Mechanics of materials in modern manufacturing methods and processing techniques 9780128182321, 0128182326

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Mechanics of Advanced Materials Series The Mechanics of Advanced Materials book series focuses on materials- and mechanics-related issues around the behavior of advanced materials, including the mechanical characterization, mathematical modeling, and numerical simulations of material response to mechanical loads, various environmental factors (temperature changes, electromagnetic fields, etc.), as well as novel applications of advanced materials and structures. Volumes in the series cover advanced materials topics and numerical analysis of their behavior, bringing together knowledge of material behavior and the tools of mechanics that can be used to better understand, and predict materials behavior. It presents new trends in experimental, theoretical, and numerical results concerning advanced materials and provides regular reviews to aid readers in identifying the main trends in research in order to facilitate the adoption of these new and advanced materials in a broad range of applications. Series editor-in-chief: Vadim V. Silberschmidt Vadim V. Silberschmidt is Chair of Mechanics of Materials and Head of the Mechanics of Advanced Materials Research Group, Loughborough University, United Kingdom. He was appointed to the Chair of Mechanics of Materials at the Wolfson School of Mechanical and Manufacturing Engineering at Loughborough University, United Kingdom in 2000. Prior to this, he was a Senior Researcher at the Institute A for Mechanics at Technische Universit¨at Mu¨nchen in Germany. Educated in the USSR, he worked at the Institute of Continuous Media Mechanics and Institute for Geosciences [both—the USSR (later—Russian) Academy of Sciences]. In 1993
94, he worked as a visiting researcher, Fellow of the Alexander-von-Humboldt Foundation at Institute for Structure Mechanics DLR (German Aerospace Association), Braunschweig, Germany. In 2011
14, he was Associate Dean (Research). He is a Charted Engineer, Fellow of the Institution of Mechanical Engineers and Institute of Physics, where he also chaired Applied Mechanics Group in 2008
11. He serves as Editor-in-Chief (EiC) of the Elsevier book series on Mechanics of Advanced Materials. He is also EiC, associate editor, and/or serves on the board of a number of renowned journals. He has coauthored four research monographs and over 550 peer-reviewed scientific papers on mechanics and micromechanics of deformation, damage, and fracture in advanced materials under various conditions. Series editor: Thomas Bo¨hlke Thomas Bo¨hlke is Professor and Chair of Continuum Mechanics at the Karlsruhe Institute of Technology (KIT), Germany. He previously held professorial positions at the University of Kassel and at the Otto-von-Guericke University, Magdeburg, Germany. His research interests include FE-based multiscale methods, homogenization of elastic, brittle-elastic, and visco-plastic material properties, mathematical description of microstructures, and localization and failure mechanisms. He has authored over 130 peer-reviewed papers and has authored or coauthored two monographs. Series editor: David L. McDowell David L. McDowell is Regents’ Professor and Carter N. Paden, Jr. Distinguished Chair in Metals Processing at Georgia Tech University, United States. He joined Georgia Tech in 1983 and holds a dual appointment in the GWW School of Mechanical Engineering and the School of Materials Science and Engineering. He served as the Director of the Mechanical Properties Research Laboratory from 1992 to 2012. In 2012 he was named Founding Director of the Institute for Materials (IMat), one of Georgia Tech’s Interdisciplinary Research Institutes charged with fostering an innovation ecosystem for research and education. He has served as Executive Director of IMat since 2013. His research focuses on nonlinear constitutive models for engineering materials, including cellular metallic materials, nonlinear and time-dependent fracture mechanics, finite strain inelasticity and defect field mechanics, distributed damage evolution, constitutive relations, and microstructure-sensitive computational approaches to deformation and damage of heterogeneous alloys, combined computational and experimental strategies for modeling high cycle fatigue in advanced engineering alloys, atomistic simulations of dislocation nucleation and mediation at grain boundaries, multiscale computational mechanics of materials ranging from atomistics to continuum, and system-based computational materials design. A Fellow of SES, ASM International, ASME, and AAM, he is the recipient of the 1997 ASME Materials Division Nadai Award for career achievement and the 2008 Khan International Medal for lifelong contributions to the field of metal plasticity. He currently serves on the editorial boards of several journals and is coeditor of the International Journal of Fatigue. Series editor: Zhong Chen Zhong Chen is a Professor in the School of Materials Science and Engineering, Nanyang Technological University, Singapore. In March 2000, he joined Nanyang Technological University (NTU), Singapore as an Assistant Professor and has since been promoted to Associate Professor and Professor in the School of Materials Science and Engineering. Since joining NTU, he has graduated 30 PhD students and 5 MEng students. He has also supervised over 200 undergraduate research projects (FYP, URECA, etc.). His research interest includes (1) coatings and engineered nanostructures for clean energy, environmental, microelectronic, and other functional surface applications and (2) mechanical behavior of materials, encompassing mechanics and fracture mechanics of bulk, composite and thin film materials, materials joining, and experimental and computational mechanics of materials. He has served as an editor/ editorial board member for eight academic journals. He has also served as a reviewer for more than 70 journals and a number of research funding agencies including the European Research Council (ERC). He is an author of over 300 peer-reviewed journal papers.

Elsevier Series in Mechanics of Advanced Materials

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques Edited by

Vadim V. Silberschmidt Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, United Kingdom

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2020 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-818232-1 For Information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisitions Editor: Dennis McGonagle Editorial Project Manager: Charlotte Rowley Production Project Manager: Debasish Ghosh Cover Designer: Matthew Limbert Typeset by MPS Limited, Chennai, India

Contents

List of contributors About the Series editors 1

2

Modeling of metal forming: a review Uday Shanker Dixit 1.1 Introduction 1.2 Modeling issues in various metal forming processes 1.2.1 Forging 1.2.2 Rolling 1.2.3 Wire drawing 1.2.4 Extrusion 1.2.5 Deep drawing 1.2.6 Bending 1.3 Various modeling techniques 1.3.1 Slab method 1.3.2 Slip-line field method 1.3.3 Visioplasticity 1.3.4 Upper bound method 1.3.5 Finite difference method 1.3.6 Finite element method 1.3.7 Meshless method 1.3.8 Molecular dynamics simulation 1.3.9 Soft computing 1.4 Inverse modeling 1.5 Modeling of microstructure and surface integrity 1.6 A note on multiscale modeling of metal forming 1.7 Challenging issues 1.8 Conclusion References Finite element method modeling of hydraulic and thermal autofrettage processes Uday Shanker Dixit and Rajkumar Shufen 2.1 Introduction 2.1.1 Hydraulic autofrettage 2.1.2 Swage autofrettage 2.1.3 Explosive autofrettage

xiii xvii 1 1 2 2 4 6 7 10 11 13 13 13 14 14 15 15 17 18 18 19 20 22 23 24 25

31 31 32 33 34

vi

3

4

Contents

2.1.4 Thermal autofrettage 2.1.5 Rotational autofrettage 2.2 Numerical modeling of elastic
plastic problems 2.2.1 Yield criteria and hardening behavior of the material 2.2.2 Approaches for numerical modeling of elastic
plastic problems 2.3 FEM formulation using updated Lagrangian method 2.3.1 Derivation of the weak form of the equilibrium equation 2.3.2 Formulation of elemental equations 2.3.3 Solution method 2.4 Typical results of FEM modeling of hydraulic and thermal autofrettage 2.4.1 Results of hydraulic autofrettage 2.4.2 Results of thermal autofrettage 2.5 Conclusion References

35 35 37 38

Mechanics of hydroforming Christoph Hartl 3.1 Introduction 3.2 Modeling of plastic deformation in tube hydroforming 3.2.1 Rotationally symmetrical tube expansion 3.2.2 Hydroforming of polygonal cross sections 3.2.3 Hydroforming of tube branches 3.3 Determination of forming limits in tube hydroforming 3.3.1 Necking and bursting 3.3.2 Wrinkling and buckling 3.4 Design of loading paths 3.5 Conclusion References

71

Electromagnetic pulse forming Verena Psyk, Maik Linnemann and Gerd Sebastiani 4.1 Process classification 4.2 Process principle and major process variants 4.2.1 General setup and process principle 4.2.2 Major process variants 4.3 Calculation of the process mechanics 4.3.1 Analytical calculation of the acting loads 4.3.2 Numerical calculation of the process 4.4 Advantages and application fields of electromagnetic pulse forming 4.4.1 Shaping 4.4.2 Joining 4.4.3 Cutting

42 44 44 46 50 52 53 58 66 67

71 74 74 81 85 89 89 95 98 101 102 111 111 112 112 113 123 123 126 131 132 136 138

Contents

5

6

vii

4.5 Prospects for future developments References

139 139

Damage in advanced processing technologies Zhutao Shao, Jun Jiang and Jianguo Lin 5.1 Introduction 5.1.1 Concepts of damage and damage variables 5.1.2 Damage mechanisms 5.1.3 Advanced manufacturing technology: hot stamping 5.1.4 Concept and features of forming limit diagram 5.2 Overview of formability evaluation 5.2.1 Forming limit prediction 5.2.2 Experimental methods for determining forming limits 5.2.3 Requirements for hot stamping applications 5.2.4 Advanced testing system for hot stamping applications 5.3 Modeling of damage evolution 5.3.1 Constitutive equations 5.3.2 Advanced damage models 5.3.3 A set of unified constitutive equations for hot stamping 5.3.4 Modeling of forming limit diagrams 5.4 Damage calibration techniques 5.4.1 Overview of damage calibration techniques 5.4.2 An example of using thermomechanical uniaxial test data 5.4.3 Examples of using thermomechanical multiaxial tensile test data 5.5 Applications of damage modeling technique for hot stamping 5.5.1 Plane stress
based continuum damage mechanics material model 5.5.2 Principal strain
based continuum damage mechanics material model 5.5.3 Prediction of formability in hot forming 5.6 Conclusion References

143

Numerical modeling of the mechanics of pultrusion Michael Sandberg, Onur Yuksel, Raphae¨l Benjamin Comminal, Mads Rostgaard Sonne, Masoud Jabbari, Martin Larsen, Filip Bo Salling, Ismet Baran, Jon Spangenberg and Jesper H. Hattel 6.1 Introduction 6.1.1 Pultrusion 6.1.2 Overview and motivation of the chapter 6.2 Resin impregnation 6.2.1 Saturated pressure-driven flow 6.2.2 Resin viscosity 6.2.3 Permeability of fiber reinforcements 6.2.4 Unsaturated impregnation flow

143 143 143 144 145 146 146 147 149 150 151 151 153 155 156 158 158 159 160 162 162 165 170 171 171 173

173 173 175 175 176 177 177 178

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Contents

6.3

7

8

Thermochemical modeling 6.3.1 Heat transfer 6.3.2 Cure kinetics and differential scanning calorimetry 6.3.3 Modeling considerations: simple models and state-of-the-art 6.4 Thermochemical
mechanical modeling and residual stress formation 6.4.1 The evolution of material properties 6.4.2 Mechanical modeling strategies 6.4.3 Assessment of the resultant residual stress fields and the verification 6.5 Pulling force 6.5.1 Before die-entrance, A0 6.5.2 Die-entrance to flow front location, A1 6.5.3 Through liquid and gel states, A2 6.5.4 Solid state and detachment from die wall, A3 6.6 Conclusion References

180 180 181 182

Modeling of machining processes Ju¨rgen Leopold Nomenclature 7.1 Introduction 7.2 Closed-loop principle of modeling 7.3 Modeling and simulation techniques 7.3.1 Slip-line method 7.3.2 Finite element modeling (finite element method) 7.3.3 Complementary methods 7.4 Modeling and simulation in the industry—selected examples 7.4.1 Cutting tool optimization 7.4.2 High-speed cutting or high-performance cutting 7.4.3 Dry machining 7.4.4 Burr formation and clean manufacturing 7.4.5 Cryogenic machining 7.5 Open issues 7.5.1 Hybrid modeling and closed-loop design 7.5.2 Multiscale modeling in machining 7.5.3 Multiscale modeling in coating-substrate simulation 7.6 Summary References Further reading

197

Mechanics of ultrasonically assisted drilling Anish Roy and Vadim V. Silberschmidt 8.1 Introduction 8.2 Drilling: theory and modeling

183 183 184 186 189 189 189 190 191 191 192

197 197 198 199 199 203 209 213 213 213 213 213 215 216 216 218 220 222 222 226 229 229 230

Contents

9

10

ix

8.2.1 Kinematic modeling 8.2.2 Finite-element modeling 8.3 Ultrasonically assisted drilling 8.3.1 Experimental setup and instrumentation 8.3.2 Case study: drilling in composites 8.4 Conclusion and outlook References

230 232 235 235 237 240 241

Machining in monocrystals Anish Roy, Qiang Liu, Ka Ho Pang and Vadim V. Silberschmidt 9.1 Introduction 9.2 Mechanics of single-crystal machining 9.2.1 Single-crystal-plasticity theory 9.2.2 Computational implementation 9.2.3 Criteria of material-removal modeling 9.3 Machining of single-crystal metal 9.3.1 Experimental procedure 9.3.2 Finite-element model and material parameters 9.3.3 Simulation and results 9.3.4 Discussion 9.4 Machining of single-crystal ceramic material 9.4.1 Experimental procedure 9.4.2 Computational modeling 9.4.3 Results and discussion 9.5 Concluding remarks References

243

Microstructural changes in machining W. Bai, R. Sun, J. Xu and Vadim V. Silberschmidt 10.1 Introduction 10.2 Microstructural evolution in machining 10.2.1 Microstructural evolution in machined surface 10.2.2 Microstructural evolution in chip 10.3 Microstructural models for machining 10.3.1 Mechanism models of microstructural evolution 10.3.2 Calculation of microstructural evolution 10.4 Microstructural evolution in ultrasonically assisted cutting 10.4.1 Microstructural evolution in machined surface with ultrasonically assisted cutting 10.4.2 Microstructural evolution in chip with ultrasonically assisted cutting 10.5 Conclusion Acknowledgments References

243 245 245 247 248 248 248 249 250 254 258 258 259 262 264 265 269 269 270 270 273 275 275 278 279 283 288 294 294 294

x

11

12

Contents

Residual stresses in machining J.C. Outeiro 11.1 Introduction 11.2 Fundamentals of machining and residual stresses 11.2.1 Metal-cutting definition and energy considerations 11.2.2 Definition and origins of residual stresses 11.2.3 Techniques for measuring residual stress 11.3 Residual stresses in machining operations 11.3.1 Origin of residual stresses in metal cutting 11.3.2 Residual stresses in difficult-to-cut materials 11.3.3 Effect of relative tool sharpness on residual stresses 11.3.4 Control of residual stresses in machining 11.4 Modeling and simulation of residual stresses 11.4.1 Modeling and simulation considerations 11.4.2 Relevance of constitutive and contact models in residual- stress prediction 11.4.3 Simulation of residual stresses for several work materials 11.4.4 Procedure for comparing predicted and measured residual stresses 11.4.5 Optimization of cutting conditions for improved residual stresses and surface roughness in machined components 11.5 Influence of residual stress on product sustainability 11.5.1 Introduction 11.5.2 Corrosion resistance 11.5.3 Fatigue strength 11.6 Conclusion References Microstructural changes in materials under shock and high strain rate processes: recent updates Satyam Suwas, Anuj Bisht and Gopalan Jagadeesh 12.1 Introduction 12.2 Shock wave and parameters 12.3 Experimental methods for investigation of shock waves 12.3.1 Taylor’s impact test 12.3.2 Explosive loading of materials 12.3.3 Flyer plate impact test 12.3.4 Split-Hopkinson pressure bar 12.3.5 Shock impact in a shock tube 12.3.6 Laser-induced shock generation 12.4 Parameters influencing material response to shock exposure 12.4.1 Types of shock-generated defects 12.4.2 Effect of material parameters

297 297 298 298 301 301 303 303 305 319 322 324 324 326 335 341

344 346 346 348 352 353 354

361 361 363 365 365 366 366 367 367 368 369 369 371

Contents

13

14

xi

12.4.3 Effect of shock parameters 12.4.4 Other factors: residual strain 12.5 Theory of defect generation under shock: past theories and new perspectives 12.6 Conclusion References

376 378

Thermomechanics of friction stir welding Mads Rostgaard Sonne and Jesper H. Hattel 13.1 Introduction 13.2 Thermal behavior 13.3 Microstructural evolution 13.4 Residual stresses and distortions 13.5 Material flow 13.6 Conclusion References

393

Modeling of friction in manufacturing processes Uday Shanker Dixit, V. Yadav, P.M. Pandey, Anish Roy and Vadim V. Silberschmidt 14.1 Introduction 14.2 History of friction modeling 14.3 Some popular friction models 14.3.1 Amontons
Coulomb’s model 14.3.2 Constant-friction model 14.3.3 Wanheim and Bay’s model 14.3.4 Asperity-based friction model 14.3.5 Plowing model 14.4 Friction in machining 14.5 Friction models in metal forming 14.6 Friction in solid-state welding 14.7 Friction models for micromanufacturing 14.8 Challenging issues and directions for future research 14.9 Conclusion Acknowledgment References Further reading

415

Index

380 384 384

393 396 399 400 403 410 410

415 416 418 419 419 420 421 434 435 437 438 439 439 440 441 441 444 445

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List of contributors

W. Bai Huazhong University of Science and Technology, Wuhan, P.R. China Ismet Baran Faculty of Engineering Technology, University of Twente, Enschede, The Netherlands Anuj Bisht Department of Materials Engineering, Indian Institute of Science, Bangalore, India Raphae¨l Benjamin Comminal Department of Mechanical Engineering, Section of Manufacturing Engineering, Technical University of Denmark, Lyngby, Denmark Uday Shanker Dixit Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India Christoph Hartl TH Ko¨ln - Faculty of Automotive Systems and Production, University of Applied Sciences, Cologne, Germany Jesper H. Hattel Department of Mechanical Engineering, Section of Manufacturing Engineering, Technical University of Denmark, Lyngby, Denmark Masoud Jabbari School of Mechanical, Aerospace & Civil Engineering, The University of Manchester, Manchester, United Kingdom Gopalan Jagadeesh Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India Jun Jiang Department of Mechanical Engineering, Imperial College London, London, United Kingdom Martin Larsen Fiberline Composites A/S, Middelfart, Denmark Ju¨rgen Leopold TBZ-PARIV GmbH, Chemnitz, Germany Jianguo Lin Department of Mechanical Engineering, Imperial College London, London, United Kingdom

xiv

List of contributors

Maik Linnemann Fraunhofer Institute for Machine Tools and Forming Technology, Chemnitz, Germany Qiang Liu Department of Materials Engineering, KU Leuven, Leuven, Belgium J.C. Outeiro Arts & Metiers Institute of Technology, Campus of Cluny, Cluny, France P.M. Pandey Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi, India Ka Ho Pang Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, United Kingdom Verena Psyk Fraunhofer Institute for Machine Tools and Forming Technology, Chemnitz, Germany Anish Roy Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, United Kingdom Filip Bo Salling Department of Mechanical Engineering, Section of Manufacturing Engineering, Technical University of Denmark, Lyngby, Denmark Michael Sandberg Department of Mechanical Engineering, Section of Manufacturing Engineering, Technical University of Denmark, Lyngby, Denmark Gerd Sebastiani imk automotive GmbH, Chemnitz, Germany Zhutao Shao Department of Mechanical Engineering, Imperial College London, London, United Kingdom Rajkumar Shufen Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India Vadim V. Silberschmidt Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, United Kingdom Mads Rostgaard Sonne Department of Mechanical Engineering, Section of Manufacturing Engineering, Technical University of Denmark, Lyngby, Denmark Jon Spangenberg Department of Mechanical Engineering, Section of Manufacturing Engineering, Technical University of Denmark, Lyngby, Denmark R. Sun Huazhong University of Science and Technology, Wuhan, P.R. China

List of contributors

xv

Satyam Suwas Department of Materials Engineering, Indian Institute of Science, Bangalore, India J. Xu Huazhong University of Science and Technology, Wuhan, P.R. China V. Yadav Department of Mechanical Engineering, Maulana Azad National Institute of Technology Bhopal, Bhopal, India Onur Yuksel Faculty of Engineering Technology, University of Twente, Enschede, The Netherlands

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About the Series editors

Editor-in-Chief Vadim V. Silberschmidt is Chair of Mechanics of Materials and Head of the Mechanics of Advanced Materials Research Group, Loughborough University, United Kingdom. He was appointed to the Chair of Mechanics of Materials at the Wolfson School of Mechanical and Manufacturing Engineering at Loughborough University, United Kingdom in 2000. Prior to this, he was a Senior Researcher at the Institute A for Mechanics at Technische Universit¨at Mu¨nchen in Germany. Educated in the USSR, he worked at the Institute of Continuous Media Mechanics and Institute for Geosciences [both—the USSR (later—Russian) Academy of Sciences]. In 1993
94, he worked as a visiting researcher, Fellow of the Alexander-von-Humboldt Foundation at Institute for Structure Mechanics DLR (German Aerospace Association), Braunschweig, Germany. In 2011
14, he was Associate Dean (Research). He is a Charted Engineer, Fellow of the Institution of Mechanical Engineers and Institute of Physics, where he also chaired Applied Mechanics Group in 2008
11. He serves as Editor-in-Chief (EiC) of the Elsevier book series on Mechanics of Advanced Materials. He is also EiC, associate editor, and/or serves on the board of a number of renowned journals. He has coauthored four research monographs and over 550 peer-reviewed scientific papers on mechanics and micromechanics of deformation, damage, and fracture in advanced materials under various conditions. Series editors David L. McDowell is Regents’ Professor and Carter N. Paden, Jr. Distinguished Chair in Metals Processing at Georgia Tech University, United States. He joined Georgia Tech in 1983 and holds a dual appointment in the GWW School of Mechanical Engineering and the School of Materials Science and Engineering. He served as the Director of the Mechanical Properties Research Laboratory from 1992 to 2012. In 2012 he was named Founding Director of the Institute for Materials (IMat), one of Georgia Tech’s Interdisciplinary Research Institutes charged with fostering an innovation ecosystem for research and education. He has served as Executive Director of IMat since 2013. His research focuses on nonlinear constitutive models for engineering materials, including cellular metallic materials, nonlinear and time-dependent fracture mechanics, finite strain inelasticity and defect field mechanics, distributed damage evolution, constitutive relations, and microstructuresensitive computational approaches to deformation and damage of heterogeneous alloys, combined computational and experimental strategies for modeling high

xviii

About the Series editors

cycle fatigue in advanced engineering alloys, atomistic simulations of dislocation nucleation and mediation at grain boundaries, multiscale computational mechanics of materials ranging from atomistics to continuum, and system-based computational materials design. A Fellow of SES, ASM International, ASME, and AAM, he is the recipient of the 1997 ASME Materials Division Nadai Award for career achievement and the 2008 Khan International Medal for lifelong contributions to the field of metal plasticity. He currently serves on the editorial boards of several journals and is coeditor of the International Journal of Fatigue. Thomas Bo¨hlke is Professor and Chair of Continuum Mechanics at the Karlsruhe Institute of Technology (KIT), Germany. He previously held professorial positions at the University of Kassel and at the Otto-von-Guericke University, Magdeburg, Germany. His research interests include FE-based multiscale methods, homogenization of elastic, brittle-elastic, and visco-plastic material properties, mathematical description of microstructures, and localization and failure mechanisms. He has authored over 130 peer-reviewed papers and has authored or coauthored two monographs. Zhong Chen is a Professor in the School of Materials Science and Engineering, Nanyang Technological University, Singapore. In March 2000, he joined Nanyang Technological University (NTU), Singapore as an Assistant Professor and has since been promoted to Associate Professor and Professor in the School of Materials Science and Engineering. Since joining NTU, he has graduated 30 PhD students and 5 MEng students. He has also supervised over 200 undergraduate research projects (FYP, URECA, etc.). His research interest includes (1) coatings and engineered nanostructures for clean energy, environmental, microelectronic, and other functional surface applications and (2) mechanical behavior of materials, encompassing mechanics and fracture mechanics of bulk, composite and thin film materials, materials joining, and experimental and computational mechanics of materials. He has served as an editor/editorial board member for eight academic journals. He has also served as a reviewer for more than 70 journals and a number of research funding agencies including the European Research Council (ERC). He is an author of over 300 peer-reviewed journal papers.

Modeling of metal forming: a review

1

Uday Shanker Dixit Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India

1.1

Introduction

Manufacturing of a product by plastic deformation of metals has been performed for ages. Plastic deformation can be accomplished with or without heating the metal. Accordingly, two broad categories of metal forming processes are hot metalworking and cold metalworking. A more refined classification includes warm metalworking in between hot and cold metalworking. In most of the metal forming processes, deformation is carried out by the application of a mechanical load; the role of heat is limited to reducing the flow stress (required to start the plastic deformation of the metal). However, there are processes, where the plastic deformation is achieved by the heat alone; an example is laser bending, where a sheet is bent by laser irradiation that creates a sufficient amount of thermal stresses to bend the sheet [1]. Based on raw material, desired final product and metal flow pattern, the metal forming processes are classified into bulk and sheet metal forming processes. Bulk metal forming processes deform high volume-to-surface area ratio raw materials resulting in a change of surface area. In the sheet metal forming processes, the raw material has a low volume-to-surface area ratio, and material deformation does not intend to change the surface area implying that the sheet thickness remains more or less unaltered. Recently, another category, namely, sheet-bulk metal forming has been introduced [2]. In sheet-bulk metal forming processes, the bulk deformation of sheet is carried out that invariably brings out the intended changes in the thickness as well. Some examples of bulk metal forming are forging, rolling, extrusion, and wire drawing. Sheet metal forming processes include deep drawing, bending, and spinning [3]. Coining, flow forming, and ironing are examples of sheet-bulk metal forming. Modeling of metal forming started since the beginning of the 20th century [4]. Initial attempts were directed to estimate the load required for plastic deformation. Prominent methods were slip-line field method, slab method, and upper bound method. These methods involved several assumptions and were incapable of providing detailed information about stress
strain distribution in the material. The technique of visioplasticity was introduced in the late 1950s to get detailed information about the deformation. In a visioplasticity method a time-dependent velocity field is Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques. DOI: https://doi.org/10.1016/B978-0-12-818232-1.00001-1 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

calculated based on a series of photographs of a grid pattern on the metal being processed. Analytical and/or numerical techniques are used to estimate the stress
strain distribution and other derivative information. The decade 1950
60 saw the emergence of finite element method (FEM). A number of articles were published in the 1970s and 1980s on the modeling of metal forming processes. These methods were perfected in the 1990s, and nowadays, there are several commercial FEM packages dedicated to modeling of metal forming processes. Since the last two decades, there have been attempts to develop advanced FEM techniques and several meshless methods. However, the requirement of huge computational time is a hindrance in the popularization of these methods. Some developments have taken place in carrying out microstructural modeling of metal forming processes. Classical plasticity theory could account for distinct material behavior based on microstructural features. Molecular dynamics simulations (MDSs) and crystal plasticity are two recently developed methods for realistic simulation of metal forming processes, particularly for modeling of microforming. However, these methods are still at a nascent stage of development for solving metal forming problems, and there are several computational difficulties to be overcome. This chapter provides an overview of modeling of well-known metal forming processes. Capabilities and limitations of various modeling techniques are highlighted. Finally, the directions for further research are provided.

1.2

Modeling issues in various metal forming processes

Why do we model metal forming processes? What is expected out of a model? In general, modeling is expected to improve the efficiency of overall manufacturing system and reduce the dependence on costly hit and trial experiments. Modeling can provide the following valuable information: (1) required deformation load; (2) energy consumption in the process; (3) stresses on the dies and tools; (4) defects in the process; (5) quality of the product, particularly in terms of dimensional accuracy and surface integrity; (6) properties of the product; (7) stress, strain, strain rate, and temperature distribution in the product as well as tooling; and (8) life assessment of the tooling and machine. Adequate modeling can help in the design and optimization of metal forming process, machines, and tooling. In the sequel, salient modeling issues of popular metal forming processes are discussed.

1.2.1 Forging Forging is a process of plastically deforming the metal by pressing and hammering. It may be performed in cold, warm, or hot state of the metal. There are mainly two types of forging processes: (1) open die or free forging and (2) closed die or impression die forging. Open die forging is the process of deforming the metal between multiple dies that do not completely enclose the material. It is often

Modeling of metal forming: a review

3

employed to preform material for subsequent metal forming. There are several different types of open die forging processes. Fig. 1.1 shows schematic diagrams of three types of open die forging—upset forging or upsetting, cogging, and orbital or rotary forging. In upsetting the complete or partial portion of the workpiece is compressed between a fixed die and moving ram in order to increase the cross section of the desired portion. In Fig. 1.1A, a rod is being upset forged to make a head. In Fig. 1.1B, cogging process is being carried out in order to make a stepped bar. It is basically an incremental forming; a portion of the workpiece is compressed between the dies, dies retract, and the workpiece is advanced axially for next compression operation. Fig. 1.1C depicts upsetting of a workpiece by orbital forging. Here the lower die is fixed. The upper die rotates about an axis slightly inclined to workpiece-axis; hence, at a time it compresses only a small portion of the workpiece. This is also an incremental forming and load requirement gets reduced. In the closed die forging or impression die forging, metal is compressed in the enclosed dies. In this process the metal is fully compacted to acquire the shape governed by the die cavities. Excess material comes out as flash and is trimmed. Die Force

Ram

Workpiece Gripping die

Workpiece

Force

(A)

(B)

2°

Upper die

Workpiece

Lower die

(C)

Figure 1.1 Open die forging: (A) upsetting, (B) cogging, (C) orbital forging.

4

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Force Moving die

Flash Workpiece

Fixed die

Figure 1.2 Closed die forging.

Usually, the mass of flash can be as much as 20%. Fig. 1.2 shows a schematic of closed die forging with flash generation. Flash-less forging is also possible but requires a careful tooling design. In forging process, strain rate may lie in the range 1023
1022 s21, depending on whether it is press forging or hammer forging. In the cold forging the effective Coulomb’s coefficient of friction ranges from 0.05 to 0.15 and 0.1 to 0.5 in hot forging. In most of the cases, Coulomb’s friction model is inappropriate. Frictional behavior may change from sticking to sliding while moving outwardly from center in a direction normal to the load. Estimation of forging load has been the focus of attention in forging. Another interest is to find outflow patterns in order to design dies. Forging process suffers from various defects that need to be controlled. Surface cracking may occur due to thermomechanical effects. Poor material flow may cause folding or overlapping of one region of metal onto another causing cold shut. It may also result in underfilling of die cavities. Surface may get roughened due to the deformation of coarse grains, which is called orange peel defect. Although forging load estimation techniques are sufficiently refined, prediction of defects is still a challenging task. The simplest open die forging, namely upsetting is often used as a benchmark test for studying the material and friction behavior. In a typical compression test the specimen is compressed between two lubricated platens to find out the deformation behavior of metals. In a ring compression test a hollow cylinder specimen is compressed between two plates and friction is estimated based on the change in the hole diameter. In the cylindrical specimen of a ring compression test, typically the hole diameter is half of the outer diameter and height is one-third the outer diameter.

1.2.2 Rolling Rolling shapes materials by passing it between counterrotating rolls. It has been in wide use since the 14th century. In this process, slabs, billets, blooms, or rods are

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Back tension Front tension

Strip Work-roll Backup-roll

Figure 1.3 A tandem rolling mill.

rolled into plates, sheets, strips, rods, and tubes. The profiles can also be produced by rolling. Because of the limitation of the maximum possible reduction in one pass, usually multipass rolls with a number of stands in series are employed. This is called tandem rolling and is schematically depicted in Fig. 1.3. It has three stands; in each stand, there are two work-rolls and two backup rolls to prevent work-roll deflection. Rolling can be employed with or without front and back tensions. Rolling process looks deceptively simple. However, a realistic simulation needs to focus on the following three complex tasks: 1. proper elastic
plastic modeling of the material to be processed, 2. modeling of friction behavior with a proper assessment of neutral zone in which the direction of frictional stresses changes from facilitating the movement of the material to opposing it, and 3. application of elasticity theory for estimating the roll flattening and roll deflection.

A proper model of rolling process estimates the roll torque, roll separating force, and roll pressure distribution accurately. The common defects in rolling process are edge cracking, wavy edge, central burst, and alligatoring. Edge cracking refers to cracking at the edges of the rolled products and occurs because of nonhomogeneous deformation due to wrong design of rolls or improper management of friction. Wavy edge occurs mainly because of roll deflection. Due to nonuniform roll gap, edges tend to elongate more than the center. To maintain continuity, edges get compressed and produce a wavy pattern. Central burst is a ductile fracture that initiates from a void at the center. In alligatoring a crack forms along the central plane and splits the ends. Excessive front or back tension may cause the tearing of the sheet. Many times, rolling process is employed for improving the material properties. Asymmetric sheet rolling, in which the surface speed of rolls or friction differs on the two sides of the sheet, has been used to improve the microstructure [5,6]. In temper or skin-pass rolling, 0.5%
4% reduction in the sheet thickness is carried out to provide a degree of hardening to sheet, to prevent stretcher strains or Lu¨ders band, and to improve the surface integrity of the sheet [7]. Accumulative roll bonding is a severe deformation process [8]. In this process a sheet is passed between two counterrotating cylinders to impart 50% reduction to it. Elongated sheet is cut into two pieces of equal length and stacked together to make it of same dimension

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

as the original sheet. It is further passed through the roll to impart 50% reduction. The procedure is repeated several times, which results in properly bonded thin stacked sheets with a large accumulated strain. With this procedure, grain size gets reformed, and strength gets improved.

1.2.3 Wire drawing Wire drawing process pulls a wire through a die to reduce its cross section. It can be carried out in the presence of a back tension that helps in reducing the die pressure and providing dimensional stability. Theoretically maximum possible reduction in one pass is 63%, but a practical limit is 45%. Hence, for getting higher reduction, multistage wire drawing is employed. Fig. 1.4 depicts a schematic of two-stage wire drawing process to reduce the cross-sectional area of wire from Ai to Af. Wire drawing is usually carried out at room temperature, but occasionally warm wire drawing is also performed. Rod drawing is similar to wire drawing. Here instead of a wire (diameter less than about 6 mm), a rod is pulled through dies. In case of tube drawing, a tube is drawn over a mandrel. In tube, sinking no mandrel is used and outer diameter reduced with increase in length; tube thickness may reduce or increase depending on the process parameters. Die design is very important in wire or rod drawing. Most of the researchers have optimized the die shape with an objective of minimizing energy. However, the shape of die has a large bearing on the quality of the product. Certain die shapes, although may increase the required power, reduce the defects and improve the mechanical properties such as tensile strength and hardness of the drawn wire/rod. There are several types of defects in a wire/rod drawing process. Some defects such as scabs (irregularly shaped flattened protrusions) occur due to defective raw material. Some defects occur due to faulty design of die and process. For example, a combination of large die angle and small reduction results in the narrowing of plastic zone in the vicinity of centerline and enhances the chance of central burst [9]. Friction is undesirable in wire drawing and usually the Coulomb’s coefficient of friction ranges from 0.01 to 0.1 with proper lubrication. In the dry wire drawing the

Back tension

Ai

Figure 1.4 A two-stage wire drawing.

Drawing force

Af Die

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lubricant is coated on the wire before entering the die. In the wet drawing, die and wire are submerged in the lubricants. It is also important to control the residual stresses in the wire by proper design of the die and process. Recently dieless drawing of wires and tubes is also gaining importance [10].

1.2.4 Extrusion In extrusion process the raw material is compressed through a die to reduce the cross-sectional area of the material or to generate special profile of the cross section. Theoretically, there is no limit to the maximum possible reduction in extrusion. The extrusion can be performed in cold, warm, or hot state of the metal. There are a lot of variants of the extrusion process. In the forward or direct extrusion the metal flows in the direction of ram motion. Fig. 1.5 shows a schematic diagram of the process. To avoid sticking of ram with raw material, a dummy block is inserted in between. Movement of material through the container causes a lot of friction between the container wall and raw material. To alleviate this problem, backward or indirect extrusion is used, in which the ram and extruded material move in different directions as shown in Fig. 1.6; however, the ram needs to be hollow, which weakens it. In lateral extrusion, material flows sideways, usually perpendicular to ram motion; it helps to reduce the frictional losses. A schematic diagram is shown in Fig. 1.7, where lateral extrusion is taking place from both sides, but it can take place from one side as well. In impact extrusion, material is extruded with the impact force of the ram. Fig. 1.8 shows a schematic of impact extrusion used to make a thin-walled tube open at one end. In hydrostatic extrusion (Fig. 1.9) the raw material is placed in a sealed chamber containing liquid and a moving ram pressurizes the liquid; the raw material is forward extruded by the pressure of the liquid. Hydrostatic extrusion reduces the friction and also enhances the ductility of the raw material. In the multi-hole extrusion (Fig. 1.10) the die has more than one opening, causing more than one product to be extruded Container

Flow lines

Ram

Force

Extruded rod Dummy block

Figure 1.5 Direct or forward extrusion.

Workpiece

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Flow lines Die Ram

F

F

Container

Workpiece

Figure 1.6 Indirect or backward extrusion.

Load

Ram Dummy block Container Workpiece

Extruded part

Die hoder

Figure 1.7 Lateral extrusion.

simultaneously. Multihole extrusion reduces the extrusion force, but improper die design can lead to poor mechanical properties [11]. Prominent defects in extrusion are piping, surface cracking, and internal cracking. In the piping defect, also known as fishtailing, tailpipe, or extrusion defect, metal flow tends to draw surface oxide and impurities toward the center of the product like a funnel. It occurs at the end of the extruded product and can be eliminated by controlling the flow pattern or using a proper dummy block. Surface cracking occurs due to thermal stresses and friction. Internal cracks are in the form

Modeling of metal forming: a review

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Load

Workpiece Ram

Die

Figure 1.8 Impact extrusion.

Container

Force

Die

Ram

Extruded part

Fluid

Workpiece

Figure 1.9 Hydrostatic extrusion.

of chevron cracking or central burst; high die angles and low reduction promote this defect [12]. Apart from forming a desired shape, extrusion is also used for improving the microstructure and mechanical properties of the product. Equal-channel angular extrusion (ECAE), invented by Segal in 1972, is widely used for producing ultrafine grained structures due to dynamic recrystallization (DRX) [13
15]. Other names for ECAE are equal-channel angular forging and equal-channel angular pressing. Fig. 1.11 depicts a schematic of ECAE. Here a billet is passed through two equal sized channels intersecting at an angle ϕ, which induces a lot of shear in the billet; however, the final size of the billet remains the same as before. The billet again passes through the cannels to undergo further straining. After several passes through the channels, the billet gets heavily strained, and its grain sized is

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Load Ram

Dummy Block Workpiece

Container

Multiple-hole die

Die holder

Extruded product

Figure 1.10 Multi-hole extrusion. Force Billet

Die j

Figure 1.11 Equal-channel angular pressing.

significantly refined resulting in the enhancement of its strength and sometimes the ductility as well.

1.2.5 Deep drawing Deep drawing process manufactures cup- or boxlike products by pushing a flat sheet through a die with the help of a punch while holding the sheet in a blank

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Punch Blank holding force Blank holder

Workpiece

Die

Partially drawn cup

Figure 1.12 Deep drawing.

holder. Fig. 1.12 shows a schematic diagram of the process. The blank holder force (BHF) and clearance between punch and die should be carefully decided. Excessive BHF causes tearing, and too less BHF causes wrinkling. Clearance is usually 7%
15% of sheet thickness. Too less clearance causes ironing; sometimes, it may be desirable. Anisotropy of the sheet has a large influence on the deep drawing ability of the sheet. Planar anisotropy, in which properties differ with direction in the plane of the sheet, causes earlike formation on the drawn product; this defect is called earing. Normal anisotropy refers to the situation in which properties in the thickness direction differ from those in the plane of the sheet. A large flow stress in the thickness direction compared to that in the plane of the sheet provides better performance in deep drawing as it avoids tearing. A measure of normal anisotropy is the plastic strain ratio, which is the ratio of true width strain to true thickness strain for a material strained in the longitudinal direction. Automobile manufacturers prefer a plastic strain ratio of 1.4 or more for steel sheets. A measure of formability is the limiting draw ratio (LDR), which is the ratio of the diameter D of the largest blank that can be successfully drawn to the diameter of the punch d. Theoretical limit for LDR is 2.7. Besides wrinkling, tearing, and earing, other common defect is the orange peeling. Orange peeling is the generation of high surface roughness in the region of the sheet that has undergone large deformation. This defect is more prominent with large grain size materials. Ironing can eliminate this defect.

1.2.6 Bending Bending is the process by which a straight object is transformed to a curve object. Bending of a tube is called tube bending, and bending of a sheet is called sheet bending. In sheet bending the bend radius is expressed in terms of sheet thickness. For example, if the minimum bend radius is 4t, then the sheet can be bent with a

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

minimum radius of four times the sheet thickness without cracking. In the tube bending, bend radius is expressed in terms of outer diameter of the tube. Bendability of a sheet can be improved by heating, applying hydrostatic pressure, or by applying compressive forces in the plane of the sheet. Sometimes, a tube is bent by filling the sand in the tube and heating it. During mechanical bending a provision has to be kept for elastic recovery, which is called springback. Sheet is bent up to the desired bend plus springback. Nowadays, laser-based bending is getting prominence. There are two types of laser-based bending processes (Fig. 1.13). One is called laser bending, where a laser beam irradiates the surface of the workpiece to be bent, which produces thermal stresses responsible for bending [16]. Fig. 1.13A shows a schematic diagram of laser bending. In the other type of processes called laser-assisted bending, the task of laser is the local heating of material to soften it, and bending is carried out by applying external load. External load can be applied by hanging a load [17], by pushing the sheet mechanically, hydraulically, pneumatically or magnetically [18
21]. Fig. 1.13B shows a schematic diagram of a laser-assisted bending. Process modeling of laser-based bending is required to get accurate bend angle.

(A)

(B)

Figure 1.13 Bending of a sheet with the help of a laser beam: (A) without any mechanical load; (B) with mechanical loading at one end.

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Due to high temperatures involved in the process, temperature-dependent properties of the sheet are needed.

1.3

Various modeling techniques

In general, the modeling of metal forming is a thermomechanical problem. A lot of heat is generated due to deformation of material, and about 95% of it is dissipated as heat raising the temperature of workpiece and tooling. Friction between workpiece and tooling also produces heat. However, in cold metalworking, temperature rise may not be enough to alter the mechanical properties of the material. In view of it, most of the times metal forming processes are analyzed without considering the heat transfer phenomenon. There are various modeling techniques differing in their abilities and capabilities. They can be classified into analytical, semianalytical, numerical, empirical, data based, and atomistic. However, often the distinction is blurred; in fact, of-late hybrid techniques are gaining more popularity. In this section, some methods are briefly described.

1.3.1 Slab method In this method the free body diagram of an infinitesimal slab of the deforming material is made, and the force balance is carried out. This produces one or more differential equations that are solved analytically or numerically. Slab method has been used for the modeling of various processes such as wire drawing, extrusion, forging, and rolling (both symmetrical and asymmetrical). Although it is a very old technique and employs several assumptions (such as uniform or linearly varying stress, and less-complicated friction law), it is still being used for analyzing the metal forming process for getting quick estimates with reasonable accuracy. For example, roll force, roll torque, and curvature of the rolled sheet in asymmetric rolling have been predicted with reasonable accuracy using the slab method [22
24]. Based on a radial gradient slab method, Hongyu et al. [25] have analyzed a dieless wire drawing process and obtained results very close to those calculated using an FEM model.

1.3.2 Slip-line field method Slip-line field method is mainly used for modeling of plane strain rigid
plastic metal forming processes. It is based on the assumption that at each point, a pair of orthogonal curves can be drawn along which the shear stress has the maximum value. These curves are called slip lines, and the material is assumed to flow along these lines. In essence, the slip line method solves the following equations: G

G

yield criterion for plane strain equilibrium equations along two orthogonal directions in the plane

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G

G

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

continuity equation (volume conservation as the density remains practically unchanged during metal forming for most of the metals) an equation indicating that directions of principal stress and corresponding principal strains coincide

Usually, slip-line field satisfying the stress boundary conditions is assumed and tested to ensure that it also satisfies the velocity boundary conditions. Construction of proper slip-line field is nontrivial. Slip-line field method is also an old technique, which is still in vogue. However, most of the engineers are disinclined to use this technique due to following reasons: G

G

It is limited in scope as it is conveniently applied only for the modeling of plane strain metal forming of rigid
plastic metals, notwithstanding that it has been applied for analyzing axisymmetric metal forming and hardening metals also by a few researchers [26
28]. Construction of slip-line field requires a lot of skill and efforts.

1.3.3 Visioplasticity The visioplasticity method, introduced by Thomsen et al. [29], combines experiments and analysis. In this method a velocity field is obtained from a series of photographs of the instantaneous grid pattern during an actual forming process. Thus the deformation field is obtained, from which strain rates and strains can be calculated. Using the equations from theory of plasticity, stress field can be obtained. This method has often been used only to study the flow pattern [30], because the calculations involved are too cumbersome. Moreover, with the advent of finite element, prominence of such type of experimental technique has diminished.

1.3.4 Upper bound method Upper bound theorem is used to find out an upper estimate of deformation field. In this a deformation field satisfying continuity equation and velocity boundary condition is assumed. Such a field is called kinematically admissible velocity field. Based on the assumed velocity field, the total energy for deformation is included. Total energy consists of energy for plastic deformation as well as energy to overcome the friction. Sometimes, discontinuous velocity field may also be assumed. However, only the tangential velocity discontinuity is permitted, not the normal velocity. This means that a portion of metal can slide against another portion, but one portion can neither penetrate into the adjoining portion nor can separate leaving a void. When the discontinuous velocity field is assumed, then the energy dissipation at the line of velocity discontinuities should also be included in the total energy. Per unit time energy dissipation at an infinitesimal surface area of velocity discontinuity is equal to the product of yield shear strength, magnitude of velocity discontinuity, and the surface area. For obtaining the friction energy, a friction factor model is assumed, in which the frictional stress is equal to some fraction of yield shear strength of the material. The advantage of using this friction model is

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that the normal stress at the interface is not required. The upper bound theorem states that the total energy calculated by an assumed velocity field will be greater than or equal to actual energy needed for deforming the metal. The upper bound theorem has been used for estimating the deformation load in processes such as forging [31
35], wire drawing [36,37], extrusion [38
40], and rolling [41
46]. Attempts have also been made to include the effect of strain hardening in this method [47,48]. A generalized upper bound theorem has also been proposed [49]. In the generalized upper bound theorem, the assumed velocity field satisfies the continuity equation but need not satisfy the velocity boundary conditions. The power imparted by the actual traction on surface due to assumed velocity field will be less or equal to actual power of plastic deformation. Using this theorem, one can use Coulomb’s friction model as well.

1.3.5 Finite difference method Finite difference method is a method for solving the differential equations. Hence, the first task while solving the metal forming problem by this method is to write down the governing differential equations. These differential equations are converted to difference equations from the basic definitions of derivatives, albeit considering approximation. This can be carried out by considering first few terms of a Taylor series. As a simple example, consider a one-dimensional case of the displacement u as a function of xAR, where R is the set of real numbers. From the basic principle of calculus, the exact first derivative at a point x is given by the following equation: du uðx 1 hÞ 2 uðxÞ 5 lim : h!0 dx h

(1.1)

In the finite difference method the same equation is used but instead of h tending to zero, a small value of h is chosen. Entire domain is discretized into a grid having lot of points (called node) separated by small distance. Difference equations can be written at each node and then assembled. Boundary conditions can be suitably employed. The assembled system of algebraic equations can be solved by a suitable numerical method giving the results at the nodes. If a result is needed at an intermediate location, interpolation can be used. Finite difference has been used for solving the metal forming problems, particularly heat transfer problem [50
52]. However, it is not convenient to make a proper grid if the domain boundaries are curved, and often the method needs a lot of computational time. It has been surpassed by FEM in many situations.

1.3.6 Finite element method Like finite difference method, FEM is also a numerical method of solving differential equations. Due to its ability to provide detailed information, this method has

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attained a large amount of popularity, and there are several commercial as well open-source software FEM packages suitable for modeling the metal forming. Hence, this method is being described relatively elaborately. There are two prominent FEM approaches for solving metal forming processes —updated Lagrangian approach and Eulerian approach [3]. In updated Lagrangian approach the domain consists of a set of particles that may keep changing their position in the space. At each time increment, the current configuration of the domain is taken as the reference configuration, that is, the reference configuration keeps updating with the time. An incremental strain tensor is taken as the measure of deformation. Accordingly, the constitutive equation is expressed in terms of incremental stress and incremental strain tensor. Following equations need to be solved using FEM: 1. Incremental strain-displacement relation: six scalar equations. 2. Incremental elastic
plastic stress
strain relations: six scalar equations. Three different sets of equations are needed as follows: (1) equations applicable during loading with plastic deformation, (2) equations applicable for loading without plastic deformation, that is, with elastic deformation alone, and (3) equations applicable during unloading. In most of the cases unloading is considered to be elastic, and there is no difference in the equations for elastic loading and unloading. 3. Incremental equations of motion: three scalar equations.

Another type of formulation is Eulerian formulation, in which the domain of the analysis is a region fixed in space. All these equations are written in rate form instead of incremental form. Thus there are following set of equations: 1. Strain rate-velocity relations: six scalar equations. 2. Elastic
plastic stress
strain relations: six scalar equations. There are different relations for elastic and elastic
plastic case. Plastic portion of deviatoric strain-rate is proportional to deviatoric stress, whereas the elastic portion of deviatoric stress is proportional to rate of deviatoric part of stress. Cauchy stress used in the analysis is objective (frame-invariant) but not its rate. Hence, a different stress rate measure has to be used such as Jaumann stress rate. 3. Incompressibility constraint: one equation stating that sum of direct strain rates is zero, that is, ε_ kk 5 ε_ 11 1 ε_ 22 1 ε_ 33 5 0:

(1.2)

The use of Eq. (1.2) yields to ill conditioning. An alternate formulation (called penalty method) transforms Eq. (1.2) in the following form: ε_ kk 1

p 5 0; λ

(1.3)

where p is called pressure (negative of hydrostatic stress), and λ is a large number called penalty number. Choosing a proper value of penalty parameter requires skill and is dependent on the computer system used. Whatever method is used for

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solving the equations (mixed or penalty formulation), pressure values usually contain errors. One way is to correct the pressure values by solving the equations again by finite difference method. Solving the metal forming problems by FEM requires a lot of skill. There are different ways to achieve the goal, and decision has to be taken carefully. For example, the deformation load can be calculated by integrating stresses over an area. However, the stresses, being a derivative quantity in FEM formation, are not as accurately determined as the primary variables. Moreover, they have generally more inaccuracy at the boundary, and the effect of local errors may introduce a significant error in the load estimation. An alternate approach is to find out total energy (or power) of deformation and from that find out the deformation load. This approach has provided significantly better accuracy. Decision to use updated Lagrangian or Eulerian FEM requires careful considerations. Updated Lagrangian method can provide the detailed information about the deformation, including the proper identification of elastic
plastic boundaries and residual stresses. However, the method is computationally intensive. Moreover, most of the reported results in the literature could not provide confidence in the accurate estimation of residual stresses. Eulerian method is computationally faster, but not very suitable for analyzing the stresses in elastic zone including residual stresses. A proper methodology for faster computation of residual stresses is the need of the hour. Arbitrary Lagrangian
Eulerian (ALE) FEM takes the advantages of both updated Lagrangian and Eulerian methods. In the ALE method the nodes of the computational mesh may be moved in an arbitrary specified fashion, such that new mesh can be generated conveniently. Updated Lagrangian and Eulerian methods become the special case of this method. The method was introduced by Trulio [53] and developed in the 1997s by Hirt et al. [54]. A lot of researchers have applied this method to metal forming problems [55
57].

1.3.7 Meshless method In the updated Lagrangian method, mesh has to be generated again and again to compensate for mesh distortion due to large deformation. Moreover, in certain situations, mesh generation requires a lot of effort. This motivated the researchers to develop meshless methods. Some meshless methods are smoothed particle hydrodynamics [58], diffuse element method [59], material point method [60], element free Galerkin (EFG) [61], HP Clouds [62], reproducing kernel particle method [63], partition of unity method [64], finite point method [65], and local boundary integration equation method [66]. Among these methods, EFG method has gained a lot of popularity. These methods require a lot of time. Mahadevan et al. [67] used a meshless method, radial basis function (RBF) collocation, for solving a problem of axisymmetric cold forging. For 20% reduction the required CPU time was about 7 hours on a Pentium IV machine.

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1.3.8 Molecular dynamics simulation MDS is a useful simulation technique for atomic and molecular simulation at small-length scale [68]. In this the evolution of an ensemble of atoms subjected to boundary conditions is simulated. System evolution is governed by Newton’s second law of motion. Force on an atom due to interaction with other atom is obtained by taking the gradient of a suitable potential function. The selection of a proper potential function is very crucial. With the net forces known on each molecule, the acceleration can be calculated using Newton’s second law. Time integration provides updated velocity and position of each particle. MDS is useful when underlying deformation mechanism is not well understood. It gives a fundamental insight into the process even in the absence of prior knowledge of system evolution. However, it requires a lot of computational time and very small domain size (of the order of 100 nm). Hence, the method is suitable for nanoforming or when specific information is needed in a small local area. MDS has been applied in the modeling of nano-forming [69].

1.3.9 Soft computing Soft computing techniques rely more on data than the physics of the process, notwithstanding that knowledge of the physics of the process augments the effectiveness of the soft computing. Three popular soft computing methods are artificial neural network (ANN), fuzzy set theory, and genetic algorithms [3]. Neural network and fuzzy set theory have been used for performance prediction in metal forming. Among these two, neural networks are good at learning from the data, and fuzzy set theory is good in tackling uncertainty. Genetic algorithms have been used for optimization of metal forming processes [70,71]. ANN is motivated by the working of the human brain, mimicking its learning behavior. Two popular neural networks are multilayer perceptron and RBF neural networks. They have been used in load prediction in metal forming [72,73]. Fuzzy set theory has been applied in three ways. One way is the use of fuzzy set operations in proper decision-making. For example, the specifications of a rolling mill were decided by treating the requirement of customers as fuzzy [74]. The second way is the use of fuzzy arithmetic for modeling uncertainty. For example, roll force and roll torques were computed by treating material and friction parameters as fuzzy numbers [75]. The third way is the use of fuzzy logic, which is useful in making inferences through fuzzy rules. Gudur and Dixit [73] have used fuzzy logic for the estimation of roll force and roll torque in rolling. Use of more than one techniques in solving a problem is called solving using hybrid methods. There are three interpretations of hybrid methods. In one interpretation, hybrid method is the method in which a method assists the main method. For example, Gudur and Dixit [76] have used neural networks for enhancing the speed of an FEM code. An RBF neural network was used by the velocity field generated by an FEM model. Trained neural network was used to provide a guess value of velocity field and location of neutral point. From the guess value the FEM model

Modeling of metal forming: a review

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was used to generate an accurate solution. Computational time got reduced by a factor of 10. The second way of hybrid modeling is the use of two methods together. The third way of using hybrid methods is to develop a method that is inspired by more than one method. One example is adaptive neuro-fuzzy inference system (ANFIS). Baseri et al. [77] used ANFIS for the modeling of springback in bending.

1.4

Inverse modeling

Metal forming models are often quite complex. They require a lot of input variables for predicting the responses. It may be difficult to get all the necessary data. Even if the data are available, it may be inaccurate. Inverse modeling helps in getting the input data based on experimental observations of the process. It also helps in finetuning the model with available experimental data. Sometimes, the inversely obtained data of input variables may not be accurate, but it may provide better results by the model than if the accurate values of the data were used. In other words, inversely obtained data compensates for modeling errors. Essentially, inverse modeling is an optimization problem. Variables to be inversely determined are treated as design variables. The objective function is the error between predicted and experimentally observed response, based on several experiments. The objective function is minimized subject to constraints and variable bounds. Sometimes, there may not be a unique solution to the problem. In that case, either the objective function should be modified by incorporating additional types of responses and/or more number of experimental data or suitable guess values should be chosen such that obtained solution is realistic. In metal forming problems, particularly in rolling, a number of researchers have carried out inverse modeling. Several researchers estimated the coefficient of friction, heat flux input into the roll, and the material properties of the strip by using an inverse analysis. Kusiak et al. [78] proposed a methodology for the evaluation of material parameters in hot forming of metals by an inverse analysis. The methodology was tested for hot compression of medium carbon tool steel. The combined effect of hardening and recovery was modeled by a hyperbolic sine equation. Cho et al. [79] proposed an inverse method to determine the flow stresses and the coefficient of friction. The inverse method was based on the minimization of the difference between the experimental loads and the corresponding FEM predictions. For bulk forming the basis for the inverse estimation was a ring compression test, and for sheet metal forming, it was a modified limiting dome height test. Inversely estimated data were tested on cylinder compression and hydraulic bulge test, as representative of bulk and sheet metal forming, respectively. Han [80] applied a modified two-specimen method for the online determination of flow stresses and the coefficient of friction in rolling. In their method the strip is rolled twice with two different sets of roll radii. Anisotropy of the sheet has been considered in this work using Hill’s 1948 criterion. Cho and Altan [81] proposed an inverse method

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

to determine flow stress and friction factor of the bulk and sheet materials under isothermal condition. First, the material parameters were determined by minimizing the difference between the experimental and calculated loads. Afterward, the shape of the deformed specimen was compared with the computed shape. If they did not match, the friction factor was adjusted in order to reduce the difference in the next iteration. The methodology was applied to the ring compression and the modified limiting dome height test. Flow stress was modeled as power law as a function of strain and strain rate. Byon et al. [82] proposed an inverse method for the prediction of flow stress
strain curve and coefficient of friction using actual mill data. The roll force and forward slip were taken as the basis for inverse estimation. Most of these models find out the parameters of a material behavior by assuming a constitutive model. Even the parameters involved in famous Johnson
Cook (JC) model can be obtained inversely, as conducting the test at high temperature, and high strain rate is time-consuming and costly. However, inverse determination of JC model parameters has been mostly carried out in machining. Recently, Yadav et al. [83] determined the material parameters as well as coefficient of friction in a warm rolling process by measuring the exit temperature and the slip. Friction has been earlier also estimated based on the measurement of forming load in rolling and/or slip. Lenard and Zhang [84] estimated the friction through an inverse technique by minimizing the error between the measured and computed roll force, roll torque, and forward slip. However, this was based only on a one-dimensional model of rolling. Lenard and Nad [85] estimated the coefficient of friction in rolling as a function of process and material parameters by an inverse analysis by minimizing the error between the measured and calculated roll force and roll torque. Friction has been inversely determined based on the measurement of curvature of rolled sheet in an asymmetric cold flat rolling process [86]. In the open die forging process the friction has been determined by the measurement of bulge [87]. In laser forming, absorptivity is determined often in an inverse manner [88]. Recently, thermal and mechanical properties of the sheet have been determined based on the measurement of bend angle and temperature profile in the laser bending process [89
92].

1.5

Modeling of microstructure and surface integrity

Microstructure describes the appearance of a material at a nm
cm length scale. It is basically the arrangement of phases and defects within a material. Consideration of microstructure is of paramount importance in hot working. Microstructure during a metalworking depends on the temperature, cooling rate, and also on the deformation. However, generally, temperature and cooling rate have more effect on microstructure. Microstructure is predicted considering phase transformation kinetics, which describes the rate of transformation of a parent phase into its microstructural constituents at different temperatures. Phase transformation is of two types—diffusional and displacive. In diffusional transformation the atomic bonds break and

Modeling of metal forming: a review

21

individual atoms move to rearrange themselves for forming new microstructures; it occurs by nucleation and grain growth. For example, in steel, diffusional transformation transforms austenite to pearlite, bainite, ferrite, or cementite. Diffusional transformation may be isothermal or anisothermal. In displacive transformation, ordered groups of atom displace without actually breaking the bonds. For example, at rapid cooling rate, austenite in steel gets converted to martensite. For predicting phase transformation, time-temperature transformation (TTT) and/or continuous cooling transformation diagrams are required. These are material specific and are obtained experimentally. Diffusional transformation under isothermal conditions is governed by the classical Johnson
Mehl
Avrami
Kolmogorov (JMAK) model, which is stated as [93] y 5 1 2 expð2 ktn Þ;

(1.4)

where y is the fraction of transformed constituent, k is a diffusion coefficient, and n is the Avrami exponent of transformation. Both k and n are constants dependent on temperature and can be obtained through TTT diagram by noting the times for 1% and 99% transformation. Displacive transformation is not dependent on the time, but it depends on temperature. An empirical equation can be used for each material. Other important phenomena in metal forming are strain hardening, dynamic recovery, static recrystallization, and DRX. Strain hardening is incorporated in classical plasticity in a phenomenological way. By dynamic recovery energy stored in the form of dislocation is released without involving the movement of high-angle boundaries. During recovery, reduction in point defect density and annihilation of dislocations takes place. Recrystallization is the formation and migration of highangle boundary grains due to stored energy of deformation. Nucleation and growth of new grains may occur during deformation, which is called DRX. DRX may be described by the JMAK equation. The temperature at which a given material almost completely recrystallizes in one hour is called the recrystallization temperature. Increasing the degree of deformation and reducing the temperature of deformation lowers the recrystallization temperature. Recrystallization temperature is less if the initial grain size is small. Recovery and recrystallization are not a phase transformation, but microstructural transformation. Cold work does not involve phase transformation. During cold work, point defects and dislocation density increase, which causes strain hardening. When a material is subjected to severe plastic deformation, new grains are formed. There is a lot of severe plastic deformation processes [94] by which ultrafine grains can be produced. By this method the strength of the metal can be enhanced by two-tothree times mainly due to Hall
Petch effect. As per Hall
Petch effect, the strength (yield as well as tensile) of the material increases with the reduction in its grain size. For a number of materials, the yield strength σY is related to its grain size in the following manner [93]: ky σY 5 σ0 1 pffiffiffi ; d

(1.5)

22

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

where d is the average grain diameter, ky is the strengthening coefficient, and σ0 is called the friction stress. As the grain size decreases, the total grain boundary area increases that causes hindrance to the movement of dislocations. This is the main reason for the appearance of Hall
Petch effect. There have been some attempts to combine phase transformation models in a classical plasticity-based FEM model. Recent work is by Shufen and Dixit [95] and Kumar and Dixit [96]. Shufen and Dixit [95] modeled a thermal autofrettage process for cylinder along with a special heat treatment to introduce compressive residual stresses at the outer surface of a thick-walled cylinder. Phase transformation kinetics was implemented through a user-defined subroutine in a commercial finite element analysis (FEA) package. By following the similar procedure, Kumar and Dixit [96] developed a model to predict the hardness of a laser bent strip.

1.6

A note on multiscale modeling of metal forming

Materials are composed of atoms. In principle, it is possible to model the behavior of each and every atom. However, this will require huge computational time. For example, MDS studies the movement of atoms and molecules under the action of various forces of interaction and relies on Newtonian mechanics. Typical time steps in MDS are of the order of femtosecond and interatomic spacing are of the order of angstrom. At this scale, classical concept of stress does not have any relevance. However, carrying out MDS for the entire domain of metal forming is next to impossible. Multiscale modeling may be a right choice for modeling the overall process. Multiscale modeling involves modeling of the processes simultaneously at different scales for predicting various parameters of the interest. For example, texture of the material can be obtained by microstructural modeling, and total load requirement can be obtained by classical FEM following continuum mechanics. There are two broad approaches to carry out multiscale modeling. In sequential multiscale modeling, some parameters of a macroscale model can be determined by carrying out a microscale modeling of the process. Parameters obtained from microscale model are passed on to macroscale model. In concurrent multiscale modeling, micro- and macroscale models run concurrently. The need for a multiscale modeling approach arises when one model is not adequate to accurately predict the behavior of the system [97]. For example, the classical theory of plasticity for macroscale analysis of elastic
plastic deformation in metals is capable of predicting the strain and stress with considerable accuracy. However, the mathematical formulations based on this approach may not be sufficient in case of thermal elastic
plastic deformations at elevated temperatures involving recrystallization. Above the recrystallization temperature, new grains appear in the material and based on the cooling rate, the different phases may be produced. Since each material phase has different properties, the overall macroscale behavior of the material is affected. In this case, one may be required to incorporate the phase transformation kinetics such as the JMAK model alongside classical

Modeling of metal forming: a review

23

plasticity. The basic features of a multiscale modeling may be described by considering the thermomechanical analysis in heat treatment of steel as that used in Shufen and Dixit [95]. A theoretical model that is capable of describing the heat treatment process in steel must involve an interaction of thermal and mechanical science at the macroscale and metallurgical science at the microscale. The thermal equation predicts the change in temperature, which in turn affects the mechanical expansion and phase change. When there is plastic deformation, the mechanical equation predicts internal heat generation due to plastic deformation and the subsequent role in phase change. When there is phase change, the microstructural model predicts the internal heat generation due to release of latent heat during phase change that affects the thermal behavior and the dilatational strain due to phase change that affects the mechanical response. There is not much literature on multiscale modeling of metal forming processes. However, some published articles indicate huge research potential in the area of multiscale modeling. McGinty and McDowell [98] have used multiscale crystal plasticity models to generate forming limit diagrams for sheet metal forming. Van Houtte et al. [99] modeled plastic anisotropy and deformation texture of polycrystalline materials following multiscale modeling. Results obtained from meso/microscale models were intended to be used in a macroscale model. Hol et al. [100] have presented a multiscale friction model for use in sheet metal forming. First, a microscale model predicts friction based on the texture of the sheet. Afterward the estimated friction is used in the macroscale model of sheet metal forming for computing other quantities of the interest. Recently, Kumar and Dixit [96] also has used a multiscale model for the estimation of hardness of strip after laser bending, although their micromodel is largely empirical.

1.7

Challenging issues

Compared to machining, metal forming processes have been accurately modeled. However, often large deformation metal forming processes are modeled by neglecting the elastic strains, that is, by rigid
plastic approach. Rigid plastic modeling is unable to predict residual stresses, for which elastic strains needs to be considered. Elastic strains are of the order of 1023, whilst the plastic strains are of the order 1. Huge difference in the values of strains creates numerical difficulties. There are very few attempts to model residual stresses in the metal forming processes [101]. Apart from numerical difficulties, there is difficulty in experimentally measuring the residual stresses. Modeling often requires a huge amount of computational time. The attempts should be directed toward minimizing the computational time. There are many ways to achieve the objective of reducing the computational time. The development of faster modeling techniques can revolutionize this research area. Evolutionary progress can be made by parallel computing, developing efficient solvers, and using

24

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

hybrid techniques. One example of hybrid techniques is to use neural networks for enhancing the efficiency of a finite element model [76,102]. Friction is an important but the least understood aspect of manufacturing processes. It is still chosen in an ad hoc manner. Although a number of models have been developed, they have not been properly applied in the modeling of manufacturing processes. The difficulty in using various friction models is that they require certain parameters that are difficult to obtain. Obtain those parameters in an inverse manner from experiments seem to be the best option available. In future, it may be possible to develop more sophisticated models using the knowledge of mechanics as well as chemistry. Study of friction becomes more pressing in the modeling of microforming processes. Modeling of microforming and nano-forming is quite different from the modeling of conventional metal forming processes. Scale effect comes into play and physical phenomena that are insignificant at macroscale gain significance at micro and nanoscale. At present, enough data are not available on the small-scale behavior of the material. For most of the materials, the conventional yield criteria may not be applicable and anisotropy may play a major role. Yield criterion has to incorporate the effect of strain gradient, which recently developed crystal plasticity-based models are able to capture. Surface roughness is an important property of a precision product. However, it is very difficult to model the surface roughness based on the physics as the number of parameters comes into play. In the area of metal forming, very few articles have been published on the modeling of surface roughness [103]. There is ample scope to investigate not only the surface roughness but also entire surface integrity in metal forming. In the past, microstructural aspects were not given enough attention while modeling of metal forming processes. Now, there is a growing realization that mechanical as well as microstructural modeling should be carried out concurrently for getting the proper benefit out of modeling. As an example, a number of papers have been published on the die design during extrusion, but all of them invariably attempt to reduce the power by minimizing the redundant work. However, the redundant work is not always wastage; it often helps to improve the mechanical properties. An integrated optimization scheme needs to trade-off between the power minimization and quality of the end product.

1.8

Conclusion

Metal forming is one of the oldest manufacturing processes. However, the modeling of metal forming processes, as a research area, is only about 100 years old. This article has summarized various analytical, semianalytical and numerical techniques employed for the modeling of metal forming. As of now, fairly accurate models are available, which can predict the forming load quite accurately. However, limited progress has been made in the prediction of residual stresses and surface integrity.

Modeling of metal forming: a review

25

There is a need to carry out research for better understanding of the physics of the metal forming processes as well as enhancing the computational efficiency of the models. Recent technological developments in microforming and development of new materials, including functionally graded materials, further demand proper characterization of the material behavior. Crystal plasticity and MDS are the promising techniques that can unfold the intricacies of metal forming processes. However, extensive computational time required in these techniques limits their domain of application. Multiscale modeling of the metal forming process may be a viable panacea in future.

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2) (2000) 293
309. [98] R.D. McGinty, D.L. McDowell, Application of multiscale crystal plasticity models to forming limit diagrams, J. Eng. Mater. Technol. 126 (3) (2004) 285
291. [99] P. Van Houtte, A.K. Kanjarla, A. Van Bael, M. Seefeldt, L. Delannay, Multiscale modelling of the plastic anisotropy and deformation texture of polycrystalline materials, Eur. J. Mech.-A/Solids 25 (4) (2006) 634
648. [100] J. Hol, M.V.C. Alfaro, M.B.D. Roolj, V.T. Meinders, Multiscale friction modeling for sheet metal forming, in: Fourth International Conference on Tribology in Manufacturing Processes, ICTMP, Nice, France, 2010, pp. 13
15. [101] U.S. Dixit, P.M. Dixit, A study on residual stresses in rolling, Int. J. Mach. Tools Manuf. 37 (6) (1997) 837
853. [102] P.P. Gudur, U.S. Dixit, A combined finite element and finite difference analysis of cold flat rolling, Trans. ASME, J. Manuf. Sci. Eng. 130 (1) (2008) 011007 (6 pages). [103] H.B. Xie, Z.Y. Jiang, W.Y.D. Yuen, Analysis of friction and surface roughness effects on edge crack evolution of thin strip during cold rolling, Tribol. Int. 44 (9) (2011) 971
979.

Finite element method modeling of hydraulic and thermal autofrettage processes

2

Uday Shanker Dixit and Rajkumar Shufen Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India

2.1

Introduction

Autofrettage is a metal working process carried out for introducing beneficial compressive residual stresses in the thick-walled vessels (usually of hollow cylindrical or spherical form). The induced compressive residual stress field is beneficial to the vessel for the enhancement of its pressure carrying capacity, fatigue life, stress corrosion resistance, and creep resistance [1
4]. This technique is incorporated in the design and fabrication of gun barrels, pressure vessels, nuclear reactors, automobile combustion chambers as well as steam and chemical pipelines, which are subjected to static/pulsating high-magnitude load and corrosive medium. Autofrettage is based on the principle of applying a plastically deforming load to the cylinder/sphere to create nonuniform through-thickness elastic
plastic deformation. The basic principle of autofrettage of a cylinder is shown in Fig. 2.1. (The subsequent discussion will be for the autofrettage of cylinders. The same basic concepts can be applied to sphere.) In the loading stage a deforming load, such as internal pressure, is applied to the inner surface of the cylinder, which creates two deformed zones, namely, an inner plastic zone from the inner surface to a certain intermediate radius and an outer elastic zone from the intermediate radius to the outer wall. The displacement of the inner surface of the cylinder with respect to the original inner surface is shown by dotted lines. When the load is removed, the elastic zone tries to shrink back to its original position while the plastic zone tends to remain in the permanently deformed state. This resistive interaction between the plastic and the elastic regions sets up a compressive residual stress field in the inner wall of the cylinder/sphere and in its vicinity. The elastic zone recovery applies pressure to the inner plastic zone and causes a slight shrinkage of inner surface of the cylinder in the final configuration of the cylinder. The origin of autofrettage may be attributed to the evolution of a series of efforts made in the design of gun barrels that dates back to the 19th century, when military engineers were assigned the task of enhancing the performance of small arms and heavy artillery. During this period, the “built up” method of fabricating gun barrels was introduced which was performed by shrink-fitting layers of wrought iron Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques. DOI: https://doi.org/10.1016/B978-0-12-818232-1.00002-3 Copyright © 2020 Elsevier Ltd. All rights reserved.

32

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Figure 2.1 The deformation stages of a thick-walled cylinder in a typical autofrettage process.

around an inner core that formed the barrel of the firearm [5]. Similar concepts were also applied for fabricating barrels from direct casting using a differential cooling technique that was performed by heating the outer surface, while cooling the bore with cold water [5]. In principle the basis of these techniques was to increase the overall strength of the gun barrel by imparting a compressive stress field to the inner wall where the impact of the forces exerted due to expansion of hot gases is the maximum. The concept of autofrettage was also based on this principle and was pioneered by Jacob [6], a French artillery officer, for prestressing monobloc gun barrels who gave the term “autofrettage” which means “self-hooping” in French. Since its inception, the application of autofrettage has gradually extended from just being a military application. There are various types of autofrettage processes, which are classified on the basis of the application of the deforming load. These different techniques of autofrettage are hydraulic, swage, explosive, thermal, and rotational. A brief description and a review of literature for each autofrettage process are discussed in the following subsections.

2.1.1 Hydraulic autofrettage Hydraulic autofrettage is the earliest concept of autofrettage that was originally conceived by Jacob [6]. In this process the bore of the thick-walled cylinder is filled with a hydraulic oil and pressurized with ultrahigh pressure. The cylinder starts yielding at the inner surface on reaching a particular pressure called yield pressure, which is the original pressure carrying capacity of the cylinder with the specific material and dimensions. As the pressure is slightly increased beyond the yield pressure, further yielding takes place in the cylinder creating an inner plastic zone that extends from the inner surface to an intermediate radius and an outer elastic zone that extends from the intermediate radius to the outer surface. At this point, if the cylinder is depressurized, the outer elastic zone recovers and compressive

Finite element method modeling of hydraulic and thermal autofrettage processes

33

Figure 2.2 A schematic of hydraulic autofrettage process.

residual stresses are induced in the vicinity of the inner surface of the cylinder. The amount of plastic-overstrain desired in the cylinder is controlled by increasing the pressure beyond the yield pressure. Care must be taken while applying the large pressure during hydraulic autofrettage because the cylinder can undergo reyielding due to the compressive residual stresses. For cylinders having wall thickness ratios (ratio of outer to inner diameters) greater than 2.22, the reyielding occurs when the applied pressure is greater than twice the yield pressure of the cylinder [7]. A schematic of a hydraulic autofrettage process is shown in Fig. 2.2. The autofrettage pressure is delivered to the cylinder by injecting the hydraulic fluid through an inlet using a powerful hydraulic pump. Seal plugs are provided at the ends of the cylinder for preventing oil leakage. To conserve the volume of the oil used in the process, a solid spacer is inserted inside the cylinder [8]. The process is very effective in terms of the maximum increase in pressure carrying capacity that can be achieved. To reach its full potential the process requires a very large magnitude of pressure to be applied at the inner wall of the cylinder, and this increases with the increase in the wall thickness ratio. For example, in a typical 10 mm-thick ring sample made of steel alloy 4333 M4 (yield strength 1070 MPa) with an inner radius of 15 mm and an outer radius of 31 mm, the pressure required for yielding up to 50% of the ring thickness is 662 MPa [9]. It is to be noted that it is the combined effect of hoop, radial, and axial stresses that causes yielding. For a large diameter thin cylinder, a small internal pressure can create yielding due to large hoop stress. However, for thick cylinders, pressure needs to be high; it may still be less than the yield strength of the material. The requirement of high pressure makes the process costly. Moreover, the use of high pressure may lead to the susceptibility for leakage of the oil used in the hydraulic circuit during the operation of the machine. The oil leakage can make the process hazardous to the environment and unhealthy to the operators. To overcome this, appropriate jointing and sealing must be installed. These special arrangements further make the process costly.

2.1.2 Swage autofrettage In this method the inner wall of the cylinder is plastically deformed by the interference of an oversized mandrel called “swage” that is mechanically forced through

34

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

(A)

(B)

Figure 2.3 (A) A schematic of swage autofrettage process and (B) mandrel geometry.

the bore of the cylinder by means of the ram of a hydraulic/mechanical press. The method was proposed by Davidson et al. [10]. A schematic of the process including the mandrel geometry is shown in Fig. 2.3. The practical implementation of swage autofrettage is relatively simpler and cheaper than that of hydraulic autofrettage [10]. Present-day gun barrels are mostly manufactured using swage autofrettage [11]. The swage mandrel used in the process is usually made of a material that has a higher strength than the material of the cylinder specimen. Chen [12] used tungsten carbide as the material of the mandrel whose elastic modulus was 610 GPa, which is about three times the elastic modulus of a typical steel alloy. The geometry of the mandrel is also an important factor in the process and has a tapered profile both at the front and the rear as shown in Fig. 2.3. The angle of approach θF that defines the front taper has a slightly lesser slope than the rear angle θR. The larger angle of taper at the rear allows the maximum plastic recovery of the inner wall of the cylinder that had just been plastically deformed by the front tapered portion. The process involves a lot of friction between the bore and the mandrel surface and hence requires a highly polished surface finish of the mandrel and use of lubricants. Davidson et al. [10] used a centerline average surface roughness of 0.05
0.38 μm for the mandrel while using copper plate and molybdenum disulfide as the lubricants. For a particular tube specimen and the same level of plastic overstrain, the fatigue lifetime of a swage autofrettaged tube is significantly higher than that for the hydraulic autofrettage. On reapplication of the working pressure, the reyielding of the autofrettaged tubes occurs at the same level of pressure implying that swage autofrettage can also achieve the same increase in the pressure carrying capacity as given by hydraulic autofrettage [13].

2.1.3 Explosive autofrettage Autofrettage can also be performed by using an explosive charge detonated inside the cylinder in a pressure propagating medium like air or water [14]. This method is called explosive autofrettage. Because of the use of explosive, the process is difficult to control and dangerous for practical implementations. A typical setup of the apparatus for the explosive autofrettage is shown in Fig. 2.4.

Finite element method modeling of hydraulic and thermal autofrettage processes

35

Figure 2.4 A schematic of explosive autofrettage process.

2.1.4 Thermal autofrettage In this method the autofrettage is carried out by using thermally induced stresses by subjecting the cylinder to a thermal gradient across its thickness. When the cylinder is cooled to room temperature, compressive residual stresses are induced in the inner wall of the cylinder due to elastic unloading of the thermal stresses. This method was suggested by Kamal and Dixit [15]. Thermal autofrettage is a very simple technique and does not require any complex machinery or special equipment for the practical implementation. Kamal et al. [16] confirmed the practical evidence of residual stresses in a thermally autofrettaged cylinder of steel alloy SS304 by using Sachs boring technique, microhardness test, and measurement of opening angle due to through-wall slicing of the autofrettaged cylinder. Despite its simplicity, the efficiency of the process in terms of the achievable maximum increase in the pressure carrying capacity is limited to only about 40%. This is mainly because the maximum temperature difference that can be used in the process for a particular cylinder dimension is limited to the maximum range beyond which material properties start to change [17]. Nevertheless, the economic advantage of the process can still be utilized and the performance of the thermal autofrettaged cylinder can be improved by shrink fitting [18]. A schematic of a setup for thermal autofrettage is shown in Fig. 2.5 [19]. Here, the outer surface of the cylinder is heated by an electric heater, and the inner surface is cooled by the flowing cold water. This creates the required thermal gradient for causing the plastic deformation of a portion of cylinder. When the cylinder is brought back to the room temperature, the residual stresses are generated.

2.1.5 Rotational autofrettage In this method of autofrettage the cylinder is applied with a centrifugal force due to rotation of the cylinder at a high angular velocity in the loading phase. In the unloading phase the angular velocity is decreased and the cylinder is gradually brought to rest that induces the residual compressive stress field in the cylinder.

36

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Figure 2.5 A schematic of thermal autofrettage process [19].

Figure 2.6 A schematic of rotational autofrettage process.

The method was proposed by Zare and Darijani [20]. A schematic of the process is shown in Fig. 2.6. The rotation of the cylinder is provided by mounting it on a bearing support. The output rotation of the motor is transmitted to the cylinder through a transmission system. For a typical 1 m long cylinder having inner radius of 0.1 m

Finite element method modeling of hydraulic and thermal autofrettage processes

37

and outer radius of 0.3 m, made of ASTM-A709 steel, the speed required for the autofrettage of the cylinder was 9780 revolutions per minute. Among various autofrettage processes, hydraulic autofrettage is the most widely employed. However, the thermal autofrettage has also a lot of potential, particularly for the situation in which the vessel encounters thermal loading apart from pressure loading. In this chapter, the finite element modeling of hydraulic and thermal autofrettage is described. The following subsection describes the numerical modeling of elastic
plastic problems.

2.2

Numerical modeling of elastic
plastic problems

In this section the phenomenological equations for incorporating material behavior and the governing equations for the formulation of elastic
plastic problems are discussed. These aspects of modeling are very important and form the basis of the finite element method (FEM) formulations for the autofrettage process. The field variables appearing in the governing equations as well as the FEM formulations involve scalars, vectors, and tensors and their operations. For the ease of representation of these variables, the index notation with Einstein’s summation convention is followed throughout this chapter. According to it, a tensor is represented by using indices. An index is designated by a small letter, say i, which can acquire the values 1, 2, and 3 which signify the x, y, and z directions in three-dimensional (3D) Cartesian coordinate system. For example, the components of a typical vector (tensor of rank 1) a are represented by ai that denotes a one-dimensional array with three components formed by the index i (1, 2, 3). Thus 8 9 < a1 = ai  a2 : : ; a3

(2.1)

The components of a typical tensor (of rank 2) a are represented by aij that denotes a matrix with nine components formed by varying the indices i (1, 2, 3) and j (1, 2, 3). Thus 0

a11 aij  @ a21 a31

a12 a22 a32

1 a13 a23 A: a33

(2.2)

If an index i appears without any repetition as in the case of Eq. (2.1) or (2.2), it is called free index and represents the number of independent components in the variable. A vector (tensor of rank 1) has one free index, and a tensor of rank 2 has two free indices. If in any term, an index is repeated, it indicates summation by varying the index over its range and adding all resulting components. The repeating index is called the dummy index, and this convention is called Einstein’s

38

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

summation convention. For example, in the 3D space, aii means the summation of three components by varying the dummy index i from 1 to 3, that is, aii 5 a11 1 a22 1 a33 :

(2.3)

Similarly ai bi with a repeating index i represents the summation of the products of the components of vector a and b, that is, ai bi 5 a1 b1 1 a2 b2 1 a3 b3 :

(2.4)

This indicates the dot product of vectors a and b. A Kronecker-delta function is represented by δij. It is defined as  δij 5

1 if 0 if

i 5 j; i 6¼ j:

(2.5)

For a tensor of second rank a with its components as aij , the following substitutional relation holds good: aij δij 5 aii :

(2.6)

2.2.1 Yield criteria and hardening behavior of the material The mathematical modeling in elastic
plastic analysis requires the accurate definition of the material behavior that will govern the deformation of the material. The important aspects in the material behavior is the selection of the yield criterion and the hardening rule. The yield criterion defines the elastic limit of a material at which it begins to yield when subjected to a gradually increasing load. Mathematically, the yield criterion is expressed as a scalar function of the components of stresses in the form f ðσij Þ 5 0;

(2.7)

where σij are the components of Cauchy’s stress tensor. The function f is called the yield function. Two yield criteria are explained here.

2.2.1.1 The von Mises yield criterion In the von Mises yield criterion the yield function is defined as a scalar function of the second invariant of the deviatoric part of Cauchy’s stress tensor. Mathematically, it is expressed as [21] f ðσij Þ  3J2 2 σ2Y 5 0;

(2.8)

Finite element method modeling of hydraulic and thermal autofrettage processes

39

where σY is the yield stress of the material and J2 is the second invariant of the deviatoric part of Cauchy’s stress tensor. The invariant J2 is defined as J2 5

1 0 0 σ σ ; 2 ij ij

(2.9)

where σ0ij is the deviatoric part of Cauchy’s stress tensor σij defined as 1 σ0ij 5 σij 2 σkk δij : 3

(2.10)

The von Mises yield criterion can also be expressed in terms of the principal stresses, σ1 $ σ2 $ σ3 ; as ðσ1 2σ2 Þ2 1 ðσ2 2σ3 Þ2 1 ðσ3 2σ1 Þ2 2 2σ2Y 5 0:

(2.11)

2.2.1.2 Tresca yield criterion The Tresca yield criterion is also called the maximum shear stress criterion. In terms of the principal stresses, σ1 $ σ2 $ σ3 , it is stated as f ðσij Þ  ðσ1 2 σ3 Þ 2 σY 5 0:

(2.12)

The yield function f ðσij Þ when plotted in the 3D stress space of σ1 , σ2 , and σ3 traces the yield pffiffiffi locus. The von Mises yield locus is a right circular cylinder of radius 2σY = 3 whose axis is oriented along the line σ1 5 σ2 5 σ3 . The Tresca yield locus is a regular hexagonal prism which is inscribed inside the von Mises cylinder. Strain hardening is the property of a material by which a material develops resistance to deformation once it has yielded, and increased load must be applied for subsequent deformation. Two models of hardening that are popular in solving elastic
plastic problems are isotropic hardening and kinematic hardening. In isotropic hardening the yield locus increases in size without any change in the shape as more strain is applied in the plastic state of the material. In uniaxial tension, isotropic strain-hardening can be expressed by a scalar function that governs the variation of true stress with the applied plastic strain as σ 5 Hðεp Þ;

(2.13)

where H is called the hardening function and σ is the true stress due to the applied plastic strain εp in the uniaxial tension. The same is assumed to hold good for 3D loading with uniaxial stress replaced by equivalent stress and longitudinal plastic strain replaced by equivalent plastic strain. Some hardening functions that have

40

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

been proposed and popularly adopted for numerical modeling of elastic
plastic problems are as follows [22]: 1. Holloman’s law: σ 5 Kðεp Þn ;

(2.14)

2. Ludwik’s law: σ 5 σY 1 Kðεp Þn ;

(2.15)

3. Ramberg
Osgood equation: (  n21 ) σ σ ; 11α ε 5 E σY p

(2.16)

4. Swift’s law: σ 5 σY ð11Kεp Þn ;

(2.17)

5. Prager’s law:  p Eε ; σ 5 tan h σY

(2.18)

where K and n are hardening parameters that are determined by fitting the function in the experimental true stress versus plastic part of the true strain curve. E is Young’s modulus of elasticity and α is a material parameter.

Kamal [19] used Ludwik’s hardening law for the thermal autofrettage analysis of a thick-walled cylinder. Wang [23] and Jahed and Dubey [24] used the Ramberg
Osgood expression for the analysis of hydraulic autofrettage. Alexandrov et al. [25] used a linear form of Swift’s expression using the parameter n as 1, which is basically a linear strain-hardening. While using isotropic strain hardening, the yield criteria defined in Eqs. (2.8) and (2.12) remain valid except that the yield stress term σY is replaced by σeq determined from the hardening function. In kinematic hardening, the size, shape, and position of the yield locus can change with the increasing plastic strain. This model is used to incorporate Bauschinger effect in the material behavior. The first kinematic hardening law was proposed by Prager [26] for the rigid translation of the yield locus without any change in its shape or size. The yield criterion for subsequent yielding in the kinematic hardening model is governed by the expression [21] f ðσij 2 αij Þ 5 0;

(2.19)

Finite element method modeling of hydraulic and thermal autofrettage processes

41

where αij is called the back stress and represents the rigid incremental translation of the yield surface in the stress space. Based on the von Mises yield function, the yield criterion for subsequent yielding in a kinematic hardening material becomes 

σ0ij 2 dαij



 2 σ0ij 2 dαij 2 σY 2 5 0: 3

(2.20)

where dαij is the incremental back stress and σ0ij is the deviatoric part of the stress tensor. Eq. (2.20) has been obtained by substituting Eq. (2.9) in Eq. (2.8) and substituting ðσ0ij 2 dαij Þ for σ0ij . Prager’s hardening law states that the incremental translation of the yield locus takes place along the direction of the incremental plastic strain dεpij as dαij 5 cdεpij ;

(2.21)

where dαij is the incremental translation of the yield locus due to the applied incremental plastic strain, c the material hardening parameter, and dεpij is the incremental plastic strain. If c is a constant, the hardening is called linear kinematic hardening, whereas if c is dependent on the deformation history, it is called nonlinear kinematic hardening. Here, only the linear kinematic hardening is discussed. For linear kinematic hardening in Prager’s model, c is equal to 2/3 times the slope of the plastic part of the bilinear stress
strain curve [21]. In a uniaxial tension, Prager’s law predicts hardening along the direction of applied stress but transverse softening. This is a slight drawback of Prager’s hardening law since transverse softening is not observed experimentally. This drawback was corrected by Ziegler [27] who presented a more widely accepted model of hardening. According to Ziegler’s hardening law, the incremental translation of the yield locus takes place along the direction of the line connecting the center of the yield locus to the point of the current stress tensor. The linear form of Ziegler’s hardening law is defined as [21] dαij 5

H0  σij 2 αij dεpeq ; σY

(2.22)

where H 0 is the constant slope of the plastic part of the bilinear stress
strain curve and ðσij 2 αij Þ represents the vector along the direction of the line connecting the center of the yield locus to the point of the current stress tensor. The symbol εpeq denotes the equivalent plastic strain and is given by the following expression: s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  2 εpeq 5 εp εp : 3 ij ij

(2.23)

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

2.2.2 Approaches for numerical modeling of elastic
plastic problems The mathematical modeling of elastic
plastic problems is generally carried out by using two approaches of formulating the governing equations for determining the stresses, strains, and displacements. These are the Eulerian formulation and the Lagrangian formulation. In the Eulerian formulation the analysis is focused on a fixed region in space called the control volume. The deforming material is assumed to flow like a non-Newtonian fluid through the control volume. Because of the assumed flowing nature of the material, this formulation is also called flow formulation. This approach is convenient for modeling the processes like rolling, wire drawing, and extrusion in which the material flows continuously through a control volume. In the Lagrangian formulation method the domain of the analysis consists of a set of particles whose position changes continuously during the deformation. The path of the particles is traced as the material changes from its original configuration to a particular state of deformation. Since the displacements of the particles in the deforming body occur incrementally with respect to time, the primary variable in this formulation may be incremental displacement. This method is suitable for processes like forging, deep drawing, and autofrettage. In this formulation the unknown field variables for the current deformed configuration may either be referenced to the original configuration that was at time t 5 0 or to the previous deformed configuration. The formulation which is based on the original frame of reference at time t 5 0 is called the total Lagrangian formulation, and the formulation based on the previous frame of reference is called the updated Lagrangian formulation. Generally, the updated Lagrangian method is followed for the elastic
plastic formulations. The updated Lagrangian method is explained in this chapter. For the analysis of solids and structures the formulation based on the updated Lagrangian approach is more convenient. Large displacement structural problems using Eulerian formulation require the creation of new control volumes because the boundaries of the solid keep changing continuously [28]. Considering the autofrettage analysis, the deformation of the cylinder takes place in the form of incremental concentric expansion of the plastic zone through the cylinder with the incremental increase in the applied load. The solution procedure requires continuous tracking of the elastic
plastic interface radius corresponding to the increase in the applied load for obtaining the stress and strain distributions. Here, the governing equations are described only for the updated Lagrangian formulation. The incremental equations for deriving the FEM formulation using the updated Lagrangian approach described here is based on two considerations. The first is the assumption that the deformations are finite. For finite deformation the incremental logarithmic strain tensor is used as the measure of deformation. The second consideration is that the equations are expressed with respect to time t 1 Δt at which the unknown variables are determined assuming that the solution at the previous configuration at time t is already known. The governing equations are as follows [21]:

Finite element method modeling of hydraulic and thermal autofrettage processes

43

1. Incremental strain
displacement relations:  L t Δεij 5

lnðt Δλi Þ 0

if i 5 j; if i ¼ 6 j:

(2.24)

2 t ΔUij

5 ðt ΔFÞTik ðt ΔFÞkj ;

(2.25)

t ΔFij

5 δij 1 t Δui;j ;

(2.26)

where t ΔεLij denote components of the incremental logarithmic strain tensor, t Δui;j is the derivative of incremental displacement t Δu with respect to position vector t x, t ΔF is the incremental form of the deformation gradient tensor, t ΔU is called the incremental right stretch tensor obtained by the polar decomposition of t ΔF, and t Δλi are the principal values of the tensor t U. The deformation gradient tensor represented by F is a measure of finite deformation and rotation at a point. Using polar decomposition, this tensor can be expressed as a product of a positive definite tensor (stretch tensor) and an orthogonal tensor (rotational component). The subscript t is used to denote that all the variables are considered with respect to time t. 2. Incremental stress
strain relations: For elastic
plastic material models, the incremental stress
strain relations of the material after and before yielding for isotropic and kinematic hardening models are as follows: a. After yielding: t Δσ ij

ð t1Δt

5

t t

 EP Cijkl d t ΔεLkl 2

ð t1Δt t t

EP Cijkl αδkl dð t ΔTÞ;

(2.27)

where the fourth-order tensor t CEP jkl is given by [29] t

CEP ijkl 5



2ν 1 2 2ν

2 Gδkl δij 1 2Gδik δjl 2 9G2 t s0 ij t s0 kl ðt H 0 13GÞt σ2eq ;

(2.28)

where t s0 ij is given by [30] ( t 0

s ij 5

t 0

σ ij σ ij 2 t αij

t 0

for isotropic hardening; for kinematic hardening;

(2.29)

and t

σeq 5 Hðt εpeq Þ:

(2.30)

G is the shear modulus, ν is Poisson’s ratio, α is the coefficient of thermal expansion, t ΔT is the temperature above the ambient temperature at time t, H is the hardening function, and H 0 is the slope of the hardening curve with respect to equivalent plastic strain.

44

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

b. Before yielding and after unloading: t Δσ ij

5

ð t1Δt t t

 E Cijkl d t ΔεLkl 2

ð t1Δt t t

E Cijkl αδkl dð t ΔTÞ;

(2.31)

where 

t

CEijkl

 2ν Gδkl δij 1 2Gδik δjl : 5 1 2 2ν

(2.32)

3. Updating scheme: The stress tensor that appears in the constitutive stress
strain relations in elastic
plastic problems must be objective. An objective stress tensor is one that remains invariant under the change of the frame of reference. In the updated Lagrangian approach for the finite deformation, the stress tensor is made objective by using the following updating procedure: t 1 Δt

σ 5 ðt ΔRÞðt σÞðt ΔRÞT 1 t Δσ;

(2.33)

where t ΔR is the finite incremental rotation tensor at time t. 4. Equilibrium equations: @t1Δt σij 5 0: @t1Δt xj

(2.34)

where t 1 Δt σij and t 1 Δt xj are the components of the stress tensor and position vector at time, respectively, with respect to time t 1 Δt.

2.3

FEM formulation using updated Lagrangian method

In this section the application of the updated Lagrangian method for the formulation of FEM equations is discussed. The general steps which are followed in the procedure are explained in the following subsections.

2.3.1 Derivation of the weak form of the equilibrium equation The equilibrium equation in Eq. (2.34) represents the strong form that is to be solved by the FEM method by making the following weighted residual as zero: ð t1Δt

V

@t1Δt σij wi dt1Δt V 5 0; @t1Δt xj

(2.35)

Finite element method modeling of hydraulic and thermal autofrettage processes

45

where t 1 Δt V is the domain volume at time t 1 Δt and wi are the weight functions. The weight function is 0 on the part of the boundary on which the incremental displacement is prescribed. Eq. (2.35) can be written as ð

@ t1Δt



t 1 Δt

@

V

σij wi

t1Δt



ð dt1Δt V 2

xj

t 1 Δt

t1Δt

σij

V

@

@wi t1Δt

xj

dt1Δt V 5 0:

(2.36)

The volume integral of the first term in Eq. (2.36) is converted to surface integral using the divergence theorem: ð

@Aij dV 5 t1Δt xj V@

ð Aij nj dS;

(2.37)

S

where Aij is any tensor function of the coordinates xj defined over the domain V, S is the boundary of the domain, and nj is the unit outward normal. Applying Eq. (2.37), the first term in Eq. (2.36), becomes ð

@ t1Δt



t 1 Δt

σij wi



@t1Δt xj

V

ð dt1Δt V 5

t1Δt

t 1 Δt

σij wi nj dt1Δt S:

(2.38)

S

The entire boundary S for the integral term in Eq. (2.38) can divided into two parts: 1. Su : The part of the boundary where the incremental displacement du is specified 2. St : The part of the boundary where the incremental traction ðdtn Þi 5 dσij nj is specified

Since the weight function is 0 at the part of the boundary where the incremental displacement du is prescribed, the surface integral term in Eq. (2.38) over Su reduces to 0. Hence, only the integral over the surface St remains. Using this reduction and applying Cauchy’s relation for stress tensor, Eq. (2.38) gets modified to the following final weak (with reduced requirement of differentiability) form: ð

@ t1Δt

V



t 1 Δt

@

σij wi

t1Δt

xj



ð d

t1Δt

V5

t1Δt

t 1 Δt St

ð σij wi nj d

t1Δt

S5

t1Δt

t 1 Δt St

ti wi dt1Δt S; (2.39)

where ti 5 σij nj :

(2.40)

46

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Considering the second term in Eq. (2.36), the term inside the integral can be modified using the symmetry of the stress tensor. To do this, it is first rewritten in the following form: t 1 Δt

! 1 t1Δt @wi 1 2 σij t1Δt 5 σij t1Δt 5 2 @ xj 2 @ xj @wi

t 1 Δt

σij

@wi @t1Δt xj

1

t 1 Δt

σij

@wi @t1Δt xj

! :

(2.41) As both i and j are dummy indices that represent summation of the terms by varying i and j over their ranges and can be interchanged. Therefore t 1 Δt

@wi

1 σij t1Δt 5 2 @ xj

t 1 Δt

σij

@wi

1

@t1Δt xj

t 1 Δt

σji

@wj @t1Δt xi

! :

(2.42)

Using symmetry of Cauchy’s stress tensor σij , Eq. (2.42) can be rewritten as t 1 Δt

σij

@wi @t1Δt xj

5

t 1 Δt

1 @wi @wj σij 1 t1Δt t1Δt 2 @ xj @ xi

! εij ðwÞ; 5 σt1Δt ij

(2.43)

where t 1 Δt

! 1 @wi @wj εij ðwÞ 5 1 t1Δt : 2 @t1Δt xj @ xi

(2.44)

Using this modification, the weighted residual form in Eq. (2.36) finally becomes ð

t 1 Δt t1Δt

V

σij

t 1 Δt

ð εij ðwÞd

t1Δt

V5

t 1 Δt t1Δt

St

ti wi dt1Δt S:

(2.45)

Eq. (2.45) represents the weak form formulation of the strong form given in Eq. (2.34). It is called weak form because the highest order of the derivative has been reduced in the equation. Compared with the strong form, a trial solution of a lesser degree will be sufficient to approximate the solution for Eq. (2.45) implying that the requirement for the continuity of the trial solution in this formulation is weaker.

2.3.2 Formulation of elemental equations To begin the FEM formulation of Eq. (2.45), it is first converted into array representation as follows:

Finite element method modeling of hydraulic and thermal autofrettage processes

ð

t 1 Δt t1Δt

where

 fεij ðwÞgT t 1 Δt σij t 1 Δt dt 1 Δt V 5

V

ð t1Δt

fwi gT St

t 1 Δt  t1Δt ti d S:

9 9 8 t 1 Δt 8 t 1 Δt εxx ðwÞ > σxx > > > > > > > > > > t 1 Δt > t 1 Δt > > εyy ðwÞ > σyy > > > > > > > > > = = < t 1 Δt < t 1 Δt



 εzz ðwÞ σzz t 1 Δt t 1 Δt ; ; εij ðwÞ 5 σij 5 t 1 Δt t 1 Δt 2 ε ðwÞ > σxy > > > > > > > > > > t 1 Δt xy > t 1 Δt > > > > > > 2 ε ðwÞ > σ > > > > ; ; : t 1 Δt yz : t 1 Δt yz > 2 εzx ðwÞ σzx 8 t 1 Δt 9 8 9 tx = < < wx = t 1 Δt fti g 5 t 1 Δt ty ; fwi g 5 wy : : t 1 Δt ; : ; wz tz

47

(2.46)

(2.47)

According to the Galerkin method, the approximation for the weight vector fwg appearing in Eq. (2.47) must be done in the same way as the incremental displacement ft Δug is approximated. The incremental displacement vector for a volume element is given by 8 9 < t Δux = Δuy 5 t ½φ fΔuge ; (2.48) t fΔug 5 :t ; Δu z t where t ½φ is the shape function matrix 2 3 T t fφ1 g 6 7 t ½φ 5 4 t fφ2 gT 5 T t fφ3 g shown in expanded form as t

fφ1 gT 5

t

N 1 ; 0; 0; t N 2 ; 0; 0; t N 3 ; 0; 0; . . .; t N n ; 0; 0g;

fφ2 gT 5 0; t N 1 ; 0; 0; t N 2 ; 0; 0; t N 3 ; 0; . . .; 0; t N n ; 0g;

t fφ3 gT 5 0; 0; t N 1 ; 0; 0; t N 2 ; 0; 0; t N 3 ; . . .; 0; 0; t N n g; t

(2.49)

(2.50)

where t N i are the shape functions for the n-node element and t fΔuge is the incremental displacement vector at the specific nodes for the n-node element and is expressed as follows: eT t fΔug 5

n

o 1 1 1 n n n Δu ; Δu ; Δu ; . . .; Δu ; Δu ; Δu t t x t y t z x t y t z :

(2.51)

When the domain t1ΔtV is discretized into volume elements, the surface domain S at the boundary is automatically discretized into area elements. For example,

t1Δt

48

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Figure 2.7 An 8-node brick element.

if the domain t1ΔtV is discretized using 8-node brick elements as shown in Fig. 2.7, the surface domain t1ΔtS is discretized into 4-node quadrilateral elements. Following this, the elemental equations are derived from Eq. (2.46). Applying the same approximation procedure for the weight vector fwi g; 8 9 < wx = fwi g 5 wy 5 t 1 Δt ½φ fwge ; : ; wz

(2.52)

where fwge is the weight vector corresponding to the specific nodes for the n-node brick element and is given as n o fwgeT 5 w1x ; w1y ; w1z ; . . .; wnx ; wny ; wnz :

(2.53)

For the term t 1 Δt εij ðwÞ, we can write t 1 Δt



 εðwÞ 5 t 1 Δt ½BL  fwge ;

(2.54)

where 3

 T φ1 ;x 7 6 7 6 t 1 Δt  T φ 2 ;x 7 6 7 6 7 6 t 1 Δt  T 7 6 ; φ 3 x 7 6 t 1 Δt ½BL  5 6 7:



 6 t 1 Δt φ ;T 1 t 1 Δt φ ;T 7 2 x 1 y 7 6 7 6 6 t 1 Δt  T t 1 Δt  T 7 φ 3 ;y 1 φ 2 ;z 7 6 5 4

 T t 1 Δt  T t 1 Δt φ 3 ;x 1 φ 1 ;z 2

t 1 Δt

(2.55)

Finite element method modeling of hydraulic and thermal autofrettage processes

49

Similarly, for the term on the right hand side of Eq. (2.37), the approximation of the weights for the area elements is given as 8 9 < wx = fwi g 5 wy 5 t 1 Δt ½φb fwge ; (2.56) : ; wz where

2

bT 3 φ1 6 t 1 Δt bT 7 t 1 Δt b 6 7; ½φ 5 4 φ2 5

 t 1 Δt φ3 bT t 1 Δt

(2.57)

where t 1 Δt ½φb is the shape function matrix for the area element shown in expanded form as bT t1Δt b N 1 ; 0; 0; t1Δt N b2 ; 0; 0; t1Δt N b3 ; 0; 0;...; t1Δt N bnb ; 0; 0g; fφ1 g 5f



 bT t1Δt φ2 5 0; t1Δt N b1 ; 0; 0; t1Δt N b2 ; 0; 0; t1Δt N b3 ; 0;...; 0; t1Δt N bnb ; 0 ;

bT

 t1Δt φ3 5 0; 0; t1Δt N b1 ; 0; 0; t1Δt N b2 ; 0; 0; t1Δt N b3 ;...; 0; 0; t1Δt N bnb ; (2.58) t1Δt

where t 1 Δt N bi are the shape functions for the boundary. The weight vector fwgb of a surface element at the boundary is given as n o b1 b1 bnb bnb bnb fwi gbT 5 wb1 : x ; wy ; wz ; . . .; wx ; wy ; wz

(2.59)

If the total number of elements used is Ne, using Eqs. (2.54) and (2.56) in Eq. (2.46) and following the elemental assembly procedure for FEM, we have Ne X

Nb X

e

e fwi geT t 1 Δt fi in 5 fwi gbTt 1 Δt fi ex ;

e51

(2.60)

b51

where t 1 Δt

e ffi gin

ð 5

t 1 Δt t1Δt

½BL T t 1 Δt fσij gdt1Δt V;

(2.61)

½φbT t 1 Δt ftgdt1Δt S:

(2.62)

Ve

and t 1 Δt

b

ff gex 5

ð

t 1 Δt t1Δt b St

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

b e where t 1 Δt ffi gin is the elemental internal P force vector and t 1 Δt fi ex is the elemental force vector. The summation symbol represents the assembling operation of the elements. The representation of Eq. (2.60) in the assembled form is given by fW gT

t 1 Δt

fF gin 5 fW gT

t 1 Δt

fF gex ;

(2.63)

where fFgin is the global internal force vector, fFgex is the global external force vector, and fWg is the global weight vector. Since fWg is arbitrary, Eq. (2.63) reduces to t 1 Δt

fF gex 2 t 1 Δt fF gin 5 f0g:

(2.64)

2.3.3 Solution method The discretized equilibrium equation given in Eq. (2.64) is a nonlinear equation of the incremental displacement vector t fΔug. To solve it, a numerical method must be applied using an iterative procedure. It is generally solved by using the Newton
Raphson method. The iterative procedure in the Newton
Raphson method is implemented by using the following equation: t 1 Δt

i fF gin

5

t 1 Δt

 fFin g ðiÞ ði 2 1Þ ; t fΔU g 2 t fΔU g @t fΔU g t fΔU g5t fΔUgði21Þ

i 2 1@ fF gin

t1Δt

(2.65) where t fΔUgði 2 1Þ is the global incremental displacement vector at time t in the previous iteration and t fΔUgðiÞ is the corresponding vector in the current iteration. The preceding equation can be rearranged in the following form: t 1 Δt

½K

ði 2 1Þ  ðiÞ t fΔU g

where t 1 Δt ½K t 1 Δt

½K

ði 2 1Þ

ði 2 1Þ

ði 2 1Þ 2 t fΔU gði 2 1Þ 5 t 1 Δt fRg ;

(2.66)

is the global stiffness matrix given by

@t1Δt fFin g 5 ; @t fU g t fΔU g5t fΔU gði21Þ

(2.67)

ði 2 1Þ

denotes the unbalanced force between the external and the vector t 1 Δt fRg ði 2 1Þ force vector t 1 Δt fF gex and the internal force vector t 1 Δt fF gin calculated in the (i 2 1)th iteration. Therefore t 1 Δt

fRg

ði 2 1Þ

ði 2 1Þ

5 t 1 Δt fF gex 2 t 1 Δt fF gin

:

(2.68)

Finite element method modeling of hydraulic and thermal autofrettage processes

51

The Newton
Raphson equation given by Eq. (2.65) is for the assumption that the internal force vector t 1 Δt fFin g is a function of fΔU g and independent of the external force vector t 1 Δt fF gex . If t 1 Δt fFin g is instead a function of t 1 Δt fFex g; the Newton
Raphson method given by Eq. (2.65) is applied with the slight modification that fΔU g is replaced with t 1 Δt fFex g. The solution procedure is initiated using a guess solution of t fΔU g, which is obtained by solving the following equation: t 1 Δt

½K tt fΔU g 5 t 1 Δt fF gex :

(2.69)

The iterative method is implemented in the following way—the global displacement vector t fΔU gði 2 1Þ in the (i 2 1)th iteration is used to update the domain to t 1 Δt ði 2 1Þ V . Corresponding to the domain t 1 Δt V ði 2 1Þ , the shape function matrix ði 2 1Þ t 1 Δt ½φ is calculated. The incremental displacement vector is approximated for an element e as t fΔug

5 t 1 Δt ½φ

ði 2 1Þ

t fΔug

eði 2 1Þ

;

(2.70)

where t fΔugeði 2 1Þ is the incremental displacement vector for the element in the (i 2 1)th iteration. Using the approximation of t fΔug, the logarithmic strain is calculated using the strain displacement equations in Eqs. (2.24)
(2.26). Subsequently, the incremental stress t 1 Δt σði 2 1Þ is determined using the incremental stress
strain relations. Then finally the internal force vector is calculated from Eq. (2.61) as t 1 Δt

eði 2 1Þ

ff gin

ð 5

t1Δt

t 1 Δt

½BL ði 2 1ÞT t 1 Δt fσgði 2 1Þ dt1Δt V;

(2.71)

V eði21Þ

During the implementation of the solution, the global internal force vector t 1 Δt

ði 2 1Þ

t 1 Δt

ði 2 1Þ

fF gin , it is assumed that the external force vector t 1 Δt fF gex is independent of t fΔU g. The iterative Eq. (2.65) is to be solved until the unbalanced force vector

fRg converges to a very small value. The convergence of the procedure can be provided using a tolerance criterion as follows: :t 1 Δt fRg :

t 1 Δt

ði 2 1Þ

fF gex :

:

# β;

(2.72)

where β is a tolerance parameter and :fai g: denotes the norm of an array defined as :fai g: 5

X

a2i :

(2.73)

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

The Newton
Raphson method given by Eq. (2.65) is called the full Newton
Raphson iterative scheme. In this scheme, the global stiffness matrix ði 2 1Þ t 1 Δt ½K is updated in each iteration. This process takes a lot of computational time. To overcome this, a modified Newton
Raphson scheme is sometimes used in which the global stiffness is kept as a constant t ½K matrix for every iteration.

2.4

Typical results of FEM modeling of hydraulic and thermal autofrettage

In this section, the typical results of hydraulic and thermal autofrettage are shown using commercial FEM package ABAQUS. The solution methods for the nonlinear FEM equation using Newton
Raphson method are all available in this package. The package has been used simply as a tool. Thick-walled cylinders subjected to hydraulic autofrettage or thermal autofrettage maybe of the plane stress or the generalized plane strain configuration. The plane stress configuration is applicable for a short cylinder with free ends where the out of plane axial stress can be taken as zero. The generalized plane configuration is applicable for long cylinder and again may have one of the three types of end conditions—closed-end, open-end, and plane strain with zero axial strain. Most of the FEM analyses of hydraulic autofrettage have been carried out for open-ended cylinders [30
33]. Gibson et al. [33] carried out the FEM analysis of hydraulic autofrettage for all the above-mentioned end conditions and compared the residual hoop stress distributions obtained from the various end conditions. It was observed that the residual hoop stress distributions obtained from plane stress and open-ended conditions were similar. Likewise, it was also observed that the residual hoop stress distributions obtained from the plane strain and closed-ended conditions were similar. The FEM simulation was carried out in ANSYS. For thermal autofrettage, the FEM analysis has been carried out by Kamal [19] only for the open-ended cylinder. Three FEM modeling approaches have been used by researchers. The first one is the 3D approach in which the cylinder is modeled as a solid part. To reduce the amount of computation the symmetry of the geometry can be utilized by modeling only a quarter of the cylinder shown in Fig. 2.8A. Kamal used the 3D approach for thermal autofrettage analysis of an open-ended cylinder [19]. The second one is the two-dimensional (2D) approach in which cylinder is modeled as a planar circular section. Like in the 3D model, symmetry of the section may be utilized by using only a quarter of the section as shown in Fig. 2.8B. Gibson et al. [33] used this approach for the analysis of the hydraulic autofrettage only for the plane stress condition. The third approach is the axisymmetric approach in which the cylinder is modeled by its longitudinal section as shown in Fig. 2.8C. This is the most popular approach and has been used by many researchers [30
33]. Gibson et al. [33] used this approach to model all the end conditions of a thick-walled cylinder except for the plane stress condition. The plane stress condition was modeled using the 2D

Finite element method modeling of hydraulic and thermal autofrettage processes

53

Z Y Z Y

X

(A)

X

Z Y

X

(B)

(C)

Figure 2.8 Schematic of (A) 3D model, (B) 2D model, and (C) axisymmetric model. The symbols a, b, and r represent the inner radius, outer radius, and an intermediate radius of the cylinder. 2D, Two-dimensional; 3D, three-dimensional.

approach. For the present discussion the results will be shown for all the three modeling approaches discussed before.

2.4.1 Results of hydraulic autofrettage The results of hydraulic autofrettage are shown for the plane stress and plane strain conditions. This is because of the similarity of results between the open-ended condition and plane stress condition and also the similarity of results between the closed-ended condition and the plane strain condition based on observation made by Gibson et al. [33]. A 5-mm thick ring with inner radius of 15 mm and outer radius of 31 mm is taken for the hydraulic autofrettage analysis for the plane stress condition. For the same radial dimensions, a 90-mm long cylinder is used for the plane strain case. The material is a high-strength low-alloy steel with designation 4333 M4 having Young’s modulus of 207 GPa with material yield strength of 1070 MPa and ultimate tensile strength of 1150 MPa. The material was taken from the experimental study carried out by Stacey et al. [9]. The results of three FEM modeling approaches are validated using analytical results of Avitzur [34]. For the validation using analytical results an elastic perfectly plastic material is taken. Wherever possible, experimental validation is also carried out based on the availability of experimental results from the literature. The type of element for the FEM analysis of the hydraulic autofrettage process is chosen on the basis of the model developed. In the 3D model the element type C3D20R, which is a 20-node quadratic brick element, is used. In the 2D model, element type CPS8R, which is an 8-node biquadratic plane stress quadrilateral

54

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

element, is used for the plane stress end condition, and the element type CPE8R, which is an 8-node plane strain quadrilateral element, is used for the plane strain model. In the axisymmetric model the element type CAX8R, which is an 8-node biquadratic axisymmetric quadrilateral element, is used. Since in autofrettage process, the yielding of the cylinder starts from the inner wall, a finer mesh is used in the vicinity of the inner wall to accurately model the elastic
plastic transition while applying the autofrettage load. This is done using an adaptive meshing scheme for radial dimension of the cylinder in which the size of the elements decreases according to a constant factor B. For n number of elements chosen, the mesh size factor B is defined as Ln 5 L1 B;

(2.74)

where Ln and L1 are the length of the nth element and the first element, respectively. The factor B was taken as 5 in all the models. A typical adaptive meshing with B 5 5 for n 5 10 elements along the radial direction of a quadrant of circular section is shown in Fig. 2.9. The boundary conditions are described in Table 2.1 for all the models. The symbols UX, UY, and UZ represent the degrees of freedom for the linear translations along the X, Y, and Z directions, respectively. For the loading condition, the autofrettage pressure is denoted by p. An autofrettage pressure of 600 MPa is used. Although the yield strength of the material is 1070 MPa, a 600 MPa pressure causes sufficient plastic deformation due to combined effect of hoop, radial, and axial stresses. The load is applied incrementally using a step size of 0.01, implying that the entire load is applied in 100 steps. This is for the accurate determination of the minimum autofrettage load at which the cylinder starts yielding during the autofrettage process. The cumulative load increment at which the von Mises stress becomes equal to the yield stress of the cylinder material gives the yield pressure of the cylinder. Smaller the size of the load increment used, the more is the accuracy of the prediction of the yield pressure.

2.4.1.1 Results for plane stress condition of hydraulic autofrettage The comparisons of the elastic-plastic and residual stress distributions from the FEM and analytical models [33] for the plane stress condition are shown in Figs. 2.10A
C and 2.11A
C, respectively. The radial stress, hoop stress, and axial stress have been symbolized by σr , σθ , and σz , respectively. A very close agreement can be observed in all the comparisons. The analytical and FEM-based results are compared and summarized in Table 2.2. From Table 2.2, it can be observed that all the FEM models give more or less the same results compared with the analytical model, but the axisymmetric model is the most computationally efficient. Hence, the axisymmetric model is selected as the representative FEM model for experimental validation. The experimental results for the same material and cylinder dimensions used in the plane stress FEM model

Finite element method modeling of hydraulic and thermal autofrettage processes

55

Figure 2.9 Adaptive meshing method in a quadrant of circular section with 10 elements along the radial direction and B 5 5.

are available from the experimental study carried out by Stacey et al. [9] for the measurement of residual stresses setup in hydraulic autofrettage process using neutron diffraction method. The autofrettage pressure in the experimental study was 662 MPa. The experimental validation is carried out for two FEM models based on a nonhardening and a linear kinematic hardening material behavior. The linear kinematic hardening model is based on Ziegler’s hardening law. The slope of the linear plastic stress
strain curve in Eq. (2.22) for the material 4333 M4 is taken as 7350 MPa. The result of experimental validation based on the nonhardening and linear kinematic hardening model is shown in Fig. 2.12A and B, respectively. In the validation, the FEM model based on the nonhardening material behavior overestimates the maximum compressive hoop stress at the inner surface by a margin of about 193 MPa compared with the experimental value. Based on the linear kinematic hardening material behavior, the FEM model predicted the maximum compressive hoop stress at the inner surface by an overestimated margin of about 166 MPa compared with the experimental value. This shows the effect of the

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Table 2.1 Boundary conditions for different FEM models of hydraulic autofrettage. 3D model Loading condition: p 5 600 MPa at ADHE Boundary conditions: 1. UY 5 0 at ABCD 2. UX 5 0 at EFGH 3. UZ 5 0 at ABFE and DCGH for pure plane strain end condition

2D model Loading condition: p 5 600 MPa at AD Boundary conditions: 1. UY 5 0 at AB 2. UX 5 0 at CD

Axisymmetric model Loading condition: p 5 600 MPa at AD Boundary condition: 1. UZ 5 0 at AB and CD for plane strain condition

2D, Two-dimensional; 3D, three-dimensional.

consideration of the kinematic hardening behavior. In the comparison shown by Stacey et al. [9] the analytical solution, which was based on Tresca yield criterion, overestimated the maximum compressive stress by a much larger margin of 350 MPa. The improved match with the experimental data in both cases of the FEM models based on the nonhardening and linear kinematic hardening material

Finite element method modeling of hydraulic and thermal autofrettage processes

57

(B)

(A)

(C)

Figure 2.10 Elastoplastic stress distribution from the (A) 3D FEM model, (B) 2D FEM model, and (C) axisymmetric FEM model of hydraulic autofrettage for the plane stress condition. 2D, Two-dimensional; 3D, three-dimensional.

behavior is because the FEM solution is based on the von Mises criterion [35] and gives a more realistic prediction of the actual material behavior. Stacey et al. [9] also used an elastic perfectly plastic material behavior and neglected the Bauschinger effect in their analytical model. Since the FEM model based on the linear kinematic hardening behavior incorporates the Bauschinger effect, this is also another cause for the better match with the experimental results.

2.4.1.2 Results for plane strain end condition of hydraulic autofrettage The comparisons of the elastic
plastic and residual stress distributions from the FEM and analytical models [33] for the plane strain condition are shown in Figs. 2.13A
C and 2.14A
C, respectively. A very close agreement can be observed in all the comparisons. The analytical and FEM-based results are compared and summarized in Table 2.3.

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

(B)

(A)

(C)

Figure 2.11 Residual stress distribution from the (A) 3D FEM model, (B) 2D FEM model, and (C) axisymmetric FEM model of hydraulic autofrettage for the plane stress condition. 2D, Two-dimensional; 3D, three-dimensional.

2.4.2 Results of thermal autofrettage For thermal autofrettage the results are shown for plane stress and open-ended condition in a long cylinder. This is based on the availability of experimental data and analytical solution for validation. A 5-mm long disc with an inner radius of 10 mm and an outer radius of 25 mm is taken for the plane stress condition. For the same radial dimensions as used in the plane stress condition, a 90-mm long cylinder is taken for the open-ended condition. The material for thermal autofrettage is stainless steel SS304 having yield strength of 215 MPa, Young’s modulus of 200 GPa, thermal conductivity of 16.2 W/m K, and coefficient of thermal expansion of

Finite element method modeling of hydraulic and thermal autofrettage processes

59

Table 2.2 Comparison of the analytical and FEM models of hydraulic autofrettage for plane stress condition. Model

Elastic
plastic interface radius (mm)

Yield pressure (MPa)

CPU time (s)

Analytical model 3D model 2D model Axisymmetric model

17.43 17.31 17.44 17.44

469 471 471 468

90 7330 81 14

2D, Two-dimensional; 3D, three-dimensional.

(A)

(B)

Figure 2.12 Experimental validation of plane stress FEM models of hydraulic autofrettage based on (A) nonhardening and (B) linear kinematic hardening material behavior. Source: Experimental results are taken from A. Stacey, H.J. MacGillivary, G.A. Webster, P.J. Webster, K.R.A. Ziebeck, Measurement of residual stresses by neutron diffraction, J. Strain Anal. Eng. Des. 20 (1985) 93
100.

17.2 3 1026/ C. The material properties have been taken from the experimental study of Kamal et al. [16]. The three models are first validated using analytical model of Kamal [19], and then wherever possible experimental validation is carried out. For the validation using analytical models, hardening is not considered. In the 3D model, the element type C3D20RT, which is a 20-node thermally coupled quadrilateral brick element, is used. In the 2D model, element type CPS8RT, an 8-node thermally coupled plane stress quadrilateral element, is used for the plane stress end condition, and the element type CPEG8RHT, an 8-node generalized plane strain thermally coupled quadrilateral element, is used for the open-end model. In the axisymmetric model the element type CAX8TR, an 8-node

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

(B)

(A)

(C)

Figure 2.13 Elastic
plastic stress distribution from the (A) 3D FEM model, (B) 2D FEM model, and (C) axisymmetric FEM model of hydraulic autofrettage for the plane strain condition. 2D, Two-dimensional; 3D, three-dimensional.

biquadratic axisymmetric thermally coupled quadrilateral element, is used. Like in the case of the hydraulic autofrettage process, a finer mesh is used in the vicinity of the inner wall of the cylinder using the adaptive meshing scheme. Here also, the mesh factor B is also used as 5 in all the models. Except for the loading conditions, the mechanical boundary conditions used in thermal autofrettage are identical to the ones used for the hydraulic autofrettage and shown in Table 2.1. The loading condition in thermal autofrettage is provided using boundary conditions for temperature. A temperature difference of 120 C was used in the FEM models of thermal autofrettage process. A temperature Ta 5 25 C is provided to the inner wall, and the outer wall is assigned a temperature Tb 5 145 C for both end conditions. In the 3D model, Ta 5 25 C at ADHE and Tb 5 145 C at BGCF are given as loading conditions. Similarly, for the 2D model, Ta 5 25 C at AD and Tb 5 145 C at BC are given and for the axisymmetric model, Ta 5 25 C at AD and Tb 5 145 C at BC given as the loading conditions. Like in the case of hydraulic autofrettage, the temperature boundary conditions are also applied incrementally by using a fixed step size of 0.01. This is for the accurate determination of

Finite element method modeling of hydraulic and thermal autofrettage processes

61

(B)

(A)

(C)

Figure 2.14 Residual stress distribution from the (A) 3D FEM model, (B) 2D FEM model, and (C) axisymmetric FEM model of hydraulic autofrettage for the plane strain condition. 2D, Two-dimensional; 3D, three-dimensional.

the minimum temperature difference at which the cylinder starts yields during the thermal autofrettage.

2.4.2.1 Results for plane stress end condition of thermal autofrettage The comparisons of the elastic
plastic and residual stress distributions from the FEM and the analytical models [19] for the plane stress condition are shown in Figs. 2.15A
C and 2.16A
C, respectively. In Fig. 2.15A
C a very good match can be seen for the radial stresses. However, in the hoop stress, slight deviation is

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Table 2.3 Comparison of the analytical and FEM models of hydraulic autofrettage for plane strain condition. Model

Elastic
plastic interface radius (mm)

Yield pressure (MPa)

CPU time (s)

Analytical model 3D model 2D model Axisymmetric model

17.27 17.31 17.31 17.27

473.1809 474 474 474

93 6669 82 15

2D, Two-dimensional; 3D, three-dimensional.

(B)

(A)

(C)

Figure 2.15 Elastic
plastic stress distribution from the (A) 3D FEM model, (B) 2D FEM model, and (C) axisymmetric FEM model of thermal autofrettage for the plane stress condition. 2D, Two-dimensional; 3D, three-dimensional.

Finite element method modeling of hydraulic and thermal autofrettage processes

63

(B)

(A)

(C)

Figure 2.16 Residual stress distribution from the (A) 3D FEM model, (B) 2D FEM model, and (C) axisymmetric FEM model of thermal autofrettage for the plane stress condition. 2D, Two-dimensional; 3D, three-dimensional.

observed. This is because the FEM model is based on von Mises yield criterion, while the analytical model is based on Tresca yield criterion. The analytical and FEM-based results are compared and summarized in Table 2.4.

2.4.2.2 Results for open-ended condition of thermal autofrettage The comparisons of the elastic
plastic and residual stress distributions from the FEM and analytical models [19] for the open-ended condition are shown in Figs. 2.17A
C and 2.18A
C, respectively. In Fig. 2.17A
C, some deviation between analytical and FEM result is observed in the plastic part. This is because

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Table 2.4 Comparison of the analytical and FEM models of thermal autofrettage for plane stress condition. Model

Elastic
plastic interface radius (mm)

Yield temperature difference ( C)

CPU time (s)

Analytical model 3D model 2D model Axisymmetric model

10.82 10.63 10.63 10.74

96.93 97.20 97.20 96.00

464.67 3070.1 44.90 19.50

2D, Two-dimensional; 3D, three-dimensional.

(B)

(A)

(C)

Figure 2.17 Elastic
plastic stress distribution from the (A) 3D FEM model, (B) 2D FEM model, and (C) axisymmetric FEM model of thermal autofrettage for the open-ended condition. 2D, Two-dimensional; 3D, three-dimensional.

Finite element method modeling of hydraulic and thermal autofrettage processes

65

(B)

(A)

(C)

Figure 2.18 Residual stress distribution from the (A) 3D FEM model, (B) 2D FEM model, and (C) axisymmetric FEM model of thermal autofrettage for the open-ended condition. 2D, Two-dimensional; 3D, three-dimensional.

the FEM model is based on von Mises yield criterion while the analytical model is based on Tresca yield criterion. The analytical and FEM-based results are compared and summarized in Table 2.5. From Table 2.5, it can be observed that the axisymmetric model is the most computationally efficient model. The axisymmetric model for the open-ended condition of thermal autofrettage is chosen as the representative FEM model for the experimental validation. The experimental results for the same material properties used in the open-ended condition of the FEM model are available from the experimental study carried out by Kamal et al. [16]. The experimental specimen was a 60 mm long cylinder with inner radius of 10 mm and outer radius of 20 mm. The autofrettage temperature difference was 130 C. For the experimental validation,

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Table 2.5 Comparison of the analytical and FEM models of thermal autofrettage for openended condition. Model

Elastic
plastic interface radius (mm)

Yield temperature difference ( C)

CPU time (s)

Analytical model 3D model 2D model Axisymmetric model

12.19 11.95 11.95 11.96

68.82 67.20 67.20 68.40

469.66 28800.53 40.200 22.200

2D, Two-dimensional; 3D, three-dimensional.

Figure 2.19 Experimental validation for the open-ended condition of the FEM model of thermal autofrettage based on the results of Kamal and Dixit [18].

a kinematic hardening model based on Ziegler’s hardening law is used. The slope of the linear plastic stress
strain curve H 0 in Eq. (2.23) for the material SS304 is taken as 10504 MPa. The result of the experimental validation is shown in Fig. 2.19.

2.5

Conclusion

In this chapter the FEM modeling of hydraulic and thermal autofrettage processes is described. The autofrettage process is first introduced. The various types of autofrettage processes are then described and briefly reviewed. After familiarizing with

Finite element method modeling of hydraulic and thermal autofrettage processes

67

the basic concept of the process, the governing differential equations and the formulation used in the modeling of hydraulic and thermal autofrettage are explained. Finally, the typical results of FEM simulation of hydraulic and thermal autofrettage processes are presented. The FEM simulations are carried out in commercial FEM package ABAQUS. Each autofrettage process is modeled using the three approaches—3D, 2D, and axisymmetric. The three models in each process are compared with existing analytical models. For this comparison, the material is assumed to behave as elastic
perfectly plastic without any hardening. From this comparison the most accurate and computationally efficient FEM model is then used as the representative FEM model to further carry out experimental validation. It is shown that the axisymmetric model approach is the most computationally efficient for both types of autofrettage processes. For the experimental validation a linear kinematic hardening model is considered to incorporate the Bauschinger effect. In the experimental validation of the autofrettage processes, it is shown that the selection of the von Mises yield criterion and the incorporation of kinematic hardening model in the FEM model resulted in a much better matching with the experimental results compared with a case of analytical model where the hardening and the Bauschinger effects are neglected. The central focus of this chapter is on the FEM modeling of hydraulic and thermal autofrettage processes. A review of the other remaining types of autofrettage processes and their modeling approaches can be found in Ref. [36]. It is to be mentioned that scant attention has been paid to the microstructural aspect of autofrettage process. In reality a material that has been cold worked or heated induces microstructural changes. This is a potential research area in autofrettage. Also most researchers have discussed the benefits of autofrettage in strengthening thick-walled vessels. However, Banks-Sills and Marmur [37] observed a slight detrimental effect of autofrettage that it reduced the fracture toughness of the cylinder. However, the authors based their experiments on a particular material (gun steels) and did not carry out extensive study covering a variety of materials; it is a worth-investigating topic. The stress corrosion cracking of autofrettaged cylinders can also be studied. For avoiding stress corrosion cracking, both internal and external surfaces of the pressure vessel should have compressive residual stresses. The traditional methods of generating stress corrosion cracking on the outer surface of the pressure vessels are shot peening and wire-winding. Recently, Shufen and Dixit [38] have proposed a heat treatment method to generate compressive residual stresses on the outer surface of a thermally autofrettaged cylinder; a detailed experimental study of the efficacy of the proposed scheme is needed.

References [1] D.W.A. Rees, A theory of autofrettage with applications to creep and fatigue, Int. J. Press. Vessels Pip. 30 (1987) 57
76. [2] D.W.A. Rees, Autofrettage theory and fatigue life of open-ended cylinders, J. Strain Anal. Eng. Des. 25 (1990) 109
121.

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[3] A.P. Parker, J.R. Farrow, Stress intensity factors for multiple radial cracks emanating from the bore of an autofrettaged or thermally stressed, thick cylinder, Eng. Fract. Mech. 14 (1981) 237
241. [4] R.H. Jones, R.E. Ricker, R.H. Jones, Mechanisms of Stress Corrosion Cracking in Stress Corrosion Cracking—Materials Performance and Evaluation, ASM International, Materials Park, OH, 1992, p. 1. [5] M.J. Bastable, From breechloaders to monster guns: Sir William Armstrong and the invention of modern artillery, 1854
1880, Technol. Cult. 33 (1992) 213
247. [6] L. Jacob, La Re´sistance et L’e´quilibre E´lastique des Tubes Frette´s, Me´moire de L’artillerie Nav. 1 (1907) 43
155. [7] J. Chakrabarty, Theory of Plasticity, McGraw-Hill Book Company, New York, 1987. [8] M.C. Gibson, Determination of Residual Stress Distributions in Autofrettaged Thick Cylinders (Ph.D. thesis), Defence College of Management and Technology, Cranfield University, UK, 2008. [9] A. Stacey, H.J. MacGillivary, G.A. Webster, P.J. Webster, K.R.A. Ziebeck, Measurement of residual stresses by neutron diffraction, J. Strain Anal. Eng. Des. 20 (1985) 93
100. [10] T.E. Davidson, C.S. Barton, A.N. Reiner, D.P. Kendall, New approach to the autofrettage of high-strength cylinders, Exp. Mech. 2 (1962) 33
40. [11] M. Perl, The change in overstrain level resulting from machining of an autofrettaged thick-walled cylinder, ASME J. Press. Vessels Technol. 122 (2000) 9
14. [12] P.C. Chen, A Simple Analysis of the Swage Autofrettage Process (No. ARCCB-TR88030), Army Armament Research Development and Engineering Center Watervliet, NY, Benet Weapons Lab, 1988. [13] A.P. Parker, G.P. O’Hara, J.H. Underwood, Hydraulic versus swage autofrettage and implications of the Bauschinger effect, ASME J. Press. Vessels Technol. 125 (2003) 309
314. [14] J.D. Mote, L.K. Ching, R.E. Knight, R.J. Fay, M.A. Kaplan, Explosive Autofrettage of Cannon Barrels, Army Materials and Research Center, Watertown, MA, 1971. No. AMMRC CR 70-25. [15] S.M. Kamal, U.S. Dixit, Feasibility study of thermal autofrettage of thick-walled cylinders, ASME J. Press. Vessels Technol. 137 (2015) 061207-1
061207-18. [16] S.M. Kamal, A.C. Borsaikia, U.S. Dixit, Experimental assessment of residual stresses induced by the thermal autofrettage of thick-walled cylinders, J. Strain Anal. Eng. Des. 51 (2016) 144
160. [17] S.M. Kamal, U.S. Dixit, A comparative study of thermal and hydraulic autofrettage, J. Mech. Sci. Technol. 30 (2016) 2483
2496. [18] S.M. Kamal, U.S. Dixit, A study on enhancing the performance of thermally autofrettaged cylinder through shrink-fitting, ASME J. Manuf. Sci. Eng. 138 (2016) 0945011
094501-5. [19] S.M. Kamal, A Theoretical and Experimental Study of Thermal Autofrettage Process (Ph.D. thesis), IIT, Guwahati, 2016. [20] H.R. Zare, H. Darijani, A novel autofrettage method for strengthening and design of thick-walled cylinders, Mater. Des. 105 (2016) 366
374. [21] P.M. Dixit, U.S. Dixit, Modeling of Metal Forming and Machining Processes: by Finite Element and Soft Computing Methods, Springer -Verlag, London, 2008. [22] U.S. Dixit, S.N. Joshi, J.P. Davim, Incorporation of material behavior in modeling of metal forming and machining processes: a review, Mater. Des. 32 (2011) 3655
3670.

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[23] G.S. Wang, An elastic-plastic solution for a normally loaded center hole in a finite circular body, Int. J. Press. Vessels Pip. 33 (1988) 269
284. [24] H. Jahed, R.N. Dubey, An axisymmetric method of elastic-plastic analysis capable of predicting residual stress field, ASME J. Press. Vessels Technol. 119 (1997) 264
273. [25] S. Alexandrov, W. Jeong, K. Chung, Descriptions of reversed yielding in internally pressurized tubes, ASME J. Press. Vessels Technol. 138 (2016) 011204-1
011204-10. [26] W. Prager, The theory of plasticity: a survey of recent achievements, Proc. Inst. Mech. Eng. 169 (1955) 41
57. [27] H. Ziegler, A modification of Prager’s hardening rule, Q. Appl. Math. 17 (1959) 55
65. [28] K.J. Bathe, Finite Element Procedures, PHI Learning, New Delhi, 1996. [29] T.T. Hsu, The Finite Element Method in Thermomechanics, Allen & Unwin, London, 2012. [30] P.C.T. Chen, G.P. O’Hara, Finite Element Results of Pressurized Thick Tubes Based on Two Elastic-Plastic Material Models (No. ARLCB-TR-83047), Army Armament Research and Development Center Watervliet NY Large Caliber Weapon Systems Lab, 1983. [31] G.H. Majzoobi, G.H. Farrahi, A.H. Mahmoudi, A finite element simulation and an experimental study of autofrettage for strain hardened thick-walled cylinders, Mater. Sci. Eng., A 359 (2003) 326
331. [32] J.M. Alegre, P. Bravo, M. Preciado, Design of an autofrettaged high-pressure vessel, considering the Bauschinger effect, Proc. Inst. Mech. Eng., E: J. Process. Mech. Eng. 220 (2006) 7
16. [33] M.C. Gibson, A. Hameed, A.P. Parker, J.G. Hetherington, A comparison of methods for predicting residual stresses in strain-hardening, autofrettaged thick cylinders, including the Bauschinger effect, ASME J. Press. Vessels Technol. 128 (2006) 217
222. [34] B. Avitzur, Autofrettage—stress distribution under load and retained stresses after depressurization, Int. J. Press. Vessels Pip. 57 (1994) 271
287. [35] H. Hibbitt, B. Karlsson, P. Sorensen, Abaqus analysis user’s manual version 6.10. Dassault Syste`mes Simulia Corp. Providence, RI, USA. 2011. [36] R. Shufen, U.S. Dixit, A review of theoretical and experimental research on various autofrettage processes, ASME J. Press. Vessels Technol. 140 (2018) 0508021
050802-15. [37] L. Banks-Sills, I. Marmur, Influence of autofrettage on fracture toughness, Int. J. Fract. 40 (1989) 143
155. [38] R. Shufen, U.S. Dixit, An analysis of thermal autofrettage process with heat treatment, Int. J. Mech. Sci. 144 (2018) 134
145.

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Mechanics of hydroforming

3

Christoph Hartl TH Ko¨ln - Faculty of Automotive Systems and Production, University of Applied Sciences, Cologne, Germany

3.1

Introduction

Hydroforming processes are used for the industrial production of thin-walled, complex-shaped components in numerous sectors, such as the automotive and aerospace industry or sanitary and piping manufacture. In particular, automotive manufacturers benefit from the advantages hydroforming offers for lightweight design of modern vehicles. Exhaust system parts, engine cradles, frame rails, hollow camshafts, and body components are some examples of typical products manufactured in larger quantities [1
4]. Hydroforming encompasses metal-forming processes applying pressurized liquids or gaseous media to plastically form a tubular blank or sheet metal blank into a desired three-dimensional shape. Depending on the type of process, additional mechanical loads are utilized and contribute to the plastic flow of the formed material or support areas of the component during expansion. According to the used form of blank, hydroforming processes are generally divided into two major categories: tube hydroforming and sheet hydroforming (Fig. 3.1) [3]. Contrary to tube hydroforming, sheet hydroforming is used in most cases for low-volume production and niche applications [5]. Fig. 3.2 illustrates the principle of hydroforming for the example of forming a short straight tubular component. The process starts with a cylindrical tube placed into a die cavity of the forming tool that corresponds to the final shape of the component. After closing the tool under the force Fc, the tube is sealed at both ends by punches that enable the filling of the tube with the pressurizing medium. The component is then expanded under the simultaneously controlled action of the internal pressure pi transmitted with the medium and the axial forces Fa applied by the movement of the punches. The loading path with the parameters Fa and pi has to be selected in a way that plastic yielding of the workpiece is maintained and that possible failures, such as wrinkling, buckling, necking, or bursting, are avoided (Fig. 3.3). Hydraulic presses are commonly applied for opening and closing the forming tool, and water
oil emulsions are typical media for pressurization with pressures pi in the range of 1200
2500 bar and up to 4000 bar in certain cases [3,4]. Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques. DOI: https://doi.org/10.1016/B978-0-12-818232-1.00003-5 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Figure 3.1 Classification of hydroforming processes.

Figure 3.2 Principle of tube hydroforming.

Figure 3.3 Range of feasible process controls for tube hydroforming.

Applicable blank materials for hydroforming can be, in general, any material that is appropriate for cold-forming processes. Common semifinished products used for the tubular blanks in automotive production are, for example, longitudinal

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73

welded tubes made from conventional unalloyed or stainless steel and from aluminum alloys or extruded aluminum profiles. Furthermore, the increasing need for weight reduction in vehicle engineering drove the application of advanced highstrength steels and lightweight alloys based on magnesium and titanium in the hydroforming technology [5,6]. Copper and brass alloys are typical materials in piping and sanitary industries for hydroformed products. Typical outer diameters, d0, of tubes in serial production range from about 20 to 140 mm with ratios of tube wall thickness t0 to outer diameter d0 in the size of about 0.02
0.05. Depending on the product, tube lengths of up to 2 m are in use. In many cases the complexity of hydroformed products requires additional preceding forming operations such as bending of the blank according to a curved main axis of the product or a mechanical forming to ensure that the blank can be inserted into the hydroforming die. These prior forming steps introduce strains and modifications of cross sections at the blank that influence crucially the formability and produce quality in the subsequent hydroforming process [7
9]. Various innovations developed in the past few years have aimed to improve the productivity and efficiency of hydroforming [3
5,10]. This concerned, for example, the integration of piercing operations into the hydroforming process to create holes in the formed component [11] and the incorporation of joining processes, for instance, used in camshaft manufacturing [4]. For hydroforming of materials with low formability, several strategies working with elevated temperatures were developed to increase the forming possibilities. This so-called warm hydroforming provides benefits to the automotive industry for the use of lightweight alloys based on aluminum, magnesium, or titanium but also of press-hardened components made from steel [12
16]. For heat-treatable aluminum alloys and press-hardened steel also the feasibility to increase the component strength with a process-integrated thermal treatment was demonstrated [17,18]. Further studies aiming to enhance formability in hydroforming were dealing with pulsed hydroforming methods, applying a slowly varying pressurization in the range of 0.5
1 Hz [19,20]. Recent work on hydroforming has also focused on the manufacturing of components for microsystem technology [21]. The results for tube hydroforming as well as for sheet hydroforming have been presented, showing the feasibility of hydroforming of tubes with outer diameters d0 in the range of 500
2000 μm [22
24] and sheet materials with thicknesses t0 between 20 and 50 μm [25,26]. The design and optimization of hydroforming processes necessitate the knowledge about essential fundamentals with regard to correlations between forming loads and forming results. The objective of this chapter is to provide an overview of the latest developments of corresponding analytical approaches and to generate an in-depth understanding of underlying mechanical relationships and particularities with a focus on tube hydroforming. Section 3.2 summarizes achievements in techniques for the modeling of hydroforming processes. Section 3.3 presents details on the state of the art in approaches to predict forming limits in tube hydroforming, and Section 3.4 introduces optimization methods applied for the determination of suitable loading paths. The chapter concludes with a brief discussion in Section 3.5

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about potential future developments and challenges in the mechanics of tube-hydroforming processes.

3.2

Modeling of plastic deformation in tube hydroforming

Over the past few years, various theoretical models were developed for tube hydroforming to determine relationships between applied forming loads and resulting stresses, strains, and nonreversible change of shape. These models can support, for example, investigations into optimized process controls of the internal pressure and the axial forces versus time, analyses of tube formability, and of material properties as well as of friction conditions in hydroforming. Existing approaches in modeling, discussed in the following sections, can be divided according to the type of analyzed workpiece shape in models for rotationally symmetrical expansions, hydroforming of parts with polygonal cross sections, and hydroforming of tube branches.

3.2.1 Rotationally symmetrical tube expansion The rotationally symmetrical free expansion of straight tubes has turned out to be the most appropriate technique to characterize mechanical properties of tubular materials for hydroforming. It enables the analysis of flow curves, plastic anisotropy, and forming limits under a biaxial stress state as well as the optimization of loading paths and the evaluation of new theoretical models. This process involves a straight tubular specimen with an outer diameter d0 and a wall thickness t0 pressurized on the inside with the internal pressure pi. While the ends of the tube are located in dies, preventing a radial expansion, the central region of the tube is free to expand in a radial direction under the effect of the forming loads. A distinction should be made between the three different kinds of boundary conditions for the axial movement of the ends of the specimen (Fig. 3.4). The tube ends can be fixed at their initial position during the entire forming process (Fig. 3.4A), or be free in their axial movement without any constraint (Fig. 3.4B), or can be subject to a controlled movement by a superimposed axial force Fa (Fig. 3.4C). The chosen type of boundary condition in conjunction with the dimensional parameters of the tube and the die, with the forming behavior of the tube material and with the amount of applied forming loads determines the shape and wall-thickness distribution of the formed specimen. In general, the wall thickness of the specimen decreases in the course of expansion and, as there is no consideration for a surrounding die surface, the radius of curvature of the bulged shape is nonuniform along the meridian. The axial force Fa can be divided, in general, into three components: F a 5 Fz 1 F f 1 F p

(3.1)

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75

Figure 3.4 Type of boundary conditions for tube-expansion tests: (A) with fixed tube ends, (B) with freely movable ends, and (C) with a superimposed axial force.

with the component Fz that is initiated into the tube wall, the component Ff as the friction force between the tube end and the surrounding die, and the sealing force component Fp, which results from the reaction force of the internal pressure pi acting on the front punch face [7,27,28]. In the case of a wall thickness of the formed part that is smaller compared to the radius of curvature, and taking the plane with the maximum expanded diameter d1 into account as plane of symmetry, the middle surface of the bulged shape of the specimen can be regarded as an element of a rotationally symmetrical shell. The membrane theory of shells can be applied to this element, making further simplifying assumptions, such as (1) shear and bending effects being negligible, (2) the normal stress in the direction being transverse to the wall, which can be disregarded, (3) points lying on the normal-to-the-middle plane remaining on the normal-to-themiddle surface after a deformation of the workpiece wall, and (4) stresses and strains being constant along the wall thickness. Fig. 3.5 shows the equilibrium of forces for a rotationally symmetrical membrane element with the distributed loads p1 along the membrane element surface and p3 perpendicular to its surface and the resulting forces per unit length Nθ in circumferential direction and Nϑ in longitudinal direction. In Fig. 3.5, r is the radius from the central axis to a given point P, perpendicular to the axis, rθ is the radius from the central axis to P, perpendicular to the meridian, and rϑ is the radius from the center of curvature of the meridian to P. The angle θ defines the position of the

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Figure 3.5 Equilibrium of forces for a membrane element of a rotationally symmetrical shell.

meridian at point P and ϑ is the angle between the central axis and the normal axis of the shell at P. The equilibrium conditions for this element under the loads p1 and p3 can be written as [29]: @ ðNϑ rθ sinϑÞ 2 Nθ rϑ cosϑ 1 p1 rϑ rθ sinϑ 5 0 @ϑ

(3.2)

Nϑ rθ 1 Nθ rϑ 1 p3 rϑ rθ 5 0

(3.3)

Both these equations can be applied to any shape of rotationally symmetrical shells subjected to the loads p1 and p3 for determining both the unknown variables Nθ and Nϑ along the shell element. Their solution requires boundary conditions to be involved, as depicted in Fig. 3.4, and the contour of the shape to be described mathematically. The derivation of the circumferential and axial stress, σθ and σϑ, then necessitates the knowledge of the wall thickness t, with σθ 5 Nθ/t and σϑ 5 Nϑ/ t. The internal pressure has to be implemented with p3 5 2pi in these equations and the force Fa is part of the constant of integration when solving Eq. (3.2). An example of a model that is appropriate to describe σθ and σϑ along the expanded region was developed by Refs. [30,31], based on the application of the equilibrium of forces at a circular circumferential section of the hydroformed specimen with fixed ends of the tube. In many cases, it has shown to be applicable in regarding, primarily, the stress state and strain state of the hydroformed part within its region of maximum expansion at the diameter d1 for the evaluation of the tube formability and material

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characteristics, for example, Refs. [32
38]. For this approach the stress state was derived from the equilibrium conditions for an element of the shell at d1, taking into account that pi does not act—unlike the assumption for Eqs. (3.2) and (3.3)— on the middle surface of the bulged shape but on its inner side. For the equilibrium condition, it then applies [32]: σϑ1

 ðra1 2 t1 Þ d1 =2 2 t1 t1 t1  1 σθ1 5 pi  ra1 2 t1 =2 d1 =2 2 t1 =2 ra1 2 t1 =2 d1 =2 2 t1 =2

(3.4)

with the wall thickness t1 at the maximum expanded diameter d1 and the local radius of curvature ra1 of the meridian of the outer surface at this point. From the force equilibrium in axial direction at the cross section through the plane that includes d1, the axial stress in case of free, movable, and closed tube ends can be derived as [38]:  2 d1 =22t1 σϑ1 5 pi t1 ðd1 2 t1 Þ

(3.5)

Combining Eqs. (3.4) and (3.5) gives the circumferential stress as [33]: 

d1 =2 2 t1  σθ1 5 pi 2ra1 2 d1 =2 2 t1 t1 ð2ra1 2 t1 Þ

(3.6)

The axial stress under an additional force Fa is often also derived from the force equilibrium similar to Eq. (3.5) with [35]:  2 d1 =22t1 1 1 Fa σϑ1 5 pi πt1 ðd1 2 t1 Þ t1 ðd1 2 t1 Þ

(3.7)

where Fa is to be inserted as a negative value in case of a compressive force. For the circumferential stress σθ at the center of the specimen, it then applies with Eqs. (3.4) and (3.7) [35]: σθ1 5 pi

ðra1 2 t1 Þðd1 2 2t1 Þ d1 2 t 1 2 σϑ t1 ð2ra1 2 t1 Þ 2ra1 2 t1

(3.8)

The plastic strains in the circumferential direction and in radial direction, εθ1 and εt1, at d1 can be determined with: εθ1 5 ln

d 1 2 t1 d 0 2 t0

εt1 5 ln

t1 t0

(3.9)

(3.10)

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

and the local axial plastic strain εϑ1 can be derived from the condition for volume constancy: εϑ 1εθ 1εt 5 0

(3.11)

with: εϑ1 52εt1 2εθ1

(3.12)

Provided that the shape of the expanded region of the tube with its values for d1, t1, and ra1 is known, the stresses σθ1,σϑ1, and the strains εθ1, εt1 can be calculated with the correlations described earlier, and the effective stress σe can be determined according to the von Mises yielding condition with: σe 5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2ϑ 1 σ2θ 2 σϑ σθ

(3.13)

for isotropic plastic deformation, as applied, for example, in Refs. [33,37,39]. The equivalent plastic strain εe is obtained under the assumption of constant strain rates from: 2 εe 5 pffiffiffi 3

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε2θ 1 ε2t 1 εθ εt

(3.14)

The determination of σe and εe in the course of the tube expansion leads to the flow curve of the tube material under the conditions of a biaxial stress state within the formed workpiece wall with σe as a function of εe: σe 5 f ðεe Þ

(3.15)

that allows describing the flow curve for the determination of the yield stress σf mathematically, based on iterative calculations in conjunction with known power laws, for example, the Ludwik
Hollomon law or the Swift law: σf 5 Kεne

(3.16)

σf 5 K ðε0 1εe Þn

(3.17)

with σe 5 σf and the material constants K and n and the prestrain ε0. Likewise, these correlations provide information about the development of stresses and strains, which enables the verification of forming limit criteria. The application of these relationships requires obtaining the values for d1, t1, and ra1 from the experimental results in the course of the forming process. The diameter d1 can be measured during the expansion with corresponding devices or when the workpiece is removed from the tool after a certain amount of forming [36].

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79

This also applies to the recording of t1 [36,38]. Also, assuming a linear relationship between the change in wall thickness and the radial expansion h1 has shown to be applicable in determining t1 [40]: t1 5 t0 2 bh1

(3.18)

where b is a constant and h1 is the amount of radial expansion (Fig. 3.4). To assess ra1, some existing approaches made the assumption that the overall shape of the meridian of the bulged shape corresponds either to a circular arc [31,39] or an elliptical arc [33,34]. Such assumptions enabled the determination of ra1 from the measured expansion d1 by the application of common geometrical relationships [31,33,34,39] and, depending on the developed model, also to determine t1 [31,39]. The consideration of the size of die radius rd (Fig. 3.4) at the fixing of the tube [33,34] improved the accuracy of the attained results for such approaches [32]. Other strategies for obtaining information about ra1 consisted in determining a spline function from several measured points along the meridian and calculating ra1 by first and second derivations of this function [41]. Assuming that at least the profile of the meridian around the maximum expansion is a circular arc, the use of a spherometer has shown to be applicable in measuring ra1 during forming [35,38]. Fig. 3.6 shows, as an example, a comparison of the result of different approaches for the determination of the stresses σθ1 and σϑ1 versus pi. Anisotropic plastic-deformation behavior in tube-expansion analyses is often considered by implementing the yield criterion from Hill’s orthogonal anisotropic theory [42], for example, in Refs. [34,38]. For a biaxial stress state where σθ and σϑ are principal stresses and where their directions are coincident with the anisotropic

Figure 3.6 Stresses at the maximum expansion diameter d1 determined with different approaches for hydroforming of tubes with d0 5 25 mm, t0 5 1 mm, material: 316 L (model 1: Ref. [33], model 2: Ref. [39], model 3: Ref. [31]). FEA, Finite element analysis. Source: Adapted from L. Vitu, N. Boudeau, P. Male´cot, G. Michel, A. Buteri, Evaluation of models for tube material characterization with the tube bulging test in an industrial setting, Int. J. Mater. Form. 11 (2018) 671
686 with permission from Springer, ©2017.

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axes of the tube, the following expression can be derived from this yield criterion for the effective stress [30,43]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R0 ð11R90 Þ 2 2R0 σe 5 σ2θ 1 σ 2 σϑ σθ R90 ð11R0 Þ ϑ 11R0

(3.19)

with the anisotropy coefficients R0 and R90, associated with the circumferential and axial direction that are usually obtained by tensile tests from specimens, cutout from the tube in longitudinal and circumferential direction or from so-called ring tests [44]. When applying tensile tests from cutout specimens, the individual anisotropy coefficients are obtained from the ratios of the strain in width direction to the strain in thickness direction, for example, R0 5 ln(b/b0)/ln(t/t0) with the widths b and b0 and the thicknesses t and t0 of the deformed and undeformed specimen. The equivalent plastic strain can be derived from the constancy of plastic work, assuming constant strain rates with: εe σe 5 εϑ σϑ 1 εθ σθ

(3.20)

It should be noted that Hill’s yield criterion cannot represent the plastic behavior of materials with anisotropy coefficients less than unity, which applies to some metals, such as aluminum [43]. Conversely, the Hosford yield criterion [45] has shown to be suitable to describe anisotropic plastic deformation in expansion processes of tubes made from aluminum alloys, for example, Refs. [32,35,40,46]. For the biaxial stress state, this yield criterion is written as [30,43]:

1 R0 R0 jσθ ja 1 jσϑ ja 1 jσθ 2σϑ ja σe 5 11R0 R90 ð11R0 Þ 11R0

1=a (3.21)

where a is related to the crystallographic structure of the material, with the best approximation of a 5 6 for body-centered cubic materials and a 5 8 for facecentered cubic materials [43]. Mathematically, the correlation is comparatively simple, but it does not consider the shear-stress components and describes only planar isotropy where the principle stress and anisotropy axes are parallel [35,43]. Furthermore, it was demonstrated that advanced phenomenological yield functions could appropriately describe the plastic flow of highly anisotropic materials in tube-hydroforming processes. This was found, for instance, for the application of the Yld2000-2d yield criterion developed by Barlat, Yoon et al. [47,48] for hydroforming of tubes made from aluminum alloys [35,49] and for the CB2001 yield criterion suggested by Cazacu and Barlat [50] for hydroforming of aluminum and mild steel alloys [51]. As an example, Fig. 3.7 depicts the results of investigations into the determination of yield loci for tubes made from an aluminum alloy with expansion tests, applying the Hosford and the Yld2000-2d yield criteria. The advanced yield criterion provided a good accordance with experimental results in

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81

Figure 3.7 Examples of yield loci of experimental results, representing various levels of constant plastic work and corresponding contours based on different yield criteria for different values of plastic strains εp0 (rotationally symmetrical hydroforming of tubes made from A5154-H112 with anisotropy coefficients of rϑ 5 0.36 and rθ 5 0.59 in the longitudinal and circumferential direction). Source: Reprinted from T. Kuwabara, K. Yoshida, K. Narihara, S. Takahashi, Anisotropic plastic deformation of extruded aluminum alloy tube under axial forces and internal pressure, Int. J. Plasticity 21 (2005) 101
117 with permission from Elsevier, ©2004.

the corresponding investigations, but it required a comparatively high number of experimental tests for the model calibration.

3.2.2 Hydroforming of polygonal cross sections A large proportion of industrial hydroforming operations involves the forming of circular tube cross sections into polygonal shapes, for example, for the manufacturing of structural members for automotive applications [3,4]. Important component areas, with regard to the required maximum forming pressure, the appearance of plastic instabilities, and the development of wall-thickness distribution, here concern the corner radii of the cross sections that are to be formed. For the hydroforming of parts with rotationally symmetrical cross sections, as discussed in Section 3.2.1, usually an approximate uniform wall thickness along the circumference is obtained. In contrast the wall-thickness distribution of hydroformed polygonal cross sections is unequal (Fig. 3.8) and decreases in general along the entire

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Figure 3.8 Decrease in wall thickness ε for the example of hydroforming a part with a square cross section. Source: Reprinted from S.J. Yuan, C. Han, X.S. Wang, Hydroforming of automotive structural components with rectangular-sections, Int. J. Mach. Tools Manuf. 46 (2006) 1201
1206 with permission from Elsevier, ©2006.

perimeter of the cross section, but with maximum reductions at the transitions from straight parts of the cross section to corner radii [52
54]. The amount of wallthickness reduction in these areas depends on factors, such as the material strainhardening behavior and the size of radii, which have to be achieved [54], but in particular, it is influenced by the friction between tube wall and forming die [52,55,56]. Due to the weakening of the wall thickness at the transitions, necking and crack initiation during the forming process predominantly occur in these areas of the workpiece [52,53]. As a result, the friction conditions crucially determine the feasible forming result in hydroforming of components with polygonal cross sections. However, this circumstance could be used in a positive way for the design of lubrication tests to obtain information about friction coefficients [57,58]. Friction between the tube wall and the forming die reduces the feasible stretching of the tube material that is already in contact with the die and thus the amount of material to support the expansion of the corner radius of the workpiece decreases. The equilibrium of forces of the straight areas that are in contact and the areas that are part of the corner and without contact with the die allowed determining the interrelationship between the internal pressure pi and the location A where the material of the straight cross-sectional part is plastically deformed (Fig. 3.9) [52,55]: pi 5

t σf ri 1 t 2 μxA

(3.22)

with the length xA from point A to point O (transition between straight area and corner radius), the friction coefficient μ, the average wall thickness t, the inner radius

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83

Figure 3.9 Model of loads at a straight part and at the corner area of a hydroformed rectangular cross section, according to Ref. [53].

ri of the formed workpiece corner, and the yield stress σf at point A, which complies with σe in case of plastic yielding. The derivation of Eq. (3.22) was based on the assumptions of plane-strain conditions without material movement in axial direction of the component, Coulomb friction, and the use of the Tresca yield criterion [53]. Due to the typically large ratios of length to the dimension of the cross section of components regarded here, friction and geometric constraints, in general, limit or prevent a material flow in the axial direction. The plastic deformation of this type of workpieces can thus be regarded as a deformation under plane-strain conditions. Eq. (3.22) showed that the effective stress is at a maximum at the transition point O and resulting in a distinct thinning in these areas [53,55]. This approach was pursued in Ref. [59] for investigations into limits in hydroforming with combined inner and outer medium pressure. Various analytical models described the plastic-deformation behavior of tube expansions in hydroforming die cavities with polygonal cross sections and have shown to be applicable in predicting the required forming pressures and the wallthickness distribution, for example, Refs. [60
64]. For the reasons mentioned earlier, such models are predominantly based on the assumption of plastic deformation under plane strain. The radial, circumferential, and axial strain, εt, εθ, εϑ, within the corner radii under that assumption can be written as: εθ 1εt 5 εϑ 5 0

(3.23)

t t0

(3.24)

εt 5 ln and

2 εe 5 pffiffiffi εθ 3

(3.25)

for the equivalent plastic strain εe. The correlation for the axial, circumferential, and radial stress, σϑ, σθ, σt in this area is:

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σϑ 5

1 ðσθ 1 σt Þ 2

(3.26)

with pffiffiffi 3 σe 5 ðσ θ 2 σ t Þ 2

(3.27)

for the effective stress σe. Common approximations for the radial stress are: σ t 5 2 pi

(3.28)

within areas of the part that are in contact with the die surface and 1 σ t 5 2 pi 2

(3.29)

for the areas under free expansion [60]. Considering plane-strain conditions and neglecting stretching of component areas with contact to the die surface by assuming sticking friction condition allowed deriving comparatively uncomplex models. Examples are approaches with closedform formulations based on correlations from the thin-walled pressure vessel theory for the determination of a biaxial stress state combined with geometric analyses to obtain information about strains, for example, Ref. [63], and comparable models that took a three-dimensional stress state into account, for example, Ref. [62]. Regarding the circular arc of the corner as a part of a thick-walled cylinder and solving the corresponding formulas for the equilibrium of forces led to the correlation [27]: 2 ra pi 5 pffiffiffi σf ln ra 2 t 3

(3.30)

that enables the prediction of the necessary internal pressure for generating plastic flow within the arc of the corner, where ra is the outer radius, t the wall thickness, and σe the effective stress. Eq. (3.30) can be used in practice for determining the maximum required internal pressure to hydroform components where radii have to be formed [27]. More complex models have been obtained when the equilibriums of forces at the corners and at the straight sections with stretching and friction were considered, for example, Refs. [60,61,64]. Such models have to be solved iteratively and can support, for example, investigations into the influence of friction in tube hydroforming [56]. Although some approaches took a nonuniform distribution of stresses across the wall thickness into account, for example, Refs. [27,62], bending effects were mostly neglected and a uniform distribution of strains along the wall thickness was

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85

Figure 3.10 Comparison of experimentally obtained corner radii with an analytical model and with simulation for the hydroforming of a part with pentagonal cross section (d0 5 50.8 mm, t0 5 0.813 mm, material: copper). Source: Reprinted from C. Yang, G. Ngaile, Analytical model for planar tube hydroforming: prediction of formed shape, corner fill, wall thinning, and forming pressure, Int. J. Mech. Sci. 50 (2008) 1263
1279 with permission from Elsevier, ©2008.

assumed, for example, Refs. [61,64]. In contrast the models described in Refs. [57,60,65,66], additionally considered the stress and strain state, resulting from bending and unbending of the formed tube cross section. For this purpose the tube wall was divided into three layers across the thickness [60]. Fig. 3.10 shows a comparison of analytically derived results from this approach with results from simulations with the finite element method and with experiments of an expansion of a pentagonal cross-sectional shape.

3.2.3 Hydroforming of tube branches Typical hydroformed tubular components with angled branches are T-, Y-, or Xshaped parts (Fig. 3.11A) with either uniform outer diameters along the entire shape or with branched outer diameters that differ from the outer diameter of the component’s main axis. Such parts are of interest, for instance, for the manufacture of exhaust system components for the automotive industry and for connecting elements for the piping industry [4,67]. Fig. 3.11B shows the process principle for the hydroforming of a T-shaped workpiece. The forming loads applied in the process are the axial force Fa, the internal pressure pi, and the counter punch force Fg. The force Fa serves to generate plastic yielding of the material within the area of the formed protrusion and to provide material for the expansion of the branch. The main purpose of the internal pressure pi is to support plastic yielding of the tube in the area of the protrusion and

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to draw the material into the corresponding part of the die cavity. Furthermore, the magnitude of pi has to be sufficient to press the tube wall against the die surface and thus to avoid the appearance of instabilities of the tube’s main axis in the form of wrinkling that can be caused by the axial forces. The counter punch with the force Fg is applied to the end of the formed branch to reduce the risk of instabilities by premature necking and bursting of this part of the workpiece, moving backward during the forming operation. The loads Fa, pi, and Fg have to be applied in the form of controlled loading paths, depending on the dimensions of the initial tube, the geometry of the die cavity, the plastic-material behavior of the tubular blank, and the interface’s friction conditions. Fig. 3.12 shows required loads and punch strokes for the example of hydroforming a Y-shaped component as well as the obtained wall-thickness distribution [67]. The availability of analytical approaches to describe and analyze the forming process and to determine necessary forming loads is considerably less for hydroforming of parts with branches, compared to hydroforming of rotationally

Figure 3.11 Hydroforming of tube branches: (A) examples of T-, Y-, and X-shaped hydroformed geometries and (B) process principle for the hydroforming of T-shaped components.

Figure 3.12 Hydroforming of a Y-shaped component: (A) forming loads and strokes versus time and (B) wall-thickness distribution (d0 5 50 mm, t0 5 1.5 mm, material: SS 304). Source: Reprinted from S. Jirathearanat, C. Hartl, T. Altan, Hydroforming of Y-shapes
product and process design using FEA simulation and experiments, J. Mater. Process. Technol. 146 (2004) 124
129 with permission from Elsevier, ©2003.

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87

Figure 3.13 Schematic end section of a hydroformed component.

symmetrical or polygonal shapes. Earlier works have shown possibilities for determining Fa for the forming of T-shaped parts, based on the equilibrium of forces of a ring element within the straight part of the tubular blank, and considering a triaxial stress state [68] or applying the upper-bound theory with an elastomer as a pressurizing medium [69]. A closed solution based on the continuum theory of plasticity to determine Fa for hydroforming of branched tube was presented in Refs. [7,70]. This latter approach employed the idealization of an axial symmetry of the plastic deformation within the axial end section of the tube when plastic yielding occurs, with a uniform wall thickness t along the circumference and in axial direction, regarded within a coordinate system with the axis r in radial, θ in circumferential, and z in the axial direction. (Fig. 3.13). Volume constancy was assumed with the following equation of continuity for the velocity field within the tube wall: @vr vr @vz 1 1 50 @r r z

(3.31)

with the local velocities in radial and axial direction vr and vz, and the following precondition was chosen for vz along the longitudinal axis: vz 5 2

vs z h

(3.32)

where vs is the velocity of the punch transferred to the end of the tubular section and h the section height. A further boundary condition considered a radial velocity

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at the contact surface with the tool that is zero, vr(r 5 r0) 5 0. Based on the solution of the earlier differential equation for the velocity field in conjunction with the flow rule, according to von Mises and the equilibrium conditions for the axisymmetric stress state, a second differential equation for the stress σr(r) was derived. With the precondition that the internal pressure pi is resulting in a radial stress at the inner radius of σr(r 5 ri) 5 2pi, the solution of this second differential equation led to the correlations for the stresses as a function of the radius r [70]: 1 σr 5 2 pffiffiffi σf ln Ar 2 pi 2 3

(3.33)

  1 4 σθ 5 2 pffiffiffi σf ln Ar 1 2 pi ar 2 3

(3.34)

  1 2br σz 5 2 pffiffiffi σf ln Ar 1 2 pi ar 2 3

(3.35)

with Ar 5

ar 2 1 c1 1 1  ar 1 1 c1 2 1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  4 2r ar 5 3 11 d0 br 5 3

 2 2r 11 d0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi   2t 4 c1 5 3 12 11 d0

(3.36)

(3.37)

(3.38)

(3.39)

From Eq. (3.33) the contact pressure σN between the tube wall and the surrounding die was derived with σN 5 2σr(r 5 r0 5 d0/2) as: 1 c1 1 1 1 pi σN 5 pffiffiffi σf ln 3 c 1 2 1Þ ð 2 3

(3.40)

The component Fz of the axial force to achieve plastic yielding within the tube section was obtained by the integration of Eq. (3.35) over the ring face of the tube end with:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   d02 π 22 ffiffiffi σf 2 2 ln Fz 5 p 3 1 c24 2 c 2 c 1 1 pi tðd0 2 tÞπ 2 2 4 3

(3.41)

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89

and c2 5 1 2

2t d0

(3.42)

The axial punch force Fa can now be determined according to Eq. (3.1) with the friction force: Ff 5 μσN d0 πh

(3.43)

where μ is the friction coefficient, and with the sealing force: π Fp 5 pi ðd0 22tÞ2 4

(3.44)

The correlations described earlier allow the determination of the required amount of Fa as a function of tube dimensions, friction conditions, the acting internal pressure, and the current material yield stress. Recent approaches to calculating the suitable internal pressure to avoid wrinkling of the component can support the design of loading paths (Section 3.3.2). However, due to a lack of analytical models for determining suitable combinations of all forming loads and process parameters involved in the forming process for hydroforming of tube branches, greater efforts in the development of the use of mathematical optimization methods have been undertaken (Section 3.4).

3.3

Determination of forming limits in tube hydroforming

As already mentioned, the main modes of failures that limit the attainable component shape in hydroforming are necking with subsequent bursting, wrinkling, and buckling of the tubular workpiece. These failures are caused by excessive stresses within the wall of the hydroformed workpiece and their onset depends on the material properties, dimensions of the formed tube and forming die, partially on friction conditions, and on the applied loading paths. The following sections discuss fundamental issues concerning these limits in tube hydroforming as well as latest achievements in failure analysis and predictions with corresponding instability criteria.

3.3.1 Necking and bursting Bursting of the workpiece wall in the course of a hydroforming process is characterized by a local crack formation that is a consequence of necking. A distinction has to be drawn between necking of the tube wall as a diffuse and as a local

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instability [71]. Diffuse necking can be described as an equilibrium phenomenon at the point where the plastic deformation starts to be nonuniform and the forming load reaches a maximum. Localized necking, preceded by diffuse necking after a certain amount of nonuniform deformation, is characterized by strain localization through the wall thickness and is followed by fracture at the thinnest point within the developing neck [71]. In conventional sheet metal forming, a large number of modeling techniques were developed over the last few years to predict these instabilities [72,73]. Concerning studies in the application of these techniques on the analysis of hydroforming processes, a classification can be made in (1) the generation of forming limit diagrams (FLDs) by the use of linear methods [73], such as the criteria developed by Swift [74] and Hill [75], for example, Refs. [76
80], or by maximum load criteria, for example, Refs. [79,81]; (2) the use of physical models [73], such as the Marciniak
Kuczynski (M
K) method [82] or the Gurson
Tvergaard
Needleman (GTN) model [83,84], to determine forming limits and generate FLDs or forming limit stress diagrams (FLSDs), for example, Refs. [71,85
95]; and (3) the use of models based on fracture mechanics [73], such as models with definitions of energy-related fracture criteria by Oyane [96], Brozzo [97], or Ayada [98], for example, Refs. [99,100] or void-growth-based models, such as the model by Rice and Tracey [101], for example, Ref. [102]. The FLD indicates the combination of the major and minor strains, ε1 and ε2, that cause necking or failure by crack formation in a formed component. It is the most widely used criterion in conventional sheet metal forming because of its simplicity in application [73]. For tube hydroforming the implementation of Swift’s diffuse necking criterion in combination with Hill’s localized necking criterion were intensively investigated to predict the forming limit curve (FLC) that is describing ε1 versus ε2 in the FLD. The two criteria were often combined to determine the right side of the FLD with the diffuse limit and the left side with the localized limit [71]. According to Swift’s diffuse necking criteria for tubes [74], plastic instability within the formed tube wall occurs when the following applies simultaneously: dpi 5 0

(3.45)

dFa 5 0

(3.46)

and

Assuming a thin-walled tubular workpiece where the length l0 is a multiple of the outer diameter d0, so that an almost cylindrical expansion with uniformly distributed stresses and without a curvature of the meridian can be supposed, the circumferential and axial stress, σθ and σϑ, within the tube wall are [80]: σ θ 5 pi

r t

(3.47)

Mechanics of hydroforming

σ ϑ 5 pi

91

r 1 1 Fa 2t 2πrt

(3.48)

with the wall thickness t and the mean radius r of the expanded tube. These simplified correlations replace Eqs. (3.7) and (3.8) for the local stress state of a bulged tube when applying the aforementioned assumptions. From Eqs. (3.45) and (3.46), one obtains for the change of stresses [74]: dσθ 5 σθ ð2dεθ 1 dεϑ Þ

(3.49)

dσϑ 5 σθ dεθ 1 σϑ dεϑ

(3.50)

where dεθ and dεϑ are the plastic-strain incremental components along the circumferential and axial direction, and εθ and εϑ correspond to the major and minor strains. For the particular case of anisotropy where R0 5 R90, the quadratic yield criterion for anisotropic materials from Hill, according to Eq. (3.19), can be written in the form of [43]: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2R σϑ σθ σe 5 σ2θ 1 σ2ϑ 2 11R

(3.51)

with R 5 R0 5 R90. With Eqs. (3.45)
(3.51) the assumption of volume constancy and the flow rule, according to von Mises, the instability condition in terms of the subtangent ZS can be derived in the form of [80]: 1 dσe 5 5 ZS σe dεe

 αð2α2ρÞ2 1 2 2 2 αρÞð2 2 αρ 1 2α 2 ρÞ 4ð12αρ1α2 Þ2=3

(3.52)

with the equivalent plastic-strain increment dεe and the ratios α 5 σϑ/σθ and ρ 5 2R/(1 1 R). For local necking of a sheet in its plane, Hill developed the criteria for the onset of plastic instability when [75]: dσ1 5 σ1 ðdε1 1 dε2 Þ

(3.53)

or, under the assumption of a constant stress ratio α: dσ2 5 σ2 ðdε1 1 dε2 Þ

(3.54)

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Analogous to the derivation for ZS, and with σ1  σθ, σ2  σϑ, ε1  εθ, and ε2  εϑ as well, the instability condition in terms of a subtangent ZH can be derived with [80]: 1 dσe ð1 2 αÞð2 2 ρÞ 5 5 ZH σe dεe 2ð12αρ1α2 Þ1=2

(3.55)

Assuming a power law for the tube material, according to Eq. (3.16), Eqs. (3.52) and (3.55) can be written as: 1 dσe n 5 5 Z εe σe dεe

(3.56)

with Z 5 ZS for the criterion according to Swift, Z 5 ZH for the criterion from Hill, and the strain-hardening exponent n. Considering an anisotropy of the tube material with R0 5 R90 5 R for the anisotropy coefficients as mentioned earlier, assuming the circumferential and axial strains, εθ and εϑ, as principal strains, according to Eqs. (3.9)
(3.12), and strain paths with a constant ratio of β 5 dεϑ/dεθ  εϑ/εθ, provides the equivalent plastic-strain increments with [80]: 11R εe 5 εθ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 2R

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 βρ 1 β 2 5 εθ ξ

(3.57)

Eqs. (3.52) and (3.55)
(3.57) with εϑ 5 βεθ now allows determining the value pairs of critical strains, εθc and εϑc, that are describing the FLC under the condition of constant ratios of α and β during expansion with: εθc 5

Z n ξ

εϑc 5 βεθc

(3.58) (3.59)

For bulge tests of tubes made from an aluminum alloy under constant ratios of α and β at the diameter d1 of maximum expansion, applying a nonquadratic yield function for anisotropic material behavior from Hill, it was demonstrated that the criterion for diffuse necking of tubes from Swift and the criterion for local necking from Hill were suitable to generate an FLC that predicted the occurrence of plastic instability [78]. However, it is necessary to take into account that hydroforming processes exhibit, in general, nonlinear strain and stress paths with varying ratios of stresses and of strains during expansion, for example, Ref. [103], and that variations of the ratio β lead to shifting of the FLC [76]. Besides the instability criteria described earlier, a variety of similar criteria were investigated to determine the forming limits by the onset of necking based on maximum load conditions, including, for example, the assumption that necking starts at

Mechanics of hydroforming

93

the peak pressure during hydroforming, such as dpi/dr 5 d(σθt/r)/dr 5 0 or dpi 5 d (σθt/r) 5 0. A comparison of several criteria from the group of linear approaches and maximum load conditions for the forming of tubes made from an aluminum alloy has shown that a combination of the internal pressure pi and axial force Fa 1 πr2pi provides the most suitable right-hand side of the FLC and a reasonably well-predicted left-hand side of the FLC [79]. Concerning the application of yield criteria for anisotropic material behavior in conjunction with the prediction of FLDs based on maximum load criteria, it was found that Hosford’s yield criterion for the plane-stress state, Eq. (3.21), provided a better estimation of the forming limit than the yield criterion proposed by Hill, according to Eq. (3.51), for an aluminum alloy with R 5 0.75 [81]. The limited validity of the FLD for forming operations under nonlinear strain paths led to the development of the concept of the FLSD that provides FLCs, which are independent from the strain path and specify the stress points in the stress space at the onset of localized necking [72,73]. This method requires the determination of an FLD, for example, by experiments, and the calculation of the corresponding stress values by either using the plastic flow rule or the finite element method to obtain the FLSD [72,73]. For tube hydroforming the determination and evaluation of FLSDs were frequently supported by the use of physical modeling methods in determining forming limits, in particular the M
K model and the GTN model. The M
K model is based on the assumption that necking occurs from zones characterized by an initial weakness, imperfection, or inhomogeneity when the strain state approaches plane strain in this region [82]. The model proposed a preexisting imperfection described by a thickness ratio f0 5 tB/tA with the thickness tA of the homogenous material and tB of the imperfected region. This ratio needs to be determined by calibrating the model to experiments. The governing equations for each region are determined based on compatibility and equilibrium equations, and it is assumed that with the start of necking, the strain development in the thinner region is greater than in the thicker regions, with an increasing difference in strain rate between both regions under increasing deformation. The limiting strains are then determined for the case where the equivalent plastic-strain increment dεeb in the groove area reaches 10 times the equivalent plastic-strain increment dεea in the adjacent areas. The GTN model includes porous plasticity models where the physical void volume fraction constitutes the primary damage variable. The damage evolution with this model is monitored through the characteristic stages of the void mechanism: nucleation, growth, and coalescence of voids, assuming that material failure occurs when a critical damage value—as a function of a critical void volume fraction—is reached [83,84]. The numerical calculation of FLSDs on the basis of the M
K model and their implementation, as a failure criterion, into finite element simulations enabled, for example, the determination of optimized loading paths in tube hydroforming, for example, Ref. [93]. However, it should be respected that the validity of the results crucially depended on the chosen thickness ratio f0 for the M
K model, for example, Refs. [71,93]. It was shown for tube hydroforming that this sensitivity could be

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

reduced when approaches based on strain-gradient theory of plasticity were combined with the M
K model [104]. Furthermore, it was found that the M
K criterion underestimated the limit strains in the case of hydroforming within a die cavity [71], caused by the discrepancy between the prediction of necking in the plane case and in the case of the pressurized tube with a geometry differing from this idealized case [71]. A certain strain-path dependency of the FLSD in hydroforming of tubes made from steel was observed for large values of prestrains, εe $ 0.35, as well as for abrupt changes in the strain path [91]. This was also observed in the case of steel tubes that showed kinematic strain-hardening behavior of the prestrained material subjected to combined axial loading and internal pressure [105]. Kinematic hardening was also determined as a possible source for deviations between simulated and experimental results within the frame of investigations into the forming limits of tubes made from an aluminum alloy, using the advanced anisotropic yield function Yld200-2d from Barlat, which considers isotropic hardening [106,107]. A further development of the FLD was an extended version derived from the FLD and the FLSD, indicating the combination of equivalent stress and mean stress as the occurrence of instability, demonstrated for the example of hydroforming of tubes made from a high-strength low-alloy steel and a dual-phase steel [94]. For the hydroforming of tube branches, an approach was developed and tested experimentally by Ref. [87] that considered the normal compressive stress for the determination of the FLD and FLSD based on the M
K model. The application of the GTN model requires the determination of several parameters for defining plastic flow and material failure under consideration of nucleation, growth, and coalescence of voids. Different strategies were applied for investigations into hydroforming to determine these values. Essential parameters that are describing void coalescence were obtained, for example, with inverse approaches by calibrating a numerical simulation model, for example, Refs. [90,92] or by deriving the relevant values from experimentally measured values for the void volume fractions at the initial specimen at the onset of necking and fracture, for example, Refs. [85,86,88]. The applicability of the GTN model was demonstrated, for example, for rotationally symmetric expansions made from unalloyed steel [85] and stainless steel [86], hydroforming of components with polygonal cross sections made from dual-phase steel and advanced high-strength steel [92] as well as from an aluminum alloy [88] and for hydroforming of T-shaped parts made from stainless steel [90]. Predictions of the onset of instabilities were achieved for investigated hydroforming operations with deviations from experimental results below about 5% [89]. The damage evolution described with the GTN approach also allowed to be implemented as an imperfection for the M
K model as shown in Ref. [95] where an anisotropic version of the GTN model was involved to determine FLCs for the hydroforming of tubes made from an aluminum alloy. Models based on fracture mechanics [73] belong to a further group of criteria for determining forming limits investigated for hydroforming of tubes. Among the fracture criteria whose formulations derive from energy considerations by assuming a porous-free deformed material, examples are to be found for the use of the models by Oyane [96] and Brozzo et al. [97] for investigations into hydroforming limits,

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for example, Refs. [99,100]. The second category of fracture criteria are voidgrowth-based models where the model by Rice and Tracey [101] was employed, for example, in combination with the consideration of kinematic hardening for the hydroforming of a hollow crankshaft component [102]. Formulations of models based on fracture mechanics generally predict the occurrence of fracture when a damage indicator variable reaches a critical value. As an example, Oyane’s ductile fracture criterion, written as [96]: ð εef  0

 σm 1 a0 dεe 5 b0 σe

(3.60)

predicts material fracture when the value of b0 is reached. In this correlation, σm is the hydrostatic stress and a0 and b0 are material constants that can be derived from bulge tests, for example, Ref. [108]. In conjunction with the implementation of the Zener
Hollomon parameter, it was possible to also apply these fracture criteria to tube hydroforming at elevated temperatures [99]. Comparisons of various models for the hydroforming of tubes made from an aluminum alloy at different temperatures and varying loading paths have shown that for such applications, the criterion developed by Ayada et al. [98]: ð εef 0

σm dεe 5 c0 σe

(3.61)

provided comparatively good predictions of forming limits [99], where c0 is a material constant (Fig. 3.14). The models based on fracture mechanics are generally easy to implement into finite element codes and do not cause great efforts in determining required material constants. However, it should be noted that their formulations do not reflect associated material softening [73].

3.3.2 Wrinkling and buckling Plastic wrinkling and buckling are instabilities resulting from excessive axial loads, which typically limit the amount of the axial stress that can be applied to the hydroformed tube to reduce the decrease in wall thickness during the expansion process. Buckling is generally understood to mean that a tube buckles in a similar way as a long column under a critical axial compressive load, whereas wrinkling is characterized by formation of symmetric corrugations of the tube wall under excessive axial loads. Buckling can typically occur in hydroforming of tubes where the length l0 is a multiple of the tube’s outer diameter d0 and where the tube is not supported by a surrounding forming die. The instability of wrinkling is to be observed in both cases, during free expansion of the workpiece and during forming of components,

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Figure 3.14 Comparison of predicted FLCs for the hydroforming of aluminum tubes at an elevated temperature of 573K (d0 5 40 mm, t0 5 1.5 mm, material: AA6063). FLC, Forming limit curve. Source: Reprinted from S.J. Hashemi, H.M. Moslemi Naeini, G.H. Liaghat, R. Azizi Tafti, Prediction of bulge height in warm hydroforming of aluminum tubes using ductile fracture criteria, Arch. Civ. Mech. Eng. 15 (2015) 19
29 with permission from Elsevier, ©2014.

such as T-shaped parts, where the workpiece wall is in contact with the surrounding tool to a large extent. Comparatively uncomplicated methods for predicting the occurrence of buckling in hydroforming are based on the determination of critical compressive buckling stresses σcr of columns studied by Euler for elastic instability, considering the plastic state by replacing the Young’s modulus E with the plastic tangent modulus Et 5 dσe/dεe, for example, Ref. [27]: σcr 5 2π2 Et

r2 l2

(3.62)

where r is the instantaneous radius from the center of the tube to the middle surface across the tube thickness and l the instantaneous tube length. With the objective of obtaining a further reaching criterion, taking the entire effect of plasticity of the tube material into account, an approach was suggested by Ref. [109], based on the use of plane-stress moduli. These moduli are part of an incremental elastoplastic constitutive equation that describes the correlation between stress rates and strain rates, and that allows incorporating advanced yield criteria [109]. Applying this approach on the prediction of buckling in tube hydroforming, the critical buckling stress can be written in the form [109]:

Mechanics of hydroforming

σcr 5

1 2 r2 1 r2 π L1 2 5 π2 L1 20 e2ðεθ 2εz Þ 2 2 l l0

97

(3.63)

with the component L1 of the plane-stress moduli and the initial radius and length of the tube, r0 and l0. Details about the component L1 will be discussed later in this section in the context with wrinkling. The strains εθ and εϑ can be obtained with Eqs. (3.9)
(3.12). Developed correlations to predict the onset of wrinkling of tubes in hydroforming processes were largely based on equations to describe deformation in cylindrical shells and to predict the critical axial compressive stress by substituting components describing elastic behavior with moduli, such as the plastic tangent modulus Et, the secant modulus Es 5 σe/εe, or the reduced modulus Er 5 4Es Et =ðEs0:5 1Et0:5 Þ2 for including plasticity, for example, Refs. [27,81,110,111]. As an example, from the differential equations for the symmetrical deflection of a cylindrical shell under axial compression and internal pressure in conjunction with the reduced modulus Er, the simplifications that Es  E and that the stress ratio α 5 σϑ/σθ is constant, with the radial stress σt considered negligible, and applying Hosford’s yield criteria according to Eq. (3.21) with R0 5 R90 5 R for the anisotropy coefficients, a criterion for wrinkling was deduced in the form of [81]:   ðn21Þ=a

1 1 1 a 11 a 1R 12 11R α α

σcr 5 2 Anc K

(3.64)

with pffiffiffi n 2t0 Ac 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffi 2 n11 r0 3ð1 2 ν Þ

(3.65)

where n and K are the material constants from the power law according to Eq. (3.16), R is the anisotropy coefficient, a the coefficient according to Eq. (3.21), and ν the Poisson ratio. Analogous to the criterion for buckling described earlier, plane-stress moduli were also applied for developing criteria for wrinkling [109] by coupling the general plane-stress moduli tensor [112] with nonlinear equilibrium equations for thin, shallow cylindrical shells [113]: σcr 5

1 t2 l2 ðmπÞ2 L1 02 e22ðεθ 22εϑ Þ 1 ðmπÞ22 L2 02 e2ðεϑ 2εθ Þ 12 l0 r0

(3.66)

where m is the wave number and L1 and L2 are plane-stress moduli. The wave number has to be found numerically by solving the previous equation iteratively to determine the wave number that corresponds to the minimum axial compressive load. For the example of a material that obeys Hill’s yield criterion, according to

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Eq. (3.51), and the application of the flow theory of plasticity, the plane-stress moduli could be deduced as [114]:  2 E σϑ L1 5 2 ð E 2 Et Þ 2 σe 12ζ

(3.67)

 2 E σθ L2 5 2 ð E 2 Et Þ 2 σe 12ζ

(3.68)

with ζ 5 R/(1 1 R) and R 5 R0 5 R90. The criteria to determine wrinkling in tube hydroforming discussed here as examples were demonstrated to be applicable for hydroforming of tubes made from aluminum alloys [81,109,114], and comparable verifications also exist for other approaches considering isotropic material behavior [110]. However, the individual correlations differ in the way these were derived and in the required calculation effort. It has to be considered that most of the discussed criteria assume a uniform distribution of the wrinkles along the workpiece axis whereby in the large majority of cases, the formation is nonuniform, which is due to the restricted movement of the tube ends in radial direction where these are sealed. The rotationally symmetrical expansion of a tubular blank typically follows a wrinkle formation with more distinct wrinkles close to the sealing areas that vanish in the course of the expansion under an appropriate control of Fa and pi, for example, Refs. [115,116]. It was found that a certain formation of wrinkles supported an increase in feasible expansion, for example, Refs. [115,116]. Further criteria for prediction of instabilities caused by wrinkling were also developed for process variants working with pulsating internal pressure to hydroform rotational symmetric shapes [117] and for the conventional hydroforming of T-shaped components [118]. For the former one, criteria were proposed based on the static equilibrium at the wrinkles and based on correlations from plastic mechanics and energy conservation [117]. The approach to predicting wrinkling in hydroforming of T-shaped components considered wrinkling at the sidewalls of the workpiece caused by excessive axial compressive loads and applied the energy method with the law of energy conservation to derive a correlation for the critical axial stress [118]. An experimental comparison of a critical loading path that was developed with this model showed an accurate prediction of the occurrence of wrinkling [118].

3.4

Design of loading paths

The result of a tube-hydroforming process is highly dependent on the applied loading path that normally consists of axial loads or axial strokes of the sealing punches

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and the internal pressure versus forming time. Usually, hydroforming of tube branches additionally involves the force or stroke of the counter punch to support the formed protrusion. The pursued objective when determining a loading path is, in general, to obtain the required component shape under avoidance of plastic instabilities and with a minimum thickness variation. In this context, nonlinear material-deformation behavior and loading-path dependency of the appearance of plastic instabilities are some essential constraints that complicate an analytical determination of suitable loading trajectories. Against this background, an extensive research work was conducted over the recent years, dealing with the use of optimization methods to facilitate the determination of loading paths. Here, a methodical overview of current investigated strategies in tube hydroforming is given without going into details of applied algorithms, but to make the reader familiar with feasible solutions for loading path determination and their particularities based on examples. Optimization strategies using classical iterative algorithms are one of the methods applied to find loading paths for tube hydroforming [119,120]. These algorithms are based on the minimization of an objective function, involving repeating numerical simulations of the hydroforming process with different values of variables. Objectives in the investigations with this method described in Ref. [121], for example, were a minimum of wall-thickness reduction of the component and high geometrical accuracy for the hydroforming of T-shaped parts. It was found that dividing the forming process in different phases, with an individual optimization of the variables, provided improved results compared to an optimization of the variables regarding the final result [121]. Optimizing internal pressure versus punch stroke with iterative techniques to obtain a minimum of wall-thickness reduction and a maximum branch length for hydroforming of a T-shaped component were the objectives in the research of Ref. [122]. The experimental validation confirmed the predicted results [122]. When applying these algorithms, it has to be considered that the determined optimum relies on the selection of the starting point and that a local optimum solution but not a global one may be obtained [119,120]. A second category of optimization methods for tube-hydroforming process design used genetic algorithms [123] in conjunction with numerical simulations [119,120]. These algorithms belong to the group of evolutionary algorithms and apply search procedures that are based on the principles of natural selection. In Ref. [124], for instance, the use of a genetic algorithms’ search method was described for developing forming parameters of a tube-hydroforming process for a component with rectangular cross section and for several advanced high-strength steel materials with the objective of maximizing the formability. FLCs were implemented to predict the occurrence of failures, and experimental results confirmed the developed load paths [124]. Another example was the optimization of the hydroforming of a nonaxisymmetric fuel filler for automotive application with genetic algorithms integrated into finite element analysis [125]. A minimum of wall thinning and the avoidance of wrinkles were the two criteria for selecting the suitable loading path in this research [125]. An advantage of these methods is that these allow finding

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global solutions [120], but a drawback consists in the comparatively high calculation effort [125]. The use of surrogate models [126] is a third category of optimization strategies for tube-hydroforming processes [119,120]. The surface response method is a comparatively often-applied technique in this group of optimization techniques. This method is based on fitting a low-order polynomial model through response data that are obtained by finite element simulations or experiments and optimizing the model by targeted variation of variables. An example of the application of the surface response method was presented in Ref. [127] for the optimization of the internal pressure versus the axial stroke of the sealing punches for the forming of a T-shaped component. The objective in this research, to influence the thickness distribution of the formed part, could be achieved and was confirmed by simulations. Uncertainties coming from variations of material properties and geometrical characteristics were considered in the investigations of Ref. [128] for the optimization of loading paths for hydroforming of T-shaped parts using the surface response method. In Ref. [129], different approaches of surrogate models were compared for hydroforming a T-shaped part with the objectives of a minimum wall thinning and a sufficient branch length. Significant differences between the individual approaches were determined, concerning the calculation effort and the reliability to find the global optimum values [129]. An example of the use of the artificial neural network method as a surrogate model in conjunction with numerical simulations was presented in Ref. [130] for the optimization of the loading path for hydroforming of an X-shaped part. The stroke of the counter punch was considered in the loading path and the results were verified by experiments in this research [130]. Implementation of adaptive algorithms can be viewed as a fourth category of optimization strategies in tube-hydroforming process design [119,120]. The basis here consists in adjusting control parameters during the process simulation for each subsequent time increment, depending on the result of the previous one [120]. Considering correlations describing the onset of plastic instabilities, the loading path can be adjusted with this method within the simulation run to avoid these instabilities. An example of application was provided in Ref. [131] for the determination of loading paths for hydroformed components. The optimized loading path determined with adaptive simulations was applied successfully to experimental hydroforming operations [131]. On the contrary, it was pointed out in this work that accurate material data and, in particular, reliable criteria to detect wrinkling were necessary to achieve these results [131]. The investigations described in Ref. [132] applied an adaptive simulation approach with an integrated fuzzy control algorithm to the optimization of the loading path for the hydroforming of a T-shaped part made from steel. Several evaluation functions were implemented for identifying the geometrical shape and part quality, and the result was confirmed by experiments [132]. Based on in-process data processing of measured data and a fuzzy control approach, the direct optimization of experimental hydroforming operations of T-shaped components was demonstrated by Ref. [133] without simulations.

Mechanics of hydroforming

3.5

101

Conclusion

Hydroforming has become well established in recent years for the manufacturing in important industrial fields, such as the automotive industry. It is constantly being developed further, concerning new process variants and modern lightweight materials, which are increasingly gaining interest, such as advanced high-strength steels, aluminum alloys, and magnesium alloys. Reliable models to support the design of hydroforming processes and to improve the characterization of tube materials are therefore of great importance. Against this background, this chapter provided an overview of latest approaches in analytical modeling of hydroforming, which are addressing these issues. The discussed models for the analysis of rotationally symmetrical expansions are useful in capturing essential material properties of specimens in a tubular form, and the implementation of advanced anisotropic yield functions has shown that these can clearly improve the accuracy of material characterization. However, considering the complex strain-path changes that can be observed in many hydroforming processes, further research on models appears necessary that goes beyond the typically assumed isotropic hardening and that considers variations and translations of the yield surface resulting from loading changes. For the hydroforming of polygonal shapes, extensive analytical models were developed enabling the prediction of the friction-dependent hydroformed shape, wall thinning and necessary internal pressure, whereby further reaching investigations should take plastic anisotropy of the tube material into account. In consequence of the complexity of the process, approaches to analyze hydroforming of tube branches are significantly less covered in the current literature compared to approaches for rotationally symmetrical expansions and for hydroforming of components with polygonal cross-sectional shapes. Optimization methods are generally used to determine suitable loading paths for the hydroforming of these components but deeper analyses of this type of process require the development of according models to support a more effective process design. Concerning investigations into the applicability of forming limit criteria to predict necking in hydroforming processes, a major trend appears to be the use of physical models and partly models based on fracture mechanics. However, the development of hydroforming models with implemented criteria for failure prediction that capture the complex strain-path changes during hydroforming is still challenging and should be intensified to improve the prediction accuracy of necking in hydroforming under any process condition. Existing approaches to determine wrinkling of tubes in hydroforming processes enable a prediction of this failure case with a good reliability, but they are predominantly based on idealized assumptions of evenly distributed wrinkle formation at cylindrical tubes. The fact that the formation of wrinkles at a hydroformed specimen is typically nonuniform still offers a potential for further investigations into advanced models to enhance the prediction accuracy of this type of instability.

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Finally, it should be noted that future investigations into the modeling of hydroforming processes for material characterization and for prediction of instabilities in hydroforming should also consider the request of the industry for modeling techniques that are easy to implement. Standardization of testing procedures for hydroforming and provisioning of material data to support model calibrations can therefore be seen as further future fields of research.

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Electromagnetic pulse forming

4

Verena Psyk1, Maik Linnemann1 and Gerd Sebastiani2 1 Fraunhofer Institute for Machine Tools and Forming Technology, Chemnitz, Germany, 2 imk automotive GmbH, Chemnitz, Germany

4.1

Process classification

It is well known that velocity effects offer significant technological and economic advantages in production technologies, specifically in sheet metal forming. This concerns processes with very low velocities such as superplastic forming as well as high-velocity forming processes [1]. In literature, there are several attempts to define high-velocity forming processes, which are also referred to as high-energyrate forming or impulse forming, but there is no general agreement. Ref. [2] classifies all processes releasing stored energy in order to form a workpiece in a short period of time as high-velocity forming without clear quantification. Ref. [3] considers all processes with cutting velocity of 0.8 m/s and higher to be highvelocity cutting, while Ref. [4] assumes a minimum velocity of 15 m/s for a high-velocity forming process. Despite these deviations, literature sources agree that high velocities in forming processes offer several advantages. The most important ones are the following: G

G

G

G

Increased formability of many materials for high strain rates. Typically, in high-velocity forming values of 102
107 s21 are reached [5]. As a consequence, the flow behavior and the strain at failure can vary significantly. However, in order to provide a precise process description and consider these effects thoroughly, it is necessary to take into account that the strain rate cannot be characterized by one single value, because it varies locally and temporally while the workpiece is accelerated from the static state to the maximum strain rate and then decelerated again. Lower localizing effects such as wrinkling or necking. Reduced springback and correspondingly higher accuracy. Short duration of the forming process in the range of several microseconds up to several milliseconds. However, auxiliary process times are sometimes significantly higher compared to forming itself. Therefore, the cycle times that can be reached in practice are not always as low as expected.

Practically, high-velocity forming, joining, and cutting processes can be based on chemically stored energy, for example, in explosive forming and cladding [6,7], mechanically stored energy, for example, in forming with pneumatically or hydraulically based high-velocity drives [8], and electrically stored energy. The latter Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques. DOI: https://doi.org/10.1016/B978-0-12-818232-1.00004-7 Copyright © 2020 Elsevier Ltd. All rights reserved.

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specifically includes electrohydraulic forming [9], forming with electromagnetically driven tools [10,11], and electromagnetic pulse forming [12]. In this group of processes, electromagnetic pulse forming excels due to the fact that no dangerous goods, such as explosives, are necessary, which significantly simplifies the implementation on the process into a typical industrial environment. Furthermore, unlike many other high-velocity forming processes (specifically explosive forming, plunger-based forming, or electrohydraulic forming), no working medium is required, and thus cleaning effort and corresponding costs are reduced. In contrast to mechanical high-velocity processes using pneumatically, hydraulically, or electromagnetically driven tools, the force applying tools in electromagnetic pulse forming can be applied much more flexibly, and the positioning requirements are much less restrictive. Thus, tryout becomes easier. However, electromagnetic pulse forming can only be used to deform electrically (highly) conductive workpiece materials such as aluminum alloys, copper, or mild steel. For other materials (stainless steel, titanium, etc.) the other high-velocity forming processes are more appropriate.

4.2

Process principle and major process variants

4.2.1 General setup and process principle The setup for electromagnetic pulse forming consists of the forming machine, the active tool, the workpiece, and further application-dependent tool components such as cutting tools, form-defining dies, or joining partners. The main function of the forming machine, which is frequently called pulsed power generator, is to store electrical energy and release it within a very short period of time. The main components of this machine are the charging unit, the capacitor bank, high-current switches, the electrical connection to the tool coil, and control and safety devices. The machine is electrically connected to the active tool, frequently referred to as inductor, which consists of the tool coil and the fieldshaper, if applicable. Here, it is important to consider that a tool coil winding does not necessarily feature a cylindrical shape, but depending on the specific forming task, it can have nearly any shape. Examples are given in Section 4.2.2.3. In the process the workpiece, which can be a tubular or sheet metal component ideally made of an electrically highly conductive material, such as aluminum or copper alloys, is positioned in direct proximity of the inductor. The capacitor battery of the pulsed power generator is charged via the charging unit up to a defined voltage and energy, respectively. Typical voltages are in the range of several hundred volts up to approximately 20 kV, depending on the size of the zone that has to be deformed. The corresponding energies are in the range of several hundred joules up to several 10 kJ. Then, the high-current switches are closed, and the energy is discharged via the tool coil. Consequently, damped sinusoidal current flows through the tool coil and induces a corresponding time-dependent magnetic field. The workpiece in proximity of the tool coil can be considered a short-circuited second

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winding of a transformer. According to the second Maxwell’s equation (law of induction), current is induced in the workpiece, which is directed opposed to the coil current according to Lenz’s law. Maximum values of the coil current are typically in the range of several 10 kA up to several 100 kA, while current rise times are usually in the range of 10 μs or slightly below in a very fast process up to 100 μs in a very slow discharging process. In such transient processes the skin effect is significant. This means that an alternating electric current tends to flow close to the surface of a conductor, so that the resulting current density is high at the surface of the conductor and decreases toward its center, thus increasing the resistance load per unit length. The reason for this effect is that the alternating fields penetrating to the conductor are damped significantly before reaching the center area. It is more pronounced in the case of higher electrical conductivity of the conductor and higher frequency of the alternating current. Local momentary electromagnetic field strength of up to several 10 T can be reached during the process. Due to the interactions of the electrical current and the magnetic field, Lorentz forces arise within the workpiece and the inductor system. These physically correct volume forces can be mathematically transformed to surface forces—the so-called magnetic pressure, which can reach maximum values of up to several 100 MPa. The magnetic pressure acting on a current-carrying conductor can be compared to gas pressure acting on a slightly porous wall. Similar to the diffusion of the gas through the wall, the magnetic field is initially largely shielded by the conductor wall but penetrates it over time [13]. However, if process parameters are well chosen, the resulting pressure difference acting on the workpiece wall is high enough to accelerate the workpiece if inertia effects are overcome; this pressure difference is also sufficient to plastically deform it if the resulting mechanical stresses in the workpiece material reach the flow stress. During this deformation, displacement velocities of up to several 100 m/s are reached as well as strain rates in the range of 102
106 s21. In general, acceleration as well as deformation of the workpiece is directed away from the inductor.

4.2.2 Major process variants Electromagnetic pulse compression and expansion of tubes or hollow profiles and electromagnetic pulse forming of flat or preformed sheets can be differentiated as major process variants, depending on the geometry and the arrangement of inductor and workpiece. The following sections will cover each of them in detail.

4.2.2.1 Electromagnetic pulse compression Electromagnetic pulse compression is the simplest process variant. Here, the tubular workpiece is positioned inside a typically cylindrical tool coil. The induced current in the workpiece is oriented in the circumferential direction, while the magnetic field lines in direct proximity of the tube are oriented in the axial direction.

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Figure 4.1 Setup of electromagnetic pulse compression processes.

According to the right-hand rule, the resulting Lorentz force acts radially between tool coil and workpiece, resulting in a reduction of the tube diameter, that is, a compression of the workpiece in the forming area as shown in Fig. 4.1. In order to withstand the high forces and guarantee an acceptable lifetime, the coil winding has to be supported by housing elements, providing sufficient mechanical strength. In the case of coils for electromagnetic pulse compression, the resulting forces acting on the tool coil are mainly directed outward. In addition, axial forces occur, especially at the ends of the coil. Thus in practice, compression coils usually feature massive housing, supporting the coil winding from the outside. Contrary, only a thin insulating layer is necessary inside the coil in the gap between the workpiece and winding, so that a small gap width can be realized, which is beneficial for the process efficiency. This setup is frequently complemented by a so-called fieldshaper or field concentrator, that is, an additional tool element featuring at least one axial slot, which is positioned between the tool coil and workpiece. In order to achieve high process efficiency, fieldshapers must be made of electrically highly conductive materials that provide sufficient mechanical strength. Frequently, high-strength copper alloys, such as CuCrZr or CuBe alloys, are used, but in the case of small lot sizes, and in processes where the required energy is not too high, also aluminum alloys, for example, from the 6000 series can be used in order to save material costs. Using a double-slotted fieldshaper, which allows separating the fieldshaper halves, can significantly facilitate handling issues. If a fieldshaper is applied, the coil current does not interact with the workpiece directly, but it induces a magnetic field and a current in the fieldshaper. Due to the skin effect, this current flows along the outer circumference of the fieldshaper in the opposite direction of the coil current. At the axial slot, it is guided to the inner circumference, where it interacts with the workpiece as shown in Fig. 4.2. Typically, here in the so-called concentration area, the fieldshapers feature significantly smaller diameters and lengths compared to the

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Figure 4.2 Setup and force F distribution of electromagnetic pulse compression processes using a fieldshaper.

Figure 4.3 Fieldshapers for adapting one and the same cylindrical compression coil for workpieces with rectangular (left) and rotationally symmetric (right) cross section.

outer surface. Consequently, the current density, the magnetic field strength, and the acting forces are much higher compared to the outer surface, which explains the name of this additional tool component. The focusing effect enables designing a forming process with significantly reduced load acting on the tool coil, and consequently coil design becomes easier, leading to cheaper tools with improved lifetime. Another important advantage of using fieldshapers is that it significantly increases the flexibility of a specific tool coil’s application. This is especially relevant in the case of manufacturing tasks with a high number of variants and small batch size. The use of different fieldshapers makes it possible to adapt one and the same tool coil to workpieces of different diameters and cross sections, as shown in Fig. 4.3, and to vary the length of the forming zone. As manufacturing a new fieldshaper is much cheaper than developing a completely new tool coil, this allows significant cost savings. Frequently, a symmetric setup, as shown in Fig. 4.2, is possible, which features symmetric loading of the equipment and the workpiece. However, especially in the case of more complex workpieces, geometric restrictions of the part and the coil

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housing might necessitate modifications of the setup. It is possible to position the concentrating area of the fieldshaper outside the axial center of the coil winding (eccentric setup) or even outside the axial length of the tool coil (out-of-coil setup). In the eccentric setup the resulting distribution of the electrical current and the magnetic field causes slightly inhomogeneously distributed force, which might be adjusted by finetuning of the geometry of the fieldshaper’s concentrating area if necessary (see Ref. [14]). The out-of-coil setup is additionally related to a reduction of the process efficiency, which means that it should be applied only if indispensable. The axial slot(s) required to enable the process with a fieldshaper cause an inhomogeneity of the force application, and clear focusing of the magnetic field on the concentration area is only possible if there is a significant increase of the gap width between the fieldshaper and workpiece outside the concentration area. At the same time the fieldshaper has to withstand the acting loads. Moreover, it has to be considered that compared to a well-adapted direct acting tool coil, the overall setup is much bigger if a fieldshaper is applied. Thus, the length of the current paths and the volume of gaps in which the magnetic field can extend are higher, resulting in additional resistive losses and reduced magnetic field density, both leading to a reduction of the process efficiency. Thus, parameter measurements (thicknesses, angles, gap widths, etc.) must be properly selected in order to guarantee highly efficient focusing of the magnetic field and the corresponding forces as well as high durability of the tool component.

4.2.2.2 Electromagnetic pulse expansion Similar to electromagnetic pulse compression, the coil winding is also usually cylindrical in electromagnetic pulse expansion processes, but in this process variant, it is positioned inside the workpiece, which is typically a tube or hollow profile (compare Fig. 4.4). Again, the induced current in the workpiece flows in the

Figure 4.4 Setup of electromagnetic pulse expansion processes.

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circumferential direction, while the magnetic field lines in direct proximity of the tube are axially oriented and the force acts radially between the tool coil and workpiece. However, due to the changed arrangement of the tool coil and workpiece, the resulting forces on the workpiece are directed outward and consequently, the workpiece is deformed outward, resulting in an increase of the tube diameter, that is, an expansion in the forming area. In the case of electromagnetic pulse expansion a minimum diameter of the tool coil is necessary, and correspondingly, a minimum workpiece diameter is required for process-related reasons. These can be explained via the following correlations: in electromagnetic pulse forming the highest forces always act in areas where the highest magnetic field strength occurs. If a cylindrical coil without any workpiece is considered, the highest magnetic field strength is generated inside the coil, causing radial forces in the outward direction on the turns of the coil. Outside the coil the magnetic field can extend freely to a nearly unlimited volume, so that the field strength and the resulting counterforces acting inward on the coil winding are correspondingly small. As a consequence, the resulting force is directed outward, and assuming the same current, it is higher the smaller the coil diameter is because the magnetic-field strength is focused on smaller volumes within the coil (see Fig. 4.5). If the coil is inserted into a workpiece made of an electrically highly conductive material, the magnetic field and the forces inside the tool coil are reduced because the current in the coil winding is drawn toward its outer surface. The distribution of the magnetic field outside the coil is changed, too. As it cannot pass the workpiece easily, the magnetic field is localized in the very small gap volume between the workpiece and coil, leading to high forces acting outward on the workpiece and inward on the tool coil. If the process and the coil geometry are properly designed, these forces are higher compared to those inside the coil and consequently, the resulting forces acting on the coil winding are directed inward. This load case can easily be compensated if the coil is equipped with a pressure-resistant core inside the coil winding. Therefore, it is indicated as load case OK (abbreviation for: okay) in Fig. 4.5. However, if the workpiece moves away from the tool coil due to the deformation, the magnetic field outside the coil can extend to a bigger volume again, and the resulting forces acting on the coil winding can change their direction from inward to outward. This is indicated as load case NOK (abbreviation for: not okay) in Fig. 4.5. Nevertheless, in order to avoid a high gap width between the coil and workpiece, leading to reduced process efficiency, it is not possible to add reinforcing housing elements here. This means that these forces must be kept at a moderate level that can be withstood by the winding itself. This effect is more critical in the case of small coil diameters because the magnetic field inside the coil is higher there leading to an increase of the forces directed outward (see Fig. 4.5). Therefore, there is a risk of early coil failure due to the process-specific load case. The critical diameter depends on various geometrical, electrical, and mechanical parameters of the setup, but usually the diameter of an expansion coil for processing electrically highly conductive workpieces should be approximately 50 mm or higher.

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Figure 4.5 Force directions acting on a cylindrical expansion coil with and without a surrounding workpiece.

Similar to electromagnetic pulse compression processes, fieldshapers can also be applied in the case of electromagnetic pulse expansion in order to adapt one and the same tool coil to different workpiece geometries. However, here, the fieldshaper surface facing the workpiece (i.e., the outer surface) features a higher diameter compared to the surface facing the tool coil (i.e., the inner surface). Thus, a concentrating effect can only be achieved if the outer length is significantly shorter than the inner one. Consequently, the typical advantages of using a fieldshaper cannot be exploited as effectively as in the case of electromagnetic pulse compression so that the corresponding drawbacks often predominate. Therefore, the use of fieldshapers is reasonable only in very specific application cases, for example, to avoid reversal of the force acting on the coil winding due to the workpiece deformation, or to level out variations in the force distribution in the area of the turns and the clearances inbetween, which can lead to nonuniform deformations, especially in the case of very thin-walled workpieces [15].

4.2.2.3 Electromagnetic pulse forming of flat and preformed sheet metal In the case of electromagnetic pulse forming of sheet metal, the spectrum of potential workpiece geometries is much wider than in the case of compression or expansion of tubes or hollow profiles. Correspondingly, various different shapes of coils can be used. Usually, flat coils are applied here. This means that the coil winding is arranged in plane and positioned in parallel and with a small distance to the workpiece, as illustrated in Fig. 4.6. The spectrum of potential winding geometries includes spiral shapes, which are frequently used for forming rotationally symmetric workpieces, rectangular shapes [16,17], super-ellipsoidal shapes [18], triangular shapes [19], trident shapes [20], U-shapes [21], and free-form contours. A particularity of most flat coils is a strong inhomogeneity in the force distribution. Fig. 4.7 shows this phenomenon exemplarily for selected winding variants. It is obvious that in direct proximity of the turns, the pressure is high, while it drops rapidly between the turns, and especially

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Figure 4.6 Setup of electromagnetic pulse sheet metal forming processes.

Figure 4.7 Exemplary flat coils and corresponding force distributions.

between turns carrying reversely directed currents (i.e., in the center of the spiral and the rectangular coil and between the central branch and outer branches of the trident-shaped coil). This can be explained via the local orientation of the magnetic field lines and the currents, that is, the discharging current in the coil as well as the induced currents in the workpiece. The magnetic field lines typically encircle the bundle of turns carrying unidirectionally oriented current. In direct proximity of the highly conductive workpiece, they are reoriented in a tangential direction, as shown in Fig. 4.6. In these areas as well the induced current is flowing in the workpiece. Taking the right-hand rule into consideration, it is obvious that this results in high

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forces acting perpendicular to the workpiece and causing out-of-plane deformation. Between the bundles of turns carrying reversely directed currents, there is no noteworthy induced current in the workpiece, and consequently, no forces arise. Thus at the beginning of the process, the deformation occurs localized in the force-actuated areas. For example, in the case of a rotationally symmetric workpiece formed with a spiral coil winding, a toroidal shape is formed, while the center of the part remains nondeformed (see Fig. 4.6). Later, in the deformation process, inertia forces might also accelerate this area of the workpiece, and it can even reach highest velocities and displacements, if this area is not too large. This nonuniform force application and the described inertia effects severely complicate the process design in electromagnetic pulse sheet metal forming. Therefore several attempts are made in order to reduce this effect. Obviously, decreasing the distance between the turns carrying reversely directed currents leads to smaller nonforce-actuated areas. However, similar to the restrictions regarding minimum diameters in electromagnetic pulse expansion processes, it also must be considered here that depending upon the geometrical and electrical parameters of the setup, critical load cases might occur during the forming process, which can significantly reduce the lifetime of the coil. The principal force distributions depicted in Fig. 4.7 represent the out-of-plane components of the forces acting on the workpiece as well as on the tool coil in the initial condition, that is, as long as the workpiece is very close to the coil winding only. Considering the mechanical loading of the coil, there are also radial forces acting between the workpiece and tool coil due to edge effects, and between the individual turns, but the main forces are directed away from the workpiece. Avoiding deformation of the coil winding due to these forces can be realized relatively easily by providing housing components, which reinforce the winding surface facing away from the workpiece and enclosing the outer coil diameter. If necessary, distance elements providing good compression strength can also be inserted between the individual turns of the coil winding in order to support them and guarantee electrical insulation. However, similar to the electromagnetic pulse compression and expansion setup between the workpiece and winding, only a thin insulation layer without noteworthy mechanical strength is usually provided in order to keep the gap width as small as possible. Nevertheless, as soon as the workpiece starts to move, the mechanical loading of the coil changes significantly toward that of an empty tool coil (i.e., a coil without workpiece in direct proximity). Fig. 4.8 exemplarily compares this load case in an idealized form, assuming rotationally symmetrical conditions, to a similar one with a sheet metal workpiece in direct proximity. It can clearly be seen that without a workpiece, the forces are focused at the inner turns of the tool coil. The out-ofplane components of the acting forces compensate each other due to the symmetry of the setup. However, in practice, this ideal symmetry is not given because of the spiral shape of the winding and the necessity to connect the ends of the winding to the pulsed power generator. As a consequence, especially in the case of coils with small inner diameters (and small distances between turns carrying reversely directed currents in the case of other coil shapes), critical components of the force can occur

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Figure 4.8 Magnetic field distribution and forces acting on a flat coil with and without a sheet metal workpiece in direct proximity.

directed out-of-plane and toward the unsupported surface of the coil. As a consequence, especially the inner turns of the coil tend to move out of the coil housing, which can lead to early failure of the coil. This so-called telescope effect is stronger for smaller inner coil radii because the forces acting on the inner turns, including the critical force components, increase with decreasing radius. It can be prevented by using a fieldshaper that stays in close proximity of the coil during the process, but this is uncommon because guiding the currents in the fieldshaper in a proper way is much more difficult compared to the tube compression setup. As an alternative, cylindrical coils with a fieldshaper can be used for sheet metal forming, as shown in Fig. 4.9. In this setup the magnetic field caused by the discharging current flowing through the cylindrical coil winding induces a circumferential current in the nearby surfaces of the fieldshaper, which is directed opposed to

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Figure 4.9 Setup and force (F) distribution for electromagnetic pulse forming of sheet metal, using a cylindrical coil and an appropriate fieldshaper.

the coil current. Due to the specific geometry of the fieldshaper, the arrangement of coil winding, fieldshaper, and workpiece, and affected by the skin effect, the current in the fieldshaper is directed toward the fieldshaper surface facing the workpiece. Thus, a magnetic field is initiated in the gap between the fieldshaper and the workpiece, which in turn induces a counter current in the sheet metal workpiece. Due to the interactions between the magnetic field and the currents, the predominantly radial forces between the coil winding and fieldshaper are transformed to mainly axial forces acting between the fieldshaper and workpiece, which can cause out-ofplane deformation of the latter. In comparison to the flat coils described earlier, the tooling concept using a cylindrical coil and a fieldshaper avoids critical loading of the tool coil due to changes in the orientation of the major force components. In addition, it allows using any desired number of turns, which might be problematic for a flat coil due to space restrictions in the case of small forming areas. However, the principle problem of a strongly inhomogeneous force distribution with zero force in the center remains the same as in the case of flat coils. An interesting option to avoid this is offered by using only those sections of the coil winding for shaping the workpiece where the turns are carrying unidirectionally oriented current. The rest of the winding is used in order to close the current loop outside the deformation zone [18]. This approach facilitates the process design significantly, but necessitates remarkably bigger, that is, more expensive tools and larger blanks for the workpiece, because the blank should cover the complete winding of a flat coil for opportune process course. Similarly, reversal of the current direction in the forming zone is also avoided in the case of the so-called uniform pressure actuator [22]. In addition to the forming of flat sheet metal, it is also possible to apply electromagnetic pulse forming of two or three dimensionally shaped sheet metal. For this purpose the surface of the tool coil or fieldshaper that faces the workpiece has to be adapted to the curvature of the preformed sheet. Except for this modification the process proceeds similarly as in the case of electromagnetic pulse forming of flat sheets. For details, see Section 4.4.1.3.

Electromagnetic pulse forming

4.3

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Calculation of the process mechanics

4.3.1 Analytical calculation of the acting loads The acting mechanical loads during an electromagnetic pulse forming process can be characterized via the Lorentz forces (volume forces) or via the corresponding magnetic pressure (surface load). The calculation of both absolutely requires knowledge regarding the evolution of the current and the magnetic field during the process. Analytical calculation approaches were developed in the 1960s. High efforts were made, for example, by the group of Professor Bu¨hler at Hanover, but others also contributed, as shown in the review by Psyk et al. [12]. According to Eq. (4.1), the calculation of the Lorentz forces acting on the workpiece is directly based on the current density J and the magnetic flux density B. F5J 3B

(4.1)

The current course can be determined by fundamental considerations of electrical engineering. Here, the setup used for electromagnetic pulse forming can be represented by an oscillating circuit. A simplified equivalent circuit diagram of a setup is shown in Fig. 4.10. Here, only the discharging circuit is shown, while the charging circuit is disregarded. In the equivalent circuit diagram the forming machine is represented by its characteristic electric properties: the capacitance C, the inner inductance Li, and the inner resistance Ri. The tool coil, characterized via its inductance Lcoil and resistance Rcoil, is connected in series with the machine parameters in the primary circuit, where the coil current Icoil flows. On the contrary, the workpiece is regarded as a separate secondary circuit, including the inductance Lwp and the resistance Rwp of the workpiece, which vary with the workpiece deformation. The two electrical circuits are inductively coupled so that current Iwp is induced in the workpiece. Via fundamental electrical approaches, the individual resistances and inductances of the machine, the coil, and the workpiece can be combined to an equivalent resistance R and inductance L of the oscillating circuit. The time-dependent current I(t) can be calculated based on the resistance R, the inductance L, and the capacitance C via the differential Eq. (4.2).

Figure 4.10 Equivalent circuit diagram of the discharging circuit in electromagnetic pulse forming processes.

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L

ð dI 1 1R1 Idt 5 0 dt C

(4.2)

With the initial conditions of no current I ð t 5 0Þ 5 0

(4.3)

and a defined voltage U0 at the capacitor, U ð t 5 0Þ 5 U 0

(4.4)

the differential equation leads to the solution for the current course I(t) I ðt Þ 5

U0 2βt e sin ωt ωL

(4.5)

with the damping coefficient β β5

R 2L

(4.6)

and the circular frequency of the current ω ω5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω20 2 β 2

(4.7)

with 1 ω0 5 pffiffiffiffiffiffi LC

(4.8)

Based on this current course I(t), the corresponding magnetic field strength H (t) can be calculated. It can easily be approximated by analytical calculations, especially for the electromagnetic pulse compression setup, taking into consideration the number of turns of the tool coil n and the length of the coil l via Eq. (4.9). H ðt; zÞ 5

nI ðtÞ kH ð zÞ l

(4.9)

It is based on the well-known equation for calculating the field strength in a long cylindrical coil (i.e., the coil length must be much higher compared to its diameter). In addition it considers edge effects via the so-called distribution factor kH, which depends on the axial position in the coil z with its origin in the middle of the coil, the coil length l, and the effective gap width a.

Electromagnetic pulse forming

kH 5

  1 2z 1 l 2z 2 l atan 2 atan π a a

125

(4.10)

This effective gap width depends on the air gap between the tool coil and workpiece aair, the skin depth in the tool coil δcoil, and the skin depth in the workpiece δwp and can be calculated via Eq. (4.11). a 5 aair 1

1 δcoil 1 δwp 2

(4.11)

The magnetic pressure p depends on the magnetic field strength at the workpiece surface facing the tool coil Hgap, the penetrated magnetic field strength at the workpiece surface facing away from the tool coil Hpenetrated, and the magnetic permeability μ. p5

 1  2 2 μ Hgap ðtÞ 2 Hpenetrated ðtÞ 2

(4.12)

If the penetrated magnetic field is negligible, this equation can be simplified as p5

1 μH 2 ðtÞ 2 gap

(4.13)

Ref. [23] shows that for tube compression processes, this is usually the case if the ratio of skin depth and inner radius of the tube is smaller than 0.2 and the ratio of wall thickness to skin depth is two or higher. In practical applications, frequently fundamental input data for the calculation of the acting local and temporal distribution of the magnetic pressure is not available. This concerns, for example, the time-dependent inductance and resistance of the equivalent circuit diagram or the position- and time-dependent variation of the air gap between the tool coil and workpiece. It is only in cases where an existing process is analyzed that this data might be available from measurements. Fig. 4.11 shows results of a study of this scenario. Here, measured current courses were used, which already include potential influences of the workpiece deformation on the electrical circuit parameters. In addition, the displacement of the workpiece surface was used in order to consider a time-dependent air gap between the tool coil and workpiece. The figure clearly demonstrates the influence of the workpiece deformation on the acting loads for a tube compression and a sheet metal forming process, and the resulting error if the retroactivity of the workpiece deformation on the parameters of the electrical circuit and on the air gap between the tool coil and workpiece are disregarded. This effect is more pronounced if the current pulse is slower (e.g., due to higher inductance of the electrical system) and if the deformation is higher. Thus, it is usually much more relevant for sheet metal forming than for tube compression or expansion processes.

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Figure 4.11 Correlations between current I, magnetic pressure p, and workpiece deformation u for tube compression and sheet metal forming processes.

4.3.2 Numerical calculation of the process As the possibilities of analytically calculating the electromagnetic pulse forming process are limited to very special cases, numerical simulation is usually applied. The full process model includes the parameters of the forming machine, the inductor unit consisting of tool coil and fieldshaper, if applicable, the workpiece, and application-dependent additional tool components such as forming dies. Especially at the process design stage, usually no measurement data is available as input data for the simulation, so this full process model is the only option to determine important time- and position-dependent process characteristics. The simulation includes four subproblems corresponding to different physical fields. Specifically, these are the electrical oscillator simulation, the electromagnetic simulation, the structural simulation, and the thermal simulation. All subproblems interact with each other via mutual influences as illustrated in Fig. 4.12. In order to consider these interdependencies, coupling of the different simulations is obligatory. Typically, a weak coupling or stepwise coupling is used. This means that the individual simulations are considered individually for short periods of time, and results are exchanged consistently until the end of the process [24
26]. Regarding, for example, the correlations between the electrical oscillator simulation and the electromagnetic simulation, this means that the electrical oscillator simulation calculates the coil current based on the capacitor charging voltage and the initial electric properties of the equivalent circuit diagram capacitance, inductance, and resistance (compare Fig. 4.10). This coil current is important input data for the electromagnetic simulation, but as shown in Section 4.3.1, the inductance and resistance of the consumer load (i.e., the inductor system including the

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Figure 4.12 Subproblems of the electromagnetic pulse forming simulation and their interactions.

workpiece) change during the process due to workpiece deformation. This change is calculated via the electromagnetic simulation and must be returned to the electrical oscillator simulation in order to allow its consideration in calculating the further current course. Similarly, the mutual influences between the acting mechanical loads and the resulting workpiece deformation are considered by coupling the electromagnetic and the structural simulation. This means that, based on the initial geometry, the coil current, and important electromagnetic material parameters, the electromagnetic process simulation is used in order to calculate the initial distribution of the Lorentz forces and the magnetic pressure, respectively. This information is transferred to the structural model. The resulting deformation is calculated for a short period of time based on this data, and additionally considering the initial geometry as well as important mechanical material parameters. Then, the updated geometry is retransferred to the electromagnetic simulation and used for calculating an updated distribution of the Lorentz forces and the magnetic pressure, respectively. The correlations between the electromagnetic and the structural simulation, on the one hand, and the thermal simulation, on the other hand, mainly comprise temperature rise in the workpiece and the tool coil due to dissipative heating and resistive losses during electromagnetic pulse forming processes. It is well known that important material properties, such as the electrical conductivity and the flow curve, change with varying temperature. Thus the temperature development during the process can have a retroactive effect on the electromagnetic and the structural simulation, which can be considered by coupling these simulation systems as well. An overview of the most important input and output data for all subproblems is given in Table 4.1. Depending on the aim of the simulation and the existing knowledge about the specific manufacturing task, simplifications can be made in order to reduce complexity and calculation time. This concerns complete elimination of subproblems as

Table 4.1 Input and output data of the different subproblems of the full process simulation. Electrical oscillator simulation

Input

Output

Electromagnetic simulation

Input

Output

Capacitor charging voltage Capacitance of the pulsed power generator Inner inductance of the pulsed power generator Inner resistance of the pulsed power generator Consumer inductance (tool coil, workpiece, and fieldshaper) Consumer resistance (tool coil, workpiece, and fieldshaper) Current course Voltage course Inductance of the electrical circuit Resistance of the electrical circuit Geometry Temperature-dependent electrical conductivity Time- and position-dependent temperature Magnetic permeability Coil current Time- and position-dependent magnetic field strength Time- and position-dependent magnetic flux density Time- and position-dependent current density Consumer inductance Consumer resistance Time- and position-dependent Lorentz force and magnetic pressure, respectively

Source: user defined Source: user defined Source: user defined Source: user defined Source: electromagnetic simulation Source: electromagnetic simulation Required for: electromagnetic simulation Required for: process analysis/evaluation Required for: process analysis/evaluation Required for: process analysis/evaluation Source: user defined (initial geometry); structural simulation (deformed geometry) Source: user defined Source: thermal simulation Source: user defined Source: electrical oscillator simulation Required for: process analysis/evaluation Required for: process analysis/evaluation Required for: thermal simulation; process analysis/ evaluation Required for: process analysis/evaluation Required for: process analysis/evaluation Required for: structural simulation; process analysis/evaluation

Structural simulation

Thermal simulation

Input

Output

Initial geometry Temperature- and strain rate
dependent flow curve Time- and position-dependent temperature Density, Young’s modulus, failure parameter, etc. Time- and position-dependent Lorentz force and magnetic pressure, respectively Deformed geometry

Input

Displacement Time- and position-dependent velocity Time- and position-dependent strain Time- and position-dependent strain rate Geometry

Output

Thermal conductivity Resistive losses Deformation heat Time- and position-dependent temperature

Source: user defined Source: user defined Source: thermal simulation User defined Electromagnetic process simulation Required for: electromagnetic simulation; process analysis/evaluation Required for: process analysis/evaluation Required for: process analysis/evaluation Required for: process analysis/evaluation Required for: process analysis/evaluation Source: user defined (initial geometry) Structural simulation (deformed geometry) Source: user defined Source: electromagnetic simulation Source: structural simulation Required for: electromagnetic simulation; structural simulation; process analysis/evaluation

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well as simplification within the specific model. In practice, the temperature rise during forming is usually of minor significance if electrically highly conductive materials, such as aluminum or copper alloys, are considered. Thus, the retroactivity on the material parameters can be disregarded, and the thermal process simulation is not necessary. Regarding the coil current, using measured data or making an educated guess based on reference data from a similar setup might be possible so that the electrical circuit simulation also becomes unnecessary. Thus the coupled electromagnetic and the structural simulation usually form the core of the process simulation. Two different principle modeling strategies, the finite element simulation (FEM) and the boundary element simulation (BEM), can be used for the electromagnetic simulation. The major difference is that in FEM, the air in the complete surrounding of the inductor system and workpiece must be meshed because the magnetic field spreads here and FEM solves the differential equations for unknowns in the volume. During the deformation process, extreme distortion of the elements occurs, for example, in the gap between the tool coil and workpiece, because the gap width increases from some hundredths or tenths of a millimeter up to several millimeters, or in sheet metal forming even up to some 10 mm so that remeshing is necessary. In contrast, there are other regions, for example, the zone between the workpiece and die, where the air must be completely eliminated during contact. These two aspects can lead to severe numerical problems, which can be avoided by using BEM. This modeling strategy uses an integral formulation of Maxwell’s equations and calculates the electric and magnetic fields directly from the source. As unknowns are determined on the boundaries only, just the active regions, for example, inductor system, workpiece, and other electromagnetically relevant (metallic) components in direct proximity must be discretized, while the surrounding air can be disregarded. As a consequence, the model size is reduced dramatically in terms of number of elements and nodes. Usually, FEM is used for the mechanical model. In many cases, only the deformation of the workpiece is of interest, and therefore only this component has to be modeled as a deformable body, while the tool modeling can be significantly simplified. Only those elements that are in mechanical contact with the workpiece have to be considered, while the inner setup, for example, of the tool coil can be disregarded. Modeling the tool elements as rigid bodies can reduce calculation time further. However, if the coil loading is of interest - for example, in order to make conclusions about coil stability and lifetime - more detailed modeling of this component is also necessary in the structural simulation. In this case, winding and housing components have to be modeled individually, and elastic or even plastic deformation must be considered. As in all simulations, the quality of the numerical result strongly depends on the input data. However, providing appropriate material parameters is very challenging here because during some process variants, strains values can reach 0.4
0.7 in larger regions and locally, they can even be in the range of 1.0 or more. At the same time, strain rates of 103
104 s21 can easily be reached and for some cases, locally, values of up to 106 s21 occur. As shown in Fig. 4.13, there are very few

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Figure 4.13 Strain rate values achievable in typical high-velocity material testing.

material characterization tests, which can provide data for such high strain rates, and frequently their soundness is limited. However, new developments in the field of material characterization, including inverse numerical simulation approaches in order to analyze measurement data, can help solve this problem [8].

4.4

Advantages and application fields of electromagnetic pulse forming

When evaluating the suitability of electromagnetic pulse forming for a special application, the process-specific advantages and limitations should be taken into consideration in order to allow exploiting the strengths of the process and avoiding its drawbacks. In addition to the typical advantages of high-velocity forming processes mentioned in Section 4.1, this process excels due to the following features: G

G

G

G

G

G

The force application is contact-free. Thus, even workpieces with sensitive surfaces can be processed without disturbance or deterioration. As a consequence of the contact-free force application, friction is avoided, and therefore no lubricants are required. This makes the process environmentally friendly and can reduce cleaning and corresponding costs. In order to clearly define the target shape of the workpiece in electromagnetic pulse forming, only one form-defining tool (die) is required, while the other one is replaced by the more flexible inductor. This can significantly reduce manufacturing and tryout costs, because the positioning of the inductor and die is much less challenging compared to, for example, a typical deep drawing tool. Only very few parameters have to be set for a designed process. The key parameter—the capacitor charging energy—is an electrical parameter, which can be set very precisely. Therefore forces can be adjusted exactly and reproducibility is high. The process is applicable with remote control and under special conditions such as in a vacuum, in a clean room, and in a radioactive environment. Apart from mere shaping applications, the process can also be used for joining by forming and for cutting by forming, and it is possible to combine such production steps. Joining by forming is possible for similar and dissimilar materials (e.g., metals and fibrereinforced plastic) and typical temperature-induced problems of conventional welding

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processes, such as formation of oxide or intermetallic phases, heat distortion, and thermal softening, are avoided.

Nonetheless, some restrictions of the process must be considered in practical applications. This means specifically the following: G

G

G

G

G

The process principle requires a closed current path in the workpiece. High efficiency can be achieved for materials with good electrical conductivity ( . 7 MS/m) only. For insulating or poorly conductive materials, a so-called driver can be used (i.e., an additional element made of electrically conductive material, which is positioned between the inductor and workpiece, and mechanically transmits the Lorentz force induced therein to the workpiece). However, this is hardly economically justifiable, so that alternative drives, for example, accelerated tools, are used [12]. To achieve high efficiency, the tool coil should be positioned close to the workpiece (initial distance # 1
3 mm). Due to the high velocities, the material flow into the forming zone is limited. This means, that the material flow mainly comes from the wall thickness, which reduces the achievable forming depth especially for small material thicknesses. Feasibility of large forming operations has to be proven for the particular case. Depending on the manufacturing task and the related inductor design, the tool lifetime is limited.

Based on the described relationships, forming, joining, and cutting operations can be differentiated, depending on the tool components used and their interaction with the workpiece. Furthermore, combinations of these operations within the same component and process step are possible.

4.4.1 Shaping 4.4.1.1 Electromagnetic pulse forming as a stand-alone process In order to control the resulting shape of the formed part, the principle setup of the process variants, illustrated in Figs. 4.1, 4.4, and 4.6, is supplemented by suitable form-defining tools, that is, mandrels in the case of tube compression processes and dies in the case of tube expansion or sheet metal forming. The highvelocity impact of the accelerated workpiece with this form-defining tool requires special attention. The reason is that the pressure distribution and the resulting deformation course as well as the material combination of workpiece and tool can hinder the achievement of the forming task. Ideally, a workpiece section touching the form-defining tool has to be stopped immediately. This means that the high kinetic energy stored in the workpiece movement at that moment has to be dissipated. If this cannot be managed, the workpiece will be reflected from the form-defining tool’s surface, resulting in severe deflections from the desired shape. This effect is frequently called rebound [27]. In contrast, workpiece sections, which have not reached the surface of the form-defining tool, must continue the deformation until they touch it. However, the movement of these areas can be severely decelerated as soon as the alignment of the workpiece

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Figure 4.14 Examples of electromagnetic pulse formed components: (A) sharp-edged bead in an aluminum sheet metal part; (B) design element formed in an aluminum sheet metal part; and (C) design elements formed in an aluminum can.

to the form-defining tool has started in a neighboring area, which can result in incomplete form filling. Both effects have to be avoided by proper process design. This is extremely complicated in the case of high forming depths, because in these cases, the workpiece is flying freely over a long distance. As explained earlier, the applied forces drop dramatically due to the increasing gap width after the first acceleration. However, especially in the case of sheet metal forming, the workpiece deformation can continue driven by inertia forces. At this stage the process can hardly be controlled any longer. Thus the initial pressure and force distribution has to be thoroughly designed so that the resulting inertia force distribution corresponds with the target geometry, and the forming task can be successfully completed. Nevertheless, in the case of small forming depths, free flying of the workpiece wall is avoided, and the complete process is mainly driven by the Lorentz forces instead of the inertia forces. This significantly facilitates the process design. As a consequence, the technology can clearly demonstrate its potential, when it comes to forming of sharp-edged details with small forming depths. Here, the increased formability due to high strain rates avoids cracking, and the reduced springback allows better geometric accuracy. Thus this process can be a proper solution for overcoming limits of conventional forming technologies. Fig. 4.14 shows examples of electromagnetic pulse formed parts correlating with the aforementioned features. Processing of the depicted precoated tubular parts is possible only due to the contact-free, that is, gentle force application.

4.4.1.2 Multiple-discharge electromagnetic pulse forming Another peculiarity of the electromagnetic pulse forming process is related to the size, because increasing energy is required with increasing area to be formed. This is related to an increasing electrical, mechanical, and thermal loading of the equipment (i.e., the tool coil and the forming machine), which can deteriorate the lifetime of these components significantly. As a consequence, electromagnetic pulse forming is preferably applied to small-sized components and component areas. An incremental

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Figure 4.15 Electromagnetic pulse incremental forming setup and chair with incrementally formed seat.

approach can be used in order to overcome the mentioned size restrictions and to extend this process to potential application cases for large components. In these cases the process is frequently referred to as sequential or incremental electromagnetic (pulse) forming or electromagnetic (pulse) incremental forming [17,28]. In this process variant, coil and workpiece are moved relatively to each other after performing a capacitor discharge and a corresponding deformation, followed by further discharges and deformations. By increasing the number of discharges and a suitable movement of coil and workpiece, larger components are deformed gradually. This is especially applicable for small-volume production of large sheet metal parts as used, for example, in 3D structural facing, classic car restoration, and machine tool housing. Fig. 4.15 shows an example of a chair seat and the corresponding tool setup. This approach of multiple-discharge forming also offers a possibility to extend electromagnetic pulse forming toward larger forming depths, because multiple small discharges allow easier control of the inertia forces, which facilitates process design as mentioned previously. However, in order to avoid extreme gap widths, which significantly reduce the process efficiency until finally no deformation can be realized any longer, it is indispensable to reposition the tool coil in direct proximity of the workpiece. Obviously, in practical applications, an automated solution is necessary for the repositioning of tools and workpieces. This can be realized, for example, via suitable robots or driven axis systems. As a matter of fact, it has to be considered that the multiple-discharge process needs multiple times the process duration of a singledischarge process so that, similar to other incremental forming processes [29], the process time is higher compared to conventional technologies such as deep drawing. However, depending on the technological and economic circumstances and requirements, other process-related advantages regarding formability, accuracy, and flexibility of the tool application might compensate this drawback of the incremental approach.

4.4.1.3 Combined electromagnetic pulse and conventional forming Another option to overcome part-size limitations of electromagnetic pulse forming and to make the process-specific advantages exploitable for bigger parts is given by

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135

combining electromagnetic pulse forming with conventional forming technologies in process chains or integrated processes. The most frequently regarded process combination considers deep drawing with integrated electromagnetic pulse calibration of sheet metal parts. Here, the overall shape of the part is generated by conventional deep drawing, while electromagnetic pulse forming is used in order to locally shape a critical area, where deep drawing alone cannot provide the desired shape, for example, due to necking, cracking, springback, or surface deflections. This can concern the forming of complex details and secondary form elements such as door handles or design elements as well as the sharpening of part radii. For this purpose a tool coil is incorporated into the deep drawing tool. Depending on the specific application, this can be the punch as well as the die. Restrictions from both processes involved have to be considered in the development of such combined tools. The electromagnetic aspects have to be taken into account regarding potential interactions between the tool coil and the surrounding metallic components of the deep drawing tools. This means that short circuits have to be avoided as well as induced parasitic currents in the surrounding tool elements, which impair the desired electromagnetic pulse calibration process. From the mechanical point of view a deliberate solution has to be found guaranteeing that the loads acting on the components of the tool coil can be withstood. Here, special high-performance materials can be used together with design strategies specifically adapted to the requirements of the combined process. Careful spotting of the surfaces is necessary to achieve high surface quality of the formed parts. In the process, the electromagnetic pulse calibration is performed at the bottom dead center of the deep drawing process, that is, under full deep drawing load [30]. Sophisticated adjustment of capacitor charging and discharging times, on the one hand, and press stroke and automated handling operations, on the other hand, enable a combined process with similar process times as in conventional deep drawing. In a similar process combination of deep drawing and electromagnetic pulse forming, repeated discharges at low energy are used to reduce friction during deep drawing in the area of the punch bottom and the blank holder in order to extend the achievable drawing ratios [31]. Apart from this currently most popular and promising process combination, there are several other attempts to combine electromagnetic pulse and conventional forming in such a way that the process-specific advantages of the technologies involved complement each other, while the respective drawbacks are compensated. Electromagnetic pulse forming can, for example, be used in order to introduce local stiffening elements during roll forming [32]. Similarly, electromagnetic pulse compression can locally form extruded profiles directly using the process heat from the extrusion in order to realize tubular components with varying cross section [33]. Combining bending and electromagnetic pulse calibration of sheets allows significant reduction of the characteristic springback [34]. In the process chain of bending—electromagnetic pulse forming (EMF)—hydroforming, optimized semifinished products for hydroforming are produced by bending and EMF. These allow significant extension of the forming limits compared to a hydroforming process using conventional semifinished parts [35].

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4.4.2 Joining In addition to mere shaping operations, electromagnetic pulse forming can also be used for joining. The resulting connections can fulfill high requirements regarding different quality criteria such as transferable load, tightness, and electrical and thermal conductivity. In principle, different joining mechanisms featuring different advantages and limits are available, so that depending on the characteristics of the components to be joined and the requirements to be made on the resulting connection, the most appropriate process variant has to be selected. These variants include interference-fit joints, which are based on an elastic
plastic bracing, form-fit joints, which are based on the formation of an undercut and metallic bonding, based on a microstructural joining mechanism. The latter one is frequently referred to as (electro-)magnetic pulse welding (see Fig. 4.16). An interference-fit joint is applicable in order to connect tubular metallic components to metal or nonmetal joining partners by electromagnetic pulse compression or expansion. Advantages of this joining technique are that only little deformation is required, which makes it especially suitable for joining brittle materials. The joint formation does not need any specific secondary design elements such as grooves. However, the joint strength is sensitive to part cleanliness, and in order to achieve high joint strength, a long joining area might be required. If the possible joint length is limited, form-fit joining might be the better solution, because this process can realize high-strength connections in a smaller space. However, for this process variant, usually one joining partner has to be equipped

Figure 4.16 Joining mechanism for electromagnetic pulse joining and corresponding examples.

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with additional geometrical features (grooves, beads, etc.) increasing the joint preparation effort. During the joint formation, material from the other joining partner flows into these geometrical elements and forms an undercut. Although the overall deformation of the workpiece is still relatively small, local strains can reach relatively high values so that ductile metals are required for the partner that is deformed. Similar to interference-fit joints, form-fit joining can also be used for metal/metal and metal/nonmetal joints. Achieving a tight connection might require some effort in both variants, and sometimes it might also be useful to combine this process with other joining techniques such as gluing. The basic setup for electromagnetic pulse welding is shown in Fig. 4.17 (left). Here, the principle of electromagnetic pulse forming is used in order to accelerate a tubular or sheet metal joining partner (the so-called flyer) toward a second static joining partner (the so-called target). Initially, a flyer and a target are positioned overlapping, with a defined distance usually in the range of 0.5
2 mm between them. The flyer is accelerated, and after overcoming the initial distance, the two joining partners collide with each other. A weld seam is formed if the impact conditions are within a process window depending on the material combination. Frequently, but not necessarily, this is characterized by a clearly recognizable wave formation with turbulences in the area of the wave junction (Fig. 4.17, right). Typical welding windows for different material combinations are presented in [36]. Typically, the weld seam is linear along the edge of tubular components or sheet metals with a length of up to several hundred millimeters [37]. However, the width is limited only to a few millimeters, and in some cases, it is necessary to have a larger welded area. Therefore, it is possible to combine electromagnetic

Figure 4.17 Principle of electromagnetic pulse welding and typical weld seam.

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pulse welding with an incremental approach, similar to the incremental process described in section 4.4.1.2 [38]. In contrast to conventional (i.e., thermal) welding operations, the weld is formed as a result of a high-speed collision without significant heating of the workpiece in magnetic pulse welding. Thus temperature-induced problems of conventional welding processes are avoided such as distortion or loss of strength in the heat-affected zone. Similarly, the formation of intermetallic or oxide phases is reduced to such an extent that they have no negative influence on the joint quality. This makes it possible to combine materials that are considered difficult-to-weld or nonweldable by conventional methods, for example, aluminum/copper or aluminum/steel (including stainless steel and press-hardened steel). No shielding gases, additives, or auxiliary materials are required for this process, offering economic and ecological advantages. The process can be used flexibly and is characterized by good reproducibility and automation, comparatively short process times, and low energy consumption.

4.4.3 Cutting In addition to joining by electromagnetic pulse forming, cutting of tubular and sheet metal components can be realized based on the principle of electromagnetic pulse forming, although it has a subordinate role and is rarely mentioned in literature. For this process the setup variants of the electromagnetic pulse forming process shown in Figs. 4.1, 4.4, and 4.6 are supplemented by a sharp-edged cutting tool positioned at the workpiece surface facing away from the tool coil. During the process, the workpiece is formed against this cutting tool. In contrast to conventional shearing operations, cutting by electromagnetic pulse forming corresponds to a bending deformation with intended fracture instead of the typical shearing process. As a result, the cutting edge differs from conventionally cut ones. Actually, it features no shear zone, there is no or hardly any burr, and the cutting zone shows a clearly visible rollover. Fig. 4.18 (left) shows an example. The coil, which replaces the conventionally used punch during cutting, allows a more flexible use of the tool. In addition, there is no tool movement, and consequently, the usually high

Figure 4.18 Demonstrator part for different cutting edges and electromagnetic pulse cut coated foil.

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tolerances related to the cutting gap are not necessary, making it possible to cut very thin foils without problems (Fig. 4.18, right).

4.5

Prospects for future developments

Electromagnetic pulse forming was already invented in the 1960s. However, the initial euphoria about the technology decreased after a few years and hence, the process disappeared again from research and industrial application. In the 1990s it was rediscovered within the activities related to the rising importance of lightweight design concepts and the correspondingly increasing desire to apply aluminum alloys. These materials provide all technological characteristics that are predestined for application in electromagnetic pulse forming, because they enable efficient use of the technology. These are high electrical conductivity, moderate strength, and low density. Since then, important progress related to the numerical simulation of the process was made, which offers an important advantage compared to the activities in the 1960s. Modern simulation tools allow assessing the feasibility of a specific manufacturing task and significantly support the process and tool design. Currently, there is a high interest again in implementing electromagnetic pulse forming into industrial manufacturing and to enable exploiting the process-specific advantages in serial manufacturing. First successful applications are in use and several more are in preparation. However, it is not reasonable to replace wellestablished and economically efficient technologies by electromagnetic pulse forming. Instead of this, electromagnetic pulse forming must be seen as a complementing technology, used to fulfill manufacturing tasks where conventional methods reach their limits. In order to achieve wider implementation and a real industrial breakthrough, additional research and development is indispensable. Currently, this especially concerns the fields of equipment design, numerical process modeling, testing of joints, and standardization issues. These fields are within the scope of the work of researchers, industrial developers, and manufacturing federations worldwide, so that important progress can be expected for the near future.

References [1] R. Neugebauer, K.-D. Bouzakis, B. Denkena, F. Klocke, A. Sterzing, A.E. Tekkaya, et al., Velocity effects in metal forming and machining processes, CIRP Ann. Manuf. Technol. 60 (2011) 627
650. [2] K. Lange, H. Mu¨ller, R. Zeller, T. Herlan, V. Schmidt, Hochleistungs-, Hochenergie-, Hochgeschwindigkeitsumformen, in: K. Lange (Ed.), Umformtechnik
Sonderverfahren, Band 4, Springer, 1993, pp. 7
46. [3] M. Schu¨ßler, Hochgeschwindigkeits-Scherschneiden im geschlossenen Schnitt zur Verbesserung der Schnitteilequalit¨at (dissertation), TU Darmstadt, 1990.

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[4] E.J. Bruno (Ed.), High-Velocity Forming of Metals, American Society of Tool and Manufacturing Engineers, Dearborn, MI, 1968. [5] H. Wielage, F. Vollertsen, Classification of laser shock forming within the field of high-speed forming processes, J. Mater. Process. Technol. 211 (2011) 953
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Karosseriefertigung im Spannungsfeld von Globalisierung, Kosteneffizienz und Emissionsschutz, Chemnitz, 2008, pp. 205
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ICHSF, Dortmund, 2006, pp. 161
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ICHSF, Dortmund, 2012, pp. 125
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ICHSF, Dortmund, 2012, pp. 45
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829. [13] D. Birdsall, F. Ford, H.P. Furth, R. Riley, Magnetic forming!, Am. Mach./Metalwork. Manuf. 105 (6) (1961) 117
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gestaltung bei der elektromagnetischen Umformung (Dr.-Ing. Dissertation), Technische Universit¨at Dortmund; Institute for Forming Technology and Lightweight Construction, Shaker Verlag, Aachen, 2005. [15] M. Linnemann, V. Psyk, C. Scheffler, W.-G. Drossel, Electromagnetic Forming of Design Elements, in: J.P. Wulfsberg, W. Hintze, B.-A. Behrens (Hrsg.) (Eds.), Production at the leading edge of technology, Springer, Heidelberg, 2019, pp. 179
188. [16] S. Golovashchenko, Material formability and coil design in electromagnetic forming, J. Mater. Eng. Perform. 16 (3) (2007) 314
320. [17] V. Psyk, P. Kurka, S. Kimme, M. Werner, D. Landgrebe, A. Ebert, et al., Structuring by electromagnetic forming and by forming with an elastomer punch as a tool for component optimisation regarding mechanical stiffness and acoustic performance, Manuf. Rev. 2 (2015) 23. Available from: https://doi.org/10.1051/mfreview/2015025. published by EDP Sciences, 2015. [18] V. Psyk, C. Beerwald, M. Kleiner, M. Beerwald, A. Henselek, Use of electromagnetic forming in process combinations for the production of automotive parts, in: Second European Pulsed Power Symposium
EPPS 2004, Hamburg, 2004, pp. 82
86, ISBN 3-8322-3217-6. [19] G.S. Daehn, V.J. Vohnout, E.A. Herman, Hybrid Matched Tool-Electromagnetic Forming Apparatus Incorporating Electromagnetic Actuator, US-Patent 6128935, 2000.

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[20] V.J. Vohnout, A Hybrid Quasi-Static/Dynamic Process for Forming Large Sheet Metal Parts from Aluminum Alloys (Ph.D. thesis), The Ohio State University, 1998. [21] I. Kwee, V. Psyk, K. Faes, Effect of the welding parameters on the structural and mechanical properties of aluminium and copper sheet joints by electromagnetic pulse welding, World J. Eng. Technol. 4 (2016) 538
561. Available from: https://doi.org/ 10.4236/wjet.2016.44053. [22] M. Kamal, A Uniform Pressure Electromagnetic Actuator for Forming Flat Sheets (Ph. D. thesis), The Ohio State University, 2005. [23] A. Mu¨hlbauer, E. von Finckenstein, Magnetumformung rohrfo¨rmiger Werkstu¨cke, B¨ander, Bleche, Rohre 8 (2) (1967) 86
92. [24] P.L. Eplattenier, C. Ashcraft, I. Ulacia, An MPP version of the electromagnetism module in LS-DYNA for 3D coupled mechanical-thermal-electromagnetic simulations, in: Fourth International Conference on High Speed Forming - ICHSF, Columbus, 2010, pp. 250
263. [25] F. Bay, J. Alves, A computational model for magnetic pulse forming processes
application to a test case and sensitivity to dynamic material behaviour, in: Eighth International Conference on High Speed Forming, Columbus, 2018. [26] M. Kleiner, A. Brosius, Determination of flow curves at high strain rates using the electromagnetic forming process and an iterative finite element simulation scheme, CIRP Ann. 55 (1) (2006) 267
270. [27] D. Risch, C. Beerwald, A. Brosius, M. Kleiner, On the significance of the die design for electromagnetic sheet metal forming, in: First International Conference on High Speed Forming - ICHSF, Dortmund, 2004, pp. 191
200. [28] M. Linnemann, V. Psyk, E. Djakow, R. Springer, W. Homberg, D. Landgrebe, Highspeed incremental forming
new technologies for flexible production of sheet metal parts, Procedia Manuf. 27 (2019) 21
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process windows, in: 20th International ESAFORM Conference on Material Forming, Dublin, 2017. [37] R. Sch¨afer, P. Pasquale, Robot automated EMPT sheet welding, in: Fifth International Conference on High-Speed Forming-ICHSF, Dortmund, 2012. [38] V. Psyk, M. Linnemann, C. Scheffler, Experimental and numerical analysis of incremental magnetic pulse welding of dissimilar sheet metals, Manuf. Rev. 6 (2019) 14.

Damage in advanced processing technologies

5

Zhutao Shao, Jun Jiang and Jianguo Lin Department of Mechanical Engineering, Imperial College London, London, United Kingdom

5.1

Introduction

5.1.1 Concepts of damage and damage variables Due to nucleation and growth of defects, such as discreet voids and cracks, in materials under various mechanical and environmental conditions, the degradation of structure leads to material failure eventually. This process is generically termed damage. Continuum damage mechanics (CDM) usually uses state variables to represent the effects of damage on material degradation at a macromechanical scale. It is usually assumed that, once the values of damage variables reach critical levels, the material is unable to undergo the applied load anymore and material or structure failure occurs. Kachanov pioneered the concept of the CDM theory in 1958 [1], and Rice and Tracey [2] applied the concept of damage in the study of material failure for the cold forming process in 1969. CDM has been rapidly developed as a branch of fracture mechanics to describe and model complex phenomena in the practical engineering analysis of advanced manufacturing processes. However, even if some state variables are measurable, experiments in the field of CDM are generally difficult, particularly under multiaxial or nonproportional loading conditions. The measurement under these conditions determines the key functions and constants of the CDM theory.

5.1.2 Damage mechanisms In metal forming processes, theories of void nucleation, growth, and coalescence occur under large plastic deformation have been developed to predict material failure. Fig. 5.1 illustrates schematically typical damage mechanisms under various conditions [3]. Fig. 5.1A shows the accumulation of mobile dislocations, which develops damage from plastic deformation in creep at high temperature. Fig. 5.1B and C show nucleation and growth of grain boundary cavity, respectively, which occur at high-temperature creep. Void growth at triple junction of grains in superplastic forming is often observed as shown in Fig. 5.1D. Fig. 5.1E shows plasticityinduced damage and voids are nucleated around second phase particles in cold metal forming. Damage in Fig. 5.1F generated at both grain boundaries and around Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques. DOI: https://doi.org/10.1016/B978-0-12-818232-1.00005-9 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Dislocations

Cavities

Cavities

Time

Time

Time

(A)

(B)

Grain boundary

Holes

Particles, holes

Slow straining

Cold straining

(D)

(E)

(C) Holes and cracks

Hot straining

(F)

Figure 5.1 Schematics showing primary damage mechanisms: (A) mobile dislocation, (B) cavity nucleation, (C) continuum cavity growth, (D) superplastic void growth, (E) ductile void growth, (F) ductile void growth and microcracking.

second phase particles is due to the grain boundary sliding and grain rotation at hot forming deformation conditions. The dominant damage mechanism depends on temperature, strain rate, microstructural evolution, and material chemical compositions. These damage mechanisms are required to be represented by physically based CDM material models.

5.1.3 Advanced manufacturing technology: hot stamping The transportation industry is facing a huge global challenge to reduce fuel consumption and minimize environmental pollution from vehicle emissions. Since a 10% decrease in the weight of a conventional automobile results in a 6%
8% decrease in fuel consumption rate, lightweighting by advanced manufacturing technology is one of the key enablers to overcome the challenge. Two feasible routes to reduce the mass of vehicle are using stronger steel sheets, which enables a thinner gage to be adopted, and using lower density sheets, such as aluminum/magnesium sheets. At room temperature, ultrahigh-strength steel is difficult to form and aluminum alloys also have low formability, which leads to high springback and poor surface quality of formed components. To deal with these problems, hot forming technologies for steel and aluminum sheets have been developed, namely, hot stamping and cold die quenching (also termed press hardening) of quenchable steel sheets [4] and solution heat treatment (SHT), forming and in-die quenching (HFQ) for sheets of light alloys [5]. A typical temperature profile of hot stamping is shown in Fig. 5.2. The hot stamping and cold die quenching process was patented and industrialized in 1977. This technology has experienced tremendous development in automotive applications and is used increasingly to obtain shapes with great complexity. In the conventional hot steel stamping processes, a metal sheet is heated to a designated temperature at which it is a solid solution with a single phase. The heated sheet is then transferred to a press and simultaneously formed and quenched in a cold tool. In the HFQ light alloy forming processes patented in 2008, a sheet is heated to an SHT temperature at which the material is in a solid solution state with

Temperature

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Heating

Soaking

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Transfer, forming/stamping and quenching

Time

Figure 5.2 A schematic showing the typical temperature profile in a hot stamping process.

a single phase, then the heated sheet is transferred to a press and simultaneously formed and quenched in a cold tool. This process enables the solid solution state microstructure virtually completely retained. The formed part can then be aged and achieved excellent mechanical properties. Both the hot stamping and the hot forming can be used to form complex-shaped components with minimal thermal distortion and springback at a relatively low cost. The forming/stamping parameters, such as the heating rate, cooling rate, temperature and strain rate, need to be controlled accurately to successfully form parts.

5.1.4 Concept and features of forming limit diagram The forming limit diagram (FLD) is a traditional and useful tool to evaluate the formability of sheet metals, which was first proposed by Keeler and Backofen [6] who developed the right side, and Goodwin [7] extended the FLD to the left side in 1968. A schematic of FLD is shown in Fig. 5.3, and the key feature of an FLD is the forming limit curve (FLC). The FLC identifies the boundary between uniform deformation and the onset of plastic instability or diffuse necking, which leads to failure. The diagram represents the combination of principal major strain ε1 and minor strain ε2 , which causes severe necking or facture. In an FLD, strain paths are described as proportional, from uniaxial through a plane strain to equi-biaxial. The region above the curve is considered to represent potential failure and the region below the curve is regarded as a safety region where uniform deformation occurs. A higher FLC indicates that under the same forming condition, such material has better formability. Compared to the FLD of a material at room temperature, at elevated temperatures, the FLD varies greatly in terms of shape and position. This is because the formability of a sheet metal depends on both intrinsic parameters, such as microstructure and constitutive properties, and extrinsic factors, that is, forming

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Figure 5.3 A schematic of forming limit diagram.

conditions, such as temperature, strain rate, and strain path. When forming is performed at high temperatures, the formability of a material is significantly affected by temperature, forming speed, and microstructural evolution of the material. It is, therefore, useful to incorporate multiple FLCs in one FLD under various temperatures and forming speed conditions. The conventional FLD is based on linear strain paths and the effects of the strain path changing must be taken into account when using FLD to analyze a forming problem because the FLC may vary with the nonlinearity of straining [8]. Thus a strain-based FLD is valid only for processes in which loading is proportional and straining path is linear, so the ratios of the principal strains should be constant before the onset of necking [9]. The effect of nonlinear strain paths undermines the utility of the traditional strain
based FLD for formability assessment of metallic materials since the strain path is usually not proportional in a multistage forming process. Kleemola and Pelkkikangas [10] first proposed a forming limit stress diagram (FLSD), which is strain path independent, as an alternative FLD, to describe the forming limits by using major stress and minor stress as coordinates, as shown in Fig. 5.4. However, it is hard to experimentally determine an FLSD of a material and the determination of FLSDs for elevated temperatures has hardly been tried.

5.2

Overview of formability evaluation

5.2.1 Forming limit prediction Various analytical and numerical models have been developed to carry out theoretical formability prediction. Primary models in the field of forming limit prediction at room

147

Major strain

Damage in advanced processing technologies

FLC after prestrain

Prestrain

Minor strain (A)

Major stress

FLC after prestrain

Yield expansion after prestrain

Initial yield surface Minor stress (B)

Figure 5.4 FLCs for linear strain paths and for a nonproportional strain path after prestrain: (A) FLD and (B) FLSD. FLC, Forming limit curve; FLD, forming limit diagram; FLSD, forming limit stress diagram.

temperature include new constitutive equations used for limit strain computation, polycrystalline models, ductile damage models, advanced numerical models for nonlinear strain path, or various process parameters. Fig. 5.5 illustrates various theoretical and numerical models associated with proposed years and they are used for formability prediction of various materials [11], in which some have been extended to applications for high temperature conditions. Since microstructural evolution of material is complex at elevated temperature and greatly affects mechanical properties, models applied for high temperature conditions are more complicated in general. These include Hora’s theory, M
K theory, and Storen and Rice’s theory. The viscoplastic CDMbased material model has also been developed for the prediction of the FLD of metals for advanced manufacturing processes, such as hot stamping. The embedded viscoplastic constitutive equations have been developed to model a wide range of timedependent phenomena at microscale, such as recrystallization, strain rate effect, and recovery. These phenomena normally occur in the forming processes at elevated temperatures greater than 0.5Tm (Tm is the melting temperature of the material).

5.2.2 Experimental methods for determining forming limits FLDs of sheet metals are usually obtained experimentally to calibrate material constants in the developed constitutive equations. Different types of testing methods have been proposed, such as varying the dimensions of specimens and shapes of elliptical dies in the hydraulic bulge test or shape of punch in the biaxial stretching test. At present, two types of formability test approaches are commonly used to determine limit strains; they are the out-of-plane test and the in-plane test, as shown diagrammatically in Fig. 5.6 [12].

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Swift’s model (1952) Models based on bifurcation theory

Models based on Geometrical imperfection theory

Swift’s and Hill’s models

Hill’s model (1952)

Storen and Rice’s theory (1975, applicable for high temperature)

Modified maximum force criterion (1996, applicable for high temperature)

Marciniak and Kuczynski (M–K) model (1967, applicable for high temperature)

Imperfection orientation modification (applicable for high temperature)

Modified M–K models Models for predicting FLDs of sheet metals

Imperfection hypothesis modification

Models taking account of void nucleation and growth Models based on continuum damage mechanics

Gurson–Tvergaard–Needleman (GTN) model (1984) A phenomenological model (2013, applicable for high temperature)

Models taking account of orthotropic damage (1997) Models taking account of viscoplascity (2013, applicable for high temperature)

NADDRG model (1977)

Others

Jones and Gills model (1984) Artificial neural network (2010, applicable for high temperature)

Figure 5.5 Examples of theoretical and numerical models used for formability prediction.

Figure 5.6 Schematics showing tooling geometries used in conventional formability tests: (A) the typical out-of-plane test setup; (B) the typical in-plane test setup.

The out-of-plane one is a commonly used method, which involves stretching specimens with different widths by a rigid hemispherical punch or hydraulic pressure. For the in-plane one, the test material is stretched over a flat-bottomed punch

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of cylindrical/elliptical cross section or by hydraulic pressure, which creates uniform in-plane biaxial strain in the center of the specimen. Nakazima method is a typical out-of-plane test to evaluate the formability of sheet metals. Multiple specimens with different widths are necessary to obtain forming limits under various strain paths. This type of test has a relatively simple procedure and good repeatability. While the in-plane test, since curvature and friction effects are absent, large strain gradients can be avoided, and forming limits are not influenced to the same extent by tooling geometry variables compared to the out-of-plane test. However, optimizing the dimensions and geometries of carrier blank and punch is necessary in order to induce strain localization and cracking in the unsupported region of the specimen, which complicates the test procedure and increases the cost of testing. For tests at high temperatures, the punch and die can be heated in order to create an isothermal environment on the specimen by heat transfer and obtain the FLD at elevated temperatures. The aforementioned out-ofplane and in-plane formability tests at ambient temperature have been standardized. In order to overcome the drawbacks of the formability testing methods described earlier, planar tensile tests utilizing a tensile test machine with cruciform specimens is an alternative method to determine FLDs of materials. Planar biaxial tensile test systems for sheet metals can be classified into two types: stand-alone biaxial testing machines and link mechanisms attached to uniaxial test machines for biaxial testing. A servo-hydraulic machine provided with four independent dynamic actuators is a typical stand-alone device for biaxial testing. Some link mechanisms based on existing uniaxial testing machines have also been designed to convert a uniaxial force to a biaxial one for the purpose of biaxial testing. But the design of cruciform specimens is a challenging and difficult issue. Neither of the conventional out-ofplane and in-plane methods of determining FLDs currently is suitable for hot stamping and cold die quenching conditions, which generally comprise subjecting the sheet specimen to a heating process, and then simultaneous cooling and deformation at elevated temperatures.

5.2.3 Requirements for hot stamping applications Accurate constitutive data relevant to processing conditions for each material are required to calibrate material and process models to optimize advanced manufacturing processes and as yet there are no generally accepted testing standard for obtaining these data. Uniaxial tensile testing is the most commonly used method to obtain mechanical properties of metals, such as ductility, yield and tensile strength, and strain hardening behavior. The uniaxial tensile test procedure and method of strain measurement and dimensions of specimens have been standardized for applications under isothermal conditions. It is noted that the existing standards are applicable only for testing under isothermal or near isothermal conditions or with a small permitted temperature deviation within the gauge region of a specimen. In order to obtain the constitutive behavior of a testing alloy subjected to hot stamping processes, rapid heating and cooling on a specimen must be the integral part of a uniaxial tensile test; therefore conventional hot uniaxial testing that is operated in an

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oven or furnace is impractical. Performing uniaxial tests in a material’s thermomechanical simulator of Gleeble can be considered as an alternative method since it has facilities to fast heat and cool a specimen at any stages in a deformation cycle. Formability tests with special tooling and test procedures are usually needed and it is not easy to obtain accurate results, especially for hot stamping applications. It is very difficult to obtain forming limits of metals under hot stamping conditions at various deformation rates by using conventional methods, since cooling occurs prior to deformation and consistent values of heating rate, cooling rate, deformation temperature, and strain rate are not easy to obtain.

5.2.4 Advanced testing system for hot stamping applications In order to determine FLDs of alloys subjected to complex forming conditions, such as hot forming and cold die quenching process, a fully integrated closed-loop control thermomechanical testing system for determining forming limits should contain a thermal system, a cooling system, an automatic feedback control system, a multiaxial mechanism, a strain rate control system, and a strain and loading measurement system. A novel multiaxial testing apparatus [13] has been invented and patented for using on the Gleeble commercial thermomechanical testing machine to enable multiaxial tensile testing, as shown in Fig. 5.7. Testing parameters, such as heating rate, cooling rate, deformation temperature, and strain rate, can be controlled precisely to reduce measurement errors of FLD results. Different proportional strain paths can be achieved by this apparatus, and

Figure 5.7 The setup of the multiaxial testing system for uniaxial tensile tests and formability tests.

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friction effects on FLD can be avoided. One type of cruciform specimen was also proposed for biaxial testing [14]. Regarding the strain measurement method, using digital image correlation (DIC) is a good contactless method for strain measurement, and it has been widespread used and accepted in the field of experimental mechanics. DIC was proposed in the 1980s and this technique enables full-field strains to be measured by determining the relative movement of successive digital images at different deformation stages. The sample surface is typically coated with various randomly distributed sprays or painting particles to maximize surface contrast. This system enables material thermomechanical behavior to be characterized for CDM-based material models.

5.3

Modeling of damage evolution

5.3.1 Constitutive equations A material flow rule represents the flow stress response to the extent and rate of plastic straining in viscoplasticity theory, incorporating factors such as initial yield and work hardening due to the interaction of dislocations. To model a hot deformation process, strain rate should be introduced into the power-law equation, such as [15]: σ 5 KεnP ε_ m P

(5.1)

where εP and ε_ P are plastic strain and plastic strain rate, respectively. Strength coefficient, K, strain hardening exponent, n, and the strain rate hardening exponent, m, are temperature-dependent material parameters. In order to model plastic yield and strain hardening, the flow stress can be expressed as a sum of initial yield stress k, strain hardening R, and viscous stress σv representing the viscoplastic effects [15], that is, σ 5 k 1 R 1 σv , as shown in Fig. 5.8.

Figure 5.8 Flow stress
strain response of viscoplastic solids.

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Then σv can be defined as, ðσ 2 R 2 k)1, where the brackets indicated that only positive results are valid. Thus the flow rule can be replaced by a new expression [16]:  ε_ P 5 ε_ 0

σ2R2k K

n1 (5.2) 1

where the value of ε_ 0 is 1.0 under tension condition and 21.0 under compression condition, n1 is a viscous exponent, which is temperature dependent. Arrheniustype functions can be used to represent the temperature-dependent parameters: 

QK K 5 K0 exp Rg T

 (5.3)

  Qk k 5 k0 exp Rg T 

Qn1 n1 5 n10 exp Rg T

(5.4)  (5.5)

where K0 , k0 , and n10 are material constants; QK , Qk , and Qn1 are activation energy; Rg is the universal gas constant; and T is the absolute temperature. The accumulation of dislocations due to plastic deformation causes work hardening of metal. As dislocation density is directly related to the hardening [17], Lin et al. [18] proposed and developed a unified equation to describe isotropic work hardening as a function of normalized dislocation density ρ (ρ 5 1 2 ρ0 =ρ, where ρ0 is the initial dislocation density, and ρ is the instantaneous dislocation density during deforamtion): R 5 Bρ nR

(5.6)

where nR is the hardening exponent, and B is a temperature-dependent parameter:  B 5 B0 exp

QB Rg T

 (5.7)

where B0 is a material constant, and QB is activation energy associated with the hardening mechanism. The mean slip distance of a dislocation L decreases with increasing dislocation density. It is the average distance over which mobile dislocations migrate before being stored [19]. The slip length L is scaled with the inverse square root of the stored dislocation density, that is, L~ρ20:5 and the work hardening is inversely proportional to the slip length L, that is, R~L21 . Therefore the constant nR is suggested

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153

to be 0.5. Then the evolutionary form of the isotropic work hardening law is expressed as: R_ 5 0:5Bρ 20:5 ρ_

(5.8)

This physically based strain hardening law enables material behavior to be captured at microscopic level. Annealing or recrystallization may occur during high temperature forming conditions, which will reduce dislocation density and thus reduce isotropic hardening or even lead to negative hardening. In this case the hardening law can account for material softening. The value of the defined normalized dislocation density varies from 0 (the initial state) to 1.0 (the saturated state of a dislocation network after severe plastic deformation). Based on this concept, a constitutive equation for describing the evolution of dislocation density can be formed as: ρ_ 5 Að1 2 ρ Þjε_ P j 2 Cρ n2

(5.9)

where A and n2 are material constants, and C is a temperature-dependent parameter. The first term in the equation represents the development of dislocation density due to plastic strain and dynamic recovery. Dynamic recovery results from the continuous reorganization of dislocations during deformation, in terms of dislocation cross-slip at low temperature and dislocation climb at high temperature. The second term represents the effect of static recovery of dislocation density. Static recovery is a time-dependent process in which annealing can effectively remove dislocations from the matrix at elevated temperature [20]. In order to introduce the effect of temperature for hot working conditions, the parameter C is presented as:   QC C 5 C0 exp 2 Rg T

(5.10)

where C0 is a material constant, and QC is the activation energy associated with static recovery mechanism.

5.3.2 Advanced damage models CDM is associated with the modeling of degradation or deterioration of materials under thermomechanical deformation or aging processes. Phenomenon-based damage constitutive equations can be used to model damage evolution in various metal forming processes, such as hot metal forming, superplastic forming, and hot rolling. Microcracks or voids generate and grow in the microstructure when metallic materials undergo continuous plastic deformation, which would lead to material failure at a macroscopic level. Failure can be introduced into predictive models by embedding damage evolution equations in elasto-viscoplastic constitutive equations.

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The development of microdamage under various deformation conditions has been reviewed and discussed by Lin et al. [3,21]. Damage model mechanisms contain void nucleation, growth and coalescence in cold metal forming and continuum cavity growth and ductile void growth/microwedge cracking at grain boundaries under hot stamping conditions. A ductile damage model proposed and applied for both cold forming and hot stamping conditions is given as: ω_ 5

σλe η1 ðε_ P Þη2 ð12ωÞη3

(5.11)

where ω_ is defined as damage evolution rate, which is a function of plastic strain rate ε_ P , flow stress level σe , and current damage value ω. The value of ω is in the range of 0 (undamaged) to 1.0 (completely failed). The value of the material constant λ is given as zero when the strain rate
induced flow stress variation does not influence damage evolution. η1 , η2 , and η3 are given in terms of the Arrhenius-law equation for hot working conditions: 

Qη1 η1 5 η10 exp Rg T 

Qη2 η2 5 η20 exp Rg T 

Qη3 η3 5 η30 exp Rg T

 (5.12)  (5.13)  (5.14)

where η10 , η20 ; and η30 are material constants, and Qη1 , Qη2 , and Qη3 are activation energy associated with damage mechanisms. The metal is considered to be subjected to increased stress over the undamaged area 1 2 ω, which creates effective stress of σ=ð1 2 ωÞ for further plastic deformation and damage evolution. Thus the flow rule is modified as: ε_ P 5 ε_ 0



n σ=ð12ωÞ 2R2k 1 K 1

(5.15)

According to Hooke’s law, flow stress can be expressed as: σ 5 EðεT 2 εP Þ

(5.16)

where E is Young’s modulus of elasticity, εT is total strain, and εP is plastic strain. By taking the damage softening effect into account, Hooke’s law can be modified as: σ 5 Eð1 2 ωÞðεT 2 εP Þ

(5.17)

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where E is a temperature-dependent parameter:  E 5 E0 exp

QE Rg T

 (5.18)

where E0 is a material constant, and QE is the activation energy.

5.3.3 A set of unified constitutive equations for hot stamping The viscoplastic-damage constitutive equations introduced previously are unified and adopted in finite element to model the thermomechanical response of metallic materials. The formulations of the unified constitutive equations based on the mechanisms of dislocation-driven evolution processes, such as work hardening, static and dynamic recovery, and plasticity-induced ductile damage, are listed next. They model time-dependent phenomena, including strain rate effects, recovery, and damage evolution. An Arrhenius-type equation is used to formulate temperaturedependent parameters in the equations. ε_ P 5

 n σ=ð12ωÞ 2R2k 1 K

(5.19)

R_ 5 0:5Bρ 20:5 ρ_

(5.20)

ρ_ 5 Að1 2 ρ Þjε_ P j 2 Cρ n2

(5.21)

η1 ðε_ P Þη2 ð12ωÞη3

(5.22)

ω_ 5

σ 5 Eð1 2 ωÞðεT 2 εP Þ

(5.23)

where the temperature-dependent parameters, K; k; n1 ; B; C; η1 ; η2 ; η3 ; and E, have been defined. Eq. (5.19) is the flow rule, in which plastic strain rate ε_ p is formulated by using the traditional power law with damage ω taken into account. The initial yield point is k, and R represents the isotropic hardening. Isotropic hardening R in Eq. (5.20) is a function of the normalized dislocation density ρ (ρ 5 1 2 ρ0 =ρ, where ρ0 is the initial dislocation density, and ρ is the instantaneous dislocation density during deformation), where ρ is given by Eq. (5.21) [22]. Eq. (5.21) represents the accumulation of dislocations due to plastic flow and dynamic and static recovery. The damage evolution ω in Eq. (5.22) was based on the growth and nucleation of voids around particles. It is a modified version of the expression set out by Khaleel et al. [22] for damage due to superplastic void growth, which is appropriate for the case where the fine-grained alloy is deformed by a significant amount at high temperature. The flow stress equation was modified to include the

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effect of damage in Eq. (5.23). This set of nonlinear ordinary differential equations can be solved with the numerical Euler integration method by giving initial values for the variables.

5.3.4 Modeling of forming limit diagrams In order to predict an FLD of sheet metals under hot stamping conditions, a general multiaxial CDM-based material model, comprising a set of equations describing viscoplastic behavior, is developed for the plane stress state [23]. These power-law viscoplastic equations are expected to capture the features of FLCs of sheet metals under various thermomechanical conditions. For an isotropic material the von Mises stress can be defined as: 

2 σe 5 Sij Sij 3

1=2 (5.24)

where Sij is the deviatoric stress component (hydrostatic stress is σkk =3): Sij 5 σij


σkk δij 3

(5.25)

where σij is stress tensor, and δij is the Kronecker delta. By considering von Mises behavior for rigid perfect viscoplasticity and ignoring work hardening and initial yield, a power-law viscoplastic potential function can be defined in the form of: ψ5

K σe n11 n11 K

(5.26)

where K and n are material constants. The following expression can be obtained by differentiating Eq. (5.26) in terms of the deviatoric stress: ε_ Pij 5

@ψ 3 @ψ Sij 3 Sij σe n 3 Sij 5 5 5 ε_ P @Sij 2 @σe σe 2 σe K 2 σe

(5.27)

where ε_ P is effective plastic strain rate, which is introduced previously. In metal forming processes, stress states of deformation are complicated. Stress state will vary depending on forming conditions and process and it can vary from one region to another. A stress state may be characterized as comprising hydrostatic stress, which does not affect plastic deformation, principal stresses, and deviatoric stress, which results in plastic deformation. The form of deformation arising under different loading conditions can be characterized by different deformation states, such as simple tension, compression, and plane strain.

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In order to describe the effects of stress states on the damage evolution of a material, stress state
dependent damage evolution equations are proposed for hot stamping applications as below. ω_ 5

  Δ α1 σ1 13α2 σH 1α3 σe ϕ η1 ðε_ P Þη2 ðα1 1α2 1α3 Þϕ σe ð12ωÞη3

(5.28)

where α1 , α2 , and α3 are weighting parameters to control the damage effects caused by the maximum principal stress σ1 , hydrostatic stress σH , and effective stress σe , respectively. The values of α1 and α3 are suggested to be in the range of 0
1.0 and the value of α2 ranges from 21.0 to 1.0. The minus value of α2 can model the negative effect on the process of compressive forming. Any values of α1 , α2 , and α3 could be set to zero for modeling the behavior of various metals, which means that the corresponding stress term does not contribute to damage during sheet metal deformation. Δ is a correction factor to adjust the global position of a curve in an FLD. It is a strain rate
dependent and also temperature-dependent parameter:  Δ 5 Δ11 exp

 Δ12 1 Δ21 expðΔ22 ε_ e Þ T

(5.29)

where Δ11 , Δ12 , Δ21 , and Δ22 are material constants, T is the absolute temperature, and ε_ e is the effective strain rate. The damage rate exponent ϕ is introduced in this equation to mathematically control the profile shape of an FLC. It is a temperaturedependent parameter:  ϕ  ϕ 5 ϕ11 exp 2 12 T

(5.30)

where ϕ11 and ϕ12 are material constants, and T is the absolute temperature. The material constants in Eq. (5.30) will be determined by fitting the computed FLD to experimental results obtained using the novel biaxial tensile testing system. The ratio of minor strain ε2 to major strain ε1 is defined as: β5

ε2 ε1

(5.31)

The range of β ratio is from 20.5 (uniaxial tension) to 0 (plane strain condition) to 1.0 (equi-biaxial stretching). A typical FLC is the plot of minor strain and major strain under different strain paths. In order to describe the effects of strain states on the damage evolution of a material, a principal strain
dependent damage evolution equation is proposed for the material under hot stamping conditions. For this set of multiaxial viscoplastic constitutive equations, the flow rule, working hardening, and dislocation density evolution law

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remain the same as discussed earlier. A new equation can be used to replace Eq. (5.28) in order to describe the effects of principal strains on the damage evolution [24]: "

#  μ1 ε1 1μ2 ε2 φ η1 ðε_ P Þη2 ω_ 5  φ γ1εP ð12ωÞη3 μ1 20:5μ2 Δ

(5.32)

where μ2 is a material constant; γ is a constant that is set to a small value to avoid zero denominator during the integration of this set of equations; η1 , η2 ; and η3 are temperature-dependent parameters; μ1 and φ are also defined as temperaturedependent parameters; and Δ is a temperature-dependent and also strain rate
dependent parameter, as presented below.  μ  μ1 5 μ11 exp 2 12 T   φ φ 5 φ11 exp 2 12 T Δ



5 Δ11 exp



  Δ12 1 Δ21 exp Δ22 ε_ e T

(5.33)

(5.34)

(5.35)

where μ11 , μ12 , φ11 , φ12 , Δ11 , Δ12 , Δ21 , and Δ22 are material constants needed to be determined using experimental data from formability tests; T is the absolute temperature; and ε_ e is the effective strain rate calculated from experimental data. The parameters μ1 and μ2 are used to calibrate the effects of major strain and minor strain on the damage evolution. The values of μ1 and μ2 are suggested to be in the range of 0
1.0. φ is introduced to control the intensity of the effects of principal strains on the damage evolution.

5.4

Damage calibration techniques

5.4.1 Overview of damage calibration techniques In general, the phenomenological elastic
plastic and elastic
viscoplastic unified constitutive equations cannot be solved analytically. Numerical integration methods are used to determine material constants from experimental data obtained from uniaxial tensile test and multiaxial formability test. Some optimization methods, such as gradient-based, generic algorithm, and evolutionary programing
based optimization techniques, have been adopted to determine the constants in constitutive equations. Taking the popular multiobjective optimization technique as an example, experimental data, including stress
strain curves and FLCs of materials, are considered as objectives and searching the best values of material constant associated with state variables in constitutive equations by using numerical integration

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methods to minimize the variation or errors between computed data and the corresponding experimental data. The large number of iterations during the optimization process is often computationally time-consuming, and commercial software, such as Matlab, can be used to implement the method for applications. The trial and error method based on the physical meaning of each state variable and material constant is a simple way to search for acceptable results. It enables the prioritization of the critical features of stress
strain curves step-by-step based on observations. The Euler method proposed by Leonhard Euler is a commonly used first-order numerical method for solving ordinary differential equations by providing the initial values for each variable. Due to the simplicity of this method, it has been used and demonstrated to determine the set of constitutive equations in the next two sections.

5.4.2 An example of using thermomechanical uniaxial test data Material constants within the set of constitutive equations introduced earlier can be determined by fitting the computed true stress
true strain curves to corresponding experimental data obtained from uniaxial tension testing at different deformation temperatures and strain rates for hot stamping applications. The first step of the fitting procedure was to determine all constants of the constitutive equations by taking the temperature-dependent parameters as constants for fitting the stress
strain curves obtained for different strain rates at one deformation temperature. All the values obtained from the first step were retained in the second step, and only the temperature-dependent parameters were adjusted. This step was to determine the preexponent and activation energy associated constants by fitting the stress
strain curves for different deformation temperatures at one designated strain rate. The fitted results of material constants may not be unique after the two steps, if by using the trial and error method. Therefore the remaining experimental stress
strain curves at other conditions were used for further validation until the best fitted results of material constants were found. The range of potential values for the constants should be chosen based on their physical meanings and previous experience. The zero value of damage ω represents the initial state of deformation. The ω value of a high value, 0.7 in this case, for example, was taken as the criterion for the material failure in order to improve the computation efficiency. It is not used to model a particular damage mechanism of the material but to be considered as an overall effect on the material during the forming process since the dominant damage mechanism may change during forming processes. For example, the calibrated material constants for aluminum alloy 6082 under hot stamping conditions are listed in Table 5.1. A comparison between the stress
strain behavior predicted by the material model and that obtained from the tensile tests are shown in Fig. 5.9. Good agreement has been obtained for AA6082 at all test conditions. This indicates that the thermal-activated mechanisms described by Arrhenius’ law are applicable for AA6082 at elevated forming temperatures. The softening in each stress
strain curve is just used to indicate how the damage factor in the constitutive equations

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Table 5.1 Material constants in constitutive equations for AA6082 under hot stamping conditions. K0 ðMPaÞ

k0 ðMPaÞ

n01 ð 2 Þ

B0 ðMPaÞ

C0 ð 2 Þ

η10 ð 2 Þ

η20 ð 2 Þ

0.0830

6.8348

0.9624

9.4004

7.3057

2.4577

0.8357

η30 ð 2 Þ

E0 ðMPaÞ

Að 2 Þ

n2 ð 2 Þ

QK ðJ=molÞ

Qk ðJ=molÞ

Qn1 ðJ=molÞ

0.2589

249.69

0.19

1.83

33488.79

5562.73

5296.43

QB ðJ=molÞ

QC ðJ=molÞ

Qη1 ðJ=molÞ

Qη2 ðJ=molÞ

Qη3 ðJ=molÞ

QE ðJ=molÞ

R [J/(mol K)]

11967.17

2112.59

16237.24

837.80

20770.87

27987.42

8.314

works for the material tested under hot stamping conditions. It can be further calibrated using data obtained by multiaxial formability tests.

5.4.3 Examples of using thermomechanical multiaxial tensile test data The rest of material constants associated with stress-based damage evolution is needed to determine as well. The procedure consists of three steps. First, the values of β (the ratio of minor strain ε2 to major strain ε1 ) under each strain path are calculated at a specified effective strain rate ε_ e , obtained from experimental data. Then the values of major strain ε1 and minor strain ε2 are outputs when the damage ω value reaches a certain level by solving the equations through the Euler integration method. Second, according to the computed curve from the first step, by taking the temperature-dependent parameters as constants, the values of material constants Δ, α1 , α2 , α3 , and ϕ are adjusted from fitting the experimental FLC at the given strain rates at one deformation temperature. Lastly, all the values obtained from the second step are retained, and the temperature-dependent constants are adjusted to determine the preexponent and activation energy associated constants from fitting FLC data for different deformation temperatures at the designated strain rate. The calibrated material constants of the damage evolution equation for AA6082 under hot stamping conditions are listed in Table 5.2. Fig. 5.10 shows a comparison between experimental (symbols) and computed (solid curves) FLD for AA6082 under hot stamping conditions. Good agreement can be seen for each FLC in the diagrams, which implies that the thermal-activated mechanisms described by Arrhenius-type equations are applicable for the plane stress
based damage evolution equation. The strain-based CDM material model can also be calibrated from the multiaxial experimental data. A similar calibration process as that for the determination of material constants in the plane stress
based CDM material model, the procedure to determine the material constants associated with the strain-based damage evolution also consists of three steps. The calibrated material constants associated with formability variables for AA6082 under hot stamping conditions are listed in Table 5.3.

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161

80 70 400°C

True stress (MPa)

60 50

450°C

40 30

500°C 20 1/s 10 0 0

(B)

0.2

0.4 True strain

0.6

0.8

80 70

True stress (MPa)

60 50 40 4/s

30 1/s

20 450°C

0.1/s

10

0 0.0

0.2

0.4 True strain

0.6

0.8

Figure 5.9 Comparison of experimental (symbols) and numerical integrated (solid curves) stress
strain curves for AA6082 at various temperatures and strain rates: (A) Different temperatures (designated strain rate is 1/s) and (B) different strain rates (temperature is 450 C).

Fig. 5.11 shows a comparison between experimental (symbols) and computed (solid curves) FLD for AA6082 under hot stamping conditions. Good agreement can be seen for each FLC in the diagrams. According to Figs. 5.10 and 5.11, both of the plane stress
based and principal strain
based CDM material model are promising for modeling the formability of metals under hot stamping conditions. The shape of an FLC can be different at a constant strain rate but various temperatures. It is noted that the stress state
dependent

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Table 5.2 Material constants associated with formability variables for AA6082 under hot stamping conditions.

AA6082

AA6082

α1

α2

α3

ϕ11

ϕ12

0.450

2 0.065

0.050

51.317

1579.47

Δ11

Δ12

Δ21

Δ22

0.343

5758.470

2.927E 2 05

2 5759.879

damage evolution equation is defined to apply to sheet metal forming process in which the plane stress situation exists, with a zero value of normal stress through the thickness direction. It has physical meaning to represent damage mechanisms by introducing stress states into the damage evolution equation. Compared with the stress state
based damage evolution equation, the principal strain
based damage evolution equation can also be used to capture the features of an FLC, and it is very useful for practical applications since formability is normally evaluated by an FLD.

5.5

Applications of damage modeling technique for hot stamping

5.5.1 Plane stress
based continuum damage mechanics material model Since material mechanical behavior, presented by stress
strain curves, and formability, presented by FLDs, varies from material to material. The developed CDMbased material model is required to be flexible to characterize the different features of material response. Here, taking the plane stress
based CDM material model as an example to analyze effects of each parameter in the constitutive equations in order to identify the feasibility and the limitations of the damage modeling technique for formability prediction. Fig. 5.12 shows the effect of the factor Δ on the position of an FLC. When the value of Δ varies from 0.5 to 2.0 with fixed other parameters (ϕ, α1 , α2 , and α3 ), the location of an FLC becomes lower, but no significant changes of shape are observed. The coverage range of the values of forming limit under each strain path becomes larger and the lowest point of the curve moves toward the left-hand side of the FLD when a lower value of Δ is adopted in the damage evolution equation. The values of forming limit may vary depending on the particular experimental test method used. Therefore Δ is also considered as a correction factor to adjust the predicted results to enable fitting of computed and experimentally determined data. Three stress state parameters, the maximum principal stress, hydrostatic stress, and the effective stress, are introduced in the damage evolution equation. The framework of damage evolution is related to multiaxial state of stress loading conditions. Damage

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163

0.5 510°C

Major strain

0.4

440°C

0.3 370°C 0.2

0.1 0.1/s –0.25

–0.15

(B)

0 –0.05 0.05 Minor strain

0.15

0.25

0.5 1/s

0.4

Major strain

0.1/s 0.01/s

0.3

0.2

0.1 440°C 0

–0.25

–0.15

–0.05

0.05

0.15

0.25

Minor strain

Figure 5.10 Comparison of experimental (symbols) and numerical integrated (solid curves) FLDs computed by plane stress
based material model for AA6082 with various deformation temperatures and strain rates: (A) FLD for different deformation temperatures at 0.1/s; (B) FLD for different strain rates at 440 C. FLD, Forming limit diagram.

mechanisms are complicated from one region to another in the material and they also vary between material, forming processes and conditions. Although these stress states are introduced in the equation, this is not used to model the specified damage mechanisms but to model an overall damage accumulation in the material. Fig. 5.13 shows the effect of the maximum principal stress on the profile shape of an FLC with a variation of α1 and fixed values of all other material constants at

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Table 5.3 Material constants associated with formability variables for AA6082 under hot stamping conditions. μ11 AA6082

0.848 Δ11

AA6082

79.785

μ12

μ2

451.863 Δ12 26.063

0.150 Δ21 2 81.107

φ11

φ12

19.094

874.552

Δ22 3.517E 2 03

the deformation temperature of 440 C and strain rate of 0.1/s. It can be seen that the values of forming limits under the uniaxial strain path (β 5 20.5) remain the same when the value of α1 varies from 0.5 to 2.0. The level of a predicted FLC differs less than 15% when the value of α1 changes from 1.0 to 2.0. Generally, for metals, the values of forming limit obtained under the biaxial straining state are lower than that obtained under the uniaxial straining state. Therefore the range of 0
1.0 for the value of α1 is applicable for metallic materials in most cases. The lowest point of an FLC is changed, and values of forming limit on the right-hand side of the FLD reduce with an increasing value of α1 , which shows that the shape of an FLC can be changed by adjusting this parameter to capture the feature of an FLC obtained from various experimental testing methods. Figs. 5.14 and 5.15 show the effects of hydrostatic stress and the effective stress on the shape of an FLC, respectively, under the condition of deformation temperature of 440 C and strain rate of 0.1/s. α2 is very sensitive to the effect of hydrostatic stress and it enables significant change of shape of an FLC on the right-hand side of the FLD with a slight variation of the value, as shown in Fig. 5.14. The position of the lowest point under plane strain state (β 5 0) can also be adjusted by varying the value of α2 . The negative value of α2 can postpone the damage evolution, which results in a high value of forming limit on the right-hand of an FLD. The parameter α3 can be used to modify the effect of the effective stress on damage evolution. The curvature of an FLC is altered significantly when the value of α3 varies from 0.5 to 1.0, as shown in Fig. 5.15. An almost rectilinear curve is obtained when the value of α3 equals 1.0, which indicates that a high value of α3 can speed up the damage evolution on the prediction of the right-hand side of an FLD. Fig. 5.16 shows the effect of the damage rate exponent ϕ on the predicted FLC at the deformation temperature of 440 C and the strain rate of 0.1/s. Exponent ϕ is the main parameter to control the shape of an FLC, and a wide range of ϕ values is able to use in order to fit mathematically different types of curves in an FLD obtained from experimental data. According to the computed results in Fig. 5.16, the shape of an FLC changes dramatically when the value of ϕ varies from 2.0 to 10.0. A parabolic shape of an FLC can be modeled with a high value of ϕ. Two intersection points can be observed at β 5 20.5 and β 5 0.3 with fixed parameters of Δ, α1 , α2 , and α3 at specified values.

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165

0.5 510°C

0.4

Major strain

440°C 0.3 370°C 0.2

0.1 0.1/s 0 –0.25

–0.15

–0.05

0.05

0.15

0.25

Minor strain (B)

0.5 1/s 0.4

0.1/s

Major strain

0.01/s 0.3

0.2

0.1 440°C 0

–0.25

–0.15

–0.05

0.05

0.15

0.25

Minor strain

Figure 5.11 Comparison of experimental (symbols) and numerical integrated (solid curves) FLDs computed by the strain-based material model for AA6082 with various deformation temperatures and strain rates: (A) FLD for different deformation temperatures at 0.1/s; (B) FLD for different strain rates at 440 C. FLD, Forming limit diagram.

5.5.2 Principal strain
based continuum damage mechanics material model Fig. 5.17 shows the effect of the correction factor Δ on the position of an FLC at the deformation temperature of 440 C and the strain rate of 0.1/s. Similar to the correction factor Δ in stress state
based damage evolution equation, when the

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1

Major strain

0.8

Δ=0.5

0.6 Δ=1.0 0.4 Δ=2.0 0.2

0

–0.4

–0.2

0 Minor strain

0.2

0.4

Figure 5.12 The effect of the correction factor Δ on the predicted FLC at the deformation temperature of 440 C and the strain rate of 0.1/s (ϕ 5 5.6, α1 5 0.45, α2 5 20.065, α3 5 0.055). FLC, Forming limit curve. 0.5

Major strain

0.4

=0.5

0.3

=1.0 =2.0

0.2

0.1

0

–0.4

-0.2

0 Minor strain

0.2

0.4

Figure 5.13 The effect of the maximum principal stress parameter α1 on the predicted FLC at the deformation temperature of 440 C and the strain rate of 0.1/s (Δ 5 1.35, ϕ 5 5.6, α2 5 20.065, α3 5 0.055). FLC, Forming limit curve.

value of Δ varies from 0.5 to 2.0, the changes in the level of an FLC are observed. Again, the results of forming limit may vary depending on testing methods. Therefore Δ is also considered as a correction factor for adjusting predicted results and fitting to experimental data.

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0.6 =–0.1 0.5

Major strain

0.4 =–0.05 0.3 0.2 0.1

=0.05

0

–0.4

–0.2

0 Minor strain

0.2

0.4

Figure 5.14 The effect of the hydrostatic stress parameter α2 on the predicted FLC at the deformation temperature of 440 C and the strain rate of 0.1/s (Δ 5 1.35, ϕ 5 5.6, α1 5 0.45, α3 5 0.055). FLC, Forming limit curve. 0.5

Major strain

0.4

=0.1

0.3

=0.5 =1.0

0.2

0.1

0

–0.4

–0.2

0 Minor strain

0.2

0.4

Figure 5.15 The effect of the effective stress parameter α3 on the predicted FLC at the deformation temperature of 440 C and the strain rate of 0.1/s (Δ 5 1.35, ϕ 5 5.6, α1 5 0.45, α2 5 20.065). FLC, Forming limit curve.

Figs. 5.18 and 5.19 show the effects of the significance of major strain and minor strain on the shape of an FLC, respectively, under the same condition of the deformation temperature of 440 C and the strain rate of 0.1/s. A small range of μ2 value causes a dramatic change of the FLC shape, which means that μ2 is a more sensitive

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0.6 0.5

=10.0

Major strain

0.4

=5.0

0.3

=2.0

0.2 0.1 0

–0.4

–0.2

0 Minor strain

0.2

0.4

Figure 5.16 The effect of damage rate exponent ϕ on the predicted FLC at the deformation temperature of 440 C and the strain rate of 0.1/s (Δ 5 1.35, α1 5 0.45, α2 5 20.065, α3 5 0.055). FLC, Forming limit curve. 1.2 1

*=0.5

Major strain

0.8 0.6 0.4

*=1.0

*=2.0

0.2 0

–0.4

–0.2

0 Minor strain

0.2

0.4

Figure 5.17 The effect of the correction factor Δ on the predicted FLC at the deformation temperature of 440 C and the strain rate of 0.1/s (Ø 5 5.60, μ1 5 0.45, μ2 5 0.15). FLC, Forming limit curve.

parameter than μ1 . The effects of the two parameters μ1 and μ2 on the shape of an FLC are quite similar. The strain level on the right-hand side of the FLD and the location of the lowest point of an FLC can be adjusted by varying the values of μ1 and μ2

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0.9 0.8

=1.0

0.7

Major strain

0.6

=0.5

0.5 0.4 0.3 0.2

=0.1

0.1 0

–0.4

–0.2

0 Minor strain

0.2

0.4

Figure 5.18 The effect of the major strain parameter μ1 on the predicted FLC at the deformation temperature of 440 C and the strain rate of 0.1/s (Δ 5 1.60, Ø 5 5.6, μ2 5 0.15). FLC, Forming limit curve. 0.7

=0.1 0.6 0.5

Major strain

=0.15 0.4 0.3 =0.2 0.2 0.1 0

–0.4

–0.2

0 Minor strain

0.2

0.4

Figure 5.19 The effect of the minor strain parameter μ2 on the predicted FLC at the deformation temperature of 440 C and the strain rate of 0.1/s (Δ 5 1.60, Ø 5 5.6, μ1 5 0.45). FLC, Forming limit curve.

to model the effects of principal strains on the damage evolution. A low value of μ1 and a high value of μ2 contribute to increasing the damage evolution rapidly. Fig. 5.20 shows the effect of the damage rate exponent φ on the predicted FLC. The shape of an FLC changes dramatically when the value of φ varies from 2.0 to 10.0. A parabolic shape of an FLC can be modeled with a high value of φ. Two intersection

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0.6 =10.0 0.5 =5.0

Major strain

0.4 0.3

=2.0 0.2 0.1 0

–0.2

0

0.2

0.4

Minor strain

Figure 5.20 The effect of the damage rate exponent φ on the predicted FLC at the deformation temperature of 440 C and the strain rate of 0.1/s (Δ 5 1.60, μ1 5 0.45, μ2 5 0.15). FLC, Forming limit curve.

points can be observed on the left- and right-hand sides of the FLD at β 5 20.5 and β 5 0.2 with fixed parameters of Δ , μ1 , and μ2 at specified values.

5.5.3 Prediction of formability in hot forming For practical applications in advanced manufacturing processes, theoretical material models need to be implemented in the finite element modeling. The developed constitutive equations can be used for process modeling and material response can be predicted. The overall framework for modeling includes (1) identify forming conditions, such as forming temperatures and strain rates; (2) formulate a set of unified constitutive equations; (3) design and conduct experimental program to generate data required for material constants determination; (4) calibrate constitutive equations from experimental data and embedded in finite element subroutines by using commercial computer-aided engineering (CAE) analysis software; (5) carry out process simulations; and (6) validate simulation results from practical forming trials. Process modeling engineers are able to use this simulation tool to model the evolution of material mechanical behavior and to develop advanced manufacturing processes in practice. Fig. 5.21 is an illustration of the results of CAE analysis to identify different failure modes, including wrinkling trend, insufficient stretching, safe, marginal zone, and cracks, in the hot forming process. This indicates how the FLDs are used in formability prediction and provides observable scenarios and useful information for engineers to evaluate the feasibility of practical forming or manufacturing processes. Using a similar framework to develop the damage modeling technique, the advanced manufacturing process will be able to be optimized in industries.

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Figure 5.21 Finite element simulation results of the hot forming process: (A) an example of simulation results for hot forming one component and (B) safety and unsafety regions identified in an FLD to assess material formability. FLD, Forming limit diagram.

5.6

Conclusion

This chapter has introduced the experimental and theoretical applications of damage mechanics in the development of advanced processing technologies. Challenges identified to be overcome are as follows: (1) different material formability measurement methods would lead to inconsistent and erroneous results, which will be different for different alloys and deformation conditions. Therefore it is necessary to standardize the formability and the thermomechanical testing methods for hot stamping applications. (2) The testing methods used to characterize material thermomechanical behavior and forming limit under hot stamping conditions are needed to be standardized because of potential temperature gradient and thus nonhomogeneous deformation of materials. (3) The effect of the nonproportionality of strain path on the determination of an FLD cannot be neglected for hot stamping applications. Material modeling for the prediction of an FLD needs to be proposed for nonlinear strain paths. In another way, FLSD could be evaluated at high temperature. (4) The anisotropic behavior of the material has not been taken into account in the material models introduced in this chapter. The anisotropy of materials at elevated temperature will be further investigated.

References [1] L.M. Kachanov, Time of the rupture process under creep conditions, Izviestia Akademii Nauk SSSR, Otdelenie Tekhnicheskikh Nauk 8 (1958) 26
31. [2] J.R. Rice, D.M. Tracey, On the ductile enlargement of voids in triaxial stress fields, J. Mech. Phys. Solids 17 (3) (1969) 201
217.

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[3] J. Lin, Y. Liu, A.T. Dean, A review on damage mechanisms, models and calibration methods under various deformation conditions, Int. J. Damage Mech. 14 (4) (2005) 299
319. [4] H. Karbasian, A.E. Tekkaya, A review on hot stamping, J. Mater. Process. Technol. 210 (15) (2010) 2103
2118. [5] A. Foster, T.A. Dean, J. Lin, Process for Forming Aluminium Alloy Sheet Component, PCT No. WO2010032002A1, 2012. [6] S.P. Keeler, W.A. Backofen, Plastic instability and fracture in sheets stretched over rigid punches, ASM Trans. Q. 56 (1) (1963) 25
48. [7] G.M. Goodwin, Application of strain analysis to sheet metal forming problems in the press shop, SAE Trans. 60 (8) (1968) 767
774. [8] A. Barata Da Rocha, F. Barlat, J.M. Jalinier, Prediction of the forming limit diagrams of anisotropic sheets in linear and non-linear loading, Mater. Sci. Eng. 68 (2) (1985) 151
164. [9] T.B. Stoughton, J.W. Yoon, Path independent forming limits in strain and stress spaces, Int. J. Solids Struct. 49 (25) (2012) 3616
3625. [10] H.J. Kleemola, M.T. Pelkkikangas, Effect of predeformation and strain path on the forming limits of steel, copper and brass, Sheet Met. Ind. 64 (6) (1977) 559
591. [11] R. Zhang, Z. Shao, J. Lin, A review on modelling techniques for formability prediction of sheet metal forming, Int. J. Lightweight Mater. Manuf. 1 (3) (2018) 115
125. [12] Z. Shao, et al., Formability evaluation for sheet metals under hot stamping conditions by a novel biaxial testing system and a new materials model, Int. J. Mech. Sci. 120 (2017) 149
158. [13] Z. Shao, Q. Bai, J. Lin, A novel experimental design to obtain forming limit diagram of aluminium alloys for solution heat treatment, forming and in-die quenching process, Key Eng. Mater. 622
623 (2014) 241
248. [14] Z. Shao, et al., Development of a new biaxial testing system for generating forming limit diagrams for sheet metals under hot stamping conditions, Exp. Mech. 56 (9) (2016) 1489
1500. [15] J. Lemaitre, J.L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, 1990. [16] J. Lin, T.A. Dean, Modelling of microstructure evolution in hot forming using unified constitutive equations, J. Mater. Process. Technol. 167 (2
3) (2005) 354
362. [17] R. Sandstro¨m, R. Lagneborg, A model for hot working occurring by recrystallization, Acta Metallurgica 23 (3) (1975) 387
398. [18] J. Lin, et al., Development of dislocation-based unified material model for simulating microstructure evolution in multipass hot rolling, Philos. Mag. 85 (18) (2005) 1967
1987. [19] E. Nes, Modelling of work hardening and stress saturation in FCC metals, Prog. Mater. Sci. 41 (3) (1997) 129
193. [20] M.S.K. Mohamed, An Investigation of Hot Forming Quench Process for AA6082 Aluminium Alloys, Imperial College London, 2010. [21] J. Lin, Fundamentals of Materials Modelling for Metals Processing Technologies: Theories and Applications, Imperial College Press, 2015. [22] M.A. Khaleel, H.M. Zbib, E.A. Nyberg, Constitutive modeling of deformation and damage in superplastic materials, Int. J. Plast. 17 (3) (2001) 277
296. [23] J. Lin, et al., The development of continuum damage mechanics-based theories for predicting forming limit diagrams for hot stamping applications, Int. J. Damage Mech. 23 (5) (2013) 684
701. [24] M. Mohamed, et al., Strain-based continuum damage mechanics model for predicting FLC of AA5754 under warm forming conditions, Appl. Mech. Mater. 784 (2015) 460
467.

Numerical modeling of the mechanics of pultrusion

6

Michael Sandberg1, Onur Yuksel2, Raphae¨l Benjamin Comminal1, Mads Rostgaard Sonne1, Masoud Jabbari3, Martin Larsen4, Filip Bo Salling1, Ismet Baran2, Jon Spangenberg1 and Jesper H. Hattel1 1 Department of Mechanical Engineering, Section of Manufacturing Engineering, Technical University of Denmark, Lyngby, Denmark, 2Faculty of Engineering Technology, University of Twente, Enschede, The Netherlands, 3School of Mechanical, Aerospace & Civil Engineering, The University of Manchester, Manchester, United Kingdom, 4Fiberline Composites A/S, Middelfart, Denmark

6.1

Introduction

6.1.1 Pultrusion Pultrusion is a continuous process used to manufacture fiber-reinforced polymer (FRP) composite profiles with a constant cross section. A composite is a composition material, typically consisting of a thermoset or thermoplastic polymer, reinforced with glass, carbon, aramid fibers, or combinations thereof. The pultrusion process originated in the early 1950s with the work of W. Brandt Goldsworthy in the United States on polyester resins and Ernst Ku¨hne in Switzerland on epoxy resin [1,2]. Since then, the industry has expanded worldwide to about 350 pultrusion companies, of which the market share of the 10 largest companies is approximately 40% [3]. Today, FRP composites are still becoming an increasingly popular material choice in many industries, and products manufactured using the pultrusion process are no exemption. For example, the European market for glass-fiber reinforced polymers (1.1 million tonnes, 2017) has experienced a steady yearly growth of 12% since 2009, of which the pultrusion industry (53,000 tonnes, 2017) was the fastest growing sector (16%) in 2017 [3]. The two most common pultrusion process designs are the resin-bath pultrusion (RBP) process and the resin-injection pultrusion (RIP) process. These pultrusion processes involve a number of following substeps (cf. Figs. 6.1 and 6.2): G

G

G

Combinations of rovings (tows of fibers), mats, and fabrics compose the layup of raw material that is drawn from fiber creels. The fiber material passes through guides that shape and organize the layup into the profile. Depending on the application, the fiber material may be preheated before entering the pultrusion die to speed up the cure step. After passing the guides the fiber material is impregnated with a resin. The impregnation step takes place before entering the pultrusion die in a resin bath, or in an impregnation

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques. DOI: https://doi.org/10.1016/B978-0-12-818232-1.00006-0 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Fiber creel

Guides and preheater Pultrusion die

Pullingmechanism Saw

Figure 6.1 A state-of-the-art resin-injection pultrusion line. Source: Courtesy Fiberline Composites A/S. Ambient or preheated material zone

Fiber guide Cooled zone(s) Dry fibers (A1)

Active and/or passive heating/cooling of the die and composite part

Resin inlet Pre fib heat e er cre r/ els

Liquid (A2)

β (x )

Heated zone(s) Convective cooling to ambient surroundings

Gel (A2)

Solid (A3)

x2

x3

Pultrusion die

Seperation from diewall

Composite part

x1 Pu

llin

gm e saw chan

ism

/

Figure 6.2 An overview of the steps that take place inside the die of a thermoset-RIP process with a tapered impregnation chamber.

G

chamber in the first part of the pultrusion die. The pultrusion die is actively heated and cooled to control polymerization (curing of thermosets) or crystallization (solidification of thermoplastics) throughout the pultrusion die. A continuous mechanism pulls the profile through the die, and the profile is cut into desired lengths.

Numerical modeling of the mechanics of pultrusion

175

Together with processes such as resin-transfer molding (RTM), vacuum-assisted RTM, and compression molding, pultrusion falls into the family of closed-mold manufacturing processes for FRP composites. Compared to open-mold processes such as spray or hand layup, closed-mold processes share the advantage that limited manual handling of raw materials is required by blue-collar workers. This is not only important due to health and safety concerns (exposure to chemicals) but is also required to allow for partial or full automation of the manufacturing process. Automation is a key in enhancing consistency in quality and production output, and pultrusion is characterized by a high degree of automation, high production output, as well as high product quality [1,2]. Despite FRP composites being generally associated with increased production cost and manufacturing complexity, when compared to conventional materials such as steel or concrete, pultrusion benefits from being one of the most cost-effective and energy-efficient manufacturing processes for the manufacture of FRPs. However, a clear drawback of pultrusion is that profiles have limited geometrical options as the cross section can only be constant in the lengthwise direction [1,2]. In addition, only very few standards for FRP composites exist, which makes it challenging to get market recognition and approval. For example, only recently in 2018, the first pultruded product received the European CE marking [4].

6.1.2 Overview and motivation of the chapter This chapter introduces the state-of-the-art tools and methods available for conducting numerical simulations of the different physics that take place inside the pultrusion die. The chapter is organized as follows: in Section 6.2 the impregnation mechanisms of RIP are described. In Section 6.3 the focus is on thermochemical (TC) modeling of pultrusion and in Section 6.4 on TC-mechanical (TCM) modeling and the associated buildup of residual stresses. Finally, in Section 6.5, methods to estimate the pulling force are discussed. While the sections are written with the general application to both RBP and RIP in mind, Section 6.2 only applies to RIP, and Sections 6.3 and 6.4 are structured for the application of thermoset resins.

6.2

Resin impregnation

Similar to other FRP composite manufacturing processes, the quality of parts produced by RIP highly depends on the resin impregnation of the fiber reinforcements. The resin impregnation is influenced by different parameters, such as the injection pressure, the location of the injection points, the chemo-physical properties of the resin, and the pulling speed of the profile. Modeling the impregnation process can be useful to improve the design of the impregnation chamber and to determine the most suitable window of process parameters, in complement to experiments.

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

6.2.1 Saturated pressure-driven flow The layup of fiber rovings, mats, and/or fabrics positioned inside the pultrusion die form a fibrous porous medium that must be impregnated by the resin. The impregnation flow is driven by the gradient of resin pressure within the porous medium. The saturated pressure-driven impregnation flow in a porous medium is modeled by Darcy’s law [5]: q52

K rp 1 θv U η

(6.1)

where q is the superficial resin velocity (also called the total discharge or Darcy’s flux), rp is the pressure gradient in the resin, K is the permeability tensor of the fibrous porous medium, and η is the resin viscosity. Furthermore, θv is the porosity (i.e., the initial volume fraction of void in the dry fiber reinforcement) and U is the bulk velocity of the porous medium, coming from the continuous pulling of the pultruded profile. The first term in Eq. (6.1) accounts for the discharge driven by the pressure gradient, while the second term accounts for the discharge related to the bulk velocity of the porous medium. The actual resin velocity is related to the superficial resin velocity by the porosity as only a fraction of the total volume is available for flow: u5

q θv

(6.2)

In a straight section of the pultrusion die the fiber velocity vector is U 5 U1e1, where U1 is the pulling speed of the profile and e1 is the unit vector of the principal direction of the die. If the die-entry has a planar taper angle β (cf. Fig. 6.2), ^ yÞÞe1 1 the fiber velocity vector in the tapered section is Uðx; yÞ 5 U1 cosðβðx; ^ yÞÞe2 , where e2 is a unit vector in the transverse direction of the die and U1 sinðβðx; ^ yÞ is the local fiber angle that varies with the (x,y) position within the planar βðx; tapered section of the die. Assuming the liquid resin and fiber material are incompressible, the superficial resin velocity follows the continuity equation: rUq 5 0

(6.3)

Inserting Eq. (6.2) into Eq. (6.3) yields a Poisson’s equation for the resin pressure: 

 K rU 2 rp 1 θv U 5 0 η

(6.4)

The boundary conditions of the pressure field are the atmospheric pressure at the flow front, the inlet pressure at the resin inlet, and zero pressure gradients normal

Numerical modeling of the mechanics of pultrusion

177

to the die walls. Finally, the pressure gradient of the resin is zero at the exit of the die, which means the resin and fiber velocity is assumed to be identical at this location. With those boundary conditions, Eq. (6.4) can be solved for the resin pressure. The gradient of the resin pressure is used to calculate the resin velocity, using Eq. (6.2). The resin velocity is used to update the position of the resin flow front (and the pressure boundary condition) with a transient surface-tracking algorithm [6
14]. This step also determines whether the profile is fully impregnated. Eqs. (6.1)
(6.4) have also been used to derive simplified one-dimensional analytical solutions of the resin pressure and flow front position inside a RIP die [8,15].

6.2.2 Resin viscosity The resin viscosity depends on the resin temperature, T, and the degree of cure (DOC), α. The resin viscosity can be modeled using an exponential relation as follows [6,16]: 

Eη 1 Kη α ηðT; αÞ 5 ηN exp RT

 (6.5)

where ηN, Eη, and Kη are empirical material parameters and R is the universal gas constant. In order to include the temperature and DOC dependencies of the viscosity into the impregnation model, T and α need to be calculated from the heat transfer and cure kinetics equations; see Section 6.3. As the resin is injected into the die, the resin viscosity initially decreases as it is heated (hence facilitating resin impregnation), until the initiation of the cure reaction. Then, the resin experiences an exponential rise in viscosity when it starts to cure. Therefore the heated and cooled regions of the die must be designed such that the liquid resin fully impregnates the fibrous medium before cure.

6.2.3 Permeability of fiber reinforcements Besides the resin viscosity, the permeability of the fiber reinforcements is one of the main parameters that influences the resin impregnation. The permeability depends on the fiber volume fraction, θf 5 1 2 θv, and the topology (fiber architecture) of the pores [17]. The permeability of fiber rovings and fabrics can be estimated through experimental characterization [18
20], computational fluid dynamics simulations, or approximate analytical solutions. In the case of unidirectional (UD) fiber reinforcements, and a small taper angle of the impregnation chamber, the permeability tensor is characterized by two components, the longitudinal permeability K|| (in the fiber direction) and the isotropic transverse permeability K\ (through the orthogonal plane to the fibers): 0

KO K5@0 0

0 K\ 0

1 0 0 A K\

(6.6)

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The longitudinal permeability is generally estimated by the empirical Kozeny
Carman model [21,22]: KO 5

R2f ð12θf Þ3 4k θ2f

(6.7)

where Rf is the radius of the fibers and k is an empirical parameter, referred to as the Kozeny constant. In the context of UD fiber reinforcements, Gebart [23] derived an approximate analytical solution of the longitudinal permeability and found that the Kozeny constant has the value k 5 1.78, for a regular quadratic fiber arrangement, and k 5 1.66, for a regular hexahedral fiber arrangement. Gebart also proposed an analytical solution for the transverse permeability [23]:

K\ 5 C1 R2f

sffiffiffiffiffiffiffiffiffiffiffi !2:5 θf ;max 21 θf

(6.8)

where θf,max is the maximum fiber volume fraction of the fully packed fiber arrangement, and C1 is a coefficient depending on the fiber arrangement. For a regpffiffiffi ular quadratic fiber arrangement, θf,max 5 π/4 and C1p5ffiffiffi 16=ð9π 2Þ, while for pffiffiffi a regular hexahedral fiber arrangement, θf ;max 5 π=ð2 3Þ and C1 5 16=ð9π 6Þ. Gebart’s permeability model was used in a finite-volume numerical code in Ref. [14], to investigate how an increase of the local permeability of the outer layer of the composite profile can improve the impregnation process. Gutowski et al. [24] proposed an alternative empirical model for the transversal permeability of aligned fiber beds: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 R2f ðθf ;max =θf Þ 21 K\ 5 0  4k ðθf ;max =θf Þ 1 1

(6.9)

where k0 is an empirical parameter that was estimated as k0 5 0.2 in Gutowski’s experimental permeability measurements. The Kozeny
Carman and Gutowski models have been used in Refs. [7,10
13,25] to study the effect of the fiber compaction, the resin viscosity, and the pulling speed on the resin impregnation step in RIP. The permeability of other fibrous reinforcements, for example, noncrimp and woven fabrics, has been estimated by computational fluid dynamics simulations in Refs. [26
31]. These numerical studies simulate the saturated resin flow through representative elementary volumes of fiber reinforcements.

6.2.4 Unsaturated impregnation flow Noncrimp and woven fabrics reinforcements typically have a bimodal distribution of pore space, with mesoscopic intertow pores and intratow micropores between

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individual fibers. This generally facilitates dual-phase flow as the resin impregnates the different pore scales with different impregnation speeds. Thus the impregnation of dual-scale fibrous reinforcements is typically an unsaturated flow. If viscous forces dominate, the resin fills the intertow pores faster than it saturates the intratow micropores [32]. Babeau et al. [33] investigated the influence of the different process parameters on the microvoid formation in pultruded profiles, due to a delayed impregnation of intratow micropores. The delayed impregnation of the tows was modeled by introducing a sink term χ(p,S) into the continuity Eq. (6.3): rUq 5 2 χðp; SÞ

(6.10)

where χ(p,S) depends on the resin pressure and the saturation level S of the micropores in the tows (S 5 0 if dry; 0 , S , 1 if partially saturated; and S 5 1 if fully saturated). Micropores are impregnated faster at high pressure, but the sink effect decreases as the saturation of the tow increases. The sink term depends on the saturation kinetics: χðp; SÞ 5 θv;micro ð1 2 θv;meso Þ

dS dt

(6.11)

where θv,micro is the volume fraction of the intratow micropores, θv,meso is the volume fraction of the intertow pores, and θv,micro 1 θv,meso 5 θv. Moreover, the saturation kinetics follows an empirical relation that depends on the fiber architecture of the material. For example, Babeau et al. [33] used multiaxial (0 and 90 degrees) plain weave glass fiber tapes, for which the following saturation-kinetics model was employed: e dS p 5 α ðebð12SÞ 2 1Þ dt η

(6.12)

where a, b, and c are curve fitting parameters. These parameters can be determined by curve fitting numerical simulations of the micropore impregnation of the fabric, see for example, Wang and Grove [34]. The model of Babeau et al. [33] showed that the micropore saturation is mainly influenced by geometrical parameters of the die (i.e., the taper angle and length), while the temperature profile, the pulling speed, and the resin viscosity only have limited influence. Unsaturated flow can also be modeled using methodologies adopted from soil science. In particular, Richard’s equation [35] for the movement of water in unsaturated soils has been adapted to describe the impregnation step in fibrous materials. The progressive saturation in this approach is related to the level of pressure and the relative permeability, through a semiempirical expression of the saturation behavior, see for example, Refs. [36
38].

180

6.3

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Thermochemical modeling

Manufacturing of high-quality thermoset FRPs is in general closely related to the thermal history of the profile during production. The high productivity, normally associated with the pultrusion process, is a result of heating of the die and exothermic curing reaction of the resin. Hence, good control of heat transfer during production is of uttermost importance to obtain high productivity as well as high-quality pultruded products.

6.3.1 Heat transfer The heat transfer during pultrusion is governed by the energy equation, where convection/advection is considered solely in the pulling direction (x1) of the composite profile. The energy equation for the composite profile becomes:   @T @T 1 U1 ρc Cp;c 2 rUðkc rTÞ 5 s @t @x1

(6.13)

where the velocity accounts for the pulling speed, U1. The energy equation for the die becomes: ρd Cp;d

@T 2 rUðkd rTÞ 5 0 @t

(6.14)

In Eqs. (6.13) and (6.14), subscript c refers to the composite and d to the die. Furthermore, T is the temperature, ρ is the density, Cp is the specific heat capacity, t is time, k is the thermal conductivity tensor, and s the volumetric source term. In the current section the focus will be on fiber-reinforced thermoset polymers. Hence, the source term in Eq. (6.13) will be used for capturing the exothermic heat released during curing of the thermoset resin. For the composite part the DOC, α, is subject to advection following the pulling speed of the profile: @α @α 1 U1 5 Rr @t @x1

(6.15)

where Rr 5 Rr (α,T) is the cure rate (Eq. 6.17). It is noted that Eqs. (6.13)
(6.15) are valid for cases where the velocity of the resin and the fiber material are identical. This assumption is valid for RBP, where the impregnation takes place before the heating die. In RIP, however, the velocity of the resin near the inlet slots is different from the fiber material, which can be taken into account by considering the convection term of the resin and fiber material separately in Eqs. (6.13) and (6.15) [39].

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181

6.3.2 Cure kinetics and differential scanning calorimetry Processing of thermosets, as opposed to thermoplastic polymers, involves a bulk chemical reaction (curing) where the polymer chains will cross-link and become set. The cure reaction is often thermally activated, hence the name thermoset. The curing behavior of thermosets is normally characterized using cure kinetics obtained from differential scanning calorimetry (DSC) as schematically illustrated in Fig. 6.3. The International Confederation for Thermal Analysis and Calorimetry gives recommendations for collection of thermal analysis data for kinetic computations in Ref. [40], with the goal of achieving the material constants for a cure kinetic equation that describes the relationship between time, temperature, and curing (DOC). The kinetic equation can then be used in numerical simulations to capture the thermal cure process via the exothermic source term, s, in Eq. (6.13). Considering a pultruded FRP with a fiber volume fraction of θf, a thermoset resin matrix system with a total heat of reaction of Htr, a density of ρm, and the DOC, α, this source term becomes: s 5 ð1 2 θf Þρm Htr Rr

(6.16)

The cure rate is typically described by the following cure kinetic equation, including an Arrhenius type temperature dependency: 

 2 Eα m Rr 5 AeUexp α ð12αÞn RT

(6.17)

Heat generation, dH (t)

where Ae is a preexponential factor, Ea is the activation energy, and m, n are reaction orders, all obtainable by DSC experiments.

Htr H (t) Time, t

Figure 6.3 Schematic description of a DSC scan for the heat generated during curing. H(t) is the generated heat until the time t and Htr is the total heat generation throughout the exothermic curing reaction. DSC, Differential scanning calorimetry.

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6.3.3 Modeling considerations: simple models and state-of-the-art Since the early development of the RIP manufacturing process, numerical models have been developed and proposed in the literature in order to simulate the evolution of temperatures and DOC of the composite and resin inside the heating die (Eqs. 6.13
6.17). In order to make a simplified model of the process, it is relevant to look at the Pe´clet number that describes the relationship between convective and diffusive terms for heat and mass transfer problems: Pe 5

uL α

(6.18)

where u is the characteristic velocity, L is a characteristic length, and a 5 k/(ρCp) is the thermal diffusivity. With respect to the heat transfer in the x1-direction the pultrusion process is dominated by convection due to the pulling speed of the profile. This can be illustrated using the Pe´clet number, where examples for pultrusion processes can be found in the range of Pe 5 50
500 [41]. In such cases the convection is much stronger than diffusion in the x1-direction, and it is therefore reasonable to neglect diffusion in the x1-direction. Following this example, heat transfer in the transversal direction only needs to be considered, and this has been utilized in the early one-dimensional models applying classical numerical schemes such as the finite difference and finite element (FE) methods [42,43]. Since then, more advanced three dimensional (3D) TC models have been developed [44] resulting in a more detailed insight into the process and development of the resin during cure. The further development of numerical models has focused on reducing the computational time in order to carry out multiobjective optimization of the pultrusion process [45], with the state-of-the-art being the steady state 3D TC model developed by Baran et al. [46]. In Fig. 6.4, typical results in terms of transient temperature and DOC from experimental characterization [47] and the state-of-the-art model are shown.

Figure 6.4 Measurements and simulated results of the development of temperature and DOC through a pultrusion die [46,47]. DOC, Degree of cure.

Numerical modeling of the mechanics of pultrusion

6.4

183

Thermochemical
mechanical modeling and residual stress formation

By coupling mechanical process modeling strategies with TC models, processinduced shape distortions and residual stresses in pultruded profiles can be predicted. Accurate mechanical modeling is of critical importance to assess shape distortions and to predict the load-carrying capacity or process-induced cracks that are affected by residual stresses. The mechanical model cannot be separated from the TC evolution of the material because the material properties depend on the temperature and DOC. As a follow-up from the TC modeling part in the previous section, this part covers the evolution of mechanical properties during the process, the coupling strategies of a mechanical model with the TC models, and examples of the predicted residual stress levels for different cross-sectional shapes.

6.4.1 The evolution of material properties While mechanical properties of fibers are stable, properties of the resin are evolving throughout the process. Different types of fibers (i.e., glass, carbon, and aramid fibers) and reinforcement (rovings, fabrics, and mats) can be used in pultrusion [1]. The effective mechanical properties together with thermal expansion/contraction and cure shrinkage of the fiber-reinforced composites are computed with a micromechanical approach [48]. For example, self-consistent field micromechanics (SCFM), which was developed in Ref. [49], is commonly used to predict the effective properties of UD composites. The effective properties of continuous filament mats (CFM), which is a common reinforcement type, can also be calculated with SCFM as a quasiisotropic (QI) laminate. The theory of the modification of SCFM for a QI laminate was presented in Ref. [50]. A good agreement between the SCFM and the experimental results on the in-plane elastic moduli of a CFM reinforced layer was reported in Ref. [51]. The temperature- and cure-dependent elastic modulus is required to predict the process-induced stresses and shape distortions. In general, for the process models of composites, different constitutive modeling approaches have been used. Some of the common approaches are (1) linear elastic models, (2) viscoelastic models, and (3) path-dependent models [48]. Due to the continuous nature and relatively short processing times, the cure hardening instantaneous linear elastic (CHILE) model is commonly used in pultrusion specific TCM models [46]. The CHILE model, which was developed in Ref. [52], captures the moduli evolution with the temperature and the DOC. The instantaneous elastic modulus of the resin can be formulated as a function of temperature and DOC as shown in Eq. (6.19). 8 0 E ; > > > Ar expðK T  Þ; > e < e  Er 5 E1 1 T 2 TC2 EN 2 E1 ; > r r > TC3 2 TC2 r > > : N Er ;

T  # TC1 TC1 , T  , TC2 TC2 , T  , TC3 TC3 # T 

(6.19)

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where TC1, TC2, and TC3 are the critical temperatures corresponding to the phase changes of the resin. Er0 , Er1 , and ErN are the matching elastic modulus values at the critical temperatures. Ae and Ke are the exponential term constants. TT, which relates the actual temperature and the cure degree with the elastic modulus, is equal to the difference between the instantaneous glass-transition temperature and the resin temperature, that is, TT 5 Tg 2 T. The instantaneous glass-transition temperature evolution can be modeled by a Di Benedetto equation [52,53]. The conversiondependent glass-transition temperature can also be estimated with a linear fitting of experimental data [54]. In Ref. [44] a linear relationship between the Tg and the conversion is used for a pultrusion specific polyester resin, that is, Tg 5 Tg0 1 αTg α. Here, Tg0 is the glass-transition temperature for DOC being equal to zero, and αTg is a constant. Besides the elastic modulus evolution through the process, the coefficient of thermal expansion (CTE) and the cure shrinkage values are also essential to predict the process-induced strains together with the residual stresses. The CTE value above Tg is known to be approximately 2.5 times higher than the CTE below Tg [55,56]. The CHILE approach with the same constants can also be used for DOC and temperature-dependent CTE [46]. However, cure shrinkage is usually assumed to be linearly changing with the DOC [49]. With a parametric study on the cure shrinkage value of a pultrusion specific resin system, 10%
12% total cure shrinkage shows a good agreement between the predicted and experimentally measured warpage values in a hollow pultruded profile [51].

6.4.2 Mechanical modeling strategies A state-of-the-art TCM model based on a sequential coupling of a quasistatic mechanical model with a 3D TC model for the pultrusion process was developed in Ref. [46]. An Eulerian frame was used in the TC part, while a Lagrangian frame was used for the mechanical part. The DOC and the temperature field were mapped into the moving frame of the mechanical model. The effect of pulling speed was implemented into the stress analysis with the speed of the moving frame. Both 2D and 3D modeling of the mechanical part have been conducted in the literature [46,57]. A schematic representation of coupling strategies for both 2D and 3D mechanical models is shown in Fig. 6.5. A plane strain assumption was made in the 2D quasistatic model since the length of the pultruded profile in the pulling direction (x1-direction in Fig. 6.5) is much larger than the cross-sectional dimensions in the transverse direction (x2- and x3-direction in Fig. 6.5). An FE analysis was carried out to solve the process-induced stresses and the displacements incrementally. The total incremental strain (Δεtot ij ) is equal to the sumth mation of the incremental mechanical Δεmech , thermal (Δε ij ij ), and chemical strain ch (Δεij ). The sum of the thermal and chemical strains is defined as the incremental process-induced strain. The incremental mechanical strain is isolated by the subtraction of the process-induced strains from the total incremental strain, which is shown in Eq. (6.20). The incremental stress tensor is calculated by the multiplication of the incremental mechanical strains with the temperature and DOC dependent

Figure 6.5 A schematic representation of the coupling of the 3D Eulerian thermochemical model with 2D and 3D Lagrangian mechanical model [57]. 3D, Three dimensional.

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stiffness matrix, C, as given in Eq. (6.21). After each time increment the instantaneous stress and strain tensors are calculated according to Eq. (6.22). In Ref. [46], only the heating die and the postdie region were considered in the mechanical model. Due to the chemical shrinkage, the composite part detaches from the die during the process which is depicted in Fig. 6.2. Neglecting the separation of the part from the die would lead to errors in the stress analysis. Therefore a mechanical contact formulation was defined between the die wall and the part which restricts the expansion beyond the die wall but allows for the separation [46]. mech ch mech 1 Δεth 1 Δεpr Δεtot ij ij 5 Δεij ij 1 Δεij 5 Δεij pr tot Δεmech 5 Δε 2 Δε ij ij ij

(6.20)

Δσij 5 CΔεmech ij

(6.21)

old σnew ij 5 σ ij 1 Δσij old εnew ij 5 εij 1 Δεij

(6.22)

6.4.3 Assessment of the resultant residual stress fields and the verification The developed TCM models have been conducted for various different crosssectional profiles, materials, and process conditions. An overview of the residual stress predictions for the pultrusion of different profiles is given in Table 6.1. The residual stresses listed in Table 6.1 are the maximum tensile and compressive residual stresses observed in the simulations. The resultant residual stress field is a result of the combination of process parameters. In general, thick profiles have higher residual stresses due to the higher nonuniformity through the cross section. However, it is noted that even for a 100 mm thick profile, the residual stresses can be reduced by means of preheating of the fibers, which was analyzed numerically in Ref. [58]. Moreover, the geometrical shape, the stacking sequence, and the process conditions are also affecting the resultant residual stresses [51]. The predicted residual stress field in the transverse directions can be explained by the nonuniform curing throughout the cross section. In general, the cure reaction starts from the outside due to the heating of the die. Peripheral heating causes faster vitrification of the outer regions than the core. During the cure evolution of the core region the vitrified outer shell restrains the process strains of the core. After the vitrification of the outer regions the hindered process strain in the core region, which is dominated by the cure shrinkage, ends up as tensile residual stress. Using hole drilling and digital image correlation, Yuksel et al. [62] found that measurements of tensile residual stresses at the core region of a square UD glass fiber
reinforced pultruded profile were in good agreement with numerical predictions. The predicted residual stress fields for a thick square profile [58] and a relatively thin L-shaped

Table 6.1 An overview of the residual stress levels in the pultruded profiles. Pultruded profile

Material

Thickness (mm)

Heater temperatures ( C)

Pulling rate (mm/min)

Flat plate [46] Rod [59]

Glass (UD)/epoxy Graphite (UD)/ epoxy Glass (CFM 1 UD)/ polyester Glass (CFM 1 UD)/ polyester Glass (UD)/epoxy Glass (UD)/ polyester

25.4 3

171
188
188 Prescribed

200 300

4.15 0.26

2 16.00 2 0.82

3

140
130

650

10.70

2 17.40

5

110
140

600

6.31

2 18.32

171
188
188 110
140

200 100

3.67 6.92

2 10.23 2 48.87

Box [51]

L-shaped [60]

NACA0018 [61] Square profile [58]

CFM, Continuous filament mats; UD, unidirectional.

18 100

Max. residual tensile stress (MPa)

Max. residual compressive stress (MPa)

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Figure 6.6 The predicted residual stress field of a thick square profile [58] (A) and an Lshaped profile [60] (B).

Figure 6.7 The predicted deformation field of a rectangular hollow profile [51] (A) and an L-shaped profile [60] and (B) compared to cross-sectional views of real parts.

profile [60] are depicted in Fig. 6.6. In addition to the geometry, reinforcement types are also different for these profiles. The L-shaped profile contains approximately 0.75 mm thick CFM layers above and below the UD fibers. The CFM layer behaves as a quasitransversely isotropic layer, which has the same mechanical properties in all in-plane directions. Higher elastic moduli in both the in-plane directions of the CFM lead to lateral constraint (the x2-direction in Fig. 6.6B), which causes higher residual stress in that direction. This phenomenon, which was investigated experimentally in Ref. [63], explains the comparable stress levels between these two profiles despite their relative difference in thickness. Some of the residual stress predictions in the literature were validated by means of shape distortion comparisons with real pultruded products. Fig. 6.7 shows the predicted deformation fields for a rectangular hollow profile [51] and an L-shaped profile [60], which are reported to be in a good agreement with experimental observations.

Numerical modeling of the mechanics of pultrusion

6.5

189

Pulling force

The pulling force is the net force required to achieve the desired profile-advancing pulling speed, which directly translates into the production output of the process. Estimations of the pulling force are of importance when choosing a suitable pulling mechanism, as well as designing the production equipment and the profile itself, which has to withstand the restoring forces. In the context of this section the pulling force is decomposed into the following force components: F 5 FA 0 1 FA 1 1 FA 2 1 FA 3

(6.23)

where each component, FAi , refers to the part of the pulling force generated over the contact area, Ai. The contact areas are bounded by the area of contact between the die and the given phase. This is illustrated in Fig. 6.2 and further explained in the following sections. The magnitude of the pulling force is strongly coupled to the physics described in the previous sections, for example, the buildup of resin pressure (Section 6.2), the temperature field and DOC (Section 6.3), as well as the process-induced mechanical deformations (Section 6.4). In addition, as the contact areas are split by the flow front location, the gelation point of the resin, and the detachment point of the profile, the location and sizes of the contact areas also depend on the physics of the process.

6.5.1 Before die-entrance, A0 The friction force generated before the fiber material enters the pultrusion die is referred to as the collimation force. The collimation force builds up as the fiber material is drawn from creels and passes through the different production equipment, for example, fiber guides, preheaters, and resin-bath. As the collimation force is closely related to circumstances of the equipment design and production layout, it is difficult to generalize. As noted in Ref. [64], process designers seek to reduce its magnitude to minimize its contribution to the net pulling force, and it is generally found to be neglected in the literature.

6.5.2 Die-entrance to flow front location, A1 In the contact area between the die-entrance and the flow front location, the bulk compaction of the dry fiber material introduces a force component, FA1 . The force component is composed of the bulk force normal to the die wall and an associated tangential friction force. Sliding against the die walls introduces friction, which is considered as mechanical sliding friction [65
68]: ð

  pf cos β sin β dA 1

FA1 5 A1

ð A1

 pf μ1 cos2 β dA

(6.24)

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

In Eq. (6.24), pf is the pressure coming from the consolidation of the fiber material, β 5 β(x) is the tapering angle of the impregnation chamber, and μ1 is the friction coefficient of the interface between the die wall and the dry fiber
material. The compaction pressure of the fiber material can be approximated by Gutowski’s model [24]: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðθf =θ0 Þ 2 1 pf 5 C0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ðθf ;max =θf Þ 21

(6.25)

where θf is the fiber volume fraction and θ0 is the fiber volume fraction at which the fiber material in unloaded conditions is at rest. While Ref. [24] suggests that θf, max is the theoretical maximum achievable fiber volume fraction, Ref. [65] advises that both the spring constant C0 and θf,max are based on consolidation experiments.

6.5.3 Through liquid and gel states, A2 The force component FA2 is composed by the hydrostatic pressure of the resin, p, and the compaction force carried by the fiber material, pf. In addition, replacing mechanical sliding friction, a viscous force coming from the thin fluid film layer between the fiber material and the die wall is considered. The force component, FA2 , acts on the contact area, A2, which is the surface area between the resin and die, where the resin is in its liquid and gel states. In other words, A2 extends from the resin flow front to the point where the resin reaches the glass-transition temperature, Tg. A common approach for the estimation of the viscous force contribution is to assume a Couette-type flow in the thin fluid film layer. This suggests that the viscous force is proportional to the profile-advancing pulling speed, U1, and the viscosity of the resin, η 5 η(α,T), and inversely proportional to the thickness of the film layer, λ [66,68
71]: ð



FA2 5

p 1 pf cosðβ Þ sinðβ ÞdA 1

A2

ð A2

U1 η cosðβ ÞdA λ

(6.26)

Assuming a hexagonal fiber stacking, the thickness of the fluid layer between the fibrous material and the die wall can be estimated as [66]:

λ 5 Rf

! sffiffiffiffiffiffiffiffiffiffiffi 2π pffiffiffi 2 2 3θ f

(6.27)

It is noted that other estimates of λ exist, see for example, Refs. [66,72,73].

Numerical modeling of the mechanics of pultrusion

191

6.5.4 Solid state and detachment from die wall, A3 When the resin solidifies, adhesive debounding to the die wall occurs and mechanical sliding friction takes over from the viscous force. This mechanical friction acts on the contact area, A3, which extends from the resin gelation point to the point where the profile physically detaches from the die wall due to chemical shrinkage and thermal contraction. The magnitude of the friction force is considered to be the product of the contact pressure to the die wall, σn, and the friction coefficient of the contact interface, μ3 [66]: ð FA3 5

σn μ3 cos2 ðβÞdA

(6.28)

A3

The contact pressure, σn, is governed by thermal contraction/expansion and chemical shrinkage of the profile, which can be predicted using TCM modeling described in Section 6.4.

6.6

Conclusion

In this chapter about the mechanics of pultrusion, state-of-the-art methods for numerical simulation of pultrusion processes were introduced. The focus was on the physics that takes place inside the pultrusion die, including (1) saturated and unsaturated impregnation mechanisms related to resin flow in straight and tapered impregnation champers; (2) simulation of heat transfer and cure, and the associated buildup of residual stresses and shape distortions; and finally (3) estimations of the pulling speed needed to achieve the desired profile-advancing pulling speed. The methods for numerical simulation of the mechanics of pultrusion introduced in the chapter illustrate that the physics taking place inside the pultrusion die is often considered separately in the established literature. This applies in particular to the impregnation step (resin flow and pressure buildup) and heat transfer (temperature and curing). These are valid simplifications if the underlying assumptions for decoupling the physics are not violated. However, it is sometimes required to consider the multiphysics of the pultrusion process, and this is not a new phenomenon. For example, as outlined in Section 6.5, early research on estimations of pulling force go back to the early 1990s, where effects of resin pressure, temperature, DOC, and the detachment point between the profile and die were considered. Today, the multiphysics of the pultrusion process is gaining more interest in the research field, as examples of coupled flow-TC models (noted in Section 6.3) and TCM models (cf. Section 6.4) are found in recent literature. To increase the understanding of the governing physics, and to increase the accuracy of numerical models, the authors see a positive outlook for a continuation of this trend in the future.

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Part 1: Thermal and chemical strain, Polym. Test. 32 (2013) 1350
1357. [55] M.W. Nielsen, J.W. Schmidt, J.H. Hattel, T.L. Andersen, C.M. Markussen, In situ measurement using FBGs of process-induced strains during curing of thick glass/epoxy laminate plate: experimental results and numerical modelling, Wind. Energy 16 (2013) 1241
1257. [56] J. Svanberg, J. Holmberg, An experimental investigation on mechanisms for manufacturing induced shape distortions in homogeneous and balanced laminates, Composites, A: Appl. Sci. Manuf. 32 (2001) 827
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[59] I. Baran, C. Tutum, J. Hattel, Probabilistic thermo-chemical analysis of a pultruded composite rod, in: Proceedings of the 15th European Conference on Composite Materials, ECCM-15, Venice, Italy, 2012. [60] I. Baran, R. Akkerman, J. Hattel, Modelling the pultrusion process of an industrial L-shaped composite profile, Compos. Struct. 118 (2014) 37
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557 (2013) 2127
2137. Main theme: The Current State-of-the-Art on Material Forming. [62] O. Yuksel, I. Baran, N. Ersoy, R. Akkerman, Investigation of transverse residual stresses in a thick pultruded composite using digital image correlation with hole drilling, Compos. Struct. (2019) 110954. [63] T. Garstka, N. Ersoy, K. Potter, M. Wisnom, In situ measurements of through-thethickness strains during processing of AS4/8552 composite, Composites, A: Appl. Sci. Manuf. 38 (2007) 2517
2526. [64] H.L. Price, S.G. Cupshalk, Pulling force and its variation in composite materials pultrusion, Polym. Blends Compos. Multiphase. Syst. 206 (1984) 18
301. ˚ stro¨m, R.B. Pipes, A modeling approach to thermoplastic pultrusion. I: [65] B.T. A Formulation of models, Polym. Compos. 14 (1993) 173
183. ˚ stroo¨m, R.B. Pipes, A modeling approach to thermoplastic pultrusion. II: [66] B.T. A Verification of models, Polym. Compos. 14 (1993) 184
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146.

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Modeling of machining processes

7

Ju¨rgen Leopold TBZ-PARIV GmbH, Chemnitz, Germany

Nomenclature ε_ εeff σ; τ k ux, vy, wz

7.1

strain rate components effective strain stress components yield shear stress displacements (respectively velocities)

Introduction

To understand the progress in manufacturing and the associated modeling and simulation methods, a short look back into the history will identify the next important steps of development and application. Starting with very simple cutting tools and hand-driven technology, the industrial revolution opened the chance to use mechanical energy in addition to man power, with the development and design of machine tools that are applied up to the present century. To the classical design a new concept—Parallel Kinematic Machine (PKM) Tools—has been developed and introduced into industry. Compared with Henry Ford’s principle of mass production, the main step forward in general manufacturing technology from “conventional machining” processes was started with the “Flexible Machine Tool System” (FMS) concept. These developments, in conventional machine tools, PKM and FMS, have been driven by the objectives of increasing product quality while retaining higher manufacturing productivity, cost reduction, and environment protection. The development and application of relevant models and simulation techniques has a very simple background: Machining is the most predominant manufacturing operation in terms of volume and expenditure. Machined components are used in almost every type of manufactured product. It has been estimated that machining expenditure contributes to approximately 5% of the GDP in developed countries, while in the United States alone it translates into approximately $250B per year [1]. In the face of worldwide competition the timeto-market of new products can be responsible for economic survival. Advanced modeling and simulation techniques can be a competitive advantage for companies. Returning to the short history of manufacturing (Fig. 7.1), the challenges of further modeling and simulation technique applications lie in the increasing of the accuracy Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques. DOI: https://doi.org/10.1016/B978-0-12-818232-1.00007-2 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Figure 7.1 Main steps in manufacturing from past to present.

for the single model on the one hand but more and more on the application of the so-called closed-loop principle [2] in modeling on the other. This will be pointed out within the next section.

7.2

Closed-loop principle of modeling

The manufacturing of a workpiece can be a very complex situation. The main goal of machining is the manufacturing of a workpiece with specified properties: geometrical properties (length, diameter, concentricity etc.); surface properties (roughness, waviness, polish etc.); subsurface properties (residual stress, crack-free, etc.); mechanical stability; etc. To manufacture a product with such properties, a complex system (Fig. 7.2) must be applied. Following Georg Schlesinger’s postulate that the “profit will be determined at the cutting tool tip”—the cutting tool is the most important component part of the closed-loop modeling principle with geometrical properties such as length, flute geometry, tool-tip geometry (rake; chip former), micro-geometry, cutting tool materials and coatings; coolant bore holes and design; mechanical stability, clamping areas of the tool within the tool holder, etc. The interaction of the cutting tool with the workpiece is responsible for the generation of the new workpiece, which is why the workpiece is one of the next main modeling part in this system. Typically single modeling systems are focusing on only this interaction—with no deformable cutting tool and also no deformable workpiece. This is commonly used but did not represent the real situation in manufacturing. The workpiece geometry can be realized in different steps (roughing and finishing)—but in the airplane industry for instance, there are often thin-walled parts manufactured. These parts are commonly produced by casting and can include previously induced residual stresses. The physical based machining residual stresses must be superposed with the residual stresses caused by previous processes and can deform the final part into an undesirable condition. A similar situation can be found for parts coming from previously forming operations. In addition to this serious problem, the simulation techniques require the

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199

Figure 7.2 Machining of predefined workpieces.

workpiece properties and the material law for this workpiece (which depend on strain, strain rate, and temperature for the contemporary homogeneous and isotropic modeling world). Both the cutting tool and the workpiece are fixed within the machine, which is why on the one hand machine tool properties such as damping and stiffness and also the same properties for the spindle and tool holder within the machine tool are influenced by the interaction of the chip-formation process at the cutting tool tip with the workpiece. On the other hand the clamping device with similar properties (stability, stiffness, and damping) also influences considerably the machining process itself. In addition, most of the machining operations are based on cooling and lubrication processes, so this liquid situation must be included with in model. Following the previous very short discussion of the simulation, the final focus for modeling and simulation must be on the development of closed-loop models. The next section will give an overview of the contemporary modeling and simulation methods, applied in machining, and will be followed by the gaps in modeling and simulation.

7.3

Modeling and simulation techniques

7.3.1 Slip-line method Consider the physically based machining situation of the tool-tip interaction with the workpiece [3]. The surface of the “old” workpiece is on the right side and that of the new one on the left side of Fig. 7.3. The original structure of the previous workpiece surface on the left side is dramatically deformed into the chip area. The line (or area) from tool-tip Point P1 to the free-surface Point P2 is called the shear line (or shear plane) and has

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Figure 7.3 Metallographic view of the chip-formation zone.

been defined by the modeling pioneers about 130 years ago (Time—first shear line equation [4]; Merchant—the so-called Merchant’s circle [5]) concerned with previous experimental investigation by Tresca [6], Mallock [7], and Svorykin [8]. With the definition of the shear line, the well-known slip-line method in chip formation was born. The principles are briefly summarized next [3]. In addition to the usual assumptions that the metal is isotropic and homogeneous, the common approach involves the following: G

G

G

G

The metal is rigid-perfectly plastic; this implies the neglect of elastic strains and treats the flow stress as a constant. Deformation is by plane strain. Possible effects of temperature, strain rate, and time are not considered. There is a constant shear stress at the interfacial boundary. Usually, either a frictionless condition or sticking friction is assumed.

The principal ways in which slip-line field theory fails to take account of the behavior of real materials are as follows: G

G

G

G

It deals only with nonstrain-hardening materials. There is no allowance for creep or strain rate effects. The rate of deformation at each given point in space and in the deforming body is generally different, and any effect this may have on the yield stress is ignored. All inertia forces are neglected and the problems treated as quasistatic (a series of static states). In the cutting operations that impose heavy deformations, most of the work done is dissipated as heat; the temperatures attained may affect the material properties of the body or certain physical characteristics in the surroundings, for example, lubrication.

The following equations are the basics for the slip-line method: Plane plastic strain: Since elastic strains are neglected, the plastic strain increments (or strain rates) may be written in terms of the displacements (or velocities) ux(x,y), vy(x,y), wz 5 0, as shown below:

Modeling of machining processes

@ux ε_ x 5 @x ε_ y 5

@vy @y

ε_ z 5

@wz 50 @z

201

0 1 1 @@ux @vy A 1 γ_ xy 5 2 @y @x 0 1 1 @@vy @wz A 1 50 γ_ yz 5 2 @z @y 0 1 1 @@wz @ux A γ_ zx 5 1 50 2 @x @z

(7.1)

State of stress: It follows from the Levy
Mises relation that τ xy and τ yz are zero and therefore that σz is a principal stress. The Levy
Mises relationship between stress and strain for an ideal plastic solid where the elastic strain are negligible is as shown below: 0

σxx σ 5 @ τ xy 0

τ xy σyy 0

0 σxx 1 0 B τ xy 0 A5B @ 0 σzz

τ xy σyy 0

0 0

1

C C 1 A ðσxx 1 σyy Þ 2

(7.2)

It follows that the plastic strain increment @εzz 5 0: The material is incompressible, ε x 5 2 ε y, and each incremental distortion is pure shear. The state of stress throughout the deforming material is represented by a constant yield shear stress k and a hydrostatic stress 2 p, which in general varies from point to point throughout the material. k is the yield shear stress in plane strain and the yield criterion for this condition is: τ 2xy 1 ðσx 2σy Þ2 =4 5 k2

(7.3)

pffiffiffi where k 5 Y/2 for the Tresca criterion and k 5 Y= 3 for the Mises criterion. The state of stress at any point in the deforming material may be represented in the Mohr circle diagram. Together with Hencky’s hydrostatic stress conditions along slip line and Geiringer velocity field equation, the initial equations can be solved graphically or analytically. Over the last 70 years this method has been used in basic research by hundreds of scientists (Merchant, 1945—Fig. 7.4A; Lee/Shaffer, 1951—Fig. 7.4B and C; Shaw, 1955—Fig. 7.4D; Palmer/Oxley, 1959—Fig. 7.4E; Weber, 1969— Fig. 7.4F; Stevenson/Oxley, 1970; Makino/Usui, 1973; Fang/Jawahir, 2002). For analytical solution the computation time compared with numerical methods are extremely short. This is responsible for the ongoing investigations, despite the main disadvantages.

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Figure 7.4 Selected examples for slip-line field solutions in machining: (A) simple shear plane; (B and C) slip-line field with shear plane and contact length; (D) slip-line fan; (E) parallel slip-line field; (F) curved slip-line field.

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Figure 7.5 Finite element application in the modeling and simulation of metal cutting.

7.3.2 Finite element modeling (finite element method) With the design and manufacturing of the computer [9], new numerical methods by mathematicians have been developed. Fig. 7.5 gives a short history of finite element method (FEM) application with respect to metal-cutting simulation. There are two main FEM concepts: the Eulerian flow formulation and Lagrance’s formulation. In a very early stage of chip-formation simulation, the Eulerian FEM principle [10] was applied to steady-state machining [3,11
15]. The plastic metal flow can be considered as an incompressible non-Newtonian flow problem with the following unknown quantities: v—flow velocity and p—hydrostatic pressure

defined by the differential equations div v 5 0(incompressibility condition) 2div½σ 5 ρg (equilibrium equation, i.e., the balance of internal and external forces)

where [σ] is the stress tensor and ρg represents the external forces. For incompressible materials, only the deviatoric stress tensor ½σ0  5 ½σ 1 p½I ([I] is the unity tensor) is influenced by the effective strain εeff and the strain rate tensor ½_ε in the form  ½σ0  5 2η ε_ eff ; εeff ½_ε; 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi εeff 5 pffiffiffi J2 ð½εÞ; ε_ eff 5 pffiffiffi J2 ð½_εÞ; 3 3

J2 2 2nd tensor invariant:

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The uniaxial flow stress σeff of the workpiece metal is assumed to be governed by the general flow rule  σeff 5 η ε_ eff ε_ eff

(7.4)

 σy 1 ð3η0 ε_ eff Þ1=m η ε_ eff 5 3_εeff

(7.5)

with

and the material parameters according to the temperature θ   ð2Þ ð3Þ σy ðθÞ 5 σð1Þ y 1 σy U exp σ y Uθ   ð2Þ ð3Þ η0 ðθÞ 5 ηð1Þ 0 1 η0 Uexp η0 U θ

(7.6)

mðθÞ 5 mð1Þ 1 mð2ÞUθ This general law includes linear Newtonian and the nonlinear Bingham and Norton
Hoff laws as special cases. The parameters in the flow rule may depend on the strains. Instead of the flow rule of type, other formulations for the thermoviscoplastic behavior of the material may be used. In general, the flow rules give a functional relation of the dependence of the flow stress on the actual effective strain rate, the actual effective strain, and the actual temperature. Typical material laws are the Johnson
Cook [16] and the Zerilli
Armstrong [17] law. Frictional stress τ F on the tool rake face may be defined using a modified model of Usui and Hoshi [18]   τ F 5 kUσeff 1 2 e2μF ðσn =σeff Þ

(7.7)

where k is the friction factor (#1.0) and μF is the friction number. Heat flux within the workpiece is modeled by the stationary heat conduction
convection equation ρcp vrθ 5 rðλrθÞ 1 w_

(7.8)

where ρcp represents the heat capacity, λ is the thermal conductivity, and w_ is the heat dissipation rate. The parameters λ and cp depend on the temperature. It may be assumed that the heat dissipated is equal to the plastic work rate w_ 5 σeffU ε_ eff . In comparison to the steady-state solution with the Eulerian approach, Lagrange’s solution can be used for a nonstationary chip-formation application. This approach must be used for the most typical industrial applications such as tool
workpiece entry or exit, burr-formation simulation, any chip thickness modifications during the machining operation, etc. This approach is more general—but is

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205

also much more time-consuming in computation. The leading industrial software code [19] is an explicit dynamic, thermomechanically coupled finite element (FE) modeling package specialized for metal cutting. The principle is pointed out in the following equations [19]. The balance of linear momentum is written as: σij;j 1 ρbi 5 ρu¨i

(7.9)

Thermal equations Heat generation and transfer are handled via the second law of thermodynamics. A discretized weak form of the first law is given by C T_ n11 1 KTn11 5 Qn11

(7.10)

where T is the array of nodal temperatures, ð Cab 5

cρNa Nb dVo

(7.11)

Bt

is the heat capacity matrix ð Kab 5

Dij Na;i Nb;j dV

(7.12)

B0

is the conductivity matrix ð Qa 5

ð sNa dV 1

Bt

hNa dS

(7.13)

Bτq

is the heat source array, with h having the appropriate value for the chip or tool. In machining applications the main sources of heat are plastic deformation in the bulk and frictional sliding at the tool
workpiece interface. Constitutive model and material characterization In order to model chip formation, constitutive modeling for metal cutting requires the determination of material properties at high strain rates, large strains, and short heating times and is quintessential for the prediction of segmented chips due to shear localization. The increase in flow stress due to strain rate sensitivity is accounted for with the relation 

   σ m1 ε_ p 11 p 5 gðεp Þ ε_ 0

(7.14)

where σ is the effective Mises stress, g the flow stress, εp the accumulated plastic strain, ε_ p0 a reference plastic strain rate, and m1 is the strain rate sensitivity

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exponent. A power hardening law model is adopted with thermal softening, which gives   εp 1=n g 5 σ0 ΘðTÞ 11 p ε0

(7.15)

where n is the hardening exponent, T the current temperature, σ0 is the initial yield stress at the reference temperature T0, εp0 the reference plastic strain, and Θ(T) is a thermal softening factor ranging from 1 at room temperature to 0 at melt and having the appropriate variation in between. Figs. 7.6
7.9 indicate the grand step forward in industrially driven machining simulations and application. In addition to this “single” numerical philosophy (Eulerian or Lagrange), the arbitrary Lagrangian
Eulerian (ALE) technique combines the unique features of Lagrangian and Eulerian formulations and adopts an explicit solution technique for fast convergence [20]. While the ALE technique with Eulerian boundaries requires a predefined chip-geometry assumption, chip formation is free of any geometrical assumption in ALE with Lagrangian boundary conditions [21]. Considering industrial demands, typical 3D computation times for a drilling operation of more than 3 months (in the year 2000) have been changed to a few hours by a new numerical algorithm and, above all, due to parallel computing (Fig. 7.10). Recently [19], the parallel computing capability in chip-formation simulation has been extended to up to 38 processors. In the case of time-consuming drilling simulation, the total elapsed time can be reduced from about 119 to about 7 hours. Thus batch jobs can run overnight with the results being available the next morning.

Figure 7.6 Euler 2D finite element simulation.

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Figure 7.7 Lagrange 3D finite element simulation of micro-cutting.

Figure 7.8 Lagrange 3D finite element simulation of drilling.

For coating-substrate simulations (CSSs), this method has been developed and applied by Refs. [22
27] in the last century. The most time- and memoryconsuming part of the numerical program is the solver for the linear FEM equation systems. Iterative methods should be preferred, because direct solvers have bad error

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Figure 7.9 Lagrange 3D finite element simulation of turning.

Figure 7.10 Example for the elapsed time reduction in parallel computing. Source: http://www.thirdwavesys.com/news/in_the_node/5-1/5-1_parallel.html.

propagation and, due to the unavoidable fill-in, they require too much storage when applied to large equation systems. Moreover, the FEM matrices’ large condition number, mainly caused by the inevitable high mesh graduation, requires special preconditioning techniques, which is why the preconditioned conjugate gradient method

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209

Figure 7.11 Domain decomposition method applied to coating insert simulation.

has been applied for the FEM solver. Additional coarse-mesh preconditioning has substantially reduced the necessary number of iterations. The accuracy of FEM simulations on standard hardware is basically limited by the available storage and the acceptable computing time. The application of parallelized software on a distributed memory computer with N processors reduces the computing time to less than the Nth part and increases the available storage by almost N times. The software developed for the simulation of coating-substrate systems on cutting tools is scalable to any number of processors and portable to a variety of hardware and operating systems. The dominating programing model for parallel FEM applications is the domain decomposition method (Fig. 7.11). For N processors the simulation domain, for example, the tool’s geometrical model, is divided into N disjunctive subdomains and each processor does all the computational work related to “its” subdomain, with some internal communication for data exchange and control purposes. Based on this parallel FEM software, deformations, strains, and stresses caused in coated tools by external loads can be computed on a microscopic scale and become available inside and between the coating layers. Based on computation results with this parallel FEM software, a cutting insert test (workpiece material: GGG-60) with an adopted new coating-substrate system (Figs. 7.12 and 7.13) [28] increased the lifetime of the tool by up to 146%.

7.3.3 Complementary methods Supplementary methods that are used successfully in other scientific applications have been transferred to machining simulations. Fig. 7.14 points out the commonly used complementary methods. Boundary element method (BEM), finite variation method, and finite difference method are used for linear elasticity and heat conductivity problems.

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Figure 7.12 Example for a coating-substrate system.

Figure 7.13 Stress concentration in the contact area of the cutting insert.

The BEM is a numerical technique to solve partial differential equations (PDEs) of a variety of physical problems with well-defined boundary conditions. The PDE over a problem domain is transformed into a surface integral equation over the surfaces that enclosed the domain. This integral equation can be solved by discretization of the surfaces into small patches—the boundary elements.

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211

Figure 7.14 Methods for machining simulations.

Figure 7.15 Principle of finite point set method.

The transformation into a surface integral is not possible generally. There is not known a global way to transform the complex thermomechanical equations describing the chip and burr-formation process to a surface integral equation. Therefore the BEM has been used only for subproblems in machining simulation, for example, for the thermal problem in steady-state chip formation [29]. New mathematical methods, for the coupling of FEs and boundary elements for problems in elastoplasticity and for contact problems with friction that yield variational inequalities, were developed and applied at the Institute of Applied Mathematics in Hannover [30]. Some years ago [31], a previously developed numerical code based on the finite point set method (FPM) (or sometimes called the grid-free method or smoothed particle hydrodynamics) (Fig. 7.15) [32], has been applied for a simple 2D chipformation simulation (Fig. 7.16). The main advantage could be to delete the second numerical step in Fig. 7.17—normally used in the FEM. For a “relatively good accuracy” [33] the CPU time can be reduced up to approximately 2.75 times less cost compared to the Eulerian model. The basic equations follow the hydrodynamics. Due to missing friction conditions between chip and cutting tool and a very simple material law, the development was stopped [31]. A later published paper [34]

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Figure 7.16 Principle of finite element method.

Figure 7.17 Finite point set method applied to metal-cutting simulation [31].

did not present any substantial progress in this direction. The authors focused on the further development in: “The FPM for the simulation of chip formation still requires considerable development effort however. Thus the algorithms for the calculation of the material behavior and for the consideration of various boundary conditions such as friction, inertia and heat transfer will be developed further” [34]. In addition, other interesting papers were published [35,36]. Molecular dynamics (MD) simulation applications for cutting have been developed in the early 1990s [37
39] and are commonly used for special material removal, such as monocrystalline silicon [40].

Modeling of machining processes

7.4

213

Modeling and simulation in the industry—selected examples

7.4.1 Cutting tool optimization State-of-the-art in the leading cutting tool companies is the daily job for tool design and tool optimization [41
44]. Investigations are already focused on the tool-tip geometry, the coolant bore design, cutting tool stability, to the chip-geometry design, mainly for deep-hole drilling, the influence of cutting tool material in relation to the workpiece material, etc.

7.4.2 High-speed cutting or high-performance cutting The influence of high speed or high loads on the stability of the cutting process and the quality of the new surface is widely investigated. There are interesting solutions available due to the holding of special international conferences [45
47].

7.4.3 Dry machining From the modeling and simulation point of view, the manufacturing without coolants or lubricants is a very welcome task. The boundary conditions of the cutting model are simpler, because no special friction conditions between the chip and cutting tool (rake face, etc.) must be included into the calculation. Due to the fact that a lot of cutting operations are not stable enough in dry conditions, leading software tools have included cooling options for coolants and lubricants.

7.4.4 Burr formation and clean manufacturing The best choice is If you do not make the burr, you do not have to remove it. In most cases this philosophy is not applicable in the industrial environment. With modeling and simulation, the generation of burrs can be investigated and good cutting conditions (including cutting tool, coatings, speed, feed, tool geometry, etc.) can be found out to prevent burrs. In the early 1990s a quantitative model of burr formation for ductile materials, which did not include fracture during orthogonal machining, had been proposed [48]. The burr-formation mechanism is divided into three parts—initiation, development, and formation of the burr with appropriate assumptions. A good overview to the state-of-the-art was published 10 years ago [49]. Based on the new mechanical principle of the “hydrostatic bowl” [50], the mechanism of burr formation can be better understood. The hydrostatic bowl is a highly negative hydrostatic pressure region immediately below the cutting edge in the workpiece (Fig. 7.18), which acts as a slipstream, may have a positive effect on burr formation as well as on the burr shear zone, assuming that chip and burr will tear off there. A bigger hydrostatic bowl leads to a better burr. The downscaling influences the size and shape of the

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Figure 7.18 Definition of the “hydrostatic bowl.”

Figure 7.19 “Hydrostatic bowl” applied to burr-formation simulation.

hydrostatic bowl [51]. The size of the pressure region increases in the course of the time up to the moment when the tool crosses the left edge of the workpiece (Fig. 7.19). The bigger the hydrostatic bowl is the smaller the rake angle is. Hence, there is an obvious influence of the rake angle on the size and depth of the pressure region [52]. In addition, there are additional investigations, concerning the failure criteria [53] to simulate burr formation, ultrasonic-assisted cutting [54], modification of the local material properties to prevent burrs [55]. Burr formation and clean ability are closely connected. In addition to preventing burrs in manufacturing, some new methods to increase the cleanliness within the machine tool have been developed [56]. With standard coatings [modified diamond like

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215

carbon (DLC)], as well as new coatings based on nanoscale metallic fluoride powders, the amount of stuck-on chips at the sheet metal surface in a machine tool after the machining of steel and aluminum alloy was reduced by up to 45% and in dry machining with nanocer coatings by up to 94%. Geometrical cutting tool design influences burr formation [57
59].

7.4.5 Cryogenic machining The most emerging needs of the modern metal-cutting operation are to increase the material removal rate with better surface finish, high machining accuracy, and acceptable tool life. These objectives can be achieved by reducing the cutting temperature in the cutting zone. In metal cutting, high cutting temperatures are generally reduced by the proper selection of process parameters, proper selection and application of coolants and lubricants, and using heat- and wear-resistant cutting tool materials—including special coatings. Conventional coolants fail to provide desirable control of cutting temperature in the cutting zone, and they also create some techno-environmental problems such as disposal and dermatological problems to the user. These problems were eradicated using liquid nitrogen as a coolant in cryogenic machining (Fig. 7.20) due its excellent cooling ability along with environmental friendliness [60
66]. Cryogenic titanium machining can increase cutting tool life by up to a factor of 10 and double the material removal rate, compared to conventional machining methods in certain applications [67]. Modeling and

Figure 7.20 Model [60] for cryogenic cooling in cutting.

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Figure 7.21 Temperature distribution around the cutting edge for cryogenic cutting.

simulation of the chip-formation process helps one to identify the best available parameters and to reduce the number of shop floor tests (Fig. 7.21).

7.5

Open issues

7.5.1 Hybrid modeling and closed-loop design Hybrid modeling is the combination (or extension) for any combination of analytical, numerical, experimental, or communication engineering methods. Well known is the combination of the slip-line approach with FEM simulations [68], the hybrid method FEM simulation with viscoplasticity [69
73] and previously discussed modeling techniques together with AI-based methods [74]. This philosophy has been introduced as “house of models” [2]. The most important deficit of knowledge is already in the system approach of modeling and simulation. As discussed in the previous sections, there is an exceptionally great step forward in physically based modeling and simulation for single solutions. The next step forward is expected to be for a closed-loop design. There are a few examples discussed in the following sections.

7.5.1.1 Clamping devices and machining The reduction of overall costs, lead times and improvement of product quality is one of the main goals in manufacturing today. At the same time, innovative materials, including titanium alloys, composites, and metal
composite sandwiches, are being used for assembling lighter and increasingly more complex parts. In manufacturing

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processes, fixturing is used for accurate location and tight holding of the workpiece. The ability to establish and secure desired positions is significant in assessing the effectiveness of a fixture system. Further, product differentiation increases the need of highly efficient, flexible, accurate, and automatic fixing systems in industry. Especially, in the aerospace and automotive sector the number of low volume, high value, and difficult-to-handle products is increasing. Specialized fixtures are economic only within mass production and cannot be reconfigured easily; otherwise the design and construction of modular fixtures are very time-consuming and expensive, with problems in repeatability and positioning accuracy. Furthermore, quality and precision cannot be controlled and influenced actively during the manufacturing process and during loading/unloading. Thus there is a need of active, efficient, flexible, and precise fixturing systems, which are the fields of research in many sectors of industry nowadays. As a basis of all development processes, the forces acting upon the fixing components during the manufacturing process have to be analyzed. An iterative improvement of fixture prototype designs is still common and causes high development costs. Consequently, an effective modeling and simulation method for the prediction of the workpiece
fixture behavior during machining is required, which would also enable the possibility of fixture optimization. Therefore the FEM is an adequate tool to be used for the simulations. In Refs. [75
77] a complex FEM model with a huge number of elements has been created followed by a parametric model with similar mechanical properties but fewer elements. The usability of this closed-loop method has been revealed by forecasting the transverse reaction forces acting on knee levers fixing a nozzle guide vane (Fig. 7.22) during shape grinding. With continuum mechanical

Figure 7.22 A nozzle-guide-vane part.

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Apparatus (CAD model) Boundary conditions Complex FE model

Material properties

• Geometric dimensions • Grinding forces

Simplified FE model

• Clamping forces • Temperature

Parameters

Analytical model

• Geometry • Forces

Grinding force control

• Material

(MARLAB/SIMULINK)

dSPACE

Figure 7.23 Closed-loop simulation system applied to a nozzle guide vane.

principles and a parametric FEM model, the calculation time has reduced dramatically (closed-loop simulation Fig. 7.23).

7.5.1.2 Closed-loop cutting tool design The current engineering of tools often proceeds as follows: a model of the tool is developed and a prototype is ground and analyzed. Results are then evaluated by development engineers together with costumers who make appropriate changes to the tool geometry. A new prototype is developed, and the cycle is repeated. Each single iteration demands high costs in terms of personnel, materials, and machine time. For a special type of cutting tool applied in high-performance cutting operations, an integrated simulation system has been developed and tested. The closed-loop system (Fig. 7.24) enables users to relocate the prototyping process from the machine tool onto a personal computer, keeping machine tests to a minimum. This system has been tested for the multivalent applicable ratio milling tool. The time-to-market was strongly reduced and the same methodology can be used for other types of cutting tools [78].

7.5.2 Multiscale modeling in machining Going back into the history, micromachining was and is an important topic within manufacturing [79,80]. The modeling task differs from other metal-cutting simulation methods in the following main areas (geometry, material structure; friction; dislocations; etc.). Fig. 7.25 demonstrates the geometrical properties for a sharp cutting tool (with a feed to cutting-edge radius of about 60—left) and a worn cutting tool (with the feed to cutting-edge radius of about 0.5—right). Looking deeper in to the material structure (Fig. 7.26) for macro-machining (the cutting-edge radius is very small compared with the chip
tool contact length and with the material structure—left) and for micromachining (the cutting tool radius is almost close to the material structure —right). In addition, the friction conditions along the cutting tool edge are more different than conventional (macro-)machining (Fig. 7.27).

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Figure 7.24 Closed-loop design of cutting tools.

Figure 7.25 Macro- and micromachining—sharp and worn tool.

It must be pointed out that up until now there has been no final model or simulation code available to investigate micro-cutting. Investigations into the material flow with the micro-viscoplasticity method have been used to verify numerical investigations— but they are very time-consuming [81]. In addition, there are a lot of investigations dealing with the application of MD—in combination with FEM simulation [82].

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Figure 7.26 Macro- and micromachining—influence of material.

Figure 7.27 Cutting-edge surface and roughness.

7.5.3 Multiscale modeling in coating-substrate simulation The geometrical differences between the well-known boundary conditions at the cutting tool clamping area, in the machine tool spindle, and at the chip
tool interaction area at the coating-substrate system are about 100 mm to 1 nm. The main four levels for investigation of CSS are as follows: G

G

G

G

the macroscopical level
continuum mechanics . macro-FEM; the microscopical level
continuum mechanics . micro-FEM; the materials science
based level . quasicontinuum method crystal plasticity and atomistic FEM; and the atomic-scale level . MD and “first-principles methods.”

The commonly used method is the well-known FE method for the macroscopically and microscopically coating-substrate investigation. The basic research at this time is focused on developing the methods separately on the one hand and finding methods to close the gap between these methods on the other (Fig. 7.28). First steps to close the gap between the four levels of simulation have been presented [8]. The goal of the European project [83,84] was to develop an integrated multiscale modeling approach to link MD crystal plasticity and continuum mechanics modeling activities for the applications. A new advanced adaptive FE code [85] has been used to investigate the continuum mechanics modeling of typical cutting tools.

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Figure 7.28 Top-down concept for a multiscale modeling of coating-substrate systems [78].

MD techniques have been applied to model atom deposition processes and the atomic structure and interfaces to achieve optimal coating microstructures. The MD method was used to simulate the formation of lattice structures of coatings and layer interfaces in thermochemical processes and the results were used as input for nano- and micro-mechanics modeling for defining material structures. Continuum diffusion models are being used to simulate such processes as plasma nitriding, plasma carburizing, and plasma nitrocarburizing. These surface modification processes have been widely used in the design of multilayered surface systems to improve their load supporting capacity. The outcomes have been used to define material structures as inputs for mechanics analysis. New physically based constitutive equations based on evolving microstructural state variables (e.g., dislocation density) were developed to predict damage. These, incorporated with the newly developed controlled Poisson Voronoi tessellation technique for generating virtual grain structures, enable the load-bearing capacity of polycrystalline films and multilayers to be predicted. In parallel, mesoscopic (called atomic) and crystal plasticity FEM modeling capabilities and facilities for predicting deformation and failure of individual layers of nano/micro-coatings have been applied. The “atomic” lattice model is solved by FE methods. Elements are used like bars or trusses. The behavior of the bar is described by an interatomic potential. This enables an atomic lattice to be modeled by a truss-like assemblage of members that deform according to a prescribed interatomic potential, which governs their relative separations under applied loading. A key feature of the atomic FE method is that it results in an FE equation with displacement degrees of freedom; hence, an atomic FE mesh can be directly coupled to other element types and larger scales. Discrete dislocation models have also been used to model at scales between the atomic and crystal scales, with concurrent coupling. The lattice structure of coatings, atomic mismatch, and dislocations at

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layer interfaces modeled at the atomistic level is used for the FEM models. Therefore an integrated technique to determine mechanical properties of nano- and microscale coatings has been developed [86
91].

7.6

Summary

This chapter presents a comprehensive summary of the state-of-the-art developments in the modeling of machining operations with special consideration of the application of models and simulation techniques in the daily job in an industrial environment as well as an outlook to the next steps in modeling. As seen, significant advances have been made in developing advanced computational tools and analytical methods for a physically based modeling at the single process level and its application in the industry. There is a lack in modeling of closed-loop design, which could be one of the main interesting topics within the next decade. Some other interesting aspects such as friction in cutting, residual stress topics, part distortion due to cutting and initial residual stresses because of previous manufacturing steps, cutting tool wear task, material law design, and aspects of machining stability have been excluded in this review—but are actually existing in the literature [21,92,93].

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[15] H. Weber, J. Leopold, Acta Tech. Acad. Sci. Hung. 86 (3
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144 (2001) 916
922. [26] J. Leopold, M. Meisel, Report on HPS-CSS (HPS-CSS has Been Supported by the ESPRIT HPCN Programme from the European Commission), 2001 (unpublished). [27] H. Oosterling, et al., Low friction, MOS2-composite coated cutting tools for dry, high speed machining of steel, in: LoFriCo Final Technical Report—BRST-CT98-5361, 2001 (unpublished). [28] J. Leopold, Mechanical and thermal behaviour of coating-substrate-systems investigated with parallel FEM, in: Proceedings of the International Conference on Metallurgical Coatings and Thin Films
ICMCTF 2002, San Diego, CA, April 22
26, 2002. [29] L.C. Cho, C. Abhijit, A boundary element method analysis of the thermal aspects of metal cutting processes, Trans. ASME, J. Eng. Ind. 113 (1991) 311
319. [30] E.P. Stephan, Chapter 13: Fundamentals, in: E. Stein, R. de Borst, T.J.R. Hughes (Eds.), Coupling of Boundary Methods and Finite Element Methods, Encyclopedia of Computational Mechanics, 1, John Wiley & Sons, 2004. [31] J. Kuhnert, A. Mattes, J. Leopold, Internal Report, FhG IWU; FhG ITWS and TU Berlin, 2005. [32] J. Kuhnert, A. Tramecon, P. Ullrich, Proceedings of the EUROPAM Conference, 2000. [33] S.S. Akarca, W.J. Altenhof, A.T. Alpas, Proceedings of the 10th International LSDYNA Users Conference, 2008. [34] E. Uhlmann et al., Proceedings of the CIRP Conference on Modelling of Machining Operations, Donostia-San Sebastian, 2009, pp. 145
151. [35] V. Gyliene, V. Ostasevicus, M. Ubartas, Proceedings of the Ninth European LS-DYNA Conference 2013 and Proceedings of the 12th CIRP Conference on Modelling of Machining Operations, 2009, pp. 145
151. [36] C. Espinosa, et al., Proceedings of the 10th International LS-DYNA Users Conference, 2008. [37] N. Ikawa, et al., Ann. CIRP 40 (1) (1991) 551
554. [38] S. Shimada, et al., Feasibility study on ultimate accuracy in microcutting using molecular dynamics simulation, Ann. CIRP 42 (1) (1993) 91
94. [39] R. Rentsch, I. Inasaki, Ann. CIRP 44 (1) (1995) 295
298.

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[40] H. Tanaka, S. Shimada, L. Anthony, Requirements for ductile-mode machining based on deformation analysis of mono-crystalline silicon by molecular dynamics simulation, Ann. CIRP 56 (1) (2007) 53
56. [41] J. Leopold, Werkzeuge fu¨r die Hochgeschwindigkeitsbearbeitung, HANSER, 1999. [42] J. Leopold, et al., Fliehkraftverhalten von Feinbohrwerkzeugen bei hohen Drehzahlen, VDI-Z III, Special “Werkzeuge,” 1996, pp. 48
50. [43] J. Leopold, et al., Festigkeits- und Verformungsanalyse von Werkzeugen fu¨r die Hochgeschwindigkeitsbearbeitung, DIMA 12 (1997) 33
39. [44] J. Leopold, G. Schmidt, A. Kieninger, FEM-Analyse modular aufgebauter HSCWerkzeuge mit 3-D einstellbaren Schneiden, DIMA 3 (2000). [45] ,http://www.hsm.tu-darmstadt.de/index_hsm/conference_program_hsm/conferenceprogram_hsm.de.jsp.. [46] ,http://www.rcmt.cvut.cz/hsm2014/en/.. [47] ,http://hpc2014.berkeley.edu/.. [48] D.A. Dornfeld, S.L. Ko, A study on burr formation mechanismTrans. ASME J. Eng. Mater. Technol. 113 (1991) 75
87. [49] J. Leopold, Prediction and verification of models of burr formation, Int. J. Mater. Prod. Technol. 35 (1/2) (2009) 89
117. [50] A. Freitag, C. Sohrmann, J. Leopold, Simulation of burr formation, in: Proceedings of the Eighth CIRP International Workshop on Modeling and Machining Operations, 2005, pp. 641
650. [51] A. Stoll, J. Leopold, R. Neugebauer, Hybrid methods for analysing burr formation in 2D-orthogonal cutting, in: Proceedings of the Ninth CIRP International Workshop on Modelling of Machining Operations, May 11
12, 2006, Bled, Slovenia, 2006. [52] J. Leopold, G. Schmidt, K. Hoyer, A. Stoll, Modelling and simulation of burr formation
state-of-the-art and future trends, in: Proceedings of the Eighth CIRP International Workshop on Modelling of Machining Operations, May 11
12, 2005, Chemnitz, Germany, 2005. [53] J. Regel, A. Stoll, J. Leopold, Numerical analysis of crack propagation during the burr formation process of metals, Int. J. Mach. Mach. Mater. 6 (1/2) (2009) 54
68. [54] A. Stoll, N. Ahmed, A.V. Mitrofanov, V. Silberschmidt, J. Leopold, Influence of ultrasonically assisted cutting on burr formation, in: Proceedings of the Ninth CIRP International Workshop on Modelling of Machining Operations, May 11
12, 2006, Bled, Slovenia, 2006. [55] M. Dix, J. Leopold, R. Neugebauer, Investigations on the influence of local material properties of burr formation, in: Proceedings of the 10th CIRP International Workshop on Modeling of Machining Operations, August 27
28, 2007, Reggio Calabria, Italy, 2007. [56] J. Leopold, A. Mucha, Non-stick coating for clean manufacturing
cleanability in high-performance cutting, in: Proceedings of the Conference NANOFAIR, Karlsruhe, 2006. [57] J. Leopold, R. Neugebauer, M. Lo¨ffler, M. Schwenck, P. H¨anle, Influence of coatingsubstrate-systems on chip and burr formation in precision manufacturing, Proc. Inst. Mech. Eng., B: J. Eng. Manuf. 219 (2005) 1
8. [58] J. Leopold, T. Matsumura, Modelling of burr formation of coated-cutting tools for clean manufacturing, in: Proceedings of the Fifth International Conference on Leading Edge Manufacturing in 21st Century
LEM21, Osaka, 2009. [59] T. Matsumura, J. Leopold, Simulation of drilling process for control of burr formation, J. Adv. Mech. Des. Syst. Manuf. 4 (5) (2010) 966
975.

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[60] S.Y. Hong, I. Markus, W. Jeong, New cooling approach and tool life improvement in cryogenic machining of titanium alloy Ti-6Al-4V, Int. J. Mach. Tools Manuf. 41 (2001) 2245
2260. [61] R. Ghosh, Z. Zurecki, J.H. Frey, Cryogenic machining with brittle tools and effects on tool life, in: Proceedings of IMECE’03, Paper No. ICMECE 2003-42232, 2003. [62] M. Dhananchezian, M. Pradeep Kumar, A. Rajadurai, Experimental investigation of cryogenic cooling by liquid nitrogen in the orthogonal machining process, Int. J. Recent Trends Eng. 1 (5) (2009) 55
59. [63] S.C. Jun, Lubrication effect of liquid nitrogen in cryogenic machining friction on the tool-chip interface, J. Mech. Sci. Technol. (KSME Int. J.) 19 (4) (2005) 936
946. [64] F. Puˇsavec, A. Stoi´c, J. Kopaˇc, The role of cryogenics in machining processes, Tech. Gaz. 16 (4) (2009) 3
10. [65] F. Puˇsavec, J. Kopaˇc, Sustainability assessment: cryogenic machining of inconel 718, Strojniˇski vestnik
J. Mech. Eng. 57 (9) (2011) 637
647. [66] T. Lu, O.W. Dillon Jr., I.S. Jawahir, A thermal analysis framework for cryogenic machining and its contribution to product and process sustainability, in: Proceedings of the 11th Global Conference on Sustainable Manufacturing
Innovative Solutions, Berlin, 2013, pp. 262
267. [67] ,http://www.mag-ias.com/en/mag/technologies.html.. [68] J. Leopold, W. Arnold, H. Gru¨ndemann, Ein integriertes Gleitlinien-Finite-ElementeModell zur Spannungsberechnung vor der Scherlinie, Wiss. Z. d. Techn. Hochschule Karl-Marx-Stadt 21 (2) (1979) 185
189. [69] G. Spur, J. Leopold, G. Schmidt, Ermittlung des Spannungs-, Verformungs- und Temperaturverhaltens spanned bearbeiteter Werkstu¨cke mit Hilfe der Visioplastizit¨at und der Finiten Elemente Methode, in: Proceedings of the DFG Priority Prpogramm “Wirkfl¨achenreibung bei inelastischer Verformung metallischer Werkstoffe”, Hannover, 1995. [70] J. Leopold, et al., Wiss. Zeitschrift der TH Karl-Marx-Stadt 21 (2) (1979) 185
189. [71] J. Leopold, FEM modelling and simulation of 3D-chip formation, in: Proceedings of the CIRP International Workshop on Modelling of Machining Operations, Atlanta, 1998. [72] J. Leopold, G. Schmidt, Proceedings of the Second CIRP International Workshop on Modelling of Machining Operations, Nantes, January 25
26, 1999. [73] J. Leopold, Proceedings of the Third International Workshop on Modelling of Machining Operations, Sydney, 2000. [74] X.P. Li, K. Lynkaran, A.Y.C. Nee, A hybrid machining simulator based on predictive machining theory and neuronal network modelling, J. Mater. Process. Technol. 89
90 (1999) 224
230. [75] M. Kl¨arner, J. Leopold, L. Kroll, Analysis of clamping within a fixing system, Lect. Notes Comput. Sci. 5315 (2008) 356
367. [76] J. Leopold, et al., Investigations to new fixturing principles for aerospace structures, in: Proceedings of the International Conference on Applied Production Technology, Bremen, 2007. [77] AFFIX consortium. Affix
aligning, holding and fixing flexible and difficult to handle components. ,http://www.affix-ip.eu., 2006. [78] J. Leopold et al., High performance cutting with optimized cutting tools, in: Proceedings of the Fourth CIRP International Conference on High Performance Cutting, Gifu, 2010. [79] M. Sato, K. Kato, K. Tuchiya, Effect of material and anisotropy upon the cutting mechanism, Trans. JIM 9 (1978) 530
536.

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[80] M. Abouridouane, F. Klocke, D. Lung, O. Adams, A new 3D multiphase FE model for micro cutting ferritic
pearlitic carbon steels, CIRP Ann. Manuf. Technol. 61 (1) (2012) 71
74. [81] J.W. Erben, Mikrovisioplastische Untersuchungen an ein- und zweiphasigen metallischen Werkstoffen als Bindeglied zur numerischen Modellierung der Mikrostruktur, in: Abschlussbericht DFG Le746/17, 1999. [82] G. Schmidt, R. Leopold, R. Neugebauer, FE-simulation of nonlinear dynamical effects in coating-substrate-systems, in: Fourth International Symposium: Investigations of Non-Linear Dynamic Effects in Production Systems, April 8
9, 2003, Chemnitz, Germany. [83] FP7 Project, “Multiscale Modelling for Multilayered Surface Systems” (M3-2S), Grant No: NMP3-SL-2008-213600, 2008. [84] ,http://ec.europa.eu/research/industrial_technologies/pdf/modelling-brochure_en.pdf.. [85] J. Leopold, et al., An advanced adaptive finite element code for coating-substrate simulation, J. Multiscale Modell. 3 (2011) 91. [86] A.V. Byakova, J. Leopold, Effect of Stress State on Failure Resistance of Brittle HighStrength Coatings, 1996 (unpublished report). [87] J. Leopold, R. Wohlgemuth, D. Shan, J. Lin, Y. Qin, Modelling and simulation of coating-substrate-systems: state-of-the-art and future trends, in: Proceedings of the Conference “TheA Coatings”, Thessaloniki, 2011. [88] J. Leopold, R. Wohlgemuth, J. Lin, S.V. Subramanian, T. Matsumura, New concepts for micro-structural simulations of coating-substrate-systems, in: Proceedings of the 12th CIRP Conference on Modelling of Machining Operations, vol. 1, May 7
8, 2009, Donostia, San Sebastia´n, Spain, 2009, pp. 117
124. [89] S. Wang, J. Lin, D. Balint, Modelling of failure features for TiN coatings with different substrate materials, J. Multiscale Modell. 03 (2011) 49. [90] D.Q. Qin, et al., Prediction of residual stress in multilayered coatings with a linear elastic model incorporating density functional theory calculations, J. Multiscale Modell. 03 (2011) 65. [91] R. Neugebauer, R. Wertheim, U. Semmler, The atomic finite element method as a bridge between molecular dynamics and continuum mechanics, J. Multiscale Modell. 03 (2011) 39. [92] J. Leopold, H. Gru¨ndemann, W. Totzauer, Kontinuumsmechanische Methoden zur Modellierung des Spanbildungsprozesses, Sitzungsberichte der AdW der DDR, 12N, 1979, pp. 51
81. [93] C.A. Luttervelt, et al., Present situation and future trends in modelling of machining operations, Ann. CIRP 47 (2) (1998) 587
626.

Further reading W. Ko¨nig, N. Spenrath, The influence of the crystallographic structure of the substrate material on surface quality and cutting forces in micromachining, in: Proceedings of the Sixth International Precision Engineering Seminar, Braunschweig, Germany, 1991, pp. 141. J.D. Kim, D.S. Kim, Theoretical analysis of micro-cutting characteristics in ultra-precision machining, J. Mater. Process. Technol. 49 (1995) 387
398.

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W.B. Lee, C.F. Cheung, S. To, Characteristics of microcutting force variation in ultraprecision diamond turning, Mater. Manuf. Processes 16 (2) (2001) 177
193. W.B. Lee, C.F. Cheung, S. To, A microplasticity analysis of micro-cutting force variation in ultra-precision diamond turning, Trans. ASME 124 (2002) 170
177. A. Simoneau, E. Ng, M.A. Elbestawi, Grain size and orientation effects when microcutting AISI 1045 steel, CIRP Ann. Manuf. Technol. 56 (1) (2007) 57
60. V. Schulze, J. Michna, F. Zanger, R. Pabst, Modelling the process-induced modifications of the microstructure of work piece surface zones in cutting processes, Adv. Mater. Res. 223 (2011) 371
380.

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Mechanics of ultrasonically assisted drilling

8

Anish Roy and Vadim V. Silberschmidt Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, United Kingdom

8.1

Introduction

Machining is one of the most common and important manufacturing processes known to man. Traditionally, it implies the removal of material from a workpiece to obtain a desired geometric shape. Thus it becomes imperative to assess the quality and performance of the surface and the machined product affected by the thermomechanical loading imposed on the surface and the subsurface of the workpiece. Hole drilling, reaming, and tapping are typical machining operations that are carried out to facilitate assembly of parts. These processes are predominantly cycletime defining operations as they are performed usually after the part is made with many parts requiring a large number of holes (e.g., aerospace components for riveting). The process of drilling typically includes large plastic deformations at low cutting speeds, but thermal effects become predominant in high-speed cutting. Drilling in polymeric composites or composite/metal stacks is particularly challenging. The need for high-accuracy and good-quality holes is particularly high as it is essential for good assembly of parts. Once the component is in use, load transfer via the bolted joints becomes critical with the holes being subjected to intense localized stress. Thus any defect during hole drilling will further exacerbate part rejection and compromise their fatigue life. This becomes particularly severe from the economic point of view since assembly occurs in the last phases of production when the manufacturing cost of the part has already been faced. Kinematically, drilling is a complex process, which requires sophisticated tooling. As a result, productivity gains in drilling lagged those made in turning and milling over the past 30 years. Drilling tools were improved upon and optimized over the years with the advent of cubic boron nitride (CBN) tipped and coated drills to improve machining productivity. To radically change the hole-making process, hybrid machining techniques were proposed, which are essentially variants of an advanced machining schemes combined with a conventional machining process in order to achieve outcomes not possible with the individual process in isolation. Several such hybrid machining processes were introduced in the industry. Ultrasonically assisted drilling (UAD) is one such process, which has a huge potential and is the subject of discussion in this chapter. Other noteworthy techniques Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques. DOI: https://doi.org/10.1016/B978-0-12-818232-1.00008-4 Copyright © 2020 Elsevier Ltd. All rights reserved.

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include abrasive jet machining, electron beam machining (typically at very small length scales), electrochemical machining, and laser beam machining [1].

8.2

Drilling: theory and modeling

Here, some of the pertinent machining theory and relevant modeling of the drilling process is reviewed. In drilling a drill-tool design is an important aspect, which significantly affects the machining performance. Usually, tools are designed to yield a long tool life with high penetration rates. In addition to a common two-flute twist drill, there are several other types in use including flat drills; drills with one, three, or four flutes; and core, shell, and spade drills. The two-flute twist drill is by far the most commonly used tool in the industry. Analysis of the machining process is crucial to improve the drilling process, especially for high-value components and intractable materials, where an extensive program of experimental trials can be prohibitively expensive [2]. First, some kinematic models (sometime referred to as mechanistic models) are reviewed, followed by numerical approaches incorporating finite-element (FE) analysis in machining studies.

8.2.1 Kinematic modeling Drilling performance depends on the materials involved, the drill geometry, the spindle speed, and the feed rate. The cutting speed V at the periphery of the drill is given by V 5 πDN; where D is the drill diameter, and N is the rotational velocity (in rpm) of the spindle. The feed rate fr , feed per revolution f , and the feed per tooth ft are related via fr 5 fN; fr 5

f ; nt

where nt is the number of flutes. The material removal rate, Q, is obtained from Q5

πD2 fr : 4

Kinematic models are used to calculate cycle times or cutting forces and/or power consumed. Estimating drilling forces are typically based on models inspired

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from oblique machining in metals and alloys [2
6]. The drilling thrust force, T, is typically represented for a material in terms of its Brinell hardness, HB as   c 2  f 0:8 T 5 D3 HB 6:962 1:2 1 0:68 ; D D where c is the length of the chisel edge of the drill bit. The drilling torque, M, is determined from   f 0:8 M 5 D HB 0:0031 1:2 : D 3

Some kinematic modeling approaches explicitly account for the complex drill geometry [7]. The tool geometry is split into parts, which consider the specific cutting angles of each. The first is the primary cutting edge, followed by the chisel edge, which consists of the secondary cutting edge and the drill tip. At the drill center an indentation process occurs, which induces strong plastic deformation in the material. The secondary cutting edge, with a negative rake angle, cuts the material. The primary cutting edge, which is responsible for most of material removal, has a changing positive rake angle associated with a variable cutting speed. For each zone an independent thrust force model is identified, which is then combined linearly to obtain the macroscopic thrust force, T. The thrust force from the primary cutting edge, F1 , is obtained from F1 5 nc K1 ðR 2 R2 Þhq1 ; where nc is the number of cutting edges, K1 is the specific cutting coefficient, q1 is a coefficient depending on the pair “drill
workpiece material,” h the uncut chip thickness, R is the drill radius (in mm), and R2 is the radius of secondary cutting edge (see Fig. 8.1). The coefficient K1 is related to the cutting speed in addition to

Figure 8.1 Schematic of a typical drill showing various cutting regions.

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the workpiece material and the tool geometry. Since the cutting speed varies from 0 at the drill center to its maximum at the periphery, an integral equation needs to be solved to determine F1 . At the secondary cutting edge, as its size is typically small (about 1/10 of the radius), a macroscopic relation is derived for the thrust force, F2 , as F2 5 nc K2 ðR2 2 R3 Þhq2 ; where K2 and q2 are coefficients, and R3 is the radius of the indentation zone. Finally, the thrust force from the indentation zone, F3 , is determined from F3 5 K3 R3 hq3 ; where K3 and q3 are coefficients. Thus the total thrust force is evaluated as, T 5F1 1 F2 1 F3 : Following this, extensive experimental studies need to be carried out to assess the parameters in the model.

8.2.2 Finite-element modeling Numerical modeling approaches have gained a significant exposure and popularity thanks to a rapidly reducing cost in performing large-scale computations. FE modeling techniques offer a framework to develop such numerical models, which can be used to predict the drilling thrust force and torque, as well as drillinginduced damage in the material. The numerical models are developed to mimic the true experimental conditions as closely as possible though some assumptions need to be made to simplify the model to speed up the analysis or make it feasible for available computational resources. Thus an appropriate balance between numerical accuracy and computational efficiency need to be made early in the model building step.

8.2.2.1 Material modeling Material modeling is a crucial and complex step in simulations of machining processes. In modeling the drilling process of carbon/epoxy composites, a user-defined 3D damage model (VUMAT) with solid elements was developed and implemented into a general-purpose FE code to predict the characteristics of damage through the composite laminate. Interface cohesive elements were inserted between the plies of the laminates to simulate delamination. Modeling damage typically starts with homogenized ply properties with interplay behavior and information about the laminate layup so that progressive failure in composites can be assessed. Intraply and interply damage mechanisms are introduced separately together with phenomenological models characterizing a complex interaction between them. An elementdeletion criterion [8] was used to represent the hole-making process based on initiation and evolution of damage in the FE mesh. That is, an FE was removed from the mesh as the threshold level of stresses, primarily in fiber direction,

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was achieved. Hashin failure criterion showed to work adequately in characterizing damage in unidirectional (UD) composite, especially fiber failure [9]. The damage model proposed by Puck and Schu¨rmann [10] demonstrated predictive capabilities of several failure criteria in UD laminates, especially matrix failure. Thus a combination of both criteria is ideal for modeling damage in the composite [8].

8.2.2.2 Finite-element model setup A 3D FE model of drilling was developed, consisting of a Jobber carbide twist drill bit, a cross-ply composite laminate and a backing plate with appropriate boundary conditions as shown in Fig. 8.2. A 2 mm-thick laminate in the model consisted of 16 plies with an individual ply thickness of 0.125 mm. The drill and the backing plate were modeled as discrete rigid bodies. The drill moved into the workpiece in the axial direction using a velocity boundary condition, representing the feed rate used in experiments. An angular velocity about the drill axis equivalent to the spindle speed of 2500 rpm was superimposed on the drill geometry. In the composite, each ply was represented by two eight-node linear brick elements through its thickness in the vicinity of the drilling zone, and two six-node triangular wedge elements away from it. A planar mesh size of 0.25 mm 3 0.25 mm (in 1
2 plane, see Fig. 8.2) in the vicinity of the drilling area was used, with a coarser mesh away from the zone of interest. At each ply interface, cohesive elements of type COH3D8, with a thickness of 10 μm, were embedded and used to model delamination initiation and growth with the failure criterion. A general contact condition was imposed between the tool and workpiece with a constant coefficient of friction of 0.3 [11].

Figure 8.2 A finite-element model of drilling in composite laminates.

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8.2.2.3 Results A comparison of the thrust force and torque is shown in Fig. 8.3; a reasonable match between experiments and modeling was achieved. This indicates that the underlying material model is correct. The delamination profiles obtained with X-ray micro
computed tomography for drill entry and exit were processed using an image-processing code developed in MATLABs , and the corresponding delamination factor was calculated. The results of FE simulations of delamination were compared qualitatively (Fig. 8.4). The delamination factors in the FE analysis were calculated using a simple methodology, where the ratio of a total number of cohesive elements before and after the drilling simulation was calculated using a python script. A reasonable match with experimental results was obtained. Thus such a modeling approach can be utilized to determine optimal machining parameters in (B) 300

(A) 200

150

100

50

Torque (experiment) Torque (FE analysis)

250

Torque (N cm)

Thrust force (N)

Thrust force (experiment) Thrust force (FE analysis)

200 150 100 50

0 0.0

0.5

1.0

1.5

Specimen depth (mm)

2.0

0 0.0

0.5

1.0

1.5

2.0

Specimen depth (mm)

Figure 8.3 Comparison of numerical results with experimental data for thrust force (A) and torque (B).

Figure 8.4 Delamination analysis for drill entry and exit: (A and B) experimental data and (C and D) numerical estimates.

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order to alleviate damage in the workpiece material. As an example, optimal drill geometry may also be determined with this approach, ensuring damage is kept to a minimum for some given predetermined drilling feed (and speed).

8.3

Ultrasonically assisted drilling

UAD is a hybrid machining process, in which high-frequency (typically .18 kHz) and low-amplitude (,20 μm) vibration is superimposed on a movement of a standard twist drill bit in the axial direction. The technique was used to improve the overall machinability of advanced materials. Prior studies indicated that vibrating the drill bit in the axial direction yielded the maximum reduction in drillinginduced damage. There are several well-documented advantages of UAD over conventional drilling (CD) techniques such as reduction in thrust forces and torque, better surface finish, low tool wear, and elimination/reduction in burr formation [12]. The thrust force has a direct effect on drilling-induced damage in composites; hence, it is considered to be the main parameter affecting quality of a drilled hole.

8.3.1 Experimental setup and instrumentation A universal lathe (Harrison M-300) was suitably modified to accommodate an ultrasonic transducer (Fig. 8.5); this allows the flexibility of conducting UAD and CD experiments using the same setup. The hybrid machining setup comprised a Langevin-type piezoelectric ultrasonic transducer that converts an imposed electric voltage into high-frequency mechanical vibrations.

Figure 8.5 Experimental setup.

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A typical transducer consists of a stack of piezoceramic elements, a front waveguide (often referred to as the horn) and a backing mass. The vibration amplitude generated in the piezoelectric element is too small for viable applications; hence, the geometry of the transducer’s components needs to be appropriately modified to amplify the vibration amplitude with an appropriate design of the waveguide [13]. Initial designs are typically developed based on well-known analytical relations, available for various horn geometries [14]. However, numerical approaches, which account for all the nuances of the geometry and material properties, are preferred to yield accurate practical results. This design is typically carried out via modal analysis to assess optimal resonant frequencies and the mode of vibration. Mode separation is an important aspect of such designs, and the goal is to have as large a separation as possible for practical use. In the experimental study the waveguide was manufactured to ensure a good acoustic coupling between the cutting tool and the horn material. The transducer was fastened and mounted on the headstock of the lathe using a four-jaw universal chuck (Fig. 8.5). In UAD experiments the frequency was tuned and set at a particular resonance level with the help of GW Instek (model SFG-2110) functional generator. The transducer was designed to resonate at a frequency of 27.8 kHz with the attached drill bit. The vibration amplitude was measured at the tooltip using a Polytec laser vibrometer (OFV-3001) with the capability of characterizing vibrations up to a velocity of 10 m/s with a resolution of 2 μm=s. The frequency and amplitude were tuned for the drill tip in free vibration as it is not possible to do this in the drilling process because of the lack of direct line of sight once the drill tip is fully engaged with the workpiece. All the experiments were conducted at the tuned frequency and amplitude. A cooling system consisting of a vortex-tube cold-air gun, a pressure regulator and a gauge meter was connected to deliver dry compressed air to the front face of the stack as shown in Fig. 8.5 and used when needed.

8.3.1.1 Measurement of drilling forces A two-channel Kistler (Model 9271A) dynamometer was bolted to an angle plate fixed on the cross slide of the lathe. The dynamometer measures the thrust force and torque during the drilling process up to 20 kN and 10 kN cm, respectively, with a maximum acquisition frequency of 3 kHz. Since this frequency is significantly lower than that of the imposed ultrasonic vibration during UAD, an average level of drilling forces was acquired in the machining process. The dynamometer and the drill bit were aligned to be coaxial. Force signals from the dynamometer were enhanced using a charge amplifier and converted and transmitted to the computer via an analog
digital converter (digital oscilloscope Picoscope). The acquired data were further processed using MATLAB.

8.3.1.2 Surface metrology Postdrilling analysis of the workpiece was conducted to evaluate the quality of the drilling process, in particular, hole circularity and/or surface roughness of the hole

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as well as delamination in the composite. In addition, tool wear was also studied. To determine hole circularity the Metris LK Ultra 627134 CMM with SP25 Ø1 mm stylus was used. The samples were positioned on the base table of the CMM with the entrance face upward. The specimens were leveled, and circularity was measured at various depths. The surface roughness of the drilled hole surface was measured using the Taylor Hobson CLI 2000 system based on noncontact measurement techniques. The drilled holes were cut in half through their thickness, and measurements were taken on the central part of each side of the cut.

8.3.2 Case study: drilling in composites Here we present a case study involving aerospace grade materials to demonstrate the efficacy of the hybrid machining process in yielding improved machinability of parts.

8.3.2.1 Workpiece material: carbon/epoxy composites The workpiece material used was an M21/T700 carbon-fibre-reinforced plastics (CFRP) provided by Airbus. The quasiisotropic composite was composed of 34 plies with the stacking sequence of [(0/45/90/ 2 45 degrees)4s/0 degree]s. The plates of CFRP (thickness: 10 mm) were cut into pieces with a length of 200 mm; and width of 10 mm. The composite material was tested, and mechanical properties were evaluated via a series of experiments. The mechanical properties of the UD 0 degree ply (all the fibers in loading direction) are given in Table 8.1 [8].

8.3.2.2 Experimental studies and machining results Cutting forces The cutting forces imposed by the drilling tool on the workpiece were recorded for different machining feed rates for both UAD and CD. Typical thrust force and torque signatures for CD and UAD drilling are shown in Fig. 8.6. A significant reduction in the level of drilling torque was observed in UAD. For feed rates of 8 mm/ min and lower (Fig. 8.7), a large reduction in the nominal thrust forces and drilling torque in UAD was observed when compared to CD. In order for UAD to be effective, the drill tool should disengage from the workpiece in each vibratory cycle. This intermittent, vibro-impact character of the resulting process yields all the benefits of ultrasonic machining [15]. At higher feed rates the drill remains fully engaged with the workpiece in UAD, thus yielding effectively CD conditions resulting in the drilling forces identical to those in CD (see Fig. 8.7 for 20 mm/min). Table 8.1 Mechanical properties of CFRP plate for ultrasonically assisted drilling. E11 (GPa)

E22 5 E33 (GPa)

ʋ12 5 ʋ13

G12 5 G13 (GPa)

G23 (GPa)

115

14

0.29

4

3.2

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Figure 8.6 Evolution of thrust force and torque feed rate of 8 mm/min.

Figure 8.7 Comparison of magnitudes of thrust force (A) and torque (B) for CD and UAD with different feed rates. CD, Conventional drilling; UAD, ultrasonically assisted drilling.

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The cause of force reduction in UAD remains an open question. Our studies indicate that frictional effects play a minimal role in force reduction. This study additionally indicates that there is a fundamental change in the physical behavior of the material subjected to ultrasonic vibration. The character of the torque trace in UAD warrants some discussion and analysis. The fact that this trace fluctuates about the axis of abscissas at lower feed rates indicates that UAD is fundamentally different from CD. Due to the very nature of the drilling process, a positive torque is expected, as the drill flutes perform an essential part of the cutting process; this was estimated to be a major contributing factor to the overall energy consumption in a CD process [16]. The fluctuating torque may be attributed to the coiling
uncoiling effect of the drill tool in a UAD-induced coupled longitudinal and torsional mode of vibration at the purely axial excitation. The helical structure of the drill-bit flutes leads to winding/unwinding motion in the torsional direction when tension/compression and vibration occur in the longitudinal direction.

Chip formation A chip-formation process is often indicative of the underlying failure characteristics of the material. In CD experiments, small-sized dust-like chips, indicating the underlying brittle nature of the composite material (Fig. 8.8A
C), were observed. However, in UAD, the formed chips were long and helical, curled continuous chips similar to those in conventional machining of ductile metals (Fig. 8.8D
F). In UAD, it was observed that the chip length varied depending on the feed rate.

Figure 8.8 Optical microscopy of chips: (A) CD at 4 mm/min; (B) CD at 8 mm/min; (C) CD at 16 mm/min; (D) UAD at 4 mm/min; (E) UAD at 8 mm/min; and (F) UAD at 16 mm/min. CD, Conventional drilling; UAD, ultrasonically assisted drilling.

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Figure 8.9 Effect of feed rate on circularity for CD and UAD. CD, Conventional drilling; UAD, ultrasonically assisted drilling.

At lower feed rates, long chips were observed (Fig. 8.8D), whereas at higher feed rates, the chips were shorter (Fig. 8.8F).

Circularity Hole circularity was analyzed for UAD and CD with various feed rates (Fig. 8.9). At a lower feed rate of 4 mm/min, the circularity was improved in excess of 50% in UAD. However, with an increase in the feed rate, this improvement decreased to 30% (at 20 mm/min). It means that improvement in circularity was consistently observed in UAD for all the used machining parameters. An interesting observation can be made for the feed rate of 20 mm/min. Although the drilling forces showed no difference in UAD and CD, considerable improvements in circularity were still measured, indicating that ultrasonic drilling has the potential to demonstrate considerable advantage even at feed rates beyond the critical value.

8.4

Conclusion and outlook

Here, some of the underlying mechanics of CD were discussed. A novel dry drilling technique, UAD, was considered. Its advantages in terms of drilling forces and quality of drilled holes were studied and qualified. Among the notable advantages is a significant reduction in drilling forces when compared to conventional dry drilling. The holes drilled with UAD also showed better circularity. A brittle-to-ductile transition occurred in the composite material subjected to ultrasonic vibration. This affected the type of chips formed during the machining process. UAD is effective if tool separation occurs in each vibration. Thus the linear feed during drilling should not exceed a critical value to ensure effectiveness of the

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method. Overall, the technique shows considerable promise in drilling of hard-tomachine materials including composite/metal stacks.

References [1] D.A. Stephenson, D.A. Stephenson, Metal Cutting Theory and Practice, third ed., CRC Press, 2016. [2] M.C. Shaw, Metal Cutting Principles, second ed., Oxford University Press, 2005. [3] D.A. Stephenson, Material characterization for metal cutting force modeling, ASME J. Eng. Mater. Technol., 111, 1989, pp. 210
219. [4] D.A. Stephenson, J.S. Agapiou, Calculation of main cutting edge forces and torque for drills with arbitrary point geometries, Int. J. Mach. Tools Manuf. 32 (1992) 521
538. [5] H.T. Huang, C.I. Weng, C.K. Chen, Prediction of thrust and torque for multifacet drills (MFD), ASME J. Eng. Ind., 116, 1994, pp. 1
7. [6] K. Sambhav, P. Tandon, S.G. Dhande, Force modeling for generic profile of drills, ASME J. Manuf. Sci. Eng. 136, 2014. 041019-1, 1-9. [7] N. Guibert, H. Paris, J. Rech, C. Claudin, Identification of thrust force models for vibratory drilling, Int. J. Mach. Tools Manuf. 49 (2009) 730
738. [8] V.A. Phadnis, F. Makhdum, A. Roy, V.V. Silberschmidt, Drilling in carbon/epoxy composites: experimental investigations and finite element implementation, Composites, A: Appl. Sci. Manuf. 47 (2013) 41
51. [9] Z. Hashin, Analysis of stiffness reduction of cracked cross-ply laminates, Eng. Fract. Mech. 5 (1986) 771
778. [10] A. Puck, H. Schu¨rmann, Failure analysis of FRP laminates by means of physically based phenomenological models, Compos. Sci. Technol. 7 (1998) 1045
1067. [11] O. Klinkova, J. Rech, S. Drapier, J.M. Bergheau, Characterization of friction properties at the workmaterial/cutting tool interface during the machining of randomly structured carbon fibers reinforced polymer with carbide tools under dry conditions, Tribol. Int. 44 (12) (2011) 2050
2058. [12] P.N.H. Thomas, V.I. Babitsky, Experiments and simulations on ultrasonically assisted drilling, J. Sound Vib. 308 (2007) 815
830. [13] M. Lucas, A. MacBeath, E. McCulloch, A. Cardoni, A finite element model for ultrasonic cutting, Ultrasonics 22 (44) (2006) 503
509. [14] D. Ensminger, F.B. Stulen (Eds.), Ultrasonics: Data, Equations, and Their Practical Uses, CRC Press, 2008. [15] V.I. Babitsky, A.V. Mitrofanov, V.V. Silberschmidt, Ultrasonically assisted turning of aviation materials: simulations and experimental study, Ultrasonics 42 (2004) 81
86. [16] R. Li, P. Hegde, A.J. Shih, High-throughput drilling of titanium alloys, Int. J. Mach. Tools Manuf. 47 (2007) 63
74.

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Machining in monocrystals

9

Anish Roy1, Qiang Liu2, Ka Ho Pang1 and Vadim V. Silberschmidt1 1 Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, United Kingdom, 2Department of Materials Engineering, KU Leuven, Leuven, Belgium

9.1

Introduction

Product miniaturization demands high-precision machining at the microscale in the component fabrication of micromechanical systems [1
3]. Unlike conventional macroscale machining, a process zone in micromachining is usually limited to only a few grains of a metallic/ceramic work-piece material. For a component at the microscale, the cutting response differs significantly from that at macroscale as the former is dominated by a grain-level crystalline structure [4,5]. A thorough analysis of cutting mechanics during the micromachining of monocrystals provides an efficient way to better understand the processes of local deformation and material removal at a tool
workpiece interface in a micromachining process [6
8]. For example, several micromachining studies of monocrystal metals were performed to reveal a significant dependence of cutting forces and chip formation on crystal orientation [6,9]. To date, the finite-element (FE) method has been widely used to model machining processes of various engineering materials, enhancing the fundamental understanding of cutting response, including analysis of chip morphology, temperature effects [10,11], imposition of ultrasonic vibration during cutting [12], influence of cutting conditions on the machined subsurface [13,14] as well as optimization of machining parameters [15,16]. For machining of monocrystals the cutting behavior not only involves complex deformation localization but also depends on crystallographic structure. The underpinning micromechanics is difficult to assess experimentally; as a result, numerical simulations play a pivotal role in this regard [17,18]. To describe the anisotropic deformation behavior induced by a crystallographic structure, a single-crystal plasticity (SCP) theory was developed [19], which introduces the concept of activated slip systems by considering the crystal orientation and loading direction. The SCP theory was used successfully to capture some of the well-known experimental observations in monocrystals, including uniaxial experiments [20,21] and nanoindentation studies [22,23]. In recent years the SCP theory was introduced into modeling schemes of micro-cutting processes. With the use of this theory, Zahedi et al. [24] and Tajalli et al. [25] revealed the dependence of chip Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques. DOI: https://doi.org/10.1016/B978-0-12-818232-1.00009-6 Copyright © 2020 Elsevier Ltd. All rights reserved.

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formation and cutting forces on the initial crystal orientations with respect to a cutting direction for a single-crystal copper workpiece. By combining the SCP theory with a strain-gradient effect, Demiral et al. [26] and Pal and Stucker [27] concluded that inhomogeneous plastic deformation could affect machinability of a workpiece. The availability of SCP-based FE simulations for monocrystal machining gained limited experimental validations. Lee et al. [28] adopted the SCP theory to analyze cutting forces and shear angles induced by crystallographic anisotropy. The FE simulations provided reasonable predictions of cutting forces relative to cutting direction. However, the magnitude of obtained cutting forces was severely underpredicted for most directions compared to experimental data. Consequently, the accuracy of FE simulations for monocrystal machining cannot be fully guaranteed by solely considering the anisotropic deformation behavior induced by a crystallographic structure. Based on macroscale machining studies, it was found that a chip-separation criterion had a significant effect on the prediction of chip morphology [29]. Similarly, in addition to accounting for anisotropic deformation behavior, the modeling of material removal, including chip separation, should be dealt with carefully for monocrystal machining in order to improve the accuracy of FE simulations. The chip separation principally reflects the failure mechanism of a work-piece material; however, it is far from easy to model the damage evolution in monocrystals due to their inherently anisotropic characteristics. Kim and Yoon [30] combined several simplified isotropic damage criteria with the SCP theory, including those based on principal strain, equivalent plastic strain, maximum shear strain, and strain energy, to describe the damage evolution in a single-crystal metal. It was claimed that the maximum-shear-strainbased damage model provided the most accurate prediction when compared to experimental data. On a related note, since extreme shear deformation usually occurs at the interface between a cutting tool and a workpiece during machining, an alternative shear-strain-based criterion, accounting for the shear strain accumulated over all the slip systems, was employed to model the material-removal process in micro-cutting by Demiral et al. [26]. Due to significant influences of different damage criteria on modeling of material removal, a comprehensive investigation in this regard is required. Liu et al. [31] compared five types of material-removal criteria in the modeling of single-crystal copper machining, which were based on the principal strain, equivalent plastic strain, maximum shear strain, the shear strain accumulated over all the slip systems in Ref. [26] and the identification of the partial and full activation of slip systems. It was found that only the criterion, accounting for both accumulated shear strain across all slip systems and strains in individual slip systems, predicted the dependence of cutting force on cutting direction accurately. For a chosen criterion of material removal, different modeling techniques can be adopted for implementation within the FE framework. To the authors’ best knowledge, there are three popular techniques for simulating material removal: the element-deletion technique [32], arbitrary Lagrangian
Eulerian (ALE) adaptive remeshing [32], and smooth particle hydrodynamics (SPH) [32,33]. For example, ¨ zel [34]; the element deletion was adopted by Demiral et al. [26] and Arısoy and O the ALE adaptive remeshing was employed by Amini et al. [35], Jin and Altintas

Machining in monocrystals

245

[36], and Parle et al. [17]; and SPH was utilized in the simulation of Zahedi et al. [24,37] and Zhang and Dong [38]. Based on the modeling of single-crystal copper machining, Liu et al. [39] compared the three modeling techniques in terms of computation accuracy and efficiency. In contrast, modeling of micromachining or microscratching in ceramic materials such as silicon carbide is severely limited [40]. This primarily stems from difficulty to carry out reliable experimental studies in such materials owing to their high stiffness and hardness.

9.2

Mechanics of single-crystal machining

9.2.1 Single-crystal-plasticity theory In this section a classical CP theory adopted in this study is reviewed. A deformation gradient, F, can be decomposed into its elastic and plastic parts: F 5 Fe F p ;

(9.1)

where the subscripts e and p denote the elastic and plastic parameters, respectively. By applying the product rule of differentiation, one can obtain the rate of the total _ as deformation gradient, F, F_ 5 F_ e Fp 1 Fe F_ p :

(9.2)

Therefore the velocity gradient, L, can be introduced following its definition _ 21 as L 5 FF _ 21 21 L 5 F_ e F21 e 1 Fe ðFp Fp ÞFe 5 Le 1 Lp :

(9.3)

It is assumed that the plastic velocity gradient, Lp , is induced by shearing on each slip system in a crystal. Hence, Lp is formulated as the sum of shear rates on all the slip systems, that is, Lp 5

N X

γ_ ðαÞ sðαÞ mðαÞ ;

(9.4)

α51

where γ_ ðαÞ is the shear slip rate on the slip system α, N is the total number of slip systems, and unit vectors sðαÞ and mðαÞ define the slip direction and the normal to the slip plane in the deformed configuration, respectively. Furthermore, the velocity gradient can be expressed in terms of a symmetric rate of stretching, D, and an antisymmetric rate of spin, W: L 5 D 1 W 5 ðDe 1 We Þ 1 ðDp 1 Wp Þ:

(9.5)

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Using Eqs. (9.3) and (9.4), it can be deduced that De 1 We 5 F_ e F21 e ; D p 1 Wp 5

N X

γ_ ðαÞ sðαÞ mðαÞ :

(9.6)

α51

Following the work of Huang [41], a constitutive law is expressed as the relationship between the elastic part of the symmetric rate of stretching, De , and the Jaumann rate of Cauchy’s stress, σr , that is, r

σ 1 σðI:De Þ 5 C:ðD 2 Dp Þ;

(9.7)

where I is the second-order unit tensor, C is the fourth-order, possibly anisotropic, elastic stiffness tensor. The Jaumann stress rate is expressed as r

σ 5 σ_ 2 We σ 1 σWe :

(9.8)

On each slip system the resolved shear stress, τ ðαÞ , is expressed by a Schmid’s law, τ ðαÞ 5 symðsðαÞ mðαÞ Þ:σ:

(9.9)

The relationship between the shear rate, γ_ ðαÞ , and the resolved shear stress, τ ðαÞ , on the slip system α is expressed by a power law proposed by Hutchinson [42]: ðαÞ n τ γ_ ðαÞ 5 γ_ 0 ðαÞ sgnðτ ðαÞ Þ; g

(9.10)

where γ_ 0 is the reference shear rate, gðαÞ is the slip resistance, and n is the ratesensitivity parameter. The evolution of gðαÞ is given by g_ðαÞ 5

N X

hαβ γ_ ðβÞ ;

(9.11)

β51

where hαβ is the hardening modulus that can be calculated in the form modified from that proposed by Asaro [43],

 ðh0 2 hs Þγ 1 hs ; τS 2 τ0 X ðt γ_ ðαÞ dt; hαβ 5 qhαα ðα 6¼ β Þ; γ 5 hαα 5 ðh0 2 hs Þ sech2

α

(9.12)

0

where h0 and hs are the initial and saturated hardening moduli, respectively, q is the latent hardening ratio, τ 0 and τ S are the shear stresses at the onset of yield and

Machining in monocrystals

247

the saturation of hardening, respectively, and γ is the accumulative shear strain over all the slip systems.

9.2.2 Computational implementation Implementation of the SCP theory in an implicit ABAQUS FE environment, by means of a user subroutine (UMAT), was introduced in the work of Huang [41], where a time integration scheme and a stress update algorithm were presented as  γ_ ðαÞ Δt 5 Δγ ðαÞ 5 Δt ð1 2 θÞγ_ ðαÞ t 1 θγ_ ðαÞ jt1Δt ; r

Δσ 5 σ Δt;

(9.13) (9.14)

where Δt is the time increment in the FE calculation; θ ranges from 0 to 1, representing different time integration schemes (as an example, setting θ 5 0 yields a simple Euler’s time integration scheme); the Jaumann stress rate, σr , was defined in Eq. (9.8). In this chapter the SCP theory is implemented employing a VUMAT subroutine in the explicit ABAQUS environment. The time integration scheme was identical to the one implemented in the UMAT; however, the stress update algorithm had to be modified due to the difference of the defined stress rate for ABAQUS/Standard and ABAQUS/Explicit formulations. The former employed the Jaumann stress rate, but the latter was based on the Green
Naghdi stress rate [32]. In contrast to the Jaumann stress rate defined in Eq. (9.8), the Green
Naghdi stress rate is defined as Δ

σ 5 σ_ 2 Ωe σ 1 σΩe ;

(9.15)

where Ω was found from a right polar decomposition of the total deformation gradient, F, as _URT ; F 5 VR; Ω5R (9.16) where R and V are the right rotation and stretch tensors, respectively. To evaluate the stress update defined by the Green
Naghdi stress rate by using the Jaumann rate, one can use the Hughes
Winget algorithm [44], as σ t1Δt 5 ΔRσ t ΔRT 1 Δσ;

(9.17)

21  1 1 I 1 ðΔW 2 ΔΩÞ ; ΔR 5 I2 ðΔW2ΔΩÞ 2 2

(9.18)

ΔW 5 WΔt;

(9.19)

ΔΩ 5 ΩΔt;

where ΔR is the relative spin increment tensor and I is the second-order unit tensor, Δσ is the stress increment obtained with the Jaumann stress rate (Eq. 9.14). Another essential difference in ABAQUS/Explicit is that the stress and strain

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tensors are defined based on the spatial coordinate system (i.e., with respect to the local coordinate system rotating with the volume), in contrast to the material coordinate system (i.e., a fixed global coordinate system) used in ABAQUS/Standard. Therefore during the conversion of UMAT to VUMAT, the stress update algorithm in VUMAT should be rewritten as σ t1Δt 5 ΔRσ t ΔRT 1 RΔσRT :

(9.20)

9.2.3 Criteria of material-removal modeling In this chapter, five different damage criteria are considered to elucidate the important role of material-removal modeling in FE simulations of monocrystal machining. The first three were based on the simplified damage criteria of principal strain, equivalent plastic strain, and maximum shear strain. The fourth one is based on the shear strain accumulated over all slip systems proposed by Demiral et al. [26]. The fifth one was proposed by Liu et al. [31], which can be expressed as follows:  max γ 2 γ cr ; γ sl; min 2 γ sl;cr $ 0; γ sl; min 5 min γ ðαÞ ; α 5 1; 2. . . N;

(9.21)

where γ sl;cr and γ cr are the critical values of shear strain on a single slip system and the accumulated shear strain on all the slip systems, respectively. According to Eq. (9.21), the reason for damage in a single crystal is divided into two categories: partial and full activation of slip systems. Material removal is activated when either the critical value for an individual slip system (γ sl;cr ) or the accumulated slip (γ cr ) is attained. Recently, Pang and Roy [45] proposed an enhanced version of these criteria by monitoring the shear strain in each slip system to capture the material removal with weak anisotropy, which can be expressed as follows: γ 2 γ cr $ 0; γ sl ðαÞ 2 γ ðαÞ $ 0; α 5 1; 2?N;

9.3

(9.22)

Machining of single-crystal metal

The single-crystal copper with a face-centerd cubic (FCC) crystallographic structure is considered in this section.

9.3.1 Experimental procedure The single-crystal copper workpiece was machined using a single-crystal diamond tool with a sharp tip. The cutting tool had a wedge angle of 60 degrees and a

Machining in monocrystals

249

clearance angle of 6.25 degrees. The cutting velocity was fixed as 10 mm/min. Two different target depths (i.e., 8 and 18 μm) were considered during micro-cutting. For a given depth the micro-cutting process consisted of two stages: the cutting depth linearly increased in Stage 1 (from 0 to a target depth) and was kept constant in Stage 2. In this study the [1 1 0] crystal orientation corresponded to a normal of the top surface of the workpiece. Three orientations were chosen as cutting directions: pffiffiffi [1 2 1 0], and the other two rotated by angles of 45 degrees (i.e., [1 2 1 2]) and 90 degrees (i.e., [0 0 1]) with respect to it. The cutting forces were defined as follows: a force along the cutting direction (the X direction) is called the principal force, and those perpendicular to the top surface of the workpiece (the Y direction), and X
Y plane are the thrust and lateral forces, respectively (see Fig. 9.1).

9.3.2 Finite-element model and material parameters Based on the experimental studies, a three-dimensional (3D) FE model was constructed to simulate the micro-cutting process of single-crystal copper as shown in Fig. 9.1. For FCC single-crystal copper, deformation slip was assumed to occur on the usual twelve f1 1 1gh1 1 0i slip systems (see Table 9.1). The cutting tool was assumed to be rigid, and the contact condition between the cutting tool and the workpiece was assumed to be frictionless. The cutting direction was in the negative X direction (Fig. 9.1A), with a cutting velocity of 10 mm/min. The groove produced by micro-cutting is shown in Fig. 9.1B, which includes two stages of micro-cutting. For a given depth the two stages of the micro-cutting process were modeled using two separated FE models in order to reduce a computational cost. As indicated in Fig. 9.1B, in the first model, the length, height, and width of the workpiece were

(A)

(B)

200

240

μm

or 3

40

μm

Groove

Y, Thrust X, Principal

l

o To

120 μm

300 μm

Z, Lateral

170 μm

60° Wo rk

pie

ce

6.25º

Y Z

X

Y X Z

Figure 9.1 (A) Finite-element model for simulation of micro-cutting and (B) groove geometry.

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Table 9.1 Twelve slip systems in face-centerd cubic single-crystal copper.

ðαÞ

s mðαÞ

1

2

3

4

5

6

7

8

9

10

11

12

011 111

101 111

110 111

101 111

110 111

011 111

011 111

110 111

101 111

110 111

101 111

011 111

Table 9.2 Material parameters of single-crystal copper. Parameter

Definition

Value

Unit

γ_ 0 n τ0 τs h0 hs q

Reference shear rate Rate-sensitivity parameter Initial slip resistance Saturated slip resistance Initial hardening modulus Saturated hardening modulus Latent hardening ratio

0.001 50 4.0 52 180 24 1.2

s21
MPa MPa MPa MPa


340, 200, and 120 μm, respectively; the respective magnitudes were 240, 200, and 120 μm in the second model. The workpiece was meshed using 8-node brick elements with reduced integration (C3D8R) available in ABAQUS. To improve accuracy a finer local mesh was used in regions near the cutting zone to a depth of 27 μm. Experimental data for single-crystal copper under compression reported by Takeuchi [46] were employed to calibrate the material parameters of single-crystal copper. The calibrated model parameters are listed in Table 9.2 and were used to simulate the micro-cutting process. The material-removal criteria were based on the relation described in Eq. (9.21) for machining of metals followed by the relation presented in Eq. (9.22) for modeling SiC and implemented as the element-deletion scheme.

9.3.3 Simulation and results 9.3.3.1 Prediction of cutting forces The variation of cutting forces with time during a full micro-cutting process for the cutting depths of 18 and 8 μm is shown in Figs. 9.2 and 9.3, respectively. For each cutting depth, two cutting directions, that is, 0 and 45 degrees were considered, with both simulation results and experimental data reported. During cutting Stage 1, the principal and thrust forces increased with time as the depth of cut increases continuously. In Stage 2 the cutting forces approached a constant value at the constant depth of cut. In the experiments the tool movement was stopped before the start of Stage 2; as a result, a clear discontinuity at the point of transition from Stage 1 to Stage 2 was observed in the experimental curves.

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251

(A) 0.4

(B) 0.4 Exp: Principal Exp: Trust

Simu: Principal Simu: Trust

0.3 Force (N)

Force (N)

0.3

Exp: Principal Exp: Trust

Simu: Principal Simu: Trust

0.2

0.1

0.2

0.1

0.0

0.0 0

3 2 Times (s)

1

4

5

0

1

2 3 Times (s)

4

5

Figure 9.2 Comparison of cutting forces from experimental data and FE simulations in micro-cutting of single-crystal copper with depth of 18 μm for 0- (A) and 45-degree (B) directions. FE, Finite element.

(A)

(B) 0.075

0.075 Exp: Principal Exp: Trust

Exp: Principal Exp: Trust

0.060

Force (N)

Force (N)

0.060

Simu: Principal Simu: Trust

0.045 0.030

Simu: Principal Simu: Trust

0.045 0.030 0.015

0.015

0.000

0.000

2

3

4 Times (s)

5

6

2

3

4 Times (s)

5

6

Figure 9.3 Comparison of cutting forces from experimental data and FE simulations in micro-cutting of single-crystal copper with depth of 8 μm for 0- (A) and 45-degree (B) directions. FE, Finite element.

For the two studied cutting depths, the simulation results agree well with the experimental data for the principal force. The principal force exhibited a significant dependence on crystal orientation in the experiments and the simulations. In Stage 2 the principal force for the 45-degree direction was about 40% higher than that for the 0-degree direction at the cutting depth of 18 μm (Fig. 9.2), and about 30% higher at 8 μm (Fig. 9.3). The good prediction on the anisotropy in the principal force also validated the suggested criterion of material removal in Eq. (9.21). The FE model provided a comparatively less accurate prediction for the thrust force. A possible reason could be due to its relatively low magnitude. It also showed less dependence on the cutting direction.

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

9.3.3.2 Prediction of chip morphology Only the results corresponding to the final cutting depth of 18 μm are given for the study of chip morphology here. The chip shapes obtained in the FE simulations and experiments for 0- and 45-degree cutting directions are given in Fig. 9.4. For cutting was performed along the 0-degree direction (i.e., [1 2 1 0] orientation), slip systems activated in the FCC crystallographic structure were symmetrical with respect to it, and, thus, the chip separated from the workpiece symmetrically. Contrarily, the cutting along the 45-degree direction deviated from the main axes of symmetry of the FCC structure and, therefore, yield an asymmetrical stress field and chip formation. For the micro-cutting in both 0- and 45-degree directions, the FE simulations show good correlation with experimental results.

9.3.3.3 Prediction of workpiece deformation The contour plots of the displacement normal to the work-piece surface are presented in Fig. 9.5 (chips were removed for clarity). The micro-cutting in the 0- and

Figure 9.4 Comparison of chip morphology from FE simulations and experiments in microcutting of single-crystal copper: (A) simulation in 0-degree direction; (B) experiment in 0degree direction; (C) simulation in 45-degree direction; and (D) experiment in 45-degree direction. FE, Finite element.

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253

Figure 9.5 Comparison of displacement fields in workpiece from FE simulation and experimental data on micro-cutting of single-crystal copper: (A) simulation in 0-degree direction; (B) experiment in 0-degree direction; (C) simulation in 45-degree direction; and (D) experiment in 45-degree direction. FE, Finite element.

45-degree directions produced symmetrical and asymmetrical deformation fields in the workpiece, respectively. Ahead of the cutting process zone corresponding to 0degree-direction cutting, a significant deformation localization was observed for directions at about 6 45 degree with respect to the cutting direction. For the microcutting in the 45-degree direction, localized deformation was more evident to the right of the groove (negative Z direction in Fig. 9.5C and D). The pile-up profiles of the deformed workpiece were extracted for two representative paths (paths A
B and C
D indicated in Fig. 9.5), which are shown in Fig. 9.6A and B, respectively. It is obvious that the cutting direction affected the residual imprint post-machining. As for the cutting in the 0-degree direction, FE simulation agreed well with the experimental data quantitatively. A higher error between the FE prediction and the experimental data was observed for the pile-up profile corresponding to the cutting in the 45-degree direction. It is necessary to point out that the experimental data obtained with a laser microscope was noisy, and the results presented are based on averaging the data from several scans. In spite of this, the FE model provides a reasonable prediction of the pile-up height for path C
D.

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

(A)

(B) 10

10

I

II

III

Height (μm)

0

Height (μm)

0

Experiment Simulation

–10

–20 –50

IV

Experiment Simulation

–10

–20 –25

0 25 Coordinate Z (μm)

50

–50

–25

0 25 Coordinate Z (μm)

50

Figure 9.6 Pile-up profiles across groove obtained with FE simulation and experiment: (A) 0-degree direction (Line A
B, Fig. 9.5A) and (B) 45-degree direction (Line C
D, Fig. 9.5C). FE, Finite element.

9.3.3.4 Prediction of misorientation angle To study the variation of crystallographic orientation of the workpiece due to micro-cutting, the concept of misorientation angle [47] was introduced, as    21 tr gA g21 B O21 θ 5 min cos 2

(9.23)

where θ is the misorientation angle, gA and gB are the orientation matrix at chosen spatial locations A and B, respectively, and O is the crystal symmetry operator. There are a total of 24 identical rotation operations in the case of FCC symmetry [47]. Here, the undeformed crystallographic structure was considered as the reference configuration for calculation of the misorientation angle. The variations of misorientation angle along the paths P
Q (for 0-degree-direction cutting) and R
S (for 45-degree-direction cutting) are presented in Fig. 9.7. For the cutting in 0-degree direction, the FE model accurately predicted the variation of crystallographic orientation compared to the experimental data, and the misorientation angle was up to 20 degrees. As expected, the misorientation angle was asymmetrical along the path R
S, with larger misorientation angles in the region with a higher pile-up height.

9.3.4 Discussion For FE simulations of monocrystal machining, computational accuracy and efficiency depend on many factors, such as mesh size, criteria, and numerical techniques of material-removal modeling. The importance of these factors was demonstrated based on the modeling of single-crystal copper with a fixed cutting depth of 18 μm.

Machining in monocrystals

255

(A)

(B) 50

50

25 20 Y

25

Misorientation angle (degree)

Misorientation angle (degree)

Simulation Experiment Simulation

Q

X

Z

10 5

P 0 –50

–40

–30 –20 Coordinate Z (μm)

–10

0

40 30 Y

20 10

S

R 0 –40

X

Z

–20

0 20 40 Coordinate Z (μm)

60

80

Figure 9.7 Misorientation angle across groove: (A) 0-degree direction, comparing FE simulation and experiment and (B) 45-degree direction, prediction based on FE simulation. FE, Finite element.

Figure 9.8 Influences of mesh number on predicted cutting forces.

9.3.4.1 Mesh-sensitivity analysis Without loss of generality, micro-cutting along 0-degree direction was considered for the mesh-sensitivity study. Mesh density was represented by the number of elements along the height of the cutting tool (i.e., the Y direction as per Fig. 9.1A), which is denoted as N. The influence of mesh on the calculated cutting forces is shown in Fig. 9.8, where both the principal and thrust forces are presented. Both cutting forces

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

decreased with increasing N. The mesh size became less significant when N reached 30, and this mesh size allowed an acceptable balance between computational accuracy and efficiency. In our study the mesh size N 5 30 corresponded to B1 3 106 brick elements in total in the FE model,

9.3.4.2 Criteria of material-removal modeling Our study indicated that the material-removal criteria determined the FE prediction of anisotropy of cutting forces with respect to the cutting directions. In order to demonstrate the difference caused by the five material-removal modeling schemes defined in Section 9.2.3, the cutting force was expressed in the normalized form, f , as f5

F : Fp;0

(9.24)

where F is the true cutting force averaged over the nominally steady cutting process (i.e., after full engagement of the cutting tool), and Fp;0 is the averaged principal force in the 0-degree direction. The variations of normalized average cutting forces with the cutting directions, obtained with different material-removal criteria, are shown in Fig. 9.9. By comparing them with the experimental data, it is clear that the choice of material-removal criterion played a pivotal role in capturing the variations of the predicted cutting

Figure 9.9 Comparison of different criteria for material-removal modeling.

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257

forces with cutting directions. The experimental data showed that the principal force in the 45-degree direction is about 40% higher than that in the 0-degree direction. However, the criteria of equivalent plastic strain and accumulative shear strain only predicted 4% and 11% increases, respectively, while the criteria of maximum principal strain and shear strain even resulted in a decrease of 5% and 7%, respectively. In this regard the proposed criterion in Eq. (9.21) yielded a rather good prediction. Thus it can be concluded that the material-removal criterion is an important factor to predict the cutting forces in monocrystal machining.

9.3.4.3 Material-removal modeling techniques The three modeling techniques for material removal, that is, element deletion, ALE adaptive remeshing, and SPH, are compared here in terms of predicted cutting forces (or accuracy) and computational time. The cutting forces obtained with the three material-removal methods are compared in Fig. 9.10 for the cutting in 0- and 45-degree directions. Compared to the sole use of element deletion, the introduction of ALE adaptive remeshing had no significant effect on the cutting forces for both directions. A more significant difference was observed between the element deletion and SPH; especially, SPH predicted a larger cutting force than the element deletion in the engagement stage. A possible reason was that the SPH particles still contributed to cutting forces after the material-removal criterion was satisfied, while the deleted elements had no such contribution. However, it is necessary to emphasize that the element deletion and SPH predicted almost the same magnitude of cutting force in the steady cutting stage of the process. The levels of average cutting forces obtained with different modeling techniques were summarized in Table 9.3, where the three methods show negligible differences in their predictions for Stage 2. Moreover, the computational cost of the three modeling techniques is also compared in Table 9.3, and the time was expressed as (A)

(B)

0.6

0.5

0.4

Force (N)

Force (N)

0.5

0.6 Element deletion ALE and elememt deletion SPH

Principal force 0.3 0.2

0.4 Principal force

0.3 0.2

Trust force

Trust force

0.1 0.0 0.0

Element deletion ALE and elememt deletion SPH

0.1

0.2 0.3 Time (s)

0.4

0.1 0.5

0.0 0.0

0.1

0.2 0.3 Time (s)

0.4

0.5

Figure 9.10 Comparison of cutting forces obtained with different material-removal methods in 0- (A) and 45-degree (B) directions.

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Table 9.3 Comparison of average cutting forces in Stage 2 and computational times for different modeling methods of material removal. Parameter

Average principal force (N) Average thrust force (N) Relative computational time

Element deletion

ALE and element deletion

SPH

0 degree

45 degrees

0 degree

45 degrees

0 degree

45 degrees

0.269

0.329

0.265

0.323

0.277

0.327

0.074

0.089

0.073

0.087

0.076

0.090

1

1.25

6.89

9.48

12.3

12.5

ALE, Arbitrary Lagrangian
Eulerian; SPH, smooth particle hydrodynamics.

normalized with that for the 0-degree-direction simulation with the elementdeletion method. The three modeling techniques show very significant differences in computational time, with the element deletion requiring the lowest computational time, while SPH was the most expensive. In the presented study, ALE adaptive remeshing and SPH were some 6 times and 10 times slower, than the elementdeletion technique. In summary, though the three modeling methods of material removal differed slightly from the viewpoint of their accuracy, the element deletion was the best in terms of computational cost.

9.4

Machining of single-crystal ceramic material

Next, micromachining in single-crystal silicon carbide with a hexagonal crystal structure (6H-SiC) is discussed.

9.4.1 Experimental procedure A micro-scratching study was conducted on a Nano Test system (Platform 3, Micro Materials Ltd.) equipped with a diamond indenter tip with a tip radius of 6 μm. Experiments were performed for two crystallographic orientations. The first specimen with dimensions 5 mm 3 5 mm 3 0.7 mm was oriented so that the microscratch test could be performed on its basal plane (0 0 0 1), hereafter referred to as “Orientation 1” (Fig. 9.11A). The scratching direction of the second sample with dimensions 2 mm 3 5 mm 3 2 mm, was oriented at 80 degrees off c-axis, henceforth, referred to as “Orientation 2” (Fig. 9.11B). For both samples the scratching experiment was performed along the [1
1 0 0] direction.

Machining in monocrystals

(A)

c

259

Orientation 1

(B)

Orientation 1 80°

c

(0 0 0 1)

(1–100)

(1–100)

(0 0 0 1)

Figure 9.11 Scratching directions with respect to crystallographic orientations: (A) Orientation 1 and (B) Orientation 2.

Table 9.4 Conditions of micro-scratching experiment. Conditions

Progressive load stage

Constant load stage

Units

Initial load Final load Scratch velocity Loading rate Scratch distance

0.5 120 5 10 120


120 5
110

mN mN μm/s mN/s μm

Both samples were polished to achieve an epi-ready surface quality with a RaB2 nm. The experiment included two stages: (1) with a progressively increasing load and (2) with a constant load; the scratching velocity remained constant at 5 μm/s throughout the test. The initial and final normal (in the negative Y direction—see Fig. 9.9) loads of the first stage were 0.5 and 120 mN, respectively, with a loading rate of 10 mN/s over a scratch distance of 60 μm. The normal load of the second stage remained constant at 120 mN, with the scratch distance 100 μm. Table 9.4 summarizes the experimental conditions.

9.4.2 Computational modeling A 3D FE model was developed to simulate microscale machining of single-crystal 6H-SiC with an hexagonal closed-pack (HCP) crystallographic structure using the general-purpose FE software ABAQUS/Explicit. The developed model is shown in Fig. 9.9. The work-piece dimensions were 9 3 20 3 8 μm3 and it was meshed with first-order hexahedral reduced-integration elements (C3D8R). To improve

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Figure 9.12 FE model of micro-scratching process. FE, Finite element.

numerical accuracy a finer mesh was used in the vicinity of the scratching tool with an effective cutting area of 9 3 12.7 3 8 μm3. The indenter made of diamond was modeled as an analytical rigid body with a conical shape and a tip diameter of 6.5 μm. Friction was ignored in this simulation and general contact conditions were assumed. Here, a constant-depth cutting simulation was performed in the X-direction in the FE model (Fig. 9.12) with a cutting velocity of 5 μm/s, cutting length of 10 μm, and cutting depth of 350 nm. The simulations were performed for crystal orientations and directions, identical to those in the experimental studies. The CP model was implemented in ABAQUS/Explicit by employing the user subroutine VUMAT. A total of 30 individual slip systems across the five slip families, including basal, prismatic, pyramidal hai, first-order pyramidal hc 1 ai, and second-order pyramidal hc 1 ai, were used in this study. Details for individual slip systems considered in the subroutine can be found in Table 9.5. The material removal was realized with the element-deletion technique as discussed in Section 9.2.3. An important difference of micromachining in ceramic materials, when compared to that of metallic materials, is that the cutting depth is typically of the order of tens to hundreds of nanometers instead of microns. The modeling of micromachining is typically implemented in two steps: the first is the indenter penetration of the material to a target depth and the second is its translation, cutting the material. The material properties, including the elastic constants and values of critical resolved shear stress for different slip systems shown in Table 9.5, were calibrated employing nanoindentation experimental and numerical studies for single-crystal 6H-SiC as discussed in Ref. [40]. Table 9.6 provides the materials parameters used in this study.

Table 9.5 Slip systems considered in the study of single-crystal ceramic. Slip system

No.

sðαÞ

mðαÞ

Slip systems

No.

sðαÞ

mðαÞ

Slip systems

No.

sðαÞ

mðαÞ

Basal

1 2 3 4 5 6 7 8 9 10 11 12

1210 2110 1120 1210 2110 1120 1210 2110 1120 1210 2110 1120

0001 0001 0001 1010 0110 1100 1011 0111 1101 1011 0111 1101

Pyramidal hc 1 ai first order

13 14 15 16 17 18 19 20 21 22 23 24

2113 1123 1123 1213 1213 2113 2113 1123 1123 1213 1213 2113

1011 1011 0111 0111 1101 1101 1011 1011 0111 0111 1101 1101

Pyramidal hc 1 ai second order

25 26 27 28 29 30

2113 1213 1123 2113 1213 1123

2112 1212 1122 2112 1212 1122

Prismatic

Pyramidal hai

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Table 9.6 Materials parameters used in finite-element model [40]. Parameters

Definition

Values

Unit

C11 C12 C13 C33 C44 γ_ 0

Elastic constants

325 60 34 367 109 0.01

GPa GPa GPa GPa GPa s21

n

Rate-sensitivity parameter

q

Latent hardening ratio

τ0

Basal and Prismatic

Reference shear rate

τs h0 τ0 τs

Pyramidal hai, first- and second-order hc 1 ai

h0

80

Initial slip resistance Saturated slip resistance Initial hardening modulus Initial slip resistance Saturated slip resistance Initial hardening modulus




1.0




9.85

GPa

15

GPa

9

GPa

11

GPa

15

GPa

9

GPa

9.4.3 Results and discussion 9.4.3.1 Surface topography The surface topography obtained from machining/scratching the surface of ceramic specimens is shown in Fig. 9.13A and B for Orientations 1 and 2, respectively. No subsurface crack was initiated under the employed loading conditions. From the extracted profile, it can be seen that the pileup on both sides of the groove in Orientation 1 was symmetric, with a pile-up height of B150 nm. In contrast, for Orientation 2, the pileup was highly asymmetric, with a maximum pile-up height on the right of the scratch equal to 101.1 nm with the other side just around 60 nm. In addition, the residual depth of the groove in Orientation 2 (B325.9 nm) was also larger than that of Orientation 1 (314.9 nm).

9.4.3.2 Comparison of cutting forces A comparison of the principal cutting force obtained in the experiments and results of numerical simulations for Orientations 1 and 2 is shown in Fig. 9.14A and B, respectively; numerical average of the cutting forces are presented with dashed lines

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263

Figure 9.13 Surface topography of scratching at Orientation 1 (A) and Orientation 2 (B). (A)

(B)

30

30 23.96

Principal force (mN)

25

25

21.25

20

20

23.34

21.22

15

15

10 5 0 0.0

10

Experiment Simulation

0.2

0.4 0.6 0.8 Normalized distance

Experiment Simulation

5 1.0

0 0.0

0.2

0.4 0.6 0.8 Normalized distance

1.0

Figure 9.14 Comparison of experimental results and simulated principal forces: (A) Orientation 1 and (B) Orientation 2.

in each figure. The results indicate that the cutting forces obtained from numerical studies correlate well with the experimental results. The cutting response along Orientation 2 was higher than Orientation 1.

9.4.3.3 Prediction of work-piece deformation The accumulative shear strain in the material post-machining is shown in Fig. 9.15. It is believed that the material removal in Orientation 1 (Fig. 9.15A) was dominated by the average response of partially active slip systems; hence, the magnitude of the residual total accumulative shear strain in the work-piece material is small. On the contrary, material removal in Orientation 2 (Fig. 9.15B) was dominated by the slip system (1 1 0 0) [11
20]. Although some slip systems were partially activated during the micromachining process the element-deletion threshold was not achieved so most partially activated systems remained in the workpiece without being removed.

9.4.3.4 Discussion For machining of the single-crystal ceramic material, the material model incorporating classical CP theory with an enhanced material-removal criterion, accounting for

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Figure 9.15 Contour plots of accumulative shear strain: (A) Orientation 1 and (B) Orientation 2.

partial and full activation of slip systems, was proposed. The new model demonstrated the ability to capture the deformation and material-removal responses in the workpiece even with weak anisotropy. The study revealed that both responses in scratching in Orientation 1 (basal plane) were dominated by the sum of partially active slip systems, whereas scratching in Orientation 2 (inclined at 80 degrees to the basal plane), this was defined by full activation of specific slip systems. The observations here also help to elucidate the differences in groove profiles observed in the experiment. The lower residual shear strain in the material machined in Orientation 1 post-scratching, as compared with Orientation 2, was observed.

9.5

Concluding remarks

Machining at small length scales is challenging, as the orientation of individual grains of work-piece material influences the machining outcome. This is in contrast to machining at the macroscopic scale, where the material response is effectively isotropic for all practical purposes. Thorough knowledge of the micromechanics of deformation in the microscale is essential to predict machining outcomes. The CP modeling framework is a viable way to assess underpinning mechanisms by varying the relevant machining parameters, including tool geometry. However, there is a need for further research in this field. Model calibration and validation should be expedited. In this regard, data-driven models incorporating machine learning techniques could make a significant contribution to progressing materials research in the context of machining and machinability of parts. Next, a need for improving crystal plasticity with physics-based modeling that incorporate the drivers of plasticity at small length scales, namely, dislocations, cannot be ignored. Some models based on robust finite-deformation field dislocation mechanics are available [48,49], which should be incorporated into numerical schemes to yield improved predictive capabilities with regard to residual stress and damage.

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Microstructural changes in machining

10

W. Bai1, R. Sun1, J. Xu1 and Vadim V. Silberschmidt2 1 Huazhong University of Science and Technology, Wuhan, P.R. China, 2Loughborough University, Loughborough, United Kingdom

10.1

Introduction

A focus of precision machining is dimensional accuracy and surface roughness of parts. Admittedly, the geometric accuracy is an important indicator for processing quality. However, the increasing demand for processing quality of machined products draws attention to other parameters such as their physical, metallographic, chemical, and biomedical properties. Microstructural changes of machined surfaces and subsurfaces affect the service performance of aerospace components, such as fatigue life as well as corrosion and wear resistance. Microhardness of a machined surface of a multiphase material is determined by the grain size and phase composition. Other microstructural features of machined surfaces can include their plasticdeformation-induced dislocation density, phase transitions, microcrack, and intergranular cracking [1]. In addition, microstructural evolution of a chip is very important, especially for the generated sawtooth chip of difficult-to-cut alloys. Shivpuri et al. [2] believed that sawtooth fracture affected chip morphology, cutting force, tool
chip contact temperature, and other dynamic behaviors of the cutting system. Therefore more in-depth investigations are needed to reveal the machining-induced microstructural evolution and the final microstructure of the produced parts. In a cutting process the tool
workpiece interaction generates three high temperature zones: the primary deformation zone, the secondary deformation zone, and the tertiary deformation zone. The first zone is caused by severe shear, one of the main heat sources in the cutting process. The secondary deformation zone is generated by friction at the tool
chip interface, with the heat transferred to the cutting tool and the chip. The last zone is a result of interaction between a flank surface of the tool and the machined surface. Thus a large amount of cutting heat is produced in the finished surface and the chip. For materials with low thermal conductivity, this causes local high temperature, resulting in intensive movements of dislocations in the metal, generation of grain boundaries, and onset of dynamic recrystallization. Thus the microstructures of the machined surface and the chip are changed. Positions and modes of microstructural evolution in the cutting process are shown in Fig. 10.1. In this chapter, microstructural evolution of machined surfaces and chips in machining processes is reviewed in Section 10.2, with microstructural models for Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques. DOI: https://doi.org/10.1016/B978-0-12-818232-1.00010-2 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Figure 10.1 Positions and modes of microstructural evolution in cutting process.

machining summarized in Section 10.3. Besides, a research case of microstructural evolution in ultrasonically assisted cutting (UAC) is provided in Section 10.3, which clearly elucidates the microstructural changes and differences of machined surfaces and chips in conventional cutting (CC) and UAC.

10.2

Microstructural evolution in machining

10.2.1 Microstructural evolution in machined surface Intense thermomechanical loading in machining process induces complex changes of material properties in the machined surface. Material strengths and hardness as well as fatigue and corrosion resistance of components are important for their applications and affected significantly by conditions of the machined surface. Depending on thermomechanical variables and material parameters, microstructural evolution of machined surface can be governed by severe plastic deformation, grain refinement, dislocation slip and glide, phase transformations, strain hardening, etc. Changes in the grain size and dislocation density are generated in a machined surface by material removal. Arısoy et al. [3] investigated the effects of tool microgeometry and coating and cutting speed on microstructural changes in machining with 3D customized finite-element (FE) simulations with a dynamic recrystallization model. Average grain sizes, phase fractions, and resultant microhardness were compared with experimental measurements revealing good agreements. (Fig. 10.2 shows SEM images and distribution of primary and secondary γ 0 -grains on machined surface.) Fernandez-Zelaia et al. [4] introduced effect of microstructure on flow stress in terms of internal state variables—dislocation density and mean grain size. A physics-based law that considers dislocation nucleation and annihilation processes was modeled as well as dynamic recrystallization. Validation against machining data showed that the model predicted machining forces and tool temperatures reasonably well over a range of feeds and speeds. Atmani et al. [5]

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Figure 10.2 (A) Microstructure of machined surface and (B) histograms of primary γ0 ðd1 Þand secondary γ 0 ðd2 Þ-grains [3].

Figure 10.3 Predicted dislocation densities and grain sizes along: (A) the primary shear zone; (B) the machined surface depth; and (C) the inner chip surface and machined surface [5].

described the thermo-viscoplastic behavior of a workpiece with a flow law with a mechanical threshold stress model and a microstructure evolution law with a dislocation density model. The predicted grain size and dislocation density are shown in Fig. 10.3. They found that the grain refinement occurred in the generated workpiece

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surface within a thin layer of about 40 μm in depth. Ding et al. [6] focused on the severe plastic-deformation behavior in high-speed machining; dislocation densities and grain sizes were predicted in the machined surface. Shen et al. [7] also developed a dislocation density
based dynamic recrystallization model for the cutting process. The microstructural changes in the primary and secondary deformation zones under dry or wet cutting conditions were simulated with a cellular automaton (CA) model. Grain refinement and phase transformation in machined surface depend strongly on thermomechanical behavior in a machining processing. Ghosh and Kain [8] observed the microstructure of AISI 304L stainless steel as a result of surface machining; the study revealed that the process caused an extensive grain refinement, a strain-induced martensite transformation and a high magnitude of plastic deformation near the surface. Zhang et al. [9] proposed an analytical model for prediction of microstructural changes in machining of 304 austenitic stainless steel. The predicted results were in a good agreement with the measured data; the resultant grain refinement and phase distribution in the machined surface were assessed (Fig. 10.4). Wang and Liu [10] investigated grain orientation and texture on machined surface with an EBSD technique in high-speed cutting. Pan et al. [11] predicted a phase transformation and grain growth in Ti
6Al
4V alloy in highspeed machining employing a dynamic recrystallization model and an Avrami model. They also conducted experiments to investigate the effects of a cutting speed and a feed rate on microstructural changes. Changes in microstructure during machining induce variation of microhardness and residual stresses, which have great impacts on initiation and propagation of corrosion cracking. Guo et al. [12] studied a deformation history of a machined surface and measured strains and microhardness in a near-surface region; the detailed results showed that deformation in the machined surface was similar to that in the chip. Zhang et al. [9] described the evolution of dislocation density and stress/strain

Figure 10.4 Grain refinement (A) and phase distribution (B) near surface of machined sample observed with EBSD [8,9].

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273

during machining analytically and predicted the levels of microhardness and residual stresses of the machined surface for different processing parameters. They concluded that a high gradient of microhardness changes in the near-surface zone as well as a higher cutting speed and a smaller depth of cut could lead to a slightly lower surface residual stresses.

10.2.2 Microstructural evolution in chip A process of chip formation during metal machining is accompanied by extensive changes in strain, strain rate, and temperature. Plastic deformation in a primary shear zone can induce microstructural changes in the chip. It is well known that severe plastic deformation was developed as an effective approach for the creation of bulk metals and alloys with ultrafine-grained microstructure and enhanced strength. Chandrasekar’s group [13
18] investigated a deformation field and microstructure evolution in large-strain machining for a long time. Chip formation with plane-strain tools with a controlled geometry and controllable process parameters (Fig. 10.5A) provides a simple experimental configuration, in which large plastic strains can be realized in a narrow deformation zone. Swaminathan et al. [14] and Shankar et al. [15] captured metallographic images of the deformation zone (Fig. 10.5B) with a high-speed CCD camera and analyzed using particle image velocimetry (PIV) to characterize the strain rates and strains associated with the

Figure 10.5 Microstructural changes in severe plastic deformation: (A) schematic of planestrain machining [16]; (B) optical micrograph of metallographic features in chip formation [15]; (C) distribution of shear strain rate in deformation zone [14]; (D) effects of chip thickness ratio and deformation-zone temperature on microstructure [18]; and (E) deformation microstructure map for various deformation conditions [16].

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deformation field (Fig. 10.5C). They demonstrated a low-cost process for the generation of nanostructured metals and alloys in large volume for steel [14], titanium [15], copper [17], and magnesium alloy [18]. Sagapuram et al. [18] controlled the texture and the grain size in an alloy by shear-based deformation processing, comparing the effect of chip thickness ratio (λ) and deformation-zone temperature (Τ) on microstructure (Fig. 10.5D). The obtained results showed that microstructure for these deformation conditions was determined primarily by temperature (Τ) in consideration of the grain size and hardness of the specimens. Brown et al. [16] investigated interaction effects of strain, strain rate, and temperature for a wide range of deformation parameters in severe plastic deformation. They summarized the deformation microstructure results in the form of maps, highlighting the mechanisms and deformation modes underlying microstructure development for various severe plastic-deformation conditions (Fig. 10.5E). Besides, Basu and Shankar [19] also performed large-strain machining with various depths of cut and observed the effect of deformation volume on microstructural evolution. Results from EBSD-based orientation imaging microscopy showed that microstructural evolution during severe plastic deformation was altered significantly in small length scales due to small deformation volumes. Severe deformation in primary and secondary shear zones generates a temperature raise and material changes in a chip. For difficult-to-machine materials, such as titanium- and nickel-based alloys, segmented chips are often observed in machining. More severe changes of thermomechanical variables in shear bands of a chip are caused by large deformation and low thermal conductivity. Fernandez-Zelaia et al. [4] considered dislocation mechanisms and grain-boundary sliding in a constitutive model for Ti
6Al
4V. A mean grain size and a slip rate of the α-phase were predicted for a segmented chips. Yameogo et al. [20] performed a numerical simulation of orthogonal cutting for Ti
6Al
4V with two models of material behavior, which took into account recrystallization. The resultant grain-size fields and chip morphologies of two models were compared, and the results suggested that dynamic recrystallization was responsible for chip morphology by softening some areas of the chip. They also proposed a model of physical behavior based on recrystallization and damage mechanisms to predict a cutting force and chip formation in the cutting process of Ti
6Al
4V [21]. Melkote et al. [22] presented a material model that accounted for microstructural-evolution-induced flow softening due to an inverse Hall
Petch effect below a critical grain size. The model predicted formation of a segmented chip and grain refinement in shear bands as shown in Fig. 10.6. Many researchers observed microstructure in shear bands and analyzed mechanisms of segmented-chip formation. Wagner et al. [23] investigated a relationship between a specific structure of Ti
6Al
4V and a direction of the shear band. They concluded that heterogeneity of a specific structure generated different types of chip formation. Sagapuram et al. [24] obtained the strain fields and microstructure in plane-strain cutting with particle image velocimetry and electron microscopy. Nanocrystalline structures were observed inside shear bands in Ti
6Al
4V with transmission electron microscope (TEM), and flow instabilities of segmented-chip formation were explained.

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275

Figure 10.6 Microstructure in shear-band region of chip: (A) optical micrograph; (B) grain size; and (C) dislocation density [22].

10.3

Microstructural models for machining

To reveal the microstructural evolution in machined surfaces and chips after machining, microstructural models are developed to understand the changes in microstructure, for example, dislocation density, dynamic recrystallization, and phase transformation.

10.3.1 Mechanism models of microstructural evolution 10.3.1.1 Dislocation density model In a severe deformation process a dislocation cell structure is assumed to form during deformation, consisting of two parts—dislocation cell walls and cell interiors— and obeying to a rule of mixtures. Different types of dislocation densities are distinguished in the model for the cell interior (ρc ) and the cell wall (ρw ). Their evolution laws are introduced as [5,6,25] 



ρ_ c 5 α

!   r 21=n 1 pffiffiffiffiffi r 6 γ_ c  r pffiffiffi _ ρw γ_ w 2 β 2 k ρc γ_ r c ; γ 0 c γ_ 0 3b bd ð12f Þ1=3

(10.1)

! pffiffiffi   r 21=n 2=3 3ð1 2 f Þ pffiffiffiffiffi r γ_ w  6ð12f Þ ρw γ_ c 1 β ρ_ w 5 β ρw γ_ r w ; γ_ r c 2 k0 fb γ_ 0 bdf 

(10.2) where α , β  , and k0 are the dislocation-evolution-rate control parameters for the material, n is the temperature-sensitivity parameter, f is the volume fraction of the dislocation cell wall, b is the magnitude of the Burgers vector of the material, d is

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the dislocation cell size, γ_ rw and γ_ rc are the resolved shear strain rates for the cell walls and interiors, respectively, and γ_ 0 is the reference resolved shear strain rate. The cell-wall volume fraction f varies with the accumulation of strain [26]: f 5 fN 1 ðf0 2 fN Þeð2γ =γ~ Þ ; r

r

(10.3)

where f0 and fN are the initial and saturation volume fractions of cell walls, respectively. γ~ r denotes the rate of variation of f with the resolved shear strain γ r . The total dislocation density is given by a rule of mixtures [27]: ρtot 5 f ρw 1 ð1 2 f Þρc ;

(10.4)

The dislocation cell size d is assumed to be inversely proportional to the square root of total dislocation density ρtot according to Holt’s formula as follows: K d 5 pffiffiffiffiffiffiffi ; ρtot

(10.5)

where K is a material coefficient. In addition to the earlier model, Melkote et al. [22] proposed a dislocation density model with a term representing the contribution of dislocation forests as follows:    D 0:5 ρtot 5 ρR 1 ρH&DRV 2 ρR ; D0

(10.6)

where ρR is the dislocation density corresponding to the fully recrystallized grain structure, D is the average grain size, D0 is the initial grain size, and ρH&DRV is the dislocation density due to slip-induced hardening and dynamic recovery processes (in the absence of dynamic recrystallization) and is expressed as: ρH&DRV 5

    A Bε 2 pffiffiffiffiffi A 1 ρ0 2 ; exp 2 B B 2

(10.7)

where ρo is the initial dislocation density, ε is the strain, and A and B are the hardening and dynamic recovery parameters, respectively.

10.3.1.2 Dynamic recrystallization model Prediction of grain changes in machining of a crystalline material was performed in a number of studies involving (1) Zener
Hollomon-based frameworks; (2) Johnson
Mehl
Avrami
Kolmogorov (JMAK) kinetics; and (3) continuous dynamic recrystallization models [28]. The JMAK model was widely used to

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277

describe the dynamic recrystallization process of a crystalline material in machining [3,21,29,30]. The volume fraction of dynamic recrystallization is defined with the Avrami equation as "   # ε2a10 εp kd XDRX 5 1 2 exp 2β d ; (10.8) ε0:5 where εp is the peak strain, and ε0:5 is the strain level for XDRX 5 0:5 and is defined as   Qact m5 ε0:5 5 a5 d0h5 εn5 ε_ m5 exp (10.9) 1 c5 ; RT where R is the universal gas constant, and T is the temperature. Dynamic recrystallization occurs when the critical strain εc 5 a2 εp is reached, with the peak strain εp given as   Qact m1 εp 5 a1 d0h1 ε_ m1 exp 1 c1 : RT

(10.10)

The recrystallized grain size is defined as  dDRX 5 a8 d0h8 εn8 ε_ m8 exp

 Qact m8 1 c8 : RT

(10.11)

The average grain size is calculated from the rule of mixtures as davg 5 d0 ð1 2 XDRX Þ 1 dDRX XDRX ;

(10.12)

where a1 , h1 , m1 , Qact , c1 , a5 , h5 , n5 , m5 , c5 , a8 , h8 , n8 , m8 , c8 , β d , a10 , and kd are the JAMK model parameters.

10.3.1.3 Phase-transformation model A phase transformation is another form of microstructural changes in the process of material deformation. Pan et al. [11] suggested a phase-transformation model according to the time-transformation-temperature profiles for α- and β-phase titanium. The volume fraction of α-to-β transformation in the heating process was expressed as "   # T 2Ts Ds fv 5 1 2 exp As ; (10.13) Te 2Ts where Ts is the phase-transformation starting temperature, Te is the temperature when the process ends, and As and Ds are material constants to be determined.

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Ramesh and Melkote [31] studied the effect of stress and strain on phasetransformation temperatures for steels. For the ferrite/martensite-to-austenite (α0
γ) transformation the equation was formulated as dP ΔHtr 5 ; dT TΔVtr

(10.14)

where ΔHtr is heat of transformation involved in the α0
γ transformation, and ΔVtr is the volume change per mole due to transformation. In addition, Zhang et al. [9] modeled a martensitic transformation based on strain-induced martensitic transformation kinetics. The evolution of martensite volume fraction is governed by the following equations:  f_α0 5 ð1 2 fα0 Þ Af ε_ p 1 Bf g_ ;

(10.15)

Af 5 αðΘÞrβ o fsbr21 ð1 2 fsb ÞP;

(10.16)

"  # β o fsbr21 1 g2g 2 Bf 5 pffiffiffiffiffiffi exp 2 H ðg_Þ; 2 sg 2πsg β0 5

c0 vα0 ; vI

(10.17)

(10.18)

where fα0 is the martensite volume fraction, f_α0 is the rate of increase in the volume fraction of martensite, ε_ p is the plastic strain rate of slip deformation in austenite, g_ is the rate of martensitic transformation driving force, αðΘÞ is a function of temperature, r models a random orientation of shear bands, P is the probability parameter, fsb is the volume fraction of shear bands, g and sg are the mean and standard deviation of the normal distribution function, respectively, H is the Heaviside unit-step function, c0 is a geometric constant, v α0 is the average volume per martensite unit, and v I is the average volume of a shear-band intersection.

10.3.2 Calculation of microstructural evolution To assess the evolution process and the final state of microstructural changes in machining, calculation models must be developed to visualize microstructural evolution. The FE method is a popular scheme used to understand the process of microstructural evolution during machining. Mabrouki et al. [32] and Yameogo et al. [21] carried out numerical simulations for microstructure evolution with Abaqus/ Explicit and used a Vumat subroutine tool to induce material flow stress and damage behaviors. The simulations were performed with a thermomechanical frame with a Lagrangian configuration. Atmani et al. [5] developed an FE model in Abaqus/Explicit based on the arbitrary Lagrangian
Eulerian (ALE) approach to predict the grain refinement during cutting. Ding et al. [33] also performed

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279

Figure 10.7 Calculation models for microstructural evolution: (A) FE model [33]; (B) cellular automaton model [7]; and (C) analytical model [9]. FE, Finite-element.

orthogonal cutting simulations in Abaqus/Explicit using the ALE technique; the coupled Eulerian
Lagrangian model setup and the predicted distribution of grain size are show in Fig. 10.7A. Fernandez-Zelaia et al. [4] and Melkote et al. [22] built an FE model of orthogonal cutting in AdvantEdge and implemented a physicsbased constitutive model employing a user-defined routine coded in FORTRAN to predict the microstructural changes. Arısoy et al. [3] and Abouridouane et al. [34] performed the FE simulations of microstructure in Deform-3D. In addition, the CA method was also applied to simulate the microstructure evolution; it can predict general microstructural characteristics and can also visibly describe the virtual microstructural evolution in detail. Shen et al. [7] developed a CA model to simulate the microstructural changes in primary and secondary deformation zones for cutting AA 1100, as shown in Fig. 10.7B. Besides, analytical models are another effective and efficient approach to predict the microstructural changes in machining. Zhang et al. [9] proposed an analytical model for prediction of the martensitic transformation, dislocation density, and microhardness variation in machining of 304 austenitic stainless steel (Fig. 10.7C). The predicted results in terms of cutting force, martensite fraction, microhardness, and residual stress were in good agreement with the measured data.

10.4

Microstructural evolution in ultrasonically assisted cutting

In this section a case of microstructural evolution in UAC is performed to compare the recrystallized process and final grain size in machined surface and chip for different machining techniques.

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A specific character of UAC, with the cutting tool separating from a workpiece in each cycle of vibration, determines strain, strain rate, temperature, and microstructure different from those in CC. Until now, no systematic investigation of microstructure was performed for UAC, except by Maurotto et al. [35]. They analyzed subsurface layers of workpieces obtained with ultrasonically assisted turning (UAT) and its conventional counterpart. No visible changes were found in the UAT workpiece with this qualitative observation; however, microstructural changes of machined surfaces might be not obvious from such observations of metallographic structures. Thus the investigation of microstructure evolution as a result of UAC is needed; so, both qualitative and quantitative comparisons are performed in this study. An FE-based model of orthogonal cutting with an enhanced material constitutive model is presented. The JMAK microstructure model is used to predict dynamic recrystallization as well as a resultant grain size of a machined surface and a chip for UAC and CC. In Section 10.4.1, microstructural changes in the surface machined in UAC and CC are compared and validated with optical microscopy; in addition, nanoindentation tests were performed to validate the distribution of grain size. Similar investigation of the chip is conducted in Section 10.4.2 to illustrate the microstructural differences in UAC and CC. In numerical simulations of processes involving irreversible deformation, a material constitutive model is required to calculate a flow stress. A modified Johnson
Cook (J
C) material model was presented by Calamaz et al. [36]; it ¨ zel [37] included flow softening at high strains and temperatures. Then Sima and O added some parameters to the equation; the modified material flow stress is expressed as follows:



  m   1 ε_ T2Tr σ 5 A 1 Bε 1 2 Tm 2Tr 1 1 C ln expðεa Þ ε_ 0

 s  1 3 D 1 ð1 2 DÞ tanh ðε1p ; r Þ n

 d D512

T Tm

(10.19)

 b ;

p5

T Tm

;

where A, B, C, a, b, d, m, n, p, r, and s are material constants, Tr is the room temperature, and Tm is the melting point of the workpiece. This approach modified a strain-hardening function of the J
C model by including flow softening at high strains, and the thermal softening function by including temperature-dependent flow softening. Therefore this flow-stress equation takes into account the strain, strain rate, temperature, and also the dynamic recovery and recrystallization mechanisms. Parameters of the Calamaz J
C material model for the Ti
6Al
4V alloy were ¨ zel et al. [38] and optimized in simulations (see Table 10.1). identified by O The updated Lagrangian formulation of software Deform-2D was used to achieve continuous remeshing in order to accommodate large deformations in the process zone. A plane-strain thermomechanical coupled analysis was performed.

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281

Table 10.1 Parameters of Calamaz-modified Johnson
Cook (J
C) material model for Ti
6Al
4V alloy. Calamaz J
C model

A

B

n

C

m

a

b

d

r

s

¨ zel et al. O

782.7

498.4

0.28

0.028

1

2

5

1

2

0.05

Table 10.2 Temperature-dependent material properties of Ti
6Al
4V alloy [38]. Properties

Ti
6Al
4V

E ðMPaÞ α (1 C21) λ (W/m/ C) Cp (N/mm2/ C)

0:7412T 1 113375 3 3 1029 T 1 7 3 1026 7:039e0:0011T 2:24e0:0007T

The temperature-dependent material properties of Ti
6Al
4V alloy were introduced; respective relationships for the modulus of elasticity (E), coefficient of thermal expansion (α), thermal conductivity (λ), and heat capacity (Cp ) are given in Table 10.2 for temperature T in  C. The serrated-chip formation was simulated by employing a fracture criterion by Cockcroft and Latham’s [39]. It is expressed as ð εf

σ1 dε 5 Dc ;

(10.20)

0

where ε f is the effective strain, σ1 is the major principal stress, and Dc is the material constant. This criterion relates the onset of fracture or chip segmentation to the integral of the major principal stress along the strain path reaching the critical value Dc ; this value was chosen as 245 MPa [40]. A schematic of orthogonal cutting with ultrasonic vibration in the direction of cutting velocity simulated below is shown in Fig. 10.8. The bottom side of the workpiece was provided with a kinematic boundary condition, while its top surface was free. The material was assumed to enter from the left-hand side of the workpiece and exit at its right-hand side and top surface. The cutting tool (rake angle of 0 degree, relief angle of 7 degrees) was assumed rigid and immovable in simulations of CC. However, vibration in the direction of cutting velocity was applied to the tool in the simulations of UAC. The vibration velocity was given by vx 5 2πfAx sinð2π ftÞ;

vy 5 0;

(10.21)

with the frequency f 5 20 kHz and amplitude Ax 5 20 μm. The maximum toolvibration speed 2πfAx (2513 mm/s) was larger than the cutting speed of 40 m/min (666.7 mm/s); thus the tool separated from the workpiece in each vibration cycle.

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Figure 10.8 Boundary conditions for orthogonal UAC. UAC, Ultrasonically assisted cutting.

Table 10.3 Johnson
Mehl
Avrami
Kolmogorov (JMAK) model parameters for Ti
6Al
4V alloy [41]. Ti
6Al
4V Peak strain DRX kinetics DRX grain size

JMAK model parameters a1 2 a5 1.21e 2 5 a8 150

h1 0 h5 0.13 h8 0

m1 0.006 n5 0 n8 0

Qact m1 1308 m5 0.04 m8 2 0.03

c1 0 Qact m5 8720 Qact m8 2 6540

a2 0.8 c5 0 c8 0

βd 0.693

kd 2

a10 0

In order to analyze microstructural evolution in UAC, the JMAK model was used. The main idea of this approach was to calculate a recrystallized volume fraction inside the material and use information on the initial grain size d0 to model the microstructure. The FE software Deform-2D provided the model of grain growth, static, metadynamic, and dynamic recrystallization (DRX) as the function of strain, strain rate, and temperature. In the cutting process, dynamic recrystallization took place. The volume fraction of dynamic recrystallization and average grain size were calculated with Eqs. (10.8)
(10.12). The JAMK model parameters for Ti
6Al
4V alloy were provided by running sensitivity analysis with Deform-2D FE simulations (see Table 10.3). The microstructure of Ti
6Al
4V alloy consists of two phases —α and β. α-Grains have a typical average grain size of d0 , while β-grains for the matrix containing net structures and are hard to assess. In the FE model the average grain size was calculated for α-grains; the as-received grain size was d0 5 20 μm, and phase transitions were not considered. Employing this JMAK model in Deform-2D allowed the assessment of microstructural evolution for the machined surface and the chip.

Microstructural changes in machining

283

10.4.1 Microstructural evolution in machined surface with ultrasonically assisted cutting In an orthogonal UAC process the cutting tool passes the machined surface back and forth; thus the magnitudes of strain, strain rate, and temperature differ from those for CC. Five points under the machined surface were determined to track the states of each parameter. Apparently, parameters change more quickly in areas close to the machined surface. So, the points were distributed unevenly in the depth: 0.005, 0.015, 0.050, 0.100, and 0.200 mm, as shown in Fig. 10.9. The calculated magnitudes of strain, strain rate, and temperature for the five points are shown in Fig. 10.10 for CC and UAC. The shaded zones represent the period that the tool passes above the points. The strain in UAC is somewhat larger than that in CC during the entire cutting duration as shown in Fig. 10.10A and B. However, evolution of the strain rate is quite different (Fig. 10.10C and D): it fluctuates sharply as the tool is engaged with the workpiece in UAC, reaching considerably higher values than that in CC. In addition, temperature of the machined surface in CC and UAC has the similar trend, increasing dramatically as the tool passes and reducing slowly afterwards to a steady state; the maximum temperature in UAC is larger than that in CC. It can be deduced from the JMAK model that the different characters of evolution of strain, strain rate, and temperature should lead to different evolutions of microstructure in CC and UAC. As shown in Fig. 10.11A and B, the recrystallized grain size for UAC is significantly different from that for CC. It is obviously smaller and eventually tends toward zero. It is because the tool separates from the workpiece in each cutting cycle in UAC, thus, resulting in the rapid change of strain rate and temperature fluctuations. However, both the strain rate and temperature have a great impact on dynamic recrystallization and dynamic recovery. As a result, the grain size in recrystallization cannot grow continuously due to the alternating dynamic recrystallization and recovery. In addition, the volume fraction of

Figure 10.9 Positions of five points under machined surface.

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

6.0 5.0 4.5

Strain (mm/mm)

(B)

CC Point 1 CC Point 2 CC Point 3 CC Point 4 CC Point 5

5.5

5.0 4.5

4.0 3.5

The period that tool passes

3.0

6.0 5.5

Strain (mm/mm)

(A)

2.5 2.0 1.5

UAC Point 1 UAC Point 2 UAC Point 3 UAC Point 4 UAC Point 5

4.0

The period that tool passes

3.5 3.0 2.5 2.0 1.5

1.0

1.0

0.5

0.5 0.0

0.0 –0.5 0.0000

0.0004

0.0008

0.0012

0.0016

–0.5 0.0000

0.0020

0.0004

Time (s) (D)

90,000

Strain rate (mm/mm/s)

80,000 70,000

The period that tool passes

60,000

CC Point 1 CC Point 2 CC Point 3 CC Point 4 CC Point 5

50,000 40,000 30,000

Maximum strain rate

20,000

0.0012

0.0016

0.0020

Time (s)

10,000

90,000

The period that tool passes

80,000 70,000

Strain rate (mm/mm/s)

(C)

0.0008

Maximum strain rate

60,000

UAC Point 1 UAC Point 2 UAC Point 3 UAC Point 4 UAC Point 5

50,000 40,000 30,000 20,000 10,000

0 –10,000 0.0000

0.0004

0.0008

0.0012

0.0016

–10,000 0.0000

0.0020

0.0004

Time (s) (F)

600

Maximum temperature

550 500

Temperature (°C)

450

The period that tool passes

400 350

CC Point 1 CC Point 2 CC Point 3 CC Point 4 CC Point 5

300 250 200 150

600

0.0016

0.0020

UAC Point 1 UAC Point 2 UAC Point 3 UAC Point 4 UAC Point 5

500 450 400

The period that tool passes

350 300 250 200 150 100

50

50

0

Maximum temperature

550

100

–50 0.0000

0.0012

Time (s)

Temperature (°C)

(E)

0.0008

0 0.0004

0.0008

0.0012

Time (s)

0.0016

0.0020

–50 0.0000

0.0004

0.0008

0.0012

0.0016

0.0020

Time (s)

Figure 10.10 Strain, strain rate, and temperature under machined surface for five points subjected to CC (A, C, E) and UAC (B, D, F): (A) and (B) strain; (C) and (D) strain rate; (E) and (F) temperature. CC, Conventional cutting; UAC, ultrasonically assisted cutting.

recrystallized grains has the same mechanism and the evolution trend. It can be concluded from Eq. (10.12) that the average grain size in a workpiece subjected to UAC is larger than that of CC as confirmed by Fig. 10.11C and D. The average grain size of surface and subsurface in CC decreases dramatically when the tool passes. After this pass the average grain size increases slowly due to the recovery, after which the grain size stabilizes. Eventually, the average grain size of the machined surface in UAC is larger than that in CC. Besides, the recrystallized grain

Microstructural changes in machining

(B)

18 16

The period that tool passes

14 12

CC Point 1 CC Point 2 CC Point 3 CC Point 4 CC Point 5

Recrystrallized grain size (μm)

Recrystrallized grain size (μm)

(A)

285

10 8 6 4 2 0 –2 0.0000

0.0004

0.0008

0.0012

0.0016

18 16

12

8 6 4 2 0

0.0004

The period that tool passes

20.0 19.5

(D) CC Point 1 CC Point 2 CC Point 3 CC Point 4 CC Point 5 19.16

19.0

18.65 18.43 18.32 18.28

18.5 18.0 17.5 17.0 0.0000

0.0004

0.0008

0.0012

Time (s)

0.0016

0.0020

Average grain size (μm)

Average grain size (μm)

20.5

0.0008

0.0012

0.0016

0.0020

Time (s)

Time (s) (C)

UAC Point 1 UAC Point 2 UAC Point 3 UAC Point 4 UAC Point 5

10

–2 0.0000

0.0020

The period that tool passes

14

20.5 20.0 19.69 19.49 19.42 19.38 19.37

19.5 19.0

The period that tool passes

18.5 18.0 17.5 17.0 0.0000

0.0004

0.0008

0.0012

UAC Point 1 UAC Point 2 UAC Point 3 UAC Point 4 UAC Point 5

0.0016

0.0020

Time (s)

Figure 10.11 Microstructure evolution under machined surface for CC (A and C) and UAC (B and D): (A) and (B) recrystallized grain size; (C) and (D) average grain size. CC, Conventional cutting; UAC, ultrasonically assisted cutting.

size reduces from P1 to P5 for both CC and UAC. On the contrary, the average grain size increases. To validate the numerical results for microstructural evolution in the machined surface, tests with orthogonal CC and UAC were performed. As shown in Fig. 10.12, the UAC device was fixed on the lathe, connecting the piezoelectric transducer and the concentrator. A cemented carbide
cutting tool was attached to the bottom of concentrator. The workpiece was machined with equidistant grooves, and the cutting edge was parallel to the cylindrical surface of the workpiece so that the cutting process was performed in a 2D plane state, the same as in the numerical analysis. Fig. 10.12 also shows the sample preparation for observation and assessment of the microstructure of the machined surface, obtained with wire-electrode cutting. After polishing and etching, microstructures of the machined surface were observed with a metallographic microscope. This study used orthogonal CC and UAC regimes with a cutting speed at 20 m/ min and a feed rate at 0.1 mm/rev. In UAC the vibration amplitude was 7.7 μm with frequency of 20,220 Hz. These parameters were also employed in numerical modeling for comparability in this section, including the original grain size of

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Figure 10.12 Experimental setup and preparation of sample of machined surface.

Figure 10.13 Microstructural analysis of machined surface of Ti
6Al
4V: (A) as-received microstructure and (B) microhardness measurement with nanoindenter.

16.22 μm discussed next. Still, the numerical analysis with a cutting speed at 40 m/ min, vibration amplitude of 20 μm, and the original grain size of 20 μm in the simulation described earlier was implemented to highlight the differences in the process of microstructural evolution in CC and UAC. Prior to machining tests, several specimens obtained from the Ti
6Al
4V cylindrical workpiece were used to analyze the as-received microstructure; Fig. 10.13A shows typical microscopic images. The microstructure of Ti
6Al
4V alloy consisted of two phases: α and β. α-Grains had a typical equiaxed structure, while β-grains formed the matrix. The average grain diameter of the cylindrical billet was found to be 16.22 μm. In addition, microhardness measurements were taken using nanoindentation to further verify the distribution of microstructure as shown in Fig. 10.13B. This allowed the indentations at much a smaller scale that is important for analysis of the chip. The microstructures obtained for areas under the machined surface are illustrated in Fig. 10.14; for this study, five points corresponding to locations studied in the

Microstructural changes in machining

287

Figure 10.14 Microstructure under machined surface: (A) CC and (B) UAC. CC, Conventional cutting; UAC, ultrasonically assisted cutting. (B)

28 26 24 22 20 18 16 14 12 10 8 6 4 2 0

CC

16.13 12.90

14.52 12.90 11.29 9.68 8.06

4.84

3.2

6.4

4.84 4.84

Frequency counts (%)

Frequency counts (%)

(A)

9.6 12.8 16.0 19.2 22.4 25.6 28.8 32.0

Distribution of grain size (μm)

27.12

28 26 24 22 20 18 16 14 12 10 8 6 4 2 0

UAC

16.95 13.56

13.56

8.47 5.08 5.08 3.39 3.39 3.39

3.2

6.4

9.6 12.8 16.0 19.2 22.4 25.6 28.8 32.0

Distribution of grain size (μm)

Figure 10.15 Distributions of grain size under machined surface: (A) CC and (B) UAC. CC, Conventional cutting; UAC, ultrasonically assisted cutting.

FE simulations under the machined surface were marked and the average grain size was measured using a linear intercept method. The measurements show that the grain size changed significantly: under the surface machined with CC it was in the range from 5.72 to 27.63 μm, while for UAC from 5.70 to 30.26 μm. The histograms for grain sizes under the machined surface for CC and UAC were obtained (Fig. 10.15); the statistical results showed that most grain sizes in CC were in the range from 8.45 to 22.25 μm. In contrast, a significant part of grains in UAC were in the size range close to the initial grain size d0 5 16.22 μm. The obtained numerical and experimental results for the average grain size for five different depths under the machined surface are compared for CC and UAC in Fig. 10.16. They demonstrate that the predicted and measured average grain sizes are very close. Generally, the average grain size in UAC is larger than that in CC and is more close to d0 . In addition, the main trend for the average grain size increased with the distance from the machined surface, that is, from P1 to P5.

288

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Figure 10.16 Comparisons of numerical and experimental results for average grain size under machined surface in CC and UAC. CC, Conventional cutting; UAC, ultrasonically assisted cutting.

In order to further validate the distribution of grain size under the machined surface for the two studied cutting techniques, microhardness of the machined surfaces was measured with the nanoindenter. Many researchers investigated a relationship between hardness and the grain size [42,43]; according to those results, hardness decreases with the increasing grain size. Indeed, the average grain size and microhardness respond differently to strain, strain rate, and temperature. A zone with severe plastic deformation stores more deformation energy, causing more intensive dynamic recrystallization. Thus the average grain size in such a zone is smaller, and it has more energy to resist the indentation deformation. Hence, the zone with a smaller grain size has higher microhardness. The microhardness tests of the workpieces under the machined surfaces were performed, with their results shown in Fig. 10.17. Generally, both CC and UAC demonstrated the same trend: the level of microhardness decreased with the depth. This decrease was rapid in CC and slow in UAC. Besides, microhardness near to the machined surface was higher in CC than that in UAC. Thus the microhardness study also confirmed the predicted microstructural changes in the workpieces machined with CC and UAC.

10.4.2 Microstructural evolution in chip with ultrasonically assisted cutting In order to investigate microstructural evolution in the chip, several points under the unmachined surface should be tracked. However, features of chip formation are not evident. So, initially, 30 points under the unmachined surface were chosen, forming 3 columns with 10 points each, distributed uniformly in columns. As shown in Fig. 10.18A, 30 tracked points moved from the undeformed chip zone to

Microstructural changes in machining

7.4 7.2 7.0 6.8 6.6 6.4 6.2 6.0 5.8 5.6 5.4 5.2 5.0 4.8 4.6 4.4 4.2 4.0 3.8 0.00

Experimental results Fitted curve

(B)

Microhardness (GPa)

Microhardness (GPa)

(A)

289

0.03

0.06

0.09

0.12

0.15

0.18

0.21

Distance from machined surface (mm)

0.24

7.4 7.2 7.0 6.8 6.6 6.4 6.2 6.0 5.8 5.6 5.4 5.2 5.0 4.8 4.6 4.4 4.2 4.0 3.8

Experimental results Fitted curve

0.00

0.03

0.06

0.09

0.12

0.15

0.18

0.21

0.24

Distance from machined surface (mm)

Figure 10.17 Microhardness under machined surface: (A) CC and (B) UAC. CC, Conventional cutting; UAC, ultrasonically assisted cutting.

Figure 10.18 Evolution of tracked points under unmachined surface: (A) evolution of tracked 30 points and (B) distribution of 10 points.

the deformation zone. It was also observed that the three columns had similar movement patterns; thus only one column is enough to track the microstructural evolution. It should be mentioned that the simulated process showed that the evolution of tracked points was similar in CC and UAC. So, 10 points under the unmachined surface were chosen for this purpose, they were distributed uniformly as shown in Fig. 10.18B. The movement of tracked points and evolution of average grain size with chip formation in CC and UAC were investigated as shown in Fig. 10.19. It was found that the upper points in the undeformed zone in both CC and UAC entered the sawtooth area in the deformation zone, the middle points moved into the primary shear zone, and the lower points entered the secondary shear zone as shown in Fig. 10.19A, B, D, and E. Eventually, all the points got into the chip (Fig. 10.19C and F). Besides, Fig. 10.19C and F indicate that the average grain size of the chip in UAC was larger than that in CC. For accurate comparisons the evolution of state variable for 10 points are studied next.

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Figure 10.19 Movement of 10 tracked points and evolution of average grain size with chip formation in CC (A, B, C) and UAC (D, E, F): (A) and (D) track points in undeformed zone; (B) and (E) track points in deformation zone; and (C) and (F) track points in chip. CC, Conventional cutting; UAC, ultrasonically assisted cutting.

The evolution of recrystallized and average grain sizes in the chip produced with CC and UAC is given in Fig. 10.20. The main trends of these processes were similar to those observed for the points under the machined surface (Fig. 10.11); still, the magnitudes of recrystallized and average grain sizes for UAC were quite different from those for CC. The process of dynamic recrystallization occurred when points moved into the deformation zone as seen in Fig. 10.20A and B. It is also apparent that it developed discontinuously in UAC, since the tool and the chip separated periodically. As a result, the recrystallized grain size in UAC was smaller than that in CC. Eventually, the average grain size in the chip in UAC was larger than that in CC (Fig. 10.20C and D). Interestingly, the final magnitudes of recrystallized and average grain sizes in CC and UAC for 10 points did not increase or decrease consistently from P1 to P10. For instance, the middle points had the highest recrystallized grain size and the minimum average grain size; the reason for this was their movement into the shear band. In order to verify the numerical simulation of chip microstructure, orthogonal cutting experiments were carried out for CC and UAC with the setup shown in Fig. 10.12 and chip samples prepared (Fig. 10.21). Since the cutting edge was parallel to the machined surface, the obtained chips were spiral columnar, convenient for the sample preparation and observation of their sections. The collected chips were mounted and polished and finally etched and observed with a metallographic microscope. The cutting speed in these orthogonal cutting experiments was set as

Microstructural changes in machining

291

Figure 10.20 Microstructure evolution in chip produced with CC (A and C) and UAC (B and D): (A) and (B) recrystallized grain size; (C) and (D) average grain size. CC, Conventional cutting; UAC, ultrasonically assisted cutting.

Figure 10.21 Sample preparation for chips.

20 m/min, and the feed rate was 0.1 mm/rev. The initial grain size was 16.22 μm. The vibration amplitude was 7.7 μm and its frequency was 20220 Hz in UAC. As mentioned earlier, the average grain size predicted for 10 points in the chip decreased or increased nonlinearly, and the distributions were complex. Thus the validation of the calculated average grain size of 10 points was cumbersome. So, another solution was found: a number of points between two sawteeth in the chip

292

(A)

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Average grain size (μm)

15.2

(B) 15.2

16.2

......

1

......

13

14.8

24 14.8

Average grain size (μm)

Average grain size (μm)

14.8 14.4

13.5

14.0

12.1

13.6 13.2

Minimum AGS AGS range

12.8

Minimum AGS AGS range

14.4 Average grain size (μm)

14.0

16.2

13.6

......

1

......

13

24

14.9

13.2 13.5

12.8 12.2

12.4

12.4 0

2

4

6

8

10

12

14

Number

16

18

20

22

24

0

2

4

6

8

10

12

14

16

18

20

22

24

Number

Figure 10.22 Calculated distributions of AGS between two sawteeth in chip: (A) CC and (B) UAC. AGS, Average grain size; CC, conventional cutting; UAC, ultrasonically assisted cutting.

Figure 10.23 Microhardness tests of chip: (A) nanoindentation tests with chip sample and (B) indentations in segmented chip.

were chosen to assess the distribution of the average grain size. The track points between these sawteeth were placed with the same spacing, with the line crossing the shear band. The predicted distribution of the average grain size between two sawteeth in the chip was obtained with the numerical model (Fig. 10.22). Apparently, this size decreased from P1 to the middle point, located in the shear band, followed by an increase to P24 in both CC and UAC. However, the minimum average grain size in UAC was larger than that in CC, and its range in UAC was smaller than that in CC; it means that average grain size was more uniform in UAC. Still, validation of the average grain size in the chip was quite complicated since it was hard to measure the grain size in the chip at the small scale. So, in this study only the microhardness tests of sawtooth chips were conducted (Fig. 10.23A);

Microstructural changes in machining

293

Figure 10.24 Microhardness distribution in chip: (A) CC; (B) UAC. CC, Conventional cutting; UAC, ultrasonically assisted cutting.

Fig. 10.23B shows positions of the indentations in the segmented chip. The zone of indentations is marked with red rectangles, with 24 indentations distributed on a line crossing the shear band. Results of the microhardness tests in the chips produced with CC and UAC are presented in Fig. 10.24. The distribution of microhardness in the chips has the same trend for both CC and UAC that is an inverse of the predicted distributions for the average grain size. The maximum microhardness in shear band in UAC was lower than that in CC. Moreover, the range of microhardness in UAC was smaller than that in CC. These results indicate that the microhardness in segmented chips produced with UAC was smaller and distributed more uniformly, which, in turn, validates the distribution of the average grain size. In general, the numerical model for microstructural changes in the machined surface and subsurface as well as in the chip of Ti
6Al
4V alloy machined with UAC was developed in this study. The distribution of average grain size in the machined surface and the chip was predicted. The recrystallized grain size for UAC was significantly different from that for CC; it was apparently smaller and tended toward zero after the cutting tool passed. Eventually, the average grain size of the machined surface in UAC was larger than that in CC. Besides, the recrystallized grain size reduced with distance from the surface for both CC and UAC; on the contrary, the average grain size increased. For the chip, 10 points under the unmachined surface were chosen to track the microstructural changes. The main trends of recrystallized and average grain sizes were similar to those observed for the points under the machined surface. But due to chip segmentation in Ti
6Al
4V, the final magnitudes of recrystallized and average grain sizes in CC and UAC for 10 points did not increase or decrease consistently. For the machined surface, experiments with orthogonal cutting demonstrated that the average grain size in UAC was larger than that in CC and was more close to the initial grain size. In addition, the main trend for the average grain size along the depth was an increase. For the chip, in the zone between two sawteeth,

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

the minimum average grain size appeared in shear bands for both CC and UAC. However, the minimum average grain size in UAC was larger than that in CC, and the range of the average grain size in UAC was smaller than that in CC; it means that the average grain size was more uniform in the former. Numerical modeling and experimental analysis both indicated that the average grain size in UAC was larger and more uniform, while the changes of the average grain size in both locations for UAC were smaller than those for CC, thus showing that the UAC is a less damaging processing technique.

10.5

Conclusion

In this chapter, microstructural changes in machined surface and chip caused by severe plastic deformation in machining were analyzed. A state-of-the-art overview of microstructural changes in machined surface and chip was presented. The main mechanism models of microstructural evolution in terms of dislocation density, dynamic recrystallization, and phase transformation were summarized. As important as mechanism models, the calculation models for microstructural evolution included FE, CAs, and analytical model were discussed. In addition, this chapter presented a case of microstructural evolution in UAC of Ti
6Al
4V, with numerical simulations and experiments performed to investigate the evolution of average grain size in machined surface and segmented chip in CC and UAC. It was concluded that the average grain size in UAC was larger and more uniform than that in CC, thus showing that the UAC is a less damaging processing technique.

Acknowledgments This work was supported by the National Basic Research Program of China (973 Program) grant no. 2013CB035805. The authors gratefully acknowledge the financial support from China Postdoctoral Science Foundation through grant no. 2019M652629 and no. 2019TQ0107.

References [1] E. Ezugwu, S. Tang, Surface abuse when machining cast iron (G-17) and nickel-base superalloy (Inconel 718) with ceramic tools, J. Mater. Process. Technol. 55 (2) (1995) 63
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6856. [19] S. Basu, M.R. Shankar, Spatial confinement-induced switchover in microstructure evolution during severe plastic deformation at micrometer length scales, Acta Mater. 79 (2014) 146
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[24] D. Sagapuram, et al., On control of flow instabilities in cutting of metals, CIRP Ann. 64 (1) (2015) 49
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J.C. Outeiro Arts & Metiers Institute of Technology, Campus of Cluny, Cluny, France

11.1

Introduction

Metal cutting, or simply machining, is one of the oldest processes for shaping components in the manufacturing industry, which must fulfill not only the dimensional and geometrical requirements but also the surface integrity. Applied to almost all materials (metals, wood, ceramics, etc.), this operation is used in almost all industrial sectors and represents nonnegligible part of the GDP of the developed countries. About 15% of the value of all mechanical components produced worldwide results from machining operations. Machining represents an important part of the cost of the final product (e.g., for a forged crankshaft the machining can represent 40% of the total cost of the piece). Therefore their cost reduction is the center of the preoccupations of industrial competitiveness. The reliability of machined components depends on a large extent on the physical state of their near-surface layers, also known as surface integrity. Surface integrity includes the alteration of material properties resulting from manufacturing operations, and its ability to withstand severe mechanical and thermal loading conditions in service (corrosion resistance, fatigue, etc.). Surface integrity can be evaluated in terms of the mechanical [residual stresses (RS), microhardness, etc.], metallurgical (grain size, phase transformations, microstructure, surface defects, etc.), chemical (bonding, reactivity, etc.), and topological states of the near-surface layers [surface roughness (SR), geometric variations, etc.] [1,2]. As abovementioned, the physical state of their surface layers also includes the distribution of RS induced by the machining process. These RS are not only the function of their machining history but also of previous processing [3]. They can enhance or impair the ability of a component to withstand loading conditions in service (fatigue, creep, stress corrosion cracking, etc.) (Fig. 11.1), depending on their nature: compressive or tensile, respectively. For example, compressive RS can increase the fatigue life of the components, as they delay crack initiation and propagation [4]. Furthermore, the residual stress distribution on a component may also cause dimensional instability (distortion) after machining. This poses enormous problems in structural assembly, and it affects the structural integrity of the whole part. Although the direct influence of the RS on the behavior (the static and dynamic strength, chemical and electrical properties, fatigue, rust, etc.) of the mechanical component in service is well known, a little is known about the origin of the Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques. DOI: https://doi.org/10.1016/B978-0-12-818232-1.00011-4 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Fatigue life Stress corrosion cracking

Fracture

Residual stresses Tensile strength

Wear

Dimensio nal stability

Coating adhesion

Figure 11.1 Effects of residual stresses.

machining RS and how these RS could be controlled during this process in order to achieve their desirable distribution. This is particularly important for the nuclear and aeronautic industrial sectors, where high-reliability levels are required. Unfortunately, due to the difficulties into control and obtain suitable RS distributions by machining, such industries are forced to apply postsurface treatments (such as low plasticity burnishing, shot peening, laser shock peening, carbonitriding) to deliberately introduce high compressive RS, thus achieving long fatigue life of critical components. To summarize, the understanding the origin of the residual stresses, their correlation with the process’s parameters and their control in machining are necessary. Moreover, despite significant experimental and modeling studies on machininginduced RS in the past, very few studies deal with design problems involving residual stress issues. With the development of different experimental and modeling techniques, the massive application of the residual stress in the design office for the integrated design of mechanical components is essential [5].

11.2

Fundamentals of machining and residual stresses

11.2.1 Metal-cutting definition and energy considerations As a science, the metal cutting has more than 100-year history of extensive studies. Unfortunately, a much smaller volume of research was devoted to discovering the fundamental mechanisms underlying metal-cutting process in general, as opposed to seeking case solutions for particular metal-cutting problem. A landmark for seeking particular down-to-earth solutions was set by Taylor [6] when he presented a

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simple way of experimental study in metal cutting by finding a simple relationship between the cutting speed and tool life. Up to date, this equation creates a false impression that the solution for any particular metal-cutting problem can be found by conducting a small series of tests. The beginning of the “modern” metal-cutting era is marked by the publication of Merchant’s vision of the metal cutting (a single shear plane model), which, even after numerous attempts to improve this theory, fails to improve its predictive ability [7]. After many years of study, theory is still lagging behind practice, which relays on time-consuming trial and error-machining tests and guided by empirical knowledge of the metal-cutting process. Therefore an urgent need is felt to develop new metal-cutting models, based on physical phenomena observed during this process. However, not many specialists in the field appreciate this importance, and therefore, an explanation of the significance of models in this field is required. In the past the components of the machining system were far from ideal for normal tool performance. Tool specialists (design, manufacturing, and application) were frustrated with old machine tools having spindles that could be rocked by hand; part fixtures that clamped parts differently every time; part materials with inclusions and great scatter in the essential properties; tool holders that could not hold tools without excessive runouts and/or assuring the proper position; starting bushing and bushing plates that had been used for years without replacement; lowconcentration contaminated coolants, manual sharpening of cutting tools by the eye; cutting speed and feed with limited range; low dynamic rigidity of machines; etc. For many years a stable balance between low-quality (relatively inexpensive tools) and poor machining system characteristics was maintained. Metal-cutting research was attributed mainly to university labs, and their results were mostly of academic interest rather than of practical significance. It is clear that the metalcutting theory and tool design based on this theory were not requested by practice. This has been rapidly changing since the beginning of the 21st century as global competition forced many manufacturing companies, first of all car manufacturers, to increase the efficiency and quality of machining operations. To address these issues, leading tools and machine manufacturers have developed a number of new products—new tool materials and coatings, new cutting inserts and tool designs, new tool holders, powerful precision machines, part fixtures, advanced controllers that provide a wide spectrum of information on cutting processes, and so on. These increase the efficiency of machining operations in industry by increasing working speeds, feed rates, tool life, and reliability. These changes can be called the “silent” machining revolution as they happened in rather short period of time. Implementation of the listed developments led to a stunning result: for the first time in the manufacturing history, the machining operating time became a bottleneck in the part machining cycle time. Therefore the development of an advanced realistic metal-cutting theory and high-productivity reliable cutting tools designed using this theory became a necessity [8]. The question is “What is metal cutting?” [9], or in other words, “What’s the physical phenomena governing metal cutting?” According to Astakhov [10], metal cutting can be viewed as a forming process, which takes place in the components of

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the cutting system that are so arranged that the external energy applied to this system causes the separation of a layer of material from the bulk workpiece. The principal difference between metal cutting and all other metal forming processes is the physical separation of the layer being removed in the form of chips from the rest of the workpiece. The process of physical separation of a solid body into two or more parts is known as fracture, and thus, metal cutting must be treated as the purposeful fracture of the layer being removed. This fracture occurs due to the continuous change of the state of stress in the first deformation zone causing a cyclic nature of the process [10]. Therefore the main goal in metal cutting should be the minimization of the external energy applied to the cutting system by generating suitable state of stresses that reduces the energy required for this material separation. Considering the metal-cutting definition described above, not only the strength of the work material but also the strain at fracture should be considered. The product of these two mechanical characteristics indicates the amount of energy that should be spent in fracturing a unit volume of the work material, allowing chip formation. As shown in Fig. 11.2, this energy is higher for difficult-to-cut materials such as Inconel 718 and AISI 316L stainless steel, when compared to the other (soft) work materials such as AISI 1045 carbon steel and AZ31B magnesium alloy. Most of this energy is converted into heat, which combined with the heat generated by friction at the tool
chip and the tool
workpiece interfaces, and the low thermal conductivity of these two difficult-to-cut materials (about 30% of the plain carbon steel), result in high localized temperatures [11]. These temperatures are particularly high at the tool
chip interface that together with the high contact stresses (both normal and shear) at this interface causes rapid tool wear and tool failures during machining of such alloys. Moreover, high cutting forces and high localized temperatures at the tool
machined surface interface may dramatically affect the surface integrity often resulting in the development of high tensile RS in the machined surfaces [12,13].

Figure 11.2 Estimation of the energy spend in plastic deformation during a tensile test until fracture for several materials.

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11.2.2 Definition and origins of residual stresses RS are multiaxial static stresses that exist in an isolated component or structure without any applied external force or moment, and they are in mechanical equilibrium. RS exist on all components, metallic or nonmetallic, and reflect their mechanical and structural history throughout their manufacturing process chain (casting, machining, surface treatments, etc.). However, they can also be produced when the component is in service due to the applications of thermal, mechanical, or chemical loadings. In both cases, RS are originated by the elastic response of the material to the heterogeneous distributions of inelastic (plastic) deformations, at any scale of the component or structure. These stresses are produced by heterogeneous (mechanical) plastic deformations, thermal contractions, and phase transformations induced during the manufacturing process such as casting, welding, machining, forming, and heat treatment. They are usually classified according to the length scale over which they equilibrate, into macro- and microstresses (Fig. 11.3). The macrostresses (or type I stresses) are those stresses equilibrated on large length scales of the order of several grains or even the components or structure size. The microstresses are equilibrated on length scales of a grain (type II stresses) or across smaller lengths of the order of the atomic scale (type III stresses).

11.2.3 Techniques for measuring residual stress Several techniques can be applied to determine the RS in engineering components/ applications. These techniques can be classified into mechanical (hole-drilling, contour, curvature, and layer removal), diffraction [X-ray diffraction (XRD),

Figure 11.3 Schematic representation of macro-(σIR ) and micro-(σIIR ; σIII R ) stresses.

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neutron diffraction, synchrotron radiation], and others (magnetic, ultrasonic, Raman spectroscopy). The selection of the proper technique is a critical issue, and the decision will depend on practical (size of the component, availability of the equipment, level of expertise required, cost, etc.), material (type of material to analyze, surface condition, etc.) and measurement issues (spatial resolution, penetration, type of stress and gradient that can be analyzed/evaluated, accuracy of the measure) [14]. Most of the residual stress measurements have been carried out in metals, including cast iron, steels, light alloys (such as aluminum, titanium, and magnesium alloys) and nickel-based superalloys. However, there is also an increasing interest to measure the RS in composite, polymer, ceramic, and other nonmetallic materials. The mechanical techniques rely on the monitoring of changes in component distortion, either during the generation of the RS, or afterward, by deliberately removing material to allow the stresses to relax [15]. Measuring these distortions using contact or noncontact techniques, the RS can be calculated using the elasticity theory. The major advantages of these techniques are their relative simplicity, quickness, low cost, and applicability to wide range of materials. The major disadvantages are their low resolution, and they are destructive. The diffraction techniques relay to the use of the radiation, such as X-rays and neutrons, to access changes in the interplanar atomic spacing of a specific family of lattice planes, and therefore, to calculate the elastic (residual) strains and stresses [16]. Indeed, the presence of residual stress within a polycrystalline material causes elastic strain and thus changes the spacing of the lattice planes from their stress-free value to a new value, which corresponds to the magnitude of the applied residual stress. Using X-ray or neutron diffraction, it is possible to measure the shift in the angular position of the diffraction peak in relation to its position when the material is without RS. The interplanar atomic spacing can be calculated knowing the angular position of the diffraction peak and applying the Bragg law. Knowing the interplanar spacing, the elastic strain can be calculated, and applying elasticity theory, the residual stress can be determined. The major advantages of these techniques are their good resolution (in particular the case of X-ray and synchrotron), they operate without contact, and, consequently, they are nondestructive (except when applying XRD and electrochemical removal process to evaluate the RS below surface). The major disadvantages are the high cost of the equipment and measurement, high level of expertise required, and they are limited to crystalline materials. The other methods are not used so frequently for residual stress evaluation. In general, such techniques such as magnetic and ultrasonic are nondestructive, cheap, simple to use, very fast, and the equipment are portable, which are suitable to routine inspections. However, low resolution is their major limitation, in particular, when compared with the diffraction techniques. The Raman spectroscopy is an exception. Its high resolution (less than 1 μm, thus higher than the diffraction techniques) makes it suitable to evaluate the RS in extremely narrow regions of a few micrometers as is the case of fiber composites, providing basic information about the RS distribution from fiber ends to centers [15]. Unfortunately, the major disadvantages of this technique relay to its calibration and limited range of materials that

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can be analyzed. Many other techniques are being developed for measuring RS most of which are still in the research and development stage. From the range of techniques described above the hole drilling and the XRD techniques are the most used in practice. According to a recent survey carried out in the United Kingdom [14], covering a representative cross section of UK industry and academia show more than 55% use the hole drilling and the XRD techniques for residual stress analysis, because they fit most of the practical, material, and measurement issues. This survey also shows that almost 50% consider the RS of high importance to their business, whilst 30% ranked them as of medium importance. A relatively good agreement between the results of both hole drilling and XRD techniques is obtained, in particular, in the interior of the samples [17]. However, the hole-drilling technique is not able to determine the RS near the surface, and also not recommended to evaluate strong residual stress gradients, such as those generated by machining of several engineering materials. The observed discrepancies between the RS measured by both techniques are often attributed to the basic shortcoming of the hole-drilling technique, which is its limitation to RS up to 60% of the material’s yield strength [18]. Because the drilling operation induces plastic deformations, the so-called plasticity effect can strongly affect the residual stress evaluation, which assumes linear elastic material behavior. In spite of these limitations, hole drilling can also be applied to determine the RS induced by machining, in particular, for evaluating the RS induced by machining of noncrystalline materials, such as the case of polymers, carbon/glass fiber reinforced polymer composites [19]. Table 11.1 compares both XRD and hole-drilling techniques. The table presents several criteria to be considered to select a suitable technique for determining the RS induced by a given machining operation.

11.3

Residual stresses in machining operations

11.3.1 Origin of residual stresses in metal cutting RS distribution in the machined components results not only from the machining history but also from the previous materials processing. The machining history consists of a sequence of machining operations (turning, milling, drilling, etc.) and corresponding cutting parameters. In this sequence the effect of successive machining passes should be also considered [20]. Defining a logical machining sequence for a given component, the resulting residual stress distribution in the machined surface layers will depend on the machining parameters used in each operation, being the strongest contribution given by the last one. Fig. 11.4 shows a schematic representation of the mechanisms of RS formation in machining. In general, RS are formed due to the gradient of plastic deformation existing in the component from the machined surface to the bulk material. This heterogeneous plastic deformation results from the complex interaction between the thermal and mechanical phenomena developed during the cutting process.

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Table 11.1 Criteria used for selecting a suitable technique for determining the residual stresses induced by a given machining operation. Criteria

X-ray diffraction

Hole drilling

Contact Size of the component Destructive Cost (equipment and measurement) Availability Portability Measurement speed Level of expertise required Type of material to analyze Surface condition

No Any No (surface); yes (depth) High

Yes Any Semi Low

Available Both Fast/medium (single measure) Medium/high

Widespread Both Fast Low/medium

Crystalline Important

Spatial resolution

10 mm depth, 1 mm laterally

Penetration

,10 mm (Fe); ,50 mm (Al); (1 mm by layer removal) Macro, micro

Crystalline, amorphous Very important (flat) 50
100 mm depth increment 1.2 3 hole diameter Macro

Strong Very good but depends on several factors

Smooth Good but varies with depth

Type of stress that can be determined Type of stress gradient Uncertainty

These two phenomena are commonly referred to the origins of the RS in machining. Phase transformation is another source of RS. However, phase transformation is a consequence of both thermal and mechanical phenomena, which can produce both tensile and compressive stresses depending on the material volume variation [21]. The heat generated in machining, which is produced by plastic deformation and friction, represents the thermal phenomenon. However, only the portion of the heat conducted to the workpiece can generate RS in the machined part. In general, this heat will contribute to the formation of tensile RS due to the thermal expansion and contraction of the surface machined affected layers. Since the core of the workpiece is not deformed plastically, heterogeneous plastic deformation in the cross section of the component is created and consequently the RS. However, this heat also influences the mechanical properties of the work material, thus indirectly the RS formation in the machined part. The mechanical phenomenon also induces heterogeneous plastic deformation due to the mechanical action of the tool over the workpiece. Liu and Barash [22] and later by Wu and Matsumoto [23] proposed a simple approach to describe that the residual stress formation in metal cutting consists of using a stress
strain curve obtained in a standard uniaxial tensile test. According to this approach, RS in the

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Thermal phenomenon

Thermal expansion

Mechanical phenomenon

Phase transformation

Heterogeneous plastic deformation

Residual stresses

Figure 11.4 Schematic representation of mechanisms of residual-stress formation in machining.

machined surface and subsurface are the consequence of a compressive load followed by a tensile load to which a micro-volume of the work material animated by the cutting speed Vc is submitted, when it follows the trajectory represented in Fig. 11.5A. According to this approach, the material element experiences deformation in compression when is located ahead of the tool, followed by yielding. Then, it experiences deformation in tension when it passed through the tooltip, followed by an eventual second yielding. Residual stresses in machining will depend on the relative magnitude of the tensile and compressive loads. As shown in Fig. 11.5B, a predominantly compressive loading over the material is ahead of the cutting tool will induce tensile residual stress, while a predominantly tensile loading results in compressive residual stress. This cyclic loading is influenced by the work material properties and chip-formation process.

11.3.2 Residual stresses in difficult-to-cut materials These RS distribution in the machined surface and subsurface depends on the mechanical and thermal properties of work material, cutting tool geometry and material, metal-working fluid (MWF), and machining operation. For the same machining operation and cutting conditions (including the same cutting tool, cutting regime parameters, and MWF), these stresses are higher for the work materials that

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Figure 11.5 Approach for explaining residual stress formation proposed by Liu and Barash, which considers the mechanical action of the tool over the surface of the workpiece: (A) micro-volume of the considered work material and its trajectory during cutting; and (B) residual stresses formation (1) by a predominant compressive loading and (2) by a predominant tensile loading.

exhibit higher energy spent in plastic deformation until the physical separation of the layer being removed in the form of chips from the rest of the workpiece occurs. Therefore the RS levels at the machined surface are higher for difficultto-cut materials such as Inconel 718 and AISI 316L stainless steel compared to the other materials shown in Fig. 11.2. Changing the machining operation and corresponding cutting conditions just changes the state of stress in the chipformation zone, thus the strain at fracture and consequently this energy. In addition, since the cutting edge is not perfectly sharp, but rather is a transition surface between the rake and flank faces, the friction generated over this surface and a part of the tool flank contact also plays an important role in the residual stress generation. Therefore the final residual stress distribution depends on both energies spend in plastic deformation until fracture in the chip-formation zone and that one generated by friction over the tool flank contact. A rapid cooling of the machined surface (e.g., using LN2) can add additional thermal stresses, which will also contribute to the resultant RS. In this section, special attention was devoted to the determination of the RS induced by machining difficult-to-cut alloys. These alloys correspond to a group of materials that requires higher cutting energy compared with low strength alloys (e.g., plain carbon steel). This group includes several alloys used in aerospace and nuclear applications, which can be classified into three major categories: nickelbased alloys (e.g., Inconel), iron-based alloys (e.g., austenitic stainless steels and case-hardened steels), and titanium-based alloys.

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11.3.2.1 Nickel-based alloys The nickel-based superalloys are heat-resistant alloys with high melting temperatures. The ability to retain high mechanical properties, good metallurgical stability, and high corrosion and creep resistances at elevated temperatures [24] makes these superalloys an ideal material for use in combustion systems, landbased power generators, and aerospace aero-engine components. These alloys, in particular, the Inconel 718, are often used in a solution-treated and aged condition. This heat treatment results in a special microstructure consisting of large grains containing a precipitated phase (γ phase), and a heavy concentration of carbides at the grain boundaries. The difficulty of dislocation motion through this microstructure is responsible for the high tensile and yield strength of the material. Moreover, Inconel 718 alloy is a highly strain-rate sensitive material, which hardens considerably. This makes Inconel 718 alloy a very difficult-to-cut material. The most critical issues in machining this alloy (and in general any nickel-based alloy) are often associated with short-tool life and poor surface integrity [25]. Fig. 11.6 shows the RS in function of the depth beneath machined surface, induced by orthogonal cutting of Inconel 690 (Fig. 11.6A) and turning of Inconel 718 (Fig. 11.6B) [13,26]. For both Inconel alloys the surface RS are always tensile in circumferential (direction of primary motion) and axial (direction perpendicular to the primary motion) directions, being these stresses are higher in the circumferential direction. As shown in Fig. 11.6, they are higher at the surface, sometimes reaching a value of around 1200 MPa in the circumferential direction. The level of RS in both directions decreases continuously with depth into the machined surface, stabilizing at a level corresponding to that found in the work material before machining. In the case of the RS generated by turning, they shift to compressive values beneath the surface before reaching the maximum in compression, while those stresses generated by orthogonal cutting are almost tensile through all the machined affected depth.

Figure 11.6 In-depth residual stress profiles obtained by (A) orthogonal cutting of Inconel 690 and (B) turning of Inconel 718.

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Fig. 11.6B shows that machining with uncoated tools when compared with coated (TiAlN coating of a 2 μm thickness) results in (1) higher surface RS; (2) lower thickness of tensile layer; and (3) lower RS (maximum) in the subsurface, with the maximum being shifted closer to the surface in machining with uncoated tools. These different residual stress profiles induced by coated and uncoated tools can be attributed to the thermal and properties of the cutting tools besides the different friction coefficients between these tools and the Inconel 718 [27,28]. As far as the influence of the cutting conditions are concerned, Fig. 11.7 shows the influence of the cutting speed and uncut chip thickness on the RS at machined surface, induced by orthogonal cutting of Inconel 690. As shown in this figure, the RS at the machined surface does not change significantly when the cutting speed increased from 60 to 175 m/min, being their variation within the error of the measurement. However, RS increase by about 200 MPa when the uncut chip thickness increases from 0.1 to 0.35 mm (Fig. 11.7B). The increase of the RS with the uncut chip thickness is reported in many studies in the literature, having this parameter the largest influence on these stresses [12,29]. In fact, increasing the uncut chip thickness increases the compressive stresses ahead of the tool cutting edge [30], which according to the approach shown in Fig. 11.5 will increase the RS. Fig. 11.8 summarizes the influence of the cutting speed, tool material (PCBN (polycrystalline cubic boron nitride) vs coated carbide), and tool wear on the nearsurface residual stress with a related uncertainty of less than 30 MPa. The results indicate that when a new tool is used, the maximum principal stress near the surface is similar for both cutting tool materials and different cutting speeds, and these are in the range of 460
660 MPa. However, in the presence of tool wear, the stress increases significantly (about 125% at Vc 5 200 and 300 m/min) when the PCBN worn tool is used, reaching about 1800 MPa, while this increase is moderate for the coated carbide tool (about 44% at Vc 5 90 m/min), reaching about 1000 MPa. Such high tensile residual stress values at the machined surface in the presence of tool wear can be mainly due to an increase in the heat generated by friction at

Figure 11.7 Influence of the (A) cutting speed and (B) uncut chip thickness on surface residual stresses induced by machining Inconel 690.

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Figure 11.8 Residual stress at machined surface of Inconel 718.

Figure 11.9 In-depth residual-stress profiles obtained under orthogonal cutting of Ti6Al4V.

the tool flank
machined surface interface. This heat generated is higher for the case of the PCBN, due to the higher cutting speeds used with this tool material when compared to the coated one. This heat combined with the lower thermal conductivity of the PCBN tools (44 W/m K for the PCBN and 100 W/m K for the coated cemented carbide [7]), will increase the temperature of the machined surface, thus the tensile RS.

11.3.2.2 Titanium-based alloys Titanium-based alloys are also difficult-to-machine materials owing to their high strength, low thermal conductivity, and strong chemical reactivity [31]. In particular, γ-TiAl is, in general, more difficult-to-machine than the standard titanium alloys (including Ti6Al4V), resulting in an extremely short-tool life [32]. Therefore machining productivity is low, the production cost is high, and the surface integrity can also be very poor. Fig. 11.9 shows some results obtained in this study, concerning the in-depth residual stress profiles induced by orthogonal cutting of Ti6Al4V alloy using

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uncoated tungsten carbide cutting tools. The RS in both directions are mainly compressive, except for the circumferential stress in a thin superficial layer where they are slightly tensile. These stresses are incomparably lower when compared with those obtained by orthogonal cutting of Inconel 718 under the same cutting conditions [33,34]. In general, RS generated by machining of Ti6Al4V alloy are almost compressive or low tensile [35].

11.3.2.3 Austenitic stainless steels Austenitic stainless steels are considered as difficult-to-cut materials because of their low thermal conductivity, and high mechanical and microstructural sensitivity to strain and stress-rate. They exhibit severe work hardening during the chipformation process compared with low alloy steels, which induces mechanical modifications and behavior heterogeneity on the machined surface. This results in unstable chip formation and may cause vibrations. Their low thermal conductivity also leads to heat concentration in the cutting zone resulting in high localized interfacial temperatures. As a result, the machining of such steels compared with machining of plain carbon steels (see Fig. 11.10) may induce [27]: (1) higher residual stress levels, (2) larger thickness of the tensile layer, (3) high work-hardening, and (4) larger thickness of the work-hardened layer. A detailed experimental analysis of the RS induced by machining of AISI 316L can be found in the literature [12,27,29,36]. This analysis includes the RS generated by both orthogonal cutting and turning operations, under different cutting conditions, including coated and uncoated cemented carbide cutting tools, cutting speed, feed and depth of cut. Fig. 11.11A
C shows the typical in-depth RS profiles in the circumferential (direction of the primary motion) directions induced by turning of AISI 316L using uncoated tungsten carbide cutting tools (reference H13) [12]. Identical in-depth RS profiles in the longitudinal (direction of the feed motion) are obtained. For the most investigated cutting parameters the tensile RS sometimes reaching 500 MPa were found on the machined surface in the circumferential

Figure 11.10 In-depth profiles of residual stresses generated by turning of AISI 316L steel and AISI 1045 steel with coated cemented carbide cutting tools.

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Figure 11.11 In-depth profiles of residual stresses generated in turning of AISI 316L steel using uncoated cemented carbide cutting tools: (A) typical in-depth residual stresses profiles in the longitudinal (direction of feed motion) and circumferential directions; (B) influence of the cutting speed; (C) influence of the feed; and (D) influence of the depth of cut.

direction, whereas the compressive RS of values sometimes reaching 2550 MPa were found in the longitudinal direction. The level of RS in both directions changes continuously with depth down to a certain maximum value in the compressive region and then gradually decreases stabilizing at the level corresponding to that found in the work material before machining. The actual depth at which the circumferential RS reach the zero stress value can be thought of as the thickness of the tensile layer due to machining. Fig. 11.11 shows the influence of the cutting speed (Fig. 11.11A), feed (Fig. 11.11B), and depth of cut (Fig. 11.11C) on the in-depth residual stress in the circumferential direction (the most critical). Fig. 11.11A shows that both surface residual stress and the thickness of the tensile layer decrease as the cutting speed increases. Fig. 11.11B shows that both surface residual stress and the thickness of the tensile layer increase with the feed. Finally, Fig. 11.11C shows that surface residual stress does not change significantly with the depth of cut, but the thickness of the tensile layer decreases as the depth of cut increases. However, if the feed is reduced from 0.2 to 0.1 mm/rev, the surface residual stress decreases as the depth of cut increases (Fig. 11.11D).

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11.3.2.4 Case-hardened steels This section deals with the RS induced by orthogonal cutting, turning and milling operations of several kinds of case-hardened steels, including molds and dies steels (AISI H13 and AISI D2) besides bearing steel (AISI 52100).

AISI H13 AISI H13 tool steel is characterized by good resistance to thermal softening, high hardenability, high strength, and high toughness. This steel has therefore been widely used to produce many different types of hot working dies, such as forging dies, extrusion dies, and die-casting dies. Complex workpiece geometries, high material hardness, and short lead times are among the main obstacles to increase productivity in machining such dies. At the same time, quality and reliability requirements are becoming more and more important, due to intensified competition and greater quality awareness. These quality and reliability are directly related to surface integrity [33]. Traditionally, SR is the principal parameter used to assess the surface integrity of machined hot working dies made from AISI 13H tool steel. However, RS are also becoming an important parameter. Their control during manufacturing will increase mold and die lifetime and their ability to withstand severe thermal and mechanical loading cycles (fatigue) in service. Significant improvements in the quality of the mold or die can therefore be achieved with the control of the RS induced during its manufacturing. Figs. 11.12 and 11.13 show the influence of the cutting conditions (cutting speed, uncut chip thickness, cutting edge preparation), and tool wear on the RS induced by dry orthogonal cutting of AISI H13 (HRC 51), using PCBN cutting tools [37]. Two tool-cutting edge preparations were used. One was a TNGN110308S (referred as T-land or chamfer) with a negative chamfer rake angle of 20 degrees 3 0.1 mm and cutting edge radius (rn) of 15 μm, while the other was a TNGN110308E (referred as hone) with the cutting edge radius (rn) of 15 μm and without chamfer. Both cutting inserts with different edge preparations presented the same tool rake and clearance angles of 28 and 8 degrees.

Figure 11.12 Influence of the (A) cutting speed and (B) uncut chip thickness on axial residual stress.

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Figure 11.13 Influence of the (A) cutting edge preparation and (B) tool flank wear on axial residual stress.

RS were measured in the circumferential (direction of primary motion) and axial (direction normal to the primary motion) directions. RS in the axial direction are always compressive, while those in the circumferential can be sometimes tensile depending on the cutting conditions [37]. Therefore the axial residual stress will be critical (tensile) for the functional performance and life of components, thus only the stress in the axial direction will be used to analyze the influence of the cutting conditions. Fig. 11.12 shows the influence of the cutting speed (Fig. 11.12A) and uncut chip thickness (Fig. 11.12B) in the in-depth axial residual stress distribution. As shown, the axial surface residual stress decreases in compression, becoming tensile, as the cutting speed and uncut chip thickness increase. Similar trend is observed concerning the maximum compressive stress below surface. This maximum decreases in compression with increasing both cutting speed and uncut chip thickness. Fig. 11.13A shows the influence of the cutting edge preparation (honed vs chamfered) in the in-depth axial residual stress distribution. Under the same cutting conditions the honed tool produces compressive surface RS, while the chamfered tool produces tensile surface residual stress. Moreover, the maximum compressive stress below surface is lower for the case of the chamfered tool. Fig. 11.13B shows the influence of tool flank wear (VB) in the in-depth axial residual stress distribution. As shown in this figure, the surface axial RS decrease in compression as VB increases. Moreover, the maximum compressive stress below surface shifts further from surface with increasing VB. Fig. 11.14 shows the RS induced by turning of AISI H13 tool steel with different hardnesses (HRC 16, 46, and 51), using coated cemented carbide (TiCN/Al2O3/ TiN, CVD coating) and PCBN cutting tools (low cubic boron nitride (CBN) contents, in this case 50% CBN), and applying different cutting parameters (cutting speed, feed, and depth of cut) [38]. This figure shows that maximum principal residual stress (σmax) is always tensile, reaching values higher than 1500 MPa, while the minimum principal residual stress (σmin) can be tensile or compressive depending on the cutting parameters, cutting tool, and work material hardness.

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1750

1250 1000

σmax –AISI H13 (HRC 51)–coated carbide σmax –AISI H13 (HRC 46)–coated carbide σmax –AISI H13 (HRC 16)–coated carbide σmax –AISI H13 (HRC 51)–PCBN σmin –AISI H13 (HRC 51)–coated carbide σmin –AISI H1 3(HRC 46)–coated carbide σmin –AISI H13 (HRC 16)–coated carbide σmin –AISI H13 (HRC 51)–PCBN

750 500 250 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 50 51 52 53 54 55 56

Principal residual stresses (MPa)

1500

–250 –500 –750

Figure 11.14 Surface residual stresses in turning of AISI H13 tool steel using both coated cemented carbide and PCBN cutting tools, for 56 different cutting conditions.

The PCBN cutting tool generates higher tensile RS when compared to those generated by the coated cemented carbide cutting tool. The tendency of the PCBN cutting tool to increase the RS was also reported in a previous study in face milling [39]. Compared with the coated cemented carbide cutting tools, the higher negative effective tool rake angle of the PCBN chamfered tools, which combined with its lower thermal conductivity (44 W/m K for the PCBN and 100 W/m K for the coated cemented carbide [7]), leads to the conduction of a greater portion of heat into the machined surface [27]. Therefore higher tensile RS are generated when machining with PCBN tools when compared with coated cemented carbide tools. Regarding the influence of the work material hardness, the tensile RS seem to be lower for the lowest workpiece hardness. Since the yield tensile stress increases with the material’s hardness, the corresponding RS are also expected to increase with hardness [38]. This RS behavior with the work material hardness was observed in Ref. [30]. Fig. 11.15 shows the RS induced by face milling of AISI H13 tool steel using both coated cemented carbide [TiN/(Ti,Al)N/Ti(C,N) coating] (Fig. 11.15A) and PCBN cutting tools (Fig. 11.15B). For both tools the RS at the machined surface in the direction of the feed motion (σxx) and perpendicular to this direction (σyy) are predominantly compressive, being less compressive (0 MPa for the PCBN and 2200 MPa for the cemented carbide) in the direction of feed motion (σxx), and, therefore, more critical to part performance in practice. The in-depth residual-stress profiles show different evolutions for the cemented carbide and PCBN inserts. In the case of the cemented carbide insert, both normal residual-stress components σxx and σyy decrease continuously in-depth down (Fig. 11.15A) stabilizing at the level corresponding to that found in the work material before machining. In the case of

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Figure 11.15 In-depth evolution of residual stresses in the direction of feed motion (σxx) and normal to this direction (σyy), for AISI H13 steel machined using: (A) coated cemented carbide tools and (B) PCBN tool.

Figure 11.16 In-depth evolution of residual stresses in the direction of feed motion (σxx) and normal to this direction (σyy), for AISI H13 steel machined using (A) new tool and (B) worn tool.

the PCBN tools (Fig. 11.15B) the normal stress components change continuously with depth down to a certain maximum value in the compressive region, decreasing then gradually until stabilized at the level corresponding to the work material before machining. These difference in the in-depth RS profiles between the cemented carbide and PCBN cutting tools is due to the different cutting speeds associated to each tool, and to their different microgeometry and thermal properties. The higher cutting speed of the PCBN tool compared to the cemented carbide, associated to its lower thermal conductivity and higher negative effective tool rake angle, leads to greater heat generation and conduction into the machined surface [27]. As a consequence, a reduction of the compressive RS at the machined surface is observed. As far as tool wear is concerned, Fig. 11.16 shows a significant increase of the maximum compressive RS below surface, its location is shifted further from surface, and an increase of the thickness of the layer affected by compressive RS (from 125 to 650 μm).

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AISI D2 AISI D2 is a cold working tool steel grade with high wear resistance combined with moderate toughness. This steel has been widely used to produce many different types of cold working tools, such as those for deep-drawing, bending, punching, extrusion die and dies for molding abrasive materials. The structure of this steel includes coarse primary carbides unaffected by heat treatment. These carbides induce the high wear resistance but lead to poor toughness of the grade. In addition, they exhibit a low machinability in hardened state, even using high performance cutting tools, such as PCBN [40]. In annealed state the presence of carbides also compromises machinability. For these reasons, steels makers are continuously looking for new metallurgical solutions to improve machinability of this steel, without compromising the primary end user requirements. Figs. 11.17 and 11.18 show the influence of the cutting speed in the in-depth maximum and minimum principal RS (σ1 and σ2, respectively) profiles, generated by milling induced by milling AISI D2 using coated cemented carbide [TiN/(Ti,Al) N/Ti(C,N) coating] and PCBN cutting tools, respectively. RS values are always compressive for both cutting tools, reaching a maximum that depends on the cutting conditions. In the case of the coated cemented carbide tool the maximum is usually located at machined surface, while for PCBN tool the maximum is located below machined surface (between 10 and 20 μm). The maximum residual stress is more compressive when the coated cemented carbide tool is used, reaching
1400 MPa, against 2800 MPa for the PCBN insert. Considering the effect of the cutting speed, Fig. 11.17 shows that when the coated cemented carbide tool is used, the thickness of the layer with compressive RS increases when the cutting speed increases from 20 to 50 m/min. However, the opposite effect is observed when the PCBN tool is used (Fig. 11.18). In this case the thickness of the compressed layer slightly decreases when the cutting speed increases from 350 to 650 m/min.

Figure 11.17 In-depth principal RS (σ1 and σ2) profiles in milling of D2 using coated cemented carbide cutting tools at: (A) Vc 5 20 m/min and (B) Vc 5 50 m/min.

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Figure 11.18 In-depth principal residual stresses (σ1 and σ2) profiles in milling of D2 using PCBN cutting tools at: (A) Vc 5 350 m/min and (B) Vc 5 650 m/min.

Figure 11.19 Effect of cutting speed on residual stress for HRC 61 along the (A) axial and (B) circumferential directions.

AISI 52100 AISI 52100 bearing steel is a high carbon, chromium containing low alloy through hardening steel. This steel has high hardness (up to HRC 67), high wear resistance (due to its high hardness), and rolling fatigue strength, which makes it specially adapted for roll bearings. Fig. 11.19 shows the influence of the cutting speed (75, 150, and 250 m/min) on the RS profiles in both circumferential (the direction of the primary motion) and axial (the direction normal to the primary motion) directions, induced by orthogonal cutting of AISI 52100 (HRC 61) using PCBN cutting tools with chamfer microgeometry [41
43]. In particular, the results highlight that with the increase of the cutting speed, a deeper compressive surface residual stress is observed in both directions. In addition, the maximum compressive residual stress below surface becomes larger when cutting speed increases and its location is shifted further from surface. These variations are more evident in the axial direction.

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Figure 11.20 Effect of initial workpiece hardness (A) and tool geometry (B) on circumferential residual stress.

Figure 11.21 Effect of microstructural changes on residual stresses profiles for (A) Vc 5 75 m/min and (B) Vc 5 250 m/min for specimens at HRC 61.

Fig. 11.20A shows the influence of the initial workpiece hardness (HRC 56.5 and 61) on the residual stress profile [41
43]. An increase of work material hardness produces more compressive residual stress profiles, and the position of maximum compressive stress shifts closer to the surface. Moreover, the thickness of the layer affected by machining RS slightly increases with the initial workpiece hardness. Fig. 11.20B reports the influence of the tool cutting edge preparation (chamfer vs honed) on the RS profiles [41
43]. It is seen that the use of a chamfered tool generates higher compressive residual stress when compared to the honed tool in terms of both surface residual stress and maximum compressive stress below the surface. In contrast, both the tool cutting edge preparations report similar location of the maximum RS below the surface in both directions. Fig. 11.21 shows a correlation between the in-depth residual-stress distributions and white and dark layers’ regions. As can be observed, at a lower cutting speed

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(Fig. 11.21A), the maximum compressive residual stress for both the axial and the circumferential directions is positioned in the white layer region. Such evidence was also observed for the specimens at HRC 56.5 machined at the same cutting speed. In contrast, at a higher cutting speed (Fig. 11.21B), the maximum compressive stress is positioned near the white-dark layer transition (circumferential direction) or in the dark layer (axial direction). Similar observations were done for specimens at HRC 56.5. The reasons for these evidences are related to the fact that both white layer thickness and location of the maximum RS below surface increase with the cutting speed, although the latter rises more than the white layer’s thickness. Moreover, it is widely assessed that in orthogonal cutting of AISI 52100, the while layer thickness increases with the cutting speed due to the increase of heat generated and conducted to the workpiece [42]. Furthermore, when the initial workpiece hardness increases, both white and dark layers thicknesses increase [42]. For all abovementioned cutting conditions the RS are compressive in both axial and circumferential directions in orthogonal cutting of AISI 52100 using PCBN cutting tools.

11.3.3 Effect of relative tool sharpness on residual stresses In order to avoid the tool breakage due to insufficient strength of the cutting tool in the vicinity of the cutting edge, practical machining operations are frequently performed with tools having large cutting edge radii. These large cutting edge radii, combined with low feeds (and thus low uncut chip thicknesses), typically, in finishing operations, represent a complex geometric and work-tool material condition signifying the strong influence of the tool edge radius on the cutting process. Even if the cutting edge radius is very small, it may have a significant influence on the cutting process, if the uncut chip thickness is of the same order or smaller than the tool edge radius. Zorev [44] suggested that the effects of the tool edge radius in the cutting process can be neglected, if the uncut chip thickness is equal or greater than 10 times the tool edge radius. However, in many practical finish machining operations, this established rule is often violated because the actual radius of the cutting edge is more than 1/10 of the uncut chip thickness. As a result, this radius significantly affects the machining performance. The sharpness of the cutting edge is a relative parameter, which depends on the ratio between the uncut chip thickness and the cutting edge radius (h/rn or relative tool sharpness). A change in the ratio h/rn changes the flow of the material around the tool edge (see Fig. 11.22), as well as the distribution of the plastic strain in the vicinity of this edge (see Fig. 11.23). In particular, as the ratio h/rn decreases, the material flow below the cutting edge of the tool is intensified. This phenomenon is generally known as plowing. Fig. 11.24 shows clearly the influence of cutting edge radius on the microstructure of the machined surface layer induced by orthogonal cutting of AISI 1045 steel using uncoated tungsten carbide tools. As can be seen in this figure, the surface grains are plastically deformed and thus elongated in the cutting direction by the cutting edge. The cutting edge radius plays an important role in deforming the grains and creating a layer of deformed grains on the finished surface. The intensity

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(B)

Figure 11.22 Simulated material flow around the cutting edge using (A) a sharp tool (edge radius less than 5 μm) and (B) a tool having an edge radius of 44 μm. Uncut chip thickness of 0.05 mm.

(A) Plastic strain 4.00 3.66 3.33 3.00 2.66 2.33 2.00 1.66 1.33 1.00 0.66 0.33 0.00

(B) Plastic strain 4.00 3.66 3.33 3.00 2.66 2.33 2.00 1.66 1.33 1.00 0.66 0.33 0.00

Figure 11.23 Simulated distribution of the plastic strain in cutting using (A) a sharp tool (edge radius less than 5 μm) and (B) a tool having an edge radius of 44 μm. Uncut chip thickness of 0.05 mm.

of the deformation process can be measured in terms of the grain inclination angle (θd) measured at the sweeping (transition) region of the deformed layer, adjacent to the heavily deformed (also known as the retarded) surface layer, and also by the thickness of the deformed layer (td). Fig. 11.24 suggests that the grain is being pulled by the larger area of contact between the rounded cutting edge and the workpiece during the machining process. The cutting edge radius also affects the microhardness along the cross section of the machined surface layer. An increased

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Figure 11.24 Microstructure of machined surface obtained using a tool having a cutting edge radius of (A) 15 μm and (B) 55 μm (Vc 5 175 m/min; h 5 0.05 mm; AISI 1045).

Figure 11.25 Effect of cutting edge radius on residual stresses at (A) the machined surface (Vc 5 175 m/min; h 5 0.05 mm; AISI 1045) and (B) subsurface.

cutting edge radius also induces higher microhardness across the machined layer due to greater plastic deformation [45]. Increases in plastic deformation and temperature in the machined surface [45] caused by large cutting edge radius also induces high residual stress levels and high thickness of tensile layer at the machined surface. As shown in Fig. 11.25A, RS are higher in the circumferential direction compared with those stresses in the axial direction. Also, for low cutting edge radius (or when the h/rn ratio is greater than 1), the RS are lower. They increase when the cutting edge radius increases up to the value of the uncut chip thickness (40
50 μm), which corresponds to a ratio h/rn equal to 1. For larger cutting edge radius (or when the h/rn ratio is less than 1), the RS do not change significantly, being almost constant. These RS are produced largely due to the plowing process. Fig. 11.25B shows the variation of the circumferential RS as a function of the distance from the machined surface. As seen in this figure, the RS are maximal at the machined surface, and then they decrease with the distance from the machined surface, stabilizing around the residual stress value found in the work material before machining (in the range of 150
300 MPa in compression). This Figure also

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shows that an increase in the edge radius from 15 to 55 μm causes an increase in the thickness of the tensile layer from 26 to 55 μm. This leads to an increase in the maximum RS at the machined surface, and thickness of tensile layer are in agreement with the observed increase in the temperature and plastic deformation developed during the cutting process [45]. To conclude the results show that the ratio between the uncut chip thickness and the cutting edge radius (h/rn) has a strong influence on the quality of the machined surface. In particular, increasing the cutting edge radius, while keeping the uncut chip thickness constant (thereby, decreasing the ratio of the uncut chip thickness and the tool cutting edge radius, h/rn), increases the cutting forces and temperatures (on the tool edge and on the flank surface) generated during the cutting process [45]. This is shown to be accompanied by a large plastic deformation on the surface and subsurface of the machined workpiece, an increase in tensile RS and in microhardness.

11.3.4 Control of residual stresses in machining The previous analyses helped us to understand the influence of cutting parameters on the RS generated on the machined surfaces of several difficult-to-cut materials, and therefore, a valuable knowledge to formulate an empirical approach to control the magnitude of residual stress. Among all the investigated cutting parameters, the feed seems to be the parameter that has the strongest influence on RS. In order to reduce the magnitude of the residual stress the feed must be kept as low as possible. However, decreasing the feed decreases the material removal rate (keeping the other cutting regime parameters unchanged). Therefore in order to increase the material removal rate without compromise the residual stresses (sometimes even improving them) the depth of cut can be increased. This finding permitted to introduce f/ap parameter, defined as the ratio between the feed and the depth of cut. This new parameter can be used in the control process of the residual stress [46]. As shown in Fig. 11.26, the residual stress increases with f/ap. Therefore if the objective is to reduce the RS, this parameter must be kept as low as possible, without compromising the productivity. Another interesting result from precedent residual stress analysis of the machined surfaces is related to the equivalent tool cutting edge angle, κeq r , defined as the tool cutting edge angle of the equivalent cutting edge. It simplifies consideration of the influence of the major and minor cutting edges and tool nose in the manner shown in Fig. 11.27A. As formulated by Colwell [47], this equivalent cutting edge is defined as a straight line that connects the end of the active parts of the major and minor cutting edges as shown in Fig. 11.27A. Once the equivalent cutting edge is constructed, Colwell [47] assumed that the direction of chip flow is perpendicular to this edge. Because the f/ap parameter and κeq r have opposite evolutions, the RS decrease with the κeq angle. Similar results are also obtained by Capello [48] who r showed that RS decreases with the major tool cutting edge angle, κr. The present work [46] suggests that the effect produced by the same cutting tool and different

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Figure 11.26 Influence of the f/ap parameter in (A) circumferential and (B) axial residual stresses in turning AISI 316L using uncoated WC
Co cutting tool.

Figure 11.27 (A) Definition of equivalent cutting edge and chip flow direction and (B) relation between chip side-flow angle and residual stresses, when turning AISI 316L using uncoated WC
Co cutting tool.

feeds and depths of cut (so different κeq r angles) is equivalent to that produced by identical cutting tools having different κr angles [46]. The chip side-flow angle (ηs) is the angle between the direction of chip flow and that of feed motion, measured on the tool rake face as shown in Fig. 11.27A. Several analytical equations can be found in the literature to calculate this angle [7]. In this work, this angle was calculated using the following equations: If ap $ rε ð1 2 cos κr Þ4f # 2rε sin κr1 , ηs 5

 π c 2 arctan ½1 2 að1 2 cos κr Þcot κr 1 aðsin κr 1 bÞ 2

(11.1)

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If ap , rε ð1 2 cos κr Þ4f # 2rε sin κr1 Þ, then ηs 5

 π c 2 arctan pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2a 2 1 1 aUb

(11.2)

pffiffiffiffiffiffiffiffiffiffiffi  where a 5 rε =p, b 5 f =2rε , c 5 1 2 a 1 2 1 2 b and κr1 is the minor tool cutting edge angle.

As can be seen in Eqs. (11.1) and (11.2), the chip side-flow angle depends on the applied cutting conditions, in particular on the feed, depth of cut, tool cutting edge angle, and tool nose radius. These parameters are among those which have a strong influence on RS [46]. Therefore it can be deduced that there is a relationship between the chip side-flow angle and the RS. Indeed, as shown in Fig. 11.27B, the RS increase as the chip side-flow angle increases. Consequently, the chip side-flow direction can be also used to control of RS.

11.4

Modeling and simulation of residual stresses

11.4.1 Modeling and simulation considerations A growing concern in several manufacturing industries (including aeronautic/aerospace, power generation, automotive and biomedical) is to build machined components in absolute reliability with maximum safety and predictability of the performance. This requires the development and deployment of predictive models for surface integrity, as well as for the functional performance of machined components (fatigue and corrosion amongst others). The ultimate objective is the optimization of the machining conditions to achieve enhanced functional performance and life of components. Therefore modeling and simulation of machining operations can be an effective way to achieve this objective, since it reduces the number of expensive and time-consuming machining tests to be performed. Over the last decades, analytical and empirical models were most commonly developed and applied to predict mainly forces and temperatures during the cutting process. However, since the early 1980s, the rapidly increased computational capabilities, with the use of computer-based modeling and simulation methods based on finite element method (FEM) and more recently the using meshless methods, have gained a significant application potential in machining. Therefore many universities, research institutions, and companies are developing numerical-based models to predict surface integrity, including RS, microhardness, SR, microstructural, and phase changes [29,49
52]. Although these models can already provide a rough estimation of the surface integrity parameters, an effort to improve the predictability of these models is required, in particular, for practical machining operations. Today, numerical-based models have a prominent place in the metal-cutting simulation. However, they are not easy to set up by someone who does not have sufficient knowledge about materials and contact modeling, and numerical methods. These models incorporate a diversity of input data, numerical methods,

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and simplistic assumptions that affect the accuracy and validity of the results. Therefore the following points must be carefully analyzed when setting up a numerical model for machining simulation. 1. Numerical methods (FEM, meshless), formulations [Lagrangian, arbitrary Lagrangian
Eulerian (ALE)] and corresponding parameters (element size, time step, etc.) and assumptions. Among the published literature [53] on the modeling of machining operations based on FEM, it can be underlined that the Lagrangian formulation is widely used in machining simulations [52]. In this formulation the finite-element (FE) mesh is composed of elements that cover exactly the whole of the region of the object under analysis. These elements are attached to the object and thus they follow its deformation. This formulation is particularly convenient when the unconstrained flow of material is involved, that is, when its boundaries are in frequent mutation. In this case the FE mesh covers the real contour of the object with sufficient accuracy. This formulation, combined with the element deletion technique, can simulate the physical separation of the layer being removed to form the chip. Nevertheless, since metal cutting involves severe plastic deformation of the layer being removed, the elements become extremely distorted. To minimize this distortion intensive remeshing is often applied. After remeshing the state variables must be interpolated from the old mesh to the new mesh. Even if the difference between the old and the regenerated mesh is small, the interpolation process generates errors, which becomes significant considering the high remeshing frequency in a single simulation. To overcome these problems related to the mesh distortion, ALE and coupled Eulerian
Lagrangian (CEL) are introduced in metal-cutting simulation. ALE formulation combines the advantages of both Lagrangian and Eulerian formulations, that is, the mesh is neither attached to the material (Lagrangian) nor fixed in space (Eulerian). Therefore this formulation can better handle the mesh distortion problems. In CEL the workpiece is modeled as Eulerian and the tool as Lagrangian. So, the workpiece mesh is fixed in space and the work material flows through the element faces allowing large strains without causing mesh distortion problems. The major drawbacks of both ALE and CEL formulations for metal-cutting simulations are (1) the impossibility to correctly simulate the chip geometry, which is a problem for serrated chip formation, observed in machining difficult-to-cut materials, such as Inconel and Titanium alloys and (2) the incorrect simulation of the physical separation of the layer being removed to form the chip. To overcome the disadvantages of the FEM mostly related to mesh distortion, meshless methods such as smoothed particle hydrodynamics and element-free Galerkin have gained the attention of several researchers working on the simulation of machining processes [54,55]. However, the relatively higher computation cost of meshless methods (when compared to FEM) and the difficulty to predict some metal-cutting outcomes with acceptable accuracy (such the RS in the machined surface and subsurface [34]), are probably the most limiting factors for their massive application to machining simulation. 2. Boundary conditions (thermal and mechanical). The size of the geometrical model of the workpiece and tool, and how the thermal and mechanical boundary conditions are applied to this model can influence the temperature, stress, and strains distributions, thus affecting the model’s accuracy. A special attention will be paid here to the thermal boundary conditions. In particular, the correct determination of the heat convection coefficients is crucial for predicting the temperature distribution in the tool and workpiece. This is critical when simulating cryogenic assisted machining. As shown by Lequien et al. [56], this coefficient is extremely variable, depending on the particular characteristics of the cryogenic fluid and surfaces of the tool/workpiece. Another parameter affecting the temperature

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distribution in the tool and workpiece is the coefficient of heat transfer between the tool and workpiece. This coefficient is often ignored in most of the metal-cutting simulations found in the literature. Its value is often calibrated based on the comparison between predicted and measured results, or adjusted in order to accelerate the convergence of the FE simulations toward steady-state conditions, despite the very short cutting time that can be effectively investigated [57]. Therefore the accurate determination of the value of this coefficient is required. For this reason, several authors [56,58] determined it experimentally for several materials, and others proposed empirical equations to estimate it under several machining conditions [59,60]. 3. Procedures for calculating the RS and for extracting them from the model. There is no standard procedure to calculate and extract the RS from the model, and unfortunately, most of the scientific publications do not describe how they have done such calculations and extractions. Therefore a new procedure is proposed in Section 11.4.4 4. Tool-chip and tool-work contact models and parameters (friction coefficient, friction factor, heat transfer coefficient, etc.). The contact model must be able to accurately predict the shear and normal stresses at tool
chip and tool
workpiece interfaces, under extreme contact conditions of sliding speed, temperature, and contact pressure. The friction coefficient is the most used parameter in the contact models, and it is used to describe the relationship between normal and contact stresses. As shown in previous publications [61
63], this coefficient is not constant along with these interfaces, and it depends on the abovementioned contact conditions. An incorrect determination/selection of the contact model and associated friction coefficients can strongly influence/affect the model’s predictions. This subject will be described in detail in Section 11.4.2. 5. Material constitutive model. This model should be able to reproduce with accuracy the mechanical behavior of the work material in metal cutting under particular loading conditions, including high strains and strain rates, complex state of stress, and very hightemperature gradients. The constitutive model should be able to reproduce with accuracy the mechanical behavior of the work material (plasticity and damage) in metal cutting under particular loading conditions, including: high strains and strain rates, complex state of stresses and very high temperature gradients. Due to the high importance of this subject, it will be described in detail in Section 11.4.2.

11.4.2 Relevance of constitutive and contact models in residualstress prediction Material constitutive and contact models are two critical points in modeling machining operations. They considerably affect the metal-cutting performance (forces, temperatures, and tool life) and the surface integrity of the machined component, including the RS. Therefore they will be discussed in detail in the following sections.

11.4.2.1 Material constitutive model Considering the metal-cutting definition presented in Section 11.2.1, the constitutive model should include not only the elastic and plastic responses of the work material but also the damage. The last one is essential to simulate the genuine physical process of the separation of the material from the workpiece to form the chip.

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327

Moreover, this model should include the most relevant parameters affecting the mechanical behavior of the work material in metal cutting, such as strain hardening, strain rate, temperature, state of stress, and the microstructure. Johnson
Cook [64,65] is probably the constitutive model most used in numerical simulations, including high velocity impact, metal forming, metal forging, and machining. This model considers the effects of strain hardening, strain-rate, and temperature in the material plasticity. However, it also includes strain-rate and temperature effects, as well the stress triaxiality in the material damage. It is available in practically all the commercial finite element analysis (FEA) software. However, as mentioned by Guo et al. [66], although this model is easy to apply and can describe the general response of material deformation, this model is deficient in the mechanisms to reflect the static and dynamic recovery, and the effects of load path and strain-rate history in large deformation processes, such is the case of metalcutting process. These effects are fundamental to proper modeling the surface integrity of machined components, including the residual stress distribution [66]. An attempt to improve the accuracy of the Johnson
Cook plasticity model led several researchers to modify it, adding additional effects that they found relevant for the correct description of the work material under investigation. For example, the so-called strain-softening term was integrated into the Johnson
Cook plasticity by Calamaz et al. [67], to better predict the segmented chip with small uncut chip thickness and low cutting speed in orthogonal cutting simulation of Ti6Al4V alloy. ¨ zel [68] did a minor modification of the Calamaz et al. [67] model to Sima and O further control the thermal softening effect. The plasticity models described above are designed as phenomenological since the material behavior is described by empirically fitted functions, which contain several macroscopic deformation variables, such as temperature, plastic strain, and plastic strain rate. Other types of plasticity model are those mentioned by Melkote et al. [69] as physical-based models, since they integrate microstructural aspects (i.e., grain size, dislocation density, and material hardness) in the material mechanical behavior. This is particularly important if the objective is to predict not only the RS but also the microstructure of the machined surface and subsurface [69]. One case is the mechanical threshold stress (MTS) model [70], involving both the thermal and athermal stresses related to the dislocation density and grain size. Melkote et al. [71] extended the MTS model to simulate chip formation of pure titanium. The effects of dynamic recovery, dislocation drag, and dynamic recrystallization were considered in their model. Although the MTS model intrinsically describe the microstructure evolution and mechanical behavior of the work material with accuracy, its complexity regarding the large number of coefficients to be determined through several types of experimental tests limits their practical application. To improve the accuracy of the Johnson-Cook plasticity model without substantial increase its complexity, some researchers proposed modified versions of this model to include some physical-based quantities, such as the hardness and/or the grain size of the work material [72]. An example is the case of machining hardened steels, such as AISI 52100 bearing steels and AISI H13 tool steels, which the

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presence of tempered martensitic or retained austenite in steels affects their hardness. These changes can occur during the machining of these steels. Since in hard machining the uncut chip thickness is very small (often lower than the tool cutting edge radius) and the cutting speed is high, the material microhardness changes in the workpiece superficial layer generated on the previous workpiece revolution or tool pass will affect the mechanical behavior of work material on the current tool pass. Therefore correct analysis and modeling of surface integrity in hard machining should be based on an appropriate constitutive model, which should also consider the tempering effect (hardness change) as a function of the material thermal history. Umbrello et al. [73] proposed a modified Johnson
Cook plasticity model for AISI H13 considering the effect of the work material hardness in the flow stress. This model is given by Eq. (11.3), where F and G are two third-order polynomial functions, Eqs. (11.4) and (11.5), that take into account the hardness effect on the material plasticity. In particular, the former, F, modifies the initial yield stress and the latter, G, the strain hardening. The other coefficients (A, B, C, D, E, n, and m), constants (_ε 0 ; Tmelt, and Troom), and variables (ε, ε_ , and T) are from the original Johnson
Cook constitutive model. The coefficients (a, b, c, d, a1, b1, c1, and d1) of the functions F and G were determined based on the evolution of the yield stress and tensile strength in function of the material’s hardness. Details about this modified Johnson
Cook model, coefficients, and constants values can be found in Refs. [73,74].       T 2Troom m ε_ σeq 5 ðA 1 F 1 GU ε 1 Bε Þ 1 1 C  ln D2E  Tmelt 2Troom ε_ 0 (11.3) n

F ðHRCÞ 5 a  HRC3 1 b  HRC2 1 c  HRC 1 d

(11.4)

GðHRCÞ 5 a1  HRC3 1 b1  HRC2 1 c1  HRC 1 d

(11.5)

As mentioned in Section 11.2.1, metal cutting must be treated as the purposeful fracture of the layer being removed. This fracture occurs due to the continuous change of the state of stress in the first deformation zone [10]. Moreover, experiments on plastic deformation of metals have shown that the state of stress has a considerable influence not only in the material damage but also in the plasticity response [75,76]. Therefore the constitutive description of the material used in machining simulations should also include the effect of the state of stress. This state of stresses is represented by the stress triaxiality and Lode angle parameter. The stress triaxiality η is defined as the ratio of the mean stress σm to the equivalent von Mises stress σ (Eq. 11.6). η 5 σm σ 5

σ1 1 σ2 1 σ3 I1 5 3σ 3σ

(11.6)

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329

Figure 11.28 (A) The space of principal stresses and (B) Lode angle definition on the π plane.

Lode angle θ is represented in Fig. 11.28 on the deviatoric plane (or π plane). It is defined as the angle between the stress tensor, which goes through the deviatoric plane and the directions of the principal stresses. This angle is given by the following equation:   27 J3 θ 5 arccos 2 σ3

(11.7)

where J3 is the third invariant of the deviatoric stress tensor. Based on this equation, the Lode angle parameter θ is calculated by the following equation: θ 512

6 θ π

(11.8)

The range of θ is [ 2 1, 1], as the range of θ is [0, π/3]. These equations show that the stress triaxiality depends on the first stress invariant I1, while the Lode angle is a function of the third deviatoric stress invariant J3 and the von Mises equivalent stress, σ. In other words the stress triaxiality controls the size of the yield surface, while the Lode angle parameter θ is responsible for its shape. Recently, several researchers working on machining simulation have included the effect of the stress triaxiality and Lode angle parameter in both material plasticity and damage. For instance, Buchkremer et al. [77] extended both the plasticity and the damage parts of the constitutive model proposed by Bai and Wierzbicki [75,78], by including the temperature and strain-rate effects in addition to the triaxiality and Lode angle parameter already included in the original Bai and Wierzbicki model. Using this model, they were able to predict the forces, temperatures, chip formation, and flow with high accuracy in turning simulation of AISI 1045 steel.

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Denguir et al. [79] proposed a constitutive model (plasticity and damage) incorporating not only the strain-hardening and strain-rate, and temperature effects but also the microstructural and state of stress effects to describe the mechanical behavior of oxygen-free high conductivity (OFHC) copper in metal cutting. This model was inspired by phenomenological models considering the strain hardening, the strain-rate, the temperature [64,65], the microstructural transformation [80], and the state of stress effects [75]. The flow stress σ is expressed by the following equation:

σ5

   m  ε_ room 1 1 C ln 1 2 TT2T  m 2Troom ε_ 0 Strain hardening effect |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Strain-rate ðviscosityÞ effect Thermal softening effect   Hðε; ε_ ; TÞ  1 2 cη  η 2 η0 |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ðA 1 Bεn Þ |fflfflfflfflfflffl{zfflfflfflfflfflffl}

Microstructural transformation effect

(11.9)

State of stress effect

where the strain hardening, strain-rate, and thermal softening effects are representing the Johnson
Cook model [64,65]. As far as the microstructural effect on the flow stress is concerned, Andrade et al. [80] proposed a function H ðε; ε_ ; T Þ introducing the microstructural transformation effect on the flow stress. Concerning the effect of the state of stress, only the stress triaxiality η effect is considered. This effect is represented by the fourth term, taken from the Bai
Wierzbicki plasticity model [75], where η0 is the reference triaxiality and the cη is a coefficient. The damage model formulation is based on the fracture initiation equation proposed by Johnson
Cook model [64,65] and fracture propagation calculated taking into account the Hillerborg et al. [81] fracture energy. Using this model, they showed a better surface integrity prediction when the proposed constitutive model was compared with that of Johnson
Cook’s model. Cheng et al. [82] proposed a constitutive model that can be used to describe the work material behavior in manufacturing process simulations involving crack propagation and neglecting the thermal effects on mechanical behavior. This is the case of Ti6Al4V titanium alloy in the first deformation zone in metal cutting, where the material separation from the workpiece is mainly caused by crack propagation. Moreover, most of the heat produced due to plastic deformation in this zone flows into the chip, and only a small amount flows into the workpiece. They proposed a constitutive model composed of two parts: plasticity and damage. The proposed plasticity model is shown in Eq. (11.10), incorporating the strain hardening, strain rate, and state of stress effects. The first two terms are based on the Johnson
Cook’s [65] and Santos et al.’s [83] models, respectively. Furthermore, the last two terms of Eq. (11.10) are related to the state of stress effect, which are characterized by the stress triaxiality and the Lode angle parameter obtained from the Bai and Wierzbicki’s [75] model.

Residual stresses in machining

σ~

5

331

  i    E 1 ε_ p A 1 mεnp 3 B 1 C ln 3 1 2 c η η 2 η0 ε_ 0 |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} h

Strain hardening

Strain rate ðviscoÞ

effect

2



s

0

effect

13

Stress triaxiality effect

(11.10)

a11 @γ 2 γ A5 3 4csθ 1 cax θ 2 cθ a11 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Lode angle effect

   γ 5 6:464 sec θ π=6 2 1  cax θ

5

ctθ ccθ

for θ . 0 for θ # 0

(11.11) (11.12)

In the previous equations (1) the coefficients A, m, and n are used in the strain hardening term; (2) the coefficients B, C; and E are used in the strain rate term; (3) the coefficient cη is used in the stress triaxiality term; (4) the coefficients ctθ , ccθ , csθ , and a are used in the Lode angle term; (5) η0 is the reference stress triaxiality and ε_ 0 is the reference strain rate; (6) γ describes the difference between Tresca and von Mises equivalent stresses on the deviatoric stress plane, as represented in Eq. (11.11); and (7) the coefficients ctθ , ccθ , csθ are interdependent and at least one of them equals 1. The temperature effect on the work material behavior is not considered in this plasticity model for the reasons mentioned by Cheng et al. [82]. The damage model includes damage initiation and damage evolution, as shown from Eqs. (11.13)
(11.15), based on Bai and Wierzbicki [75] and Abushawashi et al. [84] models. Eq. (11.13) represents the damage initiation, while Eqs. (11.14) and (11.15) represent the damage evolution. The fracture surface depends on both the stress triaxiality and Lode angle parameter. The work material strength decreases as the strain increases after damage initiation, and the fracture energy is used to estimate this material stiffness degradation.

εpi

82 3 < 1 2 5 4 D1 e2D2 η 1 D5 e2D6 η 2 D3 e2D4 η 5θ : 2 9 =  1 1 D1 e2D2 η 2 D5 e2D6 η θ 1 D3 e2D4 η ; 2 2 0 13 ε_ 3 41 1 D7 ln@ A5 ε_ 0

(11.13)

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

σ 5 ð1 2 DÞσ; ~

Gf 5

ð εp f

εpi

D5

lσdε 5 p

ð uf

1 2 expðλε Þ ; 1 2 expðλÞ

ε 5

εp 2 εpi εpf 2 εpi

σdu

(11.14)

(11.15)

0

In these equations, D1
D7, n, and Gf are the material coefficients, λ controls the material degradation rate and Gf is the material fracture energy density; ε pi is the plastic strain at damage initiation; ε pf is the plastic strain when all the stiffness and the fracture energy of the material have been lost and dissipated, respectively; σ~ is the hypothetical undamaged stress evaluated by Eq. (11.4); l is the characteristic length of the finite element; u 5 0 relates to the equivalent plastic displacement before damage initiates; and u f is the equivalent plastic displacement at complete fracture. The characteristic length is introduced to describe the fracture energy. The accuracy of the proposed constitutive model to describe the mechanical behavior of Ti6Al4V alloy under different states of stress was confirmed by a benchmark test considering the proposed model and the J
C model. The numerical simulations of mechanical tests show that when the J
C model is used, the predicted force
displacement curves are considerably overestimated in relation to the experimental curves. Therefore the J
C model is not able to represent the mechanical behavior of Ti6Al4V alloy under different states of stress, while the proposed model shows good accuracy. Cheng et al. [85] also showed an improvement in the ability of the machining model to predict not only the forces, temperatures, and chip geometry but also the surface integrity, when the constitutive model includes the state of stress.

11.4.2.2 Selection of material constitutive model One question that often arises when someone wants to develop a model of given machining operations is “What’s the material constitutive model that I should use?”. The answer is not evident and straightforward, especially if the main target of the machining simulations is to predict surface integrity, including RS. To answer this question, several factors must be considered, such as (1) the main model application (predictions) and desired accuracy; (2) the work material under analysis; (3) the variables to be included in this model (strain-hardening, strain rate, temperature, state of stress, etc.); (4) the availability of the constitutive model coefficients for a given work material under analysis; (5) the simulation time; and (6) the user experience on the implementation of new user materials models in the selected FE software. Among these factors the first three factors are critical for the selection of the material constitutive model. The constitutive model type (phenomenological or physical-based) and the variables to be included in this model strongly depend on the application. A machining model used to predict the forces does not require a

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333

complex constitutive model formulation involving many variables and corresponding coefficients to be determined through mechanical tests. A relatively simple Johnson
Cook constitutive model should be enough to predict these forces. However, if the ultimate objective of the machining model is to predict the surface integrity, a more sophisticated constitutive model, eventually a physical-based one, is the best option. In any case the constitutive model must include the most relevant variables affecting the work material behavior in machining, which will depend on the deformation mechanisms of the work material under analysis. The abovementioned last three factors are less critical for the selection of the constitutive model, but for practical reasons, they should also be considered. This is particularly relevant for end users who do not want to spend a lot of time and money to perform numerical simulation of a give machining operation. They want to obtain fast results but without significantly penalizing the accuracy. Therefore they often use constitutive models that are already implemented in the FE software packages, and they select the corresponding coefficients for the work material under analysis from the literature sources or material databases. This procedure arises another question, which is “Which set of constitutive model coefficients should I select/use?”. It is well known [49] that for the same work material, different sets of coefficients can be found in the literature for a given constitutive model. So, if one wants to identify the most suitable set of model coefficients, which produce the best fit between predicted and measured results, he or she usually performs preliminary metal-cutting simulations to compare the predicted results with those measured. Zhang et al. [86] used three kinds of cutting models, corresponding to three different numerical formulations (Lagrangian, ALE, and CEL), on the selection of the Johnson
Cook constitutive model coefficients for the orthogonal cutting simulation of the Ti6Al4V. The aim of this study was to identify the most suitable set of Johnson
Cook model coefficients that produce the best fit between predicted and measured results concerning chip geometry, chip compression ratio, forces (cutting force and thrust forces) and the distributions of temperature, and equivalent plastic strain. Two sets of Johnson
Cook model coefficients for Ti6Al4V alloy were tested as shown Table 11.2. The obtained results show that the set having lower hardening coefficients (Set 1) gives acceptable chip thickness for the ALE and CEL models, while the set having higher hardening coefficients (Set 2) presents good cutting force prediction for the Lagrangian model. This means that the best set of Johnson
Cook model coefficients is not unique for the three numerical models of metal cutting. This is because the accuracy of the existing metal-cutting models to predict the metal-cutting performance does not depend only on the constitutive Table 11.2 Johnson
Cook constitutive model parameters of Ti
6Al
4V [87]. Set

A (MPa)

B (MPa)

n

C

m

ε_ 0 (s21)

Set 1 Set 2

862 1098

331 1092

0.34 0.93

0.012 0.014

0.8 1.1

1 1

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model but also on how these metal-cutting models deal with the material separation from the workpiece to form the chip. This separation is treated differently in the three orthogonal cutting models presented in this contribution. The main difference between the Lagrangian and ALE/CEL models is the damage, which is only used the Lagrangian model to produce material separation (chip formation) and chip segmentation. This means that the selection of the most suitable set of constitutive model coefficients from the literature, based on metal-cutting numerical simulation and on the comparison between predicted and measured results, is not the best practice. This selection will be influenced by the numerical technique used to model the physical separation of the layer being removed (in the form of chips) from the rest of the workpiece. A proper selection of the constitutive model coefficients from the literature must be performed based on the comparison of the mechanical and metallurgical states of the work material under analysis with those states of the work material in the literature or database. Unfortunately, only a few literature sources and databases provide such information about the mechanical and metallurgical states.

11.4.2.3 Contact model Several contact models are available in the literature [69,88,89]. However, for matter of simplicity, Zorev’s model [44] is often used in metal-cutting simulations to describe the contact stresses at tool
workpiece interface. This model considers a sliding/sticking contact between the chip and the tool, where the contact stresses are represented in Fig. 11.29, in function of the distance from the tool cutting edge.

Figure 11.29 Distribution on the normal and shear stresses at the tool rake face according to Zorev.

Residual stresses in machining

335

Mathematically, this model can be represented by the following equation:  τ5

μσ τ lim

if if

μ  σ , τ lim μ  σ $ τ lim

(11.16)

where σ and τ are the normal and shear stresses and μ is the friction coefficient at the interface. Therefore for sliding contact conditions at the interface (μσ , τ lim), the shear stress will be calculated using the Coulomb friction model (τ 5 μσ), while for sticking contact conditions at the interface (μσ . τ lim), the sliding does not occur at the tool
chip interface but in the adjacent layers of the chip, being the shear stress equal to the limit shear stress, τ lim. This limit shear stress can be determined by σy/O3, where σy is the yield stress of the workpiece and chip materials, depending on the tool
chip or tool
workpiece contact. As shown in previous publications [61
63], the friction coefficient is not constant along the tool
chip contact, and it depends on the contact conditions (contact pressure, sliding velocity and temperature). Moreover, the value of the friction coefficient should be considered as the limiting value so that if μ . 0.577, no relative motion can occur at the interface. However, in the practice of metal cutting, this is not the case, and the reported values of this coefficient obtained from measuring the forces in metal-cutting tests are well above 0.577 [90]. On the other hand, the values of the friction coefficient, used in modeling (more often in FEM), are always below the limiting value to suit the sliding condition at the interface [90]. Interestingly, the results of FEM were always found to be in good agreement with the experimental results regardless of the particular value of the friction coefficient selected for such a modeling. So, an incorrect determination/selection of the friction coefficient can strongly influence/ affect the model’s predictions.

11.4.3 Simulation of residual stresses for several work materials In this section, metal-cutting performance, including surface integrity, obtained by numerical simulation of orthogonal cutting of four work materials will be presented. The corresponding orthogonal cutting models included different constitutive and friction models, and the simulations were performed in SFTC Deform and Abaqus FEA software.

11.4.3.1 Austenitic stainless steel AISI 316L Fig. 11.30 shows the model of orthogonal cutting of AISI 316L with the boundary conditions [29,49]. This model uses the Lagrangian formulation with remeshing technique to simulate the forces, temperatures, chip morphology, and RS using the Deform FEA software. It uses the Johnson
Cook constitutive model (only plasticity behavior), where its coefficients were taken from the literature. The contact model was based on the constant shear hypothesis (τ 5 mτ 0 ) with the shear factor, m.

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

T = 20°C Vx = Vc Vy = 0

Ux = 0 Uy = 0

S

y

x

T = 20°C

Figure 11.30 Orthogonal cutting model of AISI 316L using Lagrangian formulation and remeshing technique. Table 11.3 Coefficients of the Johnson
Cook plasticity model for AISI 316L.

Chandrasekaran and M’Saoubi [91] M1 Chandrasekaran and M’Saoubi [91] M2 M’Saoubi [92] M3 Changeaux et al. [93] M4 Tounsi et al. [94] M5

ε_ 0

A

B

C

n

m

305

1161

0.01

0.61

0.517

1

305

441

0.057

0.1

1.041

1

301 280 514

1472 1750 514

0.09 0.1 0.042

0.807 0.8 0.508

0.623 0.85 0.533

0.001 200 0.001

Since in the literature several sets of Johnson
Cook model coefficients can be found for the same work material, a preliminary study was performed to identify the set of coefficients (see Table 11.3) that gives better predictions. The main reason is that although several sets of coefficients are for the same work material reference (AISI 316L), some allowable variations in the mechanical and metallurgical states can affect these coefficients. Moreover, different test methods were used to characterize the mechanical behavior of this work material. The comparison between predicted and measured (from another study of the authors [36]) results allows us to identify the best set of Johnson
Cook model coefficients (coefficients set M5 in Fig. 11.31), which were used in further research work. This research work concentrated on determining the influence of the cutting conditions (including tool geometry and tool material) and the number of cutting passes (sequential cuts) on the RS distributions in the machined surface and subsurface [29]. The following conclusions can be drawn from this research work: G

G

RS increase with most of the cutting parameters, including cutting speed, uncut chip thickness, and tool cutting edge radius. When the uncoated tool was replaced by a coated one, the superficial RS increased by 240 MPa, when the highest cutting speed value was used (Fig. 11.32A). This is due to an increase in the heat conducted into the workpiece for the coated tool [27].

Residual stresses in machining

337

Figure 11.31 Comparison between measured and predicted residual stresses for AISI 316L along (A) axial and (B) circumferential directions (Vc 5 200 m/min, h 5 0.1 mm, ap 5 6 mm).

Figure 11.32 Circumferential residual stress for AISI 316L (A) generated by using coated and uncoated tools and (B) effects of sequential cuts on the residual stress distribution. G

G

G

An increase in tool rake angle seems to reduce the superficial RS by 140 MPa, when this angle changes from 25 to 5 degrees. Within the range of cutting parameters investigated, the uncut chip thickness seems to be the parameter having the largest influence on RS (about 390 MPa, when the uncut chip thickness increases from 0.1 to 0.2 mm), which is in agreement with most of the experimental results found in the literature [48]. The sequential cuts tend to increase superficial RS by 280 MPa from the first to the third cut (Fig. 11.32B).

11.4.3.2 Nickel-based alloys Inconel 690 A model of the orthogonal cutting of Inconel 690 identical to the model of AISI 316L (Fig. 11.30) was developed and simulated using the Deform FEA software

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Figure 11.33 Comparison between measured and predicted residual stresses for Inconel 690 along axial and circumferential directions.

[95]. This model uses the Zerilli
Armstrong constitutive model [96], where its coefficients were taken from the literature. The same friction model of the AISI 316L was used for the Inconel 690. This model was applied to simulate the forces, temperatures, chip morphology, and RS. The orthogonal cutting model was validated by comparing the predicted results with those measured, in terms of forces, temperatures, and RS. For instance, Fig. 11.33 shows the comparison between predicted and measured in-depth residual stress profiles. Some differences can be observed, although the predicted and measured stress values are very close at machined surface and for large depths. A three-dimensional cutting model was also developed and applied to simulate the turning operations of Inconel 718, but only the forces and temperatures were predicted [13]. This predicted data were used to explain the residual stress distributions in the machined surface and subsurface obtained experimentally.

11.4.3.3 Case-hardened steels AISI H13 and AISI 52100 Models of the orthogonal cutting of AISI H13 and AISI 52100 hardened steels similar to the model of AISI 316L (Fig. 11.30) were developed and simulated using the Deform FEA software [51,73,97]. In the case of AISI H13, it uses a modified Johnson
Cook constitutive model, which includes additional two parameters (F and G) that modify the initial yield stress and strain hardening to consider the work material hardness effect on the flow stress (see Section 11.4.2, Eqs. 11.3
11.5). The same friction model of the AISI 316L was used for both AISI H13 and AISI 52100 hardened steels. These models were applied to simulate the forces, temperatures, chip morphology, RS, microhardness, and white and dark layers. These models were validated by comparing the predicted results with those measured experimentally, and already

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Figure 11.34 Effect of white and dark layers on residual stress, and comparison of experimental (EXP) and numerical simulation (NUM) results with and without the effects of microstructural changes.

presented in Section 11.3.2.4. As shown in Fig. 11.34 for the case of AISI 52100, the best agreement between predicted (NUM) and experimental (EXP) residual stress profiles is obtained when the microstructural transformations (white and dark layers) are considered in the constitutive model. Fig. 11.34 also shows that the white layer has a significant impact on the magnitude and location of the maximum (peak) compressive residual stress below the surface, which is related to an increase in material hardness. In contrast, the presence of dark layer (tempered structure) reduces the compressive RS. After this validation an orthogonal cutting model was applied to investigate the influence of the cutting conditions on the white and dark layers, consequently, on RS distributions in the machined surface and subsurface. To conclude, this research work suggests that a better agreement between the experimental and predicted residual stress states in hard machining can be achieved if the residual stress model incorporates the microstructural/hardness changes generated during machining operation.

11.4.3.4 Oxygen-free high-conductivity copper Fig. 11.35 shows the model of orthogonal cutting of OFHC with the boundary conditions [79]. This model uses the ALE formulation to simulate the forces, temperatures, chip morphology, and the surface integrity using Abaqus FEA software. The following surface integrity parameters were predicted: RS, grain size, dislocation density, and microhardness. It uses a physics-based constitutive model of OFHC Cu, described in Section 11.4.2 (Eq. 11.9). Zorev contact model [44] with a friction coefficient in function of the sliding velocity at the tool-workpiece/chip interfaces was applied.

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T = 20°C

Lagrangian sliding surface Eulerian surface ALE adaptive mesh constraint Lagrangian constraint

Vx = Vc Vy = 0

y

T = 20°C

Ux = 0 Uy = 0 Forced convection

Tool

x Workpiece

T = 20°C

T = 20°C

Figure 11.35 Orthogonal cutting model of OFHC copper using ALE formulation. ALE, arbitrary Lagrangian
Eulerian; OFHC, oxygen-free high conductivity.

Figure 11.36 (A) Measured and predicted in-depth residual stress profiles and (B) predicted in-depth residual stresses profiles obtained with and without considering the damage effect.

Fig. 11.36A shows measured and predicted in-depth residual stress profiles in the cutting (σx) and transversal (σy) directions. The results also show that the predicted RS obtained using the orthogonal cutting model is reasonable for capturing the shape of the in-depth RS distribution, especially when damage is included in the constitutive model (Fig. 11.36B). Otherwise, the strains near the machined surface can reach high values when compared to the values of the facture strain of the work material for a given state of stress. As a consequence, unrealistic high stresses will be generated in this region. Considering the metal-cutting definition presented in Section 11.2.1, the fracture strain is an important mechanical property in metal cutting that depends on the stress triaxiality in the chip-formation zone. It can be suggested that fracture strain will influence surface integrity, including RS. In fact, the stresses triaxiality influences the fracture strain (see Fig. 11.37A) and, consequently, the RS (see Fig. 11.37B). An increase of the stress triaxiality, η, implies a decrease of εf, consequently, the residual stress is found less tensile or more compressive.

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Figure 11.37 Model of residual stress formation proposed by the authors: (A) influence of the stresses triaxiality, η, on the fracture strain, εf and (B) residual stresses formation based on the fracture strain, εf.

11.4.3.5 Ti6Al4V titanium alloy Fig. 11.38 shows the orthogonal cutting model of Ti6Al4V titanium alloy with the boundary conditions [85]. This model uses the Lagrangian formulation together with the element deletion (from the sacrificial layer) to simulate chip formation, thus to predict the forces, temperatures, chip morphology, and the RS using Abaqus FEA software. A constitutive model incorporating the strain hardening, strain rate, and state of stress effects is used, as described in Section 11.4.2 (Eqs. 11.10
11.15). Zorev contact model [44] with a friction coefficient in function of the sliding velocity at the tool-workpiece/chip interfaces was applied. Fig. 11.39 shows both measured and predicted RS in the longitudinal (cutting) and transversal directions for the conditions indicated in this figure. Identical results were obtained for other cutting conditions, demonstrating the accuracy of the model to predict the RS in machining Ti6Al4V titanium alloy.

11.4.4 Procedure for comparing predicted and measured residual stresses The calculation of the predicted RS and their comparison with those obtained experimentally has any meaning only when the predicted RS are simulated and extracted from the FE model under the same conditions as those applied experimentally. These conditions will not only depend on the cutting procedure but also on the experimental technique, and the procedure used to measure the RS [98]. The following methodology was adopted for calculating the RS and for extracting them from the FE model:

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Figure 11.38 Orthogonal cutting model of Ti6Al4V titanium alloy using Lagrangian formulation together with an element deletion (from the sacrificial layer).

Figure 11.39 Measured and predicted in-depth residual stress profiles in machining Ti6Al4V titanium alloy.

1. Perform metal-cutting simulation with at least two cutting passes using a tool geometry with measured flank wear. 2. Calculate the RS. 3. Extract the residual stress components from the FE model.

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11.4.4.1 Perform metal-cutting simulation with at least two cutting passes using a tool geometry with measured flank wear Only few studies have reported on the variations of in-depth residual stress profiles from one cut to another [20,99]. Furthermore, the residual stress distributions developed in the previous pass must be considered when simulating the next pass, because experimentally the RS are normally measured after performing several passes. As far as tool-wear is concerned, it should be monitored during the machining tests and it must be taken into account when modeling the RS. Because, the toolwear influences the cutting process, and the RS will be affected [39,99,100]. Therefore the RS must be simulated by considering the measured tool-wear level.

11.4.4.2 Calculate the residual stresses The procedure for calculating the RS may depend on the level of knowledge of the researcher about the RS and of the software package but, in any case, should include the workpiece’s mechanical unloading and cooling down to room temperature. This procedure can be automatic as in the case of AdvantEdge or requires the development of an additional model for unloading and cooling down the workpiece to the room temperature (Fig. 11.40). This is the case of the other software packages such as Abaqus, Deform, and LS-Dyna.

11.4.4.3 Extract the predicted residual stress from the FE model As explained by Outeiro et al. [98], if at the end of the simulation the chip is attached to the workpiece, the predicted residual stress components should be extracted from the model sufficiently far away from the chip-formation zone. As shown in Fig. 11.41, a strong residual stress variation is observed near the

Figure 11.40 Procedure for residual stress calculation after cutting simulation.

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Figure 11.41 Procedure to extract the residual stress components from the model.

chip-formation zone, which is not representative of the real RS left in the workpiece. In order to avoid this zone, Region III should be selected to evaluate the residual stress components, being its distance from the chip root a function of the cutting conditions (cutting speed, feed, depth of cut, tool geometry/material, workpiece material, etc.). In order to compare both predicted and X-ray diffraction (XRD)-measured indepth residual stress profiles, the predicted values need to be weighted using a function that takes into account the absorption of the X-ray in the material under analysis, which can be calculated by the following equation: ÐN σðzÞ  e2z=τ dz hσR i 5 0 Ð N 2z=τ dz 0 e

(11.17)

where τ is the mean penetration depth of the X-ray beam in the material [101]. Also, several residual stress profiles should be extracted from the workpiece, covering a length equal to the length/diameter of the irradiated area in the machined surface. Then, an average residual stress value can be calculated for each depth.

11.4.5 Optimization of cutting conditions for improved residual stresses and surface roughness in machined components The objective of the modeling and optimization procedures was to find the optimal cutting conditions that induce low tensile or compressive RS in the machined components and simultaneously low SR. As shown in Fig. 11.42, this procedure uses as input data the residual stress and SR values obtained experimentally. It then uses the generalization capability of the artificial neural networks (ANN) to find the residual stress and SR for a wide range of cutting conditions, and thus the residual stress and SR function. This is the multiple-objective function, which is used in the

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Figure 11.42 Modeling and optimization procedure.

genetic algorithm (GA). The optimization procedure was implemented in a computer program, developed using Matlab [38]. ANN is composed of three layers: input, hidden, and output. The input layer corresponds to the cutting (Vc, f, ap) and tool geometry (κr, rε, and rn) parameters, having six neurons (total number of cutting conditions). An optimization procedure was applied to determine the most advantageous number of neurons in the hidden layer, this number being equal to 50. The output layer is composed of two neurons corresponding to the maximum principal residual stress (σmax) and the arithmetic SR (Ra). σmax was selected due to its higher tensile values when compared to σmin; thus σmax is the most critical parameter for part performance. Regarding SR, it was established that although the three parameters (Ra, Rz, and Rmax) have different values, they follow exactly the same trend with respect to the cutting conditions. This means that the optimal cutting conditions to be identified by the optimization procedure are independent of a particular choice of SR parameter. Thus the arithmetic average SR parameter (Ra) will be used in this optimization procedure. Response surface methodology was used to obtain the correlations between the six input parameters (Vc, f, ap, κr, rε, and rn) and the predicted σmax and Ra. First, a Box
Behnken design of experiments was used to generate several combinations of the six input parameters to be simulated by the ANN. For each input parameter, three levels were selected. Each parameter was varied between the minimum and maximum values, covering the range of the experimental input data. For each combination of the six input parameters the ANN calculated the σmax and the Ra. Then the correlations between the cutting parameters and the predicted σmax and Ra were

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Figure 11.43 Influence of cutting conditions on predicted (A) maximum principal residual stress (Smax) and (B) average surface roughness (Ra) (AISI H13
HRC 51; coated cemented carbide cutting tool; dry cutting). Table 11.4 Optimal combination of cutting conditions. Work material

AISI H13 (HRC 51)

Cutting tool rn (μm)

rε (μm)

60

0.4

κr ( ) 95

Vc (m/ min)

f (mm/ rev)

ap (mm)

118

0.1

0.4

determined. Fig. 11.43 shows influence of each cutting parameter on the RS and SR. It was established that in order to decrease the magnitude of the tensile RS and SR, both the feed and depth of cut must be reduced, while the cutting edge angle must be increased. The objective of the GA is to find the optimum set of cutting conditions (Vc, f, ap, κr, rε, and rn) which result in inducing compressive residual stress in the machined component as well as low SR (minimization). This search is performed for the range of cutting conditions (including tool geometry) used experimentally and on numerical simulations. Table 11.4 shows the optimal set of cutting conditions for dry turning AISI H13 using coated tungsten carbide cutting tools.

11.5

Influence of residual stress on product sustainability

11.5.1 Introduction Sustainable products are generally defined as those products providing environmental, societal, and economic benefits, assuring public health, operational and

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347

Design for

Regional and global impact

(DFS)

al Ethic ibilty s n o resp

Design for social impact

sustainability

Energy efficiency/ Design for Power consumption resource n tio iliza utilization Material ut ble and newa of Re energy economy Use of e c r e/ sou as e rch alu Pu ket v and r t n ma tio cos lla ta ning s In rai t

d an lth ss a He etne ct w ffe e

Op

Design for recycleability/ remanufacturability

Design for environmental impact

Social impact Service life/ durability Modularity Ease of use Ma inta ser inabi lit vi Up gra ceabil y/ ity Er go deab no ilit y m ics

Design for functionability

Design for manufacturability

Assembly

ing kag Pac

ss ne ve

n

atio

ort

cti ffe

le

ge ora

na tio

lity

bi

nc Fu

lia

Re

St

nsp Tra

Op er a co tion Ma st al nu me fact tho urin ds g

er

Re

ity

ma

abil

vir o eff nme ec nta t l

Eco bala logica l n effic ce and ienc y

Disposa

ycle

bility

Rec

bly em ss le sa yc ec r Lif facto

Di

En

n re ufac us tu ab ra ilit bili sa atio y ty/ fe na ty l

personnel safety, societal welfare, and environmental protection over their full commercial cycle, from the extraction of raw materials to the final disposal. Sustainable products are increasingly gaining popularity due to their inherent societal and environmental benefits. More recent economic studies have shown the longterm economic benefits of producing sustainable products for societal use. Sustainable products can involve fewer regulatory constraints, less liability, reduced material input costs, and can offer extended product life by giving them second and subsequent life-spans, while counterintuitive, sometimes, leads to greater profits for producers. A recent analysis of product design for sustainability presents an evaluation of six product sustainability elements and the multi-life cycle, “near-perpetual” material flow highlighting the need for transformation of 3R (reduce, reuse and recycle) into 6R covering three additional Rs (recover, redesign, and remanufacture) [102]. Functional performance of a product, including the product’s service life and durability, has been shown as a significant subelement within the “Functionality” element, a major element among the six major product sustainability elements [103]—see Fig. 11.44. Sustainability improvement through increased product life is a significant focal point in designing for sustainability. Longer product life of a

Figure 11.44 The basic elements and subelements of product design for sustainability [103].

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manufactured product can be achieved in several different ways. The quality of near-surface layers, also known as surface integrity, including the RS, has a strong influence on the product’s life, because it affects the product’s operational and/or functional performance. This includes the alteration of material properties resulting from manufacturing operations, and its ability to withstand severe mechanical and thermal loading (and unloading) conditions, such as corrosion and fatigue. Residual stress was a strong influence on the corrosion resistance and fatigue strength, thus on product sustainability. This influence will be discussed in the following two sections.

11.5.2 Corrosion resistance Corrosion of metallic materials has been the core of several research works due to its impact on the functional performance and life of mechanical components, and due to its high security and economical relevance. Mechanical, physical, and microstructural properties of metals can be significantly affected by the machining process; thus the electrochemical properties and corrosion resistance are consequently altered. Pitting and intergranular corrosion can present potential initiation sites for cracks, resulting in failure. In order to control its impact on the surface life and performance, some research works explored the relation between the parameters of surface integrity and the electrochemical reactivity and/or corrosion resistance of the materials in different environments. An example of the influence of the metal-cutting process on surface integrity and its impact on the corrosion resistance is described in this section, for the case of machining the OFHC copper. Variation of cutting parameters induces variations in the surface integrity, including RS, of the machined surface and subsurface, and thus consequently the variation of corrosion resistance at local scale (electrochemical reactivity) and long term (aging). For example, Fig. 11.45A shows the influence of the cutting speed in machining OFHC copper on local corrosion, quantified through the electrochemical parameters of polarization curves. (A)

(B)

1E7

100000 1000000 10000

10000

Anodic domain

1000

1000 100

Anodic current in the passive domain Ipass

10 1 0.1 –0.6

Cathodic domain

Cathodic current density Icath I(μA/cm2)

I(μA/cm2)

100000

–0.4

–0.2

0.0

0.2

E(V/Ag/AgCl)

0.4

0.6

0.8

100

10

Pitting potential Eb

1

0.1 –600

–400

–200

0

200

400

E(mV/Ag/Ag/Cl)

Figure 11.45 (A) Polarization curves for OFHC copper surfaces obtained by orthogonal cutting using two cutting speeds and (B) electrochemical parameters of polarization curves. OFHC, Oxygen free high conductivity.

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Figure 11.46 (A) Aging under NaCl saline environment of OFHC copper surfaces obtained by orthogonal cutting using two uncut chip thicknesses and (B) the topography of the surfaces after 2500 h of aging. OFHC, Oxygen free high conductivity.

Figure 11.47 The structure of the statistical approach used to classify the relevant surface integrity and cutting parameters.

As shown in Fig. 11.45B, several electrochemical parameters can be affected by the surface integrity: the current density inside the cathodic domain Icath, the current density in the passive domain Ipass, the extent of the anodic domain, and the pitting potential Eb. As far as the long-term corrosion (aging) is concerned, Fig. 11.46A shows the aging under NaCl saline environment of OFHC copper surfaces obtained by orthogonal cutting of OFHC copper using two different uncut chip thicknesses, while Fig. 11.46B shows the topography of such surfaces after 2500 hours of aging. A statistical approach is applied to identify the relevant surface integrity parameters in the consideration of corrosion resistance of the machined surfaces, followed by the identification of the cutting parameters (including tool geometry) that generate the relevant surface integrity. Two kinds of statistical methods are used: the Pearson’s correlation and the analysis of variance (ANOVA). The statistical approach is presented in Fig. 11.47.

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Table 11.5 Classification of the most influential surface integrity parameters on corrosion resistance of the machined surfaces. Ranking

Surface integrity parameters

Electrochemical behavior affected by the parameter

First

Residual stress in the cutting direction, σx Microhardness, HV Thickness of the plastic layer, tplastic Residual stress in the transversal direction, σy Roughness at maximum height, St Grains size, dgrain

Mean oxidized depth, Eb, Ipass, and Icath

Second Third Third Fourth Fifth

Oxidized surface and Icath Mean oxidized depth, Eb, and oxidized surface Mean oxidized depth, Eb Eb

Pearson’s correlation permitted to classify the surface integrity parameters by order of importance on corrosion resistance (both aging and electrochemical reactivity) of the OFHC copper-machined surfaces. This classification is shown in Table 11.5. The residual stress in the cutting direction, σx, is one of the most influential surface integrity parameters on corrosion resistance (first ranking in Table 11.5). It is fundamental that the amplitude of the RS affects the free energy of the material that, in return, has an impact on the electrons’ work function on the surface [104]. The residual stress σx is shown as the first order influential parameter the corrosion resistance. This is because the corrosion is accelerated by the action of tensile RS. Indeed, the oxide film is locally broken under the action of tensile stresses, creating vacancies in the film [105]. For that reason an increase in the tensile RS accelerates the anodic reaction (dissolution of the surface). ANOVA permitted to classify the cutting parameters (including tool geometry), which generates the relevant surface integrity parameters by order of importance. This classification is shown in Table 11.6. The uncut chip thickness is one of the most influential cutting parameters on this residual stress (first order in Table 11.6). The residual stress measured at the machined surface in the cutting direction, σx, is mainly affected by the uncut chip thickness, h (σx decreases with the increase of h). Considering that residual stress is generated by the mechanical and thermal loadings induced by the cutting tool on the workpiece, a thermomechanical analysis is required. As far as thermal loading is concerned, the impact of h and Vc on the temperature distribution in the machined surface is analyzed by numerical simulation. When these two cutting parameters were varied, the predicted temperature distribution in the machined surface was changed insignificantly, producing a very small variation (about 1.7%) of the yield stress, thus a negligible effect on the residual stress [106]. Therefore RS are essentially generated by the mechanical action of the cutting tool that can be explained based on the approach proposed in Section 11.3.1 (see Figs. 11.5 and 11.37). This approach considers the stress triaxiality effects in the workpiece in the strain at fracture, and consequently on the

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Table 11.6 Classification of the cutting parameters (including tool geometry) that generate the relevant surface integrity parameters. Order

Cutting parameter (including tool geometry) and interactions

Affected surface integrity parameter

First First

Uncut chip thickness, h Interaction of the uncut chip thickness with the rake angle, (h 3 γ) Interaction of the clearance angle with rake angle, (α 3 γ) Interaction of the uncut chip thickness with the clearance angle, (h 3 α) Cutting speed, Vc Tool clearance angle, α Tool rake angle, γ interaction of the cutting speed with the uncut chip thickness, (Vc 3 h)

σx, tplastic, and σy tplastic

Second Second Second Third Third Third

HV, St, and dgrain HV, σy, and dgrain σy HV and dgrain σy

Figure 11.48 Distributions of simulated stress triaxiality during orthogonal cutting of OFHC, at Vc 5 120 m/min, γ 5 20 degrees, α 5 10 degrees, as a function of the uncut chip thickness, h: (A) h 5 0.05 mm and (B) h 5 0.2 mm. OFHC, Oxygen free high conductivity.

magnitude of the residual stress at the machined surface [106]. Simulation results show an increase of the area affected by high-stress triaxiality in workpiece zone ahead of the tool cutting edge with the uncut chip thickness, h (Fig. 11.48A and B), which was also confirmed by Abushawashi [107]. These stress triaxiality results

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indicate that with the increase of h, the strain at fracture should decrease, consequently the plastic deformation in chip formation and thus also the surface RS.

11.5.3 Fatigue strength The fatigue resistance is largely influenced by manufacturing processes, which can modify the mechanical, microstructural, and metallurgical characteristics of the near-surface layers and introduce a gradient of RS between the surface and the bulk material. In fatigue, compressive RS can increase the life of the components, as they delay crack initiation and propagation [4]. Attention must be paid not only to the surface RS but also to the residual stress distribution in the internal layers. For example, in contact fatigue (fretting) the maximum RS are often located below the surface, which may induce subsurface crack initiation. The significance of achieving favorable RS distribution and their control during manufacturing processes for improving product’s lifetimes has been shown in several fundamental and applied researches works [108
110]. Most of these works describe the beneficial effect of the compressive RS introduced deliberately by applying mechanical, thermal, and thermochemical surface treatment techniques, such as low plasticity burnishing, shot peening, laser shock peening, and carbonitriding. Chien et al. [111] describe how the fatigue life of automobile crankshafts is improved by fillet rolling and thereby inducing compressive RS at critical locations in the crankshaft. Also, Shaw et al. [112] have shown that the fatigue life of a carburized gear was improved up to 75% by the careful control of compressive surface RS in the gear surface material by applying shot peening. A similar study was performed by Baptista et al. [113], which studied the contact fatigue damage of gears from an automotive gearbox submitted to combined treatments of carbonitriding and shot peening. They showed that applying shot peening after carbonitriding increases significantly the compressive RS (from 2300 to 21200 MPa) at the surface. This may justify the shorter fatigue life of the carbonitrided gears when compared with those gears shot peened after carbonitriding. Sasahara [114] studied the effect of RS and surface hardness generated by machining on fatigue life on a 0.45% carbon steel (see Fig. 11.49). The relative importance of RS and SR was studied by Novovic et al. [115]. In the range of 2.5
5 μm Ra SR, the fatigue life is primarily dependent on residual stress and surface microstructure, rather than SR. Segawa et al. [116] proposed a new final pass cutting tool design that induces RS that are more compressive than what would be generally obtained from the use of conventional tools. An important issue in the context of such induced compressive stresses is their relaxation when the component is in service, mainly caused by the applied cyclic loading and/or high temperatures [113
117]. This raises questions regarding the efficiency of such surface treatments for lifetime improvement at higher temperatures and/or under fatigue conditions, which are commonly encountered in many structural components used in power generation, automotive, and aeronautic industries.

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Figure 11.49 Effect of interaction of axial residual stress and hardness on fatigue life [114].

11.6

Conclusion

The reliability of mechanical components depends to a large extent on the physical state of their surface layers, also known as surface integrity. This state includes the distribution of RS induced by machining. They can enhance or impair the ability of a component to withstand loading conditions in service (fatigue, creep, stress corrosion cracking, etc.), depending on their nature: compressive or tensile, respectively. High compressive residual stress states in the machined surfaces induce a longer fatigue life of the component, and thus, higher product sustainability. This is particularly important when critical structural components made on difficult-to-cut materials are machined, especially, if the objective is to reach high-reliability levels and long service life. This is the case of the power generation, chemical and aeronautic industries, where high levels of tensile RS are usually found after machining of stainless steel, titanium, and nickel-based alloys. RS induced by machining are due to the thermal and mechanical phenomena generated due to the interaction between the tool and the workpiece. Their study requires proper knowledge of both experimental technique/method to determine the RS and metal-cutting physics. Although many experimental techniques are available to determine the RS, only a few of them can be effectively applied to determine the strong RS gradients induced by machining operations in the surface layers of components. Several criteria should be considered to select a suitable technique for determining the RS induced by a particular machining operation and work material. The knowledge of the metal-cutting physics is of great assistance to understand the influence of the cutting conditions on the RS. It also permits to control the RS

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by selecting the cutting conditions that reduce the level of tensile RS in machining components or even induce compressive RS. The experimental approach to investigate the machining RS is time-consuming based on trial and error and guided by empirical knowledge of the metal-cutting process. Modeling and simulation represented a cost-effective approach to investigate these RS when compared to the traditional experimental approach. Many analytical and FEM-based models of the orthogonal cutting process and only few of three-dimensional cutting (turning, milling, and drilling) have been developed. The predictability of such models strongly depends on the reliability of the constitutive (material) and friction models. Therefore the development of new constitutive models that considers the most influential parameters on the mechanical behavior of the work material in metal cutting is essential. Since, in metal cutting the deformation of the work material until fracture is required, the constitutive model should not only include the metal plasticity (flow stress) but also fracture. Both phenomena are depending on the state of stress (represented by the stress triaxiality and the Lode angle parameters), in addition to strain-rate, temperature, and microstructural effects. Since the two-dimensional model of the orthogonal cutting process hardly represents complex machining operations conducted at the shop floor, a need is felt to develop and validate new analytical and FEM-based models of the threedimensional cutting. These models should be able to predict among other surface integrity characteristics, the RS formed in practical metal-cutting operations, such as turning, milling, and drilling. These models can be used in the optimization of the machining conditions for an improved functional performance and life of machined components, without compromising (or even improving) their productivity.

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[9] V.P. Astakhov, Drills: Science and Technology of Advanced Operations, CRC Press, Boca Raton, FL, 2014. [10] V.P. Astakhov, Metal Cutting Mechanics, CRC Press, Boca Raton, FL, 1998. [11] J.C. Outeiro, A.M. Dias, J.L. Lebrun, Experimental assessment of temperature distribution in three-dimensional cutting process, Mach. Sci. Technol. 8 (2004) 357
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Microstructural changes in materials under shock and high strain rate processes: recent updates

12

Satyam Suwas1, Anuj Bisht1 and Gopalan Jagadeesh2 1 Department of Materials Engineering, Indian Institute of Science, Bangalore, India, 2 Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India

12.1

Introduction

The plastic deformation of materials under extreme conditions is of immense importance due to their different response under different conditions, during service life. The shock/blast response (dynamic deformation), which is basically a high strain rate phenomenon, is one such extreme condition. Crash event is also a high strain rate phenomenon. The crash worthiness is an important parameter in designing an aircraft and automobile structure. The response of material under detonation, in case of a terrorist attack or war, is another area where the shock/blast exposure and high strain loading comes into picture. In this regard, attenuation of shock waves as it travels through the material medium is an important aspect, and the microstructural response is believed to be the key to understand shock attenuation. Further, the growing consumer demand for commodities calls for high-through output, which requires faster production lines and reduced machining time, which implies high strain rate requirement. Shock or blast waves also find application in high-velocity sheet metal forming operations. Sheet metal forming is a well-known procedure in the processing industry where plates (t . 6 mm) and sheets (6 mm . t . 0.15 mm) are deformed into coveted shapes. The rate at which these loads are applied, the range of operating temperatures and pressures and sheet thickness virtually dictate the metal forming processes. While there are established industrial methods to form metallic plates (Bfew mm thick), considerable difficulties are experienced for the adoption of conventional techniques to deform thin metal sheets/foils (thickness 0.1
1 mm) to the desired shapes. High-velocity forming of thin sheets is one of the popular methods of obtaining shapes that are difficult to form using conventional manufacturing techniques. High-velocity forming using electric or magnetic fields and explosive gases are used in the industry to form thin sheets. This acts as a frictionless punch,

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques. DOI: https://doi.org/10.1016/B978-0-12-818232-1.00012-6 Copyright © 2020 Elsevier Ltd. All rights reserved.

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and energy transfer (kinetic energy) is efficient. Hence, considerably less energy is required in high-velocity forming as compared to conventional sheet metal forming processes. Due to the high strain rates involved in such forming processes, the microstructure and texture evolution is expected to be different from conventional processing, where the strain rates are lower in order of magnitude. Further, microblast-assisted metal foil forming finds interesting biomedical applications [1,2]. In this method the energy of the microblast is used to deliver microparticles for biomedical applications such as needleless delivery. The metal foil is used as an energy-transfer media between the microblast and particles. The foil acts as a separation media between the particles and the blast combustion product, thus maintaining the pristine condition of particle. The finite element analysis using continuum approach of the deformation of the foil coupled with experiments were previously carried out [1,3]. The particle velocity, spray cone angle, and velocity distribution of jet were found to be dependent on the deformation behavior of the blast-loaded foil [1]. Moreover, the foil showed plastic hinge effect and different failure modes in finite element simulations [3]. However, the deformation behavior of the foil with respect to the material parameter was not investigated. The plastic deformation behavior of a material depends on many factors, including both material exclusive and inclusive factors. The material exclusive processing parameters include strain rate [4], strain path [5], processing temperature [6
9], and loading condition [10,11]. The material inclusive parameters include crystal structure [8,12], microstructure [13,14] (grain size, shape, precipitate, etc.), and texture (relative to the loading direction) [15
22] of the material, to name a few. Many factors are dictated by indirect influence of the crystal structure and the loading direction. This is due to the inherent anisotropic nature of crystals at atomic level [23
25]. Many of the industrially important metals have either face-centered cubic (FCC) or hexagonal closed packed (HCP) crystal structure. For the FCC crystal structure, lattice parameter a 5 b 5 c and α 5 β 5 γ 5 90 degrees. For HCP crystal structure, lattice parameter a 5 b 6¼ c, α 5 β 5 90 degrees, and γ 5 120 degrees. Stacking fault energy (SFE) is an important material parameter in metals having FCC structure, while “c/a” is an important crystal parameter in HCP metals influencing the material behavior. Also, the grain size of materials significantly influences the mechanical behavior of the material [26
28]. Understanding of material response is important for selection and design of material, which find usage in many applications. The key to understand the bulk behavior of material upon shock loading is to understand the shock compression phenomenon and local material response. With this as focus, in this chapter, first we will briefly discuss the nature of general shock waves. This is followed by a brief of experimental methods used for investigating material response under shock compression. The effect of various materials (both intrinsic and extrinsic) and shock parameters on the material response are further discussed. Finally, the process of material shock compression, the mechanism of defect formation, and the driving force for such phenomena will be discussed.

Microstructural changes in materials under shock and high strain rate processes: recent updates

12.2

363

Shock wave and parameters

The first step in understanding the material response is to understand the basic nature of shock phenomena. In this section a brief introduction of the shock wave and its characteristics along with typical shock parameters used are introduced. The condition necessary for shock waves to form in material is discussed. Any disturbance in a media results in a wave propagating through it, which carries energy. Sound wave is a classic case of elastic disturbance traveling in a media. Such waves are periodic in nature. Shock waves also propagate through media and carry energy. However, in contrast to the ordinary wave, shock wave is not periodic in nature and is characterized by an abrupt discontinuous change in pressure, density, and temperature. The shock wave travels in the form of a front, across which the discontinuity exists; the front is called “shock front.” Such waves are often characterized with the help of pressure profile (pressure
time curve). A typical pressure profile has a pressure jump (indicating the start of shock front), a dwell region, and the decay wave. Important shock parameters in the pressure profile are peak pressure, rise time, dwell time, decay time, and the associated pressure impulse. Shock waves are broadly classified as shock waves and blast waves based on whether the dwell phase is present or absent. A typical shock wave and blast wave profile with important parameters is shown in Fig. 12.1A and B, respectively. The shaded region is the impulse of the wave profile. Unlike shock wave, the blast wave has no dwell time. The rise time in a blast wave is considerably smaller than that of a shock wave. Its pressure decays exponentially after the peak value is reached and is often trailed by a negative pressure phase. As the name suggests, blast waves are typically generated in case of explosion. In a medium, shock waves are generated whenever a disturbance is produced, which moves faster than the local speed of sound. Gaseous and liquid media cannot support shearing stress. Only compressible pressure state can exist in such media. In air, formation of shock waves is typically observed in a supersonic flow such as

Figure 12.1 Typical pressure profiles of (A) a shock wave, and (B) a blast wave.

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during flight of a supersonic aircraft in the form of a Mach cone and sonic boom. A conventional shock tube can be used to generate shock wave in gaseous media in laboratory. It is generated by an abrupt creation of pressure difference (jump) between the diver tube section having high pressure and the driven tube section having low pressure with the help of diaphragm or fast opening valve. The resultant shock wave has a planar front and travels along the length of the driven section. Both shock and blast waves can be generated in a controlled and contained manner by tuning of the shock tube. Solid media can support shear stresses. Thus the condition for generation of shock waves in solid media is little different from gaseous and liquid media. When the amplitude of stress waves greatly exceeds the dynamic flow strength of a material, shear stresses can be neglected when compared to the compressive hydrostatic high-pressure state traveling in the material [29]. In such a case the solids can be safely treated as compressible fluid/gases. Velocity of a disturbance  [29] traveling 1=2 is given by ððdσ=dεÞ=ρÞ1=2 . For a gas [29], it is proportional to ðdP=dVÞ=ρ . The high-amplitude isentropic disturbances travel faster than low-amplitude disturbance in gases. For shock wave formation a disturbance front should steepen up as it travels through the material. This can happen when the high-amplitude regions of the front travel faster than the lower one. The necessary condition for shock formation in solids in the simplest form is stated as follows [29]: dσ m as σm (12.1) dV The shock waves in solids are treated as a hydrodynamic problem. Equations of state have been obtained for various materials experimentally [30
32] and theoretically [33,34]. When a solid is shock-compressed, at low shock pressures only elastic shock waves are observed [35]. Above a critical shock pressure, this single elastic shock wave splits into two waves: an elastic shock front trailed by a plastic shock front [35
37]. The transition is clear in the shock Hugoniot curve [38] (P
V curve of material under shock compression) or the surface velocity measurements using VISAR technique [35] or in shock pressure profile [39]. The two-wave structure of shock is shown in the schematic pressure profile (Fig. 12.2A) and free surface velocity profile (Fig. 12.2B) for material under plastic shock compression. A typical pressure
volume shock Hugoniot curve for metal is shown in Fig. 12.2C indicating the elastic
plastic transition point “o.” This transition point is called the Hugoniot elastic limit (HEL) in the shock Hugoniot curve. When material is subjected to shock above the HEL, defects in abundance are observed in the material [40,41]. A second transition point can be observed in the Hugoniot curve or shock pressure profile when phase transformation is observed in the material, which showed only plasticity at a lower shock pressure [42]. However, it is not necessary for all materials to show a distinct point indicating plastic and phase transition. Shock deformed materials have been studied extensively by metallurgists as they show drastically different microstructures compared to quasistatically deformed materials [43,44]. Various defects such as vacancies [41,45
47], voids

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Figure 12.2 Schematic of a typical (A) pressure profile and (B) free surface velocity under plastic shock compression showing the two-wave structure. (C) P
V shock Hugoniot curve: the point “o” is the HEL above which the elastic
plastic transition is observed. HEL, Hugoniot elastic limit.

[39,41,48
50], dislocations, twins [40,51,52], shear bands [49], and phase transformations [53
56] have been observed in shock-loaded materials. The important difference is the sheer abundance of defects generated in the material after shock loading [57]. Their generation is said to accommodate the transient strain induced by compression on the passage of shock wave, which is associated with the initial stage of shock compression [58].

12.3

Experimental methods for investigation of shock waves

A variety of experimental setups have been used in the literature to study the dynamic behavior of materials. These techniques encompass areas of high strain rate tests, shock in the material, and explosive detonation. The experimental methods used to study the material response to shock deformation are described in brief in the present section. This is important as the method employed for investigation may heavily influence the obtained results of the study. For example, the residual strain along shock direction. Moreover, with time, there have been advances in experimental facilities and simulation capabilities, which have facilitated to comprehensively investigate the minute insights related to shock compression phenomena. Thus it is crucial to understand the methodology employed for better understandability of results available in the literature. The important experimental techniques used in the literature for material shock investigation is briefly described with focus on pros and cons.

12.3.1 Taylor’s impact test Taylor et al. developed this method to determine the dynamic strength of material under compression [59
61]. A cylindrical projectile is made to strike on a flat rigid target perpendicular to it. Upon impact a compressive elastic wave travels toward

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Figure 12.3 Schematic of cylindrical sample before and after Taylor’s impact test. Li: cylinder length before impact, Lf: cylinder length after impact, and X: length of undeformed cylinder. Adoption based on earlier studies in literature [59,62].

the rear end. Once the rear end is reached, the wave is reflected back as a tension wave. If the stresses are high, the elastic wave is trailed by a plastic wave. The details of the experiment can be found elsewhere [59
61]. The cylindrical specimen before and after the test is schematically shown in Fig. 12.3. This test was popular for dynamic test as the speed of impact can be varied which made it possible to get a wide range of strain rate. There are regions in material ranging from plastic to elastic, which are helpful for understanding continuous change in the microstructure. The inaccuracy to find strain value in the sample and the nonhomogeneity of the sample are the two main drawbacks of the method.

12.3.2 Explosive loading of materials Explosive loading has been used for the dynamic testing of materials, and it relates close to the real-life scenario of blast attack on structures. There are a variety of explosives loading assemblies used in the literature [63
67]. The blast wave produced by combustion of explosives is subject to the test specimen. The main advantage of this method is the achievable high-enthalpy blast waves, which can be used for testing. This depends on the amount of explosive used. However, with increasing explosive content, the danger during the experimentation increases exponentially. A lot of safety precautions and permissions to conduct the test have to be obtained. The difficulties in obtaining explosive in considerable quantities and the permission and certification required from the government agencies for experiment pose a major drawback.

12.3.3 Flyer plate impact test The flyer plate impact test is one of the most popular methods to induce shock in the material. An accelerated plate is made to impact a plate specimen. The plate

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Figure 12.4 Schematics of flyer plate experimental setup with shock recovery fixture [69].

acceleration can be attained by means of a gas gun [68,69] or explosive lens
type setup depending on the requirement. For a small flyer plate projectile or piston the experiment is also referred as “piston-driven shock experiment.” The principal advantage of setup is the resultant planar shock produced in the material. Also, the setup can be modified to have momentum trap and specimen recovery assembly [68,69] for recovery of shocked specimen. A schematic depicting the flyer plate test with specimen recovery fixture is shown in Fig. 12.4. The flyer plate material and impact velocity are the two primary parameters that dictate the shock generated in the material. It is possible to obtain shock pressure ranging from few hundreds of kbar [70] to few tens of GPa [68,69] and shock pulse width of few μs.

12.3.4 Split-Hopkinson pressure bar The Split-Hopkinson pressure bar (SHPB) is used for the dynamic deformation of specimens at high strain rates (103
104 s21). A schematic of SHPB in compression mode [71] is shown in Fig. 12.5. The gas gun, striker bar, incident bar, and transmitter bar are the fundamental components of SHPB. In a compression test the sample is placed sandwiched between the incident and transmitted bars. The incident bar is struck with a striker bar of same diameter propelled using a gas gun. This generates a compression wave that travels down the incident bar. A part of the compression wave is reflected, and the other part is transmitted at the incident bar
specimen interface. The stress pulse continues through the specimen and into the transmitted bar. The stress pulse incident and transmitted bars are measured using strain gauges placed on the bars. The bars are designed to remain elastic during the test. The dynamic stress
strain curves of the material can be obtained from the measured pulse signal.

12.3.5 Shock impact in a shock tube This method is similar to the explosive loading of the material, however, in a contained manner. The shock wave generated in a conventional shock tube can be

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Figure 12.5 Schematic of SHPB for dynamic testing of materials [71]. SHPB, SplitHopkinson pressure bar.

Figure 12.6 Schematic layout of a conventional shock tube [73].

subjected to material [72
74]. In a conventional shock tube a normal shock front is produced by creating a sharp pressure difference in the medium. This is achieved by sudden bursting of a diaphragm, which separates a high-pressure gas (in driver section) from the low-pressure gas (in driven section). The schematic of shock tube setup is shown in Fig. 12.6. The driver section of the tube is filled with helium gas to a pressure to rupture the diaphragm. As the critical pressure reaches in the driven section, the diaphragm ruptures suddenly, and compression waves are formed. These waves travel in the driven section and rapidly steepen to form a shock front. The shock wave compresses the gas in the driven tube as it travels. The moving shock front is exposed to the disc fixed concentrically at the other end of the driven tube, deforming the flat disc into a hemispherical cap. The enthalpy of the shock generated is considerably low compared to that in explosive test [72]. It is possible to obtain blast wave and to tune the peak pressure and pulse width (ms to μs) of the shock wave in a shock tube.

12.3.6 Laser-induced shock generation In the last two decades, interest in investigation of microstructure by laser-induced shock has increased. This is primarily due to the lower limit of pulse time, which can be achieved only via laser-induced shock. The lower limit of pulse width

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achievable by other experimental techniques is of the order of μs. However, pulse width of nanoseconds [75
78], picoseconds [79,80], and femtoseconds [81,82] can easily be achieved by tuning of suitable laser source. Very high energy density and power of the order of terawatt can be achieved using focused laser, enabling to reach the limiting theoretical stresses in the material. In a laser shock experiment, laser is focused on the specimen surface. The high density of energy deposited in a very short period of time leads to spherical shock generation within the material. Many a times the surface of the specimen is coated with sacrificial material to prevent ablation of specimen. The wave produced in material is essentially a blast wave and is near spherical. The wave attenuates exponentially as it travels in the material. The VISAR technique [83] coupled with laser shock exposure of thin films has added a great dimension to the in situ measurement of the surface velocity of the material. From the velocity profile measurement, important shock parameters can be calculated. The real-time in situ X-ray diffraction (XRD) measurement of material has particularly led to interesting finding, which is possible only in laser shock experiment. Laser shock exposure of bulk sample is localized. Care must be taken while selecting the region for investigation in the specimen. Though the bulk average energy, power, and energy density can be calculated for the laser-generated shock, it is difficult to accurately relate the same at the investigated region. Also, the local strain in the region of investigation is uncertain. Accompanying simulations can help in this aspect.

12.4

Parameters influencing material response to shock exposure

Shock deformed materials show drastically different microstructures compared to quasistatically deformed material [43,44]. The microstructural response of the material upon shock loading has been studied extensively. Various defects such as vacancies [41,45
47], voids [39,41,48
50], dislocations, twins [40,51,52], shear bands [49], and phase transformations [53
56] have been observed in shock-loaded material. Multitude of parameters does influence the material response under shock loading. These range from extrinsic (grain size, texture) and intrinsic (SFE and c/a ratio) material parameters to extrinsic experimental parameters (shock parameters and residual strain in material). The behavior of the material with respect to these parameters is discussed in this section. For completeness, response of bodycentered cubic (BCC) metals to shock is also discussed.

12.4.1 Types of shock-generated defects Earlier studies on shock studies were focused on generation of all types of defects. Vacancies are observed in abnormally high concentration in shocked material. Presence of vacancies was first reported in the case of MgO single crystal when exposed to explosive shock, using electron spin resonance [66]. Later in

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molybdenum also, the vacancy concentration was found significantly higher after shock loading compared to that in material prior to shock loading, as revealed by field ion microscopy [41,46,47]. Vacancies and vacancy clusters were observed to increase by 6% with shock pressure. Field sequence showing successive atom  evaporation    layers removed from the [0 1 1]
1 2 1
1 1 2 , indicating presence of vacancy and vacancy clusters in molybdenum [46], is shown in Fig. 12.7. It is interesting to note the abundance of vacancy when compared to interstitial type point defects. Recent studies deal with the effect of vacancies on the dynamic response of Cu to shock waves [84] by molecular dynamics (MD) simulation. It was found that the shear flow strength and the spall strength decrease with increasing vacancy concentration. The effect was significant above the vacancy concentration 0.25%. With time, the focus has shifted to dislocations and twin generation during shock deformation. The dislocation loops consisting of residual microstructure after shock loading Mo [41] at 150 kbar is shown in Fig. 12.8. Microtwins and transformed phase are also observed in austenitic stainless steel [76] shock-loaded using laser with subnanosecond (sub-ns) pulse duration. Darkfield transmission electron microscopy (TEM) micrograph of austenitic steel, laser shock-loaded to 22 GPa, depicting set of twins [76] is shown in Fig. 12.9. In this study, it was observed that the microstructure of laser shock deformed steel was similar to conventional shock despite having much shorter pulse duration (sub-ns). The number of twin sets increases with increase in shock pressure accompanied by decrease in mean twin spacing, indicating overall increase in twin fraction. The embryos of α-phase were also observed in high-pressure range.

Figure 12.7  Field  evaporation   sequence showing successive atom layers removed from the ½0 1 1
1 2 1
1 1 2 . The presence of vacancy and vacancy clusters is indicated by arrow [46].

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Figure 12.8 Bright-field TEM micrograph of Mo shock-loaded at 150 kbar [41].

Figure 12.9 Dark-field TEM micrograph of austenitic stainless steel shock-loaded to 22 GPa showing set of twins [76].

12.4.2 Effect of material parameters 12.4.2.1 Grain size In quasistatic deformation, reducing the grain size increases the strength of material; the effect is known as Hall
Petch effect [26
28]. In microcrystalline metals,

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dislocation slip and twinning systems are the major modes of deformation [10,12,85,86]. In nanocrystalline (nc) metals, apart from dislocation and twin-based mechanisms [87,88], other deformation modes such as grain boundary sliding, grain rotation, and grain boundary migration become dominant [89
92], especially when the grain size is below a critical value (usually d , 20 nm). Below this critical grain size the material starts softening due to grain boundary
based mechanisms; the effect is known as Inverse Hall
Petch effect [89]. The response of material changes considerably with grain size even in quasistatic regime. In microcrystalline regime the only study pertaining to the effect of shock loading with grain size reported that the cell size did not change with grain size [93]. No further information is available on the effect of shock loading with grain size for microcrystalline material. However, the major focus of literature is on the investigation of the response of nc material to shock deformation, which has been studied experimentally and by MD simulation, with latter being abundant. Laser shock compression of Ni was done over a range of pressure [94]. Increase in hardness was observed after shock compression, unlike conventional deformation of nc Ni. The onset of slip-twinning transition pressure has shifted from 20 GPa in monocrystals to 80 GPa in nanocrystals (10 nm grain size). Accompanying MD simulation has shown that a large fraction of dislocations generated at the front are annihilated upon shock unloading. Spallation study of nc Pb under shock loading by MD was carried out by Chen [95]. It was observed that the grain boundaries had significant role in shock-induced deformation. Spallation and cavitation were observed to start at the grain boundaries. Grain boundary acted as the nucleation site for crystal plasticity and presence of voids was also reported in the MD simulation of columnar nc Cu [14]. This was attributed to the combined effect of grain boundary weakening and stress concentration. The MD study on nc Cu revealed that the nc grain size does not significantly change the pressure and temperature of the shock Hugoniot state [96]. However, the grain size, the particle velocity, and the grain orientation governs the formation of local BCC clusters in Cu between 100 and 200 GPa shock pressure [96]. The transformation was observed to be via Bain path (tetragonal transformation path) [96]. In another MD study, nc Cu was observed to show ultrahigh flow strength at high shock pressure with values about twice that observed at low shock pressure [97]. Urbassek [98] reported that the pressure-induced phase transition (BCC to HCP) reduced the probability of shock-induced spallation and crack formation in nc iron by MD simulation. However, they reported that twin growth was influenced by phase transformation and twins may provide the source for void nucleation. MD simulation of single-crystalline and nc Ni was carried out by Jarmakani et al. [99] for varying shock pressures. In the single crystal, dislocation cells were observed at low shock pressure and stacking faults at high pressure. The transition pressure was reported to be 27 GPa. In nc materials, grain boundary sliding accounted for maximum shock
induced strain (58%
90%). Dislocation partials were dominant in the nc materials. Twinning was more favorable in 5 nm grain size sample compared to 10 nm grain size sample. The critical twinning transition pressure was found to be 78 GPa.

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The nc Cu of 6 and 16 nm grain size were reported to show peak twin densities and spall strength, which were comparable to single crystal Cu [100]. This was attributed to high density of twins in the grain interior with larger fraction of coherent twin boundary segments.

12.4.2.2 Texture Earlier literature presents many contradicting views regarding changes in crystallographic orientation after shock loading. Texture changes were reported in shockloaded copper by Cohen et al. [101]. Texture changes were also observed in commercially pure aluminum after shock wave deformation by Dhere et al. [70]. However, no texture change was reported in copper and nickel by Higgins [102] and Trueb [103]. Microstructural changes were also examined for the material after shock loading [104]. Recent studies on shock-assisted deformation of Al by Ray et al. [73] and of Cu by Bisht et al. [72] showed considerable change in texture of the material. The observed texture change was found to be dependent on final strain in the material and on deformation geometry [72]. The strain path of the material is known to influence the resulting texture [15,105,106]. Thus the different reports in the literature could be a function of the experimental method used, the residual strain in shocked material, and the strain path experienced by the material (which in turn depends on the experiment).

12.4.2.3 Face-centered cubic metals: stacking fault energy SFE is a very important parameter for FCC metals and primarily governs their deformation behavior. FCC materials deform by slip as well as twinning. Twinning occurs as a dominant deformation mechanism in material with low SFE [85,87]. This is because the critical twinning stress decreases along with SFE. Shock loading at 55 kbar of a series of α-brasses was investigated by Inman [107], and a transition in dislocation configuration was observed with SFE in the range of 25 and 36 ergs/ cm2 from cell structure at higher SFE to coplanar dislocation groups at lower SFE. On explosive loading of Cu and Cu
Al alloys, it was observed that the critical pressure for slip-microtwinning transition decreased with addition of Al [63,108
110] (both under explosive loading and laser shock compression). This was attributed to the decrease in SFE with the addition of Al. The effect was explained based on threshold stress for slip-twinning transition. Cobalt addition to Ni is known to reduce the SFE of the alloy. Cobalt has very large solid solubility in Ni (up to 65 wt.%), and there is a linear but drastic decrease in SFE due to cobalt addition. Ni as well as Ni
60 wt.%Co alloy was shock-loaded by flyer plate impact technique [111]. The HEL was insensitive to SFE. However, the spallation stress increased with decrease in the SFE. Heavy deformation twinning was observed in Ni
60 wt.%Co alloy, unlike pure Ni. The effect of SFE on microstructure is summarized in the schematic (Fig. 12.10). With decrease in SFE the propensity of twin formation increases. High

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Figure 12.10 Schematic for microtwin
microband transition in FCC metals and alloys with shock structure and stacking fault energy [112]. FCC, Face-centered cubic.

SFE materials may show substructure ranging from microband to microtwin with increase in the peak pressure [112].

12.4.2.4 Hexagonal closed-packed metals: c/a ratio HCP materials are rather less investigated from shock loading perspective. Out of the available literature, the results of shock treatment on HCP metals do not appear to display consistent trend. Shock deformation of Mg [113] and Ti64 [114] led to small dislocation density. Extensive deformation twinning was observed on shock loading of Ti64 by Gray et al. [25], while a complete absence of deformation twinning in the same material was reported by Hameed et al. [115]. On shock deformation of Ti and alloys as well as Zr, some authors have reported the occurrence of only orthorhombic αʺ phase [116], while others have observed only α to ω phase transformation [117,118]. Lookman [117] observed α to ω phase transformation in Zr with the orientation relationship inconsistent with the original suggested by Silcock [119]. The existence of thermodynamically stable multiple phases could also make the response of the HCP metals and alloys under shock loading more complicated. The role of c/a ratio, which is known to affect the deformation behavior of HCP metals and alloys under quasistatic deformation condition, has not been systematically investigated for shock loading.

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12.4.2.5 Body-centered cubic metals The research on shock loading of body-centered cubic (BCC) materials has been focused only on selective important materials, namely, BCC iron [54,120
122], tantalum [123
127], and molybdenum [41,128,129]. At low shock pressure of 7 kbar, abundant defects without twins are observed in the BCC phase (α phase) of iron [130]. With the increase in pressure to 10 kbar, profuse twinning is observed in iron. The shock-induced phase transition [BCC (α) phase transforms to HCP (ε) phase] is very well reported in iron at shock pressure of 13 GPa and above [54,120,121,131,132], which is readily achieved by laser-induced shock for laser with pulse width of nanoseconds and below. The high-pressure HCP (ε) phase produced under shock compression is metastable in nature, and it reverts back to α phase after shock release. Nevertheless, indirect evidences of the α
ε phase transition have been observed. Microstructural evidence of large number of {3 3 2} h1 1 3i secondary twins (rare twin) within {1 1 2}h1 1 1i primary twins was observed by Dougherty et al. [122] in iron shock to 13 GPa has been considered as indicative of the transient α
ε phase transition. Moreover, the formation of ε phase has been verified via in situ XRD measurements under shock compression [120,131,132]. Further, the XRD whole profile analysis of in situ measurement has revealed that the ε phase formed has a grain size in the range 2
15 nm [121]. Atomic shuffle has been identified as the mechanism for the α
ε phase transition under shock compression [120,121,131], with uniaxial collapse along [0 0 1] direction accompanied by shuffling on alternate (1 1 0) atomic planes [131]. Interestingly, Sano et al. [132] observed the γ iron, which is a high-temperature FCC phase of iron, in trace amount in shock recovered sample. This led them suggest that γ phase is the intermediate metastable structure during α
ε phase transition under shock compression. However, the intermediate γ phase was not observed by Hawreliak et al. [133] in their in situ XRD measurement of shock compression of plycrystalline iron. They suggested that the discrepancy might be due to the timescale adopted for the measurement. In MD study of shock compression of iron by Kadau et al. [54], the general phase transformation from BCC to close packed phase (including both FCC and HCP phases) was observed, with FCC phase energetically favored at high shock pressures (240 GPa) [134]. The crystallographic relation observed for the transformation corresponds to standard martensitic transformation. Shock compression studies of tantalum have been primarily focused on the mechanical aspects of shock compression such as HEL, spall behavior, and damage [125,135,136]. At 7 and 20 GPa for a range of temperature up to 400 C, Ta and Ta-10W shows similar shocked substructure consisting mainly of long and straight h1 1 1i type dislocations with screw segments. Twinning starts at higher shock pressure of 25 GPa and is observed in significant volume above 55 GPa, with largest fraction of twins present for (1 1 0) crystal compared to other planes in a single crystal Ta [137]. Twinning starts at 30 GPa and is predominant above 70 GPa as reported in MD simulations of (0 0 1) Ta single crystal [138]. Short-range shuffling of alternate (1 1 2) planes by bulk reordering is identified as the twinning mechanism in Ta [124]. Nonetheless, there are no reports of phase transformation in Ta

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under shock compression unlike iron. Similarly, microstructural changes resulting in an increase in dislocation density and vacancy concentration was reported in shock-loaded molybdenum [41]. However, phase transformation has also not been reported for molybdenum.

12.4.3 Effect of shock parameters 12.4.3.1 Peak pressure and pulse width Effect of peak pressure and pulse duration on the microstructure of the material subjected to shock loading was carried out on nickel by Murr and KuhlmannWilsdorf [57]. In this study, peak pressure in range of 80
460 kbar of 2 μs duration and 0.5
2 μs duration at a constant pressure of 250 kbar was used for the study. The defect concentration in the material increased with increasing shock pressure. Dislocation cells were observed in the deformed Ni, whose cell size was inversely proportional to the square root of dislocation density and peak pressure. However, in the investigated pulse width range, no significant change occurred in the microstructure (dislocation cell size and density). The corresponding micrographs are shown in Fig. 12.11 [57]. Dislocation cell structure was not observed for 0.5 μs pulse duration, leading to the conclusion that the time required for cell formation by dislocation movement is of the order of 1 μs. For shock pressure of 350 kbar and above, twinning was prominent and the intertwin spacing decreased with shock pressure [93]. At even higher shock pressure of 30 GPa for 2 μs pulse duration, microtwins were observed in Ni [112] (Fig. 12.12). For 1018 steel, dislocations, twins, and microbands were observed after shock deformation and the defect storage increased with shock pressure as observed by greater activity around twins [139]. High defect storage led to shear localization and resulted in microstructural instability. In another study on 1018 steel, higher volume fraction of twin was observed for higher peak pressure [140]. Above 13 GPa, α
ε phase transition was observed and at 16 GPa, where secondary twins were observed [140]. In explosively loaded Cu and Cu
Al alloys, the mode of deformation changed from slip to microtwinning [63] above a critical pressure. The critical pressure was in turn dependent on SFE. In Ta, explosive loading led to formation of significant f1 1 2gh1 1 1i twins of lenticular shape preferentially along the grain boundaries [141], for both higher and lower shock pressures. It was also noted that the twins had frequent irregular morphology. In laser shock
treated copper, dislocation cell structure was observed at 12 and 20 GPa peak pressure with 5 ns pulse width [75]. At 40 GPa, twinning and stacking fault bundles were observed. At 60 GPa, microtwinning and effects of thermal recovery were present in the microstructure. It was pointed out that the observed defect structure and substructures were similar to the copper shock treated at higher pulse duration of the order of μs. Aluminum films (Bμm) when shock treated using picosecond (ps) laser showed microstructure similar to planar impact experiments [35]. The elastic
plastic wave separation decreased with increasing laser energy (increase in peak pressure). The leading elastic wave stresses were observed to be

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Figure 12.11 TEM micrographs showing residual dislocation microstructures in Ni postshock deformation at various pulse duration and peak pressure [57].

as high as 12 GPa pressure and was observed to be thickness-dependent. The elastic wave decayed rapidly with increasing sample thickness. The behavior of material and the resultant microstructure were dependent on the peak pressure. However, the shock-treated microstructure was relatively independent of pulse duration above picosecond (ps) pulse duration. This was not the case when the materials were deformed at ultrashort pulse duration of the order of femtosecond (fs). Al exposed to femtosecond laser showed shear strength of 2.7 GPa [142], while iron displayed

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Figure 12.12 Microtwins in Ni induced by plane shock of 30 GPa and 2 μs pulse duration [112].

a shear stress of 7.9 GPa [142,143], which are close to their theoretical shear strength. In MD study of propagation of femtosecond (fs) shock wave in Ni, it was reported that the shock propagated and remained as an elastic wave till a very high pressure [81], exceeding the elastic precursor pressure observed in nanosecond (ns) and piston-driven experiments [referring to the small size flyer plate (piston) experiments]. Ultrashort pulse shock in iron is reported to form dislocations with limited intersections near the surface [144]. However, dislocations far from surface had microbands with dislocation network similar to long-shock process.

12.4.4 Other factors: residual strain The residual strain in the shock-treated material can vary depending upon the method used for shock generation. It can also vary within the sample and can be inhomogeneous as in the Taylor’s impact experiment and laser-induced shock experiment. The residual strains in the experimental shock recovered samples has been reported to lie anywhere from 2% to 26% along the shock direction in a flyer plate experiment [68,69,145,146]. Study on shock-loaded copper with specimen recovery fixture by Gray et al. [69] showed that the residual stress is noticed to govern the deformation substructure, from dislocation cells at low residual stress to deformation twins at high residual stress in Cu [69] (as stress proportional to strain). The TEM micrographs have various residual strains in Cu specimen from the study are shown in Fig. 12.13. For 10 GPa shock pressure, the transition from dislocation

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Figure 12.13 TEM micrographs of Cu shock-loaded at 10 GPa and with residual strains showing various substructures (A) 2%: cellular dislocation structure, (B) 7%: cellular dislocation structure and planar dislocation arrays, (C) 26%: microbands, and (D) 26%: deformation twins [69].

cellular arrangement at 2% strain to planar dislocation array at 7% strain and finally microbands and deformation twins at 26% residual strain can be seen in Fig. 12.13A
D, respectively. Deformation twins and microbands both were observed at 10 GPa and 26% residual strain Cu sample (Fig. 12.13C and D). Also, it was reported in a shock MD simulation of Ni that the majority of dislocations generated on shock compression get annihilated upon unloading [99]. Thus the residual plastic strain considerably affects the substructure and mechanical response of the shock-loaded material, apart from peak pressure and pulse duration [69]. In a recent study on femtosecond laser-assisted shock exposure of cp-Ti, Bisht et al. [147] showed that the residual strain has a prominent effect on the presence of shock generated defect. It was reported that if the residual strain in the material is allowed to relax, the shock generated defects can revert back to the virgin parent crystal orientation without defects. This reversion due to residual strain relaxation can be observed leaving a contrast band in the image quality map as shown in Fig. 12.14A. The remnant defect (twins) can be observed along the edge of these contrast bands (indicated by arrows in Fig. 12.14A). The region which has not relaxed primarily consists of twin intersected region. It was deduced in the

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Figure 12.14 (A) IPF map superimposed with IQ of a region. The arrows indicates region where the orientation of the defect grain has reverted back to parent grain P on relaxation. (B) Magnified view (IPF map) of region L marked in (A). (C) Misorientation relation of grains marked in (B) [147]. IPF, Inverse pole figure; IQ, image quality.

investigation that the defects with large associated shear had higher driving force for reverse transformation upon residual strain relaxation. The high concentration of defect suggests that usual dislocation-based mechanism might not be operable, suggesting local atomic shuffling as the plausible active mechanism for defect formation and reversal under shock exposure. This finding supports results of previous shock MD simulations [56
58] where it was reported that majority of the defects are annihilated on shock unloading, a state with no residual strain. Thus it can be said that the defects that have formed during shock compression will revert back to the original defect-free state, provided there is zero residual strain in the material postshock passage. It was also observed that in the regions where dislocations were present the lattice misorientation buildup was near zero over long distance, implying that the statistically stored dislocation has formed during shock exposure [147]. The defect reversal during shock exposure was also observed in the in situ XRD measurement of tantalum exposed to laser-generated shock [148].

12.5

Theory of defect generation under shock: past theories and new perspectives

A number of theories have been proposed in the 20th century to explain the peculiar microstructure with abundant defects observed in shock-/blast-loaded metals. Smith [65] first proposed that the dislocation (defect) arrays are generated at the interface to accommodate the difference in lattice parameters between uncompressed and

Microstructural changes in materials under shock and high strain rate processes: recent updates

381

compressed materials. This leads to relieving of deviatoric stresses. It was proposed that the dislocations moved with shock front. Hornbogen [149] proposed that dislocation loops are formed at the compression front and the edge component moves with shock front, leaving behind the screw component; hence, a substructure can form, as observed in shock-loaded iron. These two theories require that the dislocations move at supersonic speed with shock front, which is highly unlikely. Meyers [58] proposed that dislocations are homogeneously nucleated at the shock front as a result of deviatoric stresses, leading to relaxation of the stresses. The subsonic dislocations left behind the shock front can move over short distances. Schematic of Meyers theory showing sequence of steps leading to homogeneous dislocation nucleation is shown in Fig. 12.15. Initially, the uncompressed region and the compressed region are observed across the shock front (Fig. 12.15A). To relax the high deviatoric stress, dislocations are nucleated homogeneously at the interface of the compressed and uncompressed regions (Fig. 12.15B). The further propagation of shock creates a similar situation of high deviatoric stress across uncompressed/compressed state (Fig. 12.15C). The process of dislocation nucleation repeats with the propagation of shock, resulting in layers of homogeneously nucleated dislocation left in the microstructure behind the moving shock front (Fig. 12.15D). Meyers’ theory is widely accepted; however, it does not comment upon the mechanism of homogeneous dislocation nucleation and multiplication at/behind the shock front. Mogilevsky and Newman [146] performed MD simulation of Cu for a 2D lattice having 500 atoms. The simulation cell was subjected to elevated stress (many times

Figure 12.15 Schematics showing sequence of steps (A
D described in the text) as shock front passes through the material resulting in a homogeneous nucleation of dislocation at the shock front proposed by Meyers [58]. d3 is the distance between homogenously nucleated dislocation layers.

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higher than theoretical shear strength) resulting in 18.6% strain in h2 1 1i direction. In the simulation timescale the shear stress drops to zero during the peak pressure holding time. It was also observed that the stacking faults formed spontaneously at early stages. Upon unloading, the dislocation structure reorganizes and disappears. It is to be noted that imperfections such as vacancies, interstitials, and substitutional impurities were introduced in the initial simulation cell. The dislocation preferentially nucleated at theses points. The simulation without defect did not result in significant dislocation. Weertman [150] investigated the movement of dislocations. The dislocations were treated as under weak shock and strong shock. Based on this treatment, he observed two regions: first, where Smith interface exists, and second, the region behind shock front where dislocation generation, motion, and multiplication happen by conventional mechanism. Zaretsky [151] explained the multiplication of dislocation behind shock front based on fast operated dislocation source (Fig. 12.16).

Figure 12.16 Mechanism of dislocation multiplication via motion of partially bounded by stacking fault behind shock front [151]. (A) Stretching, (B) collapse, and (C) climb with formation of double stacking fault of rectangular half-loop of partial dislocation. (D) Twin lens formed by six generations of redoubled dislocation loops.

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It was proposed that initial partial dislocation generated acts as fast dislocation source, and the motion of partials bounded by stacking fault resulted in multiplication of dislocation. The characteristic time of multiplication was obtained to be of order of sub-ns to ps. The model gives explanation for the formation of shear bands and twins. However, the model requires the formation of initial stacking fault. The stacking faults were not observed and analyzed in materials subjected to lower range of peak shock pressure. The presence of dense dislocation makes the task further difficult. Bisht et al. [152] have recently investigated the defect formation mechanism under shock compression in Cu via MD simulation, with focus on phenomenon undertaking in the elastically shock compression region and elastic
plastic shock transition. The two transition regions, namely, uncompressed-elastically shock compressed transition region and elastic
plastic shock transition region with atoms colored in accordance to their atomic volume is shown in Fig. 12.17A and B, respectively. Redistribution of atomic volume, potential energy, and momentum

Figure 12.17 (A) U-E shock compressed transition region and (B) elastic
plastic shock transition region with atoms colored in accordance with their atomic volume [152]. U-E, Uncompressed-elastically.

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was observed in the elastically shock compressed region (Fig. 12.17A). This results in breakdown of the wave front pattern resulting in a state of random distribution of high/low atomic volume. This state achieved in the elastically shock compressed state is expected to promote local atomic shuffling resulting in formation of dislocation (Fig. 12.17B). Moreover, it was deduced that local atomic shuffling is the dominant mechanism for defect formation as implied by the presence of isolated dislocation loop in the transition region and the investigation of relative displacement of atoms. The dislocation nucleation in shock-compressed state is associated with local pockets of atoms having lower potential energy attributed to a lower coordination number and relatively high atomic volume at the core of defect. Thus it is inferred that potential energy is the driving force for defect nucleation upon shock compression. This postulate is completely different from the theory proposed by Prof. Meyer stating that homogeneous defect nucleation takes place under shock compression in order to accommodate transient shear strain between uncompressed/ compressed region. The theory of energy as the primary driving force for defect nucleation under shock compression needs further validation. Also, extremely high value of local shear stress was present around the defect, as a result of lattice distortion, in the compressed lattice, which acts as the prime driving force and explains the phenomena of defect reversal upon shock unloading [56
58,147,148]. Further future studies are necessary to investigate and affirm the recent perspective findings.

12.6

Conclusion

In the present chapter the response of material to high strain and shock exposure is discussed. A wide range of discussions spanning experimental techniques employed for investigation, influence of material parameter, including both intrinsic and extrinsic parameters, and theories of defect generation are reviewed. Understanding the local phenomenon under shock compression is the key to understanding the material response under these extreme conditions. Although a large number of studies have been performed to understand the phenomenon, it is still not fully understood. Recent finding on the effect of residual strain on shock-generated defect and atomic shuffling
based defect formation mechanism is expected to shed new light on the shock compression phenomenon. The material-specific response needs to be viewed in light of the recent findings to verify these theories.

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1841. [98] N. Gunkelmann, E.M. Bringa, H.M. Urbassek, Influence of phase transition on shockinduced spallation in nanocrystalline iron, J. Appl. Phys. 118 (18) (2015) 185902. [99] H. Jarmakani, E. Bringa, P. Erhart, B. Remington, Y. Wang, N. Vo, et al., Molecular dynamics simulations of shock compression of nickel: from monocrystals to nanocrystals, Acta Mater. 56 (19) (2008) 5584
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6364. [140] E. Cerreta, L. Dougherty, C. Trujillo, G. Gray III, R. Field, R. McCabe, et al., Influence of Peak Shock Pressure on the Microstructural Evolution of 1018 Steel, 2009. [141] V. Livescu, J. Bingert, T. Mason, Deformation twinning in explosively-driven tantalum, Mater. Sci. Eng., A 556 (2012) 155
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15.

Thermomechanics of friction stir welding

13

Mads Rostgaard Sonne and Jesper H. Hattel Department of Mechanical Engineering, Section of Manufacturing Engineering, Technical University of Denmark, Lyngby, Denmark

13.1

Introduction

Friction stir welding (FSW) is an efficient solid-state joining technique originally intended for joining of, for example, similar or dissimilar high-strength aluminum alloys, which are difficult to weld with traditional welding techniques. However, today the process is applied for the joining of a variety of materials, such as steel, copper, and dissimilar combinations [1]. The FSW process has a lot of benefits compared to traditional fusion welding: a green technology as no filler material and shielding gas is required and as the process does not produce any hazardous materials or fumes, improved mechanical properties and reduced distortions and residual stresses [2]. The process though also has some shortcomings: an exit hole is generated with unfilled material at the end of the process, the large forces involved require a heavy clamping setup and machinery for holding the probe tool and finally the process is in general slower than traditional joining techniques. The FSW process, shown in Fig. 13.1, utilizes a spinning tool consisting of a pin and a shoulder, which is plunged into the metal sheet to be welded and forced forward along the weld line to create a joint. During the welding process, heat is generated due to the friction between the tool and the workpiece, as well as due to the severe plastic deformation of the material. The heat conducted into the workpiece influences the quality of the weld in terms of (1) significant changes in the weld material properties, (2) buildup of residual stresses, and (3) distortion of the welded part, with all these three phenomena obviously being interrelated. Because of the relatively low heat generation in the FSW process, it is generally believed that residual stresses are low; however, due to the nonuniform heating as well as the very rigid clamping arrangement needed in the process, considerable residual stresses can be found in FSW welds [5] and sometimes at a level of the base material yield strength [6]. It is obvious that residual stresses will influence the subsequent mechanical behavior of the structure since they act as prestresses during the in-service loading. This might lead to stresses not only exceeding the yield limit just during static loading but also reducing fatigue strength [7,8] or highly influence the buckling behavior [9]. FSW is a process that is characterized by a complex multiphysics behavior. During the plunging, dwelling, and traversing of the tool, the material flow, the Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques. DOI: https://doi.org/10.1016/B978-0-12-818232-1.00013-8 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Figure 13.1 (A) Schematic view of the FSW process [3] and (B) photograph of the real process [4].

Welding

Cooling

Heat generation Temperatures

Microstructures

Service loads

Temperatures

Microstructures

Material flow

Temperatures Stresses and strains Material properties

Mechanical loads

E.g., damage

Stresses, distortions

Stresses, distortions

Stresses and strains Material properties

Figure 13.2 The major modeling couplings in FSW during welding, cooling, and loading [10].

resulting temperatures, and the microstructural evolution in essence will be closely coupled, see Fig. 13.2 (left). It is important to emphasize that transient stresses will be produced already during this phase. Apart from the direct effects on the resulting microstructures and mechanical properties from the stirring motion and heating during rotation and traversing of the tool, once the welding has terminated and the parts cool down, the temperature fields also affect microstructures and thereby mechanical properties and these together with the thermal gradients and the clamping conditions give rise to transient and residual stresses in a semicoupled way as shown in Fig. 13.2 (middle). As discussed earlier, these properties and stresses will in turn highly influence the mechanical performance of the part during the inservice loads, see Fig. 13.2 (right).

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Figure 13.3 Lagrangian frame (fixed coordinate system) and Eulerian frame (moving coordinate system) for modeling FSW [11]. Table 13.1 Governing equations for conservation of momentum, energy, and mass, respectively, in Lagrangian and Eulerian frames [10]. Lagrangian frame

Eulerian frame

Typically CSM

Typically CFD

Energy ρcp T_ 5 ðkT;i Þ;i 1 ηsij ε_ pl ij

ρcp T_ 5 ðkT;i Þ;i 1 ηsij ε_ pl ij 2 ui ðρcp TÞ;i

Momentum ρu€i 5 σij;j 1 pi

@ðρu_i Þ=@t 5 σij;j 1 pi 2 pðu_i u_j Þ;j

Mass No explicit equation

ρ_ 5 2 ðρui Þ;i

Here, k, cp, ρ, and T are the thermal conductivity, specific heat, density and temperature, respectively. ui is the displacement vector, σij is the stress tensor, sij is the deviatoric stress tensor, εpl ij is the plastic strain tensor, pi is the external forces vector, η is the dissipation efficiency and the dot (  ) denotes time differentiation. CFD, Computational fluid dynamics; CSM, computational solid mechanics.

In general, models for FSW are categorized by either their area of application, that is, flow models or residual stress models, or by the continuum mechanics approach they are based upon, that is, computational solid mechanics (CSM) models or computational fluid dynamics (CFD) models, where the former typically are Lagrangian and the latter Eulerian, see Fig. 13.3. The equations for conservation of momentum, energy, and mass, respectively, are in the Lagrangian and Eulerian frames given in Table 13.1. These are the fundamental, governing partial differential equations, which, depending on the continuum mechanical framework used, will need to be modified

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and combined with proper constitutive laws for the particular case at hand. This will be done in the following with the particular purpose of modeling in FSW and discussed in relation to the different numerical approaches available. The following sections give a brief overview of the findings related to the different thermomechanical phenomena associated with the FSW process and a thorough description and comparison of models and simulations present in the literature.

13.2

Thermal behavior

Many of the properties of the final weld are a direct function of the thermal history of the workpiece. Furthermore, the FSW process itself is highly affected by the heat generation and heat flow. From a modeling viewpoint, thermal modeling of FSW can be considered the basis of all other models of the process, be these microstructural, CFD, or thermomechanical. In the FSW process the welding parameters are all chosen such that the softening of the workpiece material enables the mechanical deformation and material flow. However, unlike many other thermomechanical processes, the mechanisms of FSW are fully coupled, that is, the heat generation is related to material flow and frictional/contact conditions and vice versa. Experimentally, the thermal fields from the FSW process have to a very large extent been presented in the literature, see, for example, Fig. 13.4 with a thermographic image of the steady-state temperature field during the FSW process by Carlone et al. [4]. The core part of any thermal model is how the heat generation from the rotating tool is described and applied as either a boundary condition for, or a source term in, the energy equation, be it in a Lagrangian or Eulerian frame. If a fully coupled thermomechanical model is used, access to the material flow/deformation fields as well as the formation of the shear layer will be available, and this information will be reflected in the dissipation source term nsij εpl ij in the two versions of the energy equation in Table 13.1.

Figure 13.4 Thermographic image of the process [4].

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However, for the semicoupled thermomechanical models where material flow is not considered (which by far are the most frequent ones in the literature), the procedure is normally to apply a surface heat flux as representing the entire heat generation, thereby avoiding the source term in the energy equation, which calls for knowledge about the material flow. Several suggestions in the literature are given for the surface heat flux formulation, see Ref. [12] for a detailed description, but common for them all is the need for “calibration” parameters. If one has access to an experiment from which it is possible to obtain the total heat generation from measurements, one can use the well-known expression given in Eq. (13.1) to obtain the surface flux qtotal ðrÞ 3Qtotal r 5 ; A 2πR3shoulder

(13.1)

where qtotal is the total heat flux per area A, Qtotal is the total amount of heat generation, Rshoulder is the tool shoulder radius, and r is the radius variable with r is 0 at the tool center. By having information about the friction coefficient and the total downward force from the tool, and assuming full sliding, the total heat generation can be expressed as Ref. [13]. Qtotal 5 23πωR3shoulder μp;

(13.2)

where ω is the rotational speed (1/rad), μ is the Coulomb friction, and p is the normal pressure. Either way, experimental information is needed and the most utilized procedure for evaluating the total heat generation is typically to perform the actual welds and measure the applied torque and rotational speed on the tool, thereby accepting the inherent limitation of the resulting thermal model to predicting only temperatures for a known total heat generation. As a way to overcome this problem, Schmidt and Hattel [14] proposed a somewhat different thermal model in which the heat generation again is expressed as a surface heat flux from the tool shoulder (without the tool probe) into the workpiece, however, being a function of the tool radius and the temperature-dependent yield stress as follows:   qsurface 2πn σyield ðTÞ ðr; T Þ 5 ωrτ ðT Þ 5 r pffiffiffi ; 60 A 3

for 0 # r # Rshoulder

(13.3)

where n is the tool revolutions per minute and σyield (T) is the temperaturedependent yield stress which approaches zero at the cutoff temperature (typically taken as the solidus temperature of the weld material) such that once the

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Figure 13.5 Upper and lower yield stress curves, and a fitted curve used for the TPM model for aluminum alloy 2024-T3 [15].

temperature approaches this value, the “self-stabilizing effect” causes the heat source to “turn itself off,” that is, the material loses all its resistance, and the heat generation decreases automatically due to thermal softening. The model is often denoted as the “thermal-pseudo-mechanical” (TPM) model since the heat generation is expressed via the temperature-dependent yield stress, see Fig. 13.5, thus taking some mechanical effects into account; however, it should be underlined that the model is a purely thermal model involving a temperature-dependent heat generation, and in that sense, it also uses a “calibration” parameter like the more conventional procedure in Eqs. (13.1) and (13.2). Obviously, this adds nonlinearity to the thermal model, meaning that the calculation time is increased by roughly a factor of two as compared to other thermal models where the heat source is prescribed itself, like in Eq. (13.2). It should be mentioned that both of the well-accepted expressions in Eqs. (13.1) and (13.2) can be directly derived from the more general formulation of analytically modeling the heat source in FSW given by Schmidt et al. in Ref. [16] which in essence results in the following equation:  2  Qtotal 5 δQsticking 1 ð1 2 δÞQsliding 5 πω δτ yield 1 ð1 2 δÞμp 3 h i 3 3 3 ðRshoulder 2 Rprobe Þð1 2 tan αÞ 1 R3probe 1 3R2probe H

(13.4)

with Rprobe being the tool pin radius, α the tool should angle, and H the pin height. This employs a linear weighting of the contribution from sliding and sticking, respectively, in terms of the state variable δ (zero for full sliding and one for full sticking).

Thermomechanics of friction stir welding

13.3

399

Microstructural evolution

The mechanical properties of a part joined by the FSW process will in general be changed due to the metallurgical evolution during the welding process as well as the subsequent cooling. This is also the case for FSW of heat treatable aluminum alloys in which the softening is closely related to the volume fraction, the size, and the phase of the hardening precipitates, see Fig. 13.6, which in turn will be affected by the level of temperature as well as the holding time at elevated temperatures. The effect of process parameters on the microstructure has to a large extent been covered in the literature, see, for example, Ref. [17]. Some of the main findings are the dissolution or coarsening of precipitates that occurs in the heat-affected zone (HAZ) within the “mid-level hardness” region, which has a detrimental effect on the overall mechanical properties. The size of the precipitates under hot welding conditions (low advancing speed) is larger than that under cold welding conditions (high advancing speed), and as a general consequence, the hardness and yield strength of the HAZ of the hot weld are lower than in the cold weld. Basically, this development of microstructure and its related effects on the mechanical properties is modeled in the literature with two types of models. The first type expresses the volume fraction of the hardening precipitates via relatively simple kinetics of precipitate dissolution [18,19] often referred to as the Myhr and Grong model. This has been applied to FSW by several authors [15,20,21]. The basis of the Myhr and Grong model is experiments in which samples are put into an oven and kept there for a specified period of time at a specified temperature. Following this, the samples are mechanically tested for hardness from which curves for hardness versus thermal history are constructed. The model then relates the fraction of dissolved hardening precipitates Xd to the equivalent time of heat treatment, teq 5 t/t (where t is the period of time at a temperature T and t is the time for total precipitation dissolution at this temperature) the following way:

Figure 13.6 Micrograph of the AA2024-T3 alloy, with the presence of hardening precipitates.

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n Xd 5 teq

teq 5

Ntotal X Δti

t i51 i

5

Ntotal X i51

Δt   i  tref exp Qef f =R 1=Ti 2 1=Tref 

(13.5)

where tref is the time for total dissolution at the reference temperature Tref, R is the gas constant, and Q is the effective activation energy for precipitate dissolution. The right side of Eq. (13.5) reflects that the equivalent time in a numerical model is found by discretizing the thermal history into small steps, calculating the equivalent time for each step and then summing up in order to find the total equivalent time. The fraction of hardening precipitates f/f0 then relates to the equivalent time teq via the fraction of dissolved precipitates Xd in the following manner: pffiffiffiffiffi f n 5 1 2 Xd 5 1 2 teq 5 1 2 teq f0

(13.6)

where n is a material constant that is obtained experimentally. A value of 0.5 is often used as indicated in the last part of Eq. (13.6). Finally, the hardness distribution is predicted via linear interpolation between the original state and the fully dissolved state, that is, HV 5 ðHVmax 2 HVmin Þ

f 1 HVmin f0

(13.7)

where HVmax is the hardness of the material in fully hardened condition and HVmin is the hardness of the fully softened (original) material. Note that the effect of natural aging gives some strength recovery as expected, but it has very little effect on residual stresses according to Feng et al. [20]. The effect of including the Myhr and Grong softening model for the prediction of residual stresses in AA2024-T3 was investigated by Sonne et al. [15]. Here it was concluded that prediction of the residual stress field could be improved significantly by including the metallurgical model under hot welding conditions. The second approach of modeling the material softening is more complex and involves the nucleation of precipitates, their growth, coarsening, and dissolution in a coupled manner. This type of model is often referred to as the Wagner
Kampmann model originating from the original work [22]. Some examples of application of this model to FSW are given by Robson et al. [23].

13.4

Residual stresses and distortions

Many contributions have been given in the literature for measuring and calculating residual stresses in FSW. Regarding experimental measurements, application of nondestructive methods such as X-ray diffraction [5,24] and neutron diffraction [20] can be found as well as destructive methods such as cut-compliance method and the contour method [25]. Recently, Sonne et al. compared those destructive

Thermomechanics of friction stir welding

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Figure 13.7 Comparison of the longitudinal residual stress profiles from the contour method and the numerical simulation, together with the stress profile obtained by Sonne et al. [15] via the cut-compliance method from an FSW experiment comparable with the experiment in the present work [26].

methods and tried to explain the reasons for errors and deviations as a consequence of the various methods, see Fig. 13.7. In this context, it is very important to emphasize that measuring residual stresses of FSW normally involves cutting out a test piece of the real welded structure, thus leading to potential relaxation of residual stresses [27]. For this reason, Threadgill et al. [28] state that a weld length of approximately eight times the diameter of the tool must be used if 90% of the residual stresses are to be retained when cutting out the test piece and argue that not meeting this criterion might be the explanation for the low residual stresses found in some experimental studies in the literature [29]. State-of-the art in measuring stresses for FSW involves in situ neutron diffraction (ND)-measurements during the welding process [30] (thereby also avoiding cutting out test pieces) thus being able to follow the evolution of transient stress fields into the final residual stress field. A very convenient assumption used by many authors in the literature for modeling of residual stresses in FSW is to neglect the material flow during welding. In essence, this normally results in semicoupled thermomechanical models in a Lagrangian frame meaning that the thermal field somehow is calculated prior to the mechanical field either by separating the two analyses totally, that is, calculating the entire temperature history first and then applying it in the subsequent mechanical analysis, or keeping the separation within the local timestep such that the temperature field is calculated first and the mechanical analysis follows. Both approaches give the possibility of using temperature-dependent material data in the mechanical analysis. These data may come from simple “lookup tables” which are known prior to the thermal analysis and hence are history independent or from

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models for the evolution of mechanical properties, typically based on microstructural models as described previously. It is important to underline that the term “semicoupled” in this context refers to models where the flow during welding is not modeled and the thermal field is addressed by models like the ones presented in Section 13.2. In this case the Lagrangian equations in Table 13.1 for energy and momentum will reduce to the heat-conduction equation and the quasistatic equilibrium equations ρcp T_ 5 ðkT;i Þ;i

σij;j 1 pi 5 0

(13.8)

These should be solved together with Hooke’s generalized law, timeindependent plasticity theory, and small strain theory, that is (note that the yield stress in this case only depends on temperature and total equivalent plastic strain), see the following equation [10]: pl pl el th th σij 5 Cijkl ðεtot εtot ij σij ε ij 5 εij 1 ε ij 2 εij 1 δij ε Þ  E 1 ν 1 ðδik δjl 1 δil δjk 1 δij δkl Þ εtot Cijkl 5 ij 5 ðui;j 1 uj;i Þ 11ν 2 1 2 2ν 2 Ð T _ 3 sij εth ðT1 ! T2 Þ 5 T12 αðTÞdT ε_ pl ij 5 λ 2σ ; 1 2 3 σ 5 ð2sij sij Þ

3 f ðsij ; εpl ; TÞ 5 sij sij 2 σ2Y ðεpl ; TÞ 2

ε

pl

1 2 2 pl pl 5 ð3εij εij Þ

σY 5 σY ðεpl ; TÞ (13.9)

where Cijkl is the elastic fourth-order tensor, α(T) is the temperature-dependent coefficient of thermal expansion, σ and εpl are the equivalent stress and equivalent plastic strain, and σY is the yield stress. Several authors in the literature have used semicoupled thermomechanical models without taking material flow into account. The first to do this were Chao and Qi [31] who presented a 3D model in an inhouse developed code (predecessor of WELDSIM) with 1800 elements in 1998. This model was pretty advanced for its day because it employed fixture release as well as a history-dependent yield stress such that cooling down would follow a lower curve as compared to heating up. This way, some of the effects of material softening can be captured. Chao and Qi’s model was an important step forward, and all models that have followed somehow are based on or inspired by this model. Feng et al. [20] present an integrated 3D thermal
metallurgical
mechanical model and applied it for predicting the formation of the residual stress field, dissolution of precipitates, and aging in an Al6061-T6 plate which has been friction stir welded. The tool pressure was taken into account. Simulated residual stress fields for two welding speeds, that is, 280 mm/min (4.66 mm/s) and 787 mm/min

Thermomechanics of friction stir welding

403

(13.12 mm/s), were compared, and it was found that the low heat input associated with the high welding speed (cold weld) results in higher tensile residual stresses in the weld region. However, the position of peak tensile residual stress was closer to the weld centerline (narrower tensile zone in the weld). Feng et al. explained this by the less HAZ softening in Al6061-T6 in the high welding speed case. Similar results were found by Bastier et al. [32] who however employed a thermal
metallurgical
mechanical model with kinematic hardening which takes also flow into account, see Section 13.5. This effect can also be attributed to the steeper thermal gradients that are present for colder welding conditions provided combined with kinematic hardening. In this context, it is important to emphasize that Zhu and Chao [33] state that the temperature-dependent yield stress is the most governing single parameter for the formation of residual stresses in welding simulation. This is in very good agreement with the results obtained applying the TPM model as explained in the section on thermal models. Experimental confirmations of the narrowing effect with higher stress peaks for colder welding conditions are given by, for example, Feng et al. [20]. Here, it should be mentioned that Dubourg et al. [34] find the opposite trend such that higher welding speeds lead to lower residual stresses although they claim that there are some inaccuracies in their model leading to unrealistic results [3]. Recently, a thermomechanical model has been used by Sonne et al. [35] in order to assess the outcome of experimentally found residual stresses applying the contour method, highlighting the major drawbacks and influencing factors of the measuring technique. An overview of some of the most important semicoupled residual stress models without flow in the literature for FSW is given in Table 13.2.

13.5

Material flow

The 3D flow in FSW is complex as it depends on tool design, for example, threads; flutes and shoulder characteristics; welding parameters, such as rotation and welding speed and direction of rotation (clockwise or counterclockwise); tilt angle, workpiece properties; and the contact condition at the tool/matrix interface. All these factors influence the material flow around the probe, that is, how the original joint-line disrupts and the material is transported from the leading to the trailing side. In the following section a general description of the 3D flow field in FSW as well as the different thermomechanical models including material flow found in the literature will be presented. Investigations of the material flow pattern using a specific marker material (MM) have been presented previously in the literature [41,42]. Traditional metallographic investigation of the position of the MM has provided detailed information regarding the patterns, but the combination of time-consuming experiments and a limitation to a planar (2D) examination of the weld has led to the introduction of X-ray tomography to reveal the spatial position of the MM in the weld specimen.

Table 13.2 Selected thermomechanical models for residual stresses. Authors

Chao and Qi [31] Zhu and Chao [33] Chen and Kovacevic [36] Bastier et al. [32] Li et al. [37] Feng et al. [20] Richards et al. [21] Tutum and Hattel [3] Dubourg et al [34] Carney et al. [38] Yan et al. [39] Hattel et al. [27] Sonne et al. [15] Sonne et al. [40] Sonne et al. [35]

Year

1999 2004 2006 2006 2007 2007 2008 2010 2010 2011 2011 2012 2013 2015 2017

Formulation (all Lagrangian formulation) Solver

Hardening

Implicit Implicit Implicit Implicit Implicit Implicit Implicit Implicit Implicit Implicit Implicit Implicit Implicit Implicit Implicit

Isotropic Isotropic Kinematic Kinematic Isotropic Isotropic Isotropic Kinematic Isotropic Isotropic Isotropic Isotropic Isotropic Isotropic Isotropic

Metallurgical model

No No No Yes No Yes Yes No No No No No Yes Yes Yes

WELDSIM (predass.) WELDSIM ANSYS CAST3M ABAQUS/Standard ABAQUS/Standard ABAQUS/Standard ANSYS LS-DYNA LS-DYNA ABAQUS/Standard ANSYS ABAQUS/Standard ABAQUS/Standard ABAQUS/Standard

Software

Thermomechanics of friction stir welding

405

Figure 13.8 CT scan of the material flow in FSW of aluminum with embedded copper marker material [43]: (A) no transparency of the aluminum, (B) half transparency, (C) full transparency, (D) close-up on the stirred area indicating an S-shape of the markers, and (E) side view of the marker particles. CT, Computer tomography.

By embedding tracer particles with a different density than the parent material, the contrast/highlighting in the X-ray pictures indicates the MM. This was performed in the work by Schmidt et al. [43] who used a thin copper strip as MM in FSW of aluminum combined with 3D computer tomography for characterization of the 3D flow of material during the FSW process, see Fig. 13.8. Based on this experimental investigation, different characteristic flow zones were identified: G

G

G

Rotation zone—the zone in the shear layer closest to the probe consisting of the rotation flow. Transition zone—the zone in the shear layer between the rotation zone and the matrix/ shear layer border consisting of the transition flow. Deflection zone—layer characterized by a low deformation surrounding the transition layer.

The flow within these different zones and the overall material movement during the process has been simulated with different multiphysics models, which will be further covered in the following section. It should be emphasized that the main purpose of fully coupled thermomechanical models normally is to get a realistic prediction of the closely coupled material flow, heat generation, and temperature fields during welding, see Fig. 13.2 (left).

406

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

So, seen from that viewpoint, the flow models are normally not intended directly to predict, for example, residual stresses, themselves. They are normally used as a means of providing more realistic temperature fields and mechanical properties to the subsequent residual stress calculation as compared to what the semicoupled models are capable of. Flow models for FSW are based on either CFD or CSM. Whereas the former are normally not able to predict residual stresses without being coupled to a subsequent residual stress analysis based on CSM, the latter type of models have the potential of predicting residual stresses themselves. Fully coupled CSM-based flow models for FSW are typically developed from two different approaches: the simplest is to assume a rigid-viscoplastic material; hence, the total strain is equal to the viscoplastic strain, and the stress depends on strain rate only, and then typically use an implicit solver based on quasistatic equilibrium. This is done by several authors in the literature; see, for example, Refs. [44,45]. These models resemble CFD models very much. If models of this type, however, are to predict residual stresses, it calls for integration along streamlines. CSM models for doing this have been presented in the literature by Qin and Michaleris [46] who employ Anand’s elasto-viscoplastic model (it is thus not only a rigidviscoplastic model as mentioned above) in a Eulerian frame and integrate elastic strains along streamlines in order to finally obtain residual stresses. The model has the advantage like other fully coupled thermomechanical models that take the flow into account while calculating transient stresses, not only via a more realistic temperature field but also via the mechanical effect of the flow itself, and moreover since it is a Eulerian steady-state model, it is very fast and does not depend on remeshing, etc. Since then, models based on the same principles have been presented [47]. The second more general approach relies on fully coupled thermomechanical models involving a constitutive law where stresses are expressed in terms of absolute strains (as well as strain rates like viscoplasticity to get the flow right), which would be able to predict residual stresses if allowed to calculate for a sufficiently long amount of time. However, as earlier mentioned, they are typically not used for this purpose, but for calculating the flow during the welding process and coupling it with the thermal calculation via the dissipation term in the energy equation. This more comprehensive approach uses the dynamic equilibrium equations, large strain theory and takes “all” the contributions to the strain into account (including the elastic and rate-independent plastic strains, which makes residual stress prediction possible) although the viscoplastic part obviously is the dominating contribution during the flow. In a Lagrangian frame the governing equations are now the energy equation with the plastic dissipation term and the dynamic equilibrium equations as given in Eq. (13.10) [10], left (compare with the corresponding governing equations for the semicoupled case expressed by Eq. 13.9). Apart from this, the major change, as compared to the semicoupled models without flow, lies in the fact that large strain theory must be applied, the viscoplastic strain is now taken into account meaning that rate-dependent constitutive equations must be employed, which among others mean expressions for the yield stress, such as the Norton power law, the inverse hyperbolic sign expression involving the Zener
Hollomon parameter, and the Johnson
Cook expression.

Thermomechanics of friction stir welding

407

pl vp pl el th tot εtot ij 5 εij 1 εij 1 εij 1 δ ij ε or εij 5 εij

1 εtot ij 5 ðui;j 1 uj;i Þ 1 uk;i uk;j 2

pl σY 5 σY ðεpl ; ε_ ; TÞ

σY ! 0 T ! Tcut off  Tsol

Norton power law _ pl m

σY 5 Kðε Þ

Zener 2 Hollomon

Johnson 2 Cook

1 σY 5 sinh21 ðZAÞ1=n α

_ pl σY 5 ðA 1 Bðεpl Þn Þ 1 1 C ln εε_ 0

pl Z 5 ε_ eQ=RT

m5

@lnσ @lnεpl

!

(13.10)

m    ref 3 1 2 TT2T sol 2Tref

For CSM-based models the implementation is typically done in an ArbitrarilyLagrangian-Eulerian (ALE) formulation where the dynamic equilibrium equations are solved in an explicit manner which in essence results in a very simple algorithm but also calls for very small time steps, which to some extent can be overcome by the use of mass scaling. This approach has been used by several authors, see, for example, Ref. [48]. A special feature about the model by Schmidt and Hattel [49] (implemented in ABAQUS Explicit) is that the surface condition between the tool and the matrix is not prescribed but part of the solution itself. This adds not only to the generality of the model but also to the complexity and hence the need for computational power. The model by Hamilton et al. [48] certainly deserves mention in this respect also. As the model by Zhang and Zhang [50], it is also based on the model by Schmidt and Hattel [49]; however, it addresses the whole welding path and is eventually able to reach the quasisteady state that Schmidt and Hattel’s original model was not capable of. The next natural step in the development of these coupled thermomechanical models would be to use it for residual stresses by letting it take the cooling sequence into account also. By doing this, modeling of residual stresses of FSW would get substantially more general as compared to the semicoupled models, since the welding history would be better described in terms of the closely coupled phenomena of flow, temperatures, and microstructure evolution during welding. It is important to emphasize that since this approach takes the elastic strains into account during welding, the elastic deformation of the “far-field,” the pressure conditions will be described more realistically as compared to predicting the flow with a CFD model. Another way to achieve this is to couple a local, steady-state flow model in a Eulerian frame during welding with a global, transient, rate-independent residual stress model during welding and cooling. Some few works in the literature are based on this type of coupled approach. A very interesting example of such model is the work by Bastier et al. [32], which also involves a metallurgical model for the evolution of mechanical properties during and after welding. Another interesting example of model is the one provided by Grujicic et al. [51] in which a coupled thermomechanical model in ABAQUS/explicit is applied for the flow during FSW of AA5085-H131 followed by a mapping of the results to a

Table 13.3 Selected thermomechanical models applicable to FSW including flow. Authors

Year

Xu and Deng [53]

2003

Schmidt and Hattel [49]

2005

Zhang and Zhang [50]

2007

Bastier et al. [32]

2008

Qin and Michaleris [46]

2009

Hamilton et al. [48]

2010

Grujicic et al. [51]

2010

Formulation

Software

Structure

Flow

Stress

Met. model

Global

CSM-ALE

CSM-Lagrangian

No

ABAQUS/Explicit

Rigid-viscoplastic

Elastoplastic

CSM-ALE

CSM-ALE

No

ABAQUS/Explicit

Elastoplastic
viscoplastic

Elastoplastic
viscoplastic

CSM-ALE

CSM-ALE

No

ABAQUS/Explicit

Elastoplastic
viscoplastic

Elastoplastic
viscoplastic

Local/ global

CFD-Eulerian

CSM-Lagrangian

Yes

CAST3M

Rigid-viscoplastic

Elastoplastic

Global

CSM-Eulerian

Integration of elastic strains along streamlines

No

In-house

CSM-ALE

CSM-ALE

No

ABAQUS/Explicit

Elastoplastic
viscoplastic

Elastoplastic
viscoplastic

CSM-ALE

CSM-Lagrangian

No

Rigid-viscoplastic

Elastoplastic
viscoplastic

ABAQUS/Explicit and Standard

Global

Global

Elastoplastic
viscoplastic Global

Global

Buffa et al. [52]

2011

Riahi and Nazari [54]

2011

Jamshidi et al. [55]

2012

Sadeghi et al. [56]

2013

Nourani et al. [57]

2014

Global

Global

Local

Global

Global

CSM-Lagrangian

CSM-Lagrangian

Rigid-viscoplastic

Elastoplastic

CSM-ALE

CSM-Lagrangian

Elastoplastic
viscoplastic

Elastoplastic

CSM-ALE

CSM-Lagrangian

Elastoplastic
viscoplastic

Elastoplastic

CSM-Lagrangian

CSM-Lagrangian

Rigid-viscoplastic

Elastoplastic

CFD-Eulerian

CSM-Eulerian

Rigid-viscoplastic

Elasto-viscoplastic

CFD, Computational fluid dynamics; CSM, computational solid mechanics; FSW, friction stir welding.

No

DEFORM3D/ ABAQUS/Standard

No

ABAQUS/Explicit

No

ABAQUS/Explicit

No

DEFORM3D/ ABAQUS/Standard

No

Comsol

410

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

quasistatic thermomechanical model without flow in ABAQUS/standard for the calculation of residual stresses. What should be emphasized here is that even though the flow model is of CSM-type it is a rigid-viscoplastic model (total strain equal to viscoplastic strain); see Eq. (13.10), thus making the flow model resembling CFD flow models. Moreover, it should also be mentioned that Grujicic et al. suggest a version of the Johnson
Cook constitutive law that takes the dynamic recrystallization into account. Another interesting model to mention is the one by Buffa et al. [52] where the flow is predicted via a CSM model in a Lagrangian frame, applying DEFORM3D. Here the finite element flow formulation is adopted with a rigid-viscoplastic constitutive behavior for the flow. The transient temperature field is then imported into ABAQUS/Standard for the subsequently stress analysis. An overview of some important coupled thermomechanical models including flow from the literature is given in Table 13.3.

13.6

Conclusion

In this chapter the current development of models for simulating the thermomechanical conditions in FSW has been addressed. This covers the physical aspects of the very complex process in terms of thermal and microstructural behavior, development of residual stresses, and thermomechanical models including material flow. As observed from this chapter, the FSW process invented at The Welding Institute in Cambridge almost three decades ago (1991) has been investigated by a large number of researchers and is as of today to a large extent understood, especially in terms of the thermal conditions and in prediction of residual stresses. However, the material flow during the process is still poorly understood, as the flow pattern and interaction with the tool is very complex and not fully described in the literature. This calls for more detailed experimental investigations coupled with thermo-fluid CFD simulations of the FSW process.

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[5] M. Peel, A. Steuwer, M. Preuss, P.J. Withers, Microstructure, mechanical properties and residual stresses as a function of welding speed in aluminium AA5083 friction stir welds, Acta Mater. 51 (16) (2003) 4791
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689.

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Modeling of friction in manufacturing processes

14

Uday Shanker Dixit1, V. Yadav2, P.M. Pandey3, Anish Roy4 and Vadim V. Silberschmidt4 1 Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India, 2Department of Mechanical Engineering, Maulana Azad National Institute of Technology Bhopal, Bhopal, India, 3Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi, India, 4Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, United Kingdom

14.1

Introduction

Friction is one of the most important parameters to control for optimizing the quality of a product in manufacturing processes. It plays an important role in the determination of the material flow, fracture, and surface quality. Commonly, the friction behaviors are highly dependent on the following parameters: material properties, roughness of contacting surfaces, sliding length, sliding speed, temperature, and lubricant. Friction is classified into two categories—static and kinetic. The static friction is related to energy required for moving a body from rest. The kinetic friction is linked to energy required for sustaining the motion. Static friction reaches its maximum value when the body is just about to start its motion. It is crucial for restraining the motion between the objects; without friction, threaded fasteners would not be able to affix two parts together. Self-locking behavior of a jackscrew is due to static friction. Depending on the nature of motion, kinetic friction can be subdivided into sticking friction and sliding friction. The former occurs when frictional shear stress reaches the shear yield strength of one of the contacting materials. During sliding friction, this stress is less than the shear yield strengths of the contacting materials. In the case of sliding friction the frictional shear stress is positively correlated with the normal stress at the interface of the two moving bodies. In several manufacturing processes, both sliding and sticking frictions play significant roles. Generally, it is very difficult to predict friction, as it is a complex function of a number of factors. Recent studies have shown that frictional behavior at microlevel is quite different from that at macrolevel, that is, a size effect is predominant. For example, Gong and Guo [1] observed that lubrication did not affect the friction coefficient in microforming, contrary to that in macroforming. Manufacturing processes can be broadly classified into five groups—subtractive, additive, mass-containing, joining, and nanofinishing. Machining is a subtractive Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques. DOI: https://doi.org/10.1016/B978-0-12-818232-1.00014-X Copyright © 2020 Elsevier Ltd. All rights reserved.

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process that involves the removal of a material in the form of chips to obtain the desired shape, size, or features. An example of additive manufacturing is 3D printing. Metal forming is a mass-containing process; it can be divided into bulk and sheet metal forming. For example, bulk forming processes are rolling, extrusion, wire drawing, whereas deep drawing, blanking, and punching are sheet metal forming processes. Casting is another mass-containing process. Examples of joining are solid-state welding, fusion welding, and adhesive bonding. Nanofinishing is a kind of machining employed to obtain a high degree of precision and surface finish. The dependency of the friction on the accuracy of modeling of manufacturing processes ¨ zel [2] highlighted that a proper fricwas investigated by a number of researchers. O tion model was essential in a finite-element simulation of machining. Nielsen and Bay [3] presented a review of friction modeling in metal forming and emphasized the need for developing a model for predicting a real area of contact as a function of normal pressure, bulk deformation, material properties, and surface conditions. To analyze the manufacturing process, an accurate input data related to the tool
workpiece interface such as surface roughness characteristics, crystalline structure, grain size, strain hardening, and lubricants are needed. In view of it a number of friction models were developed based on some simplified assumptions to describe the friction characteristics in manufacturing processes namely, constant friction model, Coulomb’s friction model, and Wanheim and Bay’s friction model. Often, the available friction models are successful in providing the qualitative explanation of a particular behavior. For example, Lenard [4] observed that the coefficient of friction increased with reduction of the strip thickness in a rolling process and decreased with strain hardening. He mentioned that this phenomenon could be explained with an asperity-based friction model. Recently, Mao et al. [5] presented a comprehensive asperity-based friction model, considering strain hardening and junction growth.

14.2

History of friction modeling

An early study on friction was carried out by Leonardo da Vinci (1452 2 1519). Leonardo’s notebooks contain several sketches related to experimental studies on friction [6,7]. It appears that he was aware that frictional force was proportional to a normal load, and it did not depend on the apparent area of the contact. Probably he was also aware that beyond a certain load, the coefficient of friction (ratio of frictional force to normal load) reduced with the normal load. He believed that the smooth surfaces had a lower coefficient of friction compared with rough surfaces, which is a debatable concept presently. He stated that the coefficient of friction between two unlubricated dry surfaces was 0.25; it is a typical value for many material combinations, that is, wood
wood. Leonardo da Vinci studied the role of friction in inclined planes, screws, and pulleys. He also studied the effect of lubrication on sliding friction. He compared the sliding and rolling motion and attempted to explain why the frictional force in rolling was much less than that in sliding. Findings of

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Leonardo da Vinci remained unnoticed (and unavailable) for many years [8]; a first reference to his contributions in friction was made by Venturi in 1797. In 1699 Guillaume Amontons (1663
1705) proposed the following rules based on dry sliding between two flat surfaces: 1. The frictional force that resists sliding at an interface is directly proportional to the normal load. 2. The amount of a frictional force does not depend on the apparent area of contact.

These observations were verified by French physicist Charles-Augustin Coulomb (1736
1806). He put the Amontons’ first law in a mathematical form; the ratio of a frictional force to a normal force is known as Coulomb’s coefficient of friction. Coulomb also stated that the frictional force was independent of the velocity of the sliding bodies. Friction during motion is known as the kinetic friction as opposed to static friction encountered when it is attempted to move a resting body. The Amontons
Coulomb model has been the most widely used friction model notwithstanding its limitation. In the first half of the 20th century, some research groups extensively worked on the adhesion theory of friction. This theory propounds that in a case of two contacting bodies, initially the asperities only make the contact. Thus a real contact area is much lower than the apparent one. Swedish physicist Ragnar Holm (1879
1970) measured the electrical contact resistance between two contacting bodies and estimated the real area of contact based on these measurements [9]. Ernst and Merchant [10] studied the influence of hardness of the contacting bodies and normal pressure on frictional behavior. They also proposed a theory that considers the combined effect of adhesion and surface roughness. A pioneering paper on the general adhesion theory of friction was published by Bowden and Tabor [11]. They proposed that contacting asperities formed metallic junctions, which should be broken by applying a shear force. They also mentioned that during the motion, high temperature might be generated at the contacting surfaces, causing local melting. The Amontons
Coulomb model could be explained with the adhesion model. Bowden and Tabor [11] also mentioned the presence of a plowing friction term, which is often negligible. The adhesion theory can explain that for high normal pressures, the linear relation between the normal and frictional forces would be lost. In analysis of a rolling process, Orowan [12] used a friction model that followed the Amontons
Coulomb model at low pressure, but once the frictional stress attained the shear yield strength of the rolled material, it remained constant irrespective of the pressure. After the World War II, several Russian scientists carried out theoretical and experimental work on friction for boundary lubrication [13,14]. A dependency of temperature rise on the coefficient of friction was analyzed by Blok [15]. In experiments a very high local temperature between the asperities was recorded. Bowden [16] constructed a simple apparatus to measure the temperature, considering the two sliding materials as thermocouple elements. The temperatures recorded were somewhat higher than those observed by Blok [15]; however, the differences were not large. Later, Bowden and Tabor [17] studied friction by defining an extent of a real area of contact, temperature of sliding surfaces, and a nature of surface damage.

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Based on the adhesion theory, Greenwood and Williamson [18] developed an asperity-based friction model. The model considers that surfaces are composed of hemispherically tipped asperities with a constant radius of curvature. The distribution of the heights of asperities was assumed to follow a Gaussian distribution. However, the model considered only elastic deformation. Wanheim [19] used a slip-line field solution to calculate a real area of contact. This work was extended by several researchers. A model, describing a smooth transition between the Amontons
Coulomb model and the constant-frictional-stress model, was proposed [20]. Later, upper bound and finite-element methods (FEMs) were applied to understand the elastic
plastic deformation of asperities. These models took into consideration the surface roughness of the contacting surfaces besides several other parameters. A brief review of friction modeling in metal forming is available in Ref. [3]. Chang et al. [21] extended the Greenwood
Williamson contact model of rough surfaces to elastic
plastic deformation of an asperity. This model is referred as Chang
Etison
Bogy (CEB) contact model for analyzing the elastic
plastic deformation of asperities. Incremental extensions to this model were made by several researchers. However, there are only limited applications of such models in the modeling of manufacturing processes, mainly due to their complexity, apart from the incompleteness of these models due to several complex factors involved. Instead of using asperity-based models, a large number of manufacturing researchers prefer to use empirical models of friction. In these models, either a variable Coulomb’s coefficient of friction or a variable friction factor (ratio of frictional stress to yield shear strength) is used. Variability of the coefficient of friction or the friction factor with respect to a number of key factors is ascertained by carrying out some experiments in a suitable range. Recently, it was realized that the frictional behavior in micromanufacturing is quite different from that at macromanufacturing. The size effect on friction was explained by a concept of open and closed lubricant pockets at the beginning of 21st century [22]. Closed pockets on the contact surface retain the lubricants while the open pockets do not. As the size of the contacting surface decreases, the ratio of open pocket area to closed pocket area increases; hence, the ability to retain the lubricant decreases. Therefore friction of lubricated surfaces increases drastically with reduction in size. This size effect is not observed during the dry contact. Research on the modeling of friction continues. It is a challenge to develop a reliable model of friction but a greater challenge is to apply the developed models for the modeling of macro- and micromanufacturing processes. As a result, the Amontons
Coulomb model and friction factor models still dominate.

14.3

Some popular friction models

Modeling of manufacturing processes is gaining importance as a result of stiff global competition to improve surface finish and product quality in an optimal way.

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419

In particular, metal forming and machining processes were extensively modeled using both analytical solutions and numerical techniques. The efficacy of the mathematical model of the manufacturing process is highly dependent on the friction model. A good friction model should be able to consider the following aspects: (1) roughness of the contacting surfaces, (2) adhesion between the surfaces, (3) plowing of a softer surface with asperities on a harder one, and (4) effect of a mechanical load. A number of models were developed to analyze the friction in manufacturing processes. The analytical and numerical techniques are used to obtain normal and frictional shear stresses at the tool
workpiece interface. The Amontons
Coulomb model, the constant friction model, and the Wanheim and Bays model were effectively employed to simulate the manufacturing processes because of their simplicity. These models seldom consider the surface roughness. In order to account for the unevenness in the surface finish, the asperity-based friction model was developed [18]. This model is able to capture the unevenness of the contact surfaces and material properties as a function of strain, strain-rate, and temperature. Furthermore, the adhesion and plowing components can be easily incorporated into the asperity-based friction model. In the following subsections, some popular friction models employed for analysis of friction behavior at the tool
workpiece interface are summarized.

14.3.1 Amontons
Coulomb’s model In 1699, Guillaume Amontons stated that force of friction is proportional to the normal load and is independent of the contact area [8,23]. The proportionality constant is called Coulomb’s coefficient of friction μ. For bodies at rest, the Coulomb static coefficient of friction is denoted as μs, while the kinetic one is denoted as μk. This model, known as Amontons
Coulomb’s model, Amonton’s model, or Coulomb’s model, was extensively used to model manufacturing processes. Several researchers tried to improve this model by introducing the coefficient of friction as a function of various parameters instead of a constant value. A reason for the dominance of the Coulomb model is its simplicity. In a very narrow range, any friction model can be linearized in the form of Coulomb’s law. In fact, Coulomb hinted that the coefficient of friction was influenced by several factors including the sliding velocity, normal load, and characteristic size; however, often the dependencies are weak [24].

14.3.2 Constant-friction model In the constant-friction model, a frictional force is considered equal to the product of a friction factor m and shear flow stress k of the weaker material; the factor m ranges between 0 and 1. It is observed that this model is close to reality when the contacting surfaces deform plastically in the presence of a high normal load. In the constant-friction model, the friction force does not depend on the normal load. Hence, the model is conveniently used to carry out the analysis of

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manufacturing processes using an upper bound method. In this method, a velocity field is assumed that satisfies the continuity equation and kinematic boundary conditions. Based on this assumed kinematically admissible velocity field, the power needed for the plastic deformation and overcoming the friction is calculated. The total power, based on the assumed velocity field, is always higher than the actual power. However, the stresses (and hence the normal load) cannot be accurately calculated with the assumed velocity field. As the constant-friction model does not require the normal load for the calculation of friction power, it is conveniently applied in the upper bound model. There were several attempts to relate the constant friction factor m with the Coulomb coefficient of friction μ. Using a slip-line field solution, Bay [25] provided the following expression: m μ 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 27ð1 2 m2 Þ

(14.1)

Based on the ring-compression test (RCT), Rao and Sivaram [26] suggested the following form: m μ 5 pffiffiffi : 2 3

(14.2)

These relations are highly approximate in nature and can be used only for crude estimates. For m 5 0.5, Eq. (14.1) provides μ 5 0.11 and Eq. (14.2) μ 5 0.14. Although the two values of μ differ, the difference is not considerable. Recently, Molaei et al. [27] obtained a relation between μ and m based on the finite-element analysis of a double-cup extrusion process and a barrel-compression test. They also fitted the following regression model to the simulation results: μ5

m0:9 : 2:72ð12mÞ0:11

(14.3)

For m 5 0.5, Eq. (14.3) provides μ 5 0.21, a considerably different value from those obtained with Eqs. (14.1) and (14.2). It is not possible to obtain a simplified accurate relation between these two measures of friction as it depends on several other parameters.

14.3.3 Wanheim and Bay’s model In most of the early mathematical modeling of manufacturing processes, either the Amontons
Coulomb friction model or the constant-friction model was employed, thanks to their simplicity. At high normal pressure, the Amontons
Coulomb friction model is not valid. Wanheim and Bay [20] suggested an improved model for the frictional shear stress, assuming it to be a function of the normal pressure p and

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421

Figure 14.1 Variation of frictional force with normal load according to Wanheim and Bay’s model.

the friction factor m. They observed that at low normal pressure (less than 1.5 times the yield strength), the frictional shear stress was proportional to the normal stress, whereas at high normal pressure (more than thrice the yield strength), the friction was almost constant. The two ranges were combined through an intermediate transition region. Fig. 14.1 shows the relation between the normal load and the frictional force schematically according to the Wanheim and Bay’s model. The appropriateness of this model was demonstrated by Christensen et al. [28] and Zhang and Bay [29] for analyzing the plate and cross-shear plate rolling, respectively. Cross-shear rolling is a process in which each work roll rotates with different peripheral speed. Kumar and Dixit [30] used the Wanheim and Bay’s model for the analysis of a foil rolling process. They observed that the friction model had a great influence on the qualitative and quantitative predictions of this process. The level of frictional shear stress predicted with the Wanheim and Bay’s model was considerably lower than those predicted with the Coulomb model at high pressure. The Wanheim and Bay’s model relies on an empirical parameter m, which may depend on the temperature and surface roughness. However, such dependency has not been properly established.

14.3.4 Asperity-based friction model The asperity-based friction model developed by Greenwood and Williamson [18] is popularly known as Greenwood
Williamson (GW) model. In this model the distribution of heights of asperities is assumed to follow the Gaussian distribution. Chang et al. [21] extended the GW contact model to include plastic deformation. Furthermore, Chang et al. [31] extended their work by including the adhesion theory to investigate the effect of surface roughness and surface energy on the adhesion of contacting metallic rough surfaces. They found that the static coefficient of friction decreased with increased contact pressure for a specified surface roughness topography. The von Mises criterion was employed to find out the location of yield inception and the tangential force, which caused failure of contacting

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asperities; the rougher surfaces and softer materials yielded a lower tangential shear force as compared with the harder materials with good surface finish. Horng [32] extended the CEB model to a more general case of elliptic asperities. Kimura and Childs [33] studied the flattening of the asperities under bulk plastic deformation by carrying out compression tests experimentally as well as analyzing them theoretically. In their theoretical study, they used an energy method to assess the material flow during deformation. They reported that the asperity crushing took place at compressive strain of 0.02
0.08, and the real area of contact was between 75% and 95% of the apparent area. Once the initial phase of crushing was accomplished, the contact area remained constant under further bulk deformation [33]. Zhao et al. [34] studied the elastic
plastic deformation of multiple asperities to analyze the real area of contact and loads at different stages of the elastic
plastic regime. They employed the expressions for the fully elastic and plastic deformation of asperities from the GW and CEB models. However, in the elastic
plastic interference regime, the logarithm and fourth-order polynomial functions were introduced to calculate the contact area for corresponding loads. They compared their results with those obtained with the GW and CEB models for elastically and plastically deformed asperities and obtained a good agreement for different magnitudes of surface roughness with different ratios of contact area to load. Kogut and Etison [35] carried out an elastic
plastic analysis for the contact of rough surfaces with an FEM. They found that an FEM model provided the accurate results; it can be used to validate the empirical relations used in the analytical models. Some of the studies focused on the junction growth at the interface to assess the friction. Tabor [36] showed the dependency of the coefficient of friction on the junction growth based on a theoretical study. The theory proposed by Tabor [36] on junction growth was widely used by many researchers [5,37]. Brizmer et al. [37] analyzed the junction growth at the interface of the deformable sphere in contact with a rigid flat under normal and tangential loading with an FEM-based software package ANSYS. The linear strain-hardening under full stick conditions was considered. Recently, an asperity-based friction model was presented by Mao et al. [5], employing a comprehensive deformation model, considering strain hardening and junction growth. The model is referred below as Mao
Peng
Yi
Lai (MPYL) model. Dixit et al. [38] carried out a parametric study using this model, albeit considering temperature dependency of the flow stress. The MPYL model is briefly explained here along with a temperature-dependent constitutive relation. A mathematical formulation of the MPYL model employed the work of Greenwood and Williamson [18] in a fully elastic stage, that of Chang et al. [21] in a fully plastic stage and that of Zhao et al. [34] in the elastic
plastic stage to evaluate the contact area, normal load, and interfacial frictional stress during contact. A number of assumptions were based on empirical results of Tabor [36] and FEM studies of Kogut and Etison [39]. To include the effect of strain hardening, a virtual-work principle was employed. The constant friction factor and the theory developed by Tabor [36] were combined to include the effect of the junction growth

Modeling of friction in manufacturing processes

423

in the asperities. The expression for an effective-strain distribution over the surface of asperities suggested by Kogut and Etison [39] was used. Kogut and Etison [39] found that this distribution approximately followed an elliptical path. The ratio of a major axis to a minor one of the elliptical strain contour is m1. Here, the constant m1 is taken as 1.77 based on FEM results of Kogut and Etison [39]. The salient assumptions of the model are as follows: (1) the bulk deformation is neglected during the contact and friction process; (2) no interactions among asperities are considered; and (3) all the asperities are assumed hemispherical (at least near their summits), with a uniform radius R and heights varying randomly. The contact between the tools and the workpiece can be equivalently represented by a single rough surface and a rigid flat as schematically shown in Fig. 14.2. The distance between the rigid plate and the mean of asperity heights is denoted by separation d, h is the distance between the rigid flat and the mean of surface heights (RP2), ys is the separation of the two reference planes, and z is the height of the asperity measured from the mean of asperity heights. For an asperity with a height of z, the interference can be defined as ω 5 z 2 d;

(14.4)

where z is the height of asperity from the mean of asperity heights (line RP1). The separation of the reference planes is given by [40] ys 1 5 pffiffiffiffiffi ; σ 48πβ

(14.5)

where β is calculated as [40] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u3:717 3 1024 β5t  2 : 1 2 σs =σ

(14.6)

In Eqs. (14.5) and (14.6), σ and σs are the standard deviation of surface heights and asperity heights, respectively. The values of σ and σs can be obtained by a

Rigid plate

Normal compression

R

z

d h

RP1

ys

RP2 Rough surface

Figure 14.2 Schematic diagram of contact model of nominally flat surfaces. RP1, Mean of asperity heights; RP2, mean of surface heights.

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physical measurement of the surface roughness using a profilometer. The areal density of asperities is defined as [31] η5

N ; An

(14.7)

where N is the total number of asperities and An is the nominal contact area. In the literature, only three parameters are widely used to define the surface roughness topography on the basis of empirical relations [5,18,21]. It was observed that the three parameters, σ, σs, and η, are sufficient to define the characteristics of the surface roughness for any engineering material. The Gaussian distribution was employed to describe the height of asperities as [5] (  2 ) 1 z φðzÞ 5 pffiffiffiffiffiffi exp 20:5 : σ 2πσs s

(14.8)

The computational procedure to evaluate the friction as the function of the normal loads, material properties, and surface roughness parameters is as follows. The total deformation of the asperities due to the interference (ω) is divided into four stages with the limits of interference as ω1, ωb, and ω2. The asperities are deformed fully elastically if ω # ω1. Elastic
plastic deformation of asperities with nonlinear strain hardening without junction growth occurs in the interference stage ω1 # ω # ωb. The elastic
plastic deformation of asperities and their strain hardening is accompanied by the junction growth in the stage ωb # ω # ω2. If the asperities deform in a fully plastic mode, the interference is ω . ω2. The contact areas, normal loads, and tangential stresses are evaluated for each zone separately. Expressions for the interference limits contain several empirical constants and are listed in Table 14.1; they were obtained experimentally or analytically, and there is a scope to refine them. According to Chang et al. [21], the critical interferences ω1, ωb, and ω2 that mark the transition from elastically to plastically deformed asperities are given as ω1 5

  πKm H 2 R; 2EH

 ffi   12pffiffiffiffiffiffiffi 12f 2 =ð12Kc Þ ω1 ; ω b 5 ω2 ω2

(14.9)

(14.10)

and (

 2 )1=ð11ðn=2ÞÞ 200Kc Y ðπKm KH Y Þ=2EH R; ω2 5  2γ  pffiffiffi n σ0 T=Tm 2= 3 Kw Kp

(14.11)

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Table 14.1 Expressions used in Mao
Peng
Yi
Lai model. Parameter Flow stress

Hardness factor (KH) Maximum contact pressure factor (Kc) Pressure parameter (KT)

Area parameter (KS)

Initial yield strain of the workpiece material ðε0 Þ (modified in this chapter to include temperature dependency) Constant (K1)

Coefficient related to equivalent strain (Kp) Coefficient (Kw)

Expression  2γ , where T is the temperature of σf 5 σ0 εn T=Tm workpiece material (in K), Tm is the melting temperature of workpiece material (in K), σ0 is the material constant, ε is the equivalent strain, n is the strain-hardening exponent, and γ is the temperature-softening parameter h n  20:7 oi KH 5 2:84 1 2 exp 20:82 Ra , where a is the radius of the interfacing surfaces Kc 5 23 Km where Km is given by Km 5 0:454 1 0:41ν and ν is the Poisson’s ratio of the workpiece material 8 < 1 2 ð1 2 Kc Þ lnω2 2 lnω for ½ω1 ; ω2  lnω2 2 lnω1 : KT 5 : 1 for ½ω2 ; N 8 !3 !2 > < ω2ω1 ω2ω1 KS 5 1 2 2 ω2 2ω1 1 3 ω2 2ω1 for ½ω1 ; ω2  : > : 2 for ½ω2 ; N  1=ðn21Þ EH ε0 5 σ T=T ; where EH is the Hertz elastic 2γ 0ð mÞ modulus qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7m21 1 2m1 1 3 ; where m1 is the ratio of major axis K1 5 2 3 to minor axis of the elliptical strain contour. Here, m1 5 1.77 is considered based on FEM results of Kogut and Etison [39], and hence K1 5 6.16 ε0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Kp 5 pffiffi pffiffi 4=ð 3ðn 1 1ÞÞ K1 ω1 =ð 3πKC KH Rðn 1 1ÞÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Kw 5 n 12 1 pffiffi3πK KKK1 ðn 1 1Þ: T

S

H

where f is the friction factor, n is the strain hardening parameter, Y is the yield stress of the workpiece material, σ0 is the workpiece material constant, γ is the temperature-softening parameter, T is the workpiece temperature (in K), and Tm is the workpiece melting temperature (in K). In Eqs. (14.9)
(14.11), the empirical relations for Kc, KH, Km, Kp, and Kw are approximated based on the several assumptions taken by researchers [5,34,39,41] and are listed in Table 14.1. The Hertz elastic modulus of the materials, EH, is given by 1 1 2 ν 21 1 2 ν 22 5 1 ; EH E1 E2

(14.12)

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where E1 and E2 are the Young’s moduli of the tool and workpiece material, respectively, and ν 1 and ν 2 are the Poisson’s ratios of the tool and workpiece material, respectively. The total deformation of the asperity due to increase of the interference can be divided into four stages by three critical interferences ω1, ωb, and ω2. For each stage, the pressure (p), contact area (A), and frictional shear stress (τ) are calculated. The relations used for various stages are as follows: Stage 1: ω , ω1, fully elastic contact, no effect of hardening or junction growth: p1 5

4EH ω1=2 ; 3π R

(14.13)

A1 5 πRω;

(14.14)

and Y τ 1 5 f pffiffiffi ; 3

(14.15)

where f is the friction factor. The MPYL model does not suggest any procedure to determine this friction factor. It is proposed here that this factor is obtained by inverse analysis of experimental data. Stage 2: ω1 , ω , ωb, elastic
plastic contact with material hardening but without junction growth:  p2 5 KT KH σ0

T Tm

2γ  ω1=2 n 2 pffiffiffi Kw Kp ; R 3

A2 5 Ks πRω;

(14.16) (14.17)

and  2γ  ω1=2 n σ0 T=Tm 2 p ffiffi ffi p ffiffi ffi ; Kw Kp τ2 5 f R 3 3

(14.18)

where KT is the pressure parameter. Stage 3: ωb , ω , ω2, elastic
plastic contact with material hardening with junction growth:  p 3 5 KH σ 0

T Tm

2γ 

KT Ks πRω A3 5 pffiffiffiffiffiffiffiffiffiffiffiffi ; 12f2 and

ω1=2 2 pffiffiffi Kw Kp R 3

n pffiffiffiffiffiffiffiffiffiffiffiffi 1 2 f 2;

(14.19)

(14.20)

Modeling of friction in manufacturing processes

τ3 5 f

427

 2γ  ω1=2 n σ0 T=Tm 2 pffiffiffi pffiffiffi Kw Kp : R 3 3

(14.21)

where Ks is the area parameter. Stage 4: ω . ω2, fully plastic contact with material hardening and junction growth:  p 4 5 KH σ 0

T Tm

2γ 

ω1=2 2 pffiffiffi Kw Kp R 3

n pffiffiffiffiffiffiffiffiffiffiffiffi 1 2 f 2;

(14.22)

2πRω A4 5 pffiffiffiffiffiffiffiffiffiffiffiffi ; 12f2

(14.23)

and τ4 5 f

 2γ  ω1=2 n σ0 T=Tm 2 pffiffiffi pffiffiffi Kw Kp : R 3 3

(14.24)

The critical values of asperity height corresponding to critical interferences is obtained as h1 5 ω1 1 d; hb 5 ωb 1 d; h2 5 ω2 1 d;

(14.25)

where the values of ω1, ωb, and ω2 are obtained from Eqs. (14.9)
(14.11), respectively. The separation between the references plane of asperity heights and the rigid plate is denoted by d.

The total contact area for a typical separation is given by Ar 5 ηAn

ð h1

φðhÞA1 dh 1

d

ð hb h1

φðhÞA2 dh 1

ð h2

φðhÞA3 dh 1

ðN

 φðhÞA4 dh :

h2

hb

(14.26) The total normal pressure for a typical separation is given by P 5 ηAn

 ð h1

φðhÞp1 A1 dh1

d

ð hb

φðhÞp2 A2 dh1

h1

ð h2

φðhÞp3 A3 dh 1

ðN

 φðhÞp4 A4 dh :

h2

hb

(14.27) The total frictional force for a typical separation is given by T 5ηAn

ð h1 d

φðhÞτ 1 A1 dh1

ð hb h1

φðhÞτ 2 A2 dh1

ð h2 hb

φðhÞτ 3 A3 dh1

ðN

 φðhÞτ 4 A4 dh :

h2

(14.28)

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The friction coefficient is calculated as μ5

T : P

(14.29)

After evaluating out the contact area, normal load and interfacial shear stress are obtained for all four stages. The total area is obtained by multiplying the Gaussian distribution given by Eq. (14.8) and integrating between the limits of critical interferences to get the total contact area. Here, a numerical integration was performed using a trapezoidal rule [42] to obtain the total contact area. The similar procedure was employed to assess the total nominal pressure and the frictional force. The coefficient of friction was calculated by the ratio of total frictional force to total normal load. The MPYL model requires a friction factor that can be obtained in an inverse manner. A flowchart of the procedure for determining the coefficient of friction as a function of nominal pressure is given in Fig. 14.3. The surfaceroughness parameters were measured experimentally. After calculating the contact

Start

Input E1, E2, ν1, ν2, T and parameters of power law. Assume f. Experimentally measure the surface roughness parameters: σ,σs and η.

Obtain contact area, normal load, and interfacial shear stress at different stages with MPYL model. Numerically integrate to obtain normal and tangential force. Update friction factor, f.

Assess coefficient of friction as function of nominal pressure.

No

Does coefficient of friction match experimental value?

Yes Record coefficient of friction as function of nominal pressure.

Stop

Figure 14.3 Flowchart of computational procedure for multi-asperity-based modeling of friction.

Modeling of friction in manufacturing processes

429

area, normal load, and interfacial shear stress at different stages of deformation of asperities, the coefficient of friction was again evaluated for the updated friction factor. This procedure was repeated until the convergence was obtained. An extensive parametric study was carried out to analyze the effect of the following material properties and surface-roughness parameters on the coefficient of friction: the stain-hardening exponent (n), the workpiece temperature (T), the friction factor (f), the Young’s moduli of the workpiece material (E1) and of the flat die material (E2), the standard deviation of the asperities heights (σs), and the areal density of asperities (η). The coefficient of friction was evaluated as function of the normal load or nominal pressure. The workpiece material was commercially pure aluminum with 99.1% aluminum, 0.51% iron, 0.21% copper, 0.13% silicon, and 0.05% manganese. It was compressed by a rigid flat die made of steel. The input data for the material properties of the workpiece and flat die materials are as follows: E1 5 70 GPa, E2 5 210 GPa, ν 1 5 0.33, ν 2 5 0.27, f 5 0.3, and T 5 25 C. The flow stress of the workpiece material was governed by the power law [43]:  σ f 5 σ 0 εn

T Tm

2γ

;

(14.30)

where σf is the flow stress (in MPa) and ε is the equivalent strain. The material constants were determined experimentally [43] as σ0 5 84.75 MPa, n 5 0.0387, and γ 5 0.2351. The constant m1 is considered to be 1.77 based on the FEM studies of Kogut and Etison [39]. The surface-roughness parameters, namely, σ, σs, and η, correspond to the standard deviation of the surface heights, the standard deviation of the asperity heights, and the areal density of asperities, respectively. These three parameters are widely used to define the surface topography of any engineering material based on empirical relations provided in the earlier works [18,21,39]. Here, the surface-roughness parameters of the workpiece material are considered as follows: σ 5 0.0025 μm, σs 5 0.0022 μm, and η 5 0.012 μm22. The theoretical value of radius of the asperity (R) comes out to be 1148 μm based on the σ, σs, and η measurements. The effect of strain-hardening of the workpiece material on the coefficient of friction is shown in Fig. 14.4 for different nominal pressures. It is observed that the coefficient of friction decreases with increase in the strain-hardening coefficient (n) at different nominal pressures; this is due to the reduction of contact area for the strain-hardened materials. It is also observed that the rate of change of the coefficient of friction with nominal pressure decreases with increased n. It is seen from Fig. 14.4 that the coefficient of friction was higher during elastic deformation of the asperities compared with the plastically deformed ones. When the value of n increased from 0.1 to 0.2, the static coefficient of friction decreased from 0.35 to 0.25. For higher values of n the coefficient of friction remained almost constant with increasing nominal pressure. Table 14.2 shows the effect of temperature of the workpiece on the coefficient of friction at different levels of nominal pressure. It was found that the coefficient

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Figure 14.4 Effect of strain-hardening of workpiece material on friction at different nominal pressures.

Table 14.2 Effect of deformation temperature of workpiece material on friction with different nominal pressure (hardening exponent n 5 0.0387). Nominal pressure (MPa)

10 20 30 40 50 60

Coefficient of friction T 5 25 C

T 5 250 C

T 5 500 C

0.305 0.258 0.224 0.207 0.191 0.181

0.278 0.225 0.198 0.181 0.169 0.161

0.255 0.207 0.185 0.169 0.157 0.149

of friction decreased with increase in the temperature. Thus the effect of temperature should be also considered in the model. The effect of the friction factor on the coefficient of friction is presented in Fig. 14.5 for different nominal pressures. It is observed that the coefficient of friction increased with the increasing friction factor (f); other material properties and surface roughness were kept the same. The dependency of the coefficient of friction on f is more pronounced as compared with the other material properties. It is seen from Fig. 14.5 that at low magnitudes of f, the coefficient of friction remained almost constant with increasing nominal pressure. The smaller value of f represents

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Figure 14.5 Effect of friction factor of workpiece material on friction at different nominal pressures (hardening exponent n 5 0.0387).

Figure 14.6 Effect of Young’s modulus of deformable material on coefficient of friction at different nominal pressures (hardening exponent n 5 0.0387).

a surface with more impurity and weaker junctions formed between the asperities and the flat die. Hence, the coefficient of friction is reduced due to low magnitude of shear stresses. The direct measurement of f is difficult; however, an inverse

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Figure 14.7 Effect of Young’s modulus of flat die material on coefficient of friction at different nominal pressures (hardening exponent n 5 0.0387).

method can be easily employed to determine the appropriate value of f as the function of material properties. Figs. 14.6 and 14.7 show the effect of the Young’s modulus of the workpiece and flat die materials on the coefficient of friction at different levels of nominal pressure of the die. A similar trend was observed for the coefficient of friction with increasing nominal pressure for different E1 and E2. However, the effect of E1 on the coefficient of friction was more pronounced than that of E2; this can be ascertained by comparing Figs. 14.6 and 14.7. For changes in the value of E1 and E2 by 6 20%, the dependency of the coefficient of friction on E1 was stronger than that on E2. Thus the effect of E2 on the coefficient of friction is less significant. Assuming that the material properties of the workpiece and flat die are known, the effect of surface-roughness parameters on the coefficient of friction was studied. The surface-roughness topography of the engineering materials was defined by three parameters, namely, σ, σs, and η [35]. The empirical relations given in Kogut and Etison [35] were used by numerous researchers to express the surfaceroughness topography of materials in the contact modeling. Based on the empirical relations, the ratio of σs to σ should be less than 1; this can also be inferred from Eq. (14.6). The effect of the ratio of σs to σ on the coefficient of friction is shown in Fig. 14.8 for different nominal pressures. This corresponds to the values of σ 5 0.0025 μm and η 5 0.015 μm22. It was observed that as the value of σs/σ decreased from 0.9 to 0.5, the coefficient of friction diminished from 0.19 to 0.13. For further reduction of the value of σs/σ from 0.5 to 0.1, the variation in the

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433

Figure 14.8 Effect of surface roughness parameter (σs) on friction at different nominal pressures (hardening exponent n 5 0.0387).

coefficient of friction with nominal pressure became less pronounced. These results suggest that a high value of σs/σ leads to a high magnitude of the surface parameter β; therefore, the coefficient of friction increases. Hence, for a higher value of β, the coefficient of friction between the asperities and the flat die is larger. The effect of areal density of asperities (η) on the coefficient of friction at different nominal pressures was studied keeping σ 5 0.0025 μm and σs 5 0.0022 μm and the material properties constant (Fig. 14.9). The empirical relation β 5 ησR is widely used [35] to obtain the radius of the asperity (R) based on the experimental measurement of σ, σs, and η. The lower value of η represents the smooth surface; the radius of asperity R is large in this case. It was observed that as η increased the coefficient of friction decreased. The smooth surface finish results in a higher value of friction compared with the rough surface. Tests were conducted to assess the applicability of the MPYL model for assessing the level of friction in metal-forming processes [38]; a well-established RCT was performed at different temperatures for this [26]. The specimen was made of commercially pure aluminum according to ASTM-D695 and was placed between the dies. The experimentally measured surface parameters were as follows: σ 5 2.854 μm, σs 5 1.451 μm, and η 5 0.038 μm22. It was assumed that material followed the power law given by Eq. (14.30). The friction factor was found with an inverse approach by minimizing the error between the experimental and theoretical values of the friction using a bisection method [44]. The friction factor remained constant for different workpiece deformation temperatures; however, the coefficient of friction reduced with the temperature.

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Figure 14.9 Effect of surface roughness parameter (η) on friction at different nominal pressures (hardening exponent n 5 0.0387).

14.3.5 Plowing model Although adhesion and shearing of the contact between two asperities of the surfaces is the dominating cause for friction in manufacturing processes, sometimes the plowing of the softest surface by the hardest asperity also plays a significant role. In such cases, the coefficient of friction can be split into two parts as follows: μ 5 μa 1 μp ;

(14.31)

where μa is the friction due to adhesion and μp is the friction due to plowing; the former is discussed in Section 14.3.4. Bowden and Tabor [17] assumed that a conical asperity with semiangle θ indented and slid through the surface of a softer plastically deforming material. Based on this assumption, the plowing coefficient of friction was estimated as μp 5

2 cot θ: π

(14.32)

Generally, θ . 80 degrees. For θ 5 80 degrees, μp 5 0.11, a smaller value compared with adhesive friction. If the asperity has a spherical shape, the plowing coefficient of friction is given by [45] μp 5

4r ; 3πR

(14.33)

Modeling of friction in manufacturing processes

435

where R is the radius of a spherical asperity and r is the distance between the axis of the spherical asperity and its contact with the top surface of the soft material. For a known depth of penetration d, r can be calculated as r5

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Rd 2 d2 :

(14.34)

For a large depth of penetration (in comparison to R), the coefficient of friction is defined as [46,47] 2 (  )1=2 3   2 4 2R 2 21  r  2R 2 5: μp 5 sin 21 2 π r 2R r

(14.35)

When a hard transverse cylinder of radius R slides on a soft surface, the simplified expression for the coefficient of plowing friction is [48] ( 1 μp 5  2 R=d 21

)1=2 :

(14.36)

Plowing friction becomes significant in micromachining, where the ratio of chip thickness to a tool nose radius becomes very small.

14.4

Friction in machining

The study of friction in machining attracted attention of many researchers. Commonly, researchers used either the Coulomb model or the one by Zorev [49] to assess cutting forces, stresses, strain-rates, and temperatures in machining processes [2,50]. As stated earlier, the Coulomb coefficient is defined as the ratio of the frictional stress to the normal stress at the tool’s rake face; validity of the Coulomb model becomes weaker in high-speed machining. On the other hand, in the Zorev’s model, normal and tangential stresses are expressed as functions of the distance from the tool tip (Fig. 14.10). The contact length between the tool and the chip at the interface is denoted by lc and the length of a sticking zone by lst (it is usually equal to a half of the tool
chip contact length). ¨ zel [2] assessed suitability of five different friction models for orthogonal cutO ting using an FEM model at specific-cutting process parameters. The FEM model assessed cutting forces, stress distributions at the tool
chip interface, tool
chip contact length, shear angle, and maximum temperature at the tool
chip interface. The experimental results of Childs et al. [51] were used for comparison with the simulation results; the variable-friction models developed from the experimentally measured normal and frictional stresses at the tool’s rake face provided better predictions. Childs [52] critically reviewed the experimental findings on the friction in

Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

Stress

436

Normal stress σ (x)

Shear stress τ (x) x

l st lc

Distance from tool tip

Figure 14.10 Variation of stresses along tool’s rake face in machining.

metal-cutting processes. Employing the Amontons model, a modified expression for the coefficient of friction as a function of equivalent plastic strain-rate was proposed: μ 5 μ0 ð1 1 a_εp Þ;

(14.37)

where μ0 is the coefficient of friction during elastic deformation, a is the material parameter, and ε_ p is the equivalent plastic strain-rate. Joshi and Bolar [53] carried out three-dimensional FEM simulations to study the machining of a thin-wall component with varying wall constraints. In order to simplify simulations the frictional interaction between the cutting tool and the workpiece was modeled with the Coulomb model in the sliding zone and constant friction in the sticking zone. Sticking friction occurs near the cutting edge contacting the workpiece, whereas sliding occurs far away from the contacting area. A constant value (0.17) of coefficient of friction was used in the sliding zone. Milling is the process of machining by advancing a rotary cutter into a workpiece. Niknam and Songmene [54] analyzed the friction and burr formation in this process. The coefficient of friction (or friction angle) was obtained by calculating the levels of radial and tangential cutting forces. Recently, Funke and Schubert [55] studied the coefficient of static friction of microstructured surfaces of the steel using tools with a sharp and chamfered corner. For both tool types a feed per tooth was the crucial parameter with a significant influence on the coefficient of static friction. The maximum coefficient of static friction was obtained as μ 5 0.63 in case of the tool with a sharp corner, whereas it was 0.44 for the tool with a chamfered corner. A very limited research was carried out, focusing on friction modeling in the milling process. Miller et al. [56] used two unrelated coefficients of friction for calculating the forces in radial and axial directions in friction drilling. Friction drilling is a nontraditional hole-making method. These values of coefficient of friction were based on reported results in literature on friction stir welding (FSW). Qu and Blau [57]

Modeling of friction in manufacturing processes

437

modified the Miller et al.’s [56] approach and used a single, variable coefficient of friction instead of two unrelated coefficients of friction. Recently, Beirmann et al. [58] presented a review of deep-hole drilling process. Schey [59] pointed out two benefits of ultrasonic vibration in manufacturing: a reduction in strength of the material and a decrease of friction at the tool
workpiece interface. Depending upon the direction of imposed vibration, the reduction in friction can be explained either by a change in magnitude of the normal load or a change in direction of the resultant sliding velocity vector. Babitsky et al. [60] demonstrated the fundamental advantages of ultrasonically assisted turning (UAT) compared with conventional turning (CT) for Inconel 718 in terms of stresses
strains, cutting forces, and contact conditions at the interface. A number of researchers investigated friction in ultrasonically assisted machining processes [61
64]. Kumar and Hutchings [63] showed that the reduction in sliding friction between the metallic surfaces due to longitudinal vibration was greater than that due to transverse vibration; specimens made of aluminum alloy, copper, brass, and stainless steel sliding against tool steel were considered. Jamshidi and Nategh [62] presented an analytical model to estimate the friction in UAT based on the cutting forces of CT. They found that the level of friction in UAT was about 39% higher as compared with CT. Recently, Dixit et al. [61] proposed a simplified and computationally efficient inverse methodology to assess friction in dependence on cutting speeds in UAT. The methodology requires the experimentally measured cutting forces of CT at two specified cutting speeds. Zhang et al. [64] presented a review of micro/nanomachining with elliptical vibration. Nowadays, elliptical-vibration cutting gained an attention of the researchers, thanks to its excellent machining performance for precision machining of difficultto-cut materials. Generally, regarding the reduction of friction in UAT, there are conflicting reports.

14.5

Friction models in metal forming

Although highly approximate, the Amontons
Coulomb model is widely used in metal forming. Several researchers calculated frictional shear stresses using this model with an upper limit equal to a shear flow stress of the material. Most of the upper bound models employed the constant-friction model. The Wanheim and Bay’s model follows the Coulomb model at low normal pressure and the constantfriction model at high normal pressure; there is a smooth transition between them. Although this model seems to represent frictional behavior in a realistic manner, it is surprising that not many research groups used it. Still some applications of this model in the analysis of rolling are available [65]. There is hardly any scheme of metal forming that uses an asperity-based model and accounts for various factors including temperature and surface roughness. Instead, some empirical models were used. For example, Lenard and Kalpakjian [66] used the following relation to estimate the Coulomb coefficient of friction:

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

μ 5 0:41 2 0:00025 T 2 0:028 VR ;

(14.38)

where T is the temperature in  C and VR is the surface speed of the roll in m/s. Lu¨chinger et al. [67] proposed a systematic procedure for developing the friction model for a bulk metal-forming process. Experiments were performed to record the coefficient of friction as a function of sliding distance using a pin-on-disk tribometer with two different lubricants. Subsequently, regression models were used to fit the experimental data.

14.6

Friction in solid-state welding

Solid-state welding is a group of joining processes, in which coalescence of the joining surface is achieved without melting by applying pressure with or without the application of heat. Typical solid-state welding processes are friction welding, FSW, roll boding, diffusion welding, and ultrasonic welding. A few researchers attempted to study critically the friction in solid-state welding processes, which occurs mainly due to relative movement of two contacting parts. Sometimes, only one part rotates and another is kept stationary. Heat generation at contact surfaces of the rotating tool, and the weld piece is a major factor for the quality of welding. The material flow and heat generation are characterized by the contact conditions at the tool
workpiece interface. Schmidt et al. [68] investigated the friction behavior in FSW assuming the contacting surfaces as sliding, sticking, or partially sliding/ skidding. Arora et al. [69] studied a variation of the coefficient of friction in FSW and provided the following empirical expression: 

 ω r μ 5 μ0 exp 2δ ; ω 0 RS

(14.39)

where μ0 is the maximum value of the coefficient of friction assuming no slip between the tool and the workpiece, δ is the variable fractional slip between the tool and the workpiece, ω is the angular velocity of the tool, ω0 is the reference value of the angular velocity of the tool, r is the radial distance of the center of the area from the tool axis, and RS is the radius of the tool shoulder. The parameter δ depends on the characteristics of contacting surfaces defining these conditions— sticking, sliding, and combination of sticking and sliding. It can be expressed as    ω r δ 5 0:2 1 0:6 1 2 exp 2δ0 ; ω 0 RS

(14.40)

where δ0 is the constant. It follows from Eq. (14.40) that the numerical value of δ lies between 0.2 and 0.8. The lower limit corresponds to sticking when the relative velocity between the tool and the workpiece becomes zero. The upper limit is achieved when a relative velocity is very high, representing the sliding condition.

Modeling of friction in manufacturing processes

439

Assidi et al. [70] employed two different friction models, namely, the Coulomb model and the Norton’s model to analyze the effect of process parameters in FSW. The Norton’s model is commonly employed in hot metal forming. It has two adjustable parameters, determined in experiments, and considers a velocity dependence. The applicability of the both friction models was tested. The Coulomb model provided the realistic results obtained with an FEM model, and was in line with the experiments, whereas the Norton’s model showed a strong sensitivity to tool forces and temperature. He et al. [71] presented a review of numerical simulations of FSW. They emphasized that void formation in FSW depended strongly on friction, necessitating a good friction model.

14.7

Friction models for micromanufacturing

Micromanufacturing deals with creation of small parts as well as microfeatures on small and large components to enhance their capabilities and functionalities. Mathematical modeling of friction for conventional manufacturing processes is not suitable for micromanufacturing due to size effects [72]. Size effects are the deviations from the expected results that occur when the dimension of a workpiece or sample is reduced [73]. Peng et al. [74] developed friction model based on open- and closedpocket concept to analyze the friction in microforming. This approach is referred as pocket model. The pocket model allows an account for entrapment of lubricants in valleys between asperities of the workpiece material. As a result, the effect of lubricant’s performance is easily captured by this model. However, the pocket model is not suitable for the analysis of physics of friction under dry conditions, which is strongly dependent on the grain size as well. The effects of temperature and surface roughness on friction in micromanufacturing are still a challenge. There is ample scope for development of a friction models for micromanufacturing processes. Recently, a number of researchers presented concise reviews of different methods used to analyze the friction in micromanufacturing [75,76]. Wang et al. [76] emphasized that a real state of the interfacial contact between a tool and workpiece should be considered. For example, real topography of a workpiece surface should be considered rather than modeling it through isosceles triangles. With the advent of FEM-based software, the real topography of workpiece surface can be depicted with a statistical surface parameter, which provides the effect of surface roughness on friction. Dixit et al. [75] mentioned a lack of no dedicated friction model available in the literature, which can consider the effect of surface roughness, grain size, and mechanical properties.

14.8

Challenging issues and directions for future research

A major challenge in developing friction models for manufacturing process is the difficulty in acquiring the friction data, which accurately represent the manufacturing

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Mechanics of Materials in Modern Manufacturing Methods and Processing Techniques

process. Historically, highly simplified friction models (e.g., Coulomb friction) were often used in modeling of machining processes, due to a limited knowledge of complex frictional interactions at the tool
workpiece interface and a lack of suitable experimental techniques to measure the relevant friction-model parameters under conditions representative of machining process. Schemas to incorporate the dependency on temperature, surface roughness, and grain size into the friction model are not readily available in the literature. With the advent of miniaturization, there is a need to develop friction models considering the effect of surface roughness, microstructure, and mechanical properties into account. Currently, sufficient attention is not being paid to this. Based on the literature review and assessment of the challenges, the following research areas are identified: G

G

G

G

A method for evaluation of the coefficient of friction with higher accuracy should be developed by considering the real conditions at the contacting surfaces in micromanufacturing, including microforming. There is a need to develop an adequate measuring technology suitable for measuring the friction in micromanufacturing. An experimental method should be designed and implemented to assess the coefficient of friction through direct consideration of characteristics of the microforming process. Development of dedicated friction models for the manufacturing processes is needed, which may include the effect of temperature, surface roughness, grain size, and thin film of oxide. There is also a need to develop methods for identification of the parameters of the developed friction models. Improved understanding of the effects of lubricants, tool coatings, and tool wear on friction should be studied together with the ways of incorporating their effects into friction models.

14.9

Conclusion

This chapter provides an overview of the most common methods used to analyze the friction phenomenon in manufacturing processes. Among the different friction models, an asperity-based friction model is one of the promising approaches allowing an inclusion of the effects of temperature, rate of deformation, strain hardening, junction growth, and surface roughness. This model can be used to estimate the coefficient of friction in manufacturing processes. A parametric study is presented to demonstrate the influence of material properties (E1, E2, n, f, and T) and surface roughness parameters (σs and η) on the coefficient of friction. Among these parameters, the friction factor (f) is the most influencing parameter on the coefficient of friction. An inverse method is employed to identify the appropriate value of f based on the coefficient of friction. The experimentally measured coefficient of friction can be the basis of inverse estimation. It was observed that with increasing temperature, the coefficient of friction decreased in the ring-compression test. There is a scope for extensive experimental study on determining the friction to fine-tune the asperity-based model.

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Acknowledgment Funding from the Engineering and Physical Sciences Research Council (United Kingdom) through grant EP/K028316/1 and Department of Science and Technology (India) through grant DST/RC-UK/14-AM/2012 for project “Modeling of Advanced Materials for Simulation of Transformative Manufacturing Processes (MAST)” is gratefully acknowledged.

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226. [67] M. Lu¨chinger, I. Velkavrh, K. Kern, M. Baumgartner, S. Klien, A. Diem, et al., Development of a constitutive model for friction in bulk metal forming, Lubricants 42 (2018) 1
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18.

Further reading M.P.F. Sutcliffe, Surface asperity deformation in metal forming processes, Int. J. Mech. Sci. 30 (1988) 847
868.

Index

Note: Page numbers followed by “f” and “t” refer to figures and tables, respectively. A Adhesion theory, 417
418, 421
422 AISI D2 tool steel, 316, 316f, 317f AISI H13 steel, 312
315, 312f, 313f, 314f case-hardened steels, 338
339 AISI 316L steel austenitic stainless steel, 335
337 orthogonal cutting model, 336f AISI 52100 steel bearing steel, 317
319, 317f, 318f case-hardened steels, 338
339 ALE. See Arbitrary Lagrangian
Eulerian (ALE) Alloys difficult-to-cut alloys, 306 iron-based, 306 nickel-based, 307
309, 307f, 308f, 309f titanium-based, 309
310, 309f Amontons
Coulomb model, 417
421 Amonton’s model, 435
436 Anisotropic deformation behavior, 243
244 Anisotropy normal, 10
11 planar, 10
11 ANN. See Artificial neural network (ANN) Arbitrary Lagrangian
Eulerian (ALE), 325 adaptive remeshing, 244
245 method, 17, 206 Artificial neural network (ANN), 18, 345 Asperity-based friction model, 421
433 Asymmetric sheet rolling, 5
6 Atomic finite element method, 221
222 Austenitic stainless steel, 310
311, 310f, 311f AISI 316L, 335
337 Autofrettage, 31
32, 32f explosive, 34, 35f hydraulic, 32
33, 33f, 53
57

FEM modeling of, 52
66, 56t, 59f plane strain end condition of, 57, 60f, 61f, 62t plane stress condition of, 54
57, 57f, 59t rotational, 35
37, 36f swage, 33
34, 34f thermal, 35, 36f, 58
66 FEM modeling of, 52
66 open-ended condition of, 63
66, 64f, 65f, 66f, 66t plane stress end condition of, 61
63, 62f, 63f, 64t Automation, 175 Automotive manufacturers, 71 B Backward extrusion, 7
8, 8f Bai
Wierzbicki plasticity model, 329
330 BEM. See Boundary-element-simulation (BEM) Bending, metal forming, 11
13, 12f BHF. See Blank holder force (BHF) Blank holder force (BHF), 10
11 Body-centered cubic (BCC) metals, 375
376 Boundary-element-simulation (BEM), 130 Brass alloys, 71
73 C Calibration techniques, damage, 158
159 using thermomechanical multiaxial tensile test data, 160
162, 162t, 163f, 164t, 165f using thermomechanical uniaxial test data, 159
160, 160t Calorimetry, differential scanning, 181, 182f Carbon/epoxy composites, 237

446

Case-hardened steels, 312
319, 312f, 313f, 314f AISI H13 and AISI 52100, 338
339 Casting, 415
416 Cauchy’s stress tensor, 38
39, 46 CDM. See Continuum damage mechanics (CDM) CEB model, 422 CEL. See Coupled Eulerian
Lagrangian (CEL) Cellular automaton (CA) model, 270
272 Cemented carbide
cutting tool, 285 Central burst, 5 CFD. See Computational fluid dynamics (CFD) CFM. See Continuous Filament Mats (CFM) CFRP plate, mechanical properties, 237t Chang
Etison
Bogy (CEB) contact model, 418 Chao and Qi’s model, 402 CHILE model. See Cure hardening instantaneous linear elastic (CHILE) model Chip formation, 239
240, 273
274 microhardness distribution in, 293f microhardness tests of, 292
293, 292f microstructural evolution in, 273
274 microstructure in shear-band region of, 275f sample preparation for, 290
291, 291f Clamping devices and machining, 216
218 Classical plasticity theory, 2 Closed die forging, 3
4, 4f Closed-loop-design cutting tool design, 218, 219f hybrid modeling and, 216
218 Closed-loop principle of modeling, 198
199 Closed-loop simulation system, 218f Coating-substrate simulation (CSS) multiscale modeling in, 220
222 Coating-substrate-system, 210f multiscale modeling of, 221f Cobalt, 373 Coefficient of friction, 430
432 Coefficient of thermal expansion (CTE), 184 Cogging, 2
3, 3f Cold metalworking, 1 Complementary methods, 209
212

Index

Computational fluid dynamics (CFD), 396 models, 406 Computational Solid Mechanics (CSM) model, 405
407 Constant-friction model, 419
420 Contact model, 334
335 Continuous Filament Mats (CFM), 183
184 Continuum damage mechanics (CDM), 143, 153 Continuum diffusion models, 221 Conventional drilling (CD) techniques, 235 Conventional forming, 134
135 Conventional machining, 197
198 Conventional macroscale machining, 243 Conventional sheet metal forming, 90 Conventional turning (CT), 437 Copper alloys, 71
73 oxygen-free high-conductivity (OFHC), 339
340, 340f Corrosion, of metallic materials, 348, 348f Corrosion resistance of machined surfaces, 350t residual stress, 348
352, 349f, 350t, 351f Coulomb coefficient, 435 Coulomb model, 435
436, 439 Coulomb’s coefficient of friction, 418 Coulomb’s friction model, 4, 416 Coupled Eulerian
Lagrangian (CEL), 325 Cracking edge, 5 stress corrosion, 67 surface, 4, 8
9 Cross-shear rolling, 421 Cryogenic cooling, in cutting, 215f Cryogenic cutting, cutting edge for, 216f Cryogenic machining, 215
216 Crystal plasticity (CP) theory, 245 CSM model. See Computational Solid Mechanics (CSM) model CSS. See Coating-substrate simulation (CSS) Cubic metals, face-centered, 373
374 Cure hardening instantaneous linear elastic (CHILE) model, 184 Cutting cryogenic cooling in, 215f edge surface and roughness, 220f forces, 237
239 tool

Index

closed-loop design, 219f interaction, 198
199 optimisation, 213 D Damage concepts of, 143 mechanisms, 143
144, 144f Damage calibration techniques, 158
159 using thermomechanical multiaxial tensile test data, 160
162, 162t, 163f, 164t, 165f using thermomechanical uniaxial test data, 159
160, 160t Damage evolution modeling of, 151
158 advanced damage models, 153
155 constitutive equations, 151
153 unified constitutive equations for hot stamping, 155
156 Damage modeling technique applications of, 162
170 for hot stamping, 162
170 Damage variables, concepts of, 143 Darcy’s law, 176 DDC. See Domain Decomposition Method (DDC) Deep drawing, metal forming, 10
11, 11f Deformable material, effect of Young’s modulus of, 431f Deformation, 130 material, 1 plastic, 1, 273
274, 273f DIC. See Digital image correlation (DIC) Die forging closed, 3
4, 4f impression, 3
4, 4f open, 2
4, 3f Differential scanning calorimetry (DSC), 181, 182f Difficult-to-cut materials, residual stress in, 305
319 Diffraction techniques, 302
303 Diffusional transformation, 20
21 Digital image correlation (DIC), 150
151 Direct extrusion, 7
8, 7f Discrete dislocation models, 221
222 Dislocation density model, 275
276 Dislocation partials, 372

447

Distortions, residual stresses and, 400
403, 401f Domain decomposition method (DDC), 207
209, 209f Drilling, 229 circularity, 240, 240f in composites, 237
240 workpiece material: carbon/epoxy composites, 237 finite-element model of, 233f forces, measurement, 236 LAGRANGE 3D FE simulation of, 207f theory and modeling, 230
235 3D FE model of, 233 tools, 229 Dry machining, 213 DSC. See Differential scanning calorimetry (DSC) Dynamic recrystallization, 9
10, 21, 270
272 model, 276
277 E ECAE. See Equal-channel angular extrusion (ECAE) Edge cracking, 5 Einstein’s summation convention, 37 Elastic
plastic analysis, mathematical modeling in, 38 Elastic
plastic deformation, 31 of asperities, 424 Elastic
plastic problems, numerical modeling of, 37
44 Electromagnetic pulse calibration, 135 Electromagnetic pulse compression, 113
116, 114f, 115f Electromagnetic pulse forming advantages and application fields of, 131
139 and conventional forming, 134
135 cutting, 138
139 of flat and preformed sheet metal, 118
122 incremental forming, 134f joining, 136
138, 136f multiple-discharge, 133
134, 134f shaping, 132
135 as stand-alone process, 132
133, 133f Electromagnetic pulse forming

448

Electromagnetic pulse forming (Continued) general setup and process principle, 112
113 process classification, 111
112 process mechanics calculation, 123
131 acting loads calculation, 123
125, 123f, 126f numerical calculation of process, 126
131, 127f, 128t, 131f process variants, 113
122 electromagnetic pulse compression, 113
116, 114f, 115f electromagnetic pulse expansion, 116
118, 116f, 118f electromagnetic pulse forming of flat and preformed sheet metal, 118
122, 119f, 121f, 122f Electromagnetic pulse welding, 137f Elemental equations, formulation of, 46
50 Element-deletion technique, 244
245 Engineering materials, surface-roughness topography of, 432
433 Equal-channel angular extrusion (ECAE), 9
10 EULER 2D FE Simulation, 206f Eulerian formulation, 16, 42
44 Explosive autofrettage, 34, 35f Extrusion, metal forming, 7
10, 7f, 8f, 9f, 10f F Face-centered cubic (FCC) crystal structure, 362 metals, 373
374, 374f Fatigue strength, 352, 353f FCC. See Face-centered cubic (FCC) FEM. See Finite element method (FEM) Femtosecond (fs) shock wave, in Ni, 376
378 Fibre-reinforced polymer (FRP) composite, 173
175 thermoset, 180 Fibre reinforcements, permeability of, 177
178 Fibres, mechanical properties of, 183
184 Finite difference method, 15 Finite-element (FE) analysis, 186, 362 simulations, 244

Index

Finite element method (FEM), 1
2, 22, 211, 217, 243, 324, 418 elemental equations, formulation of, 46
50 formulation, 46
47 using updated Lagrangian method, 44
52 model, 435
436 modeling of hydraulic and thermal autofrettage, 52
66 software, 207
209 solution method, 50
52 Finite-element (FE) model, 203
209, 232
235 of drilling in composite laminates, 233f and material parameters, 249
250 materials parameters used in, 262t of micro-scratching process, 260f in modeling and simulation of metal cutting, 203f principle, 212f residual stresses from, 343
344, 344f setup, 233 for simulation of micro-cutting, 249f Finite-element (FE) simulation, 130, 270
272 Finite point set method (FPM), 211 applied to metal cutting simulation, 212f principle, 211f Flat die material, Young’s modulus of, 432f FLC. See Forming limit curve (FLC) FLD. See Forming limit diagram (FLD) Flexible Machine Tool System (FMS), 197
198 FLSD. See Forming limit stress diagram (FLSD) Flyer plate impact test, 366
367, 367f Forging, 2
4, 3f die forging. See Die forging estimation of forging load, 4 orbital, 2
3, 3f rotary, 2
3, 3f upsetting, 2
3, 3f Formability evaluation advanced testing system for hot stamping applications, 150
151, 150f experimental methods for determining forming limits, 147
149, 148f forming limit prediction, 146
147

Index

requirements for hot stamping applications, 149
150 Formability, in hot forming, 170 Forming limit curve (FLC), 145 Forming limit diagram (FLD), 90 concept and features of, 145
146, 146f, 147f modeling of, 156
158 of sheet metals, 147 strain-based, 145
146 Forming limit stress diagram (FLSD), 93
94, 146 Forward extrusion, 7
8, 7f FPM. See Finite point set method (FPM) Friction, 20, 24, 82
83, 415 coefficient of, 430
432 Coulomb’s coefficient of, 418 effect of deformation temperature of workpiece material on, 430t factor, effect, 430
432 kinetic, 415, 417 in machining, 435
437 size effect on, 418 static, 415, 417 Friction model, 418
435 asperity-based, 421
433 history of, 416
418 in metal forming, 437
438 for micromanufacturing, 439 Friction stir welding (FSW), 393
396, 394f, 399, 401
402, 406, 408t, 436
437 modeling couplings in, 394f 3D flow in, 403 FRP. See Fibre-reinforced polymer (FRP) FSW. See Friction stir welding (FSW) G Galerkin method, 47 Gaussian distribution, 421
422, 424 Glass-fibre reinforced polymers, 173
175 Greenwood
Williamson (GW) model, 421
422 contact model, 418 Gurson-Tvergaard-Needleman (GTN) model, 90, 93
94 H Hall
Petch effect, 21
22, 274, 371
372 Hardening behavior of material, 38
41

449

Heat-affected zone (HAZ), 399 Heat generation, in machining, 304 Heat-treatable aluminum alloys, 73 HEL. See Hugoniot elastic limit (HEL) Hencky’s hydrostatic stress conditions, 201 HERF. See High energy rate forming (HERF) Hexagonal closed-packed (HCP) metals, 362, 374 HFQ light alloy forming process, 144
145 High energy rate forming (HERF), 111 High-performance cutting, 213 High-speed cutting, 213 High-velocity forming, of sheets, 361
362 Hole drilling, 229 Hole-making process, 229 Holloman’s law, 40 Hot forming, prediction of formability in, 170 Hot stamping applications advanced testing system, 150
151, 150f requirements, 149
150 damage modeling technique for, 162
170 Hugoniot elastic limit (HEL), 364 Hybrid machining techniques, 229, 235 Hybrid modeling, and closed-loop-design, 216
218 Hydraulic autofrettage, 32
33, 33f, 53
57 FEM modeling of, 52
66, 56t, 59f plane strain end condition of, 57, 60f, 61f, 62t plane stress condition of, 54
57, 57f, 59t Hydraulic presses, 71 Hydroformed products, 73 Hydroforming, 71, 101 blank materials for, 71
73 classification, 72f design and optimization of, 73
74 loading paths design, 98
100 of materials, 73 of polygonal cross sections, 81
85 of polygonal shapes, 101 sheet, 71 tube. See Tube hydroforming of tube branches, 85
89, 86f warm, 73 Hydrostatic bowl, 213
215, 214f

450

Hydrostatic extrusion, 7
8, 9f I Impact extrusion, 7
8, 9f Impression die forging, 3
4, 4f Inconel 690, nickel-based alloys, 337
338, 338f Incremental stress tensor, 186 Indirect extrusion, 7
8, 8f International Confederation for Thermal Analysis and Calorimetry (ICTAC), 181 Inverse Hall
Petch effect, 371
372 Inverse modeling, 19
20 J Johnson-Cook law, 204 Johnson
Cook (JC) model, 20, 329
330, 332
334, 333t, 336, 338 material model, 280, 281t plasticity model, 327
328 plasticity model for AISI 316L, 304t Johnson
Mehl
Avrami
Kolmogorov (JMAK) model, 21
23, 283
285 microstructure model, 280 parameters, 282, 282t K Kinematic hardening, 40
42, 54
57, 93
94 Kinematic modeling, 230
232 Kinetic friction, 415, 417 Kistler (Model 9271A) dynamometer, 236 L LAGRANGE 3D FE simulation of drilling, 207f of micro cutting, 207f of turning, 208f Lagrangian formulation, 42 Lagrangian method, 17 FEM formulation using updated, 44
52 Laser-assisted bending, 12
13 Laser-based bending, 12
13 Laser-induced shock generation, 368
369 Laser shock experiment, 369 Lateral extrusion, 7
8, 8f LDR. See Limiting draw ratio (LDR) Limiting draw ratio (LDR), 10
11

Index

Lorentz force, 113
114, 123 Ludwik’s law, 40 M Machined components, reliability of, 297 Machined surfaces deformation history of, 272
273 grain refinement, 272, 272f microhardness of, 269 microstructural changes of, 269 microstructural evolution in, 270
273 microstructure of, 271f, 321f Machining, 297 clamping devices and, 216
218 macro- and micro, 219f, 220f microstructural models for, 275
279 of monocrystals, 243 multiscale modeling in, 218
219 residual stresses (RS) in, 297 simulations, 211f thermomechanical loading in, 270 Macro machining, 219f, 220f Macroscale machining, 244 Manufacturing technology, hot stamping, 144
145, 145f Mao
Peng
Yi
Lai (MPYL) model, 422
423, 425t, 428
429, 433 Marker material (MM), 403
405 Material constitutive model, 326
332 selection, 332
334 deformation, 1 flow, 403
410 flow rule, 151 hardening behavior of, 38
41 marker material (MM), 403
405 modeling, 232
233 properties, evolution, 183
184 Material-removal modeling criteria, 248, 256
257 MATLABs, 234
235 MDS. See Molecular dynamics simulation (MDS) Mechanical modeling strategy, 184
186, 185f Mechanical threshold stress (MTS) model, 327 Mechanistic models, 230 Metal cutting, 297, 299
300

Index

definition and energy considerations, 298
300 oxygen-free high conductivity (OFHC) copper in, 329
330 residual stresses (RS) in, 303
305 simulation, 343 Metal foil forming, microblast-assisted, 362 Metal forming, 1, 24
25, 415
416 friction models in, 437
438 modeling issues in, 2
13 bending, 11
13, 12f deep drawing, 10
11, 11f extrusion, 7
10, 7f, 8f, 9f, 10f forging, 2
4, 3f rolling, 4
6, 5f wire drawing, 6
7, 6f modeling of, 1
2 multiscale modeling of, 22
23 Metallic materials, corrosion of, 348, 348f Metals body-centered cubic (BCC), 375
376 cubic, face-centered, 373
374 face-centered cubic (FCC), 373
374, 374f forming limit diagram (FLD) of sheet, 147 hexagonal closed-packed (HCP), 362 nanocrystalline (nc), 371
372 plastic deformation of, 1 Metal-working fluid (MWF), 305
306 Meyers theory, 380
381 Microblast-assisted metal foil forming, 362 Micro cutting LAGRANGE 3D FE simulation of, 207f Microforming, modeling of, 24 Microhardness, of machined surface, 269 Micromachining, 219f, 220f of monocrystals, 243 Micromanufacturing, friction models for, 439 Micro-pores, 179 Micro-scratching FE model of, 260f study, 258, 259t Microstructural evolution, 399
400 in chip with ultrasonically assisted cutting, 288
294, 289f, 290f, 291f, 292f in machined surface with UAC, 283
288, 283f mechanism models of, 275
278

451

Microstructural model, 20
23 JMAK, 280 for machining, 275
279 calculation of microstructural evolution, 278
279, 279f mechanism models of microstructural evolution, 275
278 Microtwins, 370 in nickel, 378f Milling, 436 M-K model, 93
94 Modeling of damage evolution, 151
158 advanced damage models, 153
155 constitutive equations, 151
153 unified constitutive equations for hot stamping, 155
156 of microstructure, 20
22 and simulation techniques, 199
212 Modeling and simulation in industry, 213
216 burr-formation and clean manufacturing, 213
215 cryogenic machining, 215
216 cutting tool optimisation, 213 dry machining, 213 high-performance cutting, 213 high-speed cutting, 213 Modeling techniques, 13
19 finite difference method, 15 finite element method, 15
17 meshless method, 17 molecular dynamics simulation (MDS), 18 slab method, 13 slip-line field method, 13
14 soft computing, 18
19 upper bound method, 14
15 visioplasticity method, 14 Molecular dynamics simulation (MDS), 2, 18, 22, 212, 372 Molecular dynamics (MD) techniques, 221 Monocrystals machining of, 243 micromachining of, 243 MPYL model. See Mao
Peng
Yi
Lai (MPYL) model MTS model. See Mechanical threshold stress (MTS) model

452

Multi-asperity-based modeling of friction, 428f Multiaxial static stresses, 301 Multihole extrusion, 7
8, 10f Multiple-discharge electromagnetic pulse forming, 133
134, 134f Multiscale modeling approach, 22
23 in coating-substrate simulation, 220
222 of coating-substrate systems, 221f in machining, 218
219 of metal forming, 22
23 Myhr and Grong model, 399
400 N Nakazima method, 149 Nanocrystalline (nc) metals, 371
372 Nano-forming, modeling of, 24 Neutron diffraction, 302 Newton
Raphson equation, 51 Newton
Raphson method, 50
52 NGV. See Nozzle-Guide-Vane (NGV) Nickel alloys, 307
309, 307f, 308f, 309f Inconel 690, 337
338, 338f femtosecond (fs) shock wave in, 376
378 microtwins in, 378f Normal anisotropy, 10
11 Norton’s model, 439 Nozzle-Guide-Vane (NGV), 217
218 Nozzle-guide-vane part, 217f Numerical-based models, 324
326 Numerical modeling approach, 232 of elastic
plastic problems, 37
44 O Open die forging, 2
4, 3f Orbital forging, 2
3, 3f Orthogonal cutting model, of AISI 316L, 336f Orthogonal UAC process, 283 Oxygen-free high-conductivity (OFHC) copper, 339
340, 340f, 348f in metal cutting, 329
330 P Parallel computing, elapsed time reduction in, 208f

Index

Parallel kinematic machine (PKM) tools, 197
198 Partial differential equations (PDEs), 210 PCBN tools, 308
309, 314, 317 PDEs. See Partial differential equations (PDEs) Phase transformation, 20
21 model, 277
278 PKM tools. See Parallel kinematic machine (PKM) tools Planar anisotropy, 10
11 Planar biaxial tensile test, 149 Planar tensile test, 149 Plane plastic strain, 200 Plane stress
based continuum damage mechanics material model, 162
164, 166f, 167f, 168f Plastic deformation, 1 behavior of material, 362 in machined surface, 321 of materials, 361 of metals, 1 microstructural changes in, 273
274, 273f and temperature, 321 during tensile test, 300f in tube hydroforming, 74
89 Plasticity models, 327
328 Plasticity theory, 2 Plowing model, 434
435 Pocket model, 439 Polymers, glass-fibre reinforced, 173
175 Pragar’s law, 40
42 Precision machining, 269 Pre-Conditioned Conjugate Gradient Method (PCCG), 207
209 Principal strain
based continuum damage mechanics material model, 165
170, 168f, 170f Pulling force, 189
191 before die-entrance, 189 die-entrance to flow front location, 189
190 magnitude of, 189 solid state and detachment from die-wall, 191 through liquid and gel states, 190 Pultrusion, 173
175, 174f

Index

Q Quasi-isotropic (QI) laminate, 183
184 R Raman spectroscopy, 302
303 Ramberg
Osgood equation, 40 RCT. See Ring-compression test (RCT) Reaming, 229 Recrystallization model, dynamic, 276
277 Residual stresses (RS) boundary conditions, 325
326 calculation, 343, 343f control of, in machining, 322
324 corrosion resistance, 348
352, 349f, 350t, 351f definition and origins of, 301 in difficult-to-cut materials, 305
319 and distortions, 400
403, 401f effect of cutting edge radius on, 321f effect of relative tool sharpness on, 319
322, 321f effects of, 298f fatigue strength, 352, 353f from FE model, 343
344, 344f fields assessment of resultant, 186
188 predicted, 188f formation, 183
188 in machining, 305f, 306f influence of, on product sustainability, 346
352, 346f, 346t, 347f levels in pultruded profiles, 187t in machining, 297
298 operations, 303
324 mechanisms of, 305f in metal cutting, 303
305 modeling and simulation considerations, 324
326 optimization of cutting conditions for improved, 344
346 origin of, 303
305 prediction, constitutive and contact models, 326
335 procedure for comparing predicted and measured, 341
344 simulation of, for several work materials, 335
341 techniques for measuring, 301
303 thermomechanical models for, 404t

453

in turning AISI 316L, 323f Resin-bath pultrusion (RBP) process, 173
175 Resin impregnation, 175
179 saturated pressure-driven flow, 176
177 unsaturated impregnation flow, 178
179 Resin-injection pultrusion (RIP) process, 173
175 Resin transfer moulding (RTM), 175 Resin viscosity, 177 Richard’s equation, 179 Ring-compression test (RCT), 4, 420 Rod drawing, 6 Rolling, metal forming, 4
6, 5f Rotary forging, 2
3, 3f Rotational autofrettage, 35
37, 36f Rotationally symmetrical tube expansion, 74
81, 76f, 79f RS. See Residual stresses (RS) RTM. See Resin transfer moulding (RTM) S SCFM. See Self-Consistent Field Micromechanics (SCFM) SCP theory. See Single-crystal plasticity (SCP) theory Self-Consistent Field Micromechanics (SCFM), 183
184 Self-locking behavior, 415 Semicoupled thermomechanical models, 397 Serrated-chip formation, 281 SFE. See Stacking fault energy (SFE) Sheet bending, 11
12 Sheet-bulk metal forming process, 1 Sheet hydroforming, 71 Sheet metal electromagnetic pulse forming of flat and preformed, 118
122, 119f, 121f, 122f forming, 1, 361
362 conventional, 90 forming limit diagram of, 147 Shock deformed materials, 364
365, 369 generated defects, types, 369
370, 370f, 371f impact in shock tube, 367
368, 368f theory of defect generation under, 380
384, 381f, 382f, 383f

454

Shock/blast response (dynamic deformation), 361 Shock exposure effect of material parameters, 371
376 body-centered cubic (BCC) metals, 375
376 face-centered cubic metals, 373
374 grain size, 371
373 hexagonal closed-packed (HCP) metals, 374 texture, 373 effect of shock parameters peak pressure and pulse width, 376
378, 377f, 378f parameters influencing material response to, 369
380 residual strain, 378
380, 379f, 380f Shock tube, shock impact in, 367
368, 368f Shock wave, 363f experimental methods for investigation of, 365
369 explosive loading of materials, 366 flyer plate impact test, 366
367, 367f laser-induced shock generation, 368
369 and parameters, 363
365, 363f shock impact in shock tube, 367
368 in solids, 364 split-Hopkinson pressure bar (SHPB), 367, 368f SHPB. See Split-Hopkinson pressure bar (SHPB) SHT. See Solution heat treatment (SHT) Simulation techniques, modeling and, 199
212 Single-crystal ceramic material, machining of, 258
264 comparison of cutting forces, 262
263, 263f computational modeling, 259
261, 261t, 262t experimental procedure, 258
259 prediction of work-piece deformation, 263, 264f surface topography, 262 Single-crystal copper, material parameters of, 250t Single-crystal machining computational implementation, 247
248 material-removal modeling criteria, 248

Index

mechanics of, 245
248 Single-crystal metal, machining of, 248
258 experimental procedure, 248
249 finite-element model and material parameters, 249
250 material-removal modeling criteria, 256
257, 256f techniques, 257
258, 257f, 258t mesh-sensitivity analysis, 255
256, 255f simulation and results, 250
254 prediction of chip morphology, 252, 252f prediction of cutting forces, 250
251 prediction of misorientation angle, 254, 255f prediction of workpiece deformation, 252
253, 253f, 254f Single-crystal plasticity (SCP) theory, 243
247 Slab method, 13 Slip-line field method, 13
14 Slip-line method, 199
202, 200f, 202f Smoothed particle hydrodynamics (SPH), 211 Smooth particle hydrodynamics (SPH), 244
245 Soft computing techniques, 18
19 Solid-state joining technique, 393 Solid-state welding, 438 Solution heat treatment (SHT), 144
145 SPH. See Smoothed particle hydrodynamics (SPH) Split-Hopkinson pressure bar (SHPB), 367, 368f Stacking fault energy (SFE), 373
374 Stainless steels, austenitic, 310
311, 310f, 311f Static friction, 415, 417 Strain-based forming limit diagram, 145
146 Strain
displacement relations, 43 Strain hardening, 21, 39, 327 Stress
strain behavior, 159
160 Stress
strain relations, 43
44 Surface cracking, 4, 8
9 Surface integrity, 20
22, 297 Surface metrology, 236
237 Surface roughness, 24 in machined components, 344
346

Index

parameter, effect of, 433f, 434f topography of engineering materials, 432
433 Surface topography, 262 Swage autofrettage, 33
34, 34f Swift’s law, 40 T Tandem rolling mill, 5f Tapping, 229 Taylor’s impact test, 365
366, 366f TCM model, 175, 184
188 Tensile test, plastic deformation during, 300f Thermal autofrettage, 35, 36f, 58
66 FEM modeling of, 52
66 open-ended condition of, 63
66, 64f, 65f, 66f, 66t plane stress end condition of, 61
63, 62f, 63f, 64t Thermal behavior, 396
398 Thermal elastic
plastic deformations, 22
23 Thermal-pseudo-mechanical (TPM) model, 397
398, 398f Thermo-chemical modeling, 180
182 considerations, 182 cure kinetics and differential scanning calorimetry, 181 heat transfer, 180 mechanical modeling, 183
188 Thermomechanical loading, in machining process, 270 Thermomechanical models for residual stresses, 404t semicoupled, 397 Thermomechanical multiaxial tensile test data, 160
162, 162t, 163f, 164t, 165f Thermomechanical uniaxial test data, 159
160, 160t Thermoset FRPs, 180 Thermoset-RIP process, 174f 3D FE model, 259
260 of drilling, 233 3D thermal-metallurgical
mechanical model, 402
403 Thrust force, magnitudes of, 238f Ti6Al4V titanium alloy, 341, 342f Time-temperature transformation (TTT), 21 Titanium alloy, Ti6Al4V, 341, 342f

455

Titanium-based alloys, 309
310, 309f Tool-chip and tool-work contact models, 326 TPM model. See Thermal-pseudomechanical (TPM) model Transportation industry, 144 Tresca yield criterion, 39
41 TTT. See Time-temperature transformation (TTT) Tube bending, 11
12 Tube hydroforming, 71, 72f, 98
99 determination of forming limits in, 89
98 hydroforming of tube branches, 85
89, 86f necking and bursting, 89
95 plastic deformation in, 74
89 rotationally symmetrical tube expansion, 74
81, 76f, 79f wrinkling and buckling, 95
98 Turning, LAGRANGE 3D FE simulation of, 208f U UAC. See Ultrasonically assisted cutting (UAC) UAD. See Ultrasonically assisted drilling (UAD) UAT. See Ultrasonically assisted turning (UAT) Ultrasonically assisted cutting (UAC) boundary conditions for orthogonal, 282f microstructural evolution in, 279
294 machined surface, 283
288, 283f, 286f, 287f, 288f microstructural evolution in chip with, 288
294, 289f, 290f, 291f, 292f Ultrasonically assisted drilling (UAD), 229, 235
240 experimental setup and instrumentation, 235
237, 235f force reduction in, 239 surface metrology, 236
237 Ultrasonically assisted turning (UAT), 280, 437 Ultrasonic assisted cutting, 213
215 Uniaxial tensile testing, 149
150, 150f Unidirectional (UD) fiber reinforcements, 177
178 Upper bound method, 14
15 Upsetting forging, 2
4, 3f

456

User-defined 3D damage model (VUMAT), 232
233, 247
248, 278
279 V VISAR technique, 364, 369 Viscoplastic solids, flow stress
strain response of, 151f Visioplasticity, 1
2, 14 von Mises criterion, 38
39, 421
422 W Wanheim and Bays model, 419
421, 421f friction model, 416 Warm hydroforming, 73 Warm metalworking, 1 Welding, solid-state, 438 Wire drawing, metal forming, 6
7, 6f Workpiece material effect of deformation temperature of, 430t effect of friction factor of, 431f

Index

effect of strain-hardening of, 429, 430f X X-ray diffraction (XRD), 302
303, 369, 375 Y Yield criteria, 38
41 Tresca yield criterion, 39
41 von Mises yield criterion, 38
39 Young’s modulus of deformable material, 431f of flat die material, 432f Z Zener-Hollomon parameter, 95 Zerilli
Armstrong constitutive model, 337
338 Zerilli-Armstrong law, 204 Zorev’s model, 334, 334f, 435