DEVELOPMENT OF ADVANCED TECHNIQUES FOR IDENTIFICATION OF FLOW STRESS AND FRICTION PARAMETERS FOR METAL FORMING ANALYSIS DISSERTATION

Transcription

1 DEVELOPMENT OF ADVANCED TECHNIQUES FOR IDENTIFICATION OF FLOW STRESS AND FRICTION PARAMETERS FOR METAL FORMING ANALYSIS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Graduate School of The Ohio State University By Hyunjoong Cho, M.S. * * * * * The Ohio State University 2007 Dissertation Committee: Professor Taylan Altan, Adviser Professor June K. Lee Professor Jerald Brevick Approved by Adviser Industrial and Systems Engineering Graduate Program

2 Copyright by Hyunjoong Cho 2007

3 ABSTRACT The accuracy of process simulation in metal forming by finite element method depends on the accuracy of flow stress data and friction value that are input to FEM programs. Therefore, it is essential that these input values are determined using reliable tests and evaluation methods. This study presents the development of inverse analysis methodology and its application to determine flow stress data of bulk and sheet materials at room and elevated temperatures. The inverse problem is defined as the minimization of the differences between the experimental measurements and the corresponding FEM predictions. Rigidviscoplastic FEM is used to analyze the metal flow while a numerical optimization algorithm adjusts the material parameters used in the simulation until the calculated response matches the measured data within a specified tolerance. The use of the developed inverse analysis methodology has been demonstrated by applying it to the selected reference rheological tests; cylinder compression test, ring compression test, instrumented indentation test, modified limiting dome height test, and sheet hydraulic bulge test. Furthermore, using the determined material property data, full 3-D finite element simulation models, as examples of industrial applications for orbital forming and thermoforming processes have been developed for reliable process simulation. ii

4 As results of this study, it was shown that the developed inverse analysis methodology could identify both the material parameters and friction factors from one set of tests, simultaneously. Therefore, this technique can offer a systematic and cost effective way for determining material property data for simulation of metal forming processes. iii

5 Dedicated to my parents iv

6 ACKNOWLEDGMENTS I would like to express my most sincere gratitude to my adviser, Prof. Taylan Altan, for the opportunity to work at the Engineering Research Center for Net Shape Manufacturing (ERC/NSM) and his thoughtful advice and exceptional guidance during the course of my PhD study at the Ohio State University. Without his thrust and patience, my research would not have been successful. My great appreciation extends to Prof. June K. Lee, Prof. Jerald Brevick and Prof. Gracious Ngaile for serving on my dissertation committee and for providing valuable comments. Sincere thanks are extended to all my colleagues at the ERC/NSM, for their discussions and partial assistance in different parts of this research. My special thanks to Hariharasudhan Palaniswamy, Partchapol Sartkulvanich, Yingyot Aue-u-lan, Ibrahim Al-Zkeri, Serhat Kaya, Ajay Yadav, Manas Shirgaokar, Hui Qi, Hyunoak Kim, and Jihyun Sung. Finally, my special thanks are extended to my parents and friends, who always pray for me, for their understanding and continued support during my PhD study at the Ohio State University. v

7 VITA June 10, Born Seoul, Korea B.S. Mechanical Engineering, Hongik University, Seoul, Korea M.S. Precision Mechanical Engineering, Hongik University, Seoul, Korea Graduate Research Associate, Engineering Research Center for Net Shape Manufacturing (ERC/NSM), The Ohio State University, Columbus, Ohio PUBLICATIONS Papers Published in Peer Reviewed Journals and Proceedings 1. P. Pauskar, H. Cho, R. Shivpuri, and N. Kim, An FEM based Integrated Model for Simulation of Metal Flow and Microstructure Evolution in Hot Rolling, Trans. of NAMRI/SME, Vol. XXV, pp , H. Cho and G. Ngaile, Simultaneous Determination of Flow Stress and Interface Friction by Finite Element Based Inverse Analysis Technique, Annals of CIRP, v.1, 2003, p.221 vi

8 3. M. Shirgaokar, H. Cho, G. Ngaile, T. Altan, J.-H. Yu, J. Balconi, R. Rentfrow, and W.J. Worrell, "Optimization of Mechanical Crimping to Assemble Tubular Components", Journal of Materials Processing Technology, Volume 146, Issue 1, Pages 35-43, H. Cho and T. Altan, "Determination of Flow Stress and Interface Friction at Elevated Temperatures by Inverse Analysis Technique" Journal of Materials Processing Technology, v.170, Dec 2005, p S. Narashimhan, H. Cho, N. Kim, G. Ngaile, and T. Altan, "Inverse Analysis Method for Simultaneous Determination of Flow Stress and Interface Friction in Forming Sheet and Composite Sheets at Elevated Temperatures", (to be resubmitted after revision to Journal of Materials Processing Technology) Papers Published in Conferences or Other Journals (not Peer Reviewed) 1. H. Cho and N. Kim, J. J. Park, An Analysis of Turbine Disk Forging of Ti- Alloy by the Rigid-Plastic Finite Element Method, Trans. of The Korean Society of Mechanical Engineers, Vol. 18, No. 11, pp , H. Cho and N. Kim, A Study on the Microstructural Evolution in Hot Rolling, Proceedings of the 1995 Autumn Conference for the Korean Society for Technology of Plasticity, pp , H. Cho and N. Kim, A Study on the Microstructural Evolution in Hot Rolling, Proc. of 5th ICTP, Columbus, USA, Vol. I, pp , P. Pauskar, H. Cho, R. Shivpuri, and N. Kim, Microstructure Prediction in Multipass Bar Rolling with Interstand Cooling, 38th Mechanical Working and Steel Processing Conference, Clevelend, OH, October 13-16, vii

9 5. H. Cho and N. Kim, A Finite Element Model for Predicting the Microstructural Evolution in Hot Rolling, Journal of the Korean Society of Precision Engineering Vol. 14, No. 9, pp , H. Cho, N. Kim, and T. Altan, "Simulation of Orbital Forming Process using 3D FEM" Proceedings of the 3rd JSTP International Seminar on Precision Forging, Nagoya, Japan, H. Cho, G. Ngaile, and T. Altan, "3D FEA of Orbital Forming Process and Inverse Analysis for the Determination of Reliable Material Property", NUMIFORM 2004, June T. Altan, H. Cho, and H. Palaniswamy, "Process Modeling for Precision Forming of Discrete Parts- State of Technology and Critical Issues" Steel Research Int., No 2/3, p. 191, M. Shirgaokar, H. Cho, and T. Altan, "New Developments in FEM Based Process Simulation to predict & Eliminate Material Failure in Cold Extrusion" submitted to Int. Conference on "Developments in Bulk Forming", Stuttgart, May 30- June 1, M. Munshi, K. Shah, H. Cho, and T. Altan, 2005, FEM Analysis of Orbital Forming for Spindle/ Bearing Assembly, - Proceedings of the 8th ICTP, Verona, Italy, Oct 9/13, 2005, p.321 FIELDS OF STUDY Major Field: Industrial and Systems Engineering Studies in: Manufacturing Processes, Finite Element Method, Inverse Analysis, Material Characterization viii

10 TABLE OF CONTENTS Page ABSTRACT.....ii ACKNOWLEDGMENTS... v VITA.....vi PUBLICATIONS...vi TABLE OF CONTENTS... ix LIST OF FIGURES... xv LIST OF TABLES... xxi LIST OF SYMBOLS... xxiii CHAPTER 1 INTRODUCTION Process Simulation in Metal Forming Needs for an Inverse Analysis in Material Characterization Research Motivation Review on Inverse Analysis for Flow Stress Determination...6 CHAPTER 2 OBJECTIVES OF THE STUDY Objectives Scope of Present Study...10 ix

11 CHAPTER 3 FINITE ELEMENT MODELING IN METAL FORMING Rigid-Plastic Finite Element Formulation for Deformation Governing Equations Weak Form of the Boundary Value Problem Finite Element Discretization Procedure Treatment of Frictional Forces Finite Element Formulation for Heat Transfer Non-Steady State versus Steady State Problem Finite Element Modeling of Non-Steady State Problem Finite Element Modeling of Steady State Problem...19 CHAPTER 4 DEVELOPMENT OF INVERSE ANALYSIS METHODOLOGY Inverse Analysis Problem for Parameter Identification Flow Stress and Friction in Metal Forming Rheological Model Friction Model Inverse Analysis Methodology Using Load-Stroke Data Objective Function Minimization of Objective Function Inverse Analysis Methodology to Determine Flow Stress and Friction Simultaneously Objective Function Two-Step Minimization of Objective Function Inverse Analysis Methodology Using Geometry Change Objective Function Minimization of Objective Function Example of Inverse Analysis...41 x

12 Inverse Analysis Conditions Experiment Identified Material Parameters and Friction Factor Comparison of Load-Stroke Curve Comparison of Bulge Diameter Convergence Behavior Solution Uniqueness...50 CHAPTER 5 INVERSE ANALYSIS MODELING OF RHEOLOGICAL TESTS Rheological Test Inverse Analysis Modeling for a Rate-Dependent Material...54 CHAPTER 6 INVERSE ANALYSIS OF CYLINDER COMPRESSION TEST Introduction Inverse Analysis of Well-Lubricated Cylinder Compression Test Experiment Flow Stress Determination by Inverse Analysis Flow Stress Determination by Direct Analysis Application of Inverse Analysis Results to Simulation of Orbital Forming Introduction to Spindle/Inner Ring Assembly Operation Characteristics of Orbital Forming Process Finite Element Analysis of Spindle/Inner Ring Assembly Discussion...75 CHAPTER 7 INVERSE ANALYSIS OF RING COMPRESSION TEST Introduction Estimation of Friction from Ring Compression Test...77 xi

13 7.3. Inverse Analysis Conditions Inverse Analysis of Aluminum Ring Compression Test at Room Temperature Experiment Determination of Flow Stress and Friction Validation Inverse Analysis of Ring Compression Test at Elevated Temperatures Experiment Determination of Flow Stress and Friction at 200 o C Determination of Flow Stress and Friction at 400 o C Comparison of Flow Stress Data Determined from Cylinder and Ring Compression Tests Introduction Inverse Analysis of Cylinder Compression Test Inverse Analysis of Ring Compression Test Discussion...98 CHAPTER 8 INVERSE ANALYSIS OF INSTRUMENTED INDENTATION TEST Introduction Review on Flow Stress Determination from Indentation Test Indentation Load-Depth Curve Characteristics of Indentation Test Flow Stress Measurement on the Engineered Surfaces Inverse Analysis of Spherical Indentation Test Experiment Flow Stress Model Finite Element Model Initial Guess Generation xii

14 Inverse Analysis Results Discussion CHAPTER 9 INVERSE ANALYSIS OF MODIFIED LDH TEST Introduction to Modified Limiting Dome Height Test Inverse Analysis of Modified LDH Test at Room Temperature Preparation of Specimens Measured Hole Expansion Data Measured Load-Stroke Curve Inverse Analysis Conditions Inverse Analysis Results Validation Inverse Analysis of Modified LDH Test at Elevated Temperatures Expeirment Inverse Analysis Conditions Inverse Analysis Results Application of Inverse Analysis Results to Simulation of Thermoforming Introduction to Thermoforming Process Material Properties Simulation Results Discussion CHAPTER 10 INVERSE ANALYSIS OF HYDRAULIC SHEET BULGE TEST AT ROOM TEMPERATURE Introduction to Hydraulic Sheet Bulge Test Available Data Evaluation Technique for Flow Stress Determination Membrane Theory Analytical Method Developed at ERC/NSM xiii

15 10.3. Experiments Initial Guess Generation for Inverse Analysis Inverse Analysis Results Discussion CHAPTER 11 CONCLUDING REMARKS REFERENCES APPENDIX A EVALUATION OF SENSITIVITY OF NODAL VELOCITIES xiv

16 LIST OF FIGURES Figure Page Figure 1.1: History of practical use of FE simulation in metal forming [Kim, 2000] Figure 1.2: The direct and inverse problems... 7 Figure 3.1: Metal flow in certain forming processes [Altan, 1996] Figure 4.1: Flow chart for inverse analysis using the load-stroke curve or the geometry change Figure 4.2: The objective function: the difference between the measured and computed loads Figure 4.3: Flow chart of inverse analysis for rheological parameter identification Figure 4.4: Flow chart for determination of flow stress and friction factor by the inverse analysis Figure 4.5: The objective function: the distance between the measured and the computed geometries Figure 4.6: Non-homogeneous cylinder compression test Figure 4.7: The measured load-stroke curve from the compression test (shown with 10 data points were used for inverse analysis) Figure 4.8: Comparison between computed and measured load-stroke curves Figure 4.9: Bulge diameter vs. friction factor curve Figure 4.10: Comparison between (a) measured (top) and (b) computed (bottom) bulge radii after compression (initial sample: 0.4 in. height, in. diameter) Figure 4.11: Minimization of objective function Figure 4.12: Convergence of Newton-Raphson method at the first optimization iteration of the inverse analysis Figure 4.13: Convergence of material strength coefficient, K Figure 4.14: Convergence of strain-hardening coefficient, n Figure 6.1: Basic tool set-up for compression tests xv

17 Figure 6.2: Actual dimensions (mm) of the specimens used for the compression tests Figure 6.3: Specimen before (top) and after (bottom) compression test Figure 6.4: Initial and deformed specimens Figure 6.5: Comparison of computed and measured load-stroke curves at the height reduction of 40% Figure 6.6: Comparison of computed load-stroke curves for different interface friction factors, m (with determined flow stress data: f σ = 1093ε MPa) Figure 6.7: Predicted plastic strain distributions for different friction conditions at 42% height reduction. (with determined flow stress data: σ = 1093ε MPa) Figure 6.8: Comparison of predicted plastic strain distributions for two different flow stress data at 42% height reduction. (with friction factor, m = 0.02) f Figure 6.9: True stress-true strain curve of AISI 1070 from the tests Figure 6.10: True stress true strain curve of AISI 1070 (Logarithmic scale) Figure 6.11: Comparison of true stress-strain curves of AISI 1070 between experiment and curve fitting Figure 6.12: Comparison of bearing assembly by conventional method(left) and swaging or orbital forming (right) [NSK, 2003] Figure 6.13: Clamping force and residual stress in the spindle/bearing assembly Figure 6.14: Principle of orbital forming of wheel spindle bearing assembly Figure 6.15: Generated FE model Figure 6.16: FE model and predicted metal flow of spindle at intermediate stages Figure 6.17: Predicted stress distribution by FEM Figure 7.1: Two deformation modes in ring compression (a) initial ring geometry; (b) deformed with low interface friction, (c) deformed with high interface friction Figure 7.2: Compressed ring samples at different height reductions Figure 7.3: Comparison of ccomputed and measured load-stroke curves (ring compression test) Figure 7.4: Ring calibration curves generated by FEM simulation made with σ = 452 ε (Mpa) xvi

18 Figure 7.5: Initial (left) and compressed (right) cylinder samples at 38% height reduction Figure 7.6: Comparison of experimental load-stroke curve and the computed curve for set no.5 (T =200 o 1 C, V2=0.4 mm/s) Figure 7.7: Predicted ring shapes for various friction factors (T =200 o 1 C, V2=0.4 mm/s) Figure 7.8: Predicted self-folding when m f = 0.6 by using offset flownet in DEFORM 2D Figure 7.9: Measured (left) and predicted (right) shape of the ring at 53% compression (Inverse analysis set no.5 : m = 0. 6). The measured data was obtained from reference [Choi, 1994] Figure 7.10: Convergence of n-value (Inverse analysis set no.5) Figure 7.11: Convergence of m-value (Inverse analysis set no.5) Figure 7.12: Convergence of K-value (Inverse analysis set no.5) Figure 7.13: Comparison of the load-stroke curves between computed and measured from cylinder compression test with MoS 2 lubrication ( m f = in FEM simulation) Figure 7.14: Comparison of bulge radius between experiment and FEM simulation Figure 7.15: The predicted ring geometries for various friction conditions at 40% reduction (Initial ring dimension, OD : ID : H = 3.0 :1.5 : 1.0 in) Figure 7.16: Comparison of the load-stroke curves between computed and measured from ring compression test with MoS 2 lubrication ( m f = in FEM simulation) Figure 7.17: Ring calibration curves generated by FEM simulation made with σ = ε ( ksi) Figure 8.1: Schematic of indentation test Figure 8.2: All true stress-true strain curves that have the same stress at a true plastic strain of exhibit the same load-depth curve (Figure 8.3) for the Berkovich indenter [Morris, 2006] Figure 8.3: Simulation results for the load-depth curves from a sharp, conical indenter with half-angle of 70.3 (Berkovich indenter) from flow stresses (Figure 8.2) with identical true stress at a representative strain of [Morris, 2006] Figure 8.4: Schematic of a cylindrical AISI sample, [Morris, 2006] f xvii

19 Figure 8.5: Indentation test by Automated Ball Indentation (ABI) machine at Advanced Technology Corp., Oak Ridge, TN [Morris, 2006] Figure 8.6: Comparison of the flow stress results for the three different surfaces of an AISI steel sample [Morris, 2006] Figure 8.7: (a) stress-strain relationship of elasto-plastic material and (b) objective function defined in a typical indentation load-depth P-h curve Figure 8.8: Measured P-h curve and six data points for inverse analysis Figure 8.9: Comparison of type I and type II flow stress equations Figure 8.10: Comparison of type I and type II flow stress equations at strain up to Figure 8.11: Effective strain distribution estimated by 2D axisymmetric FEM simulation (scaled 10 times larger) Figure 8.12: The effect of strain hardening value on the normalized P-h curve (Stroke is scaled 10 times larger & load, P divided by max. load value) Figure 8.13: The effect of yield stress value on the P-h curve (Stroke is scaled 10 times larger) Figure 8.14: Determination of initial guess for yield stress ( σ =117.15MPa) Figure 8.15: Comparison of computed and measured P-h curves (1 st inverse analysis run) Figure 8.16: Convergence of σ Y -value (oscillation within search interval) Figure 8.17: Convergence of n-value (oscillation within search interval) Figure 8.18: Minimization behavior of fractional norm (oscillation) Figure 8.19: Minimization behavior of objective function (almost minimized) Figure 8.20: Comparison of computed and measured P-h curves (2 nd inverse analysis run) Figure 8.21: Comparison of computed and measured P-h curves between penetration depth of 0.0 and 0.6 mm (2 nd inverse analysis run) Figure 8.22: Comparison of computed and measured P-h curves between penetration depth of 0.8 and 1.2 mm (2 nd inverse analysis run) Figure 8.23: Convergence of σ Y -value (monotonous convergence) Figure 8.24: Convergence of n-value (monotonous convergence) Figure 8.25: Minimization behavior of fractional norm (completely minimized) Figure 8.26: Minimization behavior of objective function (completely minimized) Y xviii

20 Figure 8.27: Comparison of measured and computed P-h curves during inverse analysis (Initial guess: K=105, n=0.15) Figure 8.28: Comparison of K-value convergence behaviors Figure 8.29: Comparison of n-value convergence behaviors Figure 8.30: Comparison of the predicted indented geometries from two different initial guesses (both cases converged to K=115 & n=0.2) Figure 9.1: Schematic of modified limiting dome height (LDH) test Figure 9.2: Tooling for MLDH test and prepared sheet specimen Figure 9.3: Prepared specimen with lock bead Figure 9.4: Tested specimens for lubricants A and B Figure 9.5: Measured load-stroke curves for lubricants A and B Figure 9.6: 10 experimental data points used for inverse analysis (lubricants A) Figure 9.7: Convergence of n-value (Lubrication A) Figure 9.8: Convergence of K-value (Lubrication A) Figure 9.9: Convergence of fractional norm (Lubrication A) Figure 9.10: Convergence of objective function (Lubrication A) Figure 9.11: Comparison of measured and predicted load-stroke curves for lubricants A Figure 9.12: Comparison of measured and predicted load-stroke curves for lubricants B Figure 9.13: Schematic of sandwich panel structure Figure 9.14: Recorded load-stroke curves at temperature = 160 o C Figure 9.15: Recorded load-stroke curves at temperature = 170 o C Figure 9.16: Recorded load-stroke curves at temperature = 180 o C Figure 9.17: Recorded load-stroke curves at temperature = 190 o C Figure 9.18: Comparison of measured load at punch stroke = 10 mm Figure 9.19: Comparison of hole diameter between experiment and FEM simulation at different temperatures (m f=0.2). Figure 9.20: Comparison between experimental (EXP.) and computed (FEM) load-stroke curves at two different punch speeds ( m f = 0. 2, Temperature=160 oc) Figure 9.21: Comparison between experimental (EXP.) and computed (FEM) load-stroke curves at two different punch speeds ( m f = 0. 2, Temperature=190 oc) xix

21 Figure 9.22: Various thermoforming processes [Kalpakjian, 1991] Figure 9.23: Rate-dependent flow stress curves at temperature of 160 o C Figure 9.24: Deformation at various strokes Figure 9.25: Material flow at final mold closing stages Figure 10.1: Sketch of tooling used in hydraulic bulge test Figure 10.2: Flow stress determination by analytical method [Gutscher, 2000] Figure 10.3: Bulged steel sheet specimen (Thickness = 2.16 mm) Figure 10.4: Measured pressure vs. bulge height curves. [Palaniswamy et al., 2006] [Kim et al. 2006] Figure 10.5: Flow stress determination by analytical method. [Palaniswamy et al., 2006] [Kim et al. 2006] Figure 10.6: Effect of material parameters on the pressure vs time curve Figure 10.7: Normalized pressure vs. bulge height curves Figure 10.8: Comparison of predicted bulge heights with experiment Figure 10.9: Comparison of determined flow stress curves between analytical analysis and FEM based inverse analysis Figure 10.10: Comparison of determined flow stress curves between analytical analysis and FEM based inverse analysis Figure 10.11: Comparison of bulge heights obtained from experiment, analytical analysis, and FEM based inverse analysis Figure 10.12: Comparison of predicted % thinning between analytical analysis and FEM based inverse analysis xx

22 LIST OF TABLES Table Page Table 4.1: The identified parameters Table 4.2: The identified parameters for various initial guesses Table 5.1: Flow stress models and their associated rheological parameters Table 6.1: Measured specimens of AISI 1070 after compression (unit: mm) Table 6.2: Identified material parameters by inverse analysis for AISI Table 6.3: Identified material parameters with friction factor m = Table 6.4: Determined K- and n-values, using the load-stroke data measured up to different height reductions Table 6.5: Comparison of two simulation results (Machine: HP J6000 w/2gb Memory) Table 7.1: The measured I.D. of ring specimen at different reductions in height Table 7.2: Identified material parameters and friction factor by inverse analysis Table 7.3: The experimental results of ring compression test for Al 6061-T6 [Choi, 1994] Table 7.4: Results of inverse analysis results using the data measured at 200oC Table 7.5: Predicted ring shape after 53% reduction for different friction factors by inverse analysis (T o 1=200 C, V2=0.4 mm/s) Table 7.6: Parameter improvement over iterations Table 7.7: Results of inverse analysis by using the data measured at 400oC Table 7.8: Comparison of identified material parameters and friction factor Table 8.1: Two commonly used flow stress models Table 8.2: Load differences for different yield stresses Table 9.1: Lubricants used for the test Table 9.2: Measured hole size when lubricant A was used [mm] Table 9.3: Measured hole size when lubricant B was used [mm] Table 9.4: Summary of measured hole expansion and maximum load f xxi

23 Table 9.5: Results of inverse analysis for different friction coefficients (Lubricant A) Table 9.6: Predicted inverse analysis results after 20 optimization iterations Table 9.7: Comparison of predicted flow stress coefficients for AKDQ steel with data generated by Gutscher, Table 9.8: Measured hole diameter at 15 mm punch stroke Table 9.9: Sensitivity of friction factor at the temperature of 190 o C Table 9.10: Identified material parameters at various temperatures for friction factor, m f = Table 10.1: Determined material parameters using an analytical method based on membrane theory and FEM simulation Table 10.2: Summary of inverse analysis xxii

24 LIST OF SYMBOLS Symbol Units Description σ Y (MPa) Yield strength of the material σ (MPa) Effective stress (or flow stress) ε (-) Total effective strain & ε (1/sec) Effective strain rate T (K) Temperature K (MPa) Material strength constant n (-) Strain hardening exponent m (-) Strain rate sensitivity τ s (MPa) Frictional shear stress m f (-) Friction factor μ (-) Coefficient of friction p (MPa) Normal pressure F EXP F COM (N) (N) Measured load Computed load P k (-) Material parameters E (-) Objective function D EXP (mm) Measured geometry D COM (mm) Computed geometry σ ij (MPa) Deviatoric stress tensor xxiii

25 Symbol Units Description ε& ij (-) Strain rate tensor h ki ( N / mm sec K ) Density ( N / sec K ) Thermal conductivity 2 ρ ( N / mm K ) Heat capacity C p xxiv

26 CHAPTER 1 INTRODUCTION 1.1. Process Simulation in Metal Forming Computer simulation of metal forming processes using the Finite Element Method (FEM) are routinely practiced in today s industry for process and die design. The main goal of FE simulation in manufacturing process design is to reduce part development time and cost while increasing quality and productivity. For instance, in metal forming, process simulation can be used to develop the die design and establish process parameters by a) predicting metal flow and final dimensions of the part, b) preventing flow induced defects such as laps (forging) and excessive thinning and wrinkling (sheet forming), c) predicting temperatures (warming forming operations) so that part properties, friction conditions, and die life can be controlled. Furthermore, process simulation can be very beneficial in predicting and improving grain flow and microstructure, and optimizing product design and increasing die life. Development in FE process simulation technique in forging started about 3 decades ago as seen in Figure 1.1. The development of automatic remeshing methods made industrial application of 2D FE simulation practical. The application of 3D FE simulation started in the mid 1990 and gained rapid 1

27 acceptance. Today, 2D and 3D FE simulation techniques are widely accepted and routinely practiced in industry. Recently, FE software packages addressing machining simulation have started to emerge. Thanks to the advancements in computer technology, FEM simulation and optimization engine has been developed to automatically determine optimal process parameters or optimal die shape in the most innovative way. Using Finite Element Analysis (FEA), it is now possible to design and optimize a metal forming process to a level that could not be reached by traditional theoretical and experimental methods. A variety of FE process simulation software packages have become an integral part of manufacturing process design before conducting forming trials. Figure 1.1: History of practical use of FE simulation in metal forming [Kim, 2000]. 2

28 The success of process optimization using FEM depends on the accuracy of the workpiece material properties that are inputs to the FEM software. For example, with the weight-reduction trend in automotive industry, warm forming of Al or Mg alloys is being considered as a competitively emerging new forming technique. Development of such a warm forming technique requires a fundamental understanding of the flow, fracture, and anisotropy behaviors of workpiece materials over the entire range of warm forming conditions. Therefore, reliable material characterization becomes crucial for advanced process development through FE simulation [Keum, 2001] Needs for an Inverse Analysis in Material Characterization For a successful material characterization, a) material test and associated test conditions have to be selected appropriately and b) obtained test data should be evaluated accurately by using a reliable method. A test used to characterize the material properties such as flow stress, formability, and anisotropy should replicate processing conditions that exist in practical applications. For example, for determining the flow stress data for bulk metal forming simulation, a cylinder compression test has been used widely because a) during the test the deformation is done in a state of compressive stress, which represents well the true stress state of most bulk metal forming processes and b) the test can be done for a large strain. In this sense, for sheet metal forming where the biaxial stress state is dominating, a test similar to the limiting dome height test can provide more reliable data than the tensile test. Also, the evaluation of the test data should be able to overcome difficulties introduced by friction and inhomogeneous deformation. Even in the simple cylinder compression test, interface friction leads to an inevitable bulging of the sample and thereby to an 3

29 inaccurate flow stress determination. Thus, it is better to consider the unavoidable friction at the tool/workpiece interface in the test and to identify it together with flow stress using a reliable data evaluation method. A conventional material characterization methodology (i.e., data evaluation method), which requires analytical equations to evaluate the obtained test data in order to determine the material properties, is limited when actual test does not satisfy the ideal test conditions. The analytical equations, derived from simple plasticity theory of metals for a given material test, are based on the assumptions imposed on the specific test. Therefore, the equations can fail to accurately characterize material data when the required test conditions such as uniform deformation and constant rate of deformation are not maintained during the test. In recent years, an inverse analysis technique combining the FEM with an optimization algorithm has been introduced to identify rheological parameters for metal forming process simulation [Astakhov, 1997], [Bouzakis, 2001], [Boogaard, 2001], [Chenot, 1996], [EI-Morsy, 2001], [Gavrus, 1996], [Ghouati, 1998], [Han, 2002], [Hassan, 2001], [Hassan, 2001], [Kusiak, 1996], [Lin, 1999], [Liu, 2000], [Mahnken, 2000], [Szyndler, 2001], and [Xinbo, 2002]. In the inverse analysis, the unknown rheological parameters are determined by minimizing a least-square functional consisting of experimental data and FEM simulated data. The FEM is used to analyze the behavior of the material during the test, whereas the optimization technique allows for automatic adjustment of parameters until the calculated response matches the measured one within a specified tolerance. The use of FEA based inverse analysis technique allows characterizing material properties from various material tests under practical processing conditions. It offers advantages over the conventional data evaluation method by the selection of the a) appropriate material test, b) 4

30 test condition closer to processing condition, by eliminating the need for using the analytical equations. Any material test that can be simulated by FEM simulation can be chosen as the material test. Also, a need for a sophisticated control over test machine to achieve an ideal test condition, like maintaining constant strain rate in cylinder compression test for determining the rate sensitivity of materials at elevated temperature, is not required. This is a great advantage when the FEA based inverse analysis technique is applied to determine flow behaviors of sheet materials for warm forming applications. It is possible to characterize the rate-dependent flow stress behavior of Al and Mg alloys by conducting appropriate material tests under two different speeds, at a specific temperature, instead of several different constant strain-rates. This approach significantly reduces the number of tests required to fully describe stress-strain behavior of Al and Mg at elevated temperatures Research Motivation The FEM based computer simulation technique is being increasingly used as a cost-effective process analysis and design tool in today s manufacturing industry. However, the success of process optimization using FE simulations depends on the accuracy of the characterized material data that are provided as inputs to the FEM software. Any errors in flow stress measurements or uncertainties in the value of interface friction are expected to influence the reliability of the results of process simulation by FEM. Therefore, the development of a methodology to estimate reliable material properties for FE simulation becomes a crucial research area. Thus, the focus of the proposed study is on the development of inverse analysis methodology, which uses FEM 5

31 simulation and numerical optimization, to determine reliable material and friction data Review on Inverse Analysis for Flow Stress Determination A finite element analysis of metal forming process is regarded as direct problem, Figure 1.2(a). The required input data for direct problem (i.e. FE simulation) are geometry, process conditions, flow stress, and interface friction. In the direct problem FEM simulation predicts the metal flow, forming load and energy with the flow stress and friction provided. In the inverse problem, one or more of input data of the direct problem leading to the best fit between experimental data and FEM predictions are identified or calibrated (if initial guess is given) with experimental data, Figure 1.2(b). Therefore, the inverse problem can be formulated as a minimization problem where the difference between experimental data and FEM prediction is minimized by adjusting the input parameters iteratively. As experimental data obtained from the test, a) a measured load-stroke curve and/or b) a measured geometric change are commonly used. For systematic parameter identification, an objective function is defined as the difference between the measured experimental data and the corresponding computed values. The gradients of the objective function with respect to rheological parameters are evaluated to minimize the objective function. 6

32 Input data: flow stress, friction Identification of flow stress and friction FEM simulation FEM simulation Prediction of metal flow, forming load, and energy Measured experimental data: load-stroke curve, geometry change (a) Direct problem (b) Inverse problem Figure 1.2: The direct and inverse problems. Two major categories for the gradient evaluation are a) analytical differentiation method and b) finite difference method. The analytical differentiation is appreciated by its efficiency and its precision above all in large deformation with numerous remeshings. When a form of constitutive equation is complex and it is difficult to compute gradients by differentiating the FEM code analytically. The advantage of finite difference evaluation of the gradients is quick and easy implementation even for complex constitutive model. The main drawbacks are the bad precision of the gradients and an increase of the required number of FEM simulations as the number of rheolocigal parameters increase (it necessitates (number of parameters) + 1 direct FEM simulations per iteration for computing the gradients). Chenot [Chenot 1996] formulated an inverse problem in developing a methodology for automatic identification of rheological parameters. The inverse problem was formulated as finding a set of rheological parameters starting from 7

33 a known constitutive equation. An optimization algorithm was coupled with the finite element simulation for computing the parameter vector that minimizes an objective function representing the difference between experimental and numerical data. Boyer and Massoni [Boyer and Massoni, 2001] developed the semi-analytical method for sensitivity analysis of inverse problem in material forming domain. This method compromises between computation time and effort of analytical code differentiation and proved to be a good alternative to the finite difference method or analytical differentiation of the FEM code in conducting sensitivity analysis. As a result of this research, the identification software CART (Computer Aided Rheology and Tribology) was introduced. Pietrzyk [Pietrzyk, 2000] used inverse analysis technique to evaluate the coefficients in the friction and flow stress models for metal forming simulations. Both a conventional flow stress equation and a dislocation density based flow stress equation (internal variable model) were considered as flow stress equations. He concluded that the determination of both rheological and frictional parameters from one combined test is ideal because the interpretation of tests to determine the flow stress depends on an assumed value of the friction factor. Zhiliang [Zhiliang, 2000] introduced a new method combining compression tests with FEM simulation (C-FEM) to determine flow stress from the compression tests where inhomogeneous deformation is present due to interface friction. In this method, the flow stress obtained from the compression test is improved by minimizing the target function defined in load-stroke curves. Recently, Massoni [Massoni, 2000] conducted inverse analysis to identify the rate-sensitivity of sheet material for superplastic forming. A heated titanium sheet of 2 mm thickness was bulged through a conical die under isothermal condition. Pole height evolution-time curves were measured for two different pressure loading conditions. Finite difference method was used to evaluate gradients due to its easy implementation. 8

34 CHAPTER 2 OBJECTIVES OF THE STUDY 2.1. Objectives The overall objective of this proposed study is to develop an advanced, efficient, and robust methodology based on rigid-plastic finite element formulation to determine reliable material properties data, used as input to FEA in a variety of metal forming processes. In particular, the specific objectives of the proposed research are to: develop an inverse analysis technique to determine flow stress data (identification of rheological parameters) based on FE simulation technique in conjunction with experimental data obtained by appropriate rheological tests. develop the inverse analysis methodology to determine flow stress of workpiece material and the friction at the interface simultaneously from the single test demonstrate the use of inverse analysis by applying it to various rheological tests; cylinder compression test, ring compression test, indentation test, modified LDH test, and hydraulic sheet bulge test (for bulk forming and sheet forming applications) 9

35 apply the inverse analysis results to process simulation of orbital forming used to assemble the spindle/inner ring assembly (3-D FE Modeling of incremental forming process) apply the inverse analysis results to process simulation of warm forming of composite materials (3-D FE modeling of non-isothermal process) 2.2. Scope of Present Study In this study, development of efficient, practical and robust inverse analysis methodology has been proposed to identify rheological parameters in flow stress model. Finite element method based inverse analysis algorithm, which needs experimental quantities of a) load-stroke curve and b) geometric change of workpiece during material test, has been developed based on rigidviscoplastic finite element method (Chapter 4). The efficiency and robustness of the developed inverse analysis methodology have been demonstrated by applying it to the selected reference rheological tests; cylinder compression test (Chapter 5), ring compression test (Chapter 6), instrumented indentation test (chapter 8), modified limiting dome height test (Chapter 9), and sheet hydraulic bulge test (Chapter 10). Then, using the determined material property data, full 3-D finite element simulation models for industrial applications such as orbital forming (Chapter 6) and thermoforming (Chapter 9) processes have been developed for reliable process simulations. 10

36 CHAPTER 3 FINITE ELEMENT MODELING IN METAL FORMING The rigid-plastic FEM is generally considered as an effective and rapid numerical method for analyzing various metal forming processes. The workpiece material is assumed to be rigid-plastic and to obey Von-Mises yield criterion and its associated flow rule. Compared to the elasto-plastic material, the rigid-plastic material simplifies the solution process with less computational demand by neglecting the elastic response of deformation. It is further assumed that the material is homogeneous, isotropic and isotropically hardening. This idealization offers practical solutions in various large deformation problems in metal forming Rigid-Plastic Finite Element Formulation for Deformation Governing Equations The problems are limited to quasi-static and infinitesimal deformation based on the rigid-plastic approach. The equilibrium equations that neglect the terms of acceleration and body force in the equations of motion are given by σ ij, j = 0 in V (3.1) 11

37 where σ ij is the stress tensor. The equilibrium Equation (3.1) should be satisfied over entire domain of continuum. The boundary conditions associated with the above differential equations are given by either Dirichlet type (velocities v i are * prescribed to v on the surface S ) and/or as Neumann type (tractions σ n = t i * are prescribed to t on the surface S ). Hence, i u f ij j i v i σ n ij = v j * i = t * i on on S S u f S u + S f = S total (3.2) The flow rule (or the constitutive equations), describing the relationship between the stresses and the strain rates in plastically deforming body, is represented by 2 & ε & ε ij = σ ij 3 σ (3.3) where & ε = & ij ε & ijε ij represents the effective strain rate, and σ = σ ij σ ij is 2 the effective stress in terms of the components of deviatoric stress tensor σ ij. In case of rigid-viscoplastic material, the effective stress σ, in general, is a function of total strain, strain rate, and temperature and is expressed as σ = f ( ε, & ε, T ). To apply the finite element method to the boundary value problem, it is necessary to make an integral form of the equilibrium equations with related constraints. 12

38 Weak Form of the Boundary Value Problem A weak form of the boundary value problem Equation (3.1) used to find a kinematically admissible velocity field is stated as an integral form. σ ij, jδvidv + δvids = 0 (3.4) V * ( ti ti ) S f where the δ vi are arbitrary functions. After applying the divergence theorem and incorporating the incompressibility constraints, we obtain * σ ijδε ij ti δvids + K & ε kkδε& kk dv = V & dv 0 (3.5) S f * where S is the surface where traction is prescribed. ε& kk is the volumetric f strain rate and K is a penalty constant. σ ij is the deviatoric stress derived from 1 the relationship, σ ij = σ ij + δ ijσ m where σ m = σ kk the hydrostatic stress and 3 the kronecker delta. Then, by using the flow-rule that governs the relationship 2 σ between the strain-rate and the deviatoric stress, σ = & ε & ij, the Equation (3.5) 3 ε becomes V t i δ ij & ε ijδε& ij + K & ε kkδε& kkdv ti vids = V 2 σ dv * 0 3 & ε δ V S f (3.6) where the incompressibility is imposed by the penalty constant K (a very large positive number), which can be interpreted as bulk modulus for strain-rate sensitive material. 13

39 Finite Element Discretization Procedure The field variable in rigid visco-plastic boundary value problem is usually velocity, which can be interpolated by the matrix of shape functions v N from the nodal velocities v of 8-nodes hexahedral isoparametric element [Kobayashi, 1989]. ˆ Nv v ˆ = = v v v (3.7) Noting that the strain rate tensor is symmetric, the number of components can be reduced by representing stain as a 6 by 1 vector instead of a 3 by 3 matrix as follows. Bv ANv Av ˆ ˆ = = = = = x v x v x v x v x v x v x v x v x v ε ε ε ε ε ε ε & & & & & & & (3.8) Here, is a matrix of differential operator and the matrix A B contains appropriate spacial derivatives of the shape functions. Substitution of the above Equation (3.7) through (3.8) into Equation (3.6) yields an integral equation in matrix-vector form instead of tensor notation, i.e., 14

40 2 σ T T T T T T vˆ B DBdV + ˆ ˆ = 0 3 Kv B cc BdV t NdS δv & ε V V S f (3.9) Since δvˆ is arbitrary in Equation (3.9), the following nonlinear equations should be satisfied. 2 σ 3 & ε T T T T ( vˆ ) = B DBdVvˆ + KB cc BdVvˆ N tds = 0 Φ V V S f (3.10) This equation represents a set of nonlinear scalar elemental equations in discretized form. After the nonlinear system Φ( v ˆ ) = 0 is linearized by means of Taylor s expansion, it can be solved for the nodal velocity vector v ˆ by the iterative method. Two iterative methods are used in the actual computation. The first one is the direct iterative method and the other is the Newton-Raphson iterative method. In Equation (3.10), the first and second integrals represent the energy dissipation by shear and dilational deformation, respectively, and the third integral represents the traction boundary condition. By denoting volume and surface of each element as V ( e) S f ( e) e and respectively, and vˆ as nodal ( ) velocity vector in the corresponding element, fully descretized form of nonlinear system of equations may be written as 2 σ T ˆ t NdS = 0 ( ) 3 & e e ε ( e) ( e) V V S f T ( e) T T ( e) ( v) = B DBdVv + KB cc BdVv Φ (3.11) where e represents the assembly over all of volume and surface of each element and v ˆ is a nodal velocity vector. Equation (3.11) can be expressed in a more concise form, 15

41 { } 0 ( e) ( e) ( e) ( e) ( e) ( v) F v + G v h Φ ˆ = = (3.11.1) e ( e) 2 σ T where F = B DBdV, (3.11.2) 3 & ε ( e) ( e V ) T T G = KB cc BdV, (3.11.3) ( ) ( e V ) e T and h = t NdS. (3.11.4) ( e S ) f Treatment of Frictional Forces The contact surface between the workpiece and the die is considered as a mixed boundary. On the contact surface, the velocity is given in normal direction and the traction is given in the tangential direction in the form of friction. The direction of the traction is opposite to the relative movement of the workpiece with respect to the die. In order to include the energy dissipation at the contact surface, Kobayashi [Kobayashi, 1989] proposed a velocity-dependent frictional stress, expressed by f = 2 v v a vr 1 r r mk tan. (3.12) π where m is the friction factor and k is the local shear yield stress of deforming body at the contact interface. The relative velocity is approximated by v r = N ~ ˆ ( v ) v tool (3.13) 16

42 where N ~ is a matrix of shape function of the relative velocities and is the velocity of the tool. Substitution of Equations (3.12) and (3.13) into Equation (3.11.4) gives v tool h ( ) ( e vˆ ) = ( e ) S f mk 2 tan π 1 v r a N v r T v r ds (3.14) 3.2. Finite Element Formulation for Heat Transfer For simulation of thermo-viscoplastic deformation, a coupled analysis of transient deformation and heat transfer has to be conducted. The energy balance equation for heat transfer analysis is expressed by ( kit i ),, i ρ C p + γ = T & & 0 in V (3.15) where ( k it, i ), i is the heat transfer rate, k i is the thermal conductivity in the (i)th direction and T is the temperature. ρ C T& p is the internal energy rate and ρ, C,T & are the density, the specific heat, and the temperature rate, respectively. γ& is the rate of heat generation per unit volume in the deforming body due to plastic deformation that is expressed by p & γ = κσ ij & ε ij (3.16) where the heat generation efficiency κ represents the fraction of mechanical energy transformed into heat and usually assumed to be 0.9. The fraction is reminder of the plastic deformation energy ( 1 κ ) is expended to cause changes 17

43 in dislocation density, grain boundaries, and phases. The associate boundary conditions are; * T = T on S T (3.17) q n = q * n = k n * T n on S q (3.18) The Equation (3.15) can be solved in a weak form with the incorporation of the associate boundary conditions; temperature prescribed and heat flux prescribed as shown in Equations (3.17) and (3.18) into the same integral. The heat flux on the boundary surface can be categorized into two types in metal forming analysis: for a free surface, the heat flux is due to radiation and convection to the environment and for a tool contacting surface, the heat flux is due to convection and frictional heat generation at the tool-workpiece interface. In a non-isothermal FEM simulation, the equations for the deformation analysis and the heat transfer analysis are coupled, making a simultaneous solution of their finite element counterparts necessary. 18

44 3.3. Non-Steady State versus Steady State Problem Finite Element Modeling of Non-Steady State Problem In the nonsteady-state problems (forging, stamping), deformation is analyzed in a step-by-step manner by treating it quasi-statically (or quasilinearly) during each incremental deformation. Once a nodal velocity field is obtained as a solution of Equation (3.14), the effective strain rates are evaluated incrementally for each element to determine the total effective strains after a certain amount of deformation. The process configuration including geometry is updated with time. Therefore, the deformation behavior is kinematically transient and Lagrangian mesh, a mesh system that requires continuous updating is used to simulate deformation behavior of workpiece Finite Element Modeling of Steady State Problem Except at the start and the end of deformation, processes such as extrusion, drawing, rolling, and machining, are kinematically steady state. For these processes, steady state finite element modeling approach is more efficient for analyzing deformation behavior. In the steady state problems, the deformation zone is selected as control volume and a mesh fixed in space (Eulerian) is used to model the workpiece. Deformation behavior within the control volume is assumed as kinematically steady state condition. Once a nodal velocity field is obtained, the effective strains and strain rates are evaluated along the flow lines, passing the deformation zone, from the entrance boundary to the exit boundary. This procedure yields the effective strain distribution over the entire domain of control volume. As the iteration goes on, deformation ultimately reaches the steady state condition and simulation is complete. Steady state finite element modeling is suitable for predicting a) cutting force and chip formation in 19

45 machining process, b) extrusion force and metal flow in extrusion, c) separating, thrusting forces and roll torque in multi-rolling process at steady state condition. Figure 3.1 show the examples of non-steady state and steady state metal flows in typical metal forming operations. (a) Non-steady state upset forging (b) Steady state extrusion Figure 3.1: Metal flow in certain forming processes [Altan, 1996] 20

46 CHAPTER 4 DEVELOPMENT OF INVERSE ANALYSIS METHODOLOGY 4.1. Inverse Analysis Problem for Parameter Identification The basic concept of an inverse analysis for parameter identification is to find out a set of unknown material parameters used in flow stress equation, which represents material s flow behavior. In the inverse problem, the unknown material parameters are identified by minimizing a least-square functional consisting of experimental data and FEM simulated data. As shown in Figure 4.1, after FE simulation of rheological test is completed, the computed load-stroke curve (or geometry change) is compared with the measured one and the rheological parameters in FE simulation are adjusted in order to decrease the difference in the comparison. The FEM simulation is used to analyze the deformation behavior of the material, whereas the optimization technique allows for automatic adjustment of material parameters until the calculated response matches the measured one within a specified tolerance. In general, the procedures to identify the material parameters are: 1) Assume the material parameters as initial guess 2) Start FEM simulation of the selected material test with assumed material parameters under the same testing conditions 21

47 3) Compare the computed data with experimentally measured one 4) Find the amount of adjustment in material parameter by minimizing the deviation between the computed data and the measured one. 5) Improve the material parameters until deviation becomes within tolerance As experimental data obtained from the test, a) a measure of load-stroke curve or b) a measure of geometric change is commonly used as shown in Figure 4.1. For systematic parameter identification, an objective function is defined as the difference between the measured experimental data and the corresponding computed values. The gradients of the objective function with respect to material parameters are evaluated to minimize the objective function iteratively. Based on the gradient evaluation, the assumed material parameters are adjusted in such a way that the difference in the objective function is reduced in the next comparison. This procedure is repeated until the difference between experimental measurements and computed data decreases within a specified tolerance. The focus of this chapter is on the development of the efficient inverse analysis methodology that represents a compromise between the computation time and ease of gradient evaluation in identifying the material parameters in the flow stress equation. 22

48 Assumed rheological Initial run only! Reference rheological FEM simulation of rheological test Measured load-stroke curve geometry change Compar Computed load-stroke curve geometry change Update rheological parameters Are they matching well? Yes No Is the objective function minimized? Based on gradients of the objective function w.r.t. rheological parameters Identified rheological parameters Figure 4.1: Flow chart for inverse analysis using the load-stroke curve or the geometry change. 23

49 4.2. Flow Stress and Friction in Metal Forming Every method of metal forming analysis requires as input a) description of the material behavior under the process conditions, i.e., flow stress data, and b) a quantitative value to describe the friction, i.e., the friction factor m f, or the friction coefficient μ. These two quantities themselves - flow stress and friction - must be determined by experiments and are difficult to obtain accurately. The flow stress of metal can be expressed as a function of temperature T, strain ε, strain rate & ε, and microstructure S : σ = f ( ε, & ε, T, S) (4.1) In hot forming of metals at temperatures above the recrystallization temperature, the influence of strain on flow stress is insignificant and the influence of strain rate becomes increasingly important. Conversely, at room temperature (i.e., cold forming), the effect of strain rate on flow stress is negligible and the effect of strain on flow stress is most important. The degree of dependence of flow stress on temperature varies considerably among different materials. Therefore, temperature variations in a forming operation can have quite different effects on load requirements and metal flow for different materials. In analyzing metal forming processes, a friction condition is specified in a form of friction model represented by Equation (4.2). Here, the frictional shear stress is a function of interface pressure p, or local yield shear stress k. τ s = f ( p, k) (4.2) 24

50 Rheological Model In a large plastic deformation problem usually encountered in most metal forming applications, deformation behavior of the material can be assumed to be rigid-plastic by neglecting the elastic part. When the deformation behavior of material is also sensitive to rate of deformation at elevated temperature, it can be further assumed to be visco-plastic. In this case, the following power-law model is commonly used as flow stress equation, Equation (4.3). σ & n m = Kε ε (4.3) Where the rheological (material) parameters K, n, m are the material strength constant, the strain hardening exponent, and the strain rate sensitivity exponent, respectively. To be useful in metal forming analyses, the flow stresses of metals must be determined experimentally for the strain, strain rate, and temperature conditions that exist in actual metal forming processes. The method most commonly used for obtaining flow stress data is the simple tension or uniform compression test where state of uniform deformation is maintained. By analyzing the test data, the rheological parameters in Equation (4.3) are found. Then, with these three material parameters { K, n m} P k =,, flow stress curves are fully described. Thus, the three unknown material parameters are the design variables to be determined in the inverse analysis. 25

51 Friction Model The interface friction is expressed quantitatively in FE simulation, in terms of a factor m or a coefficient μ. The friction shear stress τ s is most commonly f expressed as τ s = μp (4.4) where p being a compressive normal stress to the interface, or as k τ s = being the shear stress of the deforming material, where m f k (4.5) 0 m 1. Studies in f forming mechanics indicate that Equation (4.5) adequately represents the friction condition in bulk forming processes while Equation (4.4) is commonly used for representation of friction in sheet metal forming [Lin, 1999]. The friction factor, is most commonly measured by using the ring compression test or by double cup m f extrusion test [Petersen, 1998], [Sofuoglu, 1999]. The friction coefficient μ, is commonly measured by strip draw test. 26

52 4.3. Inverse Analysis Methodology Using Load-Stroke Data Objective Function The difference between the measured load-stroke curve and the corresponding computed one shown in Figure 4.2 is defined as an objective function (in a least square sense) represented by the Equation (4.6). N 1 F mea Fcom( pk ) E( p = k ) N (4.6) i = 1 Fmea 2 where F is the experimentally measured load and F is the computed load. mea com N is the number of data sampling points in the load-stroke curve used to construct the objective function. When the objective function E( pk ) is minimized, the unknown rheologocal parameters p k would be determined. This objective function is a function of the rheological parameters p k because the computed load F is obtained from FE simulation made with the parameters p. During com the inverse analysis, the objective function is minimized and the rheological parameters starting from initial guesses will be converged to specific values iteratively. For the minimization of the objective function, the evaluation of gradients of the objective function with respect to the rheological parameters (i.e., design variables in optimization problem) is very important. An efficient and reliable method for minimizing the objective function for parameter identification has been developed based on the rigid-plastic finite element formulation. In this study, the analytical differentiation of rigid-plastic finite element code was conducted to calculate the required gradient values for minimizing the objective function. k 27

53 Load (N) Difference F mea F com = F com ( P k ) Stroke (mm) Figure 4.2: The objective function: the difference between the measured and computed loads Minimization of Objective Function The method for an evaluation of gradients of the objective function with respect to the rheological parameters is of major importance in the inverse analysis. For given rheological parameters p, the objective function E p ), Equation (4.6), will be minimum at: k ( k E( p p k k ) = 0 for k = 1,2... q (4.7) where q is the number of rheological parameters. The Equations (4.7) can be solved with respect to the rheological parameters p k using a gradient-based iterative algorithm, Newton-Raphson iterative method. 28

54 2 p p k E j ΔP j E = p k for k, j = 1,2... q (4.8) The first and the second gradients of the objective function with respect to the rheological parameters p k are evaluated by taking the derivatives of the objective function (Equation 4.6) with respect to : p k E p k 2 = N ( F F ) N mea com Fcom = 2 1 Fmea pk i (4.9) 2 E p p k j 2 = N ( F F ) N 2 1 F com Fcom mea com Fcom = 1 Fmea pk p j Fmea pk p j i (4.10) F 2 Fcom F are the computed load, the first, and second p p com where com,, and pk k j gradients of the computed load, respectively. The analytical expression to compute these values can be obtained from rigid-plastic FEM formulation as shown in the following derivation. When the nonlinear Equation (3.10) is solved by the Newton-Raphson iterative method, a velocity field is obtained and the nodal force vector is recovered. The expression for recovering nodal force vector on the toolcontacting elements is obtained from Equation (3.11.1). 29

55 ( e) ( e) ( e) ( e) ( e) h = F v + G v (4.11) ( e) 2 σ T ( e) T T where F = B DBdV and G = 3 & ε KB cc BdV. e ( e V ) V ( ) Substitution of the Equations (3.11.2) and (3.11.3) into Equation (4.11) yields the expression for the computed load ( e) h = F on the tool-contacting element, com 2 T ( e) σ T T F dv ˆ COM = h = B DB v + KB cc BdV e 3 & ε (4.12) e e V e V By assuming the second term is not dependent on the rheological parameters, p k its gradients (a sensitivity of nodal force vector with respect to rheological parameters) further can be evaluated by taking a derivative of flow stress equation with respect to material parameters in the first term. In summary, F COM P k = P k e e V 2σ B 3 & ε T DBdVvˆ = e e V 21 σ B 3 & ε P k T DBdVvˆ (4.13) 2 F P P k COM j 2 = P P k j e V e 2σ B 3 & ε T DBdVvˆ = e V e 21 2 σ 3 & ε Pk Pj B T DBdVvˆ (4.14) 30

56 where σ is the value of flow stress. σ and p k 2 σ pk p j are the derivatives of flow stress σ with respect to rheological parameters p k. Therefore, the rheological parameters in Equation (4.3) can be identified by evaluating the gradients of flow stress equation only. Major advantage of this proposed methodology is that only a single FEM simulation will be required to evaluate the all the gradients of the objective function per optimization iteration. A user only needs to provide analytical expression for the first and second derivatives of the flow stress equation with respect to material parameters, σ p k and 2 σ for inverse analysis. For example, if the flow stress is given by pk p j n m σ σ = Kε & ε = f ( K, n, m), the first derivative and the second derivative p k 2 σ will be: pk p j σ First derivatives, (4.15) p k σ n m 1 = ε & ε = σ, K K σ n m = K( ε lnε ) & ε = (lnε ) σ, n σ n m = Kε ( & ε ln & ε ) = (ln & ε ) σ m 31

57 2 σ Second derivatives, (4.16) Pk Pj 2 σ = 0 K 2 2 σ lnε, = σ, K n K σ lnε = σ K m K 2 & 2 ε ) 2 n = (ln m σ 2 2 σ 2 2 = (ln & ε ) σ 2 σ σ, = (lnε )(ln & ε ) σ, n m The proposed inverse analysis algorithm was programmed as a utility program running on a PC in Fortran language. The inverse analysis program consists of two major parts; numerical optimization engine and 2-D rigidviscoplastic FE simulation engine. Each part was programmed independently so that the FE simulation engine part can be replaced with commercial software like a DEFORM 2D by SFTC (Scientific Forming Technology Corporation) for handling the sophisticated problems efficiently. The program is capable of defining multiple objective functions for the problems where more than one rheological test is required to identify the rheological parameters uniquely. This capability is used to determine flow stress curves of rate-sensitive materials using two load-stroke curves obtained under two different forming speeds. The overall inverse analysis procedures are explained in Figure

58 Initial guess 0 p k σ = σ ( p k ), k = 1,..., m s = s +1 Optimization iteration Run FEM Simulation Extract simulation results Velocity, strain, temperature FE model info.(coordinates, connectivity, boundary conditions) Measured L-s curve Calculate the gradients F p com k 2 F, p p k com j 2 E E, p p p k k j Find Δp k 2 E p p k j ΔP j E = p k p ( i+ 1) ( i ) k p k + αδp No Converge? Yes Improve ( s + 1) p ( s) p + k k k βδp k Convergence criteria E p k < ε 1 Δp p k k ( s ) < ε 2 No Converge? E=. E( p k ) < ε 1 Δp p k k ( i) < ε 2 Yes Identified σ = σ ( ) p k Figure 4.3: Flow chart of inverse analysis for rheological parameter identification. 33

59 4.4. Inverse Analysis Methodology to Determine Flow Stress and Friction Simultaneously For the FE process simulation of 3D complex parts or forward extrusion of long shafts, the flow stress data described up to a large strain range is required. In most material tests, the frictional force a) generates an additional load contribution to the overall load-stroke curve and b) leads to a non-homogeneous deformation. Even at the well-lubricated cylinder compression test, when the specimen is compressed to a large extend, interface friction is significant enough to cause an inevitable bulging of the specimen regardless of lubricant. Therefore, it is desirable to determine the flow stress considering the interface friction effect. In the FEA of metal forming process, it is reasonable to use friction model to evaluate the frictional force at the tool-workpiece interface. For bulk forming, the constant shear friction model represented by Equation (4.5) is widely used. Since the frictional force affects the metal flow, as a result of that, a shape of the deformed specimen is sensitive to the friction factor. Therefore, by comparing the experimentally measured geometry and the corresponding computed geometry, the unknown friction factor can be identified. To achieve this goal, the developed inverse analysis algorithm has been modified to identify the friction factor as well as the material parameters from one set of test simultaneously. 34

60 Objective Function To form an objective function for identification of both, the material p k m f parameters and the friction factor, in addition to the measured load-stroke curve, one more measurable geometrical quantity in ring compression test, namely the bulge, is included. Because the bulge reflects a degree of inhomogeneous deformation caused by friction, identification of friction factor is possible by measuring the bulge shape of the specimen. Thus, the following objective function can be formulated: E = E 2 N Fmea Fcom( pk, m f ) + = 1 E2 + ) i= 1 Fmea i= 1 N ( Dmea Dcom( pk, m f ) (4.17) This objective function is a function of material parameters p k and friction factor m f. The first term represents the difference between the computed load F and com the experimental load F mea in a least-square sense and the second term indicates the difference between the computed bulge diameter D com and the measured bulge diameter D. The computed load F and the computed bulge diameter mea com Dcom are estimated from FEM simulation made with material parameters pk and friction factor m f. During the inverse analysis, the objective function E = E ( p k, m f ) is minimized and material parameter and friction factor are improved. 35

61 Two-Step Minimization of Objective Function In order to find out a minimum of the non-linear objective function E = E ( p k, m f ), the material parameters p are determined by minimizing the difference between the experimental and calculated loads for a given friction value, minimization of the first term of Equation (4.17). The same inverse analysis algorithm developed for identifying the material parameters without considering the friction effect is used. Then, the shape of the deformed specimen (bulge diameter) is compared with the corresponding computed one, minimization of the second term of Equation (4.17). If they do not match, the k friction factor m f of Equation (4.5) is adjusted in order to reduce the observed difference at the next iteration. If they match very well, the friction factor used in FEM simulation is regarded as an identified friction factor. This procedure is repeated until both the first and second terms are minimized completely with p k m f respect to and, as illustrated in Figure

62 Start Run FEM Simulation Identification of material Parameters Step 1 Identified σ = σ ( p k ) Experiment L-s curve E 1 1 = N t F mea F F mea com 2 Computed D-s curve Experiment D-s curve Modify friction factor No Is it good? Step 2 E 2 ( ) = D mea D com t Yes Identified m f Stop Figure 4.4: Flow chart for determination of flow stress and friction factor by the inverse analysis. 37

63 4.5. Inverse Analysis Methodology Using Geometry Change Objective Function In order to determine flow stress data using a measure of geometry evolution, some observable geometric quantities have to be defined on the basis of their sensitivity to rheological parameters to be identified. The inverse analysis problem is defined as the minimization of an objective function, representing the distance between measured and computed values of the geometry changes. Figure 4.5 shows an example of the objective function defined, using the measured bulge profile in cylinder compression test. Data measured points Measured mea xˆ l Distance δ Cylinder sample Computed com xˆ l t=t t=t t=t Figure 4.5: The objective function: the distance between the measured and the computed geometries. 38

64 The objective function is formulated as represented by Equation (4.18). 2 N mea com xˆ ˆ l xl ( pk ) = 1 ˆ mea l xl 1 E = E( pk ) = (4.18) N mea where N is the number of time increments and xˆl is a vector representing measured shape of the sheet specimen (in the case of Figure 4.5, measured bulge height) and com xˆl is a vector of the computed shape. The objective function is a function of the rheological parameters p k since the computed geometry vector com xˆ l is obtained from FE simulation made with the parameters pk. During the inverse analysis, the objective function is minimized iteratively and the rheological parameters converge to specific values Minimization of Objective Function The minimization of an objective function given by Equation (4.18) can be achieved using a Gauss-Newton iterative method that requires the first and second derivatives of the objective function with respect to the rheological parameter vector p k. 2 p p k E j Δp j E = p k for k, j = 1,2... q (4.19) Where the first and second gradients of the objective function E( pk ) with respect to the parameters p k are expressed as follow. 39

65 E p k 2 = N N l = 1 xˆ mea l mea ( xˆ ) pk l com com xˆ l xˆ l (4.20) 2 p 2 N com com N mea com com E 2 1 xˆ l xˆ l 2 xˆ l xˆ l xˆ l = + mea mea 2 k p j N l = 1 ( xˆ ) pk p j N l = 1 xˆ l pk p l j (4.21) Here, the second order derivative terms in right hand side of Equation (4.21) is simplified by assuming that the value of xˆ l mea xˆ mea xˆ l com l is small when is not too far from the optimum value. Then, the second derivative is approximated using the first derivatives only. p k p p k 2 E j 2 = + N N com com l l mea 2 l = 1 ( xˆ ) pk p j l 1 xˆ xˆ (4.22) Once Δp k is found by solving the Equation (4.19) with calculated E p k and 2 E, the rheological parameters are improved using Equation (4.23) p k p j iteratively until the objective function is minimized. p ( i + 1) k p ( i) k + αδp k (4.23) 40

66 In order to compute the gradient of nodal coordinates with respect to the rheological parameter, ˆ x com p k (sensitivity vector of the computed shape) in Equations (4.21) and (4.22), the sensitivity of nodal velocities with respect to the com v ) vˆ( pk ) rheological parameters needs to be computed. Evaluation of, p p k sensitivity of nodal velocity with respect to the rheological parameters, is done by differentiating the discretized direct model, Equation (3.10) in Chapter 3. Detail explanation is given in Appendix A. Although the analytical differentiation approach is appreciated by its efficiency, as an alternative, the derivatives in Equations (4.20) and (4.21) can be approximated by finite difference method because of its quick and easy implementation even for complex constitutive model. In Chapter 10, inverse analysis of hydraulic bulge test was done by finite difference method. k 4.6. Example of Inverse Analysis The developed inverse analysis methodology has been evaluated by applying it to the non-homogeneous (with bulging) cylinder compression test, Figure 4.6. With the measured load-stroke curve and deformed shapes from the non-homogeneous compression of a cylindrical billet from AISI 1030 steel, flow stress data and interface friction were determined simultaneously. The stability of the inverse analysis program was checked in terms of a) convergence behavior of rheological parameters over the iterations and b) uniqueness of converged solutions. Inverse analyses were conducted with different initial guesses for evaluating solution uniqueness and the convergence behavior. 41

67 Inverse Analysis Conditions The investigated material is assumed to follow strain-hardening behavior n and a power-law hardening model σ = Kε is used as flow stress equation. As a friction model, a constant shear friction model, τ s = is used to estimate an interface frictional stress. The first and second derivatives of the flow stress m f k equation with respect to material strength constant index are expressed by Equation (4.24). n K and strain-hardening 1 σ σ = K, (4.24) p k (lnε ) σ C L Initial cylinder billet Bulge profile Compressed billet Bulge diameter Figure 4.6: Non-homogeneous cylinder compression test. 42

68 Therefore, a set of material parameters defined by { K n} p k =, and friction factor m f are the unknown parameters that have to be identified. Two experimental quantities: (1) the load-stroke curve and (2) the maximum diameter of bulge at the end of stroke were measured and used as experimental data in the inverse analysis Experiment Figure 4.7 shows the measured load-stroke curve from the compression of a cylindrical billet that has a height of 0.4 in. and a diameter of in. from AISI 1030 steel. The compression speed was in./sec and the stroke was in. The bulge diameter of the specimen measured at the stroke of in. was in Load(lbf) Material: AISI 1030 Initial specimen size: (H=0.4 in./ Dia.= in.) 10 discrete pts. for inverse analysis stroke(in.) Figure 4.7: The measured load-stroke curve from the compression test (shown with 10 data points were used for inverse analysis). 43

69 Identified Material Parameters and Friction Factor The results of identification of the parameters in the flow stress equation and friction factor by the inverse analyses are summarized in Table 4.1. Total of five cases of inverse analysis were conducted by varying friction factor from 0.1 to 0.5. As initial guesses of material parameters K = and n = 0.2 were used for every case. The identified parameters in the case 2 give the best minimum for the best objective function (Equation 4.18). When the friction factor 0.15 was assumed, the inverse analysis result produced only 0.11 % underestimation in bulge diameter compared to the measured one and a set of material parameters P k { K = , = 0.138} = n was identified. Case Identified material parameters, pk Friction factor Bulge comparison K (Ksi) n m f Bulge diameter % Error (-) (-) (+) (+) (+) (Initial guesses: K=130, n=0.2) Table 4.1: The identified parameters. 44

70 Comparison of Load-Stroke Curve The computed load-stroke curve of the case 2 and the experimental loadstroke curve are plotted together to make a comparison in Figure 4.8. It is important to remember that an accuracy of identification depends on the number of data points considered in the inverse analysis. In this study, only the small amount of differences was observed and these differences between computed and experimental values could be due to a) the error generated in fitting of experimental data using improper flow stress equation, b) the measurement error, and c) the numerical error (round-off). Overall, the results indicate that the predictions are reasonable for: n (1) taking the flow stress equation of σ = Kε to describe a deformation behavior of the material tested and (2) using 10 data points taken from the measured load-stroke curve to identify the unknown parameters by the inverse analysis. Load (lbf) Material: AISI 1030 Initial specimen size: (H=0.4 in./ Dia.= in.) Experiment 2000 Simulation Stroke (in.) Figure 4.8: Comparison between computed and measured load-stroke curves. 45

71 Comparison of Bulge Diameter Comparison of bulge diameter between simulation and experiment is necessary in order to identify the interface friction. As can be seen in Figure 4.9, predicted bulge diameter increases as the friction factor in FEM simulation increases. The computed bulge radius of the case 2 and the measured bulge radius from the test are compared at the same stroke position as shown in Figure When friction factor is 0.15, they showed a good match Bulge diameter (in.) Friction factor Figure 4.9: Bulge diameter vs. friction factor curve. 46

72 C L (a) Stroke = in. Bulge radius = in in. (b) Stroke = in. Bulge radius = in. Friction factor = in. Figure 4.10: Comparison between (a) measured (top) and (b) computed (bottom) bulge radii after compression (initial sample: 0.4 in. height, in. diameter). 47

73 4.7. Convergence Behavior As can be seen in Figure 4.11, the objective function was minimized and the material parameters were found after 4 optimization iterations. Just after first optimization iteration, the big decrease in the value of objective function was observed. Figure 4.12 shows that the convergence behavior of Newton-Raphson E iterative method at the first optimization iteration to satisfy = 0 p k (Equation 4.7). Objective function, E 6.0E E E E E E E Iteration Figure 4.11: Minimization of objective function. Gradient of objective function Iteration Figure 4.12: Convergence of Newton-Raphson method at the first optimization iteration of the inverse analysis. 48

74 In each iteration, the material parameters are updated using ( i+ 1) ( i) p k p k + αδp where superscript i indicates the iteration number and α is a deceleration coefficient for improving a convergence behavior. The convergence behaviors of the material parameters p k { K, n} = were shown in Figure 4.13 and Figure 4.14, K- and n-values were monotonously converged to 113 and 0.138, respectively, only after 4 optimization iterations. 130 Strength coef. K (ksi) Iteration Figure 4.13: Convergence of material strength coefficient, K Strain hardening exponent, n Iteration Figure 4.14: Convergence of strain-hardening coefficient, n. 49

75 4.8. Solution Uniqueness During the inverse analysis for a parameter identification problem, the solution uniqueness and the convergence behavior are important. A stability of the inverse analysis code is judged by them. As discussed in the chapter 4, the inverse analysis technique is a combination of the FEM and an optimization to identify rheological parameters. However, in many optimization problems that have a highly non-linear objective function, convergence rate could be slow or solution can converge to the local minimum. In this study, various values for rheological parameters p k within reasonable range were selected as initial guess. As summarized in Table 4.2, when the inverse analyses were conducted with a good initial guess, the number of optimization iterations was reduced. However, when the initial guess was poor, the selection for p k didn t even converge. The influence of material strength parameter strain-hardening exponent. n K on the convergence was more dominating than that of 50

76 Case Initial guess of Identified value pk of pk Iterations K (Ksi) n K (Ksi) n Optimization iteration N-R iterations at each optimization step Not converged N/A N/A Table 4.2: The identified parameters for various initial guesses. 51

77 CHAPTER 5 INVERSE ANALYSIS MODELING OF RHEOLOGICAL TESTS 5.1. Rheological Test For an accurate and reliable material property characterization, the material property has to be measured from the appropriate tests that can replicate deformation conditions that exist in practical application. When FEA based inverse analysis technique is used to determine flow stress data of sheet or bulk materials, a selection of test condition becomes more flexible than in conventional data evaluation technique by eliminating the need for using the analytical equations. However, an appropriate flow stress equation that can describe the true stress-true strain behavior, existing in practical forming operations, has to be selected carefully. Also, testing conditions (a range of each testing variables) have to be reasonably selected so that flow stress data to be determined can cover the entire range of flow behavior possible in actual forming. In Table 5.1, various types of rheological tests and flow stress equations are summarized. 52

78 Rheological test (Application) Flow stress type Rheological parameters Forming condition Cylinder or ring compression test n σ = Kε τ k s = m f K, n (2 parameters) Cold (Bulk forming - forging, rolling) n σ = Kε & ε τ k s = m f m K, n, m (3 parameters) Warm or Hot Indentation test E σ = σ Y 1 + ε P σ Y n σ Y, n (2 parameters) Cold Modified LDH test (Sheet forming stamping, deep drawing) σ n σ = Kε τ s = μp & τ = μ n m = Kε ε, s p K, n (2 parameters) Cold K, n, m Warm or Hot (3 parameters) (200 o C<T<250 o C) Hydraulic sheet or tube bulge test n σ = Kε τ k s = m f K, n (2 parameters) Cold (Sheet or tube hydroforming) n σ = Kε & ε τ k s = m f m K, n, m Warm (3 parameters) (200 o C<T<250 o C) Orthogonal metal cutting test & n ε σ = K(T )( 1 + aε )( 1+ b ln ) & ε R 2 α1 T α 3 ( T T1 ) K(T ) = K ( e + α e ) o melt 2 T T T = T T room room a, n, b,, α, K o 1, α α 2 3 (7 parameters) Hot Table 5.1: Flow stress models and their associated rheological parameters. 53

79 5.2. Inverse Analysis Modeling for a Rate-Dependent Material When a material shows a rate-dependent flow behavior at elevated temperature, flow stress is generally a function of strain, strain-rate and temperature. The rheological test should be conducted at a) different temperatures, b) with forming different speeds, c) up to required punch stroke. If the rheological test is conducted at a given elevated temperature under isothermal condition, flow stress equation is represented as a function of strain and strain rate at a specified temperature, As an example, σ & n m = Kε ε T = Tspecified at (5.1) To evaluate a rate sensitivity of material, a rate-dependent response of material has to be obtained under, at least, two different rate of deformation. Therefore, the objective function consists of two terms, namely differences between the measured loads and corresponding computed ones in a least-square sense at two different forming speeds v1 and v2 as defined in Equation (5.2). 2 N N 1 F mea Fcom( pk ) 1 Fmea Fcom( pk ) E( p = + k ) (5.2) N i = 1 Fmea N i = 1 Fmea v1 2 v2 54

80 When the interface friction is significant, the objective function is modified as: E( p, m ) = k f 1 N N i= 1 F mea F F com mea 2 ( p k ) v1 + 1 N N i= 1 F mea F F com mea 2 ( p k ) v2 + N ( Dmea Dcom( pk, m f )) i= 1 (5.3) This objective function is a function of material parameters p k and friction factor mf. The third term indicates the difference between the computed shape Dcom and the measured one D mea. During the inverse analysis, the objective function is minimized and material parameter and friction factor are determined simultaneously. In the each iteration of the inverse analysis, two FEM simulations for tool speeds v1 and v2 are conducted to compute F com at the tool speed of v1 and v2 by varying the friction factor. When the material parameters p k = { K, n} are identified at various selected temperatures, their temperature dependency is estimated. Examples of identification of strain-rate sensitivity index are presented in Chapter 7 and Chapter 9. 55

81 CHAPTER 6 INVERSE ANALYSIS OF CYLINDER COMPRESSION TEST 6.1. Introduction In order to obtain reliable flow stress for FEM simulations of bulk forming processes, cylindrical samples are cut to the required dimensions. In the present example AISI 1070 material was investigated. In the test conducted at ERC/NSM lab, each specimen was well lubricated with industrial grade paraffin wax and barreling could be avoided. The measured load-stroke data were evaluated by both the developed inverse analysis technique in this study and a conventional direct analysis technique. Flow stress data obtained from each technique were compared with each other. A 3-D finite element analysis of the orbital forming operation used to assemble the spindle/inner ring assembly was conducted Inverse Analysis of Well-Lubricated Cylinder Compression Test Experiment The existing ERC/NSM compression test tooling was used to conduct the tests for this project. The tooling consists of two flat dies with carbide inserts contained in holders attached to a die set. The carbide inserts are used because of their higher resistance to elastic deflection. Figure 6.1 shows the basic tool set-up. 56

82 The tests were conducted in a Minster hydraulic press with a capacity of 160 tons. From the tests, the load-stroke data was recorded using a data acquisition system. A laser sensor was used to record the stroke, while a 200 ton Sensotec load cell was used to record the load. The cylinder specimens were cut to the required dimension from twelve forged components (AISI 1070) supplied by industry. Figure 6.2 shows dimensions and geometry of a specimen that can provide better lubrication during the compression test. Before the tests, each specimen was coated with industrial grade paraffin wax. The wax acts as a lubricant and helps in keeping the specimen from bulging. Figure 6.2 shows dimensions of cylinder specimen and Figure 6.3 shows the specimen with an applied lubrication before and after compression. Figure 6.1: Basic tool set-up for compression tests. 57

83 Figure 6.2: Actual dimensions (mm) of the specimens used for the compression tests. Figure 6.3: Specimen before (top) and after (bottom) compression test. 58

84 (a) Initial (b) 30% (left) and 40% (right) height reductions Figure 6.4: Initial and deformed specimens. In Table 6.1, measurements on the deformed specimens before and after the compression were shown. For each specimen, two different height reductions were used. (30% reduction - three specimens, 40% reduction - two specimens). During the compression, a bulging was observed due to the existing frictional force at the tool-specimen interface. Specime n Initial diameter Initial height Final diameter (top) Final diameter (middle) Final diameter (bottom) Final height Die stroke Reduction in height (%) Table 6.1: Measured specimens of AISI 1070 after compression (unit: mm). 59

85 Flow Stress Determination by Inverse Analysis In order to determine reliable flow stress, the effect of existing frictional force has to be taken into account in the inverse analysis. For this purpose, two experimental quantities: (1) the load-stroke curve and (2) the barreling shape (bulge diameter) of the cylinder specimen at the end of stroke were measured and flow stress and interface friction were determined simultaneously. As explained in the flow chart shown in Figure 4.4, flow stress is first determined by minimizing the difference in the load-stroke curves. Then, the unknown friction factor is identified by comparing the measured bulge diameter and its corresponding computed diameter. Figure 6.5 shows the measured load stroke curves from 40% reduction in height. This result indicates the true stress and true strain curves of AISI 1070 spindle (hot forged and annealed spindle) up to a strain level of 0.5 (40% height reduction). The inverse analyses were conducted with three different sets of initial guesses as shown in Table 6.2, assuming no friction at the tool-specimen interface. When initial guess p k is selected within reasonable range ( 1000 K 1200 and 0.1 n 0. 2 ), all the cases showed good convergence behaviors to a unique solution, σ = 1093ε (MPa). The effect of friction factor on the load-stroke curve is shown in Figure 6.6 and the effect of friction factor on the bulge diameter is shown in Figure 6.7. As seen here, the friction factor hardly influenced the load-stroke curve, but affected the bulge diameter. When a small interface friction factor m = was considered in the inverse analysis, the identified material parameters are slightly changed as summarized in Table 6.3. Figure 6.8 shows the comparison of predicted plastic strain distributions for two different flow stress data at 42% height reduction (with friction factor, m f = 0.02). f 60

86 Load(kN) Measured Computed 10 Data pts Stroke(mm) Figure 6.5: Comparison of computed and measured load-stroke curves at the height reduction of 40%. Iteration Initial guess #1 Initial guess #2 Initial guess #3 K (MPa) n K (MPa) n K (MPa) n Table 6.2: Identified material parameters by inverse analysis for AISI

87 Load (KN) mf=0.0 mf=0.02 mf= Stroke (mm) Figure 6.6: Comparison of computed load-stroke curves for different interface friction factors, MPa) m f (with determined flow stress data: σ = 1093ε (a) m = 0.0 (b) m = (c) f f m f = 0.08 Figure 6.7: Predicted plastic strain distributions for different friction conditions at 42% height reduction. (with determined flow stress data: σ = 1093ε MPa) 62

88 Iteration K (MPa) n Initial value Converged value Table 6.3: Identified material parameters with friction factor m f = (a) σ = 1093ε (MPa) (b) σ = 1087ε (MPa) Figure 6.8: Comparison of predicted plastic strain distributions for two different flow stress data at 42% height reduction. (with friction factor, m f = 0.02) 63

89 Flow Stress Determination by Direct Analysis The measured load-stroke curve was imported into an MS-Excel spreadsheet designed to calculate and plot the true stress-true strain curve. The flow stress curve was represented by a power-law model given in Equation (6.1). n σ = Kε (6.1) The true stress σ and true strain ε are calculated using the following equations. σ = F l, ε = ln A (6.2) l o The force F and the height of cylinder specimen l are measured continuously during the compression test and the instantaneous area A is calculated by using a volume constancy condition during plastic deformation. Figure 6.9 shows the true stress-true strain curve and Figure 10 shows it in a log-log scale. To find the K- and n-values, Equation (6.1) is linearized by taking a log on both sides. The slope of the linear line represents the n value and the intersection with the y-axis, at Log ( ε ) = 0, represents the Log(K ). The estimated K and n values are summarized in Table 6.4. Figure 6.11 shows comparison of the flow stress curves before and after curve fitting using a power-law type flow stress equation as expressed in Equation (6.1). 64

90 True stress (MPa) True strain Figure 6.9: True stress-true strain curve of AISI 1070 from the tests. Log (True Stress) - Log (True Strain) (1070, 40%height reduction) Log (stress) True stress-true strain in Log scale 2.8 y = x R = Log (strain) Figure 6.10: True stress true strain curve of AISI 1070 (Logarithmic scale). 65

91 Materials n-value K-value (MPa) AISI 1070 at 30% AISI 1070 at 40% Table 6.4: Determined K- and n-values, using the load-stroke data measured up to different height reductions. AISI 1070 (40% reduction):confrontation between true stress - true strain curve and curve plotted with n and k value True stress (MPa) Experimental data 200 Power-law curve fitting True strain Figure 6.11: Comparison of true stress-strain curves of AISI 1070 between experiment and curve fitting. 66

92 6.3. Application of Inverse Analysis Results to Simulation of Orbital Forming 3-D FEM simulation technique was applied to an incremental forming process, orbital forming used to assemble the spindle and inner ring assembly. In spite of improvement in computing speed, the time and cost associated with 3-D FEM simulation of incremental forming operation is still quite large because of its forming characteristics. The reduction in computation time is very important to make 3-D FEM simulation practical, efficient, and robust for industrial applications. In this chapter, the 3D FEM process simulation technique was applied to the design of orbital forming process, used in automotive spindle and inner ring assembly operation. Commercial FEM software DEFORM 3D v4.0 (by SFTC) that has an efficient numerical algorithm for simulating incremental forming processes was used as a process simulation tool. Flow stress data of hot forged spindle was provided by inverse analysis. As results, the amount of savings in computation time was shown and FEM predictions were compared with experiments Introduction to Spindle/Inner Ring Assembly Operation Due to advantages over the conventional nut clamping assembly process, orbital forming is now widely used by bearing manufacturers such as NSK, Delphi, and Timken. As shown in Figure 6.12(a), the wheel bearings manufactured by orbital forming process require fewer components and are therefore less costly, lighter in weight and smaller in size. The quality of spindle/inner ring assembly can be controlled by estimating the holding force and internal residual stresses created during the assembly operation. Magnitude of the holding force is determined by the formed tab (deformed flange) geometry. 67

93 Thus, the size of the flange holding the assembly of the spindle and the inner ring should be controlled in order to vary the holding force, Figure 6.13 [Cho, 2004]. Low cost Reduced size Lighter weight Nut clamping method Swaging Figure 6.12: Comparison of bearing assembly by conventional method(left) and swaging or orbital forming (right) [NSK, 2003]. Max. radius of flange edge σ θθ Max. holding force on the flange Min. tensile stress in the hoop dir. Through the inner ring Spindle Axis of center Figure 6.13: Clamping force and residual stress in the spindle/bearing assembly. 68

94 Characteristics of Orbital Forming Process In an assembly operation, the rotating tool, tilted at 3 to 6 degrees with respect to the axis of the lower die, moves axially to form the part. Unlike conventional compression forming where the process is completed in a single pass, orbital forming requires progressive forming actions (several tool revolutions with axial feeding) and typically takes 1.5 to 3 seconds to complete. Orbital forming offers great flexibility and can be used to form a wide range of materials and geometries. It has the following advantages: Internal stresses generated in assembled components are greatly reduced due to low axial load. Smooth surface finish is obtained and the possibility of crack formation is reduced. Axial load required for forming is reduced (up to 80%), As a result, press size, floor space and costs are greatly reduced. Due to lower forming forces, tool life is increased (Tooling costs are greatly reduced). Orbital forming is usually much quieter than other cold forming operations. 69

95 3 6 θ = o o Rotating orbit tool ω (rad / s ) Inner ring Spindle Figure 6.14: Principle of orbital forming of wheel spindle bearing assembly. In orbital forming the force is applied only on a small segment of the workpiece. Therefore, friction is reduced substantially and the metal can flow much easier in radial direction. Earlier studies showed that the forming force and the radius of deformed part increase with increasing axial feed. Furthermore, the axial force decreases and the formed part radius increases with increasing tool axis angle. The metal flow and tool stresses are mainly affected by: Axial feed speed (axial movement of the tool) Spindle RPM (rotational speed of the tool or the workpiece) Tool axis angle Lubrication 70

96 Finite Element Analysis of Spindle/Inner Ring Assembly In some metal forming processes such as drawing, extrusion, rolling, bending, and orbital forming, localized plastic deformation is often observed while majority of the workpiece undergoes rigid body motion. To take advantage this unique behavior, an innovative approach called Rigid Super Element (RSE) is developed in DEFORM 3D v4.0. Using this method, the equations associated with the nodes in the non-deforming zone are now reduced to six representing the rigid body motion. These equations are coupled and solved simultaneously with the equations in the deforming zones. Therefore, the number of equations and solution time can be reduced significantly. The FE model of spindle was generated with 9875 nodes and tetrahedral elements, Figure The spindle is assumed to be rigid plastic object and flow stress of σ = 1087ε (MPa) was used. The tool and the inner ring were rigid objects and shear friction factor of m f =0.15 was used, assuming lubricated condition. The time step size was determined as (sec/step) based on tool movement, represented by feed rate; axial feed per revolution. The feed rate is assumed to be 0.39 mm/rev in the simulation. This time step size provides 48 incremental steps per one revolution of the rotating tool and 500 simulation steps were assigned to complete the assembly process. 71

97 Figure 6.15: Generated FE model. In Table 6.5, the results of two simulations are summarized. When the simulation was conducted using Rigid Super Element in DEFORM 3D v4.0, 8 hrs. 30 min. was required to complete the assembly process. However, when this new method was inactivated, the CPU time was dramatically increased up to 98 hrs. Thus, around 91% saving of computation time was achieved with the improved version of the software. Activating new algorithm Using the old algorithm Overall CPU time 8 hrs. 12 min. 100 hrs. CPU Time / Iteration 0.5~1.0 sec. 30~40 sec. Table 6.5: Comparison of two simulation results (Machine: HP J6000 w/2gb Memory). 72

98 In Figure 6.16 (a), (b), and (c), the predicted deformation at different punch strokes near the tab of the spindle are shown. When the simulation results were compared with the experimental measurements, about 1% overestimation in the tab outer diameter in defined in Figure 6.16(c) and about 15% overestimation in forming force were observed. In Figure 6.17 (a), the predicted effective stress distribution is shown. The inner ring was under tensile stress in the hoop direction and the maximum 500 MPa was predicted at the top inner side of the inner ring below the spindle tab as shown in Figure 6.17(b). Tab outer diameter (a) 40% punch stroke (b) 80% punch stroke (c) 100% punch stroke Figure 6.16: FE model and predicted metal flow of spindle at intermediate stages. 73

99 1000 (MPa) 500 (MPa) (a) Effective stress (b) Hoop stress Figure 6.17: Predicted stress distribution by FEM. 74

100 6.4. Discussion In this chapter, cylinder compression test was conducted to determine flow stress of spindle materials. The cylinder specimens for the tests were prepared directly from the forged spindle parts by machining them in order to have reliable input data for FEM simulation by considering any variations of material property change during the previously involved forming processes. Then, FEM simulations of orbital forming for assembly of wheel spindle bearing were conducted using DEFORM TM 3D v4.0. It is concluded that the required computation time for simulating the incremental orbital forming process has been significantly reduced, by 91%, thanks to the newly developed Rigid Super Element (RSE). The predicted tab geometry of the spindle and maximum forming load by FEM simulation were compared with experimental measurements. This study illustrated that the use of 3-D FEM in simulating and optimizing the orbital forming seems to be a reasonably reliable tool for process and tool design in this process. Furthermore, the effect of process parameters including the axial feed rate and tool axis angle on the orbital forming process can be investigated using 3-D FEM simulation technique. 75

101 CHAPTER 7 INVERSE ANALYSIS OF RING COMPRESSION TEST 7.1. Introduction In this chapter, the developed inverse analysis methodology has been tested by using the real experimental data obtained from compression of ring specimens made of aluminum 6061-T6. Rings with 54 mm O.D. x 27 mm I.D. x 18 mm height were compressed at room temperatures. Rings with 30 mm O.D. x 15 mm I.D. x 20 mm H were compressed at temperature of 200 o C and 400 o C. At elevated temperature, the material is assumed to follow viscoplastic deformation behavior and obey the power-law hardening model including the strain rate dependency, represented by Equation (7.1). σ Kε & ε n m = (7.1) Where the parameter K, n, m are the material strength constant, the strain hardening exponent, and strain rate sensitivity index, respectively. The friction condition at the tool-workpiece interface is modeled by using a constant shear friction model expressed in Equation (7.2). 76

102 τ f = m f k (7.2) Where τ f is the frictional shear stress and k is local yield shear stress. The friction factor m f is assumed to remain constant throughout the forming operation. The objective of inverse analysis is to find the parameters K, n, m used in Equation (7.1) and the value of the friction factor m f in Equation (7.2) for a given isothermal temperature condition. Using the methodology proposed as in Figure 4.4, rheological parameters and friction factor are simultaneously determined from each compression test. Also, in order to investigate the effect of rheological test upon the flow stress determination, flow stress data of annealed 1100 aluminum was determined from two different rheological tests; ring compression test and cylinder compression test, respectively Estimation of Friction from Ring Compression Test For a successful FE process simulation of metal forming processes, it is important to estimate reliable interface friction factor m f in Equation (7.2) from the test. In bulk forming, this friction factor is assumed to remain constant throughout forming operation and can be determined by ring compression test (Figure 7.1) where the change in internal diameter of the compressed ring is sensitive to friction at the die-workpiece interface. When a ring specimen is compressed between two flat, parallel dies, the diameter of inner surface either increases or decreases as the height of the specimen is reduced. The inner diameter of the ring increases if the interface friction is low (Figure 7.1b) and it is decreased if this friction is high (Figure 7.1c). Because the change in internal diameter of the compressed ring is sensitive to friction at the die-workpiece 77

103 interface, ring compression has been widely used as a test to evaluate the friction condition in metal forming processes. (a) (b) (c) Figure 7.1: Two deformation modes in ring compression (a) initial ring geometry; (b) deformed with low interface friction, (c) deformed with high interface friction. 78

104 Since this geometrical change differs for the different lubrication condition (i.e., different friction factor m f ), in order to determine the friction condition quantitatively, the relationship between the geometrical change of the workpiece and the friction condition at the tool-workpiece interface is established. In ring compression test, this relationship is called ring calibration curves. In the ring calibration curves, a change of minimum internal diameter of ring is expressed as functions of reduction in height Inverse Analysis Conditions For determining the rate-dependent flow stress data and the interface friction simultaneously from one set of tests, the three unknown rheological parameters and the value of the friction factor are determined by minimizing the objective function, E = E ( p k, m f ), representing the difference between the experimental and corresponding computed loads and ring shapes: E( p, m ) = k f 1 N N i= 1 F mea F F com mea 2 ( p k ) v1 + 1 N N i= 1 F mea F F com mea 2 ( p k ) v2 + N ( Dmea Dcom( Pk, m f )) i= 1 (7.3) where F is the experimentally measured load and F is the computed load. mea N is the number of data sampling points in a load versus stroke curve used to construct the objective function. To evaluate the effect of rate of deformation (i.e., forming speed) on the deformation behavior of the material at a specific temperature, two different forming speeds are required under isothermal conditions. Therefore, the objective function, as a function of material parameters p k, consists of two terms, namely differences between the experimental and com 79

105 corresponding computed loads in a least-square sense at two different forming speeds v1 and v2. The third term indicates the difference between the measured inner diameter of the ring and its corresponding computed one. The first and second derivatives of the objective function with respect to the parameters p k are evaluated by taking the derivatives of Equation (7.1) with respect to material parameters p k. Thus, the procedure only requires the first and second derivatives of the flow stress equation with respect to material strength constant index. m K, strain hardening exponent n, and strain-rate sensitivity 1 σ σ K = (lnε ) σ (7.4) p k (ln & ε ) σ 2 σ p k p j = 0 ln ε σ K ln & ε σ K ln ε σ K (ln ε ) 2 σ (ln ε )(ln & ε ) σ ln & ε σ K (ln ε )(ln & ε ) σ 2 (ln & ε ) σ (7.5) 80

106 7.4. Inverse Analysis of Aluminum Ring Compression Test at Room Temperature Experiment At room temperature, aluminum 6061-T6 ring specimens with 54 mm O.D. x 27 mm I.D. x 18 mm height were compressed at various reductions. The rings were coated by spraying Teflon spray on all surfaces of the samples and on the top and bottom dies. In order to observe internal diameter variation the test was stopped at height reductions of 7.2%, 22.2% and 40%. Figure 7.2 shows the compressed ring samples. Table 7.1 shows the measured percent decrease of inner diameter (I.D.) of the ring at different reductions in height. Reduction Decrease in height [%] in I.D. [%] 7.2% -1.48% 22.2% -2.96% 40.0% -1.85% Figure 7.2: Compressed ring samples at different height reductions 81

107 Decrease in I.D. of ring [%] Reduction in height [%] Table 7.1: The measured I.D. of ring specimen at different reductions in height Determination of Flow Stress and Friction The results of identified parameters (K-value and n-value) in the flow stress equation and friction factor by the inverse analysis are summarized in Table 7.2. Three inverse analyses were conducted by varying the friction factor from 0.15 to 0.2. As initial guesses of material parameters, K = 430( MPa) and n = 0.1 were used for every case. When the friction factor was assumed, the inverse analysis prediction produced only 8.1% underestimation in I.D. comparison of the ring. Thus, a combination of friction factor = and flow stress σ = 452ε ( MPa) gives the best minimum for the objective function. Figure 7.3 shows convergence behavior of the computed load-stroke curve during inverse analysis. After four optimization iterations, computed and experimental loads are nearly identical. m f Friction m f Decrease in I.D. of ring [%] K-value (MPa) n-value Table 7.2: Identified material parameters and friction factor by inverse analysis. 82

108 th optimization iteration & experiment Load(KN) st optimization iteration Stroke(mm) Figure 7.3: Comparison of ccomputed and measured load-stroke curves (ring compression test) Validation To verify the accuracy of determined friction factor, the ring calibration curves were generated as shown in Figure 7.4. Four FEM simulations, using the determined flow stress σ = 452ε ( MPa) (together with friction factor) by inverse analysis, were conducted for various friction factors of 0.2, 0.15, and When the measured decreases of I.D. of ring at three different reductions in height were plotted on this calibration chart, friction factor slightly higher than was expected. By conducting one more FEM simulation, the friction factor m f = was identified. 83

109 Decrease of I.D. of Ring (%) Experiment FEM w /mf=0.2 FEM w /mf=0.175 FEM w /mf=0.15 FEM w /mf= Reduction in Height (%) 0.2 m f Figure 7.4: Ring calibration curves generated by FEM simulation made with σ = 452 ε (Mpa). In order to verify the flow stress determined with friction factor by the inverse analysis, the aluminium cylinders with a 30 mm diameter x 30 mm height were upset to 38% reduction in height. To minimize interface friction, the interface was lubricated with Ecoform lubricant made by Fuchs. As shown in Figure 7.5, the lubrication nearly eliminated bulging in upsetting of cylinder. Using the measured load-stroke curve, flow stress σ = 437 ε (Mpa) was obtained. The difference between the flow stress data obtained in ring and cylinder compression tests is about 3.3% in K-value and 9.5% in n-value, respectively. 84

110 Figure 7.5: Initial (left) and compressed (right) cylinder samples at 38% height reduction Inverse Analysis of Ring Compression Test at Elevated Temperatures Experiment To determine rate-dependent flow stress data of bulk material at elevated temperature, the ring compression tests were conducted without applying lubricant [Choi, 1994]. Aluminum rings made of 6061-T6 with 30 mm O.D. x 15 mm I.D. x 20 mm H were compressed on a MTS machine. With an attached electric furnace, isothermal test condition was provided. The tests were carried out at temperature of 200oC and 400 o C with two different compression speeds of 0.05 mm/s and 0.4 mm/s, as shown in Table 7.3. In the experiment, the loadstroke curves and change in internal diameter of ring specimen that is sensitive to interface friction are measured. 85

111 Load [KN] Reduction in height [%] V 1 =0.05 mm/s T 1 =200 o C T 2 =400 o C V 1 =0.05 V2=0.4 mm/s V2=0.4 mm/s mm/s Table 7.3: The experimental results of ring compression test for Al 6061-T6 [Choi, 1994] Determination of Flow Stress and Friction at 200 o C The results of identified material parameters and friction factors are summarized in Table 7.4. Six inverse analyses were conducted by varying the friction factor from 0.1 to 0.7. Initial guesses of material parameters K = 340( MPa), n = 0. 1, m = 0.02 were used for every case. At higher friction, friction factor over 0.4, larger values of material parameters K, n, m were identified. When the friction factor 0.6 was used, after seven optimization iterations, computed and 86

112 experimental loads are nearly identical as shown in Figure 7.6. Figure 7.7 shows the predicted ring shapes for inverse analysis set no. 2, 3, and 5 (i.e., different friction conditions). In Figure 7.8, the offset flownet of DEFORM 2D clearly shows the self-folding when friction factor m = 0. 6 was identified, which is observed in the tested ring specimen (left of Figure 7.9). Figure 7.9 shows the comparison of the obtained ring shape from the compression test and the computed ring shape by the inverse analysis conducted with friction factor 0.6. Due to high interface friction, internal deformations become highly non-uniform and concentration of strain in shear bands leads to undesirable self-folding of the ring specimen. The shear bands observed on the polished and chemically etched cross section of the ring specimen were in a good agreement with FEM prediction of inverse analysis result. The distances between the four points A, B, C, D and the center line of the tested ring specimen were also measured and the distances predicted by FEM simulation of inverse analysis are summarized in Table 7.5. Therefore, it is concluded that observations of (a) folding, (b) the change in internal diameter (point A), and (c) shear band are sufficient to choose the inverse analysis set no. 5, f m = 0.6, σ = 323.8ε & ε ( MPa) as the best solution f set. In this set of FEM simulation, the difference between both, the load-stroke curve and the deformed geometry comparisons, was successfully minimized. 87

113 Inverse Analysis Set no. Friction, m f K-value (MPa) Identified Material Parameters n-value m-value (Initial guess : K=340, n=0.1, m=0.02) Table 7.4: Results of inverse analysis results using the data measured at 200oC. Load (N) Experiment [3] 7th optimization 2nd optimization 1st optimization itr Stroke (mm) 10 Figure 7.6: Comparison of experimental load-stroke curve and the computed curve for set no.5 (T 1 =200 oc, V =0.4 mm/s). 2 88

114 Self-folding (a) m = 0.2 (b) = 0. 4 f m (c) m = 0. 6 f f Figure 7.7: Predicted ring shapes for various friction factors (T 1 =200 o C, V 2 =0.4 mm/s). Self-folding Figure 7.8: Predicted self-folding when m = 0. 6 by using offset flownet in DEFORM 2D. f 89

115 [mm] Initial ring profile D C B A L (Measurements [mm]: A=5.6, B=7.1, C=18.8, D=21.4, L=2.3) Figure 7.9: Measured (left) and predicted (right) shape of the ring at 53% compression (Inverse analysis set no.5 : m = 0. 6 ). The measured data was obtained from reference [Choi, 1994]. f Friction, m f A B C D Table 7.5: Predicted ring shape after 53% reduction for different friction factors by inverse analysis (T 1 =200 C, V =0.4 mm/s). o 2 90

116 In Table 7.6, the improvement of material parameters during inverse analysis is shown. The n- and m-values were converged to in 5 iterations and in 3 iterations, respectively, Figure 7.10 and Figure But the K- value was relatively slowly converged to after 7 iterations, as shown in Figure All parameters showed monotonic convergence behaviors. K-value (MPa) n-value m-value Initial Guess Iteration Convergence behavior Table 7.6: Parameter improvement over iterations. 91

117 n, strain hardening Initial guess: n=0.1 Converged to: n= Iteration Figure 7.10: Convergence of n-value (Inverse analysis set no.5). m, strain-rate sensitivity Initial guess: m=0.02 Converged to m= Iteration Figure 7.11: Convergence of m-value (Inverse analysis set no.5). 92

118 K material strength (Mpa) Initial guess: K=340 Converged to K=323.8 (MPa) Iteration Figure 7.12: Convergence of K-value (Inverse analysis set no.5) Determination of Flow Stress and Friction at 400 o C When the inverse analysis was conducted using the ring compression test data at 400 o C, strain hardening exponent, n, of was identified. From this it is concluded that Equation (7.1) is not suitable for describing a flow stress behavior of aluminum 6061-T6 at 400 o C. Thus, the strain-rate dependent flow m stress equation, σ = Kε&, was used and only the strain-rate sensitivity index m was identified, Table 7.7. Flow stress equation σ = Kε& m Friction, m f K (MPa) n-value m-value N/A N/A σ = n Kε & ε m Table 7.7: Results of inverse analysis by using the data measured at 400oC. 93

119 7.6. Comparison of Flow Stress Data Determined from Cylinder and Ring Compression Tests Introduction Lee and Altan studied the influence of flow stress and friction upon metal flow in forging of rings and cylinders. They used upper bound method for the theoretical analysis of metal flow [Lee, 1971]. A strain-hardening material, annealed 1100 aluminum was considered in their study. Cylinders with a 1.5 in. diameter x 2.25 in. height (h/d=1.5), and rings with 3.0 in. O.D. x 1.5 in. I.D. x 1 in. height were used in the tests. In cylinder compression tests, Teflon and Molybdenum Disulfide were used as lubricants to achieve two different friction conditions. They determined flow stress data σ = ε ( ksi ) from cylinder compression tests, assuming Teflon-film lubrication completely eliminated bulging in compression of a cylinder. Then, using the flow stress data obtained from cylinder compression test, the theoretical ring calibration curves for a ring of 6:3:2 size was established. In ring compression tests, they sprayed MoS 2 on all surfaces of ring samples and on the top and bottom dies, in order to study the bulged surface. The compression test was stopped at a reduction of 34% and, from the ring calibration curves, they estimated the friction factor m f = 0.25 for MoS 2 lubrication Inverse Analysis of Cylinder Compression Test Inverse analysis was conducted using the experimental data of cylinder compression test done with MoS lubrication. As results, flow stress of σ = 22.06ε ( ksi ) and friction factor m f = were simultaneously determined. In Figure 7.13, a comparison of the load-stroke curve between computed and measured from cylinder compression tests is made. In Figure 7.14, 94

120 the predicted bulge radii by FEM simulations were compared with measurements at two different locations Load(Klb) Experiment Computed w ith determined flow stress Stroke(in) Figure 7.13: Comparison of the load-stroke curves between computed and measured from cylinder compression test with MoS 2 lubrication ( m f = 0.25 in FEM simulation) C L A B Initial ring sample: (Height=2.25 in. /Radius=0.75 in.) Position Measured bulge radius FEM Predicted ( m f = ) A in in. B in in. Figure 7.14: Comparison of bulge radius between experiment and FEM simulation. 95

121 Inverse Analysis of Ring Compression Test Using the load-stroke curve measured from ring compression test done with MoS 2 lubrication, four inverse analyses were conducted by varying the friction factor 0.1 to 0.3. (i.e., m f = 0.1, 0.2, 0.25, 0.3). The predicted ring geometry by FEM for various friction conditions have been shown in Figure At low friction m f = 0.1, the outward flow dominates, but with increasing friction, the neutral point appears along the die-workpiece interface, resulting in both inward and outward barreling (i.e., bulge shape), Figure 7.15c. When friction factor m f = 0.25 was used, the predicted ring geometry by inverse analysis showed the best match with measured one. C L (a) m f = in. 0.6 in. (b) m f = in. (c) m f = in. Figure 7.15: The predicted ring geometries for various friction conditions at 40% reduction (Initial ring dimension, OD : ID : H = 3.0 :1.5 : 1.0 in). 96

122 0.249 The determined flow stress was σ = ε ( ksi) when friction factor m f = 0.25 was identified. In Figure 7.16, a comparison of the load-stroke curve between computed and measured from ring compression tests is made. In order to validate the determined flow stress data and friction factor, the ring calibration curves were generated by FEM simulations as shown in Figure As can be seen here, when m f = 0.25 was used, the predicted ring geometry change shows the best match in comparison with experiment Load(Klb) Experiment Computed w ith determined flow stress Stroke(in) Figure 7.16: Comparison of the load-stroke curves between computed and measured from ring compression test with MoS 2 lubrication ( m f = 0.25 in FEM simulation) 97

123 Decrease in Inner Diameter(%) Calibration Curve for a ring of 6:3:2 m = 0.1 m = 0.25 m = 0.3 Experiment Reduction in height(%) Figure 7.17: Ring calibration curves generated by FEM simulation made with σ = ε ( ksi) Discussion In this chapter, as a demonstration of inverse analysis technique, the inverse analysis of ring compression test was conducted. The flow stress data of aluminium 6061-T6 was successfully determined along with interface friction at elevated temperature under isothermal conditions. The results indicated that with this inverse analysis method it was possible to predict the flow stress and friction simultaneously from a single test with acceptable accuracy. Each inverse analysis performed has taken the maximum of 7~8 FEM simulations of the test. Also, a comparison of the flow stress data obtained from two different rheological tests (i.e., cylinder compression test and ring compression test) was made. The results shows that the cylinder compression test provided 98

124 approximately 10% higher K-value than the ring compression test (K=22.06 ksi for cylinder compression test and K=19.89 ksi for ring compression test) while n- values are almost same as shown in Table 7.8. The identified friction factors in both cases were the same as m f = Identified Parameters Inverse Analysis (Simultaneous determination) Direct Analysis (Individual Determination) [Lee, 1971] Cylinder compression Ring compression Cylinder compression K-value (Ksi) n-value Friction factor, m f Table 7.8: Comparison of identified material parameters and friction factor. 99

125 CHAPTER 8 INVERSE ANALYSIS OF INSTRUMENTED INDENTATION TEST 8.1. Introduction The use of instrumented indentation has become widely used as a general method to determine the mechanical properties such as Young s Modulus, yield stress, and strain-hardening exponent of various engineering materials. The idea of using instrumented indentation test for determining mechanical properties including material s flow behavior was originated from hardness test. As the indenter penetrates into the specimen, the average strain beneath the indenter increases, as does the mean contact pressure. This increase makes it possible to derive the true stress-true strain relationship (flow properties of the material) from the measured indentation load-depth curve using inverse analysis method. Since indentation test can be applied to various materials including polymers through a wide range of sizes and shapes, indentation test is a very attractive method to determine the mechanical properties of materials near the surface, especially in the case of small and thin specimens or component that exhibits a gradient in mechanical properties. Therefore, indentation test is useful a) when the material properties of the surface layer resulting from prior forming operations need to be determined and b) when the very small-size sample for material test is difficult to be prepared. 100

126 As an example, in machining operations where a large plastic deformation on the workpiece surface, it is possible that the flow stress at the surface of a material may differ from the bulk properties of the materials due to residual stresses. The difference in material s flow behavior between at the surface and the bulk material properties becomes an important consideration when dealing with FE process simulation of surface finishing operations such as hard turning and roller burnishing. In this case, indentation test is an excellent substitute for a standard tensile test for determining the material s flow behavior near the surface layer, especially in the case of property-gradient materials often found in machined surface. However, while indentation testing is quite simple, requiring only minimal sample preparation, the mechanical interpretation of indentation test is not straightforward due to the complicated stress fields beneath the indenter. Indentation involves several non-linearties such as large inhomogeneous plastic deformation, non-linear stress-strain behavior, non-linear change in real area contact, and so on. Recent technological advances in manufacturing in micro and nanoscale require measuring mechanical properties of materials on a small scale. The general ability of depth-sensing instrumented indentation machine is expanded to micro- and nanoindentation where nanoindenter provides accurate measurement of continuous indentation load down to µn as a function of indentation depth down to nm. This led to an increasing interest in the development of robust methodologies to determine material properties from depth-sensing indentation (DSI) test [Chollacoop, 2003]. In this chapter, the methodologies for determining flow behavior of various materials from instrumented indentation test have been reviewed. The developed inverse analysis methodology in chapter 4 was used to determine the 101

127 flow stress data of aluminum alloy from ball indentation tests. As a part of this, a parametric study has been conducted to investigate the effects of size of ball indenter and required indentation depth on flow stress determination by conducting FEM simulations Review on Flow Stress Determination from Indentation Test Indentation Load-Depth Curve In a depth-sensing instrumented indentation technique, the indentation load, P is measured as a function of penetration depth, h into a substrate during both loading and unloading stages. In Figure 8.1, a schematic of the profile of indentation, a stress-strain relationship of elasto-plastic material, and a typical indentation load-depth P-h curve are shown. The loading part of indentation load-depth curve is influenced by contact geometry and elasto-plastic material properties such as Young s modulus yield strength, strain-hardening exponent while the unloading curve is largely governed by Young s modulus. It is reported that the pile-up or sink-in around the indenter caused mainly by the material s strain-hardening behavior. Instrumented indentation testing has been commonly conducted using sharp (Berkovich and cone-equivalent Vickers pyramidal indenters) indenters or spherical (ball) indenters. The sharp indentation has an advantage over spherical indentation for the ability to penetrate harder materials to a greater depth. 102

128 Ball indenter Loading Unloading Workpiece Pile-up σ Y σ Indentation depth, h Flow stress Indentation Test E σ = σ Y 1 + ε P σ Y n Sink-in Indented geometry P-h curve P Loading Unloading ε Inverse analysis h Figure 8.1: Schematic of indentation test Characteristics of Indentation Test In order to extract elasto-plastic properties from depth-sensing indentation response, analytical expressions that relate indentation data to elasto-plastic properties need to be established. Then, the inverse analysis algorithm developed based on the analytical and computational studies is used. The inverse analysis methodology should determine the elasto-plastic material properties in a robust way by considering the real contact between the indenter and specimen and the pile-up/sink-in effect. Since the pile-up and sink-in behavior around the indenter influence the actual contact area, depending on the indenter shape, additional modifications on the inverse analysis algorithm is required. 103

129 Recently, Giannakopoulos [Giannakopoulos, 1999] has established the relationship between the indentation load and the true contact area using finite element analyses. He presented a step-by-step method to estimate elasto-plastic properties. Venkatesh [Venkatesh, 2000] has obtained a unique relationship between the indentation P-h response and the elasto-plastic material properties using finite element simulations and has formulated the forward problem (where the characteristics of P-h curve are predicted from the known elastoplastic properties) and backward problem (where the elasto-plastic properties are estimated from the measured loading and unloading P-h responses). He was able to determine Young s modulus E, yield stress σ, characteristics stress 29. However, he concluded that he couldn t distinguish between two materials that have identical values of Young s modulus E, yield stress σ Y, and characteristic stress σ Dao [Dao, 2001] has established a set of universal dimensionless functions for characterizing instrumented sharp indentation and derived analytical expressions using comprehensive elasto-plastic finite element computations in order to relate indentation data to elasto-plastic properties. He could successfully predict properties within the specified range. However, they found that using the power law material model for a given value of Young s modulus, within a specified range of material parameters, a representative plastic strain of ε r for a standard Berkovich indenter can be identified. The estimated value of ε r was 3.3% for Berkovich indenter. Their results indicated that, for a given value of Young s modulus, true stress-true strain responses that exhibit the same true stress at 3.3% true strain give the same indentation loading curvature. Thus, his inverse analysis method that obtained two material parameters (σ y and n) from one load-depth curve raised a question of solution non-uniqueness. Y σ

130 Morris [Morris, 2006] and Moradi [Moradi, 2004] at ERC/NSM investigated characteristics of conical indentation test. They developed the procedures using FEM simulation technique to identify the representative plastic strain for a sharp conical indenter. The estimated representative strain for Berkovich indenter, shown in Figure 8.3 matches very well with the computed value by [Dao, 2001]. Figure 8.4 shows the predicted load-depth curves by FEM simulations conducted with flow stress curves shown in Figure σy = MPa, n=0.3 True Stress (MPa) εr = σy = MPa, n=0.2 σy = 2000 MPa, n= Plastic Strain (ε p ) Figure 8.2: All true stress-true strain curves that have the same stress at a true plastic strain of exhibit the same load-depth curve (Figure 8.3) for the Berkovich indenter [Morris, 2006]. 105

131 1800 Load (N) σ y (MPa), n 2000, , , 0.3 All Curves Overlap Depth (mm) Figure 8.3: Simulation results for the load-depth curves from a sharp, conical indenter with half-angle of 70.3 (Berkovich indenter) from flow stresses (Figure 8.2) with identical true stress at a representative strain of [Morris, 2006]. Nayebi [Nayebi, 2002] has given a new method that minimizes an error between the experimental (P-h) curve and the theoretical curve that is a function of mechanical properties of the studied material. Pelletier [Pelletier, 2000] has shown the limits of using bilinear stress-strain for finite element modeling of nanoindentation response. He concluded that it is difficult to extract unique elastic-plastic properties with FEM simulations using a bilinear constitutive law. Bouzakis [Bouzakis, 2001] has used multilinear constitutive law to build stressstrain curve σ ε. In his study, stress-strain relationship in each interval was assumed to be linear and the slope of minimizing an error in comparison with the indentation load. σ ε curve was determined by 106

132 Meanwhile, the comprehensive work done on spherical indentation was presented by [Haggag, 1993], at the Advanced Technology Corporation. Their work used a specially built ball indentation tester. The flow stress was determined by iteratively solving a series of equations derived from plasticity and elasticity theories. The data were fit by regression analysis to the power law relationship of flow stress (σ = Kε n ) to obtain the values for strength coefficient (K) and strain-hardening exponent (n) Flow Stress Measurement on the Engineered Surfaces Morris et al. at ERC/NSM determined flow stress data of the surface layers for AISI hard-turned bearing steel through spherical indentation tests [Morris, 2006]. The 50 mm diameter AISI sample was hardened to the hardness of 60 HRC and subsequently hard turned and roller burnished. As shown in Figure 8.4, three different types of surfaces prepared for indentation tests are a) hard turned surface (hardening and then turning), b) roller burnished surface and c) cross section surface. Hard turning was performed using a tool holder Kennametal MDJNL124B and a ceramic insert Kennametal DNGA432T0820. Roller burnishing was conducted using a hydrostatic ball tool (6 mm diameter) at an applied fluid pressure of 36 MPa. Indentations on the AISI (60 HRC) were conducted by Advanced Technology Corporation (ATC) using the Automated Ball Indentation (ABI) system with a tip diameter of mm as shown in Figure 8.5. The measured experimental load-stroke curves from instrumented indentation tests on the three different surfaces (i.e. hard-turned, burnished and cross-section surfaces) are used for determining for the flow stress data [Morris, 2006]. As can be seen here, surface-finishing operations causes difference in load-depth responses. The determined flow stress data of AISI at ATC are shown in Figure 8.6. The 107

133 hard turned surface showed a considerably higher flow stress value than the cross-section surface. Similarly, slightly higher flow stress was estimated in the burnished surface over the hard turned surface. This demonstrates the high possibility that surface material properties will differ to the bulk material properties in most manufactured components. For adequate simulations of surface deformation processes (e.g. roller burnishing), knowledge of these material properties of hard-turned surface is essential. The technique outlined in this study provides a reasonably easy and effective method for determining the flow stress properties of the surface. Figure 8.4: Schematic of a cylindrical AISI sample, [Morris, 2006]. 108

134 Figure 8.5: Indentation test by Automated Ball Indentation (ABI) machine at Advanced Technology Corp., Oak Ridge, TN [Morris, 2006] Burnished Surface True Stress (MPa) Hard-turned Surface Non-hardened Surface True Plastic Strain Figure 8.6: Comparison of the flow stress results for the three different surfaces of an AISI steel sample [Morris, 2006]. 109

135 8.3. Inverse Analysis of Spherical Indentation Test The developed inverse analysis methodology in Chapter 4 is used to determine the flow stress data of aluminum from spherical indentation tests. The size of ball indenter and the required indentation depth were determined by conducting parametric study; the sensitivity analysis on the effect of rheological parameters on the P-h curve by FEM simulations for various ball sizes and indentation depths. During the spherical indentation test, the indentation loaddepth curve is recorded continuously and the geometric impression around the E indenter is observed. As a flow stress equation, σ = σ Y 1 + ε P listed in σ Y Table 5.1, is used. The yield stress σ Y and the strain-hardening exponent n are determined by inverse analysis. The objective function is defined as total of the least square difference between measured and predicted indentation loads at selected indentation depths as represented by Equation (8.1). n N 1 P mea Pcom( pk ) E( p = k ) N (8.1) i = 1 Pmea 2 Figure 8.7 shows that a) stress-strain relationship of elasto-plastic material and b) objective function defined in a typical indentation load-depth P-h curve. 110

136 σ Y σ E ε Y n σ = Kε (a) ε P FEM computed E = E( P k ) (b) Measured h Figure 8.7: (a) stress-strain relationship of elasto-plastic material and (b) objective function defined in a typical indentation load-depth P-h curve Experiment Indentation tests were conducted at the ERC/NSM on aluminum 6061 alloys using an Alexandra I instrumented indenter, manufactured by Quad Group Inc., Tacoma, WA. The geometry of the indenter tip is a 1.5 mm diameter ball. Figure 8.8 shows the measured indentation load-indentation depth (P-h) curve up to indentation depth of 0.12 mm. A slight slope change in P-h curve was observed at the beginning. After that, the slope remains constant (i.e., an almost linear increasing straight-line in the P-h curve). From this P-h curve six data points were taken for the inverse analysis. 111

137 Punch force(n) Experiment Experiment for IV Penetration depth(mm) Figure 8.8: Measured P-h curve and six data points for inverse analysis Flow Stress Model A simple elasto-plastic, true stress-true strain behavior can be successfully described by four material parameters p { E, ν, σ n} modulus, = where E is the Young s k Y, ν is the Poisson s ratio, σ Y is the yield stress, and n is the strain hardening exponent, Equation (8.2). Plastic behavior of many pure and alloy materials are often closely approximated by a power law description, as shown schematically in Figure 8.7 (a). Eε for σ σ Y σ = (8.2) n Kε for σ σ Y Where ε Y be rewritten, using is the yield strain, such that σ = Eε ε = ε Y + ε P, as follows. Y Y and σ K = Y ε n Y. Equation (8.2) can 112

138 σ = σ = n ( ε + ε ) = ε + ε Y n Y Y P P σ ε n Y P σ n Y σ Y 1 (8.3) εy εy εy εy n = ε + n For σ σ Y, Equation (8.2) becomes where n E σ = σy 1 + ε P (8.4) σ Y ε P is the non-linear part of the total effective strain accumulated beyond the yield strain ε Y. With above assumptions and definitions, a material s elastoplastic behavior is fully described by p { E, ν, σ,n} k =. The power-law strainhardening behavior assumption reduces the mathematical description of plastic portion of the stress-strain relation to two independent parameters p { σ,n} Similar to the Equation (8.4), Y k =. n σ = Kε p, another popular power-law type flow stress equation is considered where two independent trheological parameters are n ( 1 n ) p k = { K, n} with K = E σ. For the steel, σ Y is 457 (MPa), n is 0.1, and E is Y 210 (GPa), respectively. Comparison of flow equation type I and type II is shown in Figure 8.9 and Figure The difference between two models is almost negligible after strain is greater than Y Model type Flow stress equation Material parameters Type I Type II σ E σ = σ Y 1 + ε P σ Y n { Y1 ( E σ n ) } ε n n or p p k k = = { σ Y,n} { E, σ,n} n = Kε = p k = { K,n} Y Table 8.1: Two commonly used flow stress models. 113

139 Stress(MPa) TYPE I TYPE II Strain Figure 8.9: Comparison of type I and type II flow stress equations. 800 Stress(MPa) TYPE I TYPE II Strain Figure 8.10: Comparison of type I and type II flow stress equations at strain up to

140 For inverse analysis of spherical indentation test, the first and second derivatives of flow stress equation (type I) with respect to material parameters 115 { n}, p Y k σ = are analytically derived as follows. P k σ First derivatives, + + = + P Y Y P n P Y n P Y Y E 1 ne E 1 E 1 ε σ σ ε ε σ ε σ σ σ (8.5) + + = P Y n P Y Y E 1 ln E 1 n ε σ ε σ σ σ (8.6) j k 2 P P σ Second derivatives, (8.7) 2 P Y n P Y Y 2 2 E 1 ln E 1 n + + = ε σ ε σ σ σ (8.8) 2 P Y 2 Y 2 p n P Y 2 P Y 2 Y 2 p 2 n P Y 2 Y 2 E 1 E n E 1 E 1 E n E = ε σ σ ε ε σ ε σ σ ε ε σ σ σ (8.9) = + P Y Y P n P Y P Y Y P P Y n P Y P Y n P Y Y 2 E 1 E E 1 E 1 ne E 1 ln E 1 E 1 ln E 1 n ε σ σ ε ε σ ε σ σ ε ε σ ε σ ε σ ε σ σ σ (8.10)

141 Finite Element Model In finite element modeling of spherical indentation test, since all boundary condition as well as deformation mode is axisymmetric, 2D axisymmetric model was selected. As seen in Figure 8.8, the actual indentation depth was only 0.12 mm. However, the FE model was scaled 10 times larger to facilitate stable contact detection in FEM simulation, as shown in Figure Also, the measured experimental load-stroke curve was scaled 100 times larger. The mesh system was established with around 2000 quadrilateral elements, with smaller elements created close to the indenter where a higher deformation is expected. The spherical indenter of 15mm ball diameter was modeled as a rigid material and the workpiece material was assumed homogeneous, isotropic, hardening rigidplastic, and semi-infinite. The dimensions for workpiece were 10 mm in the axial direction and 20 mm in the radial direction. The friction factor at the interface between the indenter and the specimen was assumed to be 0.1 since the friction has negligible effect on P-h curve based on earlier study. Figure 8.11 shows the deformed mesh system with effective strain distribution. 116

142 2.5 mm 1.2 mm mm Center of Axis 20 mm Figure 8.11: Effective strain distribution estimated by 2D axisymmetric FEM simulation (scaled 10 times larger) Initial Guess Generation Since the amount of indentation is very small in the test, it is important to understand the response of P-h curve for indenter geometry and material during indentation. By looking closer at the P-h curve obtained from the indentation test, we can understand the effect of material parameters on P-h curve. First, to find out a good initial guess for the strain-hardening exponent, n, FEM simulations were performed for various n values (i.e., n=0.0, 0.2, 0.3, 0.4) while the yield stress was kept constant. ( σ = 1. Y 0 ). Then, the computed P-h curves were normalized with respect to the stroke as shown in Figure It is 117

143 observed that when the strain-hardening, n is 0.3, the calculated P-h curve does not show any slope change except the beginning of indentation. If we compare the normalized calculated curves with the measured P-h curve (Figure 8.8), strain-hardening, n=0.3 could be found as a good initial guess Normalized load (N) n=0.0 n=0.2 n=0.4 n=0.3 Decreasing n-value when n=0.3 the slope is constant If n>0.3 the slope increases If n<0.3 the slope decrease towards the end of penetration Stroke (mm) Figure 8.12: The effect of strain hardening value on the normalized P-h curve (Stroke is scaled 10 times larger & load, P divided by max. load value). To find out a good initial guess for the yield stress, σ Y, again FEM simulations were performed for various σ Y values (i.e., σ Y =234.0, 140.0, 122.0) while the strain-hardening was kept constant (i.e., n=0.3). As shown in Figure 8.13, when the yield stress decreased from 234 to 122 (MPa), the load difference, dp, was dropped to near the measured maximum indentation load. The amount of decrease of the load difference as the yield stress decreases is summarized in 118

144 Table 8.2. Using the values in Table 8.2, a linear straight line was fit in Figure 8.14 and the yield stress at the zero load difference (i.e., y-intercept of the straight line) was estimated as (MPa). Therefore, the yield stress of (MPa) and strain-hardening exponent of 0.3 were found to be the good initial guesses for the inverse analysis Experiment FEM (SIG=234,n=0.3) FEM (SIG=140,n=0.3) FEM (SIG=122,n=0.3) Linear (Experiment) Load diff. =dp Load(N) Stroke(mm) Figure 8.13: The effect of yield stress value on the P-h curve (Stroke is scaled 10 times larger). Load diff. (dp) [N] Yield stress ( σ Y ) (MPa) Table 8.2: Load differences for different yield stresses 119

145 250 Sig_Y (Mpa) y = x Load Diffefence(N), dp Figure 8.14: Determination of initial guess for yield stress ( σ Y =117.15MPa) Inverse Analysis Results First Inverse Analysis Run The first inverses analysis was carried out with the initial guess of material parameters found by sensitivity analysis of P-h curve. The results of the first inverse analysis run are shown from Figure 8.15 to Figure In the first inverse analysis run, as can be seen in Figure 8.15, the initial guess set of p = { σ = ,n 0. 3} produced a good fit of the measured P-h curve at the k Y = first inverse analysis iteration. However, it appears that more than one set of pk = { σ Y, n} yield to a good fit of the measured P-h curve during the inverse analysis. After 10 iterations within search interval (100,120) for yield stress and (0.001, 0.5) for strain hardening value, the convergence was not achieved. As shown in Figures 8.16 to 8.17, the final updated material parameters were p = { σ = ,n }. This phenomena can be attributed to (a) a linear k Y = behavior of P-h curve obtained from the indentation of aluminum alloy with the 120

146 spherical indenter and (b) a noise in FEM predicted P-h curve that showed a small jump (i.e., discontinuity) when a new node came into contact with the indenter Punch force(n) Experiment FEM - 1itr. FEM - 2nd itr. FEM - 4th itr. FEM - 10 th Itr Penetration depth(mm) Figure 8.15: Comparison of computed and measured P-h curves (1 st inverse analysis run). 121

147 Sig_Y-value(MPa) Iteration no. Figure 8.16: Convergence of σ Y -value (oscillation within search interval) n-value(mpa) Iteration no. Figure 8.17: Convergence of n-value (oscillation within search interval). 122

148 The minimization behavior during the first inverse analysis was shown in Figures 8.18 and Here, DELP / P is the fractional norm and E is the functional norm (i.e., objective function), respectively. They are defined as in the following. DELP P = 2 ( Δσ Y ) + ( Δn) ( σ ) 2 + ( n) 2 Y 2 (8.11) N 1 F EXP FCOM ( Pk ) E = (8.12) N i = 1 FEXP 2 When DELP / P drops below 10-2 and E drops below 10-3, the convergence condition is satisfied. DELP / P ) Iteration no. 10 Figure 8.18: Minimization behavior of fractional norm (oscillation) 123

149 E Iteration no. Figure 8.19: Minimization behavior of objective function (almost minimized). Second Inverse Analysis Run When the first inverse run was completed, the final updated material parameters were taken as initial guesses for the second inverse analysis run. In Figures from 8.20 to 8.22, the comparisons of measured and computed P-h curves are shown. The minimization of the objective function was achieved after 10 iterations and the yield stress converged to (MPa) and the strainhardening exponent converged to successfully. Figure 8.23 and Figure 8.24 show monotonous convergence behaviors of yield stress and strain-hardening exponent. Figure 8.25 and Figure 8.26 show the complete minimization of fractional norm, Equation (8.11) and objective function, Equation (8.12). 124

150 60000 Punch force(n) Experiment FEM - 1itr. FEM - 2nd itr. FEM - 10th itr Penetration depth(mm) Figure 8.20: Comparison of computed and measured P-h curves (2 nd inverse analysis run). Punch force(n) P-h curve comparison at points 1, 2, and 3 Experiment FEM - 1itr. FEM - 2nd itr. FEM - 10th itr Penetration depth(mm) Figure 8.21: Comparison of computed and measured P-h curves between penetration depth of 0.0 and 0.6 mm (2 nd inverse analysis run). 125

151 P-h curve comparison at points 4, 5, and 6 [Sig_Y, n ]= [ , 0.316] Experiment FEM - 1itr. Punch force(n) FEM - 2nd itr. FEM - 10th itr. [Sig_Y, n] = [ , 0.313] Experiment [Sig_Y, n] = [106, 0.3] Penetration depth(mm) Figure 8.22: Comparison of computed and measured P-h curves between penetration depth of 0.8 and 1.2 mm (2 nd inverse analysis run). 126

152 106.2 SigY-value(MPa) Converged to K= (MPa) Iteration no. Figure 8.23: Convergence of σ Y -value (monotonous convergence). n-value(mpa) Iteration no. Converged to n = Figure 8.24: Convergence of n-value (monotonous convergence). 127

153 ) DELP / P Iteration no. 10 Figure 8.25: Minimization behavior of fractional norm (completely minimized). 3.00E E E-03 E 1.50E E E E Iteration no. Figure 8.26: Minimization behavior of objective function (completely minimized) 128

154 With the identified material parameters; yield stress (MPa) & strain-hardening exponent 0.316, the flow stress data of aluminum 6061-T6 can be written using Equation (8.4) as follows. Figure 8.30 shows a plot of Equation (8.13). ( 665.3ε ) σ = p (MPa) (8.13) Max. plastic strain = 0.45 in FEM Indentation depth = 0.12mm Stress(MPa) Strain Figure 8.30: Determined flow stress curve of aluminum 6061-T6 plotted by Equation (8.13) 129

155 8.4. Discussion Based on earlier study and literature review, clearly the use of conical indentation is not preferred for flow stress determination due to the nonuniqueness of the solution. However, in this study, although the inverse analysis was conducted for spherical indentation test with a good initial guess, searching the unique solution was not robust. The converged solution was not found at first inverse analysis run and through an additional inverse analysis run the converged solution could be achieved. It is expected that main reason for this difficulty is because of a weak sensitivity of material parameters on the measured indentation load-depth curve. Therefore, it is important to examine how the sensitivity changes based on how far the workpiece material is indented. Preliminary studies on finding out the reliable indentation test conditions (i.e. appropriate values of indentation depth and ball indenter diameter) seem to be essential. The indentation conditions used in this study are the spherical indenter of 1.5 mm diameter and the maximum indentation depth h max, of 0.12 mm (scaled 10 times larger). This gave a value of h max /D equal to Thus, it is desirable to examine how the sensitivity changes based on how far the material is indented. To investigate the effect of the ratio of the maximum indentation depth to the ball indenter diameter on flow stress determination, FEM simulations of spherical indentation test were conducted using a ball indenter with 2.0 mm diameter on a workpiece material with a fictitious material data (i.e, yield stress of 115 MPa and strain-hardening exponent of 0.2). The indentation load-depth curve was generated up to 1.8 mm. The generated loaddepth curve was used as an input to inverse analysis for identifying the material parameters. In this combination, the indentation condition gave a value of h max /D equal to

156 Figures 8.27 to 8.29 show the results of inverse analysis under the condition for h max /D = 0.9. Comparison of measured and computed indentation load-depth curve is shown in Figure The computed indentation load depth curve converges to the measured curve during inverse analysis with initial guess of K=105 and n=0.15. Only after 4 iterations, the material parameters converged to target values. To check the robustness of inverse analysis, two sets of initial guesses are used and they are {K=125, n=0.3} and {K=105, n=0.15}. Figures 8.28 and 8.29 show the rapid convergence behavior of material parameters in both cases (converging to yield stress of 115 MPa and strain-hardening exponent of 0.2). In Figure 8.30, the predicted two indented geometries by inverse analysis, started from two different initial guesses were compared. The comparison showed a very good agreement with the maximum plastic strain range almost up to 3.0. Therefore, the indentation depth, 1.8 mm, was sufficient enough to make inverse analysis more robust. Consequently, it is essential to conduct tests at an adequate depth to ensure sufficient amount of plastic deformation and that the sensitivity of material parameters on the degree of indentation load-depth curvature can be noticeable. Unfortunately, higher indentation depths translate into higher required load. It would be nice to determine a criterion for the minimum h max /D value that should be attained to assure good accuracy in the results. Morris et al conducted the extensive numerical studies for better understanding of the h max /D sensitivity on flow stress determination. For example, in this case, by examining the previous graphs in Figure 8.27, it was determined that a value of h max /D about 0.9 would be the much-preferred condition for determining the reliable flow stress data. 131

157 th Iter. Indentation Load(N) nd Iter mm rd Iter. Numerical Experiment Penetration depth(mm) Figure 8.27: Comparison of measured and computed P-h curves during inverse analysis (Initial guess: K=105, n=0.15). 132

158 Material strength coef.(k) (MPa) Convergence of K-value K-value w ith {K0=125 & n0=0.3} K-value w ith {K0=105 & n0=0.15} Converged to K=115(MPa) Iteration no. Figure 8.28: Comparison of K-value convergence behaviors. Strain-hardening exp.(n) Convergence of n-value K-value w ith {K0=125 & n0=0.3} K-value w ith {K0=105 & n0=0.15} Converged to n= Iteration no. Figure 8.29: Comparison of n-value convergence behaviors. 133

159 0.38 mm 1.8 mm 4.0 mm 5.0 mm (a) Initial guess {K=125, n=0.3} (b) Initial guess {K=105, n=0.15} Figure 8.30: Comparison of the predicted indented geometries from two different initial guesses (both cases converged to K=115 & n=0.2). 134

160 CHAPTER 9 INVERSE ANALYSIS OF MODIFIED LDH TEST 9.1. Introduction to Modified Limiting Dome Height Test It is known that for sheet metal forming where the biaxial stress state is dominating, a test similar to the limiting dome height test can provide more reliable data than the tensile test. In this chapter, the modified Limiting Dome Height (LDH) test has been introduced as a reference material test to determine flow stress of sheet materials and friction at the tool/workpiece interface from one set of material tests. The principle of the original LDH test, as a friction test, is that with a good lubricant the maximum thinning should occur near the dome center while with a poor lubricant the maximum wall thinning should occur away from the dome center. In the modified LDH test, a sheet blank with a selected diameter hole at center is stretched against a hemispherical punch as shown in Figure 9.1. Since the change in hole size is sensitive to frictional force acting on contact interface, hole expansion plays its role as a good indicator for estimating interface friction condition at the sheet blank-punch interface. In the test, the load-stroke curve during the stretching is recorded and the expanded hole diameter at the end of stretching is measured. Then, the test results were evaluated using the inverse analysis technique to determine material parameters as well as friction coefficient simultaneously. For validating the flow stress data determined from the modified LDH test, the determined flow stress data were 135

161 compared with corresponding data obtained independently using the hydraulic bulge test. The developed inverse analysis technique was also used to determine the flow stress of a proprietary composite sheet material at elevated temperatures. The modified LDH tests were conducted at five different elevated temperatures (under isothermal condition) for determining rate-dependent flow stress data of a laminated composite sheet material. With the determined material property data, a thermo-coupled three dimensional finite element analysis of thermoforming process was conducted to predict thermo-mechanical behavior of composite material. Punch Clamping ring Die Initial sheet blank Stretched sheet blank Figure 9.1: Schematic of modified limiting dome height (LDH) test. 136

162 9.2. Inverse Analysis of Modified LDH Test at Room Temperature For the modified LDH test, the tooling used is the LDH test can be used as shown in Figure 9.2. The main parts of the LDH tooling are the lower die, upper die, punch, lock bead, and the load cells. The hemispherical punch is placed on top of a load cell, so the punch load can be measured during the deformation. The upper die is connected to the slide and the lower sits on cushion pins. At the beginning the tooling is open and a specimen can be placed between the upper and lower die. Then the tooling dies close and the sheet is fixed between them. The sheet blank only can be stretched and material draw-in is prevented by the draw bead. In the test, the sheet blank is stretched over a 152 mm diameter hemispherical punch. The radius of upper die is 50 mm. The press ram velocity of 60 mm/s was used. Detail A Detail A Hole Figure 9.2: Tooling for MLDH test and prepared sheet specimen. 137

163 Preparation of Specimens In the first step round sheet specimens of mm diameter and a thickness of 0.86 mm shown in Figure 9.3 were cut from a rectangular cold rolled AKDQ sheet with the dimension of mm by 1524 mm. Since a new idea of placing a hole in the middle of the sheet blank was suggested, a hole with the diameter of 8 mm was drilled in the center of the specimens. After drilling the hole, lock bead is formed with the Minster 160 ton press in the specimens. To find out an optimum hole size, which gives maximum allowable punch stroke without any fracture, preliminary FEM simulations were conducted. Two lubricants were used for the experiments as shown in Table 9.1. They are applied by using a brush on the specimens. 8 mm 355 mm Figure 9.3: Prepared specimen with lock bead 138

164 No. Name Properties/contents 1 Lubricant A Peacock Special Winter Strained Lard Oil 2 Lubricant B Fuchs Ecoform PSTT 2717x-syn Table 9.1: Lubricants used for the test Measured Hole Expansion Data Three specimens were tested for two different lubricants A and B. Due to the fact that the holes weren t an exact circle after the experiment it was necessary to take an average from different measuring points of the hole. Figure 9.4 shows the specimen tested with the two different lubricants. The values measured for the different lubricants and specimens are shown in Table 9.2 and Table 9.3. These values can be used to rank the performance of lubricants. Lubricant B with a hole expansion of 8.57 mm performs better than lubrication A with a hole expansion of 7.4 mm. A larger hole expansion indicates a better performance of the lubricant used. 139

165 Figure 9.4: Tested specimens for lubricants A and B. Lubricant A Average Specimen Specimen Specimen Initial hole size = 8 mm Avg. hole expansion = 7.40 Table 9.2: Measured hole size when lubricant A was used [mm]. Lubricant B Average Specimen Specimen Specimen Initial hole size = 8 mm Avg. hole expansion = 8.57 Table 9.3: Measured hole size when lubricant B was used [mm]. 140

166 Measured Load-Stroke Curve The effect of different lubrication conditions is also found in the measured load-stroke curves for the two different lubricants. As observed in Figure 9.5, the load for lubricant B is slightly lower than lubricant A. This means that lubricant B produced less amount of frictional force (i.e., a lower coefficient of friction) than lubricant A. Table 9.4 summarizes the measured hole diameter and maximum load for lubricants A and B at the punch stroke of 44 mm. Lubricant Measured hole diameter [mm] Measured max. load [kn] A B Table 9.4: Summary of measured hole expansion and maximum load Lub A Lub B Load(kN) Stroke(mm) Figure 9.5: Measured load-stroke curves for lubricants A and B. 141

167 Inverse Analysis Conditions To determine flow stress data of sheet material and friction coefficient simultaneously, as a flow stress equation and coulomb friction model shown in Equations (9.1) and (9.2) are selected. n σ = Kε (9.1) τ f = μp (9.2) The objective function, which is a function of material parameters p k and friction coefficient μ, is formulated as shown in Equation (9.3). The inverse analysis procedure developed in chapter 4 is used for minimizing the objective function and identifying the material parameters and friction coefficient. 2 N N 1 F mea Fcom ( pk, μ) E( p = k, μ ) + ( Dmea Dcom ( pk, μ) ) (9.3) N i = 1 Fmea i = 1 The first term of the objective function, N 1 F mea Fcom( pk, μ) Fmea N i = 1 2, is evaluated using the measured forces at 10 data points as shown in Figure 9.6 and the second term, N ( i= 1 D mea D com ( p, μ )), is evaluated using the measured hole diameter k at the end of 44mm stretching. 142

168 Lub A Data Pts for Inverse Analysis Load(kN) Stroke(mm) Figure 9.6: 10 experimental data points used for inverse analysis (lubricants A) Inverse Analysis Results Using the load-stroke curve and measured hole diameter at the final stroke, several inverse analyses were conducted by varying the friction coefficients for lubricants A and B. The results of identified parameters (K-value and n-value) in the flow stress equation and friction coefficient by the inverse analysis for lubricant A are summarized in Table 9.5. Three inverse analyses were conducted by varying the friction coefficient from 0.03 to 0.3. As initial guesses of material parameters K = 655( MPa ) and n = were used for every case. When the friction coefficient 0.15 was assumed, the prediction hole diameter by inverse analysis was mm and closest to the measured hole diameter of This is only about 0.3% difference. Thus, a combination of friction factor μ = 0.15 and flow stress K = 646σ ( MPa) gives the best minimum for the objective function, Equation (9.3). Figures 9.7 and 9.8 shows convergence behaviors of material parameters during the iterations for 143

169 minimizing the objective function. Both K-value and n-value decrease gradually as the number of iterations increases. Figures 9.9 and 9.10 shows the convergence behaviors of fractional norm and objective function during 20 optimization iterations. Case Friction Coef. Predicted hole diameter [mm] Predicted max. load [kn] K-value (MPa) n-value Experiment Measured hole diameter [mm] Measured max. load [kn] Table 9.5: Results of inverse analysis for different friction coefficients (Lubricant A) 0.18 Initial guess n=0.8 n-value Converges to n= Iteration Figure 9.7: Convergence of n-value (Lubrication A). 144

170 660 Initial guess K=655 K-value Converges to K= Iteration Figure 9.8: Convergence of K-value (Lubrication A). Fractional Norm Iteration Figure 9.9: Convergence of fractional norm (Lubrication A). 145

171 Objective Function 1.43E E E Iteration Figure 9.10: Convergence of objective function (Lubrication A). Table 9.6 summarizes the predicted inverse analysis results for lubricants A and B after 20 optimization iterations. The measured expanded hole diameters and corresponding inverse analysis predictions showed a good agreement for both lubricants. The identified friction coefficient for lubricant B was It is noteworthy that, while these two different lubricants gave two different friction coefficients, the determined K and n values are almost identical. This result is expected since sheet material was the same in both tests. Comparison of measured and predicted load-stroke curves for lubricants A and B are shown in Figure 9.11 and Figure As seen here, FEM predictions match very well with experimental measurements. Lubricant Friction Coef. Predicted hole diameter [mm] Predicted max. load [kn] K-value (MPa) n-value A B Table 9.6: Predicted inverse analysis results after 20 optimization iterations. 146

172 Lub A - Measured Lub A - Predicted Load(kN) Stroke(mm) Figure 9.11: Comparison of measured and predicted load-stroke curves for lubricants A Lub B - Measured Lub B - Predicted Load(kN) Stroke(mm) Figure 9.12: Comparison of measured and predicted load-stroke curves for lubricants B. 147

173 Validation To further evaluate the accuracy of the flow stress data obtained by inverse analysis a comparison was made with published information. Table 9.7 gives the material parameters K and n of the flow stress equation ( σ = Kε ), determined by inverse analysis, with corresponding values obtained in an earlier study [Gutscher, 2001]. This comparison indicates that the predictions made with the method, developed in the present study, are comparable to results obtained directly with bulge tests, conducted in an earlier investigation. n Parameter Inverse Analysis Reference [Gutscher, 2001] Difference (%) K (MPa) ~ ~ 12.1 n ~ ~ 32.2 Table 9.7: Comparison of predicted flow stress coefficients for AKDQ steel with data generated by Gutscher,

174 9.3. Inverse Analysis of Modified LDH Test at Elevated Temperatures In thermoforming, a polymeric composite material is heated to its softening point and formed against the contour of male and female molds. For successful numerical modeling of thermoforming process by FEM, an accurate determination of thermal and mechanical properties of the laminated composite material is very important. The laminated composite material used in thermoforming has a typical sandwich panel structure where the multi-layered skins are separated by lighter-weight core material, Figure It is manufactured by laminating the thermoformable polyurethane foam core with two fiber reinforced multi-layered. The overall property of the sandwich composite depends on the properties of each lamina (the skins and core foam) and its stacking sequence. Material properties change through the thickness of the sandwich panel and show a normal anisotropy. Randomly oriented discontinuous fiber Skin layer Core Void Figure 9.13: Schematic of sandwich panel structure 149

175 The core represents the largest part of entire composite and overall material property will strongly depend on that of the core material. When a foam material is used as the core material, because of the volume change and compressibility, the mechanical and thermal properties of the core depend on the relative density that is continuously changing during thermoforming. In this sense, a description of thermal and mechanical properties of the composite layers as a function of temperature and relative density is necessary. However, it is not easy to determine reliable material properties of separate layers from the tensile tests. Tensile test data of individual layer, recovered from testing tensile specimens in the oven, was not generally consistent in most cases. In this study, the deformation behavior of the composite at high temperature is assumed to have a homogeneous viscoplastic behavior. The completely laminated sandwich composite is considered as one isotropic material. In other words, the laminated composite is modeled as one homogeneous layer and material property of this model is assumed as average of all layers. As a reference material test, the modified LDH test at elevated temperature has been used and flow stress data of the composite material was determined by the inverse analysis technique developed in chapter 4. This practical approach only requires material tests of the completely laminated sandwich composite only. 150

176 Expeirment A composite specimen is clamped on its periphery to prevent material drawing into the cavity and then is stretched over a hemispherical punch under biaxial stress state at elevated temperatures. Testing at high temperatures was conducted with the use of a heating chamber that rested inside the testing area of the MTS machine. The prepared specimen was first preheated in the oven and then was placed in the modified LDH tooling mounted entirely inside the heating chamber. The upper arm is stationary and the punch mounted on the lower arm moves until it reaches the designated stroke position. In the test, loadstroke curve is measured and the change in diameter of hole was also measured at the end of punch stroke. In Figures 9.14 through 9.17, the recorded load-stroke curves for two different punch velocities (10mm/s and 80mm/s) at four different temperatures (160 o C, 170 o C, 180 o C, and 190 o C) are shown. The recorded load-stroke curves at same conditions always showed some deviations. This is because of a relaxation behavior of polymeric material at high temperature. Although the same preheating temperature was specified for all specimens, specimens exposed to high temperature for longer time had more stress relaxation. Table 9.8. The results show that the measured hole diameter decreased with increase in temperature. This indicates that the interface friction between the punch and the composite specimen increases with increase in temperature. The difference in hole diameter between 160 o C and 200 o C was about 1.8 mm. 151

177 Load(N) V = 80 mm/s V = 10 mm/s Stroke(mm ) Figure 9.14: Recorded load-stroke curves at temperature = 160oC Load(N) V = 80 mm/s V = 10 mm/s Stroke(mm ) Figure 9.15: Recorded load-stroke curves at temperature = 170oC. 152

178 Load(N) V = 80 mm/s V = 10 mm/s Stroke(m m) Figure 9.16: Recorded load-stroke curves at temperature = 180oC Load(N) V = 80 mm/s V = 10 mm/s Stroke(m m) Figure 9.17: Recorded load-stroke curves at temperature = 190oC. 153

179 Load (N) o C 170 o C V1 V2 Velocity increase 80 mm/s 10 mm/s 180 o C 190 o C 200 o C Temperature increase Figure 9.18: Comparison of measured load at punch stroke = 10 mm. Temperature ( o C) 1st / 2nd measurements Average mm / mm mm mm / mm mm mm / mm mm mm / mm mm mm / mm mm Table 9.8: Measured hole diameter at 15 mm punch stroke. 154

180 Inverse Analysis Conditions For the inverse analysis, a rate-dependent flow stress equation is considered to describe stress-strain diagram of the composite material at elevated temperatures as shown in Equation (9.4). σ & n m = Kε ε for a given temperature (9.4) The existing friction between the punch and the composite is modeled by using a constant shear friction model, per Equation (9.5). k τ f = m f k (9.5) Here, is the local material shear stress and is the friction factor which indicates the friction condition at the interface between the punch and the composite sheet. The objective function, as a function of material parameters, consists of two terms, representing the difference of forming load at two different forming speeds v1 and v2 in a least-square sense, Equation (9.6). m f p k 2 N N 1 F mea Fcom( pk ) 1 Fmea Fcom( pk ) E( p = + k ) (9.6) N i = 1 Fmea N i = 1 Fmea v1 2 v2 155

181 Inverse Analysis Results Using the load-stroke curves obtained from the modified LDH test at 190 o C, inverse analysis was first conducted to identify the unknown material parameters of the flow stress equation and friction factor. Table 9.9 shows the effect of friction factor on the identified material parameters (i.e., K, n, and m) at the temperature of 190 o C by inverse analyses. As seen here, as friction factor increases from 0.0 to 0.6, only the K and n values decrease. The predicted hole diameter was hardly influenced by the friction condition and none of the predicted hole diameter by FEM simulations with different friction factors could not match the measured hole diameter of mm, Table 9.9. This can be attributed to the simplifications in material characterization in present method; the consideration of the laminated compressible composite as a homogeneous single layer material and assuming its deformation behavior as viscoplastic in FEM simulation. Therefore, in the rest of inverse analyses, rather than identifying the friction factor, a constant interface friction factor of m f =0.2 was assumed. Friction factor K (Mpa) n m Predicted diameter (mm) Avg. measured diameter (mm) Table 9.9: Sensitivity of friction factor at the temperature of 190 o C. 156

182 The identified material parameters at various temperatures are shown in Table As temperature increases, the K and n values decrease while the m value increases. This means that the effect of strain-hardening on flow stress decreases while the effect of rate-sensitivity increases with increasing temperature (from 160 o C to 190 o C). Also, the decrease in hole diameter with increase in temperature (from 170 o C to 200 o C) indicates tribological condition at the sheet blank/punch interface is worse at 190 o C as observed in the test. Temperature ( o C) K (Mpa) n m Predicted hole diameter (mm) Table 9.10: Identified material parameters at various temperatures for friction factor, m f =0.2. The measured hole diameter and the computed hole diameter, using the determined flow stress data, are compared in Figure The hole diameters are compared at 15mm punch stroke position. As temperature increases, the hole diameter decreases both in experiment and FEM simulation. This means that at lower temperature the composite material is easier to be stretched than at higher temperature. Therefore, it is concluded that interface friction increases as temperature increases. When friction factor m f =0.2 was used, the predicted hole 157

183 diameter was underestimated by 4.1% and 4.7% at 160oC and 170 o C, respectively. They are overestimated by 0.7% at 180oC, 4.7% at 190oC and 3.4% at 200oC. Hole diameter (mm) Experiment FEM Prediction o C o C o C 1904 o C 2005 o C Temperature Figure 9.19: Comparison of hole diameter between experiment and FEM simulation at different temperatures (m f =0.2). The load-stroke curve computed using the determined flow stress data and the experimental load-stroke curve at the temperatures of 160 o C and 190 o C are plotted together in Figure 9.24 and Figure 9.25, respectively. The increase in maximum forming load from V 1 =10mm/s to V 2 =80mm/s at 160 o C is about 37% and it is increased to about 118% at 190 o C. This means that the effect of m-value is dominant at 190 o C. 158

184 Load(N) Temperature=160 o C EXP. V1=10mm/s FEM V1=10mm/s EXP. V2=80mm/s FEM V2=80mm/s Stroke(mm) Figure 9.20: Comparison between experimental (EXP.) and computed (FEM) load-stroke curves at two different punch speeds ( m = 0. 2 Temperature=160oC). f, Load(N) Temperature=190 o C EXP. V1=10mm/s FEM V1=10mm/s EXP. V2=80mm/s FEM V2=80mm/s Stroke(mm) Figure 9.21: Comparison between experimental (EXP.) and computed (FEM) load-stroke curves at two different punch speeds ( m = 0. 2, Temperature=190oC). f 159

185 9.4. Application of Inverse Analysis Results to Simulation of Thermoforming Introduction to Thermoforming Process DEFORM-3D v.4.0 that is capable of simulating complex threedimensional material flow with heat transfer analysis has been used to simulate thermoforming processes (i.e. preheating, forming and cooling) as shown in Figure Heat transfer analysis is an important part of the non-isothermal FEM simulation because the change in the temperature of the composite material affects the material flow. In heat transfer analysis, a cooling of the deforming material due to (a) convection and (b) radiation (on environment composite boundary), and (c) conduction (on mold composite boundary) was taken into account. Also, temperature increases due to internal heat generation of plastic deformation are considered. Figure 9.22: Various thermoforming processes [Kalpakjian, 1991]. 160

186 Material Properties Using the obtained flow stress equation ( σ = 7.19ε & ε refer to Table 9.10) at 160 o C, flow stress curves (stress-strain relations) are plotted as shown in Figure At 160 o C, the stress-strain curves were defined for three different strain rates ( & ε = 1.0, 10.0, and 100.0), and at six different strain values ( 0. 0 ε 3. 2). Thus, for a particular temperature, a total of 18 data points were used to describe rate-dependent stress-strain behavior of the composite material. When 18 discrete flow stress data points were assigned to DEFORM 3D, interpolating the existing data generates continuous curves at every instantaneous value of strain, strain-rate, and temperature. Stress (Mpa) Strain rate, m=1.0 Strain rate, m=10.0 Strain rate, m= Temperature = 160 o C Strain Figure 9.23: Rate-dependent flow stress curves at temperature of 160oC. 161

187 The density of the foam within the composite material has the major effect on the value of thermal conductivity, along with other variables such as foam size and geometry. In the temperature range between glass transition and melting, thermal conductivity decreases as temperature increases. The thermal conductivity of the typical composite material is a very low value when compared with that of base polymer. Closed-cell type foams have the lowest thermal conductivity of any conventional non-vacuum insulation. Interface heat transfer coefficient is an important factor that determines heat exchange rate between two objects during heat transfer analysis. This value is generally regarded as a function of (a) pressure, (b) temperature, and (c) density variations. The density of the foam has the major effect on the value of thermal conductivity, along with other variables such as foam size and geometry Simulation Results Figure 9.24 shows the deforming composite material with solid elements at various stroke positions. The forming simulation took approximately 29 hrs. CUP time on HP J6000 engineering workstation and could be completed without remeshing. The consumed CPU time for cooling simulation was about 2~3 hrs. As simulation results, temperature, strain stress distributions were predicted successfully. The material flow at four different stroke positions during a final mold closing stage was shown in Figure

188 (a) Stroke = mm (b) Stroke = mm (c) Stroke = mm Figure 9.24: Deformation at various strokes. 163

189 (a) Deformed geometry at the stroke = mm (b) Stroke = mm (c) Stroke = mm (d) Stroke = mm (e) Stroke = mm Figure 9.25: Material flow at final mold closing stages. 164

190 9.5. Discussion In this chapter, the Modified Limiting Dome Height (LDT) test (sheet blank with a hole at the center stretched with hemispherical punch) has been introduced to simultaneously determine the flow stress and interface friction from a biaxial stretch test for sheet materials at room temperature as well as at elevated temperatures. By measuring the change in hole diameter and the loadstroke curve during the experiment, it was possible to determine a) flow stress of rectangular cold rolled AKDQ sheet material and b) friction at the interface from a single test by inverse analysis technique. The results were compared with that of determined from other method, viscous pressure bulge test and indicated that with this method it was possible to predict the flow stress and friction with acceptable accuracy. Also, flow stress data of the proprietary composite material was successfully determined by conducting the modified LDH test at elevated temperatures. Then, using a full 3D FEM code, DEFORM 3D v4.0, the nonisothermal finite element modeling of forming process has been done successfully. The composite material was assumed to have a homogenized material property, determined by inverse analysis of the modified LDH test. The experience gained in this work indicates that the use of 3D FEM simulation technique to improve non-isothermal composite material forming process is promising. It is concluded that when the results of FEM simulation are utilized together with the results of experimental investigations, a more reliable process design procedure can be developed. 165

191 CHAPTER 10 INVERSE ANALYSIS OF HYDRAULIC SHEET BULGE TEST AT ROOM TEMPERATURE Introduction to Hydraulic Sheet Bulge Test The hydraulic bulge test is widely used to determine formability, incoming sheet quality, and flow stress curves of sheet materials. The stress state in hydraulic bulge test is equi-biaxial similar to stamping conditions. Consequently, the maximum achievable strain without localized necking is much larger than in tensile test. Therefore, the flow stress curve can be determined up to larger strains than in tensile test, which is important in using flow stress data for reliable FE analysis. At ERC/NSM, a tooling for hydraulic bulge test was built and used to characterize material properties (flow stress determination and formability evaluation) of sheet at room temperature. Figure 10.1 shows the sketch of the tooling used for hydraulic bulge test at ERC. As can be seen here, using the integrated position transducer in the tooling, the bulge (or dome) height could be continuously measured and pressure of viscous medium was measured using pressure transducer. Using the measured data, true stress-strain curve is determined from the data analysis method, developed at ERC/NSM, based on membrane theory and FEM simulation. This methodology in determining flow stress data from 166

192 hydraulic bulge test was proved to be a robust method. The details of this method are given in the ERC report [Gutscher, 2000]. Flow stress determination using this tooling consists of the following steps: Step 1: expand the sheet blank with internal pressure while the ends of the sheet are held firmly to avoid drawing Step 2: measure the hydraulic pressure and bulge (dome) height during expansion Step 3: convert these data into true stress-strain data using analytical model based on membrane theory Step 4: fit the data into known and widely used equation forms. In this chapter, the developed inverse analysis technique was applied to the hydraulic bulge test. The determined flow stress data obtained by inverse analysis technique was compared with that of determined by analytical technique developed by [Gutscher, 2000]. Figure 10.1: Sketch of tooling used in hydraulic bulge test. 167

193 10.2. Available Data Evaluation Technique for Flow Stress Determination Membrane Theory To determine flow stress curve by using the hydraulic bulge test, the membrane theory is most commonly used. The membrane theory neglects bending stresses and thus it is only applicable for thin sheets. In the membrane theory, when σ1 and σ 2 are the principle stresses on the sheet surface, R 1 and R 2 are the corresponding radii of the curved surface, p is the hydraulic pressure, and t is the sheet thickness, the Equation (11.1) is satisfied. σ 1 σ 2 + R R 1 2 = p t (11.1) For the axi-symmetric case of the hydraulic bulge test (Figure 10.2), equi-biaxial stress state is achieved during bulging (σ 1 =σ2) and the radius of the dome is R d = R = R. Therefore, Equation (11.1) can be simplified to: 1 2 2σ R = p d t d (11.2) pr d σ = (11.3) 2t d Pressure is applied on the internal sheet surface and no normal forces acting on the outer sheet surface. Therefore, the average stress in the sheet normal to the sheet surface is: p + 0 p σ n = = (11.4) 2 2 The effective stress can be calculated by Tresca s plastic flow criterion: 168

194 prd p σ = σ max σ min = (11.5) 2t 2 p R d σ = + 1 (11.6) 2 td The effective strain can be calculated for the axi-symmetric hydraulic bulge test using the sheet thickness: d t ε = ε = ln d t (11.7) t 0 To determine the flow stress curve with Equations (11.6) and (11.7), it is necessary to have the hydraulic pressure p, the instantaneous radius at the top of the dome R d, the instantaneous thickness at the top of the dome t d, and the initial thickness of the sheet t 0. The hydraulic pressure p is usually measured with a pressure transducer. However, measuring the instantaneous radius at the top of the dome R d and the instantaneous thickness at the top of the dome t d require a complicated special instrumentation Analytical Method Developed at ERC/NSM The working equations, Equations (11.6) and (11.7), derived from the membrane theory, require a) radius of dome curvature, b) wall thickness at the apex of the dome, and c) internal pressure. At ERC/NSM, a correlation between the radius and the thickness at the top of the dome, and the dome height [R d = f(hd), t d = f(h d )] was established by using FEM simulation technique. A series of FEM simulations of hydraulic bulge tests with various n-values were conducted 169

195 to generate the database, using PAMSTAMP, a dynamic explicit code for 3D simulation with shell elements. Figure 10.2 shows the diagram for these calculations. The radius and the thickness at the top of the dome are calculated using the initial sheet thickness, geometrical data of the tooling and the continuously measured dome height. However, the calculation of flow curve (a conversion of the measured pressure and bulge height curves to stress-strain curve) is based on the membrane theory, which holds only for thin sheets. Hydraulic bulge test Biaxial stress state True strain up to 0.7 t 0 : initial thickness of the sheet t d : thickness at the top of the dome h d : dome height R d : radius at the top of the dome d c : diameter of the cavity R c : radius of the fillet of the cavity F c : clamping force P : hydraulic pressure Measurement: Bulge height, h d Pressure, p Database ( h,n) r = d f d t f ( h,n) d = d Flow Stress ( σ ε ) p R d σ = td t ε = ln t d 0 Figure 10.2: Flow stress determination by analytical method [Gutscher, 2000]. 170

196 10.3. Experiments Palaniswamy et al [Palaniswamy, 2006] determined flow stress and anisotropy of sheet material AKDQ steel, DDS steel, DP600 steel and AL5754-O using the hydraulic bulge test. Kim [Kim, 2006] also conducted hydraulic bulge test for steel sheet material of thickness 2.16 mm (Figure 10.3) and determined flow stress curve using the analytical technique developed at ERC/NSM. Figure 10.4 shows the measured pressure versus bulge height curves obtained from hydraulic bulge test experiment for AL5754 (t=1.3 mm), DP 600 (t=0.6 mm), DDS (t=0.77 mm), AKDQ (t=0.83), and Steel (t=2.16 mm) sheet materials. As shown in Figure 10.4, the pressure required to bulge the 2.16 mm thickness steel sheet up to 27 mm was about 230 bar and about 120 bar was required for a high strength steel, DP600. In Figure 10.5 and Table 10.1, flow stress data for the sheet materials determined using an analytical method based on membrane theory and FEM simulation are shown [Palaniswamy, 2006]. As seen here, the DP600 show the highest strength but shows the minimum n-value of AL5754 shows the maximum n-value of Figure 10.3: Bulged steel sheet specimen (Thickness = 2.16 mm). 171