DETERMINATION OF MATERIAL CONSTITUTIVE MODELS USING ORTHOGONAL MACHINING TESTS. A Dissertation By. Mahmoud Al Bawaneh

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1 DETERMINATION OF MATERIAL CONSTITUTIVE MODELS USING ORTHOGONAL MACHINING TESTS A Dissertation By Mahmoud Al Bawaneh Master of Science, Jordan University of Science and Technology, 21 Bachelor of Science, Jordan University of Science and Technology, 1998 Submitted to the Department of Industrial and Manufacturing Engineering And the faculty of the Graduate School of Wichita State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy December 27

2 Copyright 27 by Mahmoud Al Bawaneh All Rights Reserved

3 DETERMINATION OF MATERIAL CONSTITUTIVE MODELS USING ORTHOGONAL MACHINING TESTS The following faculty members have examined the final copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy with a Major in Industrial Engineering. Viswanathan Madhavan, Committee Chair Janet Twomey, Committee Member Hamid Lankarani, Committee Member Jamal Sheikh-Ahmad, Committee Member Krishna Krishnan, Committee Member Accepted for the College of Engineering Zulma Toro-Ramos, Dean Accepted for the Graduate School Susan Kovar, Dean iii

4 DEDICATION To my beloved family: Father Mother Hakam Hekmat Zein Ahamd Mohammad Katherine Ibrahim iv

5 ACKNOWLEDGEMENTS I would like to express my gratitude to Dr. Viswanathan Madhavan for his wisdom and guidance over the last several years. I would like to thank, Dr. Janet Twomey, Dr. Hamid Lankarani, Dr. Jamal Sheikh-Ahmad and Dr. Krishna Krishnan, for their advice and support. Finally I would like to thank Dinesh Mahadevan, for his work in providing the experimental data used in this work. v

6 ABSTRACT The aim of present research is to fine-tune and improve Johnson-Cook (JC) constitutive model parameters for American Iron and Steel Institute (AISI) 145 carbon steel and Oxygen-free highconductivity (OFHC) copper using orthogonal machining tests. This dissertation is divided into two parts. Part one is concerned with improving Johnson-Cook (JC) constitutive model parameters for AISI 145 carbon steel by comparing results of finite element (FE) simulations of orthogonal machining with experimental results. A large number of FE simulations are carried out using different values for each of the five unknown parameters of the JC model. Second-order polynomials are fitted to the results of FE simulations to describe how the peak strain rate, strain rate distribution in the primary shear zone (PSZ) and chip thickness vary with the changes in the material parameters. The peak strain rate and strain rate distribution in the PSZ obtained experimentally using Digital Image Correlation (DIC) of the high-speed microphotographs of the PSZ and the measured chip thickness are available. The material constants that minimize the error between experimental observations and FEA results for each of these quantities are found by non-linear optimization using the sum-squared errors of the objective function. The efficacy of this approach is assessed by comparing FEA results obtained using this set of parameters to experimental measurements under Assessment of Machining models (AMM) cutting condition 2. Part two first quantitatively evaluates and compares the performance of different material models for OFHC copper widely used in machining with reference to the experimental results obtained from cutting tests. Unlike other research, the strain and strain rate along the nominal shear plane are experimentally measured using Digital Image Correlation (DIC) of high-speed photographic images of the side surface of the chip. Since the average strain and strain rate directly measured from cutting tests, an analytical estimate is required only for the temperature, hence reducing the uncertainty in the proposed approach for identification of the constants used in different material models. The determination of the material constants used in the Johnson-Cook and Zerilli-Armstrong constitutive models for OFHC copper from carefully designed machining experiments is the focus of part two. vi

7 TABLE OF CONTENTS PART ONE Page DETERMINATION OF JOHNSON-COOK CONSTITUTIVE MODEL PARAMATERS IN MACHINING AISI 145 USING FINITE ELEMENT ANALYSIS OF ORTHOGONAL MACHINING SUMMATION BACKGROUND JOHNSON COOK (JC) CONSTITUTIVE MODEL FINITE ELEMENT MODEL AND INPUTS USED DESIGN OF EXPERIMENTS Response Surface Methodology Central Composite Design Second Order Polynomials RESULTS AND DISCUSSION Typical FEA Results ANOVA JC MODEL PARAMETER SENSITIVITY DETERMINATION OF JC MODEL PARAMETERS SENSITIVITY STUDY OF THE OPTIMIZATION PROCESS SUMMARY AND CONCLUSION...4 LIST OF REFERENCES 42 APPENDIX A 92 vii

8 TABLE OF CONTENTS (Continued) Part Two Page DETERMINATION OF CONSTITUTIVE MODELS FOR OFHC COPPER USING ORTHOGONAL MACHINING TESTS.44 SUMMATION INTRODUCTION HIGH STRAIN RATE CONSTITUTIVE MODELS FOR OFHC COPPER JC Material Model Other Material Models COMPARISON OF MATERIAL MODELS CALCULATION MACHINING MODELS FOR CALCULATION OF FLOW STRESS AND TEMPERATURE METAL CUTTING EXPERIMENTS DETERMINATION OF FLOW STRESS, STRAIN, STRAIN RATE AND TEMPERATURE ALONG NOMINAL SHEAR PLANE Flow Stress Temperature Strain and Strain-Rate RESULTS AND DISCUSSION Results of Cutting Tests Assessment of Different Material Models Determination of JC Model Parameters Comparison of JC Model with MTS Model Temperature Sensitivity Analysis SUMMARY AND CONCLUSION LIST OF REFERENCES...89 APPENDIX B 138 viii

9 LIST OF FIGURES PART ONE Figure Page 1.1 Lagrangian approach along with mesh adaptivity used in MSC. Marc Model of chip formation showing AB and CD at which the strain rate is extracted from FE simulation Final mesh of FE simulation #27 using MAT1 set of constants showing the fully developed chip and the effective stress in the machined surface, PSZ and the chip Cutting force versus time predicted by FE simulation #27 using MAT1 set of JC model constants Thrust force versus time predicted by FE simulation #27 using MAT1 set of JC model constants Contours of (a) Equivalent plastic strain rate in 1/s, (b) Velocity in mm/s, (c) Equivalent plastic strain, (d) Temperature in o C and (e) Equivalent von Misses stress in MPa. (FEA run #27 using MAT1) Plot of residuals versus predicted values for peak strain rate Normal probability plot of the residuals for peak strain rate Plot of residuals versus predicted values for chip thickness Normal probability plot of the residuals for chip thickness Decrease of chip thickness with increase in A Decrease of chip thickness with increase in B Decrease of chip thickness with increase in C Increase of chip thickness with increase in n Decrease of chip thickness with increase in m Change of peak strain rate with increase in A Change of peak strain rate with increase in B Change of peak strain rate with increase in C Decrease of peak strain rate with increase in n Change of peak strain rate with increase in m Optimization process of the objective function defined by equation (1.7).35 ix

10 LIST OF FIGURES (Continued) PART ONE Figure Page 1.22 Comparison of experimental strain rate with strain rate predicted from FEA simulation along CD using the new optimum set of JC model constants (MAT7) under AMM cutting condition 2 (CD direction from left to right) Comparison of experimental strain rate with strain rate predicted from FEA simulation along AB using the new optimum set of JC model constants (MAT7) under AMM cutting condition 2 (AB direction from left to right)..37 x

11 LIST OF FIGURES PART TWO Figure Page 2.1 Comparison of flow stress predicted from JC model using different constants Comparison of five different material models for OFHC copper Model of chip formation showing the primary and secondary shear zones Decrease in chip thickness with increase in cutting speed (Depth of Cut =.1 mm) Increase of chip thickness with increase in depth of cut (Cutting Speed = 1. m/s) Decrease of strain with increase in cutting speed (Depth of Cut =.1 mm) Increase of strain with increase in depth of cut (Cutting Speed = 1. m/s) Increase of strain rate with increase in cutting speed (Depth of Cut = 1. m/s) Decrease of strain rate with increase in depth of cut (Cutting Speed = 1. m/s) Variation of average strain rate along shear plane AB obtained by DIC with ratio of shear velocity to length of shear plane AB Increase of temperature with increase in cutting speed (Depth of Cut =.1 mm) Increase of temperature with increase in depth of cut (Cutting Speed = 1. m/s) Percentage error in predicted flow stress using different jc model constants with reference to machining test data Percentage error in predicted flow stress using different material models with reference to machining test data Optimization process of the objective function defined by equation (2.17) Comparison of machining flow stress data with flow stress predicted from different material models using the optimum sets of constants shown in Table Comparison of machining flow stress data with flow stress predicted from JC model (using optimum sets of constants given in Table 2.17) and MTS model (using material constants given in Table 2.2) Comparison of predicted temperatures of Boothroyd s model with Weiner s model in machining OFHC copper 87 xi

12 LIST OF TABLES PART ONE TABLE PAGE 1.1 High-Strain Rate and Machining Tests Used to Obtain Flow Stress Data for AISI 145 Carbon Steel Material Constants for Johnson-Cook Material Model for AISI 145 Carbon Steel Material Properties of the Workpiece and Tool Used in FEA Simulations of AISI 145 Steel The Locations in the PSZ Along AB and CD At Which Strain Rate is Extracted from Both Machining Test and FEA Simulation Factors and Levels of Central Composite Design (CCD) Experiments Summary of FEA Results Summary of Analysis of Variance (ANOVA) for Chip Thickness and Peak Strain Rate Summary of JC Model Parameter Sensitivity Optimum Set of JC Model Parameters Summary of FEA Predictions Using the Optimum Set of JC Model Constants (MAT7) Under AMM Cutting Condition Summary of Strain Rate Along AB and CD Predicted Using the Optimum Set of JC Model Constants (MAT7) Under AMM Cutting Condition Summary of the Sensitivity Analysis of the Optimization Process..4. xii

13 LIST OF TABLES PART TWO TABLE PAGE 2.1 Material Constants Proposed to be Used in Johnson-Cook s Model for OFHC Copper The Ranges of Strain, Strain Rate and Temperature at Which JC Model Constants Were Determined Ranges of JC Model Constants Different Material Constitutive Models for OFHC Copper Authors of Different Material Constitutive Models for OFHC Copper Material Constants for Five Different Material Models of OFHC Copper The Ranges of Strain, Strain Rate and Temperature at Which Different Model Constants Were Determined Experimental Data for OFHC Copper Obtained from High Strain Rate Tests Comparison of Experimental Data With Flow Stress Predicted from JC Model Using Different Sets of Constants Comparison of Experimental Data With Flow Stress Predicted from Different Material Models Using Different Sets of Constants Cutting Conditions in Orthogonal Machining Tests performed for OFHC Physical Properties of the OFHC Workpiece Material Cutting Conditions and Measured Process Variables in Orthogonal Machining of OFHC Copper Computed and Measured Process Variables in the Primary Deformation Zone in Orthogonal Machining of OFHC Copper Percentage Error in Predicted Flow Stress Using Different JC Model Constants with Reference to Machining Test Data Percentage Error in Predicted Flow Stress Using Different Material Models With Reference to Machining Test Data Optimum Material Constants for Different Material Models The Ranges of Strain, Strain Rate and Temperature at Which the Optimum Material Constants are Determined Percentage Improvement in the Performance of Different Material Models to Predict Work Material Flow Stress Obtained from Cutting Tests MTS Model Parameters for OFHC Copper 84 xiii

14 PART ONE DETERMINATION OF JOHNSON-COOK CONSTITUTIVE MODEL PARAMETERS IN MACHINING AISI 145 USING FINITE ELEMENT ANALYSIS SUMMATION This research is aimed at improving Johnson-Cook (JC) constitutive model parameters by comparing results of finite element (FE) simulations of orthogonal machining with experimental results. A large number of FE simulations are carried out using different values for each of the five unknown parameters of the JC model. Second-order polynomials are fitted to the results of FE simulations to describe how the peak strain rate, strain rate distribution in the primary shear zone (PSZ) and chip thickness vary with the changes in the material parameters. The peak strain rate and strain rate distribution in the PSZ obtained experimentally using Digital Image Correlation (DIC) of the high-speed microphotographs of the PSZ and the measured chip thickness are available. The material constants that minimize the error between experimental observations and FEA results for each of these quantities can be found by non-linear optimization using the sum-squared errors of the objective function. A global optimizer that uses combinations of branch-and bound based global search (BB), global adaptive random search (GARS), random multi-start based global search (MS), and generalized reduced gradient (GRG) algorithm based local search (LS) is used to find the optimum set of the JC model parameters that minimize the objective function. The efficacy of this approach is assessed by comparing FEA results obtained using this set of parameters to experimental measurements under Assessment of Machining models (AMM) cutting condition 2. 1

15 1.1 Background In the past decade, the need for accurate material constitutive equations covering wide ranges of flow stress, strain, high strain rate and elevated temperature has been increasingly felt in impact and crash simulation as well as in the simulation of high speed forming processes such as strip rolling, wire drawing and machining. In the impact physics area, high-strain rate tests such as the split Hopkinson pressure bar (SHPB) test have been widely used to produce flow stress data and fit that data to different material constitutive equations. In the machining area, the use of cutting tests to obtain the material model parameters is widely prevalent. A survey of the literature summarized in Table 1.1 indicates that flow stress data for metals have been mainly obtained using four methods: high-speed compressions (HSC) tests, split Hopkinson pressure bar (SHPB) tests, practical machining tests and reverse engineering using FEA simulation (Allen et al., 1997; Cho and Altan, 25). Table 1.1 shows that SHPB tests can generate high strain rates and elevated temperatures but not at the levels observed in machining. Machining tests can raise the achievable strain rate up to 6.8 x 1 5 1/s, which is two orders of magnitude higher than that achieved by SPHB tests (Jaspers and Dautzenberg, 22; Sartkulvanich et al., 24). The SPHB test is widely used for intermediate and high strain rate measurements and can be adapted to several modes of loading such as tensile, compressive, flexure and torsion. A great advantage of machining tests is that wide ranges of large strain and high strain rate can be conveniently achieved by adjusting the cutting conditions composed of rake angle, cutting speed and depth of cut. However, the measurements of low and moderate strain rate using machining tests become difficult. Additionally, machining tests coupled with analytical models are limited to cases of continuous chips. Therefore, several researchers have combined both machining and SHPB tests to optimize the material model constants in which the SPHB tests were used to obtain the flow stress at moderate strain rates and machining tests were used for high strain rates (Ozel and Zeren, 26; Mathew and Arya, 1993). In earlier work using machining to obtain material model parameters, small number of cutting conditions was used to fine-tune and optimize the material constants (Ozel and Zeren, 26; Mathew and Arya, 1993). The approach was primarily based on measurements of cutting forces and chip thickness. The flow stress was calculated from measured cutting forces and chip thickness. Analytical models are 2

16 used to estimate the strain rate and temperature along the nominal shear plane AB, but the estimates depend on the material model constants. This coupling causes problems in the techniques used to finetune and optimize the material model constants by fitting the model predictions of the flow stress to the experimental measurements of the flow stress and results in increased uncertainty in the material model constants obtained from machining tests. If the average strain and strain rate along the nominal shear plane AB can be directly measured from cutting tests, analytical estimates will be required only for the temperature, hence reducing the uncertainty in the material model constants (Madhavan et. al, 27). Most previous orthogonal machining experiments were conducted by tube turning on a lathe or slot milling. These cutting tests do not necessarily represent true orthogonal cutting due to the curvature surfaces of the chips (Sartkulvanich et. al., 25). This could potentially cause discrepancy between the measured cutting forces and the cutting forces predicted by the FEA simulations. It has been concluded that results of FEA simulations of machining are sensitive to the flow stress data of the work material and the friction characteristics between the cutting tool face and the chip (Sartkulvanich and Altan, 25; Deshpade et al., 26; Umbrello et al., 27). Thus, the majority of researchers have agreed that there is a need to develop more accurate material constitutive equations applicable for wide ranges of flow stress, strain, high strain rate, and elevated temperatures occurring in machining. It is also recognized from previous published studies that the material constants used in various constitutive equations were obtained by extensive and expensive experiments including high strain rate and orthogonal machining tests. Not every academic institution and company can afford the time and the cost of these experiments. Similarly the experimental approaches of determining the friction along the chip-tool interface under different cutting conditions during machining are difficult, expensive, and time consuming. In summary, there is a need for accurate work material constitutive equations, which are prerequisites for FEA of machining (Arsecularatne and Zhang, 24). FE simulation of metal cutting processes is one of the state of the art developments in manufacturing engineering. Both academia and industry have extensively used commercial FEA packages as a reliable prediction tool for metal cutting process design due to the numerous advantages of FEA models over other analytical and empirical models (Dirikolu et al., 2). Though FEA modeling of machining can accurately predict the global (e.g. 3

17 cutting forces and chip thickness) and local variables (e.g. stress, strain, strain rate and temperature in the deformation zones), it requires highly accurate constitutive material models for it to be useful as a reliable prediction tool. It was found by previous published studies that: 1. The numerical predictions of the typical machining outputs are sensitive to the material models used and that not one of the material models used is best in all aspects (Adibi-Sedeh et al., 25; Sartkulvanich et al., 25). 2. Using flow stress data obtained from different methods and tests produced different sets of the constitutive model constants (Holmquist and Johnson, 1991). 3. For a specific material model such as JC model, FEA predictions of the typical machining outputs are sensitive to different sets of JC material constants (Deshpade et al., 26; Umbrello et al., 27). 4. The imperative prerequisite for simulation of work materials is determination of their flow stress covering wide ranges of flow stress, strain, high-strain rate, and elevated temperature under machining conditions (Shirakashi and Obikawa, 1998). TABLE 1.1 HIGH-STRAIN RATE AND MACHINING TESTS USED TO OBTAIN FLOW STRESS DATA FOR AISI 145 CARBON STEEL Type of Test Authors (-) Strain (-) Strain Rate (1/s) Temperature ( o C) HSC SHPB Machining Shida, 1969 up to.7 up to Oxley, 1989 up to 1. up to Zerilli and Armstrong, 1987 up to 1. up to Lim et al., 23 up to.4 up to Jaspers and Dautzenberg, 22 up to.3 up to El-Magd et al., 23 up to.5 up to Ozel and Zeren, up to Sartkulvanich et al., up to

18 1.2 Johnson-Cook (JC) Constitutive Model The material model most commonly used in metal cutting is the Johnson-Cook (JC) material model. This model considers the flow stress to be a product of three terms representing the effect of strain, strain rate, and temperature. It is defined by the following equation: σ = & ε T T r A 1 + C ln 1 & ε Tm T n [ + Bε ] r m (1.1) where σ is the material flow stress, ε is the plastic strain, έ is the strain rate (s -1 ), έ o is the reference plastic strain rate (s -1 ), T is the temperature of the work material ( C), T m is the melting temperature of the work material and T r is the reference temperature. A, B, C, n and m are five empirical material constants. These five empirical material constants can be obtained from various tests. Table 1.2 shows different sets of material constants proposed to be used in the JC model for AISI 145. The MAT1 set of material constants was obtained from mechanical testing at high-strain rate (up to 75 1/s) and elevated temperatures (up to 5 o C) on cylindrical specimens of AISI 145 having different lengths using SHPB tests (Jaspers and Dautzenberg, 22). The MAT2 set of material constants was determined from mechanical testing on cylindrical samples of AISI 145 covering a wide spectrum of strain rates ranging from a very low strain rate of 1-4 to a very high strain rate of 1 5 1/s. The servo-hydraulic compression, drop weight, SHPB, and flyer plate impact tests were used to measure different regimes of the strain rates. The MAT2 set is only applicable for initial flow stresses in which strain hardening is not considered (Meyer et al., 27). The MAT3 set of material constants was identified using inverse technique based on Oxley analysis of orthogonal metal cutting process along with experimental data obtained from machining and SPHB tests conducted on AISI 145. Then the value of C (i.e. the material constant in Oxley theory of machining) was determined through iterative approach which satisfied the hydrostatic stresses at points A and B calculated from the measured cutting forces and the stress condition at point A. The termination criterion of the iteration procedure was satisfied when the relative change in the strain rate constant between successive iterations was small (Ozel and Zeren, 26). Points A and B are defined such that 5

19 the PSZ extends from the tip of the cutting edge denoted as B to the intersection point of the free surface and the workpiece, denoted as A. Finally, the M4, M5 and M6 set of material constants were obtained by using one single objective function to describe the difference (error) between the measured flow stress from SPHB compression tests plus the predicted flow stress from orthogonal machining tests and the flow stress calculated using the JC constitutive model. Then the difference (error) was minimized by changing the five material constants to yield a more accurate set of material constants using evolutionary computational algorithms. Three different evolutionary computational algorithms were used, namely; particle swarm optimization (PSO), modified particle swarm optimization (PSO-s) and cooperative particle swarm optimization (CPSO), which yielded the M4, M5 and M6 sets of material constants respectively (Ozel and Yigit, 27). TABLE 1.2 MATERIAL CONSTANTS FOR JOHNSON-COOK MATERIAL MODEL FOR AISI 145 CARBON STEEL Material A (MPa) B (MPa) n C m Set of JC model constants Type of test MAT1 SPHB MAT2 SPHB + Flyer-plate impact AISI x MAT3 Machining MAT4 Machining MAT5 Machining MAT6 Machining 6

20 The aim of this research is to fine-tune and optimize the material constants used in the JC constitutive model for AISI 145 by matching the experimental machining measurements for cutting force, chip thickness, and strain rate distribution in the PSZ with the corresponding results of the FEA simulations under AMM cutting condition 2 (Ivester et al., 2). Since no other optimization effort for the material model constants has used direct measurements of the strain and strain rate, our effort is likely to be more accurate than all other previous approaches. The JC constitutive model is purposefully chosen because it has enjoyed great success in the past decade due to its simplicity and the availability of its material model constants for various metallic material and aerospace alloys. It is understood that more complicated and physically based constitutive models may provide better predictions of the work material response during high strain rate deformation processes. However, most commercial FEA packages, which are available in today s market, cannot automatically support these diverse constitutive models. The complexity of these material constitutive models and the difficulty to obtain their material model constants have limited their use in the FEA arena. American Iron and Steel Institute (AISI) 145 carbon steel is purposefully chosen in this study because it has a great application in the manufacturing processes due to its characteristic of high machinability and low cost. Moreover, this work material has been extensively studied by other researchers especially in the context of machining and a large amount of flow stress data through machining and high-strain rate tests can be found in the literature (see Tables 1.1 and 1.2). A series of 2D virtual experiments are carried out on a computer using FEA package MSC. MARC in which JC material parameters are systematically varied. For each FEA experiment, the coefficient of friction is tuned iteratively until the cutting force predicted by the FEA simulation best matches the measured cutting force. The coefficient of friction for each computer experiment is tuned since the friction is unknown and matching cutting force is key since flow stress, chip formation, cutting temperature, tool wear and surface finish all depend on the cutting force. The strain rate distribution obtained experimentally using Digital Image Correlation (DIC) of the high-speed microphotographs of the PSZ, the shear angle observed in the photographs and the chip thickness are available (Madhavan et al., 27). The peak strain rate, strain rate in the PSZ and chip thickness are extracted under steady state condition for each FEA run that produced the best match between the measured and predicted forces. 7

21 1.3 Finite Element Model and Inputs Used The FEA models consist of an elastic tool and a deformable workpiece. The left and bottom boundaries of the workpiece are fixed. The cutting tool is pushed into the domain of interest (workpiece) with assigned velocity equal to the cutting velocity, thus making the workpiece stationary throughout the analysis. The FEA simulations of orthogonal machining of AISI 145 under AMM cutting conditions 2 are carried out as 2D coupled thermo-mechanical analysis using MSC. MARC under plane strain assumption by repeatedly remeshing the workpiece during the analysis. In Lagrangian approach along with adaptive remeshing capability used in MSC. MARC the cutting tool approaches the workpiece and then engages without prior assumptions of the chip geometry as in Figure 1.1(a), initially indents the workpiece as in Figure 1.1(b), and then the chip begins to form and finally curls over ahead of the cut as in Figure 1.1(c). The work material is modeled using the JC material model, defined by equation (1.1). The physical properties of AISI 145 steel are modeled as functions of temperature using the values and equations shown in Table 1.3. Simple frictional behavior is applied with Coulomb friction model. The coefficient of friction for each simulation is tuned to match the cutting force predicted by the simulation to the measured cutting force. The workpiece and tool materials are AISI 145 carbon steel and carbide (Kennametal K68) respectively. The assumed value of the cutting edge radius of the tool is 1µm, similar to the inserts used in AMM orthogonal experiments. The workpiece is modeled as block of size 4 mm x 2 mm with initial temperature of 2 o C. Extra thermal properties of the workpiece and the contact interface are: contact heat transfer coefficient = 4 N/s/ o C/mm and heat transfer coefficient with air =.44 N/s/ o C/mm. Automatic global remeshing capability that uses advancing front mesher in MSC. MARC is used to repeatedly remesh the deformed workpiece during the FEA simulation. The workpiece is initially meshed with only 3 elements and then the number of elements is allowed to grow to 3, during the analysis using remeshing capability. The intensity, frequency, curvature control and smoothing ratio parameters are adjusted to most optimum setting for maintaining a successful mesh during each metal cutting simulation. 8

22 TABLE 1.3 MATERIAL PROPERTIES OF THE WORKPIECE AND TOOL USED IN FEA SIMULATIONS OF AISI 145 STEEL (Deshpade et al., 26) Material property Workpiece Cutting tool Thermal conductivity (k, W/m o C) Specific heat (Cp, J/Kg o C) Thermal expansion coefficient (α, I o C) Young s Modulus (E, Gpa) T T x x Poisson ratio (ν).3.2 Density (ρ, Kg/m 3 ) cutting tool workpiece (a) (b) (c) Figure 1.1 Lagrangian approach along with mesh adaptivity used in MSC. MARC (a) Initial mesh of the workpiece (b) Initial tool indentation (c) Chip formation (fully developed). 9

23 The typical machining outputs are extracted from FEA simulations in a manner similar to the measurements obtained from AMM orthogonal cutting experiments (Ivester and Kennedy, 24). The cutting and thrust forces are averaged over time over steady state region to obtain single values for each FEA simulation. The FEA machining process is considered to be at steady state when the outputs do not change with time significantly. The chip thickness is extracted from the final mesh of the FEA simulation in the region where chip leaves contact with the tool. The peak strain rate and strain rate distribution in the PSZ are extracted in a manner similar to the measured strain rate obtained using Digital Image Correlation of the high-speed photographs of the PSZ conducted at Wichita State University (Madhavan et. al., 27). The nominal shear plane AB is determined by drawing a straight line connecting points A and B with point A located as the intersection of the workpiece free surface with the back surface of the chip, and point B located at the intersection of the rake and flake faces of the cutting tool. The CD is located at the middle of shear plane AB by drawing a straight line perpendicular to AB as shown in Figure 1.2. The strain rate along AB and CD is extracted from each FE simulation at nine locations shown in Figure 1.2 and defined in Table 1.4. The strain rate is extracted at the same nine locations regardless of the variations in the length of shear plane AB and thickness of the PSZ with the variations in JC model parameters. The measured strain rates at these nine locations are also available. Since the experimental strain rate distribution along CD shows double peaks in the middle of the PSZ and predicted strain rate does not, the locations designated by 6, 7, 8, and 9 are chosen farther away from the middle of the PSZ. TABLE 1.4 THE LOCATIONS IN THE PSZ ALONG AB AND CD AT WHICH STRAIN RATE IS EXTRACTED FROM BOTH MACHINING TEST AND FEA SIMULATION Points along AB (see Figure 1.2) Distance between points (µm) Points along CD (see Figure 1.2) Distance between points (µm) (3 2) 25 (3 7) 2 (3 4) 25 (3 8) 2 (1 2) 5 (6 7) 2 (4 5) 5 (8 9) 2 1

24 Chip Depth of cut A 1 2 C D B Cutting tool Figure 1.2 Model of chip formation showing AB and CD at which the strain rate is extracted from FE simulation. 1.4 Design Of Experiment (DOE) Response Surface Methodology (RSM) A standard approach for experimental design is to use a full factorial design method. However, a full factorial approach is acceptable only when a few variables (usually not more than three variables) are to be investigated. For example, a full factorial design with three levels of each variable requires a total of 3 k experimental runs if there are k variables to be investigated; each consists of three different levels. The aim of this research is to fine-tune and optimize five material constants used in the JC constitutive model for AISI 145. Application of a full factorial approach to the present study requires a total of 243 FEA simulations, and therefore is not practical. An alternative approach to a full factorial design is a fractional factorial design such as response surface methodology (RSM). RSM is a robust design technique widely used in both academia and industry for product and process optimization. RSM is a tool to explore the relationships between several input variables and one or more output (response) variables. The main idea of RSM is to use a set of designed experiments to obtain an optimal 11

25 response. Then a Central Composite Design (CDD) can be implemented to estimate a second-order polynomial model, which is still only an approximation at best. However, the second-order model can be used to optimize (maximize, minimize, or attain a specific target for) a response. A response with two input variables describes a surface while a response with more than two variables describes hypersurface Central Composite Design A Box-Wilson Central Composite Design often called Central Composite Design (CCD) is an experimental design, which is useful in response surface methodology, extensively used for estimating second order response surfaces. The CCD contains fractional factorial design with center points that is augmented with a group of star points that allow estimation of curvature. It is perhaps the most popular class of second order designs. Thus, CCD is used to systematically study the effect of, and the interactions among the following material constants used in the JC constitutive model: (1) initial yield strength of the material at room temperature and a strain rate of 1 1/s A, (2) hardening modulus B, (3) strain rate sensitivity coefficient C, (4) hardening coefficient n, and (5) thermal softening coefficient m. The levels of interest for each constant are presented in Table 1.5. These levels of each constant are purposefully chosen to cover a wide range of all possible values of JC material constants found in the literature (see Table 1.2). The design included a total of: 2 k-1 + 2k + s runs, where k is the number of variables studied (five JC material constants i.e. k =5), 2 k-1 are the points from a factorial experiment (i.e. sixteen for cube experiments) and 2k are the number of star points carried out on the axes at a distance (ω) equal to ±2 from the center, A five-variable CCD is rotatable if ω = ±2. The number of runs at the centered conditions is fixed at s = 1. These levels for JC model parameters given in Table 1.5 are also used to establish the limits (ranges) for the optimization process of the objective function described later. A statistical analysis software (MINITAB) is used to generate the 27 DOE points at which FEA simulations are carried out. The results of FE simulations (i.e. chip thickness, peak strain rate, strain rate at other nine locations along AB and CD) are also input into MINITAB for further analysis as it is described in the following sections. The following sections are concerned with second-order polynomials, typical FEA results, JC model parameter sensitivity, determination of JC model parameters and finally concluding remarks. 12

26 TABLE 1.5 FACTORS AND LEVELS OF CENTRAL COMPOSITE DESIGN (CCD) EXPERIMENTS Coded Factors Level A (Mpa) B (Mpa) C n m Second-Order Polynomials This research assumes that three-, four-, and five- factor interactions are negligible, because these high order interactions are normally assumed impossible in practice. The method presented here is to use second-order polynomials to describe how the chip thickness, peak strain rate and strain rate at nine other locations along AB and CD (see Figure 1.2) vary with the changes in the material constants used in the JC constitutive model. These second-order polynomials are defined by: t FEA = λ + λ A + λ B + λ C + λ n + λ m + λ A + λ B + λ C + λ n + λ m + λ AC + λ An + λ Am + λ BC + λ Bn + λ Bm + λ Cn + λ Cm + λ nm λ AB 11 (1.2) & ε FEA 2 2 ( peak) = ψ + ψ A+ ψ B+ ψ C+ ψ n+ ψ m+ ψ A + ψ B + ψ C + ψ n + ψ AC+ ψ An+ ψ Am+ ψ BC+ ψ Bn+ ψ Bm+ ψ Cn+ ψ Cm+ ψ nm ψ m + ψ AB (1.3) & ε FEA ( i) = β ( i) + β ( i) A + β ( i) B + β ( i) C + β ( i) n + β ( i) m + β ( i) A + β ( i) n β ( i) Bn + β β ( i) m 2 + β ( i) AB 11 ( i) Bm + β ( i) Cn + β + β ( i) AC + β ( i) Cm + β 2 6 ( i) An + β ( i) nm β ( i) B 7 2 ( i) Am + β 15 + β ( i) C 8 ( i) BC 2 (1.4) and (i) =1, 2, 3, 4, 5, 6, 7, 8 and 9. 13

27 where i is the index of the location along AB and CD at which the strain rate is extracted from FE simulation as shown in Figure 1.2, β 1 through β 2, ψ through ψ 2 and λ 1 thorugh λ 2, are the coefficients of the second-order polynomials that need to be determined. t FEA, έ FEA (peak) and έ FEA (i) are the predicted chip thickness, peak strain rate and strain rate (i.e. at other nine locations along AB and CD) respectively. The least square technique is used to estimate the coefficients of the second-order polynomials by minimizing the residual error measured by the sum of square deviations between the actual and the estimated responses. After finding the optimal set of the coefficients of the second-order polynomials, the second-order polynomials are obtained. However, the estimated coefficients of the second-order polynomials need to be tested for statistical significance. This involves performing the following statistical tests: (1) test for significance of the regression model, (2) test of significance on individual model coefficients, (3) test for pure error and lack-of-fit. Additionally, the normal probability plot of residuals and residuals versus fitted values plots should be examined to validate that the residuals are normally distributed and to detect non-constant variance, missing high-order terms and outliers. Another useful check is the coefficient of multiple determination (R 2 ) and adjusted coefficient of multiple determination (R 2 ad). These coefficients have values between and 1. Generally the higher these coefficients, the better the model fits the actual data. A complete description on DOE, Analysis of Variance (ANOVA), and regressions techniques can be found in (Montgomery, 2). 1.5 Results and Discussion Typical FEA Results The FEA commercial software MSC. Marc is used to simulate two-dimensional orthogonal machining. The 27 different sets of JC model parameters presented in Table 1.6 are used to simulate one cutting condition (AMM cutting condition 2). A sample of the results extracted from FEA simulations as per experimental plan shown in Table 1.5 are presented in Table 1.6. The strain rates at other nine locations along AB and CD are not shown in table 1.6 for the sake of briefness. Typical print out (from MSC. MARC) of the final mesh for FEA simulations as per experimental plan shown in Table 1.5 are provided in Appendix A The simulations are conducted and the chip formation process (from the time the tool indents the workpiece to the steady state) is fully developed as shown in Figure 1.3. The FEA machining process is 14

28 considered to be at steady state when the outputs do not change with time significantly. The cutting and thrust forces stabilize and reach steady state early in the analysis ad even before a fully developed chip is achieved as shown in Figures 1.4 and 1.5. The average cutting force is N and the average thrust force is 169 N when the friction coefficient is.375. The average predicted cutting forces are matched very well with experimental cutting forces (575 N) that represent less than.67% error. The predicted thrust forces are always lower than the experimental force (38 N) by approximately 25% to 5%. A similar discrepancy in the thrust forces predicted by FE simulation was observed even though three different commercial software (MSC. MARC, Deform2D, and Advant-Edge) and various values for friction coefficient were used (Halil et al., 25). A similar discrepancy in the predicted thrust forces was also observed using Abaqus to simulate machining under AMM cutting conditions 2, 5, 6 and 8 (Adibi-Sedeh et al., 25). This issue needs to be addressed in future research as it is discussed later. A summary of cutting and thrust forces, chip thickness, peak strain rate and strain rate at other nine locations along AB and CD are provided in appendix A. Figure 1.6 shows the distribution of the equivalent plastic strain rate, velocity, equivalent plastic strain, temperature, and the equivalent von misses stress. The high rate deformation during metal cutting process is concentrated in two regions close to the cutting tool edge. These regions are a parallel-sided primary shear zone and a triangular secondary shear zone near the rake face of the cutting tool as shown in Figure 1.6(a). The strain rate reaches its maximum value inside the primary shear zone and close to the cutting tool edge. It can be seen from Figure 1.8(b) that the chip thickness is uniform and the chip velocity is also uniform through the chip thickness. Graphical plots for strain rate along AB and CD for FEA simulations as per experimental plan shown in Table 1.5 are provided in Appendix A. Figure 1.8(c) shows the variations of the equivalent plastic strain, which gradually increases as the material enters and passes the primary deformation zone. The plastic strain is uniformly high along the rake face and close to the tip of cutting tool edge due to the plastic deformation, and in addition, due to friction at the tool-chip interface. Similar trends are observed for the temperature where the maximum temperature occurs along the chip-tool interface as shown in Figure 1.8(d). From figure 1.8(e), it can be seen that effective stress in the PSZ increases due to increase in both the strain and strain rate and then decreases due to decrease in the strain rate and increase in temperature. 15

29 TABLE 1.6 SUMMARY OF FEA RESULTS Run # Friction coefficient Design variables A B C n m Chip thickness (µm) Responses Peak strain rate (1/s)

30 Figure 1.3 Final mesh of FE simulation #27 using MAT1 set of constants showing the fully developed chip and the effective stress in the machined surface, PSZ and the chip Cutting force (N) FE simulation time (second) Figure 1.4 Cutting force versus time predicted by FE simulation #27 using MAT1 set of JC model constants. 17

31 Thrust force (N) FE simulation time (second) Figure 1.5 Thrust force versus time predicted by FE simulation #27 using MAT1 set of JC model constants. (a) Equivalent plastic strain rate in 1/s. 18

32 (b) Velocity in mm/sec. (c) Equivalent plastic strain. 19

33 (d) Temperature in o C. (e) Equivalent von misses stress in MPa. Figure 1.6 Contours of (a) Equivalent plastic strain rate in 1/s, (b) Velocity in mm/s, (c) Equivalent plastic strain, (d) Temperature in o C and (e) Equivalent von misses stress in MPa. (FEA run #27 using MAT1) 2

34 1.5.2 ANOVA It is initially assumed that full second-order polynomials can be used to describe how the chip thickness, peak strain rate, and strain rate at nine other locations along AB and CD vary with the changes in the JC model parameters. However, best subsets regression identifies the best-fitting regression models that can be constructed with as few variables as possible. Subset models may actually estimate the regression coefficients and predict future responses with smaller variance than the full model using all variables. Therefore, the FEA results of the chip thickness, peak strain rate and strain rate at other nine locations are input into a statistical commercial package (MINITAB) to find the best subset that includes only the important variables of the response under investigation. This results in improved models with as few variables as possible as it is described later and summarized in appendix B. The chip thickness and peak strain rate are explored in this section as an example of how the statistical tests are performed to asses the second-order polynomials: equations (1.2) and (1.3). The statistical tests are also carried out to asses the second-order polynomials for strain rates at other nine locations (equation 1.3) and not discussed since it is too lengthy to be included here (see appendix B for details). The FEA results of the chip thickness and peak strain rate (see Table 1.6) are input into a statistical commercial package (MINITAB) for further analysis following the steps outlined in the previous sections. Examination of ANOVA, which is shown in Table 1.7 along with the P-values that are smaller than 5% for the chip thickness and peak strain rate show that the reduced second-order polynomials are statistically significant. The R 2 values are high which is desirable and is in reasonable agreement with adjusted R 2. A check on the normal probability plots of the residuals and the plots of the residuals versus the predicted values for the chip thickness and peak strain rate (see Figures 1.7, 1.8, 1.9 and 1.1) reveal that the residuals generally fall on a straight line and are scattered randomly around zero and there is no obvious pattern. This implies that the proposed reduced second-order polynomials are adequate and there is no reason to suspect any violation of normal distribution or constant variance assumption. Summary of analysis of variance (ANOVA) for strain rate at other nine locations along AB and CD are summarized in Appendix B. 21

35 TABLE 1.7 SUMMARY OF ANALYSIS OF VARIANCE (ANOVA) FOR CHIP THICKNESS AND PEAK STRAIN RATE Source Response: chip thickness Response: peak strain rate F-test P-value F-test P-value Regression R R 2 adj Response: chip thickness Response: peak strain rate Model terms Coefficients P-value Model terms Coefficients P-value Constant Constant A n B m C C*C n n*n n*n A*C A*B..9 A*n A*m.13.2 A*m B*C B*C C*n B*n n*m B*m C*n C*m

36 Residuals Versus the Fitted Values (response is strain rate) 2 Standardized Residual Fitted Value Figure 1.7 Plot of residuals versus predicted values for peak strain rate. 99 Normal Probability Plot of the Residuals (response is strain rate) 95 9 Percent Standardized Residual 2 3 Figure 1.8 Normal probability plot of the residuals for peak strain rate. 23

37 2 Residuals Versus the Fitted Values (response is chip thickness) Standardized Residual Fitted Value Figure 1.9 Plot of residuals versus predicted values for chip thickness. 99 Normal Probability Plot of the Residuals (response is chip thickness) 95 9 Percent Standardized Residual 2 3 Figure 1.1 Normal probability plot of the residuals for chip thickness. 24

38 1.6 JC Model Parameter Sensitivity Figures 1.11 through 1.2 show plots of the chip thickness and peak strain rate predicted from the FE simulations with the changes in the JC model parameters (see Table 1.6). It is clear from Figures 1.11, 1.12, 1.13 and 1.15 that the chip thickness decreases with increase in JC model parameters: A, B, C and m. From equation (1.1), increasing the value of any of JC model parameters (A, B, C or m) while keeping all other parameters constant will result in increase in the flow stress. In this paper as it is mentioned previously, the coefficient of friction for each FE simulation is tuned such that the cutting force predicted by FE simulation best matches the experimental cutting force. This results in decrease in the coefficient of friction with increase in JC model parameters (A, B, C and m) so that the cutting force predicted from FE simulation best matches the experimental one. This decrease in friction explains the decrease of chip thickness with increase in JC model parameters. Conversely, Figure 1.14 indicates that the chip thickness increases with increase in strain hardening exponent n due to decrease in the friction to keep the cutting forces predicted by FE simulation best matches the experimental one. Figures 1.16, 1.17 and 1.2 indicate that peak strain rate changes with changes in A, B and m. The peak strain rate decreases when JC model parameters (A, B and m) are less or greater than the nominal values found by Jaspers (Jaspers and Dautzenberg, 22). It is clear from Figure 1.19 that the peak strain rate decreases with increase in n and C. In addition to the graphical plots of the chip thickness and peak strain rate with the changes in the JC model parameters shown in Figures 1.11 through 1.2, a sensitivity analysis of FE predictions for chip thickness and peak strain rate is carried out with reference to chip thickness and peak strain rate predicted from FE simulation #27 using the nominal values of JC model constants (MAT1). The lowest (- 2) and highest (+2) levels of each parameter given in Table 1.5 are used to study the percentage change in FE predictions with the percentage change in JC model parameter. The lowest (-2) and highest (+2) levels of each parameter represent 5% decrease or increase in each parameter respectively. The following equations are used to calculate the percentage change in the chip thickness and peak strain rate respectively: tno min al tfea # % change in chip thickness = 1 (1.5) tno minal 25

39 & ε % changein peak strain rate = no min al & ε & ε no min al FEA# 1 (1.6) where t nominal and έ nominal are the chip thickness and peak strain rate predicted respectively from FE simulation # 27 shown in Table 1.6 using MAT1 set of JC model constants. On the other hand t FEA# and έ FEA# are the chip thickness and peak strain rate predicted respectively from FE simulations shown in Table 1.6 corresponding to the lowest (-2) and highest levels (+2) of each parameter shown in Table 1.5. Table 1.8 shows a summary of JC model parameter sensitivity on the chip thickness and peak strain rate. Some conclusions based on Figures 1.11 through 1.2 as well as table 8 are as follows: 1. With increasing A, B, C and m, chip thickness decreases. 2. On the contrary with increasing strain hardening exponent n : a. Chip thickness increases. b. Peak strain rate decreases. 3. The 5% reduction in each of A, B, C, n and m has more significant influence on the chip thickness than 5% increase in each of A, B, C, n and m. 4. The 5% reduction in each of A, B, C and m has more significant influence on peak strain rate than 5% reduction in each of the same parameters. 5. The 5% reduction in strain hardening exponent n results in highest percentage change in both the chip thickness of 22.89% and peak strain rate of %. 6. The influence of thermal softening parameter m is not very significant on the chip thickness compared to the influence of A, B, C and n. The effect of m can be very significant at slow cutting speeds and high feed rates since under these cutting conditions extremely high temperatures may be present. 26

40 TABLE 1.8 SUMMARY OF JC MODEL PARAMETER SENSITIVITY JC model parameter % Change in JC model parameter Coded levels % Change in chip thickness % Change in peak strain rate A B C n m Chip thickness (micrometer) A (MPa) Figure 1.11 Decrease of chip thickness with increase in A 27

41 Chip thickness (micrometer) B (MPa) Figure 1.12 Decrease of chip thickness with increase in B Chip thickness (micrometer) C Figure 1.13 Decrease of chip thickness with increase in C 28

42 Chip thickness (micrometer) n Figure 1.14 Increase of chip thickness with increase in n Chip thickness (micrometer) m Figure 1.15 Decrease of chip thickness with increase in m. 29

43 36 34 Peak strain rate along CD (1/s) A (MPa) Figure 1.16 Change of peak strain rate along CD with increase in A Peak strain rate along CD (1/s) B (MPa) Figure 1.17 Change of peak strain rate along CD with increase in B. 3

44 36 34 Peak strain rate along CD (1/s) C Figure 1.18 Decrease of peak strain rate along CD with increase in C Peak strain rate along CD (1/s) n Figure Decrease of peak strain rate along CD with increase in n. 31

45 Peak strain rate along CD (1/s) m Figure 1.2. Change of peak strain rate along CD with increase in m. 1.7 Determination JC Model Parameters The procedure used to fine-tune and optimize JC model parameters for AISI 145 is to use a single aggregate objective function defined by: N & ε ( A, BC, n, m) = min i= 1 FEA ( i) & ε & ε ( i) EXP EXP 2 ( i) t + W 1 FEA t t EXP EXP 2 + W 2 & ε FEA 2 ( peak) & ε EXP( peak) & ε EXP( peak) (1.7) where i is the index of the nine locations along AB and CD at which strain rates are obtained from both cutting tests and FE simulations (see Figure 1.2), t EXP and έ EXP (i) are the measured chip thickness and strain rate respectively. On the other hand t FEA and έ FEA are second-order polynomials defined previously by equations (1.2), (1.3) and (1.4). W 1 and W 2 are scalar weights account for the relative importance of chip thickness and strain rate on achieving the optimal set of JC model parameters as it is described later. The objective function given by equation (1.7) does not incorporate the cutting forces because it 32

46 was mentioned earlier that cutting forces predicted by FEA matched measured forces by tuning coefficient of friction for each DOE point. Several commercial optimization software available in today s market can be used to optimize various engineering problems. Microsoft Excel Solver, MathOptimizer for Mathematica, Matlab, and Maple are just few examples of commercial optimization packages. These codes share the same unique feature-iteration. Local or global search algorithms are employed to find the optimum solution. Local search methods use first and second derivatives to find local optimum while global search methods systematically search the entire general region in which the global optimum lies. In this research, Global Optimization Toolbox (GOT) of Maple is used to perform the optimization of the objective function defined by equation (1.7). The core of GOT is the Maple-specific implementation of the Lipschitz Global Optimizer (LGO) solver. LGO offers Local and global search algorithms. Moreover the GOT optimizer currently includes the following search methods that can be optionally selected by the user: 1. Branch-and-bound based global search (BB). 2. Global adaptive random search (single-start) (GARS). 3. Random multi-start based global search (MS). 4. Generalized reduced gradient (GRG) algorithm based local search (LS). A combination of the above four search methods is used to minimize the objective function with the result being an optimum set of JC model parameters. A complete description on Global Optimization Toolbox (GOT) of Maple can be found on as well as an excellent book written by Janos (Janos, 26). The flow chart shown in Figure 1.21 outlines the optimization process of the objective function defined by equation (1.7). The JC model parameters identified using the methodology described above are presented in Table 1.9. This optimum set of JC model parameters (designated by MAT7) is then used to simulate machining under AMM cutting condition 2 for which experimental measurements for chip thickness, peak strain rate and strain rate distribution along AB and CD are available. The FEA predictions obtained with this new optimum set of constants (MAT7) are summarized in Table 1.1 and The coefficient of friction used in FEA simulation with this new optimum set of constants (MAT7) is fine-tuned iteratively as 33

47 described previously until the cutting forces predicted by the FEA simulation best matches the measured cutting forces. The average predicted cutting forces using a value of (.25) for coefficient of friction are matched very well with experimental cutting forces (575 N) that represent less than 5 % error. Figures 1.22 and 1.23 also provide a comparison of the strain rate obtained experimentally from machining test with the strain rate predicted from FEA simulation using MAT7 (new optimum set of constants given in Table 1.9) and MAT1 (Jaspers and Dautzenberg, 22) sets of JC model constants. The MAT1 set of JC model constants is chosen purposefully for comparison purposes with MAT7 performance since MAT1 set of constants has been widely used by numerous researchers to simulate machining as well as Taylor impact test for AISI 145 carbon steel (Deshpade et al., 26; Adibi-Sedeh et al., 25; Allen et al., 1997). It can be concluded from Figures 1.22 and 1.23 as well as Table 1.1 and 1.11 that the results of FEA simulation when MAT7 set of JC model constants is used are in close agreements with the experimental measurements and outperforms the performance of MAT1 set of constants obtained from SPHB tests. This indicates the potential success of this methodology. 34

48 Start DOE points: A, B, C, n, m (27runs presented in table 6) Run FEA Simulation: 27 runs (MSC Marc) Begin with an initial value of friction µ µ = µ ± µ (Fc)p : FEA predcition of cutting forces. (Fc)m: measued cutting forces from experiment. NO (Fc)p =(Fc)m ± 1 (Newton) YES Extract FE simulation results: (1) Peak strain rate. (2) Strain rates at 1, 2, 3, 4, 5, 6, 7, 8 and 9 defined in figure 2. (3) Chip thickness. Estimate the coefficients of the second-order response surfaces: Equations 2, 3 and 4 (MINITAB) Determine the lower and upper limits for A, B, C, n & m Optimization process of the objective function: Equation 8 (GOT of Maple) Optimum set of JC model parameters End Figure 1.21 Optimization process of the objective function defined by equation (1.7). 35

49 TABLE 1.9 OPTIMUM SET OF JC MODEL PARAMETERS Optimum set of JC constants A B C n m MAT TABLE 1.1 SUMMARY OF FEA PREDICTIONS USING THE OPTIMUM SET OF JC MODEL CONSTANTS (MAT7) UNDER AMM CUTTING CONDITION 2 Method Chip thickness (µm) Average strain rate (1/s) Machining test FEA (MAT1) FEA (MAT7) TABLE 1.11 SUMMARY OF STRAIN RATE ALONG AB AND CD PREDICTED USING THE OPTIMUM SET OF JC MODEL CONSTANTS (MAT7) UNDER AMM CUTTING CONDITION 2 JC constants ε& 1 ε& ε& 2 3 ε& ε& 4 5 ε& 6 ε& 7 ε& 8 ε& 9 DIC FEA (MAT1) FEA (MAT7)

50 3 25 Experimental (DIC) FEA (MAT1) FEA (MAT7-optimum) 2 Strain rate (1/s) Distance along CD (micrometer) Figure 1.22 Comparison of experimental strain rate with strain rate predicted from FEA simulation along CD using MAT1 and MAT7 sets of JC model constants under AMM cutting condition 2 (CD direction from left to right) Experimental (DIC) FEA (MAT1) FEA (MAT7-optimum) 7 Strain rate (1/s) Distance along AB (micrometer) Figure 1.23 Comparison of experimental strain rate with strain rate predicted from FEA simulation along AB using MAT1 and MAT7 sets of JC model constants under AMM cutting condition 2. (AB direction from left to right) 37

51 1.8 Sensitivity Study of the Optimization Process A sensitivity analysis of the chip thickness and peak strain rate on the optimization process of the objective function is carried out by changing the relative importance of W1 and W2 given in equation (1.7). This sensitivity analysis measures the effect each output (chip thickness and peak strain rate) has on the process of minimizing the objective function defined by equation 1.7 as the root mean-squared (RMS) error between the data predicted from FE simulation and data obtained from machining test. The optimum set of JC model constants presented in Table 1.9 is achieved by assigning equal relative importance of one to both W 1 and W 2 (W 1 = W 2 = 1.). In the present sensitivity analysis, the relative importance of W 1 and W 2 is allowed to vary between. and 1 and the objective function is again minimized producing different sets of JC model constants. Table 1.12 summarizes the sensitivity study performed as it is described above showing the root mean-squared (RMS) error and the percentage changes in the optimal values of A, B, C, n and m. For example, the percentage change in A is calculated by: ASEN # AMAT 7 % change in A = 1 (1.8) AMAT 7 where A MAT7 is the optimal value of A shown in Table 1.9 obtained when W 1 = W 2 = 1. and A SEN# is the new optimal value of A obtained by minimizing again the objective function when W 1 and W 2 are allowed to vary as shown in Table The percentage changes in B, C, m and n are calculated in a manner similar to the calculation of percentage change in A. When the % change in A is equal to zero (A MAT7 = A SEN# ), this indicates that the objective function is independent of the relative importance of W 1 and W 2 since they have no effect on achieving the optimal value of the objective function initially obtained when W 1 = W 2 = 1. On the contrary, if the percentage change in A is equal to 1%, this indicates that the objective function is strongly dependent on the combination of W 1 of W 2. Some conclusions based on Table 1.12 are as follows: 1. With increasing the relative importance of (W 1 ) or (W 2 ), the RMS error increases. 2. The relative importance of W 1 and W 2 is not significant on achieving the optimal value for each of the JC model parameters (A, B and m). 38

52 3. The relative importance of W 1 and W 2 has a significant effect on achieving the optimal values for C and n. Moreover, the relative importance of W 1 and W 2 has more effect on the strain rate sensitivity C than strain-hardening exponent n. 4. With increasing relative importance of W 1, the percentage change in both C and n decreases. 5. When assigning high values for the relative importance of chip thickness (W 1 = 5 and 1), the RMS error increases significantly. 6. The dependence of achieving the optimal values of JC model parameters on peak strain rate is expected. The strain rate sensitivity C represents and accounts for the strain rate in the PSZ. This indicates that combining peak strain rate in the objective function may be necessary to achieve the optimal values of strain rate sensitivity C. 7. The significant dependence of achieving the optimal values of JC model parameter n on the chip thickness can be explained as follows. The parameter n affects the flow stress of the work material. Higher values of n represent higher flow stress. This results in increase in the work required to form the chip. Since the predicted cutting forces obtained from FE simulation is matched with the corresponding experimental one, higher flow stress (higher values of n ) allows the cutting tool to initially indent the work piece and then begins to form the chip more difficulty causing a thinner chip. 8. The optimum set of JC model parameters shown in Table 1.9 obtained when W 1 = W 2 = 1. is believed to give more accurate results since the chip thickness and the strain rate distribution in the PSZ have equal significant effect on the flow stress. 39

53 TABLE 1.12 SUMMARY OF THE SENSITIVITY ANALYSIS OF THE OPTIMIZATION PROCESS SEN # W1 W2 MSE % Change in A % Change in B % Change in C % Change in n % Change in m Summary and Conclusion In the present study, we propose a framework as a first attempt to fine-tune and optimize JC model parameters beyond cutting forces based on direct measurement of strain rate in the PSZ under only one orthogonal machining test (AMM cutting condition 2). Design of Experiment (DOE) approach and FEA simulations are used to conduct computer experiments to explore the design space of all possible values of JC model parameters. A Second-order response surfaces are fitted to the FEA predictions to 4

54 describe how chip thickness, peak strain rate and strain rate at five points along AB and four points along CD vary with the changes in JC model parameters. The objective is to determine the material constants that minimize the error between experimental observations and FEA results for each of these quantities and the objective function is taken to be the sum-squared error. A global optimizer that uses combinations of branch-and bound based global search (BB), global adaptive random search (GARS), random multistart based global search (MS), and generalized reduced gradient (GRG) algorithm based local search (LS) is used to find the optimum set of JC model parameters that minimize the objective function. This optimum set of JC model constants are then used to simulate machining test under AMM cutting condition 2 using MSC. MARC. The FEA predictions using this optimum set of constants (chip thickness, shear angle, shear plane length AB, peak strain rate and strain rate distribution along AB and CD) are found to be in close agreements with the experimental measurements and indicates the potential success of this methodology. The accuracy of this optimum set of JC model constants needs to be further validated by simulating machining under other cutting conditions for which the experimental data (chip thickness, shear angle and strain rate distribution in the PSZ) are available. Another validation and sensitivity study are recommended by simulating machining under AMM cutting condition 2 using other commercial software such as Abaqus. Abaqus can also be used to carry out computer simulations at 27 different sets of JC model parameters presented in Table 1.6. Second-order polynomials equations (1.2), (1.3) and (1.4) can be again fitted to the FEA predictions (chip thickness, peak strain rate and strain rate at five points along AB and four points along CD). Then, the optimization process can be carried out using GOT of Maple with the result being a new optimum set of JC model constants. This future work will check the sensitivity of JC model parameters to the type of FEA commercial software used. It will also explore the capability of using different commercial software on achieving approximately and reasonably the same optimum set of constants. 41

55 LIST OF REFERENCES Adibi-Sedeh, A.H., M. Vaziri, V. Pednekar, V. Madhavan, and R. Ivester (25). Investigation of the Effect of Using Different Material Models on the Finite Element Simulations of Machining. 8th CIRP International Workshop on Modeling of Machining Operations, Chemnitz, Germany. Allen, D.J., W.K. Rule, and S.E. Jones (1997). Optimizing Material Strength Constants Numerically Extracted from Taylor Impact Data. Experimental Mechanics. Vol. 33. pp Arsecularatne, J.A. and L.C. Zhang (24). Assessment of constitutive equations used in machining. Key Engineering Materials, Vols , p Cho, H. and T. Altan (25). Determination of flow stress and interface friction at elevated temperatures by inverse analysis technique. Journal of Materials Processing Technology, Vol. 17, p Deshpade, A., V. Madhavan, V. Pednekar, A.H. Adibi-Sedeh, and R. Ivester (26). Dependence of Cutting Simulations on the Johnson Cook Model Thermal Softening Parameter. Transaction of the North American Manufacturing Research Institute of SME, Vol. 34, pp Dirikolu, M.H. and H.C. Childs. (2) Modelling Requirements for Computer Simulation of Metal Machining. Turkish Journal of Engineering and Environmental Science, Vol. 24, pp El-Magd, E., C. Treppmann, and M. Korthauer (23). Constitutive Modelling of CK45N AlZnMgCu1.5 and Ti-6AL-4V in a Wide Range of Strain Rate and Temperature. Journal De Physique IV, Vol. 11, pp Halil, B., A. Tekkaya, and S. Engin. (25) 2D FE Modeling of Machining: A Comparison of Different Approaches with Experiments. VIII International Conference on Computational Plasticity, Vol. 3, pp. 1-4, 25. Holmquist, T.J. and G.R. Johnson (1991). Determination of Constants and Comparison of Results for Various Constitutive Models. Journal de Physique IV, Vol. I, pp Ivester, R.W., and M. Kennedy (24). Comparison of Machining Simulation for 145 Steel to Experimental measurements. IMTS Conference, Chicago, Illinois, USA. Ivester, R.W., M. Kennedy, M. Davies, R. Stevenson, J. Thiele, R. Furness, and S. Athavale (2). Assessment of Machining Models: Progress Report. Machining Science and Technology, Vol. 4, pp János D.P. (26). Global Optimization with Maple, Pintér Consulting Services Inc., Canada, 26. Jaspers, S.P.F.C. and J.H. Dautzenberg (22). Material Behavior in Conditions Similar to Metal Cutting: Flow Stress in the Primary Shear Zone. Journal of Materials Processing Technology, Vol. 122, pp Lim, K., P.A. Manohar, D. Lee, Y.C. Yoo, C.M. Cady, G.T. Gray III, and A.D. Rollett (23). Constitutive Modeling of High Temperature Mechanical Behavior of a Medium C - Mn Steel. Materials Science Forum, Vols , p Madhavan, V., D. Mahadevan, A. Belur-Sheshadri, K. Yegneswaran, AH. Adibi-Sedeh, A. Hijazi and M. Saket-Kashani (27). Experimental Determination of Velocity And Strain Rate Fields in Metal Cutting. Submitted to Journal of the mechanics and physics of solids. 42

56 Mathew, P., and N.S. Arya (1993). Material Properties from Machining Proceedings of the Conference on Dynamic Loading in Manufacturing and Service, Melbourne Australia. pp Meyer, L.W., T. Halle, N. Herzig, L. Kruger, and S.V. Razerenov (26). Experimental investigations and Modelling of Strain Rate and Temperature Effects on the Flow Behavior of 145 Steel. Journal De Physique IV, Vol. 134, pp Montgomery, D.C. (2). Design and Analysis of Experiments. John Wiley and Sons, Inc., New York. Oxley, P.L.B. (1989). The Mechanics of Machining: An Analytical Approach to Assessing Machinability. Ellis Horwood, Chichester, England. Ozel, T. and K. Yigit (27). Identification of Constitutive Material Model Parameters for High-Strain Rate Metal Cutting Conditions Using Evolutionary Computational Algorithms. Materials and Manufacturing Processes, Vol. 22, pp Ozel, T. and E. Zeren (26). A Methodology to Determine Work Material Flow Stress and Tool Chip Interfacial Friction Properties by Using Analysis of Machining. Journal of Manufacturing Science and Engineering, Vol. 128, pp Sartkulvanich, P., T. Altan, and A. Gocmen. (25). Effects of Flow Stress and Frictions Models in Finite Element Simulation of Orthogonal Cutting- A Sensitivity Analysis. Machine Science and Technology, Vol. 9, pp Sartkulvanich, P., F. Koppka, and T. Altan (24). Determination of Flow Stress for Metal Cutting Simulation-A Progress Report. Journal of Material Processing Technology, Vol. 146, pp Shida, S. (1969). Empirical Formula of Flow Stress of Carbon Steels-Resistance to Deformation of Carbon Steels at Elevated Temperature. 2nd Report. JTSP, Vol. 1. pp Shirakashi, T. and T. Obikawa (1998). Recent progress and some difficulties in computational modeling of machining. Machining Science and Technology, Vol. 2, p Stevenson, M.E., Jones, S.E., and Bradt, R.C (23). The High Strain Rate Dynamic Stress-Strain Curve for OFHC Copper. Materials Science Research International, Vol. 9, pp Umbrello, D., R. M Saoubi, and J.C. Outeiro. (27). The Influence of Johnson Cook Material Constants of finite element simulation of Machining of AISI 316L Steel. International Journal of Machine Tools and Manufacture, Vol. 47, pp

57 PART TWO DETERMINATION OF CONSTITUTIVE MODELS OF OFHC COPPER USING ORTHOGONAL MACHINING TESTS SUMMATION Many experimental and numerical research in metal cutting proved that the predictions obtained from both the analytical and Finite Element Analysis (FEA) based models are highly dependent on the material models (and their constants) used. It also shows that both empirical and physically based models may give erroneous results when used to predict the flow stress over their extended ranges of strain, strain rate, and temperature. Despite the uncertainty in the material models (and their constants), these models have been extensively used in FEA by academia and industry to provide a reliable and accurate prediction of metal cutting processes. Therefore, performance validation, fine-tuning, and optimization of the material models in machining is an important effort. The present study first quantitatively evaluates and compares the performance of different material models widely used in machining with reference to the experimental results obtained from cutting tests. Unlike other research, the strain and strain rate along the nominal shear plane are experimentally measured using Digital Image Correlation (DIC) of high-speed photographic images of the side surface of the chip. Since the average strain and strain rate directly measured from cutting tests, an analytical estimate is required only for the temperature, hence reducing the uncertainty in the proposed approach for identification of the constants used in different material models. The determination of the material constants used in the Johnson-Cook and Zerilli-Armstrong constitutive models for Oxygen-free highconductivity (OFHC) copper from carefully designed machining experiments is the focus of this research. 44

58 2.1 Introduction In the past decade, the need for accurate material constitutive equations covering wide ranges of flow stress, strain, high strain rate and elevated temperature has been increasingly felt in impact and crash simulation as well as in the simulation of high speed forming processes such as strip rolling, wire drawing and machining. In the impact physics area, high-strain rate tests such as the split Hopkinson pressure bar (SHPB) tests have been widely used to produce flow stress data and fit that data to different material constitutive equations. In the machining area, the use of cutting tests to obtain the material model parameters is widely prevalent. A survey of the literature (see Table 1.1 of part one) indicates that flow stress data for metals have been mainly obtained using four methods: high-speed compressions (HSC) tests, split Hopkinson pressure bar (SHPB) tests, practical machining tests and reverse engineering using FEA simulation (Allen et al., 1997; Cho and Altan, 25). SHPB tests can generate high strain rates and elevated temperatures but not at the levels observed in machining. Machining tests can raise the achievable strain rate up to 6.8 x 1 5 1/s, which is two orders of magnitude higher than that achieved by SPHB tests (Jaspers and Dautzenberg, 22; Sartkulvanich et al., 24). The SPHB test is widely used for intermediate and high strain rate measurements and can be adapted to several modes of loading such as tensile, compressive, flexure and torsion. A great advantage of machining tests is that wide ranges of large strain and high strain rate can be conveniently achieved by adjusting the cutting conditions composed of rake angle, cutting speed and depth of cut. However, the measurements of low and moderate strain rate using machining tests become difficult. Additionally, machining tests coupled with analytical models are limited to cases of continuous chips. Therefore, several researchers have combined both machining and SHPB tests to optimize the material model constants in which the SPHB tests were used to obtain the flow stress at moderate strain rates and machining tests were used for high strain rates (Ozel and Zeren, 26; Mathew and Arya, 1993). In earlier work using machining to obtain material model parameters, small number of cutting conditions was used to fine-tune and optimize the material constants (Ozel and Zeren, 26; Mathew and Arya, 1993). The approach was primarily based on measurements of cutting forces and chip thickness. The flow stress was calculated from measured cutting forces and chip thickness. Analytical models are 45

59 used to estimate the strain rate and temperature along the nominal shear plane AB, but the estimates depend on the material model constants. This coupling causes problems in the techniques used to finetune and optimize the material model constants by fitting the model predictions of the flow stress to the experimental measurements of the flow stress and results in increased uncertainty in the material model constants obtained from machining tests. If the average strain and strain rate along the nominal shear plane AB can be directly measured from cutting tests, analytical estimates will be required only for the temperature (Madhavan et al., 27). In summary, many experimental and numerical research proved the following with regard to metal cutting and material models: 1. The considerable sensitivity to strain rate, strain hardening, and thermal softening. 2. The numerical predictions of the typical machining outputs are sensitive to the material models used and that not one of the material models used is best in all aspects (Adibi-Sedeh et al., 25; Sartkulvanich et al., 25). 3. The numerical predictions of the typical machining outputs are sensitive to different constants used in the same material model (Deshpade et al., 26; Umbrello et al., 27). 4. Different methods and test data produced different material model constants (Holmquist and Johnson, 1991) 5. A highly reliable and accurate material model covering wide ranges of flow stress, strain, high-strain rate, and high temperature under machining conditions is perquisite for the Finite Element (FE) simulation of machining (Shirakashi and Obikawa, 1998). 6. Machining experiments under a small number of cutting conditions in conjunction with analytical models were used to identify different material model constants based primarily on the measured cutting forces and chip thickness (Mathew and Arya, 1993; Ozel and Zeren, 26; Tounsi et al., 22). 7. Direct measurement of high strain rate by experiment under machining conditions is needed to develop and validate the material models. This paper is organized in the following main sections: first, we introduce Johnson-Cook (JC) material model and present the JC model constants found in the previous published studies. Next, based 46

60 on the published values of JC model constants found in the literature, a comparison of JC model with different model constants over a wide range (strain = 2., strain rate = 1 5 1/s and temperature = 1 1 o C) typically observed in machining is presented. The second section is concerned with different material models used to describe the mechanical behavior of OFHC. The material constants for different models are summarized followed by a comparison of these models over a wide range of strain, strain rate, and temperature typically observed in metal cutting. The third section consists of a brief introduction to theory of machining as well as the mathematical formulas used to predict the flow stress and temperature in machining. The fourth section is concerned with the orthogonal machining experiments that are carried out on a high-speed linear slide. It presents a novel experimental capability for obtaining the velocity, strain, and strain rate fields in the PSZ. The fifth section provides an assessment of different material models with reference to the calculated flow stress and temperatures as well as measured strains and strain rates obtained directly from the machining experiments. The sixth section consists of the optimization problem with a brief introduction to the commercial global optimizer used to fine-tune and optimize the JC model constants. Finally, the last section consists of a brief discussion and concluding remarks for the research. 2.2 High Strain Rate Constitutive Models for OFHC Copper The material model most commonly used in metal cutting is the Johnson-Cook (JC) material model. This model considers the flow stress to be a product of three terms representing the effect of strain, strain rate, and temperature. It is defined by the following equation: n m ( A + Bε )( 1+ C lnε )( 1 T ) σ = & (2.1) H where T H T T T T r = and m r & ε = & ε & ε where σ is the material flow stress, ε is the plastic strain, έ is the strain rate (1/s), έ o is the reference plastic strain rate (1/s), T is the temperature of the work material ( C), T m is the melting temperature of the work material and T r is the reference temperature. A, B, C, n and m are five empirical material constants. These five empirical material constants can be obtained from various methods and tests. 47

61 Table 2.1 shows different sets of material constants proposed to be used in the JC model for OFHC. The MAT1 set of constants was obtained from mechanical testing on OFHC using quasi-static tests as well as SHPB tension and torsion tests (Johnson and Cook, 1983]. While MAT2 set of constants was determined from SHPB tension tests only, the MAT3 set of constants was identified from both SHPB tension and torsion tests (Holmquist and Johnson, 1991). The approach of determining MAT2 and MAT3 sets of constants was to identify five data points from the tension and torsion/tension data. Direct substitutions of stress, strain, strain-rate, and temperature into the JC model for each of the five data points produced five equations with five unknown parameters. Then the task was to solve the five equations with five unknowns (JC model constants) with the result being an optimum set of material model constants (Holmquist and Johnson, 1991). The MAT4, MAT5, MAT6, and MAT7 sets of constants were extracted by using a second orderpolynomial to describe the volume difference between a deformed Taylor test cylinder and the FEA simulations (EPIC code) of the test. Then the volume difference was minimized using standard optimization technique by changing the design variables to produce a more accurate set of the material model constants. Four different sets of the JC model constants were initially used to start the optimization procedure, which produced four different sets of the material parameters namely: MAT4, MAT5, MAT6 and MAT7 (Allen et al., 1997). MAT8 set of constants was obtained by matching the experimental measurements of the projectile deformation with the corresponding results of the FEA simulations under Taylor impact tests (Jian et al., 26). Finally, MAT9 set of constants was identified from both quasistatic ( /s) and high strain rate tests (52 and 6 1/s) for temperature ranges from 25 and 269 C (Tanner et al., 1999). An optimizer using combination of genetic, gradient, and simplex algorithms was used to obtain the optimum set of JC model constants that minimized the least square between the predicted and experimental measurements of flow stress (Tanner et al., 1999). Table 2.2 provides a summary of the ranges of strain, strain rates, and temperature at which different JC model constants (as given by Table 2.1) were determined. 48

62 The ranges (minimum and maximum) of interest for each JC model constant are presented in Table 2.3. These ranges of each constant cover a wide range of all possible values of JC model constants found in the literature (see Table 2.1) TABLE 2.1 MATERIAL CONSTANTS PROPOSED TO BE USED IN JOHNSON-COOK S MODEL FOR OFHC COPPER Material Reference A (MPa) B (MPa) n C m JC model constants Type of test n m ( A + Bε )( 1+ C lnε )( 1 T ) σ = & H [1] MAT1 SPHB [2] MAT2 SPHB MAT3 SPHB OFHC MAT4 Taylor impact/fea [3] MAT MAT6 Taylor impact/fea Taylor impact/fea MAT7 Taylor impact/fea [4] MAT8 Taylor impact/fea [5] MAT9 SPHB 49

63 TABLE 2.2 THE RANGES OF STRAIN, STRAIN RATE AND TEMPERATURE AT WHICH JC MODEL CONSTANTS WERE DETERMINED JC model constants Strain Strain rate (1/s) Temperature ( o C) MAT MAT MAT MAT4 (*) x x MAT5 (*) n.a 6 n.a MAT6 (*) n.a 6 n.a MAT7 (*) n.a 6 n.a MAT8 (*) n.a 1. x1 4 n.a MAT (*) The values of strain, strain rate, and temperatures are approximately estimated using the velocity of projectile and the length of Taylor impact specimen. TABLE 2.3 RANGES OF JC MODEL CONSTANTS Material Level A (MPa) B (MPa) n C m Low OFHC High The shortcomings of JC model are: 1. It does not consider blue brittleness in plastic deformation of plain carbon steel. 2. It does not represent any thermal or strain rate history effects. 3. It does not include enhanced strain rate sensitivity at strain rate higher than 1 3 1/s. 4. It decouples the strain rate and temperature effects. 5. It is empirical based model (not a physically based model). This means cautious must be exercised when it is used for extrapolation to region outside the experimental region. 5

64 The strengths of JC model are: 1. It is simple and primarily intended for use in commercial FEA packages. 2. It uses few variables that are already available in most of commercial FEA codes. 3. It provides a basis of comparison with other constitutive models. 4. It has been widely used in analytical and numerical modeling of metal cutting. 5. Its parameters can be easily obtained from limited number of experiments. 6. Availability of its material model constants for various metallic material and aerospace alloys JC Material Model Constants Figure 2.1 shows a comparison of the material flow stress as function of strain, strain rate and temperature obtained from JC material model using different model constants given in Table 2.1. Different symbols shown in Figure 2.1 do not represent any experimental data but they are used for the sake of clarity. It is clear from Figure 2.1 that different JC model constants (MAT1 through MAT9) give different predictions of the work material flow stress. It is mentioned previously that MAT4, MAT5, MAT6 and MAT7 sets of constants were determined by minimizing the volume difference between a deformed Taylor test cylinder and the FEA simulations of the test. The MAT4, MAT5, and MAT6 sets of constants are not included in this comparison study while MAT7 is since MAT7 was found to predict the final volume of a deformed Taylor test cylinder the closest to the experimental data (Allen et al., 1997). The MAT8 set of constants gives consistently the greatest predicted values of flow stress while MAT3 set of constants gives the lowest predicted ones. Figure 2.1 indicates that special care must be exercised when using different JC model constants to perform extrapolations to region outside the experimental region. The predictions of flow stress, strain, strain rate, and temperature are highly dependent on the JC model constants as shown in Figure 2.1. This indicates that caution must also be exercised when the FE simulation is used to predict rather than interpret the typical machining variables. 51

65 6 5 Material flow stress (MPa) MAT1 MAT2 MAT3 MAT7 MAT8 MAT Strain (a) Strain rate = 1 3 and Temperature = 3 o C Material flow stress (MPa) MAT1 3 MAT2 MAT3 25 MAT7 MAT8 MAT Strain-rate (1/sec) (b) Strain =. 75 and Temperature = 3 o C 52

66 6 5 Material flow stress (MPa) MAT1 MAT2 MAT3 MAT7 MAT8 MAT Temperture (c) Strain =.75 and Strain rate = 1 1/s Figure 2.1 Comparison of flow stress predicted from JC model using different constants Other Material Models Several authors have revised the JC material model to better represent the strain rate and temperature effects on the material flow stress. Five revisions of the JC material model designated RJC1, RJC2, RJC3, RJC4, and RJC5 are presented in Table 2.4. Table 2.5 provides a summary of authors for each material constitutive model shown in Table 2.4 The RJC1 model constants (MAT1 and MAT11) were obtained from tension and tension/torsion data respectively which are presented in Table 2.6 (Holmquist and Johnson, 1991). The RJC2 model constants (MAT12) was identified from Taylor impact tests and the FE simulations of the test as given in Table 2.6 (Stevenson et al., 23). The RJC3 model was revised from the original JC model to describe the flow stress of three metals: AISI P2 mold steel (3 HRC), AISI H13 tool steel (46 HRC), and A127 aluminum (1 HB). Then 2D orthogonal slot milling experiments along with OXCUT code was used to determine the RJC3 model constants (Sartkulvanich et al., 24). 53

67 The shortcomings of the original JC model are that it does not include enhanced strain rate sensitivity at high strain rates nor consider blue brittleness in plastic deformation of plain carbon steel, Therefore, RJC4 model has an expanded temperature term for materials such as AISI 145 plain carbon steel that exhibit blue brittleness (Shatla et al., 21). Moreover RJC5 has also an expanded strain rate term for materials that exhibit high rate sensitivity at strain rate higher than 1 3 1/s (Johnson et al., 26). The simplicity of the JC material model and the availability of its constants for various metallic material and aerospace alloys are advantageous for FEA modeling. Physically based material models such as Zerilli-Armstrong (ZA) model can be used to describe the variation of flow stress with strain, strain rate, and temperature (Armstrong and Zerilli, 1987). The ZA material model for FCC (Faced-centered cubic) metals and its constants are given in Tables 2.4 and 2.6 (Armstrong and Zerilli, 1987). The tension and tension/torsion data used to obtain JC model constants (MAT1 and MAT11) were used to obtain ZA model constants (MAT13, MAT14 and MAT15) that are given in Table 2.6 (Holmquist and Johnson, 1991; Armstrong and Zerilli, 1987). The JC model does not consider the coupling effect between strain rate and temperature whereas ZA model does, Thus JC and ZA models were combined into one flow stress equation designated JC-ZA to incorporate the coupling behavior observed in Oxley s model (Holmquist and Johnson, 1991). The combined JC-ZA model and its constants are presented in Tables 2.4 and 2.6 respectively. Table 2.7 summarizes the ranges of strain, strain rate, and temperature at which different model constants were determined. The comparison of four material models (JC, RJC1, ZA and ZA-JC) over a wide range of strains, strain rates, and temperatures (strain =.5 2., strain rate = 1 5 1/s and temperature = 1 1 o C) typically observed in metal cutting are shown in Figure 2.2. Different symbols shown in Figure 2.2 do not represent any experimental data but they are used for the sake of clarity The following sets of constants: MAT3 for JC model, MAT11 for RJC1 model, MAT14 for ZA model and MAT17 for ZA-JC model are used purposefully for models comparison because they were obtained from the same tension/torsion test data. It is evident from Figure 2.2 that the material models predict the flow stress of OFHC differently. The ZA-JC model gives consistently the greatest predicted values of flow stress as 54

68 shown in Figure 2.2. Additionally the predicted flow stress obtained from JC model is very close to predicted flow stress obtained from RJC1 model. TABLE 2.4 DIFFERENT MATERIAL CONSTITUTIVE MODELS FOR OFHC COPPER Material Model Multiplicative Strain Rate Flow Function: Johnson-Cook Constitutive Model Work hardening Strain rate Temperature Parameters n JC ( A + Bε ) ( + ε& m 1 C ln ) ( 1 ) n RJC1 ( A Bε ) n RJC2 ( A Bε ) + ( ) C C lnε C 1 C2 ln & ε C3 T H m ε& ( 1 ) T H m & ( 1 ) T H n RJC3 ( A + Bε ) ( + ε& m 1 C ln ) ( D ) ET H A, B, C, n, m A, B, C, n, m A, B, C, C 1, C 2, C 3, n, m A, B, C, D, E, m n RJC4 ( Bε ) ( + ε& m ( T 7) 1 C ln ) ( 1 H ) + ae T B, C,n, a, m n C3 RJC5 ( A + Bε ) ( 1 C ln C 2(ln ) ) m + & ε + & ε ( 1 ) T H A, B, C, C 2, C 3, m Physically-based Strain Rate Flow Function: Zerilli-Armstrong Constitutive Model n ZA C C ε [ EXP C T + C T ln &)] + FCC metals 2 ( 3 4 ε C, C 2, C 3, C 4, n Combined Johnson-Cook and Zerilli-Armstrong: ZA-JC Constitutive Model A EXP C T n ZA-JC ( + ε ) + ln ) B C T ( 3 4 ε & A, B, C 3, C 4, n 55

69 TABLE 2.5 THE AUTHORS OF DIFFERENT MATERIAL CONSTITUTIVE MODELS FOR OFHC COPPER Material model Authors JC Johnson, 1985 RJC1 Holmquist and Johnson, 1991 RJC2 Stevenson et al., 23 RJC3 Sartkulvanich et al., 24 RJC4 Shatla et al., 21 RJC5 Johnson et al., 26 ZA Armstrong and Zerilli, 1987 ZA-JC Holmquist and Johnson, 1991 TABLE 2.6 MATERIAL CONSTANTS FOR FIVE DIFFERENT MATERIAL MODELS FOR OFHC COPPER Material Model Model Constants A B C C 1 C 2 C 3 C 4 n m JC RJC1 MAT MAT MAT MAT RJC2 MAT MAT ZA MAT MAT ZA-JC MAT MAT

70 TABLE 2.7 THE RANGES OF STRAIN, STRAIN RATE AND TEMPERATURE AT WHICH DIFFERENT MODEL CONSTANTS WERE DETERMINED Material model JC [2] RJC1 [2] Reference Model constants Strain Strain rate (1/s) Temperature ( o C) MAT MAT MAT MAT RJC2 [6] MAT up to n.a MAT [2] ZA MAT [7] MAT MAT ZA-JC [2] MAT JC (MAT3) RJC1 (MAT11) ZA (MAT14) ZA-JC (MAT17) Flow stress (MPa) Strain (a) Strain rate = 1 3 and Temperature = 3 o C 57

71 45 4 Flow stress (MPa) JC (MAT3) RJC1 (MAT11) ZA (MAT14) ZA-JC (MAT17) Strain rate (1/s) (b) Strain =.75 and Temperature = 3 o C 6 5 JC (MAT3) RJC1 (MAT11) ZA (MAT14) ZA-JC (MAT17) Flow stress (MPa) Temperature (C) Strain =.75 and Strain rate = 1 1/s Figure 2.2. Comparison of different material models for OFHC copper. 58

72 2.3 Comparison of Material Models Calculation The performance of different material models using different sets of constants should be evaluated and compared to the experimental data obtained from various tests and methods. The experimental data of Johnson (Johnson, 1985; Armstrong and Zerilli, 1987) and Tanner (Tanner et al., 1999) summarized in Table 2.8 are used as a basis for material models evaluation. Tables 2.9 and 2.1 provides a comparison of flow stress predicted from different material models using different sets of constants with experimental data given in Table 2.8. It is clear from Tables 2.9 and 2.1 that RJC1 model using MAT1 set of constants gives the lowest root mean-squared (RMS) error between predicted and experimental flow stress while RJC2 model using MAT12 gives the highest RMS error. The predictions of flow stress are highly dependent on the material models used (and their constants used) as shown in Table 2.9 and 2.1. Tables 2.9 and 2.1 also indicate that special care must be exercised when using different JC model constants to perform extrapolations to region outside the experimental region. TABLE 2.8 EXPERIMENTAL DATA FOR OFHC COPPER OBTAINED FROM HIGH STRAIN RATE TESTS Data point # Authors T ( o C) ε& (1/s) ε σ EXP (MPa) Johnson, 1985; Armstrong and Zerilli, Tanner et al., Johnson, 1985; Armstrong and Zerilli, Tanner et al., Johnson, 1985; Armstrong and Zerilli,

73 TABLE 2.9 COMPARISON OF EXPERIMENTAL DATA WITH FLOW STRESS PREDICTED FROM JC MODEL USING DIFFERENT SETS OF CONSTANTS n m ( A + Bε )( 1+ C ln ε )( 1 T ) σ = & H Data point # Flow stress (MPa) MAT1 MAT2 MAT3 MAT4 MAT5 MAT6 MAT7 MAT8 MAT9 σ EXP RMS error

74 TABLE 2.1 COMPARISON OF EXPERIMENTAL DATA WITH FLOW STRESS PREDICTED FROM DIFFERENT MATERIAL MODELS USING DIFFERENT SETS OF CONSTANTS Data point # Flow stress (MPa) RJC1 RJC2 ZA ZA-JC MAT1 MAT11 MAT12 MAT13 MAT14 MAT15 MAT16 MAT17 σ EXP RMS error

75 2.4 Machining Models for Calculation of Flow Stress and Temperature The high rate deformation during metal cutting processes is concentrated in two regions close to the cutting tool edge. These regions are often referred to as primary and secondary shear zone and denoted as PSZ and SSZ respectively. In order to predict the flow stress and temperature in the primary deformation zone, it is first necessary to determine the chip thickness ratio (r) and shear angle (Φ), which can be calculated using the following equations: t1 r = t 2 r cosα tanφ = 1 r sinα (2.2) (2.3) The shear force can be directly obtained using the following equation: F s = F cosφ F sinφ c t (2.4) where Fs is the shear force and can be calculated from measured cutting force Fc and feed force Ft. The measurements of cutting and feed forces can be achieved through several types of force dynamometers. The shear flow stress designated by K AB along AB can be determined from the following equation: K AB = F S sinφ t w 1 (2.5) where w is the width of cut and all other variables are defined before. The relationship between uniaxial stress and the corresponding effective stress can be obtained using Von Misses criterion as follows: σ = 3 AB K AB (2.6) The average temperature in the shear plane AB is found from the following equation: 1 β FS cosα T AB = TW + η (2.7) ρst1w cos( φ α ) where ρ is the density of the work material, S its specific heat, Tw is the initial workpiece temperature, η is a factor has a value between and 1 and β is the proportion of heat conducted into the workpiece which is estimated by Boothroyd s temperature model given by: β =.5.35log( R tanφ) if.4 R T tan Φ 1. (2.8) T 62

76 β =.3.15log( R tanφ) if R T tan Φ > 1. (2.9) T where R T is a dimensionless thermal number given by: ρsut = K R T 1 (2.1) There have been several attempts by some researchers to predict β theoretically but Oxley and Hastings [Oxley and Hastings, 1976) and Ozel and Zeren (Ozel and Zeren, 26) found that the most reliable estimates of β and hence the temperature can be made based on Boothroyd s temperature model. 2.5 Metal Cutting Experiments Oxygen-free high-conductivity (OFHC) copper is purposefully chosen in this study because it has a great application in the manufacturing processes, it has been extensively studied by other researchers and a large amount of flow stress data obtained from various methods and tests can be found in the literature. The orthogonal machining experiments are carried out on a high-speed linear slide. The cutting conditions composed of the rake angle, depth of cut, and cutting speed are summarized in Table Figure 2.3 shows a schematic of the cutting geometry as well as the cutting conditions given in Table The cutting and thrust forces are measured using a Kistler three-component force dynamometer. The cutting and thrust forces are averaged over a steady-state region to obtain single values for each cutting experiment. Several sections of chips are collected for each cutting condition after each cutting experiment and the chip thickness is measured. The measurements of the chip thickness are validated by direct measurement of the chip thickness from the high-speed micrographs of the PSZ. Multiple data sets, obtained from several experiments, are used to obtain a single value for each of the average cutting and thrust forces, chip thickness, average strain and average strain rate in the PSZ for each cutting condition. A complete description on the experimental setup along with the data processing procedure of high-speed photographic images using the DaVis software package can be found in MS Thesis of Mahadevan (Mahadevan, 27). The following sections explain the procedure implemented to estimate the flow stress and temperature and experimentally measure the strain and strain rate along the nominal shear plane for each cutting test under different cutting conditions given in Table

77 .1 V (m/s) 1. 3 t 1 (µm) 3 Rake angle = +3 (deg) Figure 2.3. Model of chip formation showing the primary and secondary shear zones. TABLE 2.11 CUTTING CONDITIONS IN ORTHOGONAL MACHINING TESTS PERFORMED FOR OFHC Test # Depth of cut (µm) Cutting Speed (m/s) Rake angle (deg)

78 2.6 Determination of Flow Stress, Strain, Strain Rate and Temperature in the PSZ Flow Stress The procedure to determine the flow stress involves obtaining the experimental values of cutting forces (Fc), thrust forces (Ft), and chip thickness (t 2 ), for each cutting condition, which is defined by the cutting speed (V), the rake angle (α), and depth of cut (t 1 ), These experimental values are used in equations (2.2), (2.3), (2.4) and (2.5) respectively to determine the chip thickness ratio (r), shear angle (Φ), shear force (Fs) and shear flow stress (K AB ) Then the shear flow stress is used in equation (2.6) using Von Misses criterion to calculate the effective flow stress (σ AB ) Temperature The average temperature along nominal shear plane AB is obtained using equations (2.7), (2.8), (2.9) and (2.1) plus the physical properties of the workpiece material. The physical properties of OFHC are modeled as functions of temperature using the values and equations shown in Table This involves determining first the dimensionless thermal number (R T ) and the proportion of heat conducted into the workpiece (β) and then determining the average temperature (T AB ) along the shear plane AB using equation (2.7). The work material temperature is taken as 2 o C and the factor η is taken as 1.. In order to solve for temperature, the values of thermal conductivity (K) and specific heat (S) must be known. Therefore the starting values of temperature (T) used in thermal conductivity (K) and specific heat (S) calculations (see Table 2.12) are the work material temperature. Then the final value of the average temperature along nominal shear plane is determined through iterative approach. The termination criterion of the iteration procedure was satisfied when the relative change in average temperature along nominal shear plane between successive iterations is less than.5 o C. TABLE 2.12 PHYSICAL PROPERTIES OF THE OFHC WORKPIECE MATERIAL Thermal conductivity (k, W/m o C) T Specific heat (Cp, J/Kg o C) T Thermal expansion coefficient (α, I o C) 5. x 1-5 Density (ρ, Kg/m 3 )

79 2.6.3 Strain and Strain-Rate A novel experimental capability for obtaining the velocity and strain rate fields in the PSZ in metal cutting has been developed and used to obtain the average strain rate along the mid-plane of the PSZ under Assessment of Machining Models (AMM) cutting condition 2 (Madhavan et. al, 27). Moreover, initial data obtained using this capability has been previously presented (Madhavan and Hijazi, 24). Digital Image Correlation (DIC) software is used to obtain the velocity fields in the PSZ by 2D and 3D stereoscopic correlation of high-speed photographic images of the side surface of the chip. The gradient of the velocity fields yields the strain rate and the integration of the strain rate yields the strain. Multiple data sets, obtained from several experiments, are used to obtain a single value of the average strain and strain rate along shear plane AB for each cutting condition. A complete description on the patent pending multi-channel non-intensified ultra high-speed camera system used in this research to obtain a sequence of images usable for Digital Image Correlation (DIC) of the side surface of the chip can be found in the previously published study presented by Madhavan and Hijazi (Madhavan and Hijazi, 24). Additionally, a complete description on the experimental setup along with the data processing procedure of high-speed photographic images using the DaVis software package can be found in currently submitted paper presented by Madhavan and his co-workers and MS Thesis of Mahadevan (Madhavan et. al, 27; Mahadevan, 27). 2.7 Results and Discussion Results of Cutting Tests The measured data (cutting forces, thrust forces and chip thickness) obtained from cutting tests as per experimental plan shown in Table 2.11 are presented in Table The other known parameters are cutting speed (V), rake angle (α), depth of cut (t1) and width of cut (w). Following the procedure described above, the estimated flow stress and temperature as well as the measured strain and strain rate in the PSZ under different cutting conditions are summarized in Table Figures 2.4 through 2.12 show plots of the calculated and measured process variables (chip thickness, strain, strain rate and temperature) with the changes in the cutting conditions (cutting speed and depth of cut). Figure 2.4 indicates that chip thickness decreases with increase in cutting speed. Additionally it can be seen that chip thickness is proportional to unperformed chip thickness as shown in 66

80 Figure 2.5. While strain in the PSZ measured experimentally by DIC technique decreases with increase in cutting speed, it increases with increase in unperformed chip thickness as shown in Figures 2.6 and 2.7 respectively. These observations are in agreement with the predictions obtained from analytical models of machining. Figure 2.8 indicates that measured strain rate is proportional to cutting speed and the following equation: ε& V (2.11) can be fitted when unreformed chip thickness equals to 1 (µm/rev). On the other hand, measured strain rate decreases with increase in unperformed chip thickness as shown in Figure 2.9, which is in agreement with other experimental observations as well as strain rate predicted by analytical models. In Oxley theory of machining, the shear plane AB is assumed to be a direction of maximum shear strain rate which was validated by experimental measurements of strain-rate in the primary deformation zone during orthogonal machining of carbon steels (Stenvensen and Oxley, 1969). The experimental results show that for a wide range of strain rate, the average strain rate along AB can be fitted to the following empirical equation (Stenvensen and Oxley, 1969): = C V 3 L S ε& (2.12) AB where C is an empirical constant for a particular work material, V S is shear velocity and L AB is length of shear plane AB. The shear velocity and length of shear plane AB are defined by: cosα Vs = V (2.13) cos( φ α) t L 1 AB = (2.14) sinφ To compare experimental strain rates measured by DIC of high speed images of the PSZ in machining OFHC copper with equation (2.12), the average strain rate measured by DIC is plotted against shear velocity divided by length of shear plane AB as shown in Figure 2.1. From the results in Figure 2.1, the line of best fit passing through origin is: 67

81 V = L S ε& (1/s) (2.15) AB where C can be calculated to be Figure 2.1 indicates that the variation of average strain rate along AB measured experimentally by DIC for OFHC copper with shear velocity divided by length of shear plane AB are in close agreement with experimental measurements of strain-rate in the primary deformation zone during orthogonal machining of carbon steels (Stenvensen and Oxley, 1969). Finally, the calculated temperature along the nominal shear plane AB increases with increases in cutting speed and unreformed chip thickness as shown in Figures 2.11 and These temperature observations are in close agreements with the experimental and FEA results of metal cutting (Liu et al., 1999). The measured and calculated data shown in Table 2.14 are used in subsequent sections as a reference data to asses different material models used in machining. Moreover, these data shown in Table 2.14 are used to obtain the material parameters for different material models through optimization approach that is discussed later. TABLE 2.13 CUTTING CONDITIONS AND MEASURED PROCESS VARIABLES IN ORTHOGONAL MACHINING OF OFHC COPPER Test # Depth of cut (µm) Cutting Speed (m/s) Rake angle (deg) Fc (N) F t (N) t 2 (mm)

82 TABLE 2.14 COMPUTED AND MEASURED PROCESS VARIABLES IN THE PRIMARY DEFORMATION ZONE IN ORTHOGONAL MACHINING OF OFHC COPPER Test # Stress in MPa (Machining Theory) Strain (Machining tests) Strain rate in 1/s (Machining tests) Temperature ( o C) (Machining Theory) Boothroyd s model Chip thickness (mm) Depth of cut =.1 mm Cutting speed (m/s) Figure 2.4 Decrease of chip thickness with increase in cutting speed (depth of cut =.1 mm). 69

83 Chip thickness (mm) Cutting speed = 1. m/s Depth of cut (mm) Figure 2.5 Increase of chip thickness with increase in depth of cut (cutting speed = 1. m/s) Strain Depth of cut =.1 mm Cutting speed (m/s) Figure 2.6 Decrease of strain with increase in cutting speed (depth of cut =.1 mm 7

84 Strain Cutting speed = 1. m/s Depth of cut (mm) Figure 2.7 Increase of strain with increase in depth of cut (cutting speed = 1. m/s) Strain rate (1/s) Depth of cut =.1 mm Cutting speed (m/s) Figure 2.8 Increase of strain rate with increase in cutting speed (depth of cut =.1 mm). 71

85 Strain rate (1/s) Cutting speed = 1. m/s Depth of cut (mm) Figure 2.9 Decrease of strain rate with increase in depth of cut (cutting speed = 1. m/s) Strain rate (1/s) Vs/L AB (1/s) Figure 2.1 Variation of average strain rate along shear plane AB obtained by DIC with ratio of shear velocity to length of shear plane AB. 72

86 14 12 Temperature (Degree celsius) Depth of cut =.1 mm Cutting speed (m/s) Figure 2.11 Increase of temperature in o C with increase in cutting speed (depth of cut =.1 mm) Temperature (Degree celsius) Cutting speed = 1. m/s Depth of cut (mm) Figure 2.12 Increase of temperature in o C with increase in depth of cut (cutting speed = 1. m/s). 73

87 2.7.2 Assessment of Different Material Models The assessment of the material models and their constants presented in Tables 2.1, 2.4 and 2.6 is carried out with reference to the calculated flow stress and temperatures as well as measured strains and strain rates that are presented in Table The quantitative comparison of the flow stress calculated directly from experimental cutting forces and chip thickness and flow stress predicted from the material models using strain, strain rate and temperature given in Table 2.14 forms the basis for the material models evaluation. The assessment criterion is the percentage error in predicted flow stress which is defined by: σ PRED σ EXP % error = 1 σ EXP (2.16) where σ EXP is the experimental flow stress calculated from equations (2.5) and (2.6) using the experimental cutting forces and chip thickness. On the other hand σ PRED is the flow stress predicted by direct substitutions of the strains, strain rates, and temperatures given in Table 2.14 into the material models given in Table 2.4 using the model constants presented in Tables 2.1 and 2.6. Tables 2.15 and 2.16 present a comparison between the flow stress predicted from different material models and the experimental flow stress obtained from machining tests using equation (2.16). Table 2.15 shows that JC model with MAT5 set of constants gives the lowest percentage error of.1% at strain rate of 79 (1/s). Additionally, the ZA model using MAT14 set of constants gives the lowest percentage error among other models of.5% at strain rate of 275 (1/s) as shown in Figure Some conclusions based on Tables 2.15 and 2.16 as well as Figures 2.13 and 2.14 are as follows: 1. All material models with different material parameters show discrepancies with reference to machining data. They are deficient and have poor performance to predict the work material flow stress under machining test # 1 (Ozel and Yigit, 27). 2. The material model predictions of the flow stress obtained from machining tests are sensitive to the material models used and that not one of the material models used is best in all aspects (Adibi-Sedeh et al., 25; Sartkulvanich et al., 25). 3. The material model predictions of the flow stress obtained from machining tests are highly dependent on the material constants used (Deshpade et al., 26; Umbrello et al., 27). 74

88 4. Both empirical (JC model) and physically (ZA model) based models may give erroneous results when used to predict the flow stress over their extended ranges of strain, strain rate, and temperature (Johnson and Holmquist, 1988). 5. The use of MAT15 set of ZA model constants results in the lowest root mean square error (RMS) between the experimental and predicted flow stress. On the contrary, the use of MAT12 set of RJC2 model constants results in the highest RMS error and gives highly poor performance. Therefore, improving the work material constitutive equations from carefully designed machining tests at high strain rates is an important effort in metal cutting research. It provides a methodology to validate and verify the performance of the material model predictions under machining. These improved constitutive models can be used later in the FEA simulations of machining so that the FEA simulations can predict more accurately the typical machining outputs. The following section is concerned with finetuning and optimizing the constants of different material models based on the results of cutting tests presented in Table TABLE 2.15 PERCENTAGE ERROR IN PREDICTED FLOW STRESS USING DIFFERENT JC MODEL CONSTANTS WITH REFERENCE TO MACHINING TEST DATA n m ( A + Bε )( 1+ C lnε )( 1 T ) σ = & Test # MAT1 MAT2 MAT3 MAT4 MAT5 MAT6 MAT7 MAT8 MAT RMS error H 75

89 TABLE 2.16 PERCENTAGE ERROR IN PREDICTED FLOW STRESS USING DIFFERENT MATERIAL MODELS WITH REFERENCE TO MACHINING TEST DATA Test # RJC1 RJC2 ZA ZA-JC MAT1 MAT11 MAT12 MAT13 MAT14 MAT15 MAT16 MAT RMS error % error in predicted flow stress MAT1 MAT2 MAT3 MAT7 MAT8 MAT Machining test No Figure 2.13 Percentage error in predicted flow stress using different JC model constants with reference to machining test data. 76

90 2 JC (MAT3) RJC1 (MAT1) RJC2 (MAT11) ZA (MAT13) ZA-JC (MAT16) % error in predicted flow stress Machining test No Figure 2.14 Percentage error in predicted flow stress using different material models with reference to machining test data Determination of JC Model Parameters The aim of the present research is to fine-tune and optimize JC model parameters for OFHC copper. Machining experiments are carried out using a linear cutting setup under a small number of different cutting conditions as described previously. The results of cutting forces and chip thickness are then used to estimate the average temperature along shear plane AB using analytical model (Boothroyd s temperature model). Most importantly, the average strain (ε AB ) and strain rate (έ AB ) along AB are directly measured using Digital Image Correlation (DIC) of high-speed photographic images of the side surface of the chip. Subsequently, the least square technique is applied to determine JC model parameters as defined by the following equation: N ( A, BC, n, m) = min σ i= 1 AB ( i) ( A+ Bε AB ) n ( i) & ε AB( i) TAB( i) Tr 1+ Cln 1 & ε AB Tm Tr m 2 (2.17) 77

91 where N is number of machining experiments and i is the index of the experiment. The minimization of the objective function (equation 2.17) can be conveniently accomplished using several commercial optimization software by adjusting and changing JC model parameters with the result being an optimum set of material constants. The JC model is explored in this section as an example of how the optimization problem is formulated and can be applied to other material models. The ranges of JC model parameters (A, B, C, n and m) given in Table 2.3 are used in conjunction with Equation (2.17) to perform the optimization process. The same optimization approach in conjunction with machining tests data given in Table 2.14 is used to fine-tune and optimize the parameters for the following material models: RJC1, RJC2, ZA and ZA- JC that are presented in Table 2.4. The ranges of each material model (RJC1, RJC2, ZA and ZA-JC) parameter are established based on previously published values that are summarized in Table 2.6 as follows. First, the lower and upper limit of each parameter are determined (from Table 2.6) and then the average and range of each parameter are calculated. Subsequently, the new lower and upper limit of each parameter that are used in conjunction with Equation (2.17) are determined by subtracting or adding the calculated average to the calculated range respectively. If the new lower limit of any material parameter is calculated to be less than zero, the lower limit is taken as zero since negative values cannot be assigned to the parameters used in different constitutive material models ((RJC1, RJC2, ZA and ZA- JC). Several commercial optimization software available in today s market can be used to optimize various engineering problems. Microsoft Excel Solver, MathOptimizer for Mathematica, Matlab, and Maple are just few examples of commercial optimization packages. These codes share the same unique feature-iteration. Local or global search algorithms are employed to find the optimum solution. Local search methods use first and second derivatives to find local optimum while global search methods systematically search the entire general region in which the global optimum lies. In this research, Global Optimization Toolbox (GOT) of Maple is used to perform the optimization of the objective function defined by equation (2.17). The core of GOT is the Maple-specific implementation of the Lipschitz Global Optimizer (LGO) solver. LGO offers Local and global search algorithms. Moreover 78

92 the GOT optimizer currently includes the following search methods that can be optionally selected by the user: 5. Branch-and-bound based global search (BB). 6. Global adaptive random search (single-start) (GARS). 7. Random multi-start based global search (MS). 8. Generalized reduced gradient (GRG) algorithm based local search (LS). A combination of the above four search methods is used to minimize the objective function given by equation (2.17). A complete description on the Global Optimization Toolbox (GOT) of Maple can be found on as well as an excellent book written by Janos (Janos, 26). Figure 2.15 shows a flow chart of the optimization process of the objective function defined by equation (2.17). The material constants identified using this methodology are presented in Table 2.17 for the following material models: JC, RJC1, RJC2, ZA, and ZA-JC. The ranges of strain, strain rate, and temperature at which these constants are determined are summarized in Table Table 2.19 shows the percentage improvement in the material model performance to predict the flow stress under machining tests. It is clear from Table 2.19 that the performance of all material models is significantly improved. The performance of the material models shows approximately 34-86% improvement from the previous published values. Figure 2.16 also provides a comparison of the flow stress obtained from machining tests with the flow stress predicted from different material models using these optimum sets of constants presented in Table It can be seen from Figure 2.16 that flow stress predicted by JC model is the closest in agreement with machining data while the flow stress predicted by RJC2 model is the farthest in agreement. Moreover, all four models (JC, RJC1, ZA and ZA-JC) show good agreement with cutting tests data when using these data to improve the material models as shown in Figure Thus, it indicates the potential success of this methodology. The next section is concerned with comparing experimental machining flow stress with flow stress predicted from JC model and mechanical threshold stress (MTS) model proposed and implemented by Follansbee and Kocks (Follansbee and Kocks, 1988). The following second section provides a sensitivity analysis of temperature estimation using both Weiner s and Boothroyd s models on the optimum values of JC constants by minimizing the objective function defined by equation (2.17). 79

93 START DOE: AMM cutting conditions (6 machining tests) Conduct machining tests (WSU) Extract machining results (1) Measured cutting forces (2) Measured thrust forces (3) Measured chip thickness Extract machining results (1) Measured average strain rate (έ AB) (High-speed photographs using DIC) Perform calculations: (1) σ AB (2) T AB (Oxley s Model) Perform calculations: (1) Strain (ε AB) = έ AB dt σ AB, ε AB, έ AB, T AB Least square method JC model RJC1 model RJC2 model ZA model ZA-JC model Perform optimization process (Equation 2.17) GOT of Maple Optimum values of material constants END Figure 2.15 Optimization process of the objective function defined by equation (2.17). 8

94 TABLE 2.17 OPTIMUM MATERIAL CONSTANTS FOR DIFFERENT MATERIAL MODELS Material Model A B C C 1 C 2 C 3 C 4 n m JC RJC RJC ZA ZA-JC TABLE 2.18 THE RANGES OF STRAIN, STRAIN RATE AND TEMPERATURE AT WHICH THE OPTIMUM MATERIAL MODEL CONSTANTS ARE DETERMINED Material model constants Strain Strain rate (1/s) Temperature ( o C) JC RJC1 RJC2 ZA ZA-JC TABLE 2.19 PERCENTAGE IMPROVEMENT IN THE PERFORMANCE OF DIFFERENT MATERIAL MODELS TO PREDICT WORK MATERIAL FLOW STRESS OBTAINED FROM CUTTING TESTS Material Model Optimum values RMS in flow stress Previously published values % Improvement JC RJC RJC ZA ZA-JC

95 8 Machining data JC model RJC1 model RJC2 model ZA model ZA-JC model 7 6 Flow stress (MPa) Cutting test # Figure 2.16 Comparison of machining flow stress data with flow stress predicted from different material models using the optimum sets of constants shown in Table Comparison of JC Model with MTS Model Since JC material model using optimum set of constants listed in Table 2.17 is found to be the closest in agreement with machining data, its performance is further validated and compared with mechanical threshold stress (MTS) model. The mechanical threshold stress (MTS) model proposed and implemented by Follansbee and Kocks (Follansbee and Kocks, 1988) is selected for comparison purposes with JC model performance under machining data. The MTS model is purposefully chosen since it represents thermal and strain rate history effects while all other material models (including JC model) described and discussed previously are history independent on temperature and strain rate. The MTS model is defined by the following equations: 82

96 p q 3 µ σ ˆ σ ε ε exp p b g a & = & 1 sgn( σ ) (2.18) KT ˆ σ ˆ σ a ˆ σ ˆ σ a tanh 2 ˆ σ ˆ σ ˆ s σ a = θ 1 ε tanh(2) (2.19) where saturation stress or stress at zero strain hardening rate ( σˆ s ), athermal stress ( σˆ a ) and initial strain hardening rate ( θ ) can be defined respectively as follows: ˆ σ s & ε = σ s & ε s 3 µ b A KT (2.2).278 ˆ σ a = d (2.21) θ & = ε (2.22) where p ε& is the plastic strain, σ is flow stress, σˆ is reference threshold stress, d is initial average grain size and all other variables are constants for a particular work material. These constants are summarized and presented in Table 2.2. The MTS model constants given in Table 2.2 were identified from both quasistatic ( /s) and high strain rate tests (52 and 6 1/s) for temperature ranges from 25 and 269 C (Tanner et al., 1999). An optimizer using combination of genetic, gradient, and simplex algorithms was used to obtain the optimum set of MTS model constants shown in Table 2.2 that minimized the least square between the predicted and experimental measurements of flow stress (Tanner et al., 1999). Figure 2.17 provides a comparison of machining flow stress data given in Table 2.14 with flow stress predicted from JC model using optimum set of constants given in Table 2.17 and MTS model using optimum set of constants given in Table 2.2. It can be seen from Figure 2.17 that flow stress predicted by JC and MTS models are in close agreement with experimental machining data. This indicates that JC model using the optimized constants given in Table 2.17 outperforms all other constitutive material models discussed in the present study including MTS model under machining conditions. 83

97 TABLE 2.2 MTS MODEL PARAMETERS FOR OFHC COPPER Material Authors Range MTS model parameters Optimum set of constants g 2.98 P.318 OFHC Copper Tanner et al., x /s o C q 1.2 ε& (1/ ) 1 7 s K/b 3 (MPa K -1 ).848 A.633 σ s 17 ε& 4. x 1 23 s 75 7 Experimental machining data JC model MTS model 65 Flow stress (MPa) Machining test # Figure 2.17 Comparison of machining flow stress data with flow stress predicted from JC model (using optimum sets of constants given in Table 2.17) and MTS model (using material constants given in Table 2.2). 84

98 2.7.5 Temperature Sensitivity Analysis There have been several attempts by some researchers to predict β (proportion of heat conducted into the workpiece) theoretically but Oxley and Hastings [Oxley and Hastings, 1976) and Ozel and Zeren (Ozel and Zeren, 26) found that the most reliable estimates of β and hence the temperature can be made based on Boothroyd s temperature model. On the contrary, among different analytical models, it has been found that Weiner s model predicts β and hence the temperature the closest to the FEA results (Pednekar, et al., 25). Table 2.21 shows a summary of the equations used for calculation of average temperature along nominal shear plane AB using both Weiner s and Boothroyd s models. All variables used in the equations given in Figure 2.21 are defined before. TABLE 2.21 DIFFERENT EQUATIONS PROPOSED FOR CALCULATION OF AVERAGE TEMPERATURE ALONG SHEAR PLANE AB AND PROPORTION OF HEAT CONDUCTED INTO THE WORKPIECE Source Equation Oxley, 1989 T AB = T W 1 β FS cosα + η ρst1w cos( φ α) β =.5.35log( R tanφ) if.4 R T tanφ 1. T Oxley, 1989; Boothroyd, 1963 β =.3.15log( R tanφ) if R T tan Φ > 1. where ρsut = K R T 1 T Weiner, β = 4Y L erf Y L + (1 + Y L ) erfc Y L YL e π 2 1 Y L + Y L where ρsvt tan Y L = 4K 1 φ 85

99 Therefore, a sensitivity analysis of temperature estimation using both Weiner s and Boothroyd s models on the optimization process of the objective function (equation 2.17) is carried out. This sensitivity analysis measures the effect each temperature model has on the optimum values achieved by minimizing the objective function defined by equation (2.17). The temperature estimated using both Weiner s and Boothroyd s models under different cutting conditions given in Table 2.11 are summarized in Table 2.22 and shown in Figure The average temperatures along nominal shear plane AB estimated from equation (2.7) (using both Weiner s and Boothroyd s models given in Table 2.21 to provide an estimate for β) are calculated following the methodology described in details in section While the maximum difference in temperature estimation is 13.7, the minimum difference is.22 as shown in Table Since JC material model using the optimum set of constants listed in Table 2.17 is found to be the closest in agreement with machining data, it is chosen for the temperature sensitivity analysis. Table 2.23 summarizes the sensitivity study performed as it is described above showing the root mean-squared errors (RMS) and the percentage changes in the optimal values of A, B, C, n and m. It is clear from Table 2.23 that variations (changes) in temperature estimation as much as 14 o C is not significant on the achieving the optimal values of JC model parameters (A, B, C and m). On the contrary, the work hardening parameter n is somewhat sensitive to small variations in temperature estimation as it is shown in Table It can be concluded from Table 2.23 that JC model gives slightly better performance when Boothroyd s model is used. TABLE 2.22 COMPARISON OF PREDICTED TEMPERATURES IN O C OF BOOTHROYD S MODEL WITH WEINER S MODEL IN MACHINING OFHC COPPER Test # Weiner s model ( o C) Boothroyd s model ( o C) Difference in temperature

100 14 12 Boothroyd's model Weiner's model Predicted temperature (degree celsius) Test No Figure 2.18 Comparison of predicted temperatures in o C of Boothroyd s model with Weiner s model in machining OFHC copper. TABLE 2.23 COMPARISON OF OPTIMIZED JC MODEL CONSTANTS WHEN BOOTHROYD S MODEL AND WEINER S MODEL ARE USED TO ESTIMATE THE TEMPERATURE RJC2 model constants Boothroyd s model Weiner s model A B C n m RMS

101 2.8 Summary and Conclusion This paper presents an explicit technique to fine-tune and optimize the material constants used in different material models based on small number of cutting tests. Unlike other research, the strain and strain rate along the nominal shear plane are experimentally measured using Digital Image Correlation (DIC) of high-speed photographic images of the side surface of the chip. Since the average strain and strain rate directly measured from cutting tests, an analytical estimate is required only for the temperature, hence reducing the uncertainty in the material model constants. A summary of previously published values of the constants used in different material models for OFHC is also provided. Then an assessment of these material models is made under the ranges of strain, strain rate, and temperature typically observed in machining, Subsequently, the calculated flow stress and temperature as well as measured strain and strain rate from machining tests are used to determine the optimum set of constants of different material models based on the least square technique. Some conclusions are as follows: 1. All material models with previously published values of their parameters show discrepancies with reference to machining data (Ozel and Yigit, 27). 2. The material model predictions of the flow stress obtained from machining tests are highly dependent on the material constants used. 3. Both empirical (JC model) and physically (ZA model) based models may give erroneous results when used to predict the flow stress over their extended ranges of strain, strain rate, and temperature (Johnson and Holmquist, 1988). 4. Cautious should be taken when the material models are used in different FEA commercial packages to predict the typical machining outputs (Deshpade et al., 26; Umbrello et al., 27) 5. All four models (JC, RJC1, ZA and ZA-JC) show good agreement with cutting tests data when using these data to improve the material models. 6. JC model shows the closest agreement with machining data while RJC2 model shows the farthest. 7. The optimization approach discussed in the present study shows significant improvement in the performance of all the material models investigated. 8. More experimental machining tests under different cutting conditions for the purpose of material constitutive models verification are recommended. 88

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104 APPENDIXES 91

105 APPENDIX A Figure 1. Print out of MSC. Marc for the final mesh of FE simulation #1. Figure 2. Print out of MSC. Marc for the final mesh of FE simulation #2. 92

106 Figure 3. Print out of MSC. Marc for the final mesh of FE simulation #3. Figure 4. Print out of MSC. Marc for the final mesh of FE simulation #4. 93

107 Figure 5. Print out of MSC. Marc for the final mesh of FE simulation #5. Figure 6. Print out of MSC. Marc for the final mesh of FE simulation #6. 94

108 Figure 7. Print out of MSC. Marc for the final mesh of FE simulation #7. Figure 8. Print out of MSC. Marc for the final mesh of FE simulation #8. 95

109 Figure 9. Print out of MSC. Marc for the final mesh of FE simulation #9. Figure 1. Print out of MSC. Marc for the final mesh of FE simulation #1. 96

110 Figure 11. Print out of MSC. Marc for the final mesh of FE simulation #11. Figure 12. Print out of MSC. Marc for the final mesh of FE simulation #12. 97

111 Figure 13. Print out of MSC. Marc for the final mesh of FE simulation #13. Figure 14. Print out of MSC. Marc for the final mesh of FE simulation #14. 98

112 Figure 15. Print out of MSC. Marc for the final mesh of FE simulation #15. Figure 16. Print out of MSC. Marc for the final mesh of FE simulation #16. 99

113 Figure 17. Print out of MSC. Marc for the final mesh of FE simulation #17. Figure 18. Print out of MSC. Marc for the final mesh of FE simulation #18. 1

114 Figure 19. Print out of MSC. Marc for the final mesh of FE simulation #19. Figure 2. Print out of MSC. Marc for the final mesh of FE simulation #2. 11