free surface. The approximation allows one to consider that the stresses acting on the shear plane are uniform. Oxley used the modified Hencky relationships with a work-hardening term to describe the stress condition in the plastic zone as: 2  Literature Survey  15       = ∂ ∂ +− = ∂ ∂ ++ ∫ ∫ βψ αψ linealong2 linealong2 2 1 1 2 constds s kkp constds s kkp                          (2.8) where p is the hydrostatic stress, k is the yield stress, ψ is the slip line counter-clockwise rotation angle, ds1 and ds2 are elemental length along the slip lines α and β, respectively, and ψ is the angle between a tangent to the slip line α at any point and a reference axis. The method first involves finding two expressions for the hydrostatic stress at point B; one applying the modified Hencky relationships along the shear plane, the other considering the stress between the chip and tool. The assumed stress distribution is shown in Figure 2.3b. From these expressions, the relationship for the angle θ, between the resultant cutting force and the shear plane, is obtained as ( ) ( ) γβφθ γφ β γφφpiθ −+=       − − −       ′ ++−+= − and 2 2sin tan2 2cos1 42 1 tan 1 h h                    (2.9)  (a) Slip-line field  (b) Assumed stress distribution on the rake face Figure 2.3 Oxley’s thin zone model  Kudo (1965) suggested a slip-line field solution that included the possibility of curl by replacing the straight slip-lines of Lee and Shaffer with curved slip-lines (Figure 2.4). The suggested field satisfies the kinematic requirements of chip flow but the 2  Literature Survey  16 solution becomes inadmissible for machining without chip breaker, as static force equilibrium is not realized. Kudo (1965) later suggested another slip-line field that replaced the concave slip-line PR with a convex slip-line identical to the circular arc OP to satisfy static force equilibrium. Unfortunately, a convex slip-line suggests a decrease in normal stress on the rake face from the end of rake face contact to the tool tip; this of course is in contrast to expectation, (and with the experimental observations). h0 h1  (a) Kinematically inadmissible solution with curved boundary line of chip  (b) Kinematically admissible but statically inadmissible solution producing curled chip Figure 2.4 Kudo’s slip line field  Oxley’s later parallel zone model (Oxley and Hastings 1976, Oxley 1989) assumes that the deformation is concentrated in two zones as shown in Figure 2.5, a primary shear zone centered about AB (the nominal shear ‘plane’) and a secondary shear zone along the tool-chip interface. The actual shapes of the two zones may be approximately depicted as in Figure 2.5(a). Oxley’s model assumes that the primary and secondary shear zones are parallel-sided and of constant thickness, (as shown in Figure 2.5(b)). The maximum strain rate in the primary shear zone, along AB, is considered to be cVs/L, where c is a scale factor that depends upon the cutting conditions, L is the length of AB. The ratio of the thickness of the secondary shear zone to the chip thickness is represented by δ.  2  Literature Survey  17  (a) Actual extent of the deformation zone   (b) Simplified representation of the deformation zones Figure 2.5 Oxley’s parallel zone model  In the configuration shown in Figure 2.5, the slip-line in the primary shear zone along the direction AB is an α slip-line and the slip-line in the secondary shear zone along the chip face is a β slip-line. Assuming that the strain along AB is uniform, equal to one half the strain in the primary shear zone, and further assuming that the temperature and strain rate are uniform along AB, the shear stress along AB is calculated from the material model. This is used to obtain the shear force as well as the power dissipated in the primary shear zone. The partition of heat between the chip and the workpiece is obtained by using the results of Boothroyd (1965). Again assuming a uniform distribution of the normal stress along the rake face, the tool-chip contact length is obtained so that the moment of the normal force about the point B equals to the moment of the resultant force along the shear plane. The normal stress σn and the shear stress τint on the rake face are obtained from the corresponding forces and the contact length. The normal stress on the rake face can be obtained in a different manner by assuming that the α slip-line in the primary shear zone turns to meet the tool perpendicular to the rake face. Assuming that 2  Literature Survey  18 there is sticking friction at the tool-chip interface, the shear strength of the chip (kchip) at the average temperature of the tool chip interface should be equal to τint is taken to be correct shear plane angle. Dewhurst (1978) proposed a non-unique solution for machining with chip curling. Dewhurst noted that other investigators, notably Oxley and coworkers, have taken the view that an understanding of the process can only be attained by considering the distribution of temperature in the deformation zone as well as the influence of strain-rate and strain hardening of the deforming material. With this premise however the problem becomes analytically intractable so that theoretically unjustifiable assumptions about the mode of deformation are needed in order to obtain solutions. For example, in their investigation of the friction conditions along the tool face, Oxley and Hastings (1976) were forced to make assumptions about the thickness of the deformation zone and the contact length between the tool and the chip. Moreover, the known solutions ignore one of the most basic experimental facts that the chip emerges from the deforming zone with significant angular velocity. Deriving and analyzing the mixed stress and velocity boundary conditions in the deforming zone, Dewhurst was able to construct his slip-line field justifying these comments. The physical model and corresponding hodograph are shown in Figures 2.6a and 2.6b respectively. Analyzing this velocity hodograph, Dewhurst concluded that the slip-line field could satisfy a range of values of rake angles. Therefore only three conditions F1 = F2 = M =0 (the force components that maintain chip equilibrium) remain to determine the value of four parameters ψ, θ, η, and pA (which is the hydrostatic pressure). As a result, infinite numbers of solutions exist for every value of the rake angle and shear stress. Further, Dewhurst concluded that the major theorems 2  Literature Survey  19 of engineering plasticity couldn’t be applied to analyze machining due to the lack of defined boundary conditions. Also, if strain hardening is included, the solution is in principle poorer since the uniqueness theorem then does not apply to any steady-state process. Maity and Das (2001) applied Dewhurst’s model to consider both slipping and sticking friction zones on the tool rake face. This model is possible to obtain statically as well as kinematically admissible solutions to the Kudo’s slip line field (Kudo 1965) with the assumption of an elastic zone at the chip-tool interface. They showed that the stress boundary conditions can be satisfied by assuming an exponential distribution of normal stress in the elastic region. The method can introduce more realistic friction effect into the plastic slip line field.  (a) Slip-line field  (b) Hodograph Figure 2.6 Dewhurst model  Dewhurst and Collins (1978) also developed a matrix technique for solving a certain class of mixed boundary value problems of slip line field. The procedure is based on power series representation of the solution of governing equations first used by Ewing (1967), and a vector representation of slip-lines and a system of matrix operators developed by Collins (1968, 1970). A universal slip line model and its associated hodograph for restricted contact machining have been presented by Fang et. al. (2001). It has been shown that the previously developed slip line models (Johnson (1962), Lee and 2  Literature Survey  20 Shaffer (1951), Mechant (1945), Kudo (1965) and Shi and Ramalingam (1991)) are all special cases of the universal slip line field.  Some unknown parameters should be determined previously to use this model. This will lead non-unique solution for machining processes. A simple predictive model of orthogonal metal cutting was developed by Yellowley (1983, 1987). The model assumes that the rake face can be divided into two zones, in the first zone (AB), there is a plastic contact with close-to-sticking friction prevailing, in the second zone (BC), and an elastic sliding with an average friction angle (βs) is assumed (Figure 2.7). The boundary of the primary deformation zone (DEFA), is taken to be a direction of maximum shear stress, meeting the free surface at pi/4 and being parallel to the simple shear plane between E and F, then curving to meet the rake face at A at pi/2. The line BF represents, for the force equilibrium purposes, the boundary of the secondary deformation zone. The force equilibrium and moment equilibrium of the chip in Yellowley’s model are given in Figure 2.8.  Figure 2.7 Geometry of the basic model 2  Literature Survey  21    (a) Force equilibrium of chip L R B L/2 R/2 wk(1+p/2-2 )L wkL wk(1+p/2-2 )R FN (Normal force component in elastic zone) Tool wkR  (b) Moment equilibrium of chip Figure 2.8 Equilibrium of chip   The relationship of R, L and shear angle φ is given by Yellowley (1987) as ))(221(1 )()221()tan( LR LR s φpi φpiγβφ −+− −−+ =−+                                     (2.10) In which βs is the angle of friction in elastic contact zone on the rake face, R/L is the ratio of the height of the secondary deformation zone to the primary deformation zone. The model is based on the fundamental mechanics of the process, (it satisfies force equilibrium, moment equilibrium, the known tribology of the sticking and sliding zones and stress continuity on the rake face). 2.2.3 Upper Bound Method The upper bound theorem states that the rate of total energy associated with any kinematically admissible velocity field defines an upper bound to the actual rate of total energy required for the deformation (Bower 2010). Hence, for a given class of kinematically admissible velocity fields, the velocity field that minimizes the rate of total 2  Literature Survey  22 energy is the lowest upper bound, and therefore is nearest to the actual solution. Here, the kinematically admissible velocity field is used to denote a velocity field that satisfies the incompressibility requirement for a rigid-plastic material and the prescribed velocity boundary conditions. The best-known approach to machining using a similar methodology  is that due to Merchant (1945) hence leading to a shear angle equation in the form (2.11). 224 βγpiφ −+=                                                       (2.11) Rowe and Spick (1967) avoided the introduction of an elastic friction coefficient and were thus able to take a more conventional upper bound approach where the resultant force acting on the rake face is assumed to derive from plastic work. The resulting shear angle relationship is stated as ( ) 0sin2coscos 2 =⋅⋅−−⋅ φχγφγ m                             (2.12) where m is the ratio of shear stress on rake face over the shear yield stress of the chip material during cutting, χ is the ratio of uncut chip thickness and contact length on the rake face. DeChiffre (1977) in similar fashion, attempted to find a simpler expression by replacing the shear angle with the chip compression ratio and the friction coefficient by the chip contact length. This simplification yielded an expression that is equivalent to the one obtained by Rowe and Spick. Mittal and Juneja (1982) proposed a theoretical solution for the determination of shear angle for controlled contact orthogonal cutting based on Merchant’s minimum energy approach. Stephenson and Agapiou (1997) provided a detailed discussion about minimum work and uniqueness assumptions applied to the cutting problem where they conclude that minimum work is an acceptable line of attack. 2  Literature Survey  23 2.2.4 Material Separation In the traditional approach, machining models account only for the energy consumed in the primary and secondary deformation zones. The energy used to separate the chip from the metal just ahead of the cutting tool tip is almost always ignored as being small, (based on simple surface energy calculations, Shaw 1997), that relating to flow around and under the edge is also usually ignored but has been addressed by several authors, (Masuko 1953; Albrecht 1960; Palmer and Yeo 1963; and Johnson 1967).  Recently, Atkins (2003, 2005, and 2006) has contended that the fracture terms are important in any cutting analysis. Atkins argued that ductile fracture mechanics can be used to explain chip formation in ductile metals by incorporating fracture toughness, (as the specific work of surface creation) into the machining model. This approach adds a constant term in the force expression that gives rise to the “so called” size effect in force, (as opposed to material strength). This approach was also shown to account for the observed trends in shear angles with decrease in uncut chip thickness. Atkins has gone further and suggested cutting as a way to establish fracture toughness values of materials (2005). Atkins (2005) cited the work associated with the chip separation criterion in finite element simulations of machining to be orders of magnitude different from that associated with chemical surface energy or solid surface tension, and comparable to fracture toughness values. Following this approach then the cutting force can be formulated as follows, (the final constant term in the equation corresponds to the work of surface creation) ( ) Q wG Q wkh F cC + −⋅       = γφφ γ cossin cos0                                  (2.13) 2  Literature Survey  24 where Gc is the fracture toughness, Q is a friction correction factor given by ( ) ( )( )( )[ ]γφγβφβ −⋅−⋅−= coscos/sinsin1Q                            (2.14) Wyeth (2009) applied Atkins’s theory to investigate the orthogonal cutting of Nylon 66. It was shown that a better correlation was found with the inclusion of separation energy as proposed by Atkins (2003) with that of models did not consider the separation energy. Williams et al. (2010) and Patel et al. (2009) extended the fracture mechanics to study the case where the tool tip touches the crack tip and so obviate the need for cracks to precede the tool. Cutting analysis incorporating shear yielding on a slip plane in addition to fracture toughness Gc have been applied to cutting polymers. The results also show that the inclusion of the fracture energy Gc form a good base for describing data for cutting polymers. 2.2.5 Finite Element Modeling of Machining The application of Finite Element (FE) models of the machining process aids in both qualitative and quantitative understanding of metal cutting behavior. The main advantage in using the finite element method is that various material models and complex boundary conditions can be simulated. The information delivered by the finite element simulation is more detailed than that obtained from analytical or mechanistic models. The FE models can simultaneously provide: 1) Detailed information on chip formation, shear angle, flow stress and interaction with the cutting tool; 2) Strain and strain rate during machining; 3) The temperature distribution in the chip/tool/workpiece interface; 4) Cutting forces; 2  Literature Survey  25 5) The stress distribution within the tool itself. The difficulties of accurate FE modeling of cutting processes, even for conventional machining derive from the need to have detailed constitutive equations for the work material, and to have realistic contact conditions on the rake face. Clearly there will often be the need to iterate, and often effects such as strain gradients and history will be difficult to accommodate.  The researcher must also decide upon a suitable chip separation criteria and FE modeling will be time consuming, because of the remeshing process. The accuracy of the results depends largely upon the representation of workpiece and tool material properties. Several authors have examined the extent of these difficulties. Jaspers and Dautzenberg (2002), Shi and Liu (2004) and Sartkulvanich et al. (2004) studied the effects of different material constitutive models. Sartkulvanich et al. (2005) published a flow stress database, known as Material Database for Machining Simulation (MADAMS), established with inputs from several research groups. Guo et al. (2005) proposed an internal state variable plasticity-based approach to predict the strain rate history and temperature effects. Pantale et al. (2004) adopted the damage constitutive law in models allowing defining advanced simulations of the tool’s penetration into the workpiece and chip formation. Ozel and Zeren (2004, 2006) proposed an improved Oxley model to characterize work material flow stress and friction in the primary and secondary deformation zones around the cutting edge by utilizing orthogonal cutting tests. Iwata et. al. (1984) proposed an equivalent strain criterion for simulating the crack propagation at the tool-tip. Ng et. al. (2002) employed the conditional link element to simulate the tool- tip crack propagation. They showed importance of the failure and crack propagation criteria in FEM modeling of machining processes. 2  Literature Survey  26 2.3  Oblique Cutting Models Although the idealized orthogonal process is a close approximation to many actual machining operations, it is clear that many practical operations dictate the consideration of obliquity. The three-dimensional plastic flow in oblique cutting is considerably more complex than orthogonal cutting. Noteworthy attempts to extend the mechanics of orthogonal cutting to the oblique cutting process have been made by Merchant (1944), Stabler (1951), Shaw et al. (1952), Brown and Armarego (1964), Pal and Koenigsberger (1968), Zorev (1966), Spaans (1970), Morcos (1980), Lin et al. (1982), Hu et al. (1986), Rubenstein (1983), Lau and Rubenstein (1983), Vinuvinod and Jin (1995).  Most of these analyses attempt to predict three important variables- chip flow angle, shear angle and forces. These in turn are functions of tool geometry and machining conditions such as rake angle, inclination angle, coefficient of friction etc. There has been an ongoing debate on the equivalent rake angle that would influence the mechanics of oblique machining similar to the rake angle in orthogonal machining. Some consider the velocity rake to be significant (Kronenberg 1966), while some others consider the effective rake to be significant (Shaw et al. 1952), while a third group considers the normal rake to be significant (Brown and Armarego (1964), Armarego and Brown (1969), and Armarego (2000)). The thin shear zone model for “classical” oblique cutting is an extension of the orthogonal case using similar assumptions. To derive relations for forces FC, FT, and FR in terms of stress in the shear plane, the following assumptions were made (Armarego and Brown 1969) 1) The tool tip is sharp and no rubbing or ploughing forces act on the tip; 2) The stress distributions on the shear plane are uniform; 2  Literature Survey  27 3) The resultant force RTool acting on the chip at the shear plane is equal, opposite and collinear to the force acting on the chip at the rake face. As for orthogonal cutting the resultant force can be considered to act as two components in the shear plane (Fs and Ns) and two components on the rake face (Ff and N). The shear force Fs is inclined at the same direction of shear flow velocity (ηs) in the shear plane. Similarly, the friction Ff is at the same direction of chip flow angle (ηc) on the rake face. By resolving the resultant force in a plane perpendicular to the cutting edge and along the cutting edge, the following equations have been developed (Armarego 2000) ( ) ( ) ( ) ( ) ( ) ( )         ⋅+−+ ⋅−⋅− =         ⋅+−+ − ⋅ =         ⋅+−+ ⋅⋅+− = NcNNN NcNN N R NcNNN NN N T NcNNN NcNN N C iwkh F i wkh F iwkh F βηγβφ βηγβ φ βηγβφ γβ φ βηγβφ βηγβ φ 222 0 222 0 222 0 sintancos sintantancos sin sintancos sin cossin sintancos sintantancos sin                (2.15) The shear angle and friction angle are given by cN N c N c N i i r ηββ γη γη φ costantan, sin cos cos1 cos cos cos tan ⋅=       −       ⋅ =                  (2.16) where r=lc/l is chip length ratio, lc is the chip length, and l is the length of cut (Song 2006). The collinearity condition provides a means of relating the force and velocities in oblique cutting. The condition assumes the friction force on the rake face is collinear to the chip velocity direction and that the shear force in the shear plane is collinear to the 2  Literature Survey  28 shear velocity direction. When the collinearity conditions are satisfied, the following equations must be met to find the chip flow angle ηc (Armarego 2000) ( ) i i Nc N NN tansintan costan tan ⋅− ⋅ =+ γη γβφ                                     (2.17) 0h cnV ctV cV cη snV stV sV sη sF 0V TF CFRF  Figure 2.9 Force in oblique cutting with single edge tool  Predictive oblique cutting analyses have been proposed by Lin and Oxley (1972), Lin (1978), Lin et al. (1982) and Hu et al. (1986). They assumed that the flow in the plane normal to the cutting edge could be treated as plane strain deformation with the Oxley orthogonal cutting model being used to predict the chip geometry and the associated forces referred to this plane. To relate these values to the three-dimensional values, collinearity is assumed. The forces FC and FT can be determined from the 2  Literature Survey  29 orthogonal theory; based on Stabler’s rule, the radial force FR is then found as (Oxley 1989) ( ) ii FiiF F cN cNTcNC R costansinsin tancostansincossin +⋅⋅ ⋅−⋅⋅− = ηγ ηγηγ                 (2.18) Eqn. (2.18) must be satisfied in order that the resultant cutting force lies in the plane normal to the tool cutting face that contains the resultant frictional force acting in the chip direction. A model of oblique cutting for viscoplastic materials was developed by Moufki et al. (2000, 2004). The thermomechanical properties and the inertia effects were accounted for so as to describe the material flow in the primary shear zone. The results show that a good correlation with the experimental measurements. Some authors have also pursued an energy approach or traditional upper bound methods to oblique cutting (Usui et al. 1978; Usui and Hirota 1978; Tai et al. 1994; Chang 1998; Fuh and Change 1995; Stephenson and Wu 1988; Shamoto and Altintas 1999; Seethaler and Yellowley 1997 and Sedeh et al. 2002, 2003, 2004). Traditional upper bound analyses of oblique cutting generally make an assumption to calculate the friction area at the chip-tool interface. Seethaler and Yellowley (1997) proposed a methodology to calculate the area of cut on the rake face by obeying force equilibrium, which results in ( )eee es f A A βγφ β +− = cos sin                                            (2.19) where Af is the area of chip plastic deformation on the rake face, As is the area of shear plane. The effective rake angle γe and effective shear angle φe are measured in the plane containing the cutting direction and chip flow direction. The friction angle βe depends on 2  Literature Survey  30 the material properties and on the amount of work hardening present. In the absence of work-hardening and the presence of sticking friction, it can be written as (Seethaler and Yellowley 1997) e e γpi β 221 1 tan −+ =                                              (2.20) The optimized unknown parameters are the magnitude of chip velocity and chip flow angle ηc. Sedeh et al. (2002, 2003, 2004) extended Seethaler and Yellowley’s approach to sharp corner and flat face nose radius tools and introduced a robust approach for calculation of the uncut chip area. 2.4 Analysis of Practical Milling Tools The two most common milling processes found in production are face milling and peripheral milling (or end milling). Figure 2.10 and 2.11 show the difference between these two types. The symbols for face milling shown in Figure 2.10 are described as  Ft Tangential force R Tool radius Fth Thrust force a Axial depth of cut Fr Radial force St Feed per tooth Fz Axial force ψ Cutter rotation angle Fx Feed force ψ1 Entry angle Fy Normal force ψ2 Exit angle Tc Cutting torque ψs Swept angle   Ψe Chip flow angle  2  Literature Survey  31   Figure 2.10 Face milling geometry Figure 2.11 shows the workpiece geometry for end milling with the following symbols: i Helical angle D Tool diameter a Axial depth of cut d Radial depth of cut St Feed per tooth Ω Spindle speed   ψ Cutter rotation angle   Figure 2.11 End milling geometry 2  Literature Survey  32 Face milling generates a surface that is perpendicular to the axis of cutter rotation; in the usual case the cutter both enters and exits the workpiece with a finite chip thickness and the axial depth of cut is relatively low. End milling cutters have cutting edges located on both the end of face of the cutter and (usually in helical form) on the periphery of the cutter body; they usually have relatively high depths of cut and may often be used with little or no end/face cutting. Normal peripheral or end milling can be classified as up milling and down milling (see Figure 2.12). In up milling, the cutter enters with zero chip thickness and exits with a chip thickness which is dependent upon the width of cut and cutter radius. In down milling, the cutter exits with zero chip thickness. For small widths of cut this leads to very different force directions and the mechanics of cutting can be quite different, as one approaches widths close to the cutter diameter then the processes become similar.   (a) Up milling   (b) Down milling  Figure 2.12 Milling types  The earliest analysis of the kinematics of milling processes was carried out by Martellotti (1941). Martellotti showed that the true path of milling cutter path is trochoidial and the equations describing the tooth path are given by 2  Literature Survey  33 ( )    −= +±= ψ ψψ pi cos1 sin 2 Ry R SN x tf                                                   (2.21) where x and y are the coordinates of a point along the cutter path, Nf is the number of cutter teeth, the plus and minus signs apply to up and down milling, respectively. Martellotti showed that the tooth path is almost circular for light feeds, and the circular tooth path approximation may be invoked if the radius of the cutter is much larger than the feed per tooth. This approximation simplifies the analysis of the process, and in practice, the necessary conditions of the radius being larger than the feed per tooth is usually satisfied. Using this approximation the formulation of the chip thickness can be derived as (Martellotti 1941) ψsin⋅= tSh                                                         (2.22) 2.4.1 Modeling of Milling Forces The forces acting upon a single straight tooth in up and down milling are shown in Figure 2.13. It is customary to resolve the radial and tangential force components parallel to the feed direction (x) and perpendicular to the feed (y). In the simplest case it is assumed that the tangential component may be calculated by multiplying the instantaneous area of cut by the specific cutting pressure (K) and that the radial component will be proportional to the tangential component (the constant r is often used to denote the ratio of radial to tangential components). Some authors have assumed these parameters to be functions of chip thickness (Koenigsberger and Sabberwal 1961; Fu et al. 1984; Kapoor et al. 1998; Ranganath et al. 2007; Feng and Su 2001). Since the chip thickness in peripheral milling is generally very low then it would seem logical, as a first 2  Literature Survey  34 improvement on a linear model of cutting forces, to consider the edge forces, (nose and flank), which are known to constitute a significant proportion of total forces at such conditions (Zorev 1966; Spaans 1970; Yellowley 1985; Armarego 2000). For steady state cutting, the instantaneous tangential and radial forces acting on a straight single tooth- milling cutter are given by (Yellowley 1985, 1988)              +=       += t tr t tt S h rrKaSF S hKaSF * 21 * sin sin ψ ψ                                         (2.23) where r1 and r2 are the tangential to radial cutting forces ratios of the cutting and parasitic components, respectively. The parameters K, r1, r2 and h* are assumed to be constant for a tool-workpiece pair. The empirical parameters K and r1 contain the information regarding the work material’s shear yield stress, coefficient of friction, shear angle and the tool geometry, while parameters r2 and h* are related to the ratio of energy consumption in the nose area and the remainder, as well as the stress situation which is quite different from rake to clearance faces. The torque is related to the tangential force component through the radius of the cutter tC FRT ⋅=                                                       (2.24)  (a) Up milling d Vf RPM x y Ft Fr Fx Fy  (b) Down milling Figure 2.13 Milling forces 2  Literature Survey  35  To decompose the cutting forces Ft and Fr into the x and y directions, the instantaneous cutting forces in the x and y directions with a single cutting edge can be expressed as (a) Up milling ψψ ψψ sincos sincos try rtx FFF FFF −= +=                                             (2.25) (b) Down milling ψψ ψψ sincos cossin try trx FFF FFF += −=                                              (2.26)  The periodic, intermittent nature of the cutting force in milling lends itself to frequency domain analysis. The convention for representing cutting forces in milling operations can be written as (Yellowley 1985) ( ) ( ) ( )      ++=       ++=       ++= ∑ ∑ ∑ ∞ = ∞ = ∞ = 1 0 1 0 1 0 sincos sincos sincos k ykykyty k xkxkxtx k kktt kbkaaKaSF kbkaaKaSF kbkaaKaSF ψψ ψψ ψψ                              (2.27) The equivalent Fourier series for the cutting force Fx and Fy in helical end milling cutter can be expressed in the form of 2  Literature Survey  36 ( ) ( ) [ ]       ++= ++++=       ++= ++++= ∑ ∑∫ ∑ ∑∫ ∞ = ∞ = ∞ = ∞ = 1 0 111 1 00 1 0 111 1 00 sincos )](sin)(cos[{1 sincos ))](sin()(cos[{1 k ykyky yk k ykyy k xkxkx xk k xkxx kbhkahaha dkbkaa k F kbhkahaha dkbkaa k F ψψ ααψαψ ψψ ααψαψ α β α β              (2.28) where R ik R a tan , tan = ⋅ = β η α                                                (2.29) These equations indicate that it is possible to relate the Fourier coefficients of helical cutter to the Fourier coefficients of an equivalent straight cutter as follows ( ) )cos1(sin cos1sin,00 α α α α α α α α k k ak k bbh k k bk k a ahaah xkxk xk xkxk xkxx −−= −+==                                 (2.30) and ) cos1 sin( 2 ) cos1 sin( 2 ,00 α αα α αα k kbakbh b k k a k ahaah ykykyk ykykykyy − += − − ==                                  (2.31) It should be pointed out that the period of the fundamental frequency of the Fourier series is at the spindle rotation frequency. Gygax (1980) developed a convolution operator for the multi-tooth milling force analysis in the frequency domain. Yellowley et al. (1992), Seethaler (1997) have used frequency domain analysis of milling forces in order to arrive at efficient identification techniques for swept angles of cut and wear from in-situ measured cutting forces. 2  Literature Survey  37 Seethaler and Yellowley (1999), Zheng et al. (1997) and Wang and Huang (2004) applied frequency domain analysis to estimate the cutter runout in milling operations. 2.4.2 Modeling of Ploughing Forces  In practice no tool is perfectly sharp and the cutting edge always has a finite radius with corresponding forces acting at the cutting edge. In addition wear and deformation of the cutting edge will cause additional forces to exist in this region. Because these forces are not considered to contribute to the chip removal process they may be collectively referred to as the ploughing force. Since it is extremely difficult to measure the ploughing force directly, indirect methods have been used to estimate its magnitude, e.g. the extrapolation method (Zorev 1966). Figure 2.14 shows a scheme of forces acting on the rake face and flank surfaces of a free-cutting tool, the normal force N and the tangential force F, which together give the resultant force RTool,s, act on rake face OB. The normal force Ncf and the tangential force Fcf, which together give the resultant RTool,cf, act on the clearance surface OA.  Figure 2.14 Scheme of forces acting on the face and flank surface of the tool  2  Literature Survey  38 Force RTool,s and RTool,cf added together give the cutting  and thrust forces as cfTTST cfCCSC FFF FFF , , += +=                                                               (2.32) where FCS and FTS are the projections of the force RTool,s onto the vertical and horizontal force on the tool face; FC,cf and FT,cf are the projections of the force RTool,cf on the vertical and horizontal projections of the force acting on the clearance surface, or so called edge force. It is possible to find the tangential and normal forces on the clearance surface by extrapolating the experimentally obtained relationships for the vertical and horizontal projections of the cutting force with varying depths of cut. It is noted that the extrapolating methods require maintaining shear angle φ constant when varying uncut chip thickness to find edge forces. Endres et al. (1995) proposed a dual-mechanism approach to address the chip removal and edge ploughing mechanisms. The geometry local to the cutting edge and along the clearance face is shown in Figure 2.15a, which includes primary, secondary and tertiary shear zones. The geometric relations of the dual-mechanism approach can be visualized using a diagram similar to that of Merchant’s (1944) force circle diagram represented by Masuko (1953) and Albrecht (1960). The dual-mechanism approach is shown in Figure 2.15b and displays the forces as follows NFAKN fcf µ== ,                                          (2.33) and cfcfcficfcf NFVKN µ== ,                                         (2.34) where Kf and µ are the rake face force coefficient and friction, respectively. Kcf and µcf are the normal and friction coefficients on a clearance face; Vi is the interference volume. 2  Literature Survey  39  (a) Model diagram FsNs Rtool,s Ff N Fcf Ncf Rtool,cf FT Fc RTool  (b) The duel-mechanism force circle diagram Figure 2.15 Endres’ dual-mechanism approach  Shi and Ramalingam (1991) proposed a slip-line field for orthogonal cutting with chip breaker and flank wear. For a worn tool, the slip-line field includes a primary deformation zone with finite thickness; two secondary shear zones, one along the rake face and the other along the flank face; a predeformation zone; a curled chip and a flank face system. Fang (2003) presented a comprehensive slip-line model for machining with a round edge tool based upon the flat material flow theory (without stable built-up edge). In this model, the entire region of chip formation is composed of three shear zones (termed as the primary, secondary, and tertiary shear zones), three transition regions, and a material “pre-flow” region as shown in Figure 2.16. The tertiary shear zone is directly caused by the rounded edge of the tool. The round cutting edge BN is approximated by two straight chords SB and SN to simplify the mathematical formulation. Point S is the stagnation point of flow of material. The entire slip-line field is divided into 27 sub- regions; since the model accounts for the chip up-curling effect; the slip-lines are curved with different slip-line angles that introduce numerous model coefficients that need to be 2  Literature Survey  40 determined. In order to mathematically formulate the problem and solve for the coefficients, Dewhurst and Collins’ matrix technique is employed.  (a) Three major shear zones  (b) A total of 27 slip-line sub-regions Figure 2.16 The division of slip-line field (Fang 2003)  Ren and Altintas (2000) presented an analytical model to investigate the influence of a chamfer tool based on the dead zone model developed by Oxley (1989) and Zhang et al. (1991), which is shown in Figure 2.17. The plastic deformation zones divided into primary shear zone, secondary deformation zone where the chip moves over the regular rake face, and the chamfered edge zone where the metal is trapped by forming a dead metal zone. An extrusion model is applied to the trapped metal zone.  Figure 2.17 Ren and Altintas’s model (2000) 2  Literature Survey  41  Waldorf et al. (1998) developed a plane strain, slip-line plasticity model to predict the machining forces during orthogonal machining incorporating the effect of tool edge radius. The slip-line field is shown in Figure 2.18. In this model, a stable built-up region occurs and is shown shaded with point A at the point of material separation. The model made the assumption of straight chip formation and a parallel-sided shear zone to simplify the formulation. The line AB is an α slip-line and represents the shear angle inclined at angle φ to the uncut workpiece surface. The field below AB can be determined based on geometrical and friction considerations as (Waldorf et al. 1998) ( )ηριηδ φρpiξ η sinsin2sin 4 )(cos5.0 1 1 − − −+= −−= ⋅= m                                (2.35) m is the friction factor. The radius (R) of the circular fan field center at A is solved for using the following equation [ ] ηρ γpi ργpi sinsin2)2tan( sin2 24 tan 2 2 ×+      + +      += RRrR e             (2.36)  Figure 2.18 Schematic of Waldorf’s ploughing force model (1998) 2  Literature Survey  42 The forces in this model are determined from the forces acting along the shear plane (line AB), and the bottom of the build-up region (line CA). The stresses in the slip- line field are found based on the equilibrium conditions that require constkp =+ ψ2                                                 (2.37) along an α slip-line, and constkp =− ψ2                                                  (2.38) along a β slip-line where p is the hydrostatic pressure and ψ is the counter-clockwise angle through which the slip-line has rotated. Since no normal or tangential stresses act on the free surface, p is equal to k on that surface. The slip-line has rotated by -ξ on the shear plane, resulting in ( )ξ21+= kp                                                      (2.39) Along CA, the slip-lines have rotated by –(ξ+δ) resulting in ( )δξ 221 ++= kp                                                 (2.40) The shearing forces in the cutting (parallel to V0) and thrust (normal to V0) directions are determined from the shear angle φ and Eqn. (2.37) ( )[ ] ( )[ ]φφξφ φξφφ sinsin21 sin sin21cos sin 0 0 −+= ++= wkh F wkh F Ts Cs                                     (2.41) The forces along CA are found to be ( ) ( ) ( )[ ] ( ) ( ) ( )[ ]ηδφωηδφηδξ η ηδφωδξηδφη η +−⋅−+−⋅+++= +−⋅+++++−⋅= sin2coscos2sin221 sin sin2sin221cos2cos sin kwRF kwRF TP CP     (2.42) The total cutting and thrust forces can then be found 2  Literature Survey  43 TPTsT CPCsC FFF FFF += +=                                                  (2.43) Based on Connolly and Rubenstein’s (1968) cutting model, Manjunathaiah and Endres (2000) set up a simple geometrical and machining force model to include the edge radius effect by conducting a force balance on the lower boundary of the deformation zone, as shown in Figure 2.19. It employs a shear zone instead of a shear plane and includes the deformation region below the tool edge radius. This model calculates strain and strain rate, and in the force prediction it includes an explicit effect of the ploughing component due to edge radius. The force prediction, in the model, does not treat the chip formation force component and ploughing force component as independent of each other like many researchers (Masuko, 1953; Albrecht, 1960; Endres et al., 1995; Waldorf et al. 1995), but couples the two components through the shear angle φ. Therefore, an increase of the force, due to the increase in edge radius, is not completely attributed to the ploughing mechanism, but partly also because of an increased chip formation force arising from a smaller shear angle.  φpi /4 h0 δ ξ r e λ s A D B P Chip Tool Worpiece  Figure 2.19 Manjunathaiah and Endres’s geometric model of edge radius tool (2000) 2  Literature Survey  44 2.5 Conclusions The effects of ploughing or rubbing due to rounded cutting edges and worn tools have been shown to significantly affect process outputs such as part quality and cutting forces. The ploughing phenomenon has generated considerable research aimed at gaining a fundamental understanding of the mechanisms involved. The majority of the modeling efforts have concentrated on the application of slip line fields. The basic problem with most slip line field applications to metal cutting has been the need to either ignore elastic contact or to incorporate this in an approximate way with the plastic zone. The extension of fields around the nose of the tool is even more challenging since in this region one has a singularity due to separation, and the strain, strain rate and temperature distributions on each side of the boundary between chip and finished surface are difficult to estimate. An early attempt to model the nose region was proposed by Oxley (1989) and further developed by Zhang et al. (1991), and Ren and Altintas (2000) for the analysis of chamfer tools. The metal flow resulting from the field, originally developed for the analysis of polishing operations, leads to the necessity to allow a second “chip” to exit below the tool, (the field exit velocity is not that of the work). Waldorf et al. (1998) developed a later slip line field which accommodates the ploughing action at the tool edge. The geometry of the field includes a raised prow and a stable build-up of workpiece material adhered to the cutting edge which acts to divert the coming flow either into the chip or into the workpiece. The model requires extensive experiments to calibrate unknown parameters. Manjunathaiah and Endres (2000) extended Connolly and Rubenstein’s (1968) sharp tool model to the analysis of rounded edge tools. The geometry of the zone in this case is very approximate, (it is not a slip line field), and 2  Literature Survey  45 hence some of the assumptions regarding (normal) stress on the boundaries can be questioned. Not surprisingly then, (again), significant experimental data is required to calibrate the model. Given the previous comments, the author believes that there is value in examining the extension of a thick zone slip line field model which includes elastic contact forces to incorporate flow under the tool edge. Most of the oblique cutting models in the literature concentrated on force equilibrium, using orthogonal cutting as a base and very often by defining the equivalent parameters as a convenience in understanding this more complex process. Some researchers have also pursued an upper bound approach. There is a split between the more conventional UB approach and those that use the notion of apparent friction coefficient to give solution that parallels the Merchant solution of orthogonal cutting. If one wishes to obviate the need to use the apparent friction coefficient in the UB approach, one must use a force balance to obtain the area of cut on the rake face.  The disadvantage to the latter approach is the need to estimate normal stress on the tool rake face. It would seem that the increased complexity of oblique cutting could be used to advantage here through the balancing of shear stresses along the tool edge to achieve the same result with more realistic assumptions. Considerable progress has been made in the calculation of the forces acting upon helical milling cutters based on both the empirical and mechanistic approaches. The author believes that there is still scope to examine the important entry and exit processes and to examine the role that the non-straight cutting edge will exert through the added kinematic constraint on chip flow.   46 3  A New Approach to the Modeling of Oblique Cutting Processes  3.1 Introduction The majority of practical cutting tools have both non-straight cutting edges and non-planar rake faces. It is of great importance to build an oblique cutting model to be able to analyze such complex kinematically constrained tools (primarily milling, drilling, and tapping operations). This chapter describes a new upper-bound model that incorporates force equilibrium parallel to the cutting edge for the oblique cutting operations. In addition to the normal shear angle, two new fundamental variables RATIO (force-based variable) and SLIP (kinematic variable) are introduced to characterize the energy requirements of the oblique cutting processes. Calibration of the model for either real time identification purposes or for process planning/optimization requires experimental data but no shear angle data. This is very important for complex milling, drilling and tapping operations, since these operations will not yield geometry data for determining shear angle. The relationships among major parameters over a wide range of obliquity and normal rake angles are discussed. Finally, this chapter demonstrates the success of the model using existing experimental data. 3.2 Analysis Traditional Upper Bound analyses of oblique cutting utilize the normal shear angle, the angle of chip flow in the rake face and the apparent coefficient of friction on the rake face. The energy approach to be presented here utilizes force equilibrium to determine the friction area/force on the rake face; the approach was first suggested by 3  A New Approach to the Modeling of Oblique Cutting Processes  47 Seethaler and Yellowley (1997) and extended by Sedeh et al (2002, 2004). Fortunately the oblique cutting process allows force equilibrium to be applied directly in the direction parallel to the cutting edge, (in essence now comparing shearing forces on the shear and rake faces and so there is no need to make uncomfortable assumptions regarding rake face normal stress as is necessary in the simpler case of orthogonal cutting). The energy approach is framed in terms of the normal shear angle and the following two variables that derive directly from force and kinematic considerations: SLIP is the ratio of the edgewise shear velocity imparted to the chip on the shear plane parallel to the cutting edge versus the incoming velocity along the cutting edge.  It is expected that improving efficiency on the rake face will reduce SLIP and lead to higher chip velocities; a SLIP of unity implies that the chip flows perpendicular to the tool edge. RATIO is the ratio of the friction force on the rake face to the resultant shearing force on the shear plane. RATIO is a complicated variable that needs to be determined experimentally. RATIO is dependent upon material properties at the relevant conditions on each plane, as well as the stress distributions on these planes and the elastic contact conditions of the rake face.  s fst F F RATIO iV VSLIP == , sin0                                                (3.1) The power consumed in oblique cutting is given by csss VARATIOkVAkP ⋅⋅⋅+⋅⋅=                                               (3.2) where As is the area of the shear plane, Vs is the resultant shear velocity on the shear plane, Vc is the resultant chip velocity on the rake face. The shear area As is derived as i wh A N s cossin 0 ⋅ = φ                                                (3.3) 3  A New Approach to the Modeling of Oblique Cutting Processes  48 where w is the width of cut, h0 is the depth of cut. 0h cnV ctV cV cη snV stV sV sη sF 0V TF CFRF  Figure 3.1 The oblique cutting process  The resultant velocities and components on the two planes (see Figure 3.1) are given as follows:  ( ) ( ) ( ) iVSLIPV iVV iVSLIPViVV st NN N sn ct NN N cn sin, cos coscos sin1, cos sincos 0 0 0 0 ⋅= − ⋅ = ⋅−= − ⋅ = γφ γ γφ φ                    (3.4) where i is the inclination angle, φN is the normal shear angle, and γN is the normal rake angle. The power consumed in the process is assumed to be a minimum and an upper bound solution for the normal shear angle in terms of tool geometry, SLIP and RATIO obtained. In this case however it is necessary to be able to estimate either the area of contact on the rake face, or the RATIO directly. Rather than make an estimate based on 3  A New Approach to the Modeling of Oblique Cutting Processes  49 approximate normal stress as suggested by Seethaler and Yellowley (1997), one can simply balance force in the direction of the tool edge, this results in ccss AkAkF ηη sinsintan ⋅⋅=⋅=                                            (3.5) Thus RATIO = sinηs sinηc = VstVc VctVs = SLIP (1− SLIP) Vc Vs                                   (3.6)  where As and Ac are the area of plastic deformation on the shear plane and rake face, respectively and where ηs and ηc are the shear flow and chip angle angles, respectively. Given the relationship between forces and velocities derived in equation (3.6) then the power requirements of the process may be rewritten as follows: ))1(1( 2RATIO SLIP SLIPFVVFVFP sscfss − +=+= ⋅⋅                           (3.6)  The ratio of rake face work to shear plane work is then given by Rw = (1− SLIP) SLIP RATIO2                                          (3.7) It is well known that in orthogonal cutting the ratio of work is remarkably constant. The only influence being seen through normal rake angle (Zorev 1966). The upper-bound problem can be mathematically defined as the dual problem which seeks the smallest value of P(SLIP, φN). The problem was formulated as a constrained minimization problem as follows 10 24 0 :toSubject Minimize ≤≤ +≤≤ ⋅ ⋅ = SLIP γpi φ VV VV RATIO )P(SLIP, N N sct cst Nφ                                               (3.8)  3  A New Approach to the Modeling of Oblique Cutting Processes  50 where the constraint 24 0 NN γpi φ +≤≤  was suggested by Brown and Armarego (1964); the maximum normal shear angle ( 24 Nγpi + ) corresponds to the frictionless case. The nonlinear optimization was carried out using Maple (2010).The tolerance allowed on the objective function was 10-9, while the tolerance on the state variables (SLIP and φN) was 10-6. It would be advantageous, for real time applications, if it was possible to obtain a solution for the Upper Bound problem without resort to a general purpose non-linear optimization process. The major difficulty with seeking a closed form solution is that of the defining of variables in such a manner as to be able to define their relationship with the normal shear angle. In the model described here it is assumed that the SLIP is constant for one workpiece-tool pair but that RATIO will have a relationship with both normal rake angle and with the shear angle. (The data available reveals that RATIO varies relatively little with shear angle but one should still account for the non-zero nature of the derivative in the formulation). Since the lateral force balance is always an active constraint to the optimization then in this case the best manner of modeling RATIO through the examination of this constraint may be determined. At small values of inclination angle the resultant velocities on the shear plane and rake face are approximately equal to the normal components, thus in this case N N sn cn S c V V V V γ φ cos sin =≈                                                       (3.9) or N N SLIP SLIPRATIO γ φ cos sin )1( −≈                                            (3.10) 3  A New Approach to the Modeling of Oblique Cutting Processes  51 Then friction force Ff can be approximated as  N f i hwk SLIP SLIPF γcoscos)1( 0 ⋅ ⋅⋅ − ≈                                        (3.11) So it is expected that the rake face resultant force must be simply proportional to the uncut chip area. For large values of inclination angle i (i≥300), the situation is more complex, the relationship between Vc and Vs can be written as [ ] ( ) ( )[ ] ( )                     − ⋅ +⋅−       − ⋅ +⋅ =      2 2 2 2 2 cos coscos sin1 cos sincos sin NN N NN N s c iiSLIP i iSLIP V V γφ γ γφ φ                    (3.12) In this case there are large and constant tangential force components acting on both the shear plane and rake face, as a result the ratio of resultant velocities varies little and the assumption can be made as follows Constant V V s c ≈                                                      (3.13) Hence ConstantRATIO SLIP SLIP ≈ −1                                    (3.14) It is then assumed that RATIO remains approximately constant and that the derivative of RATIO with respect to normal shear angle in searching for a minimum value may be ignored. The rake face resultant force in this case can be formulated as N f i khwCONST SLIP SLIPF φsincos1 0 ⋅ ⋅⋅ ⋅⋅ − =                                  (3.15) 3  A New Approach to the Modeling of Oblique Cutting Processes  52 The validity of the approximations presented are easily seen by substituting the two approximations for resultant rake face force into the power equation, differentiating with respect to normal shear angle and comparing the shear angles at which a minimum power requirement is expected with the actual power curve. Figure 3.2 shows the results for a small and large inclination angle. In each case the overall power is shown together with the derivative of power with respect to normal shear angle calculated according to the two simplified approaches described. As expected the two approximations each work well in predicting the value of shear angle at which power is minimized provided they are used in the correct circumstances. It should also be noted that at small values of SLIP and RATIO there is little difference between the two results, (since rake face work is small).  Figure 3.2 Power and derivative with different assumptions regarding the formulation of rake face force  3  A New Approach to the Modeling of Oblique Cutting Processes  53 3.3 Special Cases  The boundaries of oblique cutting are reached when SLIP is zero (and RATIO is zero) and when SLIP is unity, (and RATIO is infinite). It is instructive to examine these boundaries to ascertain the range of behaviours to be expected from oblique cutting. 3.3.1 Zero SLIP When SLIP is zero, then to satisfy force equilibrium, there can be no force on the rake face parallel to the tool edge. The velocity of the chip in that direction is V0 sin(i) and hence non-zero for any degree of obliquity. It must be concluded that the RATIO is zero and there is no resultant friction force on the rake face when SLIP is zero. Given the preceding conclusions from force equilibrium along the tool edge, the final solution is trivial, being simply equivalent to minimising the shear energy in orthogonal cutting, (rake face power is zero).  The complete solution in this case is given by: 2 2,0,0 piγφ =−== NNRATIOSLIP                                         (3.16) The chip flow angle in the rake face is given as ( )       −⋅ = − N NN c i φ γφη sin costan tan 1                                             (3.17) Substituting Eqn. (3.16) into Eqn. (3.17), then NNN φpiγφ −=− 2/                                                      (3.18) and ii N N c =      = − φ φη sin sin tantan 1                                           (3.19) 3  A New Approach to the Modeling of Oblique Cutting Processes  54 Therefore the Stabler solution (Stabler 1951) is always valid for this special case, (this has been indicated by other authors (Seethaler and Yellowley 1997, Sedeh et al. 2002) as friction approaches zero but here the result and the reasons for the result are easily proven once force equilibrium is applied). 3.3.2 SLIP Approaching Unity In the case where SLIP approaches unity then, by definition, the chip must flow perpendicular to the cutting edge, and from force equilibrium one must then conclude that the RATIO approaches infinity as the SLIP approaches unity. In the normal case of 1<      sn st V V , one may then approximate the power as follows ( ) ( ) ( ) N NN N NN N NN N N sn st snC N iVwh V RATIOVwh V VVRATIOV i wh P γ γφ φ γφ γ γφ φ φ φ cos cossin sin2 1 cos cos cos sin sin 5.01 sincos 2 00 000 2 0 −⋅⋅ +       − ⋅ +⋅ − ⋅ =                       ++⋅ ⋅ =                           (3.20) The first term in equation (3.20) represents the power requirements of the equivalent orthogonal process; the second term is much smaller and is generally decreasing with increasing φN in the region where the orthogonal model shows a minimum energy condition. One thus expects the minimum energy to be found at slightly higher values of normal shear angle than that found in the equivalent orthogonal case. Figure 3.3 shows the variation of the normal shear angle with respect to inclination angle for normal rake angles of 00, +200 and -200, respectively. It is seen that the shear angle in oblique cutting, (even for large values of inclination), takes on values that are only 3  A New Approach to the Modeling of Oblique Cutting Processes  55 slightly higher than those for orthogonal cutting. This finding is consistent with the discussion of equation (3.20) and very useful in establishing the expected trends between the practical variables. 0 10 20 30 40 50 60 0 10 20 30 40 50 60   Oblique Cutting γN=20 0  Orthogonal Cutting γN=20 0  Oblique Cutting γN=0 0  Orthogonal Cutting γN=0 0  Oblique Cutting γN=-20 0  Orthogonal Cutting γN=-20 0 N o rm a l S he a r An gl e  φ N 0 Inclination Angle i0  Figure 3.3 Normal shear angle versus inclination angle (RATIO=1)  3.4 Relationship between the Major Parameters 3.4.1 SLIP and RATIO There is a close relationship between the RATIO and amount of SLIP; this occurs as a results of force equilibrium where the degree of SLIP is related to the force available on the rake face to balance the parallel component on the shear plane causing the slip. 3  A New Approach to the Modeling of Oblique Cutting Processes  56 The relationship is reinforced by the fact that the optimal value of shear angle is related to the RATIO, (and hence SLIP), also. Figure 3.4 demonstrates just how little variation there is in the relationship between these variables for large changes in tool geometry. 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0  i=600,γN=-30 0  i=450,γN=-30 0  i=300,γN=-30 0  i=150,γN=-30 0  i=600,γN=0 0  i=450,γN=0 0  i=300,γN=0 0  i=150,γN=0 0  i=600,γN=+30 0  i=450,γN=+30 0  i=300,γN=+30 0  i=150,γN=+30 0  Fitted Curve for γN=-30 0  Fitted Curve for γN=0 0  Fitted Curve for γN=+30 0 SL IP RATIO  Figure 3.4 The relationship between SLIP and RATIO with varying inclination and normal shear angles  3.4.2 SLIP and Angle of Chip Flow on the Rake Face The angle of chip flow in the rake face is given by  3  A New Approach to the Modeling of Oblique Cutting Processes  57 ( ) ( )NN Ncn ct c iV iVSLIP V V γφ φη − ⋅ ⋅− == cos sincos sin1 tan 0 0                                          (3.21) thus ( ) ( ) N NNc SLIP i φ γφη sin cos1 tan tan −− =                                             (3.22)  It can be expected that the ratio of the angle tangents will be approximately constant as inclination angle changes (normal shear angle varies little), and to be determined almost uniquely by the SLIP. Fortunately the influence of normal rake angle is also small over the practical range. Figure 3.5 shows the relationship that result over the entire practical range (including a 60 degree range of both normal rake angle and inclination angle). The fitted curve is given by  2 53039.047799.000928.1       −      −= ii SLIP cc ηη                            (3.23)  and is accurate enough for most practical purposes. (The relationships for individual normal rake angles have been calculated where accuracy is essential; this is only necessary in exceptional circumstances or at extreme values of rake and inclination angle).  3  A New Approach to the Modeling of Oblique Cutting Processes  58 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0     i=600,γN=-30 0 ,    i=600,γN=-15 0 ,   i=600,γN=0 0 ,   i=600,γN=15 0 ,    i=600,γN=30 0   i=450,γN=-30 0 ,    i=450,γN=-15 0 ,   i=450,γN=0 0 ,   i=450,γN=15 0 ,   i=450,γN=30 0  i=300,γN=-30 0 ,   i=300,γN=-15 0 ,  i=300,γN=0 0 ,  i=300,γN=15 0 ,   i=300,γN=30 0  i=150,γN=-30 0 ,   i=150,γN=-15 0 ,  i=150,γN=0 0 ,  i=150,γN=15 0 ,   i=150,γN=30 0  i=100,γN=-30 0 ,   i=100,γN=-15 0 ,  i=100,γN=0 0 ,  i=100,γN=15 0 ,   i=100,γN=30 0  Fitted Curve of SLIP versus ηc/i SL IP η c /i   Figure 3.5 SLIP as a function of normalized flow angle   3.5 Model Calibration from Measurement of Cutting Forces To validate the model described in this paper, SLIP, RATIO, normal shear angle, φN, and shear yield stress, k, need to be obtained from measured values of cutting force. 3  A New Approach to the Modeling of Oblique Cutting Processes  59 Cutting forces are usually measured with a three-axis dynamometer that provides magnitude and direction of the resultant cutting force on the work piece. By resolving the cutting force into the plane of the rake face, the chip flow direction on the rake face, ηc, is obtained.  One particularly appealing feature of the model presented here is that if the chip flow angle, ηc, with the inclination angle, i, is normalized, then a unique relationship between ηc/i and SLIP can be obtained.  This relationship is shown in Figure 3.5. Once SLIP has been found, the shear angle can be derived as follows ( ) ( ) ( ) ( )1sintantan 1costan tan cos sincos sin1 tan 0 0 −⋅⋅+ −⋅⋅ =∴ − ⋅ ⋅− == SLIPi SLIPi iV iVSLIP V V Nc N N NN Ncn ct c γη γφ γφ φη                            (3.24)  Subsequently, RATIO is obtained from the force balance in the tangential flow direction sct cst c s VV VV RATIO == η η sin sin                                               (3.25) Finally, the shear yield stress is found from the power consuming force, FC ( )css C VRATIOVA VFk . 0 + =                                                (3.26) 3  A New Approach to the Modeling of Oblique Cutting Processes  60 3.6 Comparison with Existing Experimental Tests The model is intended to be used in the analysis of practical processes, particularly milling.  Some approximations are necessary to allow a basic set of data to be used for calibration then the resulting equations reused to cover a variety of applications over a wide range of cutting conditions. At the same time the practical applications envisaged usually allow some tolerance for reasonable approximations (tool wear and end condition may influence the forces at small chip thickness quite significantly). The assumptions made are stated as follows: a) The shear yield stress on the shear plane may be assumed constant for one work/tool pair over a reasonable range of cutting conditions and tool geometry. b) SLIP may be assumed constant for one work tool pair with a constant value of normal rake angle over a reasonable range of cutting conditions. This may be seen when examining typical chip flow data where the normalized chip flow angle is constant as the inclination angle changes. The behavior depicted in Figures 3.5 and 3.6 implies almost constant SLIP with only small changes in SLIP resulting from large changes in normal rake angle. Since there are only small changes in normal shear angle, one also expects relatively small changes in RATIO as shown in Figure 3.4. c) Edge forces in particular would defeat these assumptions if small chip thicknesses were considered during the calibration stage with no accounting made of their presence. The data examined here will then utilize large values of chip thickness so that the edge forces will not play a large role. Clearly in milling operations methods must be devised to compensate for these components. 3  A New Approach to the Modeling of Oblique Cutting Processes  61 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 45 50    Stabler (1951)  Russell and Brown (1966)  Present Model Ch ip - flo w  A n gl e η c 0 Rake Angle γN 0   Figure 3.6 The influence of normal rake angle on chip flow angle  Table 3.1 and 3.2 show values of SLIP and RATIO calculated from previously published data ( Brown and Armarego (1964) for SAE 1008 steel with a normal rake angle of 20 degrees and Pal and Koenigsberger for Aluminium Alloy and a normal rake angle of 10 degrees). It is seen that there is a fair amount of scatter in the data, this is specially evident in the calculated values for low inclination angles. This occurs as a result of the relatively small velocity vector along the edge. At the same time of course any errors introduced will lead to only small changes in force at the same conditions, (and for the same reason). 3  A New Approach to the Modeling of Oblique Cutting Processes  62 Table 3.1 SLIP and RATIO from experimental data (Brown and Armarego (1964)) h0 (mm) i(deg) FC (N) FT (N) FR (N) ηc (deg) SLIP RATIO k (MPa) 10 1112.1 609.4 112.1 5.0 0.6083 1.1296 601.1 20 1112.0 609.4 161.0 13.6 0.4185 0.6054 795.6 30 1156.6 589.4 224.2 22.3 0.3461 0.4711 876.7   0.2032 40 1223.3 645.0 302.5 29.2 0.3742 0.5736 837.7  Table 3.2 SLIP and RATIO from experimental data (Pal and Koenigsberger (1968)) i (deg) FC (N) FT (N) FR (N) ηc (deg) SLIP RATIO k (MPa) 10 977.25 555.12 62.83 8.6 0.1883 0.1776 495.6 20 966.18 499.25 151.18 16.0 0.2656 0.2813 454.9 30 971.72 482.83 194.02 26.5 0.1632 0.1661 489.7 40 972.79 420.89 273.10 35.9 0.1478 0.1649 483.2 50 976.71 428.39 330.75 43.0 0.2112 0.2815 426.2 Table 3.3 shows the average values of SLIP and RATIO from published data (Brown and Armarego (1964), Russell and Brown (1966), Moufki et al. (2000), and Pal and Koenigsberger (1968)) for different materials and rake angles; the average values of SLIP are then used to recalculate the chip flow angle. The results are shown in Figures 3.7, 3.8 and 3.9 where it is seen that the influence of averaging on the chip flow angle is negligible. Table 3.3 Average values of RATIO and SLIP extracted from experimental data Material Experimental Data Rake Angle γN (deg)  SLIP RATIO SAE 1008 Steel Brown and Armarego (1964) 20 0.4368 0.6955 15 0.1628 0.1580 AA65S-T6 Aluminum  Russell and Brown (1966) 30 0.2573 0.3426 AISI 4142 Steel Moufki et al. (2000) 0 0.2694 0.2528 Aluminum Alloy Pal and Koenigsberger (1968) 10 0.195 0.234  3  A New Approach to the Modeling of Oblique Cutting Processes  63 0 10 20 30 40 0 10 20 30 40    Present γN=25 0  Moufki et al. (2000) γN=25 0  Present γN=-25 0  Moufki et al. (2000) γN=-25 0  Present γN=0 0  Moufki et al. (2000) γN=0 0  Stabler Rule (1951) η c =i Ch ip  Fl o w  A n gl e η c 0 Inclination Angle i0  Figure 3.7 Comparison of the predicted chip flow angle with experimental results (Moufki et al. (2000))  0 10 20 30 40 50 0 10 20 30 40 50   Present Model γN=10 0  Pal and Koenigsberger (1968)  Stabler Rule (1951) η c =η Ch ip  Fl o w  An gl e  η c 0 Inclination Angle i0  Figure 3.8 Comparison of the predicted chip flow angle with experimental results (Pal and Koenigsberger (1968)) 3  A New Approach to the Modeling of Oblique Cutting Processes  64 0 10 20 30 40 0 5 10 15 20 25 30 35 40 45 50    Present Model γN=20 0  Present Model γN=30 0  Present Model γN=15 0  Present Model γN=0 0  Experiment γN=20 0  (Brown and Armarego (1964))  Experiment γN=10 0  (Brown and Armarego (1964))  Experiment γN=30 0  (Russell and Brown (1966))  Experiment γN=15 0  (Russell and Brown (1966))  Experiment γN=0 0  (Russell and Brown (1966))  Experiment γN=0 0  (Zorev (1966)) Ch ip  Fl o w  A n gl e η c 0 Angle of Inclination i0  Figure 3.9 Comparison of the predicted chip flow angle at various normal rake angles with experimental data from several investigators  3.7 Conclusions This chapter has described the development of a new upper bound model for oblique cutting. The analysis replaces the normal variables of chip flow angle and apparent coefficient of friction with basic variables relating to the kinematics (SLIP) and forces in the two main deformation zones, (RATIO). The relationship between SLIP and RATIO is easily found through force equilibrium parallel to the tool edge thus avoiding any assumptions regarding rake face stress or the ratio of stresses on the shear plane. The 3  A New Approach to the Modeling of Oblique Cutting Processes  65 model avoids the need for shear angle data, relying instead on experimental force data for calibration and an internal optimization routine to find the normal shear angle. The decision to avoid the necessity of having shear angle data is driven by the realization that the normal shear angle is difficult to measure when tools have obliquity and non-straight cutting edges. The influence of constraint and obliquity on the expected value of normal shear angle along the tool edge is assessed. The model allows the prediction of chip flow angle and in this regard it is proved that the Stabler’s law (1951) must result when RATIO is zero. This chapter also demonstrates the use of the model with the simplifying assumptions of constant SLIP, in the prediction of chip flow angle. Good agreement with measured chip flow angle from the published data is demonstrated.    66 4  The Modeling of Tool Edge Ploughing Forces in Milling Operations  4.1 Introduction As mentioned previously, the milling process exhibits small average values of chip thickness and, in the case of peripheral milling, a chip thickness that goes to zero at either entry or exit. It is likely then that one must examine both the influence of edge effects during cutting and the process of entering and exit where one may encounter elastic contact and ploughing as well as cutting. It is to be expected that the previous history of cutting or rubbing will influence the plastic field and forces, hence up milling and down milling should not yield similar results in the area where ploughing and cutting are to be expected, (based on chip thickness alone). This chapter describes two simple models to examine the entry and exit stages of milling. The models are approximated and intended to provide a better understanding of the shape and magnitude of the forces during these periods. This chapter also introduces a model to estimate the ploughing components during actual cutting; the model extends a previous slip-line field to include flow under the tool edge. And although phrased as an Upper Bound approach it preserves both force and moment equilibrium of the chip. The author will first introduce the basic theoretical analysis of both the transient rubbing/cutting process and that of cutting with a radiused edge before making comparison with experiment and finally attempting to quantify the conditions for the rubbing cutting transition. 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  67 4.2 Predictive Model for Orthogonal Cutting to Include Flow below the Cutting Edge This section deals with the development of an upper-bound method coupled with a simple predictive model proposed by Yellowley (1987) to analyze flow around and under the tool edge. Optimization techniques are applied to optimize the velocity field and simple equilibrium is used to obtain the normal and tangential forces acting on the chamfer land.  The chip force and moment equilibrium are satisfied and a realistic elastic/plastic rake face contact situation is incorporated in this approach. 4.2.1 Upper-Bound Formulation Upper-bound models for orthogonal cutting have been examined by DeChiffre (1977), and Rowe and Spick (1967). Perhaps the best-known approach which (essentially) results in an upper-bound is that due to Merchant (1945). In this section, a new upper bound solution is proposed; the physical model and hodograph are shown in Figures 4.1 and 4.2, respectively.  Figure 4.1 Proposed slip-line field for orthogonal cutting chamfer tools 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  68   η φ1 φ ξ ξ λ α pi/2 −φ δ pi/2 −γ 2 pi/2 −γ1 γ2 V0 VEF V1,6 V2,1 V4,2 V4,3 V3,5 V3,0 V6,5 V5,1 0 6 1 24 35  Figure 4.2 Hodograph of the proposed slip-line field  The chamfer land length (EF) is expressed as Lchmf, and the rake angles for the chamfer land and tool-interface are stated as γ1 and γ2, respectively. The flow is assumed to occur without a built-up edge or a stable stagnation zone. The uncut chip thickness is h0, the chip separation point defined by η and φ1, respectively, the penetration depth under the chamfer land is assumed to be a2. The shear plane angle is φ. The upper part of this field is the one proposed by Yellowley (1987). The lower part of the slip-line field is an extension of the upper part which reaches the chamfer land, and is consistent with the experimental observation (Challen, Oxley and Doyle 1983; Challen and Oxley 1983 and Kopalinsky and Oxley 1995). Based on force equilibrium, the relationship between R, L and φ is given by Yellowley (1987) as ))(221(1 )()221()tan( 2 LR LR s φpi φpiγβφ −+− −−+ =−+                                  (4.1) and based upon moment equilibrium of the chip [ ] 0)/(1)22/1()2/()/( 22 =+⋅−+⋅−⋅+⋅ LRkLRkL zFN φpi                   (4.2) 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  69 in which k is the shear yield stress of work material, βs is the angle of friction in elastic contact zone on the rake face, FN and z are the normal component of force acting upon elastic region of the rake face contact and distance between the plastic point B and FN respectively, (see Figure 4.3). L R B L/2 R/2 wk(1+p/2-2 )L wkL wk(1+p/2-2 )R FN (Normal force component in elastic zone) Tool wkR  Figure 4.3 Moment equilibrium of chip  Assuming a normal stress distribution in the elastic region given by following expression nxconst ⋅=σ                                              (4.3) where x is the distance from the end of the chip-tool contact. Then Eqn. (4.2) is rewritten in the form [ ] 0)/(1)22/1()2/()/( 2 1 2 2 =+⋅−+⋅−⋅++ + ⋅ LRkLRk n n L F B N φpi σ               (4.4) in which σB is the normal stress at point B. Based on the proposed slip-line field, the geometry relationships can be written as follows: 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  70 ( ) , sin )sin( ,2/,)cos( )sin( tan sin , cos sincos , sin , ,sin),/(tan )cos(cos)/(sin,cos 2 2 21 1 21 2 0 01 1 2 0 1 ε γφ εφpiδ γφ γφ ε φγ φφ λ φλ φγφγ −⋅ =+−=      −⋅− −⋅ = + = ⋅−+−⋅ === =⋅−== −⋅+ == − − FG CF FG FG CD cf FGAGCG AC chmfcf L L LR L aa L LhaR LLLRL LLLhaRL LR h LLa   (4.5) In the previous slip-line field analysis, the angle η is assumed to be friction angle (Shi and Ramalingam 1991; Waldorf et al. 1998 and Fang 2003), which is needed to be given, here there is no need to make an assumption, and the angle η will be determined by the upper-bound optimization method. The formulation of angle η is stated as         ⋅+−⋅+⋅ = − 1211 21 cot)(sincostan φγδη aaLL a chmfCF                      (4.6) and the angle ξ is given as         + ⋅−⋅+ = − 2 1211 coscot)(tan aa Laa cf CF δφξ                         (4.7) Finally the lengths of FD, BG and EF are given as RL aa LLL BG cf FDchmfEF = + == , cos , 2 ξ                                 (4.8) It should be noted that the above geometry relation is based on the angles δ and ξ remaining positive; once the angles δ or ξ change the sign, one needs to keep the geometry length positive. In order to obtain the admissible slip-line field, the same idea is applicable to the velocity relationships. Based on the hodograph in Figure 4.2, the bottom part velocities of V6,0 and V1,6 are given as 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  71 )cos( sin ,)cos( cos 1 0 0,6 1 10 6,1 ηγ η ηγ γ − = − = VVVV                           (4.9) in which V0 is the cutting velocity. The transition part velocities V5,1, V6,1 and V3,5 are given as )cos( cos)cos( ,)cos( )sin( ,)cos( )cos( 2 20211,5 5,3 1 16,1 5,6 1 6,1 1,5 δγ γγφ φξ ηφ ξφ ηξ + −− = + +⋅ = + −⋅ = VV V V V V V  (4.10) The upper part chip velocity Vc, shear velocity Vs, and velocities V3,1, V4,3, V4,2 and V2,3 are written as )sin( )sin(),sin(sin )sin( sin )cos( sin)sin( )sin( sin),cos(sin )cos( sin)sin( 2 23,2 2,421,301,2 2 3,2 2 011,5 3,4,3 2 3,2 3,42,303,2 2 011,5 .3 γλφ γφγφφ γλφ λ δγ δδφ γλφ λγφφ δγ δδφ −+ − =−+== −+ + + −+ =+= −+ =−−= + −+ = V VVVVV VVV VVV V VVVV VV V s OC O O              (4.11) 4.2.2 Optimization Procedure Having obtained the geometry of an upper bound field and the magnitudes of velocity discontinuities from the associated hodograph, power consumption PT can be calculated as Celasticf CFFDOFGOEFDE cdCGAGsACCT VF VLVLVLVLVL VLVLVLVLwkVFP ⋅+ ⋅+⋅+⋅+⋅+⋅+ ⋅+⋅+⋅+⋅== , 5,35,6,3,66,1 1,53,22,40 ) (         (4.12) where w is the width of cutting, Ff,elastic is the elastic friction force acting on the tool-chip interface, which is given as 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  72 ( ) ( )[ ] ( ) ( )[ ] s elasticf RpLRLpk RLpRpLkF βγφφ γφφ tan}sinsincos cossincos{ 2 2, ⋅⋅⋅⋅−−⋅−⋅+ ⋅⋅−⋅+⋅⋅−=              (4.13) in which       −+= φpi 2 2 1p                                                    (4.14)  For each elastic coefficient of friction the shear angle is incremented and the values of R and L are determined from the model, this continues until moment equilibrium is satisfied (Yellowley 1987). There is then a unique solution to the upper part of the field for each elastic coefficient of friction, the energy minimization in essence determines only the position of point D in Figure 4.1, which is depended upon the parameters of a2 and φ1, once a2 and φ1 are determined, all the slip-line field and associated hodograph are determined.  Thus, the upper-bound problem can be mathematically defined as the dual one which seeks the smallest value of ),( 12 φaPT  as 1120 1 12 sincoscot)sin( 4 0:toSubject ),(Minimize γδφφ φφ piφ φ chmfCF T LLaLh aP +≤⋅+⋅− ≤ ≤<         (4.15a-d)  Eqn. (4.15d) is added to guarantee the position of point D allows the construction of the external slip-line field (ABCDE) with a convex geometry. 4.2.3 Force Prediction It has been observed by many researchers that flank wear does not affect the shear angle (Smithy et al. 2001; Song 2006; Spaans 1970; and Huang and Liang 2005); 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  73 therefore the superposition of force due to force flank wear and shear forces is widely accepted. The forces acting on the elastic region at the tool-interface (shown in Figure 4.4) can be given as φ+βs-γ2 βsφ-γ 2 wk[L-(1+pi/2-2φ)R] wk[(1+pi/2-2φ)L-R] FC,elastic FT,elastic R'tool Rtool  Figure 4.4 Force equilibrium on the tool-rake face  ( ) ( )[ ] ( ) ( )[ ]φφ φφ sincos sincos , , ⋅⋅−−⋅−⋅= ⋅−⋅+⋅⋅−= RpLRLpwkF RLpRpLwkF elasticT elasticC                               (4.16) where ( )φpi 221 −+=p                                                         (4.17) FC,elastic and FT,elastic are the tool-interface elastic forces on the cutting and thrust directions, respectively. The normal stress acting the plastic region FG based on the slip- line field can be stated as       −+= 2, 22 1 γpiσ GFN                                                    (4.18) The forces acting on the cutting material by tool are shown in Figure 4.5. The decomposed forces at the region of GF and chamfered land EF to the cutting and thrust directions can be written as 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  74 22, 22, sincos cossin γγ γγ ⋅−⋅= ⋅+⋅= GFGFGFT GFGFGFC NFF NFF                                          (4.19) at the contact length of GF, and 11, 11, sincos cossin γγ γγ ⋅+⋅−= ⋅+⋅= EFGFEFT EFEFEFC NFF NFF                                         (4.20) at the chamfered land EF. Based on the force equilibrium, one can obtain the forces acting on the chamfered land as 1 1,, cos sin γ γ⋅−−− = = EFGFCelasticCC EF EFEF wkLFFF N wkLF                             (4.21) Once the forces acting on the chamfered land are known, the thrust force can be given as EFTGFTelasticTT FFFF ,,, ++=                                                (4.22)    Figure 4.5 Force equilibrium for chamfered tool 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  75 4.2.4 Adapting the Predictive Model for Radiused Edge Tools The effective rake angle of the tool is a very important parameter especially for honed tools. Based on the work of Manjunathaiah and Endres (2000), the actual rake angle in the presence of an edge radius or chamfer is quite different from the nominal rake angle  γ ground on the tool. The actual rake angle henceforth referred to as the effective rake angle γavg, can be computed from the relative size of the undeformed chip thickness to the tool edge radius h0/re. The expressions (Manjunathaiah and Endres 2000) given in Eqn. (4.23) have been used in this thesis to compute the effective rake angle. ( ) ( ) ( ) ( )γ θ θγγ γ θ θγγγγ sin1 cos1/ sin//2 tan sin1 cos1/ sinsectan1/ tan 0 0 001 2 0 0 01 2 +≤⋅         +−⋅ +⋅⋅⋅− −== +>⋅      +−⋅ +−−⋅ == − − e e ee ave e e e ave rhcfor rhc rhcrhc rhcfor rhc rhc     (4.23) Typically it is assumed that the contact length c at the tool-interface is twice the undeformed chip thickness (Manjunathaiah and Enres 2000). Hence, the predictive slip- line model incorporated with the round edge tool is shown in Figure 4.6.  Figure 4.6 Upper bound field for radiused edge 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  76 4.2.5 Case Studies of Predictive Cutting Model 4.2.5.1 Optimized Upper Bound Geometries Having assumed that the chamfer length does not influence stresses on the rake face, it is instructive to examine the shape of the flow zones as chamfer width is increased. Consider the case of γ1=-700, γ2=00, and βs=400, the typical influence of chamfer length with Lchmf/h0=0.1, 0.3 and 0.5 are plotted in Figures 4.7(a), (b) and (c), respectively. It can be seen that large chamfers exhibit a lower boundary that is almost parallel to the incoming velocity, (indicating an almost stagnant zone) and an apparent increase in the curvature of the leading boundary of the shear plane. The fields for chamfer angle with -800, -700 and -600 at the friction angle βs=400 are plotted in Figures 4.8(a), (b) and (c), respectively. From Figure 4.8, it can be observed that the angle δ has negative  and increases with the chamfer angle. The optimized flow lines around the chamfer land are similar with Shi and Ramalingam (1991) and experimental observation (Oxley 1989). The latter suggests an increasing hydrostatic stress at the tool edge. The methodology used in this thesis to calculate forces on the chamfer is relatively robust in terms of the potential influence of varying slip-line curvature. To demonstrate this, the expected normal stress levels on the chamfer calculated assuming that the leading slip- line either intersects the rake face at right angles or in the extreme case the effective slip line meets at the angle of intersection of CF and the rake face are examined to see if the final solution is sensitive to this assumption. Figure 4.9 shows that there is remarkably little influence between these extremes.   4  The Modeling of Tool Edge Ploughing Force in Milling Operations  77    (a) Lchmf/h0=0.1 (b) Lchmf/h0=0.3 ( c) Lchmf/h0=0.5 Figure 4.7 Upper bound solutions for the case of γ1=-700, γ2=00, βs=400      (a) γ1=-800, γ2=00 (b) γ1=-700, γ2=00 ( c) γ1=-600, γ2=00 Figure 4.8 Upper bound solutions for different chamfer angles  1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0 1 2 3 4 5 6    Shear Boundary Meets           Tool at Right Angle  Shear Boundary Has Increased           Curvature as Indicated by UB Model Ch am fe r La n d N o rm al  St re ss  σ EF /w 2 k h 0 h0/Lchmf  Figure 4.9 Calculation of normal stress on chamfer using two extreme cases of field geometry 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  78 4.2.5.2 Validating the Proposed Model with Experimental Data The force predictions from the model were compared with recent experimental data. The first series of experiments were conducted by Smithy et al. (2001). The parameters in this predictions are used as γ1=-850, γ2=50 and βs=420 (which is the same friction coefficient as Smithy et al. (2001) with mp=0.9). Figures 4.10(a) and (b) show the comparison of the predicted cutting forces with the experimental results given in Smithy et al. (2001). A good agreement is observed between the predictions and the measured values. 0.0 0.5 1.0 0 1 2 3 4 4.07 3.59 3.21 3.85 3.583.41 Sharp tool Cu tti n g Fo rc e F C /w kh 0 L chmf/h0  Experimental Results  Present Simulation  (a) Cutting force FC  0.0 0.5 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 2.49 2.80 2.182.28 2.051.99 Sharp tool Th ru st  Fo rc e F T /w kh 0 L chmf/h0  Experimental Results  Present Simulation  (b) Thrust force FT Figure 4.10 Comparison of predicted cutting forces with experimental results 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  79 The second set of experiments concern the cutting of 0.2% carbon steel using a tool with a 0.1 mm radius tool edge and were reported by Kim et al. (1999). In order to characterize the edge radius, the approach suggested by Manjunathaiah and Endres (2000) is used, and the actual critical angle chosen at point (F) is 300. Figures 4.11(a) and (b) present the comparison of the cutting and thrust forces with experiment and FE simulations for various depths of cut. Good correlation is found for the cutting and thrust forces over the entire range of depths of cut tests. Figure 4.12 shows the ratio of thrust force to cutting force. As depth of cut decreases, the ratio increases.   0.05 0.10 0.15 0.20 0.25 0.30 0.35 0 200 400 600 800 1000 1200 1400 1600 1800 2000   Present Model  Experiment (Kim et al. 1999)  FEM (Kim et al. 1999) Cu tti n g Fo rc e F C (N ) Uncut Chip Thickness h0(mm)  (a) Cutting force FC  4  The Modeling of Tool Edge Ploughing Force in Milling Operations  80 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300    Present Model  Experiment (Kim et al. 1999)  FEM (Kim et al. 1999) Th ru st  Fo rc e F T (N ) Uncut Chip Thickness h0(mm)  (b) Thrust force FT Figure 4.11 Variation of cutting and thrust forces versus uncut chip thickness  Cutting Conditions Workpiece Material: 0.2% Carbon Steel Width of Cut: 3.2 mm Rake Angle: 12 deg Cutting Velocity: 2.16 m/s Shear Yield Stress: 622.62 MPa 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9    Present Model  Experiment (Kim et al 1999)  FEM (Kim et al. 1999) F T /F C Uncut Chip Thickness h0 (mm)  Figure 4.12 Change in force ratio as uncut chip thickness increases 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  81 4.2.6 Parameter Studies of Predictive Cutting Model The validation of the predictive cutting model has been conducted in Section 4.2.5, which shows a good correlation with the experiments. This section attempts to investigate the influences of the chamfer parameters on the cutting forces. 4.2.6.1 Influence of Chamfer Angle on Cutting Forces The influence of chamfer angle on the cutting forces is investigated here with the consideration of three different rake angles γ2 at -100, 00 and 100, respectively.  The friction angle βs is taken as 400. The shear angles for the rake angles at -100, 00 and 100 are calculated as 24.50, 29.50 and 340, respectively. The relations of the normalized cutting force FC, and thrust force FT with respect to the chamfered angles are given in Figures 4.13, and 4.14, respectively. From Figures 4.13, and 4.14, it can be observed that the cutting force Fc is decreased as the chamfered angle is increased; while the thrust force FT is increased as the chamfer angle is increased. -85 -80 -75 -70 -65 -60 -55 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0    Rake Angle γ2=10 0  Rake Angle γ2=0 0  Rake Angle γ2=-10 0 Cu tti n g Fo rc e  F C /w kh 0 Chamfer Angle γ1 0  Figure 4.13 Cutting force Fc versus chamfer angle for the rake angle of -100, 00 and 100 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  82 2 Model Parameters Lchmf/h0=0.2 s=40 deg -85 -80 -75 -70 -65 -60 -55 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5    Rake Angle γ2=-10 0  Rake Angle γ2=0 0  Rake Angle γ2=10 0 Th ru st  Fo rc e  F T /w kh 0 Chamfer Angle γ1 0 1 Chamfer Land Lchmf  Figure 4.14 Thrust force FT versus chamfer angle for the rake angle of -100, 00 and 100      4.2.6.2 Influence of Chamfered Land Length on Cutting Forces The influence of chamfer length on the cutting forces is examined in this section. The geometry parameters are considered as: γ1=-700, γ2=00 and βs=400, the ratio of chamfer length to the uncut chip thickness is varied in the range of 0.1 to 0.6. Figure 4.15 plots the relationship of normalized cutting forces FC and FT with respect to the ratio of chamfer land length to the uncut chip thickness, from the figure, it can be seen that the forces FC and FT are increase in nonlinear fashion with the ratio of chamfer land length to uncut chip thickness. 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  83 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 1 2 3 4 5 6    Cutting Force FC  Thrust Force FT Cu tti n g Fo rc e s F C /w kh 0 an d F T /w kh 0 Chamfer Land Length L chmf/h0  Figure 4.15 Variation of cutting force FC and thrust force FT with respect to chamfer length  4.3 Ploughing Models The ploughing process where majority of workpiece material is displaced without being removed is studied in this section. Two ploughing models which include the effect of build up of a bulge ahead of the tool during sliding are presented to predict the forces during ploughing. 4.3.1 Single Edge Ploughing Model The single edge ploughing model is based on the wedge indentation model discussed by Rowe and Wetton in respect to grinding, (1969). The wedge is moved to the right with its apex at a constant depth h0 below the surface, the volume of metal displaced and the height of bulge will increase.  In the absence of leakage, the volume displaced will increase by lslideh0. 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  84  ρ φ h 0 A BC D E 0 1 2 3 Tool workpiece γ1 Feed Direction F pi/42δ  Figure 4.16 Slip-line field for single edge ploughing model (Rowe and Wetton 1969)  Based on the proposed slip-line field, the geometry relationships can be given as follows: 1 1 2, 28 γδφγρpiδ −=−−=                                              (4.24) The angle ρ can be obtained as Figure 4.16       − = − L hL 2 cos sin 011 γρ                                                   (4.25) δsin2,2,,, LLLLLLLLLL BCADABBDCD =====                        (4.26)  where γ1 is the rake angle, h0 the uncut chip thickness, L the length of fan radius. For the case of sliding distance lslide=0, L is given as 1 0 cosγ h L =                                                            (4.27) for the case of sliding distance lslide>0, L can be obtained once one knows the bulge area (DAEF) in the field (shown in Figure 4.16), therefore ( ) ( ) ( ) 02tancoscos2cos 01201201201 =⋅−⋅−⋅+−⋅−−⋅ slidelhhLhLLhL γγγγ     (4.28) 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  85 The solution should keep the slip-line field convex and correspond to minimum energy consumption. It should be noted that the above geometry relation is based on the angles ρ and φ being positive values. Based upon the slip-line field, the normal and shear stresses can be given as ( ) ( ) ( ) ( ) ( ) kCkCpkBkBp =+=== τδτ ,41,,                                  (4.29) Therefore the ploughing force acting on the horizontal and vertical directions can be stated as ( ) ( ) 1111 cossin41,sincos41 γγδγγδ kLkLFkLkLF TC ++−=++=        (4.30) and the ploughing power can be expressed by ( ) 10100 sincos41 γγδ kLVkLVVFP C ++==                                     (4.31) where V0 is sliding velocity. 4.3.2 Double Edge Ploughing Model A double edge ploughing model is needed to more accurately simulate the effect of tool edge radius. The proposed slip-line field and hodograph for the double edge ploughing model are given in Figures 4.17 and 4.18, respectively. φ κ λ κ 1 ε ξ pi/4 ρ 2δ γ1 γ2 h0 A B C D E F Tool workpiece 0 1 2 3 4 Feed Direction  Figure 4.17 Double edge slip-line field 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  86 κ 1 1 4 0 4',2 3 V0 γ1 φ γ2  Figure 4.18 Hodograph  Based on the slip-line field, the geometry relationships are given as follows 21 2 ,, 4 ,2 γξpiεξφφpiργδξ −−==−=−=                                 (4.32) [ ])cos()sin( 2,,),cos( ),sin(),cos(, 22 2 22 γξγξ γξ γξγξ +++⋅+= ===++= =+=+== EFEDAC BFAFEIBCBFABEFEDBF EIBCEFEIEFFIEDEC LLL LLLLLLLLL LLLLLLLL          (4.33) κ λκpiκ λκ sin , 2 tan,tan 1 11 EC AE EC BC AJ EC L L L L L L =−−=       =      = −−                                (4.34) and ( ) 0ldistanceslidingfor)cos( tan)2cos1(5.02sin5.0 0ldistanceslidingfor sincos2cos sin2cos 2 2 0 22 10 > +         − ⋅−+ = = +− +⋅− = slide ED slide EF slide EDED EF L lh L LLh L γξ γρρ ργξγ ργ  (4.35) 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  87 Based upon the hodograph in Figure 4.18, the velocity relationships can be stated as ( ) ( ) ( ) ( ) 121421 21220 32 21 11210 03 10121004 , cos coscos cos sinsin cos,sin VVVVV VVV VVVV = + +− = + −+ = == γκ γφγ γκ κφκ γγ                                (4.36) In order to calculate the plastic dissipation of the fan field, a cylindrical-polar coordinate system is adopted. The strain rates in the fan region can be expressed by r V rrr 14 2 1 ,0 === θθθ εεε &&&                                             (4.37) Then the equivalent strain rate is given as r V ijij p 33 2 ˆ 14 == εεε &&&                                                 (4.38) Thus the plastic dissipation is δ θθε δδ 4 3 3]][[ˆ 14 14 2 0 142 00 ⋅⋅= +=+Ω= ∫∫ ∫∫∫∫Ω CD S Lp fan LkV dkVrdrd r VkdsVkdYP CD&       (4.39) Finally the total power dissipation can be obtained as )4( 03321214040 VLVLVLVLVLkwVFP EFAEACEDEDC ++++== δ              (4.40) Using the upper-bound technique, the optimization algorithm is stated in the form as 0 4 sin constant,0,0:toSubject Minimium 2 0 ≥      ++ =≥≥ γξpi φρ h FC                                    (4.41) 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  88 The normal force acting on the ploughing face EF and the vertical ploughing force are then derived as ( ) ( ) 2121 2 121 coscossinsin41 cos cos41sinsin γγγγδ γ γδγγ EFEDNEDT EDEFEDc N kLkLFLkF LkkLkLF F ++−+−= +−−− =            (4.42)  The MAPLE global optimization toolbox (Maple 2010) has been used to solve the above nonlinear equations. In the numerical calculation the accuracy is chosen as 10-9 when the minimization of power consuming is satisfied. 4.3.3 Ploughing Model with Rounded Edge Tool The geometry of rounded edge tools is idealized to achieve a deformation zone and equivalent tool that is made up of straight boundaries. As shown in Figure 4.19, the edge radius is approximated by one and two line segments for the single and double edge model, respectively. The equivalent tool geometry is formulated as follows. α 2 λs = θ/ 2 h0 re θ α1  Figure 4.19 Round edge tool ploughing geometry  For the single edge model, the ploughing angle is approximated as: se s r h λλ sin2sin 0 =                                                          (4.43) 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  89 where re is the edge radius, h0 is the uncut chip thickness.  Thus the attack angle λs can be obtained as         = − e s r h 2 sin 01λ                                                         (4.44) On the other hand, for double edge ploughing model, the geometry can be derived as       − +== − s s s s λ λλαλα sin cos1 tan, 2 1 21                                        (4.45) 4.3.4 Ploughing Model Applications 4.3.4.1 Effect of Attack Angle In this section, the effect of attack angle (λs=pi/2-γ1) on chip formation has been examined using the single edge ploughing model for the zero sliding distance, (lslide=0). The ploughing angle γ1 is changed over what is expected to be a typical range. The resulting values of the specific cutting power are presented in Figure 4.20 and it is clear that the specific power increases very rapidly as the attack angle decreases towards zero. 10 20 30 40 50 6 7 8 9 10 11 12   Specific Cutting PowerS pe ci fic  Cu tti n g Po w er  P/ kh 0V 0 Attack Angle (λ s )0  Figure 4.20 Specific cutting power versus attack angle 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  90 4.3.4.2 Edge Radius Effects The ploughing process for a tool having 0.5 mm tool edge radius has been simulated using both the single and double edge models. Figure 4.21 presents the comparison of the specific cutting power consumption using the single edge model and double edge model for various ploughing depths with zero sliding distance. Reasonable agreement is seen for the single edge and double edge models over the range of depths of ploughing testes. As expected, the force increases with increasing depth of ploughing. 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0 2 4 6 8 10 12 14    Single Edge Ploughing Model  Double Edge Ploughing Model Sp ec ifi c Cu tti n g Po w er  P/ kh 0V 0 h0/re  Figure 4.21 Specific ploughing power with respect to ploughing depth  4.4 Transition of Ploughing to Cutting Processes This section deals with the determination of the transition point from the ploughing process to the cutting processes. It is assumed that when the specific energy of ploughing reaches the critical specific energy of the cutting model (including the edge radius effect), the cutting process occurs. This assumption has been discussed by Felder and Bucaille (2006) who used the Merchant model to determine the critical point. 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  91 4.4.1 Effect of Sliding Distance on the Single Edge Ploughing Model The growth of the frontal bulge pattern as sliding motion proceeds was discussed by Rowe and Wetton (1969).  If the wedge is moved to the right with its apex at a constant depth h0 below the surface, the volume of metal displaced will increase and the height of the bulge will increase, also. In the absence of leakage, the volume displaced will be increased by lslideh0, (for unit width). Figures 4.22, 4.23 and 4.24 plot the specific power with respect to the sliding distance for attack angles of λs=350, λs=450 and λs=550, respectively. The cutting model uses the predictive model (Section 4.2) with zero elastic friction angles. Figures 4.22-4.24 show that, as the attacking angle increases, the ploughing to cutting transformation will occur sooner. Figure 4.25 plots the slip-line field which occurs as the bulge varies in size with different sliding distances, (for a constant attack angle of 30 degrees). Figure 4.26 plots the slip-line field variation with different attack angles and a sliding distance equal to twice the uncut chip thickness. 0 2 4 6 8 10 0 2 4 6 8 10 12 14 16   Ploughing Model  Predictive Cutting ModelS pe ci fic  En er gy  P/ kh 0V 0 Sliding Distance l slide/h0  Figure 4.22 Specific power versus sliding distance for attack angle λs=350 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  92 0 2 4 6 8 10 0 2 4 6 8 10 12 14 16    Ploughing Model  Predictive Cutting Model Sp ec ifi c En er gy  P/ kh 0V 0 Spindle Sliding Distance l slide/h0  Figure 4.23 Specific power versus sliding distance for attack angle λs=450  0 2 4 6 8 10 0 2 4 6 8 10 12 14    Ploughing Model  Predictive Cutting Model Sp ec ifi c En er gy  P/ kh 0V 0 Sliding Distance l slide/h0  Figure 4.24 Specific power versus sliding distance for attack angle λs=550  4  The Modeling of Tool Edge Ploughing Force in Milling Operations  93  (a) lslide=0  (b) lslide=6h0  (c) lslide=12h0 Figure 4.25 Slip-line field variation with sliding distance for attack angle λs=350  (a) λs=100  (b) )λs=300  (c) λs=400 Figure 4.26 Slip-line field variation with attack angle for sliding distance lslide=2h0  4.4.2 Effect of Sliding Distance on Double Edge Ploughing Model The influence of sliding distance on the energy requirements predicted by the double edge model is examined in this section. Figure 4.27 plots the specific energy with respect to the sliding distance for the case of γ1=800, γ2=550, LED=0.5h0. The result is also compared with the cutting model with a chamfer Lchmf=0.5 and βs=0. From Figure 4.27, it can be seen when sliding distance is approximately equal to the chip thickness, (lslide=h0), the cutting process will occur. Figure 4.28 plots the slip-line field for the case of sliding distances equal to 0 and 4h0, respectively. 0 2 4 6 8 10 0 2 4 6 8 10    Double Edge Ploughing Model  Predictive Cutting ModelSp ec ifi c En er gy  P/ kh 0V 0 Sliding Distance (l slide/h0)  Figure 4.27 Specific power versus sliding distance for γ1=550, γ2=800 and LED=0.5 h0 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  94   (a) lslide=0  (b) lslide=4h0 Figure 4.28 Slip-line field with different sliding distance for ploughing angle γ1=550, γ2=800 4.4.3 Discussion on the Entry and Exit Milling Processes The purpose of the preceding simplified analyses has been to demonstrate that the transition from ploughing to cutting is strongly influenced by the sliding distance; there is then no simple formulation of a single critical depth of cut at which the transition will always occur. Previous authors have noticed an abrupt change in the force pattern in cutting particularly in the entry stages of up-milling. They have usually then referred to the chip thickness at which this occurs as the minimum chip thickness, (Chae et al. 1997; Miao et al. 2007; Jun et al. 2006; and Bissacco et al. 2008). On the other hand, there is little evidence of the same phenomenon occurring in down milling exit conditions where the chip thickness variation is the mirror image to that observed in the entry to up milling. The author believes that this is due to the delay in the ploughing/cutting transition due to the build up that occurs with ploughing distance. In the case of down milling, then at the critical chip thickness, if ploughing begins, it is likely that the process will prefer to revert to cutting as the field develops and that this will continue until a condition is reached that physically precludes cutting, (This will 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  95 depend upon both the effective attack angle, including the bulge and the adhesion or friction properties between chip and tool). 4.4.4 Validating Entry and Exit with Worn Tool Milling Operations Experiments have been conducted previously within the Manufacturing Engineering Laboratory of UBC by Oyawoye (1993); the tests were conducted on an Adcock and Shipley universal milling machine. Cutting forces were measured with a three directional Kistler model 9257A dynamometer. A four tooth 300mm diameter face milling cutter with only one insert was used in these tests and the workpiece was a titanium alloy. (The large diameter of the cutter and large effective approach angle minimize the surface slope seen by the equivalent cutting edge. To convert the facemilling operation with a non straight cutting edge to an equivalent straight edge orthogonal process the concepts of equivalent chip thickness and effective approach angle are used, (see Yellowley and Lai 1993). The equivalent chip thickness is defined as the uncut area of the chip over the engaged cutting length of the tool insert ete laSh =                                                               (4.46) where le is the equivalent length, St is feed per tooth, and a is the depth of cut. For cutter with a finite nose radius r, the expression is approximated ( ) 22cos sin1 t e S r ral +      −+ −− = ψpi ψ ψ                                       (4.47) where ψ is the approach angle. The effective approach angle is given as ( )( )       +⋅+⋅−− = − a Srra t e 2/costansin1 tan 1 ψψψψ                        (4.48) 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  96 Therefore the measured axial force can be given as eTAxial FF ψsin⋅=                                                         (4.49) where FT is the thrust force. For the milling processes, the instantaneous chip thickness can be approximated as ϕϕ sin)( ⋅= ec ht                                                            (4.50) where φ is the angle of immersion.  In the section that follows, the author intends to show that the proposed predictive cutting and ploughing models described earlier are capable in at least a qualitative manner to explain the influence of the cutting/poughing transition on the force pattern observed with worn tools. At the entry of the cut before the ploughing/cutting critical transition point is reached, the single edge ploughing model is used for the force simulation. The modeling of force in the cutting process which follows the transition is simulated using the chamfer tool model discussed in Section 4.2. The ploughing to cutting transition occurs when cutting is feasible and the energy consumed is less than that due to ploughing, (it should perhaps be noted that energy was always less for cutting when this was feasible in the cases examined here). The forces in the exit “down milling” region are again calculated using the ploughing model; in this case however the process is assumed to operate continuously with the critical value of effective rake angle, (i.e., that above which cutting is feasible). A comparison of the numerical simulation against the experimental result is shown in Figure 4.29. The plots are fairly close as a result of calibrating the value of shear yield stress for the simulation from the experimental data. The points to be made, however, pertain to the qualitative assessment of the shape of the plots as follows: 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  97 a)  The peak in force versus the later cutting force and point of transition are seen to be in close correspondence. b)  The exit force shows no tendency for the process to revert to ploughing. Finally it should be noted that the difference in shape of the force during exit and entry are likely due primarily to the inability of the model to include elastic contact forces in these regions. 0 50 100 150 200 250 0 50 100 150 200 250 300    Experiment  Calculated Results Ax ia l F or ce  (N ) Cutting Time (ms)   Figure 4.29 Simulation of micromilling processes  4.5 Conclusions The very low chip thickness in milling forces one to examine the basic mechanics of cutting with respect to the incorporation of edge influences into the normal cutting process and in the ploughing process that must inevitably occur before cutting commences. The major contribution in this chapter is the creation of a new model of orthogonal cutting that incorporates the edge flow processes and can be used to model 4  The Modeling of Tool Edge Ploughing Force in Milling Operations  98 rounded edge and worn tool forces. It might be noted that the author has not explicitly included the energy required for surface generation. It is believed difficult to do this because the majority of that energy is due to the plastic field accompanying the crack; such work is inevitably related to and already partially accounted for by the plastic flow field included here. The cutting model is novel being an upper bound solution which is based upon a complete slip line field solution on the chip side. One is thus able to guarantee chip equilibrium and maintain realistic rake face boundary conditions while optimizing the field below the tool and ascertaining the likely influence of chamfer, wear or edge roundness on the process at low chip thicknesses. The model does not need extensive calibration, the only unknown is the elastic friction angle and, if forces are required, a mean shear yield stress is also required. The predicted results are in good agreement with the experimental results and FEM simulations. This chapter also presents two simple fields to model the ploughing process that may be expected in the entry and exit stages of milling with a finite edge radius or wear land. The ploughing models include the effect of sliding distance. It is found that as the tool slides, the energy will continue to rise and it becomes preferable for the process to move. The models have been used as a tool to qualitatively explain the entry and exit force characteristics during milling operations.   99 5  A New Predictive Model for Peripheral Milling Operations  5.1 Introduction Milling is one of the most important manufacturing operations; it differs from most other cutting operations in the magnitude of typical chip thickness, the variation in chip thickness, the surface slope at the free surface and the discontinuous nature of the process. Typical milling practice most often leads to widths of cut that are less than cutter radius and hence to relatively few teeth in cut. One may argue that larger widths are much more economic but that is not the normal approach adopted; moreover in finish or semi finish milling this is likely to be the case, even with good practice.  The finished surface is then produced most often with just a single tooth in contact and with extremely low chip thickness. It is seen then that where force prediction and material flow patterns are most important than the use of typical force modeling approaches and equations from turning or similar sources are unlikely to be particularly useful. The increasing importance of meso and micromilling operations has led to more interest in such matters, however it needs to be remembered that the conventional milling process will always exhibit similar complexity during the time that the surface is actually being generated. A lot of attention has been concentrated on the prediction of cutting forces over the last few decades; the mechanistic approach has gained by far the most amount of popularity (Koenigsberger and Sabberwal 1961; Tlusty and MacNeil 1975; Fu et al. 1984). This is because of the simplicity involved in building empirical coefficients in the model which result in accurate force predictions.  A mechanistic force model relates cutting forces to uncut chip sectional area through a set of coefficients. These cutting 5  A New Predictive Model for Peripheral Milling Operations  100 force coefficients, which are comprised of the specific cutting pressure coefficients and the chip flow angle, are extracted from the measured forces. Two basic calibration approaches are often used in the literature: an empirical parameter fitting approach (Koenigsberger and Sabberwal 1961; Kline et al. 1982; Engin and Altintas 2001), and an orthogonal to oblique transformation approach (Armarego and Deshpande 1991; Budak et al. 1996; Armarego 2000). Generally, both calibration procedures provide only one set of average cutting force coefficients from each cutting tests. In order to obtain a robust application range from the related cutting force model, a great number of tests must be performed for different feeds, depths of cut, etc. This chapter is concerned with the building of a model of oblique cutting operations which is suited to the analysis of tools with non straight cutting edges and non planar rake faces, (primarily milling, drilling, and tapping operations). The application of the newly developed oblique cutting model (Zou, Yellowley and Seethaler 2009) to the milling processes is described in detail. The model is provided with a suitable material constitutive equation, to allow the consideration of the influence of chip thickness on yield strength and the edge force components are also estimated allowing some degree of confidence at low values of chip thickness. Also the potential influences of surface slope as well as the complex kinematics are examined. 5.2 The Development of a Methodology to Estimate Milling Forces As described earlier, a theoretical model of the milling process is made difficult by the fact that the milling process creates chip thicknesses that are both small and variable; moreover during surface creation the rate of change of chip thickness is high. The author believes that any theoretical approach must then consider the following: 5  A New Predictive Model for Peripheral Milling Operations  101  a) The influence of chip thickness on both the shear yield stress in the shear plane and the corresponding normal stress which is influenced by both geometry and work hardening across the shear zone; b) The influence of surface slope on the equilibrium value of shear angle; c) The estimation of the likely magnitude of edge forces that may be attributed to the flow of the work material under the cutting edge; d) For very small chip thickness there should be some mechanism to allow a switch between ploughing and cutting models as well as to estimate forces in the regime; e) The helical edge complicates the kinematics of the process and it is not sufficient to simply estimate forces on elemental lengths of edge and then integrate to estimate total forces acting upon the cutter. As the chip flows inward there is evidently additional deformation which, though small, needs to be estimated in any scheme that tries to avoid such issues through calibration rather than through the use of basic physical properties.   The block diagram of the proposed predictive approach is given in Figure 5.1. The basic equations needed to address the concerns enumerated above are outlined in the next sections; following this the overall software model developed to calculate cutting forces is described. Finally, an experimental program and verification of the model are presented in the following chapter.  5  A New Predictive Model for Peripheral Milling Operations  102  Prediction of Force Components in Milling Uncut Chip Thickness Model Cutter Type Helical End Mill Nf (Teeth Number) Feed per Tooth Cutter Fault Runout  ρ, λ 3D Oblique Cutting Model Discretization  Ne Work Material Properties Modified Johnson-Cook  Model Strain Strain Rate Temperature Material  Constants ρ, S, K Tool Parameters γN, i, R Process Parameters Tool Geometry Feeding  Speed Spindle Speed Depth  of Cut Width  of Cut Up or  Down Milling SLIP Tool Edge Force Model Ploughing Model Edge Radius Nose Radius Wear Up or Down Milling Kinematics Parameters Velocity Profile Surface Parameters Surface Slope  Figure 5.1 Diagram of predictive model  5.3 The Estimation of Oblique Cutting Forces The first step in the development of the force model is to estimate the force acting upon an element of cutting edge engaged in an oblique cutting operation with essentially constant chip thickness. This requires the use of a suitable constitutive equation and knowledge of strain, strain rate and temperature, as well as the edge force attributed by the surface generation at the tool edge. This problem is addressed here while later sections address the influence of the remaining process variables on the force components. 5.3.1 Deformation Energy Dissipation on the Shear Plane The rate of plastic energy dissipation in the shear plane can be expressed as 5  A New Predictive Model for Peripheral Milling Operations  103 εσ ε dwVhVFP sss ⋅== ∫ 0 00                                          (5.1) where ε  is the equivalent oblique cutting plastic strain, which can be written as (Atkins 2006) ( ) s NNNoblique η γφγγ ε cos3 tancot 3 −+ ==                                  (5.2) According to the Johnson-Cook model, the von Misses flow stress σ, is given by (Jaspers and Dautzenberg 2002) ( ) ( )mmn TCBA * 0 1]ln1[ −      ++= ε ε εσ & &                            (5.3) where 0εε &&  is the dimensionless strain rate for 1 0 0.1 − = sε&  and A, B, Cm, n, and m are material constants. The shear strain rate is estimated as follows shear sSoblique l VConst y V 333 = ∆⋅ == γ ε & &                             (5.4) where ∆y is the width of the shear zone, which is assumed to have a constant relationship with shear plane length (lshear). The relationship between the absolute temperature T and thermal softening term T* is expressed as )/()(* roommeltroom TTTTT −−=                             (5.5) where Troom and Tmelt are the room temperature and the melting point of the specific alloy, respectively. The Johnson-Cook model assumes the slope of the flow stress curve to be independently affected by strain, strain rate and temperature represented by the terms in each set of brackets. 5  A New Predictive Model for Peripheral Milling Operations  104 Substituting Eqn. (5.3) into Eqn. (5.1), the rate of plastic energy dissipation in the shear plane can be derived as [ ] [ ]mmnS TC n BAwVhP **100 1ln11 −⋅+⋅    + += + εεε &                        (5.6) The increased temperature in the shear zone can be calculated as follows (Loewen and Shaw 1954, Lei et al. 1999) ( ) ξ ρ εσ η ρ S d x S P xT s ∫=−=∆ 1                                 (5.7) where x is the ratio of plastic energy converted  to heat, and ξ=1-η is the fraction of the heat remaining in shear zone. According to Lei et al. (1999), ξ can be estimated from the expression 00 328.11 1 hSV Koblique ρ γ ξ + =                                          (5.8) where ρ, S and K are the workpiece material density, specific heat, and thermal conductivity, respectively. 5.3.2  Friction Energy Dissipation at the Tool-chip Interface The rate of energy dissipated by friction at the tool-chip interface can be written as cff VFP ⋅=                                                (5.9) where Ff is the friction force. The shear velocity Vs and chip velocity Vc can be given in terms of the SLIP on the rake face in the oblique cutting process (Zou, Yellowley and Seethaler 2009) 5  A New Predictive Model for Peripheral Milling Operations  105 ( ) ( ) ( ) iVSLIPV iVV iVSLIPViVV st NN N sn ct NN N cn sin, cos coscos sin1, cos sincos 0 0 0 0 ⋅⋅= − ⋅ = ⋅⋅−= − ⋅ = γφ γ γφ φ                (5.10) and ( ) ( )       =      = +=+= −− cn ct c sn st s stsnsctcnc V V V V VVVVVV 11 2 1 222 1 22 tan,tan , ηη                               (5.11) where φN is the normal shear angle, γN the normal rake angle, ηs  the shear plane flow angle, and ηc the chip flow angle, shown in Figure 5.2. 0h cnV ctV cV cη snV stV sV sη sF 0V TF CFRF  Figure 5.2 Coordinate system of oblique cutting  5  A New Predictive Model for Peripheral Milling Operations  106 The total internal work rate is given by the summation of Eqns. (5.6) and (5.9). The external work rate is given by FCV0. Thus, the energy equation for oblique cutting is given as [ ] [ ] cfmmnC VFTC n BAwVhVF +−⋅+⋅    + += + **1000 1ln11 εεε &         (5.12) 5.4 The Influence of Work Hardening on the Average Normal Stress in the Shear Plane The influence of work hardening on the average normal stress in the shear plane is estimated based on the application of force equilibrium in the normal shear plane as shown in Figure 5.3.  Figure 5.3 Normal shear plane for oblique cutting  The angle θN between the shear force and the normal force on the normal shear plane is calculated as follows (Lin et al. 1982, Oxley 1989) 5  A New Predictive Model for Peripheral Milling Operations  107 N AB N AB ABN N NN k l s k 24 21tan 2       ∆ ∆ −      −+= φpiθ                                 (5.13) where ABN N s k       ∆ ∆ 2  is the variation of the shear flow stress across the width at the parallel sided shear zone (Oxley 1989). This term is evaluated in the following manner: ABNAB N AB N N ABN N ds dt dt d d dk ds dk       ⋅      ⋅      =      22 γ γ                           (5.14) where )cos( coscos , sin 00 NN NN s N N iVVhl γφ γ φ − ⋅ ==                                     (5.15) γN is the rake angle. The strain along the normal shear plane is given as (Oxley 1989, Lin et al. 1982) )cos(sin cos 32 1 NNN NN AB γφφ γ ε −⋅ =                                       (5.16) The strain rate along the shear zone AB is given as N snN AB l VConst 3 =ε&                                                 (5.17) Therefore, the shear flow stress along AB using the Johnson-Cook model can be written as ( ) ( )mABNABmnNABNAB TCBA * 0 1]ln1[ −⋅      +⋅+= ε ε εσ & &                           (5.18) Each of the above terms may be evaluated independently. The derivatives in Eqn. (5.14) can be expressed as 5  A New Predictive Model for Peripheral Milling Operations  108 ( ) ( ) N N sn ABNAB N NABN N AB N AB N AB N AB AB N N iVl VConst ds dt dt d iVds dt d d d d d dk φ γ φε σ ε σ γ sincos 1 sincos 1 , 3 1 3 3/ 02 02 ⋅ =            ⋅ =      ==                     (5.19) Then eqvN AB N ABN N nConst k l s k ⋅=      ∆ ∆ 22                                         (5.20) where ( )       ⋅⋅+=            = N AB N AB N AB N AB N AB N AB eq cccd d n σ ε σ ε ε σ 321                            (5.21) in which ( ) ( ) N AB lu AB roommelt m AB N AB m nN AB m roommelt ABmelt N AB m nN AB TTT c TT TmCBAc TT TTCnBc εε ε ε ε ε ε ε 2 ln1 ln1 3 1* 0 2 0 1 1 − =      ∂ ∂ =         − − ⋅      +⋅+=       − − ⋅      += − − & & & &                     (5.22) where Tu and Tl are the temperatures at the upper and lower boundaries along the normal shear plane, respectively. Tu and Tl can be obtained by substituting the strain and strain rate at the upper and lower boundary of shear zone as well as the Johnson-Cook model into Eqn. (5.7). Thus the normal friction angle can be given as NNNN γφθβ +−=                                                       (5.23) and the friction angle along the chip flow direction is       = − c N η ββ cos tan tan 1                                                    (5.24) 5  A New Predictive Model for Peripheral Milling Operations  109 where ηc is the chip flow angle. 5.5 The Estimation of Edge Force Components The edge force occurring in orthogonal cutting analysis is discussed in Chapter 4, the orthogonal edge forces need to be transformed into a three-dimensional coordinate system. It is assumed that the orthogonal edge forces act on the velocity plane in this case. The velocity rake angle is given as i N v cos tan tan γγ =                                                  (5.25) Then the orthogonal cutting force, CEF ′ , and thrust force, TEF ′ , become the 3D force, FCE and FTE, respectively, which can be expressed min0 hhfor 0 ' ' ≥      = = = RE TETE CECE F FF FF                                    (5.26) where hmin is the minimum chip thickness. As discussed previously, when the uncut chip thickness is less than the minimum chip thickness at the entry or exit region, the total forces are mainly caused by either the ploughing processes (entry) or the high negative rake cutting process (exit). The 2D ploughing forces should also be transformed to the force components in oblique cutting as min0forexit 0 ' ' or  entry, 0 ' ' hh F FF FF F FF FF RE THTE CHCE RE TPTE CPCE <      = = =      = = =            (5.27) where CPF '  and TPF '  are the ploughing forces at the entry process, while  CHF '  and THF '  are the high negative rake cutting forces occurring at the exit process. 5  A New Predictive Model for Peripheral Milling Operations  110 5.6 The Effect of Work Surface Slope The effect of surface slope on milling has been studied by Pandy and Shen (1972) and Altintas (1986). Pandy and Shan (1972) examined the slope change due to the advancement of shear plane, while Altintas (1986) only considered the chip thickness variation. The work surface slope caused by both the relative angle of the tool edge and the free end of the shear plane as well as the variation of uncut chip thickness is examined in this section. The relationship between the surface slope and shear angle is examined as well. 5.6.1 Surface Slope Formations Figure 5.4 shows the surface slope change caused by the relative position of the tool edge and the free end of the shear plane during up milling operations. At the tooth rotation angle ψ, the work material is assumed to be sheared along the plane AC and the corresponding shear angle is represented by φN0. The uncut chip thickness at the point of cutting is BC. Surface slope caused by the curvature of the workpiece is represented by ∆ψ, which is a function of the cutter diameter and feedrate in milling operations, and can be approximated as R S Nt 0cotsin φψψ ⋅≈∆                                               (5.28) On the other hand, the surface slope caused by the variation of uncut chip thickness ( ψsintSh = ) can be derived as R S R S Rd Sd ds dh ttt ψψ ψ ψλ coscostan)sin(tantan 111 ≈      =      == −−−                 (5.29) 5  A New Predictive Model for Peripheral Milling Operations  111     (a) Normal plane of oblique cutting with surface slope  (b) Plan view of normal plane in milling  Figure 5.4 Slope variation in milling Therefore the total surface slope δs in milling can be taken a simple summation of Eqn. (5.27) and (5.28) [ ]ψφψλψδ coscotsin 0 +⋅=+∆= Nts R S                           (5.30) In wave removing, it is generally assumed that the shear angle is a linear function of work surface slope, i.e. ssNN C δφφ ⋅+= 0                                             (5.31) where 0Nφ  is the normal shear angle when the work surface slope δs is zero and Cs is a constant. Shaw and Sanghani (1963) performed experiments to measure shear angle during the cutting of a wavy surface and indicated that a Cs of close to unity results. Wallace and Andrew (1965) found the value of Cs to be 0.75 from their wave removing experiments. The purpose of the following analysis is to investigate the influence of surface slope variation on the shear angle. Two cases are examined: 1) frictionless orthogonal cutting, and 2) oblique cutting process. 5  A New Predictive Model for Peripheral Milling Operations  112 5.6.2 Frictionless Orthogonal Cutting The geometry of the orthogonal cutting model having surface slope δs is shown in Figure 5.5. From Figure 5.5, the geometry relationship can be written as ( ) ( )ss s s s hll δφδδpi −==+ sincos2/sin 0                            (5.32) Then the shear plane length is given by ( )s s s hl δφ δ − = sin cos0                                                 (5.33) The total cutting power for the frictionless case is given by ( ) ( ) ( ) 000 sinsin coscos sin cos wVhkwVklVFP s s sss γφδφ γδ γφ γ −− = − ==                      (5.34)  Tool Chip h0 δs φ−δs pi/2+δs φ ls  Figure 5.5 Orthogonal model with surface slope  The derivative of P with respect to φ, yields ( )[ ] 02sincoscos =−−⋅⋅= γδφγδφ ssd dP                               (5.35) The relationship of shear angle with surface slope can be stated as 5  A New Predictive Model for Peripheral Milling Operations  113 02 =−− γδφ s                                                      (5.36) Thus, Cs for the frictionless case is equal to 0.5, i.e. 5.0=sC                                                             (5.37) 5.6.3 Oblique Cutting Model The power consumed in oblique cutting is given by csss VARATIOkVkAP ⋅⋅⋅+=                                      (5.38) where As is the area of the shear plane having a surface slope ( ) i wh A sN s s cossin cos0 ⋅− ⋅ = δφ δ                                            (5.39) Further, if one adopts the optimization algorithms developed for the oblique cutting process (Zou, Yellowley and Seethaler 2009), the relationship between the shear angle change with respect to slope) and RATIO is easily calculated. The result is shown in Figure 5.6 for an inclination angle of 30 degrees. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2  (φN-φN0)/δs versus RATIO  Linear Regression (φ N - φ N 0)/ δ s RATIO 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2   (φN-φN0)/δs versus Slope  Linear Regression (φ N - φ N 0)/δ s RATIO  (a) i=300, γN=100 (b) i=300, γN=50 Figure 5.6 Shear angle relationship with RATIO  5  A New Predictive Model for Peripheral Milling Operations  114 It can be observed from Figures 5.6 (a) and (b) that the linear fitted parameter Cs changes very little for both γN=100 and γN=50, which can be approximated as 5.0159.0 +⋅= RATIOCs                                        (5.40) 5.7 Force Prediction Based on the Shear Plane Forces After one obtains the work surface slope, the shear force along the shear plane can be rewritten in the form ( )sN s s i kwh F δφ δ −⋅ = sincos cos0                                                 (5.41) where φN is the normal shear angle given by Eqn. (5.31). The normal resultant angle θN and mean nominal friction angle βN based on Oxley’s machining theory (see Section 5.2) are NNNN eqvsNN nConst γφθβ δφpiθ +−=       ⋅−      +−+= − 4 21tan 1                        (5.42) The friction angle along the chip flow direction is       = − c N η ββ cos tan tan 1                                           (5.43) Thus the resultant force Rtool can be obtained based on the force balance between the shear plane and the tool rake face as follows ( ) ( )[ ]NNcNN ss tool F R γφηβγφβ η −⋅⋅−−⋅ = sincostancoscos cos               (5.44) Based on the force balance on the plane perpendicular to the normal shear plane, the normal force in the shear plane can be derived as 5  A New Predictive Model for Peripheral Milling Operations  115 ( ) ( )[ ]NNcNNtool Nsss R FN γφηβγφβ θη −⋅⋅+−⋅⋅= = coscostansincos tancos             (5.45) The force diagram at the shear plane is shown in Figure 5.7 and the global force based on the shear plane can be given as iNiFiFF FNF iNiFiFF NsNssssR NssNsT NsNssssC sinsinsincoscoscossin sincoscos cossincoscoscossinsin ⋅−⋅⋅−⋅= ⋅−= ⋅+⋅⋅+⋅= φφηη φηφ φφηη              (5.46)  (a) 3-D shear plane forces    iF Nss coscoscos ⋅⋅ φη NssF φη sincos ⋅ iF Nss sincoscos ⋅⋅ φη ssF ηcos NssF φη coscos ⋅  (b) Shear force perpendicular to cutting edge (Part A) 5  A New Predictive Model for Peripheral Milling Operations  116 sN NsN φsin iN Ns sinsin ⋅φiN Ns cossin ⋅φ NsN φcos  (c) Shear plane normal force (Part B)  ssF ηsin iF ss sinsin ⋅η iF ss cossin ⋅η  (d) Shear force parallel to cutting edge (Part C) Figure 5.7 Force diagram for the shear plane  5.8 Kinematics of Helical End Milling Tools This section is concerned with the integration of the previous material into a model that is suited to the analysis of tools with helical cutting edges and non-planar rake faces, (primarily milling, drilling, and tapping operations). 5  A New Predictive Model for Peripheral Milling Operations  117 5.8.1 Coordinate System and Velocity Relationship The helical flute cutter coordinate system and the corresponding oblique cutting coordinate system are shown in Figure 5.8.  The coordinate transformation matrix for the coordinate (x, y, z) to the coordinate (x’, y’, z’) can be obtained through two angular transformations: 1) Rotate along the x axis with angle i; the transformation matrix can be written as           −= ii iiiRx cossin0 sincos0 001 )(                                            (5.47) 2) Rotate along y axis with angle γN; the transformation matrix can be written as ( )           − = NN NN NyR γγ γγ γ cos0sin 010 sin0cos                                    (5.48) t n b a i ze x' y' z' x y z V0 Helical Tool Oblique Cutting Coordinate N  Figure 5.8 Geometry of helical tool system  Thus the coordinate transformation from (x, y, z) to (x’, y’, z’) can be written as 5  A New Predictive Model for Peripheral Milling Operations  118 ( )[ ] [ ]                     ⋅⋅− ⋅−⋅ =                     −           − =⋅=           ′ ′ ′ z y x NNN NNN z y x NN NN T x T Ny z y x u u u ii ii ii u u u ii iiiRR u u u coscossincossin sincos0 cossinsinsincos cossin0 sincos0 001 cos0sin 010 sin0cos )( γγγ γγγ γγ γγ γ        (5.49) where γN is the normal rake angle, and i the inclination angle. Coordinates x, y, and z denote the uncut chip plane and coordinates x’, y’, and z’ denote the rake face coordinate system. Considering the velocity field along the tool radius direction (ur ) and along the tangential direction on helical surface (ut), the velocity relationship based on the above- defined coordinate systems is expressed as      = = = ′ ′ 0z ty rx u uu uu                                                           (5.50) and    = −= ′ cnx ctt Vu Vu                                                          (5.51) in which Vcn, Vct, Vsn and Vst are the velocities of straight edge oblique cutting in the rake face and shear plane, respectively. They can be written as (Zou, Yellowley and Seethaler 2009) ( ) ( )    ⋅⋅−= − ⋅ = iVSLIPV ViV ct NN N cn sin1 cos cossin 0 0γφ φ                                          (5.52) and ( )     ⋅⋅= − ⋅ = iVSLIPV ViV st NN N sn sin cos coscos 0 0γφ γ                                             (5.53) 5  A New Predictive Model for Peripheral Milling Operations  119 where V0 is the magnitude of the incoming velocity. Based on the Eqns.5.49, 5.50 and 5.51, the following relations can be obtained 0coscossincossin sincos cossinsinsincos ' ' =⋅⋅+⋅⋅−⋅= ⋅+⋅== ⋅⋅−⋅⋅+⋅== ′ zNyNrNz zyty zNyNrNcnx uiuiuu uiuiuu uiuiuVu γγγ γγγ                      (5.54) Solving Eqn. (5.54), the velocities uy and uz are given by iii uui u i uiu u NN rNtN z zt y sintancoscoscos sintancos cos sin ⋅⋅+⋅ ⋅−⋅⋅ = ⋅− = γγ γγ                                         (5.55) Substituting Eqn. (5.55) into Eqn. (5.49), the relationship of ux’ and ur is derived as r NN N r NN N t N t N tNNrx u iii i u iii ii u iii ii u iii ii uiuu sintancoscoscos cossin sintancoscoscos tansinsin sintancos tancossin sintancos tansinsin tansincos 2 2 2 ⋅⋅+⋅ ⋅ + ⋅⋅+⋅ ⋅⋅ + ⋅+ ⋅⋅ − ⋅+ ⋅⋅ − ⋅⋅+⋅= ′ γγ γ γγ γ γ γ γγ                                 (5.56) Simplifying Eqn. (5.56), the relationship of ux’ and ur is obtained as ( ) N r N N Nrx u uu γγ γγ coscos sin cos 2 =      += ′                                (5.57) 5.8.2 Helical Tool Velocity Profile The basic geometry of helical pitch during the milling process is shown in Figure 5.9. 5  A New Predictive Model for Peripheral Milling Operations  120 S S-δS i z (R-δR)ψL RψL  Figure 5.9 Basic geometry of helical pitch  From Figure 5.9, the helical pitch relationship can be written as [ ] [ ] iRSRSSS LL sin2222 ⋅⋅⋅−⋅+=− ψδψδδ                              (5.58) where pitch rotation angle is i R z L tan=ψ                                                            (5.59) Simplifying Eqn. (5.58), the following relationship can be obtained as iu dt dS Lr sin⋅⋅= ψ                                                        (5.60) Satisfaction of the condition of Eqn. (5.60) requires the velocity profile to be in the form ( )Lt r CCu Cu ψ⋅+= = 32 1                                                        (5.61) where C1, C2 and C3 are constants. ψL is the angle of global rotation; ur and ut denote the velocity in radial and tangential direction to the helix curve, respectively. It is noted that 5  A New Predictive Model for Peripheral Milling Operations  121 the direction of tangential velocity defined in Figure 5.8 is opposite to the chip sliding direction; therefore, Eqn. (5.60) can be rewritten in the form of ( )01 tt uudt dS −−=                                                     (5.62) δR ut0 ψL S ur ut1  Figure 5.10 Tangential velocity of the segment  where ut0, and ut1 are the tangential velocities at the start and the end of the segment shown in Figure 5.10. Substituting Eqn. (5.62) into Eqn. (5.60) yields [ ]01 sin 1 tt L r uui u − ⋅ −= ψ                                          (5.63) The helical tool boundary velocity conditions are given by Lt t CCu Cu ψ⋅+= = 321 20                                                (5.64) Substituting Eqns. (5.61) and (5.64) into Eqn. (5.63), the relationship between C1 and C3 is derived as iCC sin13 ⋅−=                                                 (5.65) Based upon Eqn. (5.57), the velocity ux’ is obtained N x C u γcos 1 = ′                                                 (5.66) 5  A New Predictive Model for Peripheral Milling Operations  122 Thus the parameter C1 is written as NcnVC γcos1 ⋅=                                              (5.67) Then the parameter C3 is derived as iViCC Ncn sincossin13 ⋅⋅−=−= γ                                    (5.68) After the constant parameters C1 and C3 are determined, the only unknown is C2. Considering the constant velocity gradient in tangential direction, it should have a height that meets the straight edge oblique cutting velocity ( ( ) 0sin1 ViSLIP ⋅⋅− ) along the tangential direction, that height is defined as zh with a rotational angle Rizhh /tan=ψ . Thus, using Eqn. (5.51) together with the relationship mentioned above yields ( ) iVSLIPi R zCC h sin1tan 032 ⋅−−=+                                (5.69) Then the parameter C2 is derived as ( ) ( ) i iV R ziVSLIPC Ncnh cos sin cossin1 2 02 γ⋅+⋅−−=                         (5.70) Finally, the radial and tangential velocities are written as ( ) LNcnhNcnt Ncnr iV R z i iViVSLIPu Vu ψγγ γ ⋅⋅⋅−       ⋅⋅−−−= = sincos cos )(sin cossin1 cos 2 0       (5.71) Substituting Eqn. (5.71) into Eqn. (5.52), the velocity field on the rake face can be derived as ( ) ( ) LNcnhNcnct NN N cn iV R z i iViVSLIPV ViV ψγγ γφ φ ⋅⋅⋅+       ⋅⋅⋅−−= − ⋅ = sincos cos )(sin cossin1 cos cossin 2 0 0      (5.72) The tangential velocity in the shear plane can be obtained by 5  A New Predictive Model for Peripheral Milling Operations  123 iVVV ctst sin0=+                                                       (5.73) Then the velocity field in the shear plane can be derived as ( ) LNcn h Ncnst NN N sn iV R z i iViVSLIPV ViV ψγγ γφ γ ⋅⋅⋅−       ⋅⋅+⋅= − ⋅ = sincos cos )(sin cossin cos coscos 2 0 0          (5.74)  5.8.3 Continuous Shear The plan view of a chip is given in Figure 5.11. R is the chip radius and δR is the change of radius. R734 , 37 161 ,01 R ur S S+δS . . . δR  Figure 5.11 Plan view of chip    Since the velocity in radial direction (ur) is constant, there is no velocity jump on radial lines, but there is velocity change in the circumferential direction. The chip arc length change at radius (R-δR) can be given as 5  A New Predictive Model for Peripheral Milling Operations  124 Lr RR RiuSS ψδδ ⋅     − ⋅⋅=+ sin&&                               (5.75) Eqn. (5.75) can be rewritten as ( )( ) Lr RiuSSRR ψδδ ⋅⋅⋅=+− sin&&                              (5.76) or R SRS & & δδ =                                                  (5.77) Thus S&  changes linearly with radius. If the inside chip radius is Ri and the outside chip radius is Ro, the change of S&  along the chip can be written as [ ] L i o r R R iuS ψ⋅⋅⋅=∆ sin&                                        (5.78) Then the volume shear power can be given as [ ] Loavshear RhkSP ψ⋅⋅⋅⋅∆= &                                      (5.79) where hav is the average chip thickness.  5.8.4 Special Case: Frictionless As discussed in Section 3.3.1, in the case of SLIP being zero, the complete solution of basic oblique cutting parameters is given by 2 2,0,0 piγφ =−== NNRATIOSLIP                                         (5.80) The power can be written as ss dAVkP ⋅⋅= ∫                                                        (5.81) where 5  A New Predictive Model for Peripheral Milling Operations  125 dz R idd i SRdA V VVVVV N t s sn st snstsns tan , sinsin 2 11 2 22 = ⋅ ⋅ =               +=+= φφφ                                  (5.82) For the frictionless case, the tangential velocity profile is linear and asymmetric along the depth of cut, the mean value of the velocity being at the mid depth ( )     − ⋅ ⋅ = z a iR iVV Ncnst 2cos sincos 2γ                                      (5.83) Then the power can be written as ( ) ( ) ( ) PPdzz a V V iR i i VkS P sn cnNa N snt ~ 2cos2 sincos1 cossin 22 22 42 0 +=               −      ⋅ ⋅ + ⋅ = ∫ γ φ              (5.84) where i aVkS P V V N snt N N sn cn cossin , cos sin ⋅ == φγ φ                                            (5.85) where P  is the power with the constant velocity profile, and P~  is the extra power related the linearly changed tangential velocity as ( ) ( ) ( ) ( ) ( ) iiaVkS RV V R aiiVkS dzzaza V V iR i i VkS P Nsnt sn cn N Nsnt a sn cnN N snt sintansin 24 1 sin24 sin)(tancos 4cos2 sincos cossin ~ 33 2 2 2 332 2 2 0 2 2 42 ⋅⋅=     ⋅⋅⋅ =       +−      ⋅ ⋅ ⋅ = ∫ φφ γ γ φ (5.86) Figure 5.12 plots the power versus depth of cut and helical angle for the case of inchDN 16/1,0 0 ==γ . Figure 5.13 plots the cutting power with respect to radius R and helical angle i for the case of mmaN 0.1,0 0 ==γ . From Figures 5.12 and 5.13, it can be observed that the power consumption is increased with an increased depth of cut or helical angle, but decreased with an increased tool diameter. In other words, for the 5  A New Predictive Model for Peripheral Milling Operations  126 large depths of cut and small diameter tools the kinematics will have a significant influence on the helical tool milling operation.  Figure 5.12 Power change versus depth of cut a and helical angle i  P o w er  ( k S tV 0 )  Figure 5.13 Power change versus tool radius R and helical angle i 5  A New Predictive Model for Peripheral Milling Operations  127 5.9 Cutting Force Prediction for a General Helical Tooth Milling Cutter In order to incorporate the above-mentioned physical processes in the model, a numerical approach is developed in this thesis. The velocity field is assumed to be constant along the radial direction and linear along the tangential direction. An external optimization process determines the velocity profile. Then the velocity profile is taken as an input for the further computations. Five steps are employed in the proposed methodology:  1) The position of the individual incremental helical cutting edges are calculated by cutting edge discretization; 2) The uncut chip thickness is calculated by performing the coordinate transformation; 3) The velocity profile (constant along the radial direction, linear along the tangential direction) is determined by the optimization processes; 4) The flow stress and edge force are calculated based on the local oblique cutting system and the new velocity profile; 5) The resulting cutting force is estimated by using the shear plane prediction methodology plus extra continuous shear caused by kinematics constraint.  The chip formation process and elementary geometry of a helical toothed milling cutter are shown in Figure 5.14.  5  A New Predictive Model for Peripheral Milling Operations  128  (a) 3D helical tooth flute j=2 flute j=3 flute j=4 flute j=1flute j=1 flute j=2 flute j=3 flute j=4 Cutter at z=0 Cutter at z=ze ψ  (b) Immersion angle at different height flute j=4 flute j=3 flute j=2 flute j=1 ρ κ o o'  (c) Cutter runout  Figure 5.14 Geometry of helical tooth  The helical tool has Nf flutes and Ne discrete elements are created on each flute. Each element (e) can be considered as an elemental oblique cutting edge which has a width of ∆a, a normal rake angle γN, an inclination angle i, and the three local coordinates (n, t, b). It is assumed that the position of the middle section of the element is the height 5  A New Predictive Model for Peripheral Milling Operations  129 of the element (ze). The uncut chip thickness, immersion angle and height of each element are given as ( ) ze z zi R z N j z zRzRzSmzh e e f ej emjejejtjej j ∆−+∆=−−⋅−= −+= − )1( 2 ,tan 12)( )()()(sin)( piψψ ψ                   (5.87) where St is the feed per tooth, mj means that the current tooth or flute (j) is removing the material left by the mjth tooth (Sutherland and DeVor 1986, Wang and Chang 2003, and Wang and Huang 2004), and ( )         − −−⋅+= f e ej N ji R z RzR piκρ 12tancos)(                        (5.88) where ρ and κ are the magnitude and angle shift of runout (shown in Figure 5.13c). By resolving Eqns. (5.88) and (5.87) together, Eqn. (5.87) can be rewritten as ( )                 −− −−⋅         −+= f ie f j jejtjej N mji R z N m mzSmzh piκ pi ρψ 22tansinsin2)(sin)(  (5.89) The kinematics constraint assumed for helical tools requires that the chip flow velocity is constant along the radial direction and linear changing along the tangential direction. The velocity fields in the rake face and the shear plane for element (e) are given as (see Section 5.8) ( ) ( ) ( )       −−⋅−= − ⋅ = R z R z i iViVSLIPV ViV eh N e cn e ct NN Ne cn cos sin cossin1 cos cossin 2 0 0 γ γφ φ          (5.90) ( ) ( )       −+⋅= − ⋅ = R z R z i iViVSLIPV ViV eh N e cn e st NN Ne sn cos sin cossin cos coscos 2 0 0 γ γφ γ             (5.91) 5  A New Predictive Model for Peripheral Milling Operations  130 The power consumption is expressed as (see Chapter 3 and Section 5.8.3)       + − +=∑ = e shear e s e c e e s e ce s e s e N e PAVk V V SLIP SLIPAVkP e 11                        (5.92) where esA  is the shear plane area for element e i azh A N eje s cossin )( ⋅ ∆ = φ                                                (5.93) The velocity profile is obtained by minimizing the power consumption. However, Eqn. (5.92) is a highly nonlinear function that is difficult to solve directly and the highly nonlinear effect will not affect the velocity profile (see Section 5.8) significantly. For simplicity then, it is assumed that the shear yield stress and shear angle may be assumed constant at an average value for the incut flute. An external optimization algorithm is then used to determine the velocity profile. After obtaining the velocity profile and uncut chip thickness, the shear yield stress can be determined exactly from the Johnson-Cook model for each individual element, and then the shear force at the shear plane including the extra continuous shear caused by kinematics constraint for element (e) can be given as ( ) es N e shear e sN e sej e e s i V P i zahk F η φ δφ δ cos coscos sincos cos)( 0 ⋅ + −⋅ ∆ =                          (5.94) in which the shear angle Nφ  can be obtained from centerline milling tests with the same feedrate expressed in Section 6.4. Then following the procedures discussed in Sections 5.5 and 5.7, the oblique cutting forces acting on each element (e) can be derived as      += += += )()()( )()()( )()()( jFjFjF jFjFjF jFjFjF e RE e RS e R e TE e TS e T e CE e CS e C                                       (5.95) 5  A New Predictive Model for Peripheral Milling Operations  131 It is noted that when the element (e) is at the entry or the exit region, the forces )(),(),( jFjFjF eRSeTSeCS  will be set to zero and only the edge forces ( )(),(),( jFjFjF eREeTEeCE ) are left and calculated by Eqn. (5.27). Then the oblique cutting forces in each element in the local principal directions (C, T, R) shown in Figure 5.14a are transformed to the global elemental forces in x, y and z directions as                     −=           )( )( )( 100 0)(cos)(sin 0)(sin)(cos )( )( )( jF jF jF zz zz jF jF jF e R e T e C e j e j e j e j e z e y e x ψψ ψψ                        (5.96) These forces must now be summed for each edge j of the elementary disk and for each elementary disk along the engaged length of flute on each tooth of the milling cutter. The instantaneous forces on the whole cutter are a function of mill angular position ψ and given by                   =           ∑∑ ∑∑ ∑∑ == == == )( )( )( )( )( )( 11 11 11 jF jF jF F F F e z N e N j e y N e N j e x N e N j z y x ef ef ef ψ ψ ψ                               (5.97) 5.10 Conclusions The mathematical formulations for the analysis of peripheral milling operations are presented. The Johnson-Cook constitutive model is used to allow the consideration of chip thickness, strain, strain rate, work hardening and temperature on yield strength and normal stress on the shear plane. The ploughing force model for orthogonal cutting developed in Chapter 4 is applied for the prediction of edge forces which are resolved 5  A New Predictive Model for Peripheral Milling Operations  132 from the velocity plane to provide the required oblique force components. A new approach for the consideration of surface slope on the shear angle and cutting forces is presented and the influence of slope on shear angle and forces are investigated. An improved but still simple kinematic arrangement that satisfies the plastic incompressibility for the helical tool is proposed. The model allows the prediction of the influence of helical angle, radius and depth on the degree of constraint and provides estimates of the additional energy requirements in the process.   133 6  Experimental Validation of the Predictive Model  6.1 Introduction  A GUI-based Matlab program based on the theoretical formulations presented in Chapter 5 has been developed to estimate forces in the milling process using helical toothed cutters. In order to verify the model and to estimate the overall predictive capability of the model, it is necessary to assess the model qualitatively and quantitatively with respect to experimental results. This chapter describes a series of experiments with different workpiece materials, tool diameters, milling types, and a wide range of feeding velocities which allow the examination of the accuracy of the proposed model. The edge radius of the cutters used in the experiments has been measured using a scanning electron microscope (SEM) to allow edge forces to be estimated, and an innovative algorithm is used to identify the constitutive parameters of the work materials using a central cut milling test. The proposed algorithm does not need shear angle data for calibration, relying instead on the measured force data. This chapter also presents the constitutive equations and primary shear zone parameter identifications for steel, titanium alloy and aluminum alloy workpieces. 6.2 Experimental Setup  All experiments were conducted on a vertical milling machine center using end mills and dry cutting. The three orthogonal milling force components were measured using a three component dynamometer which incorporates piezoelectric load cells; signals were low pass filtered with a cutoff frequency of 200 Hz and data was recorded at 6  Experimental Validation of the Predictive Model  134 sampling rates ranging from 1×104 (1/s) to 2.5×104 (1/s) using a Tektronix Oscilloscope. The testing program included the use of 3 workpiece materials with very different physical properties. The tests examined both up and down milling operations; additional tests used centerline face milling to reduce the variation in chip thickness and to maintain surface slope at a lower value (note centerline milling also allows easier identification of tool position). The cutters used were HSS end mills to cut an aluminum alloy and solid carbide end mills with TiN coating to cut a titanium alloy and high strength steel. The cutter diameters varied from 3/16 inch to 1/2 inch and the feedrates were chosen to best allow the identification of physical parameters while avoiding tool edge or shank breakage. It should be noted that the use of small end mills on the hardened steel led to the adoption of small feeds and in this case cutters have a single edge (of two) ground away to avoid the need to deal with cutter runout in the identification procedure. The range of experiments, tools and workpiece materials are shown in Table 6.1, a full list of experiments is given in Appendix A. Table 6.1 The range of cutting tests undertaken Workpiece Material Tool Manufacturer Cutter Diameter (in) Feeding Velocity (in/min) Cutting Type Uncut Chip Thickness (mm/per tooth) Spindle Speed (RPM)  Aluminum Alloy Niagara Cutter: High Speed Steel  1/2  1.28 -7.9 Up/Down /Center/ Slotting 0.01626- 0.10033  1000  Titanium Alloy SGS Tools: Solid Carbide TiN Coated  3/8  0.75-4.4  Up/Down/ Center  0.01656- 0.09718  575  Titanium Alloy SGS Tools: Solid Carbide TiN Coated  3/16  0.75-2.56  Up/Down/ Center  0.00827- 0.0282  1152  Steel SGS Tools: Solid Carbide TiN Coated  3/8  0.75-3.95  Slotting  0.0191- 0.10033  1000 6  Experimental Validation of the Predictive Model  135 6.3 Edge Radius Measurement Cutting tool edges are critical to cutting performance (Filiz et al. 2007). Edge sizes range from microscale (including cutting tools for plastic web materials and diamond tools for precision applications) to mesoscale for traditional cutting tool inserts (Aramchareon et al. 2008). In this thesis, three types of helical end-milling cutters (1/2 inch HSS end mill cutter with helical angle 30 deg, 3/8 inch solid carbide coated TiN end milling cutter with 30 deg helical angle and 3/16 inch solid carbide coated TiN end milling cutter with 30 deg helical angle) have been measured using the scanning electron microscope (SEM). The cutters have been cut on a diamond-slicing machine and encapsulated in an epoxy resin using a mould cylinder. Before examining under the microscope, the samples were polished using abrasive paper in four stages (P180, P400, P600 and P1200 grades) and finally polished by 6 µm and 1 µm diamond grinding as shown in Figure 6.2. Detailed information regarding cutters A-E is given in Table 6.2.   Cutter A   Cutter B   Cutter C   Cutter D  Cutter E  Figure 6.1 Mounted cutter samples 6  Experimental Validation of the Predictive Model  136 Table 6.2 Cutter information Cutter Type Manufacture Company Tool Material Tool Diameter Number of Teeth Helix Angle Radial Rake Angle Cutter A Niagara Cutter HSS 1/2 inch 2 300 100 Cutter B SGS Tools Solid Carbide TiN Coated 3/8 inch 2 300 50 Cutter C SGS Tools Solid Carbide TiN Coated 3/8 inch 2 300 50 Cutter D SGS Tools Solid Carbide TiN Coated 3/8 inch 1 300 50 Cutter E SGS Tools Solid Carbide TiN Coated 3/16 inch 2 300 50 Note: In Cutter D one tooth is ground away and only one tooth left Figures 6.2-6.6 show the SEM images of the HSS cutter used to cut the aluminum alloy, the 3/8” SGS carbide coated cutter used to cut the titanium alloy and steel, and the 3/16” SGS carbide coated cutter used to cut the titanium alloy, respectively.  The cutting operations and the measured edge radiuses are given in Table 6.3. Table 6.3 The cutting operations and measured edge radii Cutter Type Tool Diameter (inch) Cutting Operations Mean Edge Radii (µm) Cutter A 1/2 Aluminum Alloy (Table AA.1) (Up/Down/Center) 3.425 Cutter B 3/8 Titanium Alloy Table AA.2: Test 1-5), (Center Cut) 5.3026 Cutter C 3/8 Titanium Alloy (Table AA.2: Test 6-10), (Up/Down Milling) 6.6332 Cutter D 3/8 Steel (Table AA.4: Test 1-6) (Slotting) 4.4873 Cutter E 3/16 Titanium Alloy (Table AA.3: Test 1-12), (Up/Down/Center) 3.5473 6  Experimental Validation of the Predictive Model  137  Figure 6.2 SEM image for Cutter A   Figure 6.3 SEM image for Cutter B   Figure 6.4 SEM image for Cutter C 6  Experimental Validation of the Predictive Model  138  Figure 6.5 SEM image for Cutter D     Figure 6.6 SEM image for Cutter E 6  Experimental Validation of the Predictive Model  139 6.4 Identification of the Constitutive Parameters The advantage of using a constitutive model over following a purely empirical method is that, for a known material, the experimental data required to determine the dependence of flow stress on temperature, strain and strain rate can be dramatically reduced.  Ideally, theoretical relationships derived from the physical processes at the atomic level should be used to describe the macroscopic flow behavior of material. However, a soundly based theoretical approach of good accuracy is still some way from being realized (Jaspers and Dautzenberg 2002). Consequently, it is inevitable that one must be satisfied with an empirical model of the material behavior itself. In this thesis, the Johnson-Cook constitutive model has been selected as the basis for the prediction of material yield strength and work hardening over the shear plane. The values of the constants in the constitutive equation are available in the literature for conditions approaching those found in metal cutting. To allow for material variations due to slight changes in composition, structure and treatment, it is assumed that the Johnson-Cook constitutive parameters, (m, n, Cm) are the same as those quoted in the literature but that parameters A and B within the term ( )nBA ε+  may be scaled. This methodology will maintain the relationships between the variables but vary the magnitude of shear yield stress. The identification process is carried out using the one set of forces collected at the center point of a centered face milling cut. The process adopted for identification minimizes the errors due to “timing” of the force signal; it minimizes edge force components by using the largest chip thickness during the cut and eliminates some of the error due to slope considerations. The identified process is based on the largest feed of central cut, the uncut chip thickness is more than an order of magnitude 6  Experimental Validation of the Predictive Model  140 larger than the edge radius in all of these tests therefore the edge force effect to the total force will be limited. Based upon the measured three forces for large feed, the forces acting on the rake face (shown in Figure 6.7) are given as [ ] [ ]    −⋅+= −= +⋅+= NTNRC RCft NTNRCfr FiFiFN iFiFF FiFiFF γγ γγ sincossincos cossin cossinsincos                              (6.1)  The friction force Ff and chip flow angle can then be obtained as         =+= − fr ft cftfrf F F FFF 122 tan, η                                         (6.2) Based on the work of Zou, Yellowley and Seethaler (2009), the kinematic parameter SLIP is given as 2 53039.047799.00928.1       −      −= ii SLIP cc ηη                             (6.3) The normal shear angle is then estimated based on the energy minimization of the oblique cutting processes discussed in Section 5.3 as [ ] [ ] cfmmn VFTC n BAwVhP +−⋅+⋅    + += + **100 1ln11 εεε &            (6.4) In order to apply the optimization technique, the constitutive parameters are selected from the literature for the initial step, the parameters in the following steps use the parameters obtained from the previous step. The cutting velocity is 60000/4.250 ⋅Ω⋅= DV pi                                      (6.5) where D: diameter of tool (inch); Ω: spindle speed (rev/min); V0: cutting velocity (m/s). The model of helical tool kinematics adopted here requires that the chip velocity is constant along the radial direction and changes linearly along the tangential direction (see 6  Experimental Validation of the Predictive Model  141 Section 5.7). After obtaining the normal shear angle, the forces acting on the shear plane (shown in Figure 6.7) may be calculated as follows [ ] [ ]    +⋅+= +−= −⋅+= NTNRCs RCst NTNRCsr FiFiFN iFiFF FiFiFF φφ φφ cossinsincos cossin sincossincos                           (6.6) Then the total shear force and the shear flow angle can be given as       =+= − sr st ssrsts F F FFF 122 tan, η                                        (6.7) So the shear yield stress and normal pressure on the shear plane are 00 cossin , cossin wh iN A N p wh iF A Fk Ns s s AB Ns s s AB ⋅ == ⋅ == φφ                      (6.8) The mean inplane stress and the normal stress at point A on the shear plane are estimated as (Oxley 1989)             +−+= + = sNABA BA AB kp pp p δφpi 4 21, 2                                (6.9)   Figure 6.7 Forces acting on the rake face and shear plane 6  Experimental Validation of the Predictive Model  142 Therefore, from Eqns. (6.8) and (6.9), the pressure at point B can be obtained as A Ns B p wh iN p − ⋅ = 0 cossin2 φ                                       (6.10) Applying force equilibrium along AB, the following relations result (Oxley 1989) N N AB ds ds dkdp 1 2 =                                                 (6.11) where s1N and s2N are the slip-line length along AB and the width of the zone at the normal plane. By substituting Eqn. (5.21) into Eqn. (6.11), the following relationship can be derived as ABeqBA knConstpp ××=− 2                                          (6.12) where pA and pB are the hydrostatic stresses at points A and B, respectively. Then the value Const is obtained as ABeq Ns sNAB kn wh iNk Const 0 cossin 4 21 ⋅−            +−+ = φδφpi                           (6.13) The optimization technique used to identify these parameters is then phrased as AccuracyConstConst kk measure AB measure AB ≤− − )abs(:toSubject Minimum                            (6.14) where the shear yield stress kab is calculated based on the strain, strain rate and temperature. It is noted that during the first iteration it is not necessary to consider the edge force effect on the large feed. In order to predict the constitutive parameters and shear angle accurately, the edge force effect can be considered in the following iteration rounds. The edge forces at large feed are approximated based on the methodology 6  Experimental Validation of the Predictive Model  143 discussed in Chapter 4 and Section 5.5 as well as the shear yield stress obtained from Eqn. (6.14). Then the measured forces with edge forces subtracted are used for the following round of identification procedures, (i.e., FC, FR, and FT are replaced by (FC- FCE), (FR-FRE), and (FT-FTE), respectively). The whole identification process stops when the required accuracy of shear angle is obtained, (i.e., ( ) εφφφ ≤− −1jNjNabs , where εφ is the prescribed shear angle accuracy). The above methodology avoids the need for shear angle data, relying instead on experimental force data for calibration and an internal optimization routine to find the normal shear angle. This is important since actual milling operations do not yield chip geometries which may be analyzed for shear angle. Three materials, AL-6061 T6, Ti-6AL-V4 and AISI 4140 Steel have been tested in the milling experiments. The material properties are given in Table 6.4. Table 6.4 Workpiece material properties Material Al-6061-T6 (Johnson et al. 1996) Ti-6Al-4V (Meyer and Kleponis 2001) AISI 4140 Steel (Arrazola et al. 2008) Thermal Conductivity (K) 167 (w/m-K) 7.3 (w/m-K) 46 (w/m-K) Specific Heat (S) 896 (J/kg-0C) 580 (J/kg-0C) 450 (J/kg-0C) Melting Temperature (Tmelt) 582 (0C) 1605 (0C) 1460 (0C) Young Modulus (E) 68.9GPa) 113.8 (GPa) 210 (GPa) Poisson Ratio (ν) 0.33 0.342 0.3 Density (ρ) 2700 (kg/m3) 4430 (kg/m3) 7862 (kg/m3)  The forces measured at the midpoint of a large feed central cut are shown in Table 6.5. Based on the measured forces, shear angles expected to be 25.60, 23.70, and 24.90 for the cutting of aluminum alloy, titanium alloy and steel, respectively. As mentioned previously, the identification process is conducted to keep the Johnson-Cook constitutive parameters, (m, n, Cm), the same as found in previous literature (Johnson et al. 1996, Meyer and Kleponis 2001, and Arrazola et al. 2008 for aluminum alloy, titanium alloy, 6  Experimental Validation of the Predictive Model  144 and steel, respectively), on the other hand the magnitude of ( )nBA ε+  is allowed to vary to fit the experimental data. The identified Johnson-Cook parameters, width of shear zone and thermal transfer fraction are given in Table 6.6. Table 6.5 The measured forces at the midpoint of central cuts Measured Forces Material Cutting Operation FC (N) FT (N) FR (N) Aluminum Alloy Table AA.1: Test 5 195.1 122.4 27.1 Titanium Alloy Table AA.2: Test 5 351.2 235.3 114.8 Steel Table AA.4: Test 6 276.4 111.9 40.9  Table 6.6 Identified J-C constitutive parameters for aluminum alloy, titanium alloy and steel Johnson-Cook Constitutive Equation Parameters Identified Parameters for Aluminum Alloy Identified Parameters for Titanium Alloy Identified Parameters for Steel A(MPa) 422.5 728.6 648.3 B(MPa) 148.7 279.8 832.6 Cm 0.002 0.012 0.0137 n 0.42 0.34 0.2092 m 1.34 0.8 0.807 const 7.47 7.95 7.94 x 0.681 0.692 0.801  Based on the parameters determined for the Johnson-Cook equation, the cutting forces are predicted using the oblique cutting force model (described in Chapter 5) and compared with the measured cutting forces in centerline milling operations. The comparison between calculated and measured forces is shown in Figures 6.8-6.10 for aluminum alloy, titanium alloy and steel, respectively. It is noted that the calculated ploughing force has been subtracted from the measured force and that the calculated force does not include the ploughing component. The force versus chip thickness is 6  Experimental Validation of the Predictive Model  145 concave and so any attempt to use a linear relationship on the full force to estimate the edge forces will lead to an overestimate. The shear angles predicted for the centerline milling operations are shown in Figures 6.11-6.13 for aluminum alloy, titanium alloy and steel, respectively. It can be observed that the shear angle decreases with a reduction in uncut chip thickness. 0.00 0.02 0.04 0.06 0.08 0.10 0 50 100 150 200 250 300    Measured Force FC  Measured Force FT  Predicted Force FC  Predicted Force FT Fo rc es  F C  an d F T  (N ) Uncut Chip Thickness h0 (mm)  Figure 6.8 Centerline cutting force comparison as feedrate changes for central cutting aluminum alloy (cutter A) 0.00 0.02 0.04 0.06 0.08 0.10 0 50 100 150 200 250 300 350 400   Measured Force FC  Measured Force FT  Predicted Force FC  Predicted Force FT Fo rc es  F C  an d F T  (N ) Uncut Chip Thickness h0 (mm)  Figure 6.9 Centerline cutting force comparison as feedrate changes for central cutting titanium alloy (cutter B) 6  Experimental Validation of the Predictive Model  146 0.00 0.02 0.04 0.06 0.08 0.10 0 50 100 150 200 250 300    Measured Force FC  Measured Force FT  Predicted Force FC  Predicted Force FT Fo rc es  F C  an d F T  (N ) Uncut Chip Thickness h0 (mm)  Figure 6.10 Centerline cutting force as feedrate changes for slotting of steel (cutter D)  0.00 0.02 0.04 0.06 0.08 0.10 0 5 10 15 20 25 30 35 40    Shear Angle  Second Order Fit Sh ea r A ng le  φ N 0 h0 (mm)  Figure 6.11 Predicted shear angle versus uncut chip thickness for central cutting aluminum alloy (cutter A)  6  Experimental Validation of the Predictive Model  147 0.00 0.02 0.04 0.06 0.08 0.10 0 5 10 15 20 25 30    Shear Angle  Second Order Fit Sh e a r An gl e  φ N 0 Uncut Chip Thickness h0 (mm)  Figure 6.12 Predicted shear angle versus uncut chip thickness for central cutting titanium alloy (cutter B)  0.00 0.02 0.04 0.06 0.08 0.10 0 5 10 15 20 25 30    Shear Angle  Second Order Fit Sh ea r A n gl e φ N 0 Uncut Chip Thickness h0 (mm)  Figure 6.13 Predicted shear angle versus uncut chip thickness for slotting steel (cutter D) 6  Experimental Validation of the Predictive Model  148 6.5 Detailed Validation of the Proposed Model The results predicted by the proposed model are compared with the experimental test results in this section. (The details of the test have been described earlier in this chapter and a full list of tests is given in Appendix A). It is known that the kinematic constraint imposed by the helical tooth is a minor factor in almost all practical operations and especially here where axial depths are relatively low. The author has simplified the process by approximating the expected change in velocity along the edge for constant shear angle and yield stress using the UB method described earlier. The estimated velocity profile is then input to the general force calculation method discussed in Chapter 5. Figures 6.14-6.17 show the estimated single flute radial and tangential velocity components for cutting aluminum alloy (Cutter A), titanium alloy (Cutter B and C), titanium alloy (Cutter E), and steel (Cutter D), respectively. The dashed black line represents the radial velocity component; the solid blue line represents the expected tangential component of velocity for a straight edge oblique cutting process, while the dotted red line represents the tangential component of velocity along the helical tool cutting edges (the component changes linearly along the helix cutting edge as dictated by Eqn (5.71). The incoming velocities for cutting aluminum alloy (Cutter A), titanium alloy (Cutter B and C), titanium alloy (Cutter E), and steel (Cutter D) are 0.665 m/s, 0.2868 m/s, 0.2872 m/s and 0.4987 m/s, respectively. The ratios of constant tangential velocity versus radial velocity for the cutting aluminum alloy (Cutter A), titanium alloy (Cutter B and C), titanium alloy (Cutter E), and steel (Cutter D) are 0.6013, 0.4010, 0.6886, and 0.5503, respectively. The following sections present the complete force validation results. It should be noted that runout is not considered. 6  Experimental Validation of the Predictive Model  149  Figure 6.14 Velocity profile for cutting aluminum alloy (Cutter A)   Figure 6.15 Velocity profile for cutting titanium alloy (Cutter B and C)  6  Experimental Validation of the Predictive Model  150  Figure 6.16 Velocity profile for cutting titanium alloy (Cutter E)   Figure 6.17 Velocity profile for cutting steel (Cutter D)  6  Experimental Validation of the Predictive Model  151 6.5.1 Validating Simulation Results for Aluminum Alloy The tests were all conducted using cutter A and cover a wide range of feeds and a variety of milling modes. The test conditions were selected so as to allow the assessment of the influence of the second order effects that have been included in this model but are rarely addressed by other authors within actual model development. The force measurements are presented in Figures 6.18-6.28 which portray the feeding and normal forces Fx and Fy as a function of cutter rotational angle in up milling, down milling, and central cut modes. A good agreement between the predictive model (all force component modules included) and experimental results is found excepting in those cases where the feeding velocity is very low, (Vf=1.28 in/min), and runout distorts the expected result.  Figure 6.18 Measured and predicted forces Fx and Fy for up milling of aluminum alloy (Vf=3.95 in/min, Cutter A)  6  Experimental Validation of the Predictive Model  152  Figure 6.19 Measured and predicted forces Fx and Fy for up milling of aluminum alloy (Vf=7.9 in/min, Cutter A)   Figure 6.20 Measured and predicted force Fx and Fy for down milling of aluminum alloy (Vf=1.28 in/min, Cutter A)  6  Experimental Validation of the Predictive Model  153  Figure 6.21 Measured and predicted force Fx and Fy for down milling of aluminum alloy (Vf=2.56 in/min, Cutter A)   Figure 6.22 Measured and predicted force Fx and Fy for down milling of aluminum alloy (Vf=3.95 in/min, Cutter A)  6  Experimental Validation of the Predictive Model  154  Figure 6.23 Measured and predicted force Fx and Fy for down milling of aluminum alloy (Vf=6.75 in/min, Cutter A)   Figure 6.24 Measured and predicted forces Fx and Fy for central cutting aluminum alloy (Vf=1.28 in/min, Cutter A) 6  Experimental Validation of the Predictive Model  155  Figure 6.25 Measured and predicted forces Fx and Fy for central cutting aluminum alloy (Vf=2.56 in/min, Cutter A)   Figure 6.26 Measured and predicted forces Fx and Fy for central cutting aluminum alloy (Vf=4.4 in/min, Cutter A) 6  Experimental Validation of the Predictive Model  156   Figure 6.27 Measured and predicted forces Fx and Fy for central cutting aluminum alloy (Vf=6.75 in/min, Cutter A)   Figure 6.28 Measured and predicted forces Fx and Fy for central cutting aluminum alloy (Vf=7.9 in/min, Cutter A)  6  Experimental Validation of the Predictive Model  157 To investigate the importance of the ploughing effect, force predictions with and without considering the ploughing force are shown in Figures 6.29 and 6.30 for feeds of Vf=1.28 in/min and 2.56 in/min respectively in the central cut operations. The figures confirm the strong influence of ploughing force; these of course are especially significant at low feed.  Figure 6.29 Force comparison with and without considering ploughing force on central cutting aluminum alloy (Vf=1.28 in/min, Cutter A)  Figure 6.30 Force comparison with and without considering ploughing force on central cutting aluminum alloy (Vf=2.56 in/min, Cutter A) 6  Experimental Validation of the Predictive Model  158 The influence of surface slope variation on the cutting forces is shown in Figures 6.31 and 6.32 for both central cutting and up milling of aluminum alloy at a feed Vf=7.9 in/min. It can be seen that the surface slope changes both the force magnitude and shape.  Figure 6.31 Force comparison with and without considering slope effect on central cutting aluminum alloy (Vf=7.9 in/min, Cutter A)   Figure 6.32 Force comparison with and without considering slope effect on up milling aluminum alloy (Vf=7.9 in/min, Cutter A) 6  Experimental Validation of the Predictive Model  159 Figure 6.33 shows the force comparison between the experimental data and the predicted results for slotting aluminum alloy at the feeding speeds 2.2 in/min and 4.4 in/min. It should be noted that in this case, there is a jump in depth from 0.05 inch to 0.165 inch at the center of the cut. This depth jump means that there is a kinematics change around the jump locations, which can be used to further validate the accuracy of the proposed predictive model. As expected, a good agreement can be observed for these two cutting processes.  Vf=2.2 in/min  (b) Vf=4.4 in/min Figure 6.33 Measured and predicted forces Fx and Fy for slotting (center jump of depth of cut) aluminum alloy at different feeding velocities (Cutter A) 6  Experimental Validation of the Predictive Model  160 6.5.2 Validating Simulation Results for Titanium Alloy with 3/8” SGS Cutter Experiments were conducted at a spindle speed 575 RPM and included down milling, up milling and central cutting of the titanium alloy at five different feed rates ranging from 0.75 in/min to 4.4 in/min. Figures 6.34-6.44 plot the force comparison between the predicted results and measured values. The dotted line indicates the predictions made from the present model; the solid line represents the experimental results. Overall, the present model is shown to accurately predict the cutting forces in both x and y directions.  Figure 6.34 Measured and predicted forces Fx and Fy for down milling of titanium alloy (Vf=0.75 in/min, Cutter C)  Figure 6.35 Measured and predicted forces Fx and Fy for down milling of titanium alloy (Vf=1.5 in/min, Cutter C) 6  Experimental Validation of the Predictive Model  161  Figure 6.36 Measured and predicted forces Fx and Fy for down milling of titanium alloy (Vf=2.2 in/min, Cutter C)   Figure 6.37 Measured and predicted forces Fx and Fy for up milling of titanium alloy (Vf=0.75 in/min, Cutter C)  6  Experimental Validation of the Predictive Model  162  Figure 6.38 Measured and predicted forces Fx and Fy for up milling of titanium alloy (Vf=1.5 in/min, Cutter C)    Figure 6.39 Measured and predicted forces Fx and Fy for up milling of titanium alloy (Vf=4.4 in/min, Cutter C) 6  Experimental Validation of the Predictive Model  163  Figure 6.40 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=0.75 in/min, Cutter B)   Figure 6.41 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=1.5 in/min, Cutter B) 6  Experimental Validation of the Predictive Model  164  Figure 6.42 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=2.2 in/min, Cutter B)    Figure 6.43 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=3.95 in/min, Cutter B) 6  Experimental Validation of the Predictive Model  165  Figure 6.44 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=4.4 in/min, Cutter B)  The predicted contributions of ploughing force to overall force in milling operations have been investigated. The predicted forces for up milling, down milling, and central cutting of titanium alloy at a low feed of Vf=0.75 in/min are shown in Figures 6.45, 6.46 and 6.47, respectively. The solid line in these figures represents the measured forces, the dotted line indicates the force prediction including the ploughing forces, and the dashed line indicates the predicted results without considering the ploughing forces. The figures confirm the strong influence of ploughing force at low feed. The proposed upper bound model does an excellent job of accounting for the additional forces due to ploughing. The slope effect was examined previously for the high feed Vf=4.4 in/min in up milling and central cutting, it is interesting to compare the influence of slope at the lower feed and to contrast this with the influence of ploughing force. Figures 6.48 and 6.49 show the predicted results with and without considering the slope effect. It can be 6  Experimental Validation of the Predictive Model  166 seen that the surface slope does influence the overall shape of the force magnitude, but to a much lesser extent than the ploughing effect at these low chip thicknesses.  Figure 6.45 Force comparison with and without considering ploughing force for central cutting titanium alloy (Vf=0.75 in/min, Cutter B)   Figure 6.46 Force comparison with and without considering ploughing force for up milling of titanium alloy (Vf=0.75 in/min, Cutter C) 6  Experimental Validation of the Predictive Model  167  Figure 6.47 Force comparison with and without considering ploughing force for down milling of titanium alloy (Vf=0.75 in/min, Cutter C)   Figure 6.48 Force comparison with and without considering slope effect for up milling of titanium alloy (Vf=4.4 in/min, Cutter C) 6  Experimental Validation of the Predictive Model  168  Figure 6.49 Force comparison with and without considering slope effect for central cutting titanium alloy (Vf=4.4 in/min, Cutter B)  6.5.3 Validating Simulation Results for Titanium Alloy with 3/16” SGS Cutter The identified constitutive parameters of titanium alloy are now used in the predictions of cutting forces for the data collected (Table AA.3) using a 2 teeth 3/16 inch SGS Ti-N coated carbide end mill cutter (Cutter E). In order to keep the same cutting velocity with the 3/8 inch SGS cutter, the spindle speed is doubled from the 3/8” tool tests to 1152 RPM for 3/16” cutter. The tool edge radius for 3/16” cutter is re=3.5473 µm. Figures 6.50-6.61 show the comparison of the predicted results with measured values for the up milling, down milling and central cutting of titanium alloy. These comparisons clearly show that the predictions are in good agreement with the experiments; there is an 6  Experimental Validation of the Predictive Model  169 obvious but slight difference between the measured and the predicted cutting force that is caused by the tool run-out.  Figure 6.50 Measured and predicted forces Fx and Fy for down milling of titanium alloy (Vf=0.75 in/min, Cutter E)   Figure 6.51 Measured and predicted forces Fx and Fy for down milling of titanium alloy (Vf=1.5 in/min, Cutter E) 6  Experimental Validation of the Predictive Model  170  Figure 6.52 Measured and predicted forces Fx and Fy for down milling of titanium alloy (Vf=2.2 in/min, Cutter E)    Figure 6.53 Measured and predicted forces Fx and Fy for down milling of titanium alloy (Vf=2.56 in/min, Cutter E) 6  Experimental Validation of the Predictive Model  171  Figure 6.54  Measured and predicted forces Fx and Fy for up milling of titanium alloy (Vf=1.5 in/min, Cutter E)    Figure 6.55  Measured and predicted forces Fx and Fy for up milling of titanium alloy (Vf=1.5 in/min, Cutter E) 6  Experimental Validation of the Predictive Model  172   Figure 6.56  Measured and predicted forces Fx and Fy for up milling of titanium alloy (Vf=2.2 in/min, Cutter E)   Figure 6.57  Measured and predicted forces Fx and Fy for up milling of titanium alloy (Vf=2.56 in/min, Cutter E)  6  Experimental Validation of the Predictive Model  173  Figure 6.58 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=0.75 in/min, Cutter E)    Figure 6.59 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=1.5 in/min, Cutter E)  6  Experimental Validation of the Predictive Model  174  Figure 6.60 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=2.2 in/min, Cutter E)   Figure 6.61 Measured and predicted forces Fx and Fy for central cutting titanium alloy (Vf=2.56 in/min, Cutter E)   6  Experimental Validation of the Predictive Model  175 Figures 6.62-6.64 show the force comparisons between the measured values and the predicted results when the ploughing force is neglected at the relatively low feed of Vf=0.75 in/min. The red solid line indicates the measured forces, the blue dotted line represents the predicted forces including the ploughing force, and the cyan dash-dotted line represents the predicted forces neglecting the ploughing force. There is a large error when the ploughing component is neglected (generally greater than 15%). Figures 6.65- 6.67 show the force components Fx and Fy as a function of rotation angle at the higher feeding velocity of Vf=2.56 in/min, for the down milling, up milling, and central cutting of the titanium alloy. It is expected that there is a larger range of slope here with the smaller diameter tool. The slope effect is examined in these three figures and as expected, the figures clearly show that the surface slope has a significant influence on the shape of the force profiles.  Figure 6.62 Comparison of the predicted forces with and without considering ploughing force for down milling of titanium alloy (Vf=0.75 in/min, Cutter E) 6  Experimental Validation of the Predictive Model  176  Figure 6.63 Comparison of the predicted forces with and without considering ploughing force for up milling of titanium alloy (Vf=0.75 in/min, Cutter E)   Figure 6.64 Comparison of the predicted forces with and without considering ploughing force for central cutting titanium alloy (Vf=0.75 in/min, Cutter E) 6  Experimental Validation of the Predictive Model  177  Figure 6.65 Comparison of the predicted forces with and without considering slope effect for down milling of titanium alloy (Vf=2.56 in/min, Cutter E)   Figure 6.66 Comparison of the predicted forces with and without considering slope effect for up milling of titanium alloy (Vf=2.56 in/min, Cutter E) 6  Experimental Validation of the Predictive Model  178  Figure 6.67 Comparison of the predicted forces with and without considering slope effect for central cutting titanium alloy (Vf=2.56 in/min, Cutter E)  To show the relative influence of the chip “crowding” that results from the kinematic constraint, the author has calculated cutter forces at the larger feed (Vf=2.56 in/min) for two cases. The first being the case where it is assumed there is no constraint and the tangential component of velocity is constant, the second where the constraint is acknowledged and an appropriate linear variation in tangential velocity applied. Figures 6.68-6.70 show the resulting force predictions and it is seen that the kinematic constraint changes the force shape, but does not make a large amount in magnitude.  Figure 6.68 Comparison of the effect of velocity profile for down milling of titanium alloy (Vf=2.56 in/min, Cutter E) 6  Experimental Validation of the Predictive Model  179  Figure 6.69 Comparison of the effect of velocity profile for up milling of titanium alloy (Vf=2.56 in/min, Cutter E)   Figure 6.70 Comparison of the effect of velocity profile for central cutting titanium alloy (Vf=2.56 in/min, Cutter E) 6  Experimental Validation of the Predictive Model  180 6.5.4 Validating Simulation Results for Steel As seen in Sections 6.5.1-6.5.3 cutter runout has a very large influence on force at feeding velocities. Since the steel workpiece dictated very small chip thicknesses, then to avoid the problem of runout, the cutter had one tooth ground down to ensure that only a single tooth was in cut. All tests were performed with 3/8 inch SGS end milling cutters (Cutter D) with a spindle speed 1000 RPM. The work material is AISI 4140 steel and the feed ranges from 0.75 in/min to 3.95 in/min as shown in Table AA.4. The measured edge radius and the identified constitutive parameters are given in Tables 6.3 and 6.5 respectively. A comparison of the predicted forces with experimental results for the feeds varying from 0.75 in/min to 3.95 in/min is given in Figures 6.71-6.76. The red solid line indicates the experimental results, while the blue dotted line represents the results predicted by the present model. From these comparisons between the measured and predicted cutting forces, it was confirmed that the proposed cutting force model could effectively predict the cutting forces in helical milling operations.  Figure 6.71 Measured and predicted forces Fx and Fy for slotting of steel (Vf=0.75 in/min, Cutter D) 6  Experimental Validation of the Predictive Model  181  Figure 6.72 Measured and predicted forces Fx and Fy for slotting of steel (Vf=1.28 in/min, Cutter D)   Figure 6.73 Measured and predicted forces Fx and Fy for slotting of steel (Vf=1.5 in/min, Cutter D) 6  Experimental Validation of the Predictive Model  182  Figure 6.74 Measured and predicted forces Fx and Fy for slotting of steel (Vf=2.2 in/min, Cutter D)   Figure 6.75 Measured and predicted forces Fx and Fy for slotting of steel (Vf=2.56 in/min, Cutter D) 6  Experimental Validation of the Predictive Model  183  Figure 6.76 Measured and predicted forces Fx and Fy for slotting of steel (Vf=3.95 in/min, Cutter D)  Again for illustration, force predictions are made by neglecting the ploughing force. Figure 6.77 shows the force comparison with the present model, neglecting the ploughing component, and the corresponding experimental values for the low feed Vf=0.75 in/min. The figure clearly confirms the significant influence of ploughing effect at low feed.  Figure 6.77 Comparison of force prediction with and without considering edge force for slotting of steel (Vf=0.75 in/min, Cutter D) 6  Experimental Validation of the Predictive Model  184 6.6 Conclusions  Milling operations present small varying chip thickness, they have relatively large cutting edge radii and the edges are generally non-straight. The practical parameters in milling then dictate that any approach to modeling should include the variation in yield stress and normal stresses on the shear plane as well as the influence of ploughing forces at the edge. The author also suspected that the surface slope induced in the process and the kinematic constraint imposed by the helical tooth form should be investigated further. The experimental work described in this chapter has shown that the material model and ploughing force model appear to work well when compared to a wide range of practical data. Differences due to the influence of surface slope (more pronounced at lower radii) have also been identified and illustrated. The influence of kinematic constraint over the range of parameters studied has been shown to be relatively small.     185 7  Conclusions  7.1 Dissertation Overview This thesis presents a theoretical modeling methodology to allow the prediction of cutting forces in typical milling operations with multiple helical teeth. The approach is based upon the fundamental physics of cutting and makes an effort to include many of the more complex and second order influences that are normally neglected. The author begins with the analysis of the basic oblique cutting process in an attempt to provide a base which avoids the necessity of defining equivalent orthogonal geometries and to provide a simple energy based model that can be used on an element by element basis along the flute of the cutter. The oblique cutting model development was critical to the work since it allows predictions of shear angle and the extraction of material parameters directly from milling operations; this is a major advantage over methods which require separate tests with straight edged tools at identical conditions. The integrated force model is based upon a fairly traditional approach to the calculation of cutting conditions and the estimation of the material properties which are expected at these conditions. To this base is added a consideration of ploughing forces, a simplified analysis of the influence of “chip crowding” induced by the kinematics of the process, a simple model of the influence of slope and finally consideration is given to the expected transitions between cutting and ploughing at very low chip thicknesses. Despite using as much basic material information as possible, the model still needs calibration on specific work-tool pairs. To minimize this process, the author has developed and tested an innovative identification methodology that combines the central 7  Conclusions  186 cutting milling tests and an optimization algorithm to determine the flow stress data, and other required process information. Finally, the model has been integrated within a simple GUI based modeling package to allow easy calculation of forces in practical processes and a comparison with experimental data has been conducted to prove the validity of the approach. 7.2 Major Contributions The main contributions of this research to the modeling of helical end milling operations can be summarized as follows: 1.  A new upper bound model for oblique cutting has been developed. The analysis replaces the normal variables of chip flow angle and apparent coefficient of friction with basic variables relating to the kinematics (SLIP) and forces in the two main deformation zones, (RATIO). The relationship of SLIP and RATIO is easily found through force equilibrium, thus avoiding any assumptions regarding rake face stress or the ratio of the stresses in the shear plane. 2.  The upper-bound method coupled with the predictive method developed by Yellowley (1987) has been applied to the analysis of chamfered and rounded edge tools. The model utilizes an upper bound approach combined with primary and secondary boundaries that guarantee both force and moment equilibrium of the chip. The model does not need extensive calibration, the only unknown is the elastic friction angle and if forces are required a mean shear yield stress is also needed. The calculated results are in general agreement with the experimental observations. 3.  The characteristics of the ploughing process and the ploughing/cutting transition have been studied in this thesis. Single edge and double edge ploughing models 7  Conclusions  187 based on slip line fields for wedge indentation have been developed and a transition formulation of ploughing to cutting process based on force equilibrium and critical specific energy has been developed. 4.   The author has investigated the influence of the total surface slope on forces in milling operations. A relationship between the shear angle, surface slope and the force parameter RATIO has been derived. 5. A new, kinematically admissible, velocity field that satisfies the incompressibility criterion has been developed for helical end mills. This leads to changes in predicted forces that are normally neglected but can be significant for very small tool diameters and high helical angles. 6.  An innovative algorithm is developed for the identification of material constitutive equation as a function of strain, strain rate and temperature. The model avoids the need for shear angle data, relying instead on experimental force data for calibration and an internal optimization routine to find the normal shear angle. 7.   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J. of Machine Tools and Manufacture, 49, 701-707.  Zou, G. P., Yellowley, I., and Seethaler, R. J., 2010. The extension of a simple predictive model for orthogonal cutting to include flow below the cutting edge, 36 MATADOR Conf., Manchester, UK.    200 Appendices  Appendix A: Experimental Tests  Full list of experimental tests for aluminum alloy, titanium alloy and steel are given in Table AA.1-AA.4. Table AA.1  Milling tests for aluminum alloy Test No. Milling Type Tool Diameter (in) Feed Velocity (in/min) Spindle Speed (RPM) Depth of Cut a (in) Width of Cut w (in) 1 Center Cut 1/2 1.28 1000 0.08 0.283 2 Center Cut 1/2 2.56 1000 0.08 0.283 3 Center Cut 1/2 4.4 1000 0.08 0.283 4 Center Cut 1/2 6.75 1000 0.08 0.283 5 Center Cut 1/2 7.9 1000 0.08 0.283 6 Slotting 1/2 2.2 1000 a1=0.05, a2=0.165 ___ 7 Slotting 1/2 4.4 1000 a1=0.05, a2=0.165 ___ 8 Down Milling 1/2 3.95 1000 0.08 0.266 9 Down Milling 1/2 2.56 1000 0.08 0.266 10 Down Milling 1/2 1.28 1000 0.08 0.266 11 Down Milling 1/2 6.75 1000 0.08 0.266 12 Up Milling 1/2 7.9 1000 0.08 0.266 13 Up Milling 1/2 3.95 1000 0.08 0.266 Note for test 6 and 7, the depth of cut jump at the center of slotting.  Table AA.2  Milling tests for titanium alloy with 2 teeth 3/8” end mill cutter Test No. Milling Type Tool Diameter (in) Feed Velocity (in/min) Spindle Speed (RPM) Depth of Cut a (in) Width of Cut w (in) 1 Center Cut 3/8 2.2 575 0.08 0.21 2 Center Cut 3/8 0.75 575 0.08 0.21 3 Center Cut 3/8 1.5 575 0.08 0.21 4 Center Cut 3/8 3.95 575 0.08 0.21 5 Center Cut 3/8 4.4 575 0.08 0.21 6 Down Milling 3/8 0.75 575 0.073 0.154 7 Down Milling 3/8 1.5 575 0.073 0.154 8 Down Milling 3/8 2.2 575 0.073 0.154 9 Up Milling 3/8 1.5 575 0.073 0.158 10 Up Milling 3/8 0.75 575 0.073 0.158 Appendices   201 Table AA.3  Milling tests for titanium alloy with 2 teeth 3/16” end mill cutter Test No. Milling Type Tool Diameter (in) Feed Velocity (in/min) Spindle Speed (RPM) Depth of Cut a (in) Width of Cut w (in) 1 Down Milling 3/16 0.75 1152 0.037 0.1482 2 Down Milling 3/16 1.5 1152 0.037 0.1482 3 Down Milling 3/16 2.2 1152 0.037 0.1482 4 Down Milling 3/16 2.56 1152 0.037 0.1482 5 Up Milling 3/16 2.56 1152 0.037 0.152 6 Up Milling 3/16 2.2 1152 0.037 0.152 7 Up Milling 3/16 1.5 1152 0.037 0.152 8 Up Milling 3/16 0.75 1152 0.037 0.152 9 Central Cut 3/16 0.75 1152 0.037 w1=0.08675 w2=0.06275 10 Central Cut 3/16 1.5 1152 0.037 w1=0.08675 w2=0.06275 11 Central Cut 3/16 2.2 1152 0.037 w1=0.08675 w2=0.06275 12 Central Cut 3/16 2.56 1152 0.037 w1=0.08675 w2=0.06275  Figure AA.1 Geometry of central cutting titanium alloy by 3/16” end mill cutter  Table AA.4  Milling tests for AISI 4140 steel Test No. Milling Type Tool Diameter (in) Feed Velocity (in/min) Spindle Speed (RPM) Depth of Cut a (in) Width of Cut w (in) 1 Slotting 3/8 (One Tooth) 0.75 1000 0.04 3/8 2 Slotting 3/8 (One Tooth) 1.28 1000 0.04 3/8 3 Slotting 3/8 (One Tooth) 1.5 1000 0.04 3/8 4 Slotting 3/8 (One Tooth) 2.2 1000 0.04 3/8 5 Slotting 3/8 (One Tooth) 2.56 1000 0.04 3/8 6 Slotting 3/8 (One Tooth) 3.95 1000 0.04 3/8 Appendices   202 Appendix B: Dynamometer Calibration  The dynamometer calibration procedure was performed at the Manufacturing Engineering Laboratory of UBC. The 3D dynamometer linked to three charge amplifiers (Kistler Model 5004) was mounted at the XY motion table developed by Oldknow 2004. The cutting forces were measured by the Tektronix TDS 2024B scope. The measured data was postprocessed by using the MATLAB program. The load was applied through a wheel and linked by small diameter strong plastic wire mounted at the Bridgeport machine in order to reduce the friction effect in the calibration processes. The piezotransducer sensitivity and cross sensitivity were identified by using a Sensortronics 60001-256 load cell. The identified parameters of the charge amplifier for x, y and z directions are given in Table AB.1.  Table AB.1  Identified parameters used for charge amplifiers (Kistler Model 5004) Parameter x y z Mechanical Unit  (N/V) 20 20 20 Sensitivity Range 1-11 1-11 1-11 Sensitivity  (mV/N) 86 86 73  A series of incremental loads ranging from 5.8 N to 85 N were applied as steady weight through the plastic wire. Figures AB.1-AB.3 plot the calibration of the dynamometer in x, y and z directions, respectively. The response of the dynamometer is quite linear in all three directions. The calibrated relationships of the force and measured voltage for the x, y and z direction are stated as following Appendices  203      +⋅= +⋅= +⋅= 1284.0133.19)( 1018.0077.19)( 2091.095.18)( VNF VNF VNF z y x                                             (AB.1) 0 1 2 3 4 5 0 20 40 60 80 100   Force F x  Linear Fit A pp lie d Fo rc e (N ) Measured Voltage (V)  Figure AB.1 Calibration of the dynamometer in x direction  0 1 2 3 4 5 0 20 40 60 80 100    Force Fy  Linear Fit A pp lie d Fo rc e (N ) Measured Voltage (V)  Figure AB.2 Calibration of dynamometer in y direction Appendices  204 0 1 2 3 4 5 0 20 40 60 80 100    Force F z  Linear Fit Ap pl ie d Fo rc e  (N ) Measured Voltage (V)  Figure AB.3 Calibration of dynamometer in z direction       

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